PREFACE
These two parts bring together a number of authoritative, state-of-the-art reviews and contributions, written by well recognized experts in the field of"flow and rheology of non-Newtonian fluids." Knowledge of non-Newtonian behavior is of vital importance to a variety of manufacturing processes including, for example, mixing, shear-thickening, fibre spinning, coating, and molding. This work covers areas such as bio- and food- rheology, electro-rheological fluids, polymers, flow in porous media, and suspensions. Complex and industrial flow situations are dealt with via analytical, as well as, numerical methods. In Chapter 1, a critical account of advances made in the area of flow-induced interactions in circulation is presented. Chapters 2 & 3 deal with shear-thickening in biopolymeric systems, and with the rheology of food emulsions, respectively. The next six chapters are on complex flows, in particular, Chapter 4 discusses worm-like micellar surfactant solutions. Chapter 5 covers time periodic flows. Chapter 6 communicates on secondary flows in tubes of arbitrary shape. Chapter 7 relates effects of non-Newtonian fluids on cavitation. Chapter 8 discusses viscoelastic Taylor-Vortex flow. Chapter 9 deals with non-Newtonian mixing. This is followed by two chapters on computational methods relevant to homogeneous viscoelastic fluids at the macro-level. The next major section is on constitutive equations and viscoelastic fluids. Chapter 12 discusses recent advances in transient network theory. Chapter 13 deals with theories based on fractional derivatives and Chapter 14 involves kinetic theory. Chapters 15 and 16 put forward new concepts approaching the constitutive structure of polymeric melts. The next chapter communicates the theory of flow of smectic liquid crystals. Part A ends with an overview of extensional flows. Volume B starts with a section on electro-rheological fluids. The first two chapters in the section summarize the constitutive theories for electro-rheological fluids from the continuum and molecular points of view. Chapter 21 relates a comprehensive approach to the constitutive structure of electromagnetic fluids, and the following two chapters deal with the properties of electro-rheological fluids. The next section covers some industrial flows related to drag reduction, and paper coating. Polymer processing and the related rheology are discussed in Chapters 26-29. In particular, the rheology of long discontinuous iber thermoplastic composites, thermo-mechanical modelling of polymer processing, injection molding and flow of melts in channels with moving boundaries are covered in Chapters 26-29 respectively. Free surface viscoelastic and liquid crystalline polymer fibers and jets, and numerical sinmlation of melt spinning of polyethylene fibers are the subject of Chapters )0 and 31, respectively.
vi Section 9 contains two chapters dealing with foam flow and non-Newtonian flow in porous media. Section 10 discusses four chapters on various aspects of suspension.Chapter 34 reviews and puts forth new ideas on the fluid dynamics of fine suspensions. Chapter 35 deals with concentrated suspensions. This section ends with a discussion on fiber suspensions and fluidized beds. The last section of Part B contains a discussion on transport-phenomena involving heat and mass transfer in rheologically complex systems and an account of a new onedimensional model for viscoelastic diffusion in polymers. With the publication of this work we hope to update and complement earlier work in a diverse range of topics. These two volumes should be of interest to all those engaged in basic, as well as applied research. The information presented herein is equally valuable for practising engineers who are constantly dealing with complex situations involving non-Newtonian materials. The contents of the two volumes are accessible to those with a background in engineering and/or pure sciences. We would like to take this opportunity to thank the contributors who, despite their busy schedules, kindly agreed to participate.
Dennis A. Siginer New Jersey Institute of Technology Newark, NJ, USA Daniel DeKee Tulane University New Orleans, LA, USA Raj P. Chhabra Indian Institute of Technology Kanpur, UP, India
vii
LIST OF CONTRIBUTORS Advani, Suresh G. University of Delaware Department of Mechanical Engineering Spencer Laboratory Newark, Delaware 19716, USA
Cairncross, R. A. Department of Chemical Engineering Drexel University Philadelphia, Pennsylvania 19104, USA
Agassant, J. F. Centre de Mise en Forme des Mat6riaux Ecole des Mines de Paris URA CNRS 1374 BP 207 06904, Sophia Antipolis, FRANCE
Carreau, Pierre J. Center for Applied Research on Polymers CRASP Department of Chemcical Engineering Ecole Polytechnique, Montreal QC H3C 3A7, CANADA
Bagley, Edward B. 756 S. Columbus Morton, Illinois 61550-2428, USA
Chhabra, R. P. Department of Chemical Engineering Indian Institute of Technology Kanpur, INDIA 208016
Bakhtiyarov, Sayavur I. Space Power Institute 231 Leach Center Auburn University Aubum, Alabama 36849-5320, USA
Co, Albert Department of Chemical Engineering University of Maine Orono, Maine 04469-5737, USA
Bechtel, Stephen E. Department of Aerospace Engineering Applied Mechanics and Aviation The Ohio State University Columbus, Ohio 432 I0, USA
Conrad, Hans Department of Materials Science and Engineering North Carolina State University Raleigh, North Carolina 27695, USA
Blumen, A. Theoretical Polymer Physics Freiburg University Rheinstr. 12, 79104 Freiburg, GERMANY
Couniot, A. Siemens-Nixdorf Information Systems S.A. LoB "Major Projects" Chaussee de Charleroi 116_ B- 1060 Brussels, BELGIUM
Bousfield, Douglas W. Department of Chemical Engineering University of Maine 5737 Jennes Hall Orono, ME 04469-5737 USA
Coupez, T. Centre de Mise en Forme des Materiaux Ecole des Mines de Paris URA CNRS 1374 BP 207, 06904, Sophia Antipolis, FRANCE
Brito-De La Fuente, Edmundo Food Science and Biotechnology Department Chemistry Faculy "E" National Autonomous University of Mexico UNAM, 04510 Mexico, D.F., MEXICO
Creasy, Terry University of Southern California Center for Composite Materials VHE 602 MC0241 Los Angeles, California 90089-0241, USA
Brunn, Peter O. Universitat Erlangen-Nurnberg Lehrstuhl fur Stromungsmechanik Caueerstr. 4 D-91058 Erlangen, GERMANY
De Kee, D. Department of Chemical Engineering Tulane University New Orleans, Louisiana 70118, USA
Buyevich, Yuri A. Center for Risk Studies and Safety University of California Santa Barbara 6740 Cortona Dr. Santa Barbara, Califomia 93117, USA
Demay, Y. Centre de Mise en Forme des Materiaux Ecole des Mines de Paris URA CNRS 1374 BP 207, 06904, Sophia Antipolis, FRANCE
viii
Dintzis, Frederick R. USDA ARS National Center for Agricultural Utilization Resarch Peoria, Illinois 61604, USA Dulikravich, George S. Aerospace Engineering Department 233 Hammond Building The Pennsylvania State University University Park, Pennsylvania 16802, USA Dunwoody, James Department of Applied Mathematics & Theoretical Physics The Queen's University Belfast BT7 INN NORTHERN IRELAND Dupret, F. CESAME Unite de Mecanique Appliquee Universite catholique de Louvain Avenue G. Lemaitre 4-6 B- 1348 Louvain-la-Neuve, BELGIUM Duming, Christopher J. Department of Chemical Engineering and Applied Chemistry Columbia University New York, New York 10027, USA Fong, C. F. Chan Man Department of Chemical Engineering Tulane University New Orleans, Louisiana 70118, USA Forest, M. Gregory Department of Mathematics University of North Carolina Chapel Hill, North Carolina 27599-3250, USA Franco, J.M. Departamento de Ingenieria Quimica Universidad de Huelva Escuela Politecnica Superior La Rabida, 21819 Palos de la Ftra (Huelva), SPAIN Friedrich, Chr. Freiburg Materials' Research Center Freiburg University Stefan-Meier-Str.21 79104 Freiburg, GERMANY Fruman, Daniel H. Groupe Phenomenes d'Interface Ecole Nationale Superieure de Techniques Avancees 91761 Palaiseau Cedex - FRANCE
Gallegos, C. Departamento de Ingenieria Quimica Universidad de Huelva Escuela Politecnica Superior La Rabida, 21819 Palos de la Ftra (Huelva), SPAIN Goldsmith, Harry L. Department of Medicine The Montreal General Hospital 1650 Ave Cedar Montreal, Quebec H3G 1A4, CANADA Hoyt, Jack W. 4694 Lisann Street San Diego, Califomia 92117, USA lsayev, A. I. Institute of Polymer Engineering The University of Akron Akron, Ohio 44325-0301, USA Kanu, Rex C. Department of Industry and Technology Ball State University Muncie, Indiana 47306, USA Khayat, Roger E. Department of Mechanical & Materials Engineering The University of Western Ontario London, Ontario, CANADA N6A 5B9 Kim, Kyoung Woo Fiber Research Center Sunkyong Industries Su Won, 440-745, KOREA Kim, Sang Yong Department of Fiber and Polymer Science College of Engineering Seoul National University San 56-1, Shinlim-Dong, Kwanak-Ku Seoul 151-742, KOREA Kornev, Konstantin G. Institute for Problems in Mechanics Russian Academy of Sciences 101 (1) Prospect Vemadskogo Moscow 117526, RUSSIA Kwon, Youngdon Department of Textile Engineering Sung Kyun Kwan University Su Won, 440-746, KOREA
Lavoie, Paul Andre Center for Applied Research on Polymers CRASP Department of Chemcical Engineering Ecole Polytechnique, Montreal QC H3C 3A7, CANADA Leonov, Arkadii I. Department of Polymer Engineering The University of Akron Akron, OH 44325-0301, USA Leslie, Frank M. Mathematics Department University of Strathclyde Livingstone Tower Richmond Street Glasgow G 1 1XH, SCOTLAND Letelier, Mario Universidad de Santiago de Chile Santiago CHILE Mal, O. CESAME Unite de Mecanique Appliquee Universite catholique de Louvain Avenue G. Lemaitre 4-6 B- 1348 Louvain-la-Neuve, BELGIUM Neimark, Alexander V. TRI/Princeton 601 Prospect Ave. P.O. Box 625 Princeton, New Jersey 08542-0625, USA
Prakash, J. Ravi Department of Chemical Engineering Indian Institute of Technology Madras, INDIA, 600 036 Rajagopal, K. R. Texas A&M University College Station, Texas, 77842-3014, USA
Rozhkov, Aleksey N. Institute for Problems in Mechanics Russian Academy of Sciences 101 (1) Prospect Vernadskogo Moscow 117526, RUSSIA Schiessel, H. Theoretical Polymer Physics Freiburg University Rheinstr. 12, 79104 Freiburg, GERMANY Shaw, Montgomery T. Department of Chemical Engineering and Polymer Program University of Connecticut 97 North Eagleville Road U-136 Storrs, Connecticut 06269-3136, USA Siginer, Dennis A. Department of Mechanical Engineering New Jersey Institute of Technology Newark, New Jersey 07102, USA Steger, R. Rheotest Medingen GmbH RodertalstraBe 1, D-01458 Medingen b. Dresden, GERMANY
Overfelt, R. A. Space Power Institute 231 Leach Center Auburn University Auburn, Alabama 36849-5320, USA
Tang, P. H. Department of Chemical Engineering and Applied Chemistry Columbia University New York, New York 10027, USA
Padovan, J. Department of Mechanical Engineering The University of Akron Akron, Ohio 44325-0301, USA
Tanguy, Philippe A. Department of Chemical Engineering Ecole Polytechnique Montreal P.O. Box 6079 Station Centre-ville Montreal, H3C 3A7 CANADA
Petrie, Christopher J. S. Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne NE1 7RU, UNITED KINGDOM Phan-Thien, Nhan Department of Mechanical and Mechatronic Engineering The University of Sydney NSW 2006, AUSTRALIA
Tanner, R. I. Department of Mechanical and Mechatronic Engineering The University of Sydney NSW 2006, AUSTRALIA Tao, Rongjia Department of Physics Southern Illinois University at Carbondale Carbondale, Illinois 62901, USA
Vanderschuren, L. Shell Research S.A. Avenue Jean Monnet 1, B-1348 Louvain-la Neuve, BELGIUM
Zhang, Y. Institute of Polymer Engineering The University of Akron Akron, Ohio 44325-0301, USA
Vergnes, B. Centre de Mise en Forme des Materiaux Ecole des Mines de Paris URA CNRS 1374 BP 207, 06904, Sophia Antipolis, FRANCE
Zhou, Hong Department of Mathematics University of North Carolina Chapel Hill, North Carolina 27599-3250, USA
Verhoyen, O. CESAME Unite de Mecanique Appliquee Universite catholique de Louvain Avenue G. Lemaitre 4-6 B- 1348 Louvain-la-Neuve, BELGIUM Verleye, V. TECHSPACE AERO Route de Liers 121 B-4041 Milmort, BELGIUM Vincent, M. Centre de Mise en Forme des Materiaux Ecole des Mines de Paris URA CNRS 1374 BP 207, 06904, Sophia Antipolis, FRANCE Vossoughi, Shapour Department of Chemical and Petroleum Engineering University of Kansas 4006 Learned Hall Lawrence, Kansas 66045-2223, USA Wang, Qi Department of Mathematical Sciences Indiana University-Purdue University at Indianapolis Indianapolis, Indiana 46202, USA Wu, C. W. Research Institute of Engineering Mechanics Dalian University of Technology Dalian 116024 People's Republic of China Yoo, Jung Yul Department of Mechanical Engineering College of Engineering Seoul National University Seoul 151-742, KOREA Yziquel, F. Center for Applied Research on Polymers CRASP Department of Chemcical Engineering Ecole Polytechnique, Montreal QC H3C 3A7, CANADA
Zook, C. Institute of Polymer Engineering The University of Akron Akron, Ohio 44325-030 l, USA
FLOW-INDUCED
INTERACTIONS
IN THE CIRCULATION
Harry L. Goldsmith McGill University Medical Clinic, Montreal General Hospital, Montreal, Quebec H3G 1A4, Canada 1. I N T R O D U C T I O N Many years ago, it was suggested to the English physiologist L.E. Bayliss by the eminent Cambridge authority on fluid mechanics, G.I. Taylor, that rather than work on the highly complex rheology of blood, he would be better spending his time studying the flow of marmalade through sewage pipes. That advice did not deter Dr. Bayliss [1,2] and a host of physiologists, rheologists, chemical and mechanical engineers from engaging in the problem, which had, after all, an honourable parentage in the person of Jean Leonard Marie Poiseuille, who published as early as 1839 [3,4], and in this century, the Swedish pathologist, Robin F~thraeus [5,6]. Quite apart from the clinical value of discovering how blood cells travel in the vessels of the circulation, there is, for the rheologist, a fascination about this fluid, since it is able to flow, even at hematocrits (packed red blood cell (RBC) volume concentration) well above 40%, with such ease compared to other suspensions and emulsions [7]. Thus, it is possible to pack a column of mammalian red cells containing less than 2% trapped plasma into a 2 mm diameter glass tube, and then suck it out quite easily into a micropipette whose tip has a diameter of only 0.5 mm. By comparison, oil-in-water emulsions reach the quasi-solid consistency of margarine at much lower concentrations. Clearly, in such a concentrated suspension, interactions between the blood corpuscles and between the corpuscles and the vessel walls play an important role in determining the mechanics of the motion. This chapter is concerned with a description of these interactions. We begin by taking the reader through a brief description of blood, seen as a dispersion of charged colloidal-size particles in a Newtonian suspending medium, and the main flow regimes to which it is subjected. To understand the cell and cell-wall interactions, one must begin with the macroscopic, overall non-Newtonian rheology of blood,
and then to correlate these rheological properties with the flow properties of the individual cells, studied first in isolation, then at increasing concentrations as interactions become increasingly important, i.e. the microrheology of the blood. This part of the chapter will mainly involve the erythrocytes or red cells, since they occupy >99% by volume and -96% by number of the particulate phase of blood, and thus effectively determine both the macro- and microrheology of the suspension. However, the most recent advances in the field have been made at the submicroscopic level, where the molecules and the forces involved in cell-cell and cell-wall adhesion have been investigated, thus entering the realm of what has been called molecular rheology. At this level, the chapter focusses not on the red cells but on the platelets and leukocytes, since it is they that are involved in key physiological and pathophysiological events through their role in blood coagulation and a g g r e g a t i o n - the platelet, and in i n f l a m m a t i o n - adhesion of leukocytes to the vessel wall and their passage into the extravascular space. In fact, adhesion processes between cells and between cells and the vessel wall are particularly critical in the circulation for all three classes of cells. The biophysics and rheology of certain of these adhesive reactions and the means of measuring forces of adhesion are dealt with in Sections 4 and 5. Where possible, the phenomena described are discussed in terms of fluid mechanical and colloid chemical theory. 1.1 B l o o d as a colloidal d i s p e r s i o n Mammalian blood is a dispersion of three classes of negatively charged particles (the cells or corpuscles; Table 1) in an aqueous solution (the plasma) containing dissolved salts and proteins (Table 2). As pointed out above, from the point of view of its rheological properties, blood is essentially a concentrated suspension of red blood cells 1.1.1 Cell membrane The biconcave red cell has a very thin lipid bilayer membrane (-10 nm) containing proteins linked to a submembraneous skeleton of actin/spectrin. The membrane encapsulates a 33% Newtonian salt solution of hemoglobin having a viscosity ,--6x that of the plasma at 37~ The ease of deformation of the red cell is mainly due to the properties of the membrane acting as a twodimensional incompressible material, which deforms at constant surface area and constant volume [8], and when osmotically swollen does not increase its surface area. The deformed cell remembers its undeformed shape and recovers it within a fraction of a second upon removal of the forces of deformation. By means of micropipette aspiration techniques, the viscoelastic properties of the membrane have been extensively characterized. The elastic deformation is described by three fundamental, independent deformations [9]3 elongation or shear of the membrane having a shear modulus of 6-9 • 0mN m -~, an isotropic expansion of the membrane surface without shear or
Table 1 Mean values of physical parameters of human blood cells Cell
Shape
Major Axis gm
Volume 3 gm
Number g1-1
Volume Fraction
Erythrocyte (Red Cell)
Biconcave disc
8.3
88
5.4x106 4.8xl 0 6
0.46 (m) 0.42 (f)
Thrombocyte (Platelet)
Oblate spheroid
3.1
7.5
2.5x105
1.9x10 -3
Leukocytes (White Cells)
Ruffled spheres
7.0x103 (total)
1.2x10 -3 (total)
Neutrophil
7.8
240
3.7x103
8.8x10 -4
Lymphocyte
6.0
115
2.5x103
2.9x10 -4
Table 2 Mean values of concentrations of ions and proteins in plasma PROTEINS
SALTS Ion
Concentration meq./litre
Name
Molecular Weight
Concentration gm/litre
Na +
150
Albumin
69,000
44.1
K+ Ca++ Mg++
4 5 2
Fibrinogen
340,000
3.0
C1HCO3PO 4-
111 28 3
t~l-globulin ~2-globulin
200,000
2.7 5.9
SO4--
1
13-globulin ),-globulin
90,000 160,000
8.8 14.2
bending, having an area expansion modulus of 450 mN m-', and a bending of the membrane with a very low bending modulus of 1.8 + 0.2x10 -19 N m. The resistance of the membrane to the rate of deformation is characterized by its
viscosity; the area viscosity has been found to have a value of 0.6-1.2x10 -3 mN s m -1 [9].
1.1.2 Charge and stability. The negative charge on the red cell membrane is due to N-acetyl and Nglycolneuraminic acid (sialic acids) residues attached via a gycosidic linkage to other molecules. Measurements of the electrophoretic mobility of red cells in buffers of physiological ionic strength = 0.145 yield values of the ~-potential -15 mV [10]. When treated with the enzyme neuraminidase to remove the sialic acid residues, the amount of charge lost, as calculated from the change in electrophoretic mobility, is underestimated 2-3x, when compared with the amount of sialic acid released. The reason for the discrepancy is thought to be due to the presence of a layer of polyelectrolytes on the surface of the lipidprotein bilayer through which the solvent can flow [11]. The sialic acid charges are distributed within this layer, and not anchored to a rigid surface as in solid colloidal particles such as charged latex spheres. The layer of polyelectrolyte, called the 'glycoprotein calyx' is believed to extend out to -- 710 nm beyond the bilayer, and its thickness is a function of the ionic strength. The presence of such a layer would in part account for the fact that red cells suspended in physiological saline (e.g. Ringers, a solution of ions mimicking those in plasma, to which albumin may be added) do not aggregate. If treated as a rigid colloidal charged particle, calculations of the double layer thickness at physiological ionic strength yield a value o f - - 8 nm, i.e., within the glycocalyx boundary and so close to the membrane that one would have expected the cells to be subject to van der Waals attractive forces. In fact, application of the DLVO theory of colloidal stability [12,13] to red cells clearly indicates that the cells should aggregate in physiological saline [11,14]. However, red cells do aggregate in plasma in the presence of fibrinogen, as well as in buffers containing plasma expanders such as dextran [14,15] and polyvinylpyrrolidone. Within the primary aggregate, or rouleau, the cells lie in a regular array as shown in Figure 1, discoid face to discoid face, and deform so that the adjoining faces are locally parallel and have maximum contact. Studies of aggregation in dextran solutions have been interpreted in terms of cross-bridging by polymer molecules long enough to simultaneously adsorb on the surfaces of two neighbouring cells [16]. In the absence of reliable evidence that dextrans are adsorbed on red cells, this interpretation has been questioned and an alternate mechanism, based on depletion flocculation, proposed [17]. In this scenario, surfaces are initially repulsive and stabilized by an electrostatic stress IIo~exp(-z/2~), where z is distance and ~ is the electrostatic decay constant. With addition of non-adsorbing polymer, an osmotic pressure difference builds up between the mid point of the intermembrane gap and the external bulk solution. The osmotic pressure induces an attractive depletion stress in which the decay constant is now the
Figure 1: A network of rouleaux of human red blood cells photographed at rest on a microscope slide. correlation length of the polymer ~, i.e., IIpeXp(-z/~), where ~, characterizes the range of the depletion zone. At low [polymer] there is no adhesion since the electrostatic stress is less than the depletion stress. With increasing [polymer], the osmotic pressure difference increases, and so does the attractive depletion stress, becoming larger than the electrostatic stress, and aggregation occurs. This continues with increasing polymer concentration. However, at the same time, ~ decreases thereby reducing the range and magnitude of the depletion term. Ultimately, the repulsive electrostatic stress once again wins out and the tendency to aggregation is reduced resulting in electrostatic stabilization. These two competing stresses account well for the observed aggregation vs [dextran] curves [15]. 1.2 Blood vessels and flow regimes Table 3 gives values of the vessel and flow parameters in the human circulation. On leaving the left ventricle of the heart, blood enters the ascending aorta having a resting diameter of-.25 mm, and is accelerated to mean linear velocities as high as 0.9 to 1.4 m s-~. Almost immediately, the blood is subjected to branching flows, the vessel diameters decrease and their numbers increase rapidly, and hence the cross-sectional area of the vessel bed. Values given for the dog show that the area increases from 5 c m 2 in the large arteries, to 20 cm 2 in the small arteries to 1,360 c m 2 in the capillaries [18]. Broadly speaking, one can distinguish three flow regimes:
(i)
A rapid, pulsatile flow in the aorta and larger arteries where inertial effects due to acceleration and deceleration of the fluid predominate. The mechanics of the flow have been modelled regarding blood as a homogeneous fluid, neglecting its particulate character [19]. (ii) A much weaker pulsatile or quasi-steady viscous flow in the smallest arteries and arterioles, in which interactions between the blood cells largely determine the mechanics of the flow. (iii) A viscous flow in the capillaries whose diameter is smaller than that of the red cell, and which can be described as a bolus flow of a train of cells surrounded by a thin lubricating film of plasma at the vessel wall, with pockets of plasma (boluses) between cells [21]. The red ceils readily deform into complex shapes [22,23], and their motion has been modeled [24]. The poorly deformable white cells travel more slowly, causing red cells to accumulate behind them [25,26]. Due to the multiplicity of branching, the non-uniformity of vessel diameters and the existence of curved vessel segments, sudden changes in velocity and direction of the bloodstream result in secondary flows having radial components, and not infrequently in flow separation and the formation of recirculation zones. Such flows have been designated by the term "disturbed" [27], to distinguish them from laminar and turbulent flows, the latter observed near normal aortic and pulmonary valves, increasing markedly in disease when the valves become partially stenosed [28]. Branching occurs with such frequency that the disturbance created by the flow at one bifurcation has not had time to dissipate before that due to another branch comes into play. It is well known that these effects are particularly important with regard to the genesis of thrombosis and atherosclerosis, and clearly that they involve interactions between the corpuscles and fluid with the vessel wall. Such interactions, which will be dealt with in Section 6 below, can result in injury to the vessel wall and to the blood cells themselves. 2. M A C R O R H E O L O G Y OF BLOOD The rheological properties of blood resemble that of a pseudoplastic fluid, being a strong function of the rate of deformation. At low shear rates the apparent viscosity decreases with increasing shear rate as the network of rouleaux (Figure 1) breaks up. At moderate and high shear rate, the apparent viscosity decreases as shear-induced deformation of the red cells increases.
2.1 Viscometry in Couette Flow 2.1.1 Effect of shear rate on apparent viscosity The role of red cell deformation and aggregation is shown in Figure 2 in a plot comparing the apparent relative viscosity, T/r at 37~ of 45% red cell
suspensions in plasma and in albumin-saline with that of aldehyde-hardened (still biconcave) red cells in albumin-Ringer solution [29]. All suspending media had the same viscosity, 1.2 mPa s, and measurements were made in a concentric cylinder Couette device. Since there was no fibrinogen or other aggregating protein in the albumin-Ringer suspension, no rouleaux were present and the difference between rh in plasma and albumin-Ringer suspensions in the range 10-z < G < 5 s-~ was due to the progressive break-up of aggregates. The further decrease in rh is then ascribed to cell deformation, the curve eventually flattening out as the blood assumes a quasi Newtonian character at G > 10 3 s -1. By contrast, r/, for the rigidified cells is almost constant with increasing G, there being no aggregation or deformation. 10 3
>., r,o o
o
10 2
~
..=.,
"
'
~
'
~
R B C in
Aggregation
>
plasma
~
Hardened cells in alb.-Ringer
(D >
...=.
o......-o.........~ ......~ ...... ~.
_ ~'~... ........ v ......,~...... o......-':~..
R B C in
alb Ringer
Deformation
/
~ C> '----',.--~ " - ~ 0
"'"
"--"
(~)
rr
.L,,I
10-2
........
I
1 0 "1
........
I
1
........
I
10
......
A,I
10 2
,
, 9 , .... I
10 3
Shear Rate, s ~
Figure 2. Relative apparent viscosity vs shear rate in suspensions of normal RBC in plasma (aggregation and deformation), and in Ringers-albumin (no aggregation) and hardened RBC in Ringersalbumin (no aggregation or deformation). From Chien [29] with permission. 2.1.2 Constituitive equations A comparison of 11 equations which have been used to describe the rheological behaviour of human blood was made by Easthope and Brooks [30]. The equation which best represented their data obtained in a concentric cylinder measuring system was that of Walbum and Schneck [31]:
~'=a 1 exp a2H +
G -1-a4H
(1)
where H = hematocrit, and a~-a4 are adjustable constants, functions of temperature, macromolecular composition of the suspending phase etc. The
equation was developed from a power law relationship between shear stress and shear rate, using regression curve fitting of trial equations built by successive inclusion of variables of decreasing influence. The model does not allow for a yield stress. Because of the complexity and high concentration of blood, it has not been possible to develop constituitive equations based on a mechanistic model of blood flow, without the use of empiricism or approximations. Such an equation is that due to Casson [32] first used by Scott Blair [33]" ,~.1/2._ aol/2 + a l G a ~ 2 (2) Here, the suspension is modelled as containing particles which can aggregate at low shear rates to form rod-like particles, whose length increases with decreasing shear rate. The similarity between these aggregates and rouleaux (Figure 1) makes the model attractive. However, the yield stress, a0 does not become zero except when H = 0. Nevertheless, Equation (2) represents the rheological data on blood over a limited shear rate range with one set of constants. Also, the experimental determination of yield stress poses real problems, since aggregation at very low shear rates leads to migration of rouleaux away from the walls of the Couette cylinders (syneresis), and sedimentation is then also a problem [34]. Another equation based on mechanistic modelling is that due to Quemada [35,36]" ~r
_
(1
_
0.5kH) -2,
where
k
__
k0 + k o o t-~rl/2 --r 1 + G" " F 1/2
and
Gr-
G
Gr
(3)
Here, the suspension properties are represented by the coefficients k0 and k.., the intrinsic viscosities at zero and infinite shear stress of the particles which predominate at those shear rates, and G~, the critical shear rate, which can be considered to be the inverse of the relaxation time for the dominant structural unit causing the suspension to be non-Newtonian. The coefficients are functions of hematocrit, suspending phase composition, etc. The model has been found to be almost as good as the Walbum-Schneck equation [31] with the advantage that the coefficients have a physical meaning. Further discussion of this model and its application may be found in reference [37], and a comprehensive discussion of blood rheology in reference [38]. 2.2 Blood flow in cylindrical tubes The use of macroscopic rheological data obtained in Couette instruments and continuum models to predict flow behavior in l a r g e vessels appears satisfactory as long as there is no appreciable cell aggregation resulting in syneresis and sedimentation. Both effects will occur in horizontally positioned tubes at mean linear velocities < 1 tube diameter/second [39]. In vertically positioned tubes,
10
aggregation leads to syneresis and the two-phase flow of a core of aggregates surrounded by a cell-depleted peripheral layer, with a reduced pressure gradient at a given volume flow rate [40]. With these exceptions, continuum mechanics appears to be satisfactory for vessels having diameters > 500 ktm. As the tube diameter decreases, however, and approaches the dimensions of the red cells, blood no longer acts as a continuum and the following effects arise.
2.2.1 The Fdhraeus effect As the tube diameter decreases below 500 gm, the measured instantaneous hematocrit in the tube, Hr, is found to be smaller than the hematocrit, HR, in the stirred inflow reservoir, or even in the discharge, Ho (providing no screening effects occur at the tube entrance). The ratio Hr/Ho decreases with decreasing diameter, as shown in Figure 3, until the tube diameter falls below - 15 gm, when it increases again. The effect is due to the existence of a slower moving cell-depleted peripheral layer of low hematocrit, surrounding a faster moving central core of higher hematocrit. When mixed, the tube hematocrit is smaller than the reservoir or discharge hematocrit. It can be shown that: Hr
HD = ( 1 - H r )
+H r
(4)
where is the average tube blood velocity, and and are the average plasma and blood cell velocities [6].
!
o,
v
,
i
HD 0.6
04 i ~ Critica,,lDiameter 92
I~)0
I000
Diameter, l~m
Figure 3. F~thraeus effect for human red cells, HR = 40-45%, for all literature data for suspensions in tubes, at flow rates ensuring no RBC aggregation (cross-hatched region; also 9 F~hraeus data [6]). Critical diameter (-2.7 g m ) ~ the smallest tube through which a shuman RBC can flow. From [6], with permission.
11
2.2.2 The Fdhraeus-Lindqvist effect The effect refers to the original observation [5] that the hydrodynamic resistance of blood and other red cell suspensions decreases as vessel diameter decreases below 300 gm diameter. The most obvious explanation for a lower resistance or a lower effective viscosity (computed using the Poiseuille-Hagen equation to distinguish it from the apparent viscosity, determined from the measured shear rate and shear stress, as in Couette instruments [37]), is that it is a consequence of the F~ihraeus effect, i.e. a lower tube viscosity due to a lower hematocrit. It may also be due to the rheological effect of a non-uniform distribution of cells across the vessel lumen or to the failure of the continuum model of the suspension, as will be discussed in the next section. 3. M I C R O S C O P I C C O R R E L A T E S OF M A C R O S C O P I C F L O W BEHAVIOUR The general problem of microrheology is the prediction of the macroscopic rheological properties of a material from a detailed description of the elements of which it is composed. In the case of blood, the elements are the individual corpuscles (effectively the red blood cells) each surrounded by the suspending fluid, the plasma. What is most striking about the mechanics of the motion is the fact that, under physiological conditions, the cell finds itself subjected to shear stresses and considerable particle crowding, such that it is continually distorted from the biconcave resting shape. The flow behaviour and interactions of individual red cells has been studied using microrheological techniques, in particular the travelling microtube, a device for tracking the motions of cells and colloidal particles through vertically mounted precision bore glass tubes of 50-200 gm diameter, while photographing or videotaping through a high resolution microscope [41-43]. The microscope axis is fixed, and the tube is mounted within a chamber attached to a vertically mounted sliding platform supporting a syringe infusion-withdrawal pump. Both the platform and the syringe pumps are driven hydraulically by continuously variable speed electronically-controlled DC motor drives. 3.1 Rotation and deformation of red cells At shear stresses z < 0.03 Nm -2, isolated human RBC rotate with periodically varying angular velocity, but for small perturbations due to Brownian diffusion, maintaining their biconcave shape [43]. The rotational orbits are similar to those previously found for rigid discs [44,45], in accord with theory [46] applied to rigid spheroids, as illustrated in Figure 5 for the angular motion of the axis of revolution of an RBC and a 4-cell rouleau:
dt
= 1G[ 1 + B(re)]COS 2q~
(5)
12
where ~ is the azimuthal angle of the axis of revolution with the diametrical, X2-axis (Figure 4; = + 90 ~ in positions 1 and 5, and 0 ~ in position 3 in Figure 5), B(re)= (r 2 - 1)/(r 2 + 1), r~ is the equivalent ellipsoidal axis ratio (axis of revolution/diametrical axis) and T the period of rotation through 2n, given by [47]: T - 27r (re -
G(r----~
+
re 1)
(6)
Integration of Equation (5) yields: tan~=re(
Gt
re + l / r e
)
(7)
As predicted for an oblate spheroid having = 0 . 3 8 , at any given instant, the largest fraction of red cells in the tube (49%) are found with their major axes within + 20 ~ of the direction of flow (r = 90 + 20 ~ [43]). Similar results have been obtained with isolated platelets, except that here, the effects of Brownian rotary motion are appreciable [48], as has been found with other colloidal-size particles, such as doublets of latex spheres [49]. With increasing shear stress, the rotational motion progressively deviates from that predicted by theory, as cells spend more time aligned with the flow. At "t"> 0.1 N m -2, a large fraction are seen lying in the median plane of the tube without apparently rotating, instead aligning themselves at a constant angle to the flow; moreover, they are deformed with an increase in the major, diametrical axis [43]. Such behaviour, resembling that of the deformation of liquid drops in immiscible viscous fluids [7,50], has been studied with red cells suspended in higher viscosity media such as buffered low molecular weight dextran [43,51]. It has been shown that there exists a critical shear stress above which the membrane begins to rotate about the interior of the cell, in what has been called a tank-treading motion, an unfortunate term, since the membrane motion is likely transmitted into the interior of the cell resulting in circulation patterns within the hemoglobin solution. As predicted by a two-dimensional theory applicable to Couette flow [52], an increase in the ratio of external to internal viscosity promotes the stationary orientation of the particle.
3.2 Lateral migration of cells The redistribution of blood cells in narrow tubes, which is at the root of the Fhhraeus and Fhhraeus-Lindqvist effects, is due to a net lateral migration of red cells away from the tube wall. There exists a substantial body of theoretical and experimental knowledge on the effect of the vessel wall on the motions of suspended rigid or deformable model particles (the reader is referred to reviews by Brenner [53] and Leal [54]) and of blood cells, including platelets and leukocytes [55,56].
13
I I l
X
3
r '
X
"-1'
U ~
GX
2
2
Figure 4. Rotation of the axis of revolution of a spheroid (heavy line) defined by the Cartesian (Xi) and polar (0, ~0) coordinates constructed at the particle centre of rotation and origin of a Couette shear field.
360 A
i
_
B,| I
......
I
270 _
i
! 4~~ i
i/1
i1) -o
Ot~" r
~.'~_._.
180
~ I
red cell
i
~r:;0;35 .....
Zol i
90
1~ -
,~
4-cell
"-
i I
rouleau
r'i 1"1
-
i I
0.25
0.50 t T
0.75
1.00 FLOW
Figure 5. Rotation of a single human RBC and 4-cell rouleau in Poiseuille flow at G < 20 s-~. A: Variation of the angle ~ with time t during an orbit having the period T. The line drawn through the solid circles was computed from Equation (7); the dashed line corresponds to uniform angular velocity. B" The same particles, drawn from cinemicrographs at orientations corresponding to positions 1 to 5. From [43], with permission.
14
The vessel boundary exerts its influence on the flowing blood cells not only by retarding their translational and rotational velocities, an effect appreciable within one or two cell diameters from the wall, but also by generating radial components in the cell velocity. There are two established mechanisms by which migration across the streamlines can occur [45,47,53-55,57]: migration due to particle deformation at low Reynolds number, and migration due to inertia of the fluid at moderate and high Reynolds number. Here, we concern ourselves mostly with the former mechanism. Deformable particles (liquid drops, flexible fibers), suspended in Newtonian media undergoing Poiseuille flow in the creeping flow regime, migrate away from the wall towards the axis [45]. In the case of a fluid drop, it has been shown to be an effect of the particle disturbance flow which generates a flow with a radial component in the neighbourhood of the drop. In the absence of fluid inertia, rigid spheres and spheroids do not migrate laterally across the streamlines [45]. Under conditions of negligible fluid inertia, particle Reynolds number < 10 -6, the latter defined in terms of the particle translational slip velocity (particle [u] - fluid [U]) of a sphere, radius b at the axis of a tube, radius R, in Poiseuille flow [58]:
Single human red cells and rouleaux also migrate radially inward, as illustrated in Figure 6, for red cells in Ringers solution and in buffered dextran solutions [55]. As previously found with fluid drops [45], the rate of migration increases with particle deformation (it is much greater in dextran than in Ringer solution; Figure 6) and increases rapidly with increasing ratio of particle to tube diameter. Such migration, although severely inhibited by the particle-crowded conditions in normal blood, nevertheless appears to result in a thin (-- 4 ktm wide) cell-depleted layer at the wall and a significant lowering of the hydrodynamic resistance in many blood vessels. 3.3 Cell interactions at normal hematocrits Long before hematocrits approach those normally present in the circulating blood (N 40-45%), even at 0.1% there are frequent two- three- and some multibody collisions, and these, as will be seen in Section 4, can be studied to obtain information on the forces at play during such shear-induced interactions. As the hematocrit exceeds 10%, there are constant collisions between all cells, and above 30%, particle crowding becomes a factor and actually contributes to the deformation of the red cells from their resting biconcave shape. Such deformation can be observed even at shear stresses < 0.03 N m -2 at which the isolated single cell in plasma would have rotated as a rigid disc (as in Figure 5). Since one cannot observe the motions of individual blood cells in the interior of whole blood flowing through narrow tubes, due to continual reflection and
15
~ O or)
~,
~ ~
RBC in 40~ Dextran
! r
E
-;
Ceins
0.2
I,,,,X X
A 0.4 (.9 V 0.6
0.4
,
1
,
0.6
I
0.8
....
1.0
r/R
Figure 6. Measured inward migration of RBC in plasma Ringers solution compared to that in 25% buffered dextran having 35x the plasma viscosity. is the mean tube shear rate, x is axial distance from the tube entrance. From [55], with permission. refraction of the transmitted light, it was necessary to render the blood transparent. This was achieved by preparing "ghost red cells" by osmotic hemolysis of normal red cells in buffer of 1/10 the ionic strength of plasma. The cells were washed to remove the hemoglobin solution, and then slowly returned to their normal salt and protein environment, whereupon they assume their former biconcave shape [59,60]. Over a range of volume concentrations from 10-80%, the measured apparent relative viscosities of red cell and derived ghost cell suspensions did not differ significantly. Tracer normal blood cells were added to the ghost cell suspensions and the motions of these, clearly visible in the interior of the tube, photographed and analyzed.
3.3.1 Deformation due to shear and particle crowding Figure 7 illustrates the continually changing deformation of a red cell in a 55% ghost cell suspension. The cells no longer undergo angular rotation but spend much of their time aligned and deformed in the direction of flow. The membrane likely rotates about the interior in an irregular fashion. Deformation of rouleaux in ghost cell suspensions is also observed [60]. The ability of red cells to deform and squeeze past each other in flow is the microscopic correlate of the macroscopic flow behaviour of blood in Couette viscometers which exhibits such remarkably low viscosity at moderate and high shear rate, relative to that of concentrated suspensions of rigid particles and even that of concentrated emulsions.
16
0 sec.
0.36sec.
0.72 sec.
1.08 s e c .
O.072sec.
' - - - - 10 ~ - - - - ~ I v"
II - 5 0 p m ~--
I I I E
Figure 7. Tracings from photomicrographs of the deformation of a tracer RBC at 7.2 ms intervals in a 55% ghost cell suspension being tracked in robe flow. From [61], with permission.
3.3.2 Velocity distributions at high concentrations The effects of particle crowding on the velocity distribution had previously been explored in the tube flow of concentrated suspensions of macroscopic size rigid spheres and discs. Here, the suspensions were made transparent to transmitted light by matching refractive indices of particle and suspending phase, and then adding visible tracer quantities of particles of the same size and shape but having a different refractive index [62]. It was shown that for 0.05 < b/R < 0.15, when particle volume concentrations exceeded 20%, the velocity distributions became blunted in the centre of the tube with a core of radius r~ in which particle velocities, u(r) were maximum and constant, = UM, and < U(O), the centreline velocity in Poiseuille flow (as illustrated in the upper panel of Figure 8). This region was designated as partial plug flow although this did not imply that the profile was mathematically flat, only that there was no
17
1.0 ~
.
l
l
l
'
R
" ~ ........... -
Complete
Par
Plug Flow , PlugFlow____~____;t~ 0
0.5
-
L....
1.0 '
"-
:
......~
I 1.0
I
-I
,~
o
I
Spheres ~
0.5
r
]
~
Rigid
......~
.....~
................i ~
i
;
00000fO~176176
~ e u , , , e
.o~"
-
Flow
F'o'~
- Drops
~"
9
-
9I
o
,
s
I t Cells~
0.5- 9
-~
c = 32%
....~'0~
0
Plug Flow -
0.5
~
I .__L_
~]__t_~~ /
.(7""
D
iscs Q
1.0 ........ O-
........ 0.25
I 0.50
I 0.75
1.0
u(r)
U(O) Figure 8. Dimensionless velocity profiles of visible tracer particles in 32% transparent suspensions of model rigid particles and ghost red cells: plot of relative radial distance vs. particle translational velocity + centreline velocity in Poiseuille flow at the same volume flow rate. Upper panel: Rigid spheres (b/R = 0.11) exhibiting complete plug flow compared to emulsion droplets (b/R = 0.09) with only partial plug flow. Lower panel: Rigid discs (b/R = 0.08) compared to ghost cells (b/R = 0.11). From [62], with permission. measurable velocity gradient. Nevertheless, these suspensions were still quasiNewtonian in that the velocity profile was independent of flow rate at low R%, and the pressure drop per unit length, AP, was directly proportional to volume
18 flow rate, Q [62]. In fact, it was shown that, despite the continual erratic (Brownian diffusion-like) radial displacements of the spheres in those regions of the tube in which shear-induced interactions occurred, over relatively short time periods, the displacements were reversible in time and space [62,63]. At a given b / R , the degree of blunting increased with increasing volume fraction, c, of the disperse phase, and at a given c, it increased with increasing b/R. Blunting of the velocity distribution was not due to aggregation of the spheres or discs whose interactions are due solely to hydrodynamic forces, nor due to a redistribution of particles in the tube which could have resulted in the two-phase flow of a more viscous and concentrated central core of suspension surrounded by a less concentrated and less viscous peripheral layer. With the exception of the exclusion layer closer than one particle radius from the wall, sphere centres were shown to be uniformly distributed in the tube [62]. Rather, the effect is due to particle interactions in the crowded suspension, and has been treated by Skalak [64] who has interpreted the interactions of the particles in terms of passing versus non-passing motions of adjacent particles. It should be noted, however, that more recent work has shown that redistribution of rigid particles in variable shear fields can and does occur in the creeping flow regime providing particle interactions proceed for long enough times [65-69]. As shown in the lower panel of Figure 8, at comparable particle volume concentrations, the ghost cell suspensions exhibited blunting of the velocity profile, similar to that Of the discs, but with significantly lower r c [59,70]. As with suspensions of rigid particles, r c increased with increasing b / R and concentration. However, as expected, given the shear and concentrationdependent deformation of the ghost and red cells, there is no quasi Newtonian behaviour: the degree of blunting decreased with increasing Q as the parabolic Poiseuille velocity profile was approached, accompanied by a decrease in effective viscosity [71 ], and the motions were not reversible in time and space. In this respect, the non-Newtonian behaviour of red cells and ghost cell suspensions resembled that of concentrated emulsions of deformable liquid droplets in which the droplets are also deformed not only by the shear stress, but also through particle crowding. As shown by work in transparent oil-in-oil emulsions, the droplets were distorted from prolate ellipsoids into irregular shapes [72]. Moreover, the shear-induced deformation led to some migration of the droplets away from the wall (opposed by an outward dispersive force due to the continual interactions of droplets in the core of the emulsion), and as with the red blood cells, such migration was not reversible in time and space. In addition (see upper panel of Figure 8) the velocity distribution was less blunted than that of a suspension of rigid spheres at the same particle volume fraction and b / R , and was flow rate dependent: as the flow rate increased and the droplets became increasingly deformed, the velocity distribution gradually approached the parabolic Poiseuille velocity profile [72,73].
19
3.3.3 Convective dispersion of cells As pointed out above, the resistance to crowding of the blood corpuscles at normal hematocrits opposes any substantial inward radial migration of red cells, such as observed at low hematocrits. At the microscopic level, this resistance is seen as a series of continuous collisions between red cells which results in a marked dispersion of all blood cells and the surrounding plasma, as illustrated in Figure 9 for a tracer red cell and a leukocyte in a 40% ghost cell suspension. Measurements of the mean square radial fluctuations over small time intervals At of the paths of tracer red cells and 2 ktm diameter latex microspheres were used to compute dispersion coefficients D~ defined as:
Dr= <AP>/2At
(9a)
At shear rates from 2-20 s -~, Dr for RBC and latex spheres ranged from - 1 0 -12 to >10 -11m 2 s -1 [55,59], values 2 to 3 orders of magnitude greater than the
0.9 _ 0.8
/ - 40
"~
~PMN
o.7
"l ~
5O
2
7O
Q Q
"
5
~ ~ "0
0.
I
1,0~
u
u
,
r
.-i. Q I
"~
n
35
"-
0.9
"1: 3
tr
0.8
30
20
,
0
5
lO
15
20
Time,s Figure 9. Radial dispersion of the paths of a polymorphonuclear leukocyte (PMN) and an RBC in 40% ghost cell suspensions in tube flow. Asterisks denote times when particles collided with the wall. The slightly deformed PMN and markedly deformed RBC are shapes seen while tracking the cells. From [74], with permission.
20
Brownian translational diffusion coefficient, Dr, for the isolated red cell suspended in plasma, viscosity 7/- 1.8 mPa s at 23~
Dr = kBTr/K, ri = 4.4
x 10 -14 m E S -1
(9b)
where kB is the Boltzmann constant, TK the absolute temperature and K, = 5.23x10 -5 m, the translational resistance coefficient for motion along or transverse to the axis of revolution (2.4 l-tm) of a rigid spheroid, re = 0.38 [75]. The above values of radial diffusion coefficients in ghost cell suspensions are in fair agreement with blood platelet translational diffusion coefficients obtained from measurements of their dispersion in flowing blood, which increased from < 1 0 -12 m 2 s -1 in plasma to 2.5 x 10 -~ m E s 1 at 50% hematocrit in flowing blood at a mean tube shear rate - 100 s -~ [76]. As a consequence of the enhanced radial dispersion of the other blood cells by the red cell motions, is an increase in the two-body collision rate between platelets in shear flow, which is discussed in Section 4. More important is the effect of the red cells in increasing the frequency of platelet-wall collisions with increasing hematocrit, an effect of considerable significance in hemostasis (the normal plugging of cut or damaged vessel wall, a process initiated by platelets) as well as in thrombosis (abnormal, pathological adhesion and growth of platelet thrombi on the walls of arteries), particularly in regions of disturbed flow, as described in Section 6. It has been shown that adhesion of platelets to various artificial surfaces [77,78] as well as artery subendothelium [79,80] (artery stripped of the endothelial layer thereby exposing collagen fibres to which platelets strongly adhere) in flowing blood is markedly enhanced by the introduction of red cell or ghosts cells into the suspending medium. The extent to which red cell deformation or size may affect the increase in the effective diffusion constant of cells and solute has also been studied. The transport of molecular solute was found to increase slightly with cell size and rigidity [81], and the wall adhesion of platelets observed to increase substantially with red cell size [81]. Augmented transport in the shear flow of concentrated suspensions of model particles and red blood cells has been thoroughly reviewed by Zydney and Colton [82]. They proposed a model of augmented solute transport based on shear-induced particle migrations and the concomitant dispersive fluid motion induced by these particle migrations. Augmented solute transport was defined as (D~e/DsF ) -- 1, where Dee is the effective solute diffusion coefficient measured in the sheared suspension, and D SF i s the solute diffusion coefficient in the absence of flow. If particle rotations are assumed to be unimportant, D ~ - DSF + Dp, Op being the particle diffusion constant. Augmented diffusion is predicted to vary as the Peclet number, Pe - b2G/OsF.
3.3.4 Redistribution of blood cells in tube flow At normal hematocrits and in vessels of diameters > 1 mm, the thin
21
peripheral layer of--4 gm thickness will likely have a negligible effect on the distribution of red cells across the lumen. As mentioned in Section 2.2, however, at low flow rates (corresponding to mean velocities < 1 tube diameter s-~) aggregation of red cells becomes an important factor. Paradoxically, it is at low, and not high flow rates, that inward migration of red cells in plasma or high molecular weight dextran buffer is most pronounced in tube flow. Here, the formation and rapid inward drift of rouleaux of red cells results in the two phase flow of an inner core of a network of rouleaux surrounded by a peripheral cell-depleted layer in which can be seen single red cells, small rouleaux and an apparently large number of platelets and leukocytes [40,56,60,83-85]. Such two-phase flow has also been induced in small vessels in the microcirculation of animals by intravenous injection of gelatin or fibrinogen and of dextran. Both in vivo and in vitro it has been shown that the formation of the red cell core is associated with the 'margination' (displacement to the periphery) of white blood cells [56,84,85]. Cine films of blood flow taken in tubes of 100-340 gm diameter clearly show that the effect is due to the outward displacement of white cells by the inwardly migrating network of packed red cell rouleaux [56,85]. Here, an "inverse" F~hraeus effect occurs: since is less than [Equation (4)], the leukocyte concentration in the tube is greater than that in the infusing reservoir [56]. That such two-phase flow results in a decrease in hydrodynamic resistance, R - AP/Q, had earlier been shown in studies of the oscillatory flow of concentrated suspensions of rigid neutrally buoyant spheres where inertial effects led to a small but significant inward migration of the particles from the tube wall [86]. The effect was later demonstrated in the flow of mammalian blood, where it was also shown that in vertically positioned tubes, enhanced red cell aggregation in the presence of 250 kDa dextran resulted in a lower effective viscosity at < 2 s-~ [39]. By contrast, in horizontally positioned tubes, where there is an asymmetric distribution of red cells and aggregates due to sedimentation of the core, the effective viscosity continues to increase with decreasing . The relation between the radius of the core, r~, and hydrodynamic resistance has been studied in buffered 110 kDa dextran suspensions [40], and as shown in Figure 10 in a tube, R = 172 gm, hydrodynamic resistance at first increased with decreasing until aggregation brought about syneresis and a shrinking diameter red cell core. That the effect is due to aggregation was confirmed by experiments in 10% buffered albumin in which rouleau formation was totally absent, and R continued to increase with decreasing down to < 0.2 s-~ [40]. As expected, in buffered albumin suspensions there was no margination of leukocytes [56]. Redistribution of platelets in flowing blood has also been observed. In the case of platelets in arterioles [87,88], and 1-2.5 gm diameter latex microspheres in robes of-- 200 gm diameter [89,90], particle number concentrations near the vessel wall have been shown to be higher than in the core of the flowing
22
1.0
~
,
,
0.9 re I-I 0.8
0.7
i
i
i
i
9
Citrated Blood
O
Heparinized Blood
9
RBC in 1.5% Dextran
i
i
i
i
3.0 0
co ,2.5
E t~ Q- 2.0
.4
1.5 ----it 1.0
i
9 1.5% Dextran i
i
i
1
~ i
i
i
0.05 0.1 0.2 0.5 1.0 2.0 5.0 10 20 , Tube Diameters s 1
50
Figure 10. Development of a peripheral cell depleted layer in a 34% suspension of RBC in 1.5% buffered dextran 110 in a 172 gm diameter vertically mounted tube. The decrease in the relative width, r JR, of the red cell core of aggregates (upper panel) occurs in parallel with a decrease in hydrodynamic resistance (lower panel) as mean tube velocity decreases below 1 s -~. For RBC in albumin buffer (no aggregation) R continues to increase with decreasing . From [40], with permission. suspension ("near wall excess"). The effect, which requires the presence of red cells, has been modelled by adding a lateral drift term to the convective diffusion equation for platelet transport in flowing blood [91]. The reason for the net outward drift is believed to arise from the inward migration of the more rapidly migrating red cells resulting in a marginal layer of lower red cell concentration, whose width continuously fluctuates and should be viewed in a statistical sense. The outward lateral drift of platelets or microspheres occurs
23 because of a net flux of particles from a region of higher red cell concentration and hence higher red cell collision rate to a region of lower red cell concentration and lower collision rate. Such a drift has been shown to occur in alloys [92]. The fact that the location of the near wall excess of microspheres occurs a few microns away from the wall is due to the fact that the particles are physically repelled, and at sufficiently high Rep, fluid dynamically repelled from the wall through inertial effects. It should be noted that the above hypothesis does not attribute the motion of platelets toward the wall to their exclusion by red cells in the interior of the tube, as is the case for leukocytes.
4. SHEAR-INDUCED TWO-BODY I N T E R A C T I O N S : F R O M CHARGED C O L L O I D A L P A R T I C L E S TO BLOOD CELLS
Microrheological techniques, both in tube (travelling microtube [41-43]) and in Couette flow (Rheoscope [93]) have been used to study two-body interactions between charged colloidal particles and between blood cells. The left panel of Figure 11 illustrates a two-body collision between equal-sized rigid spheres as it appears when the particles are tracked by moving the tube upward with a velocity equal to that of the downward flowing fluid at the mid-point of the axis between the two spheres. The collision shown is one in which the trajectories of the spheres are symmetrical, as previously observed in the case of neutral macroscopic particles in viscous media [47,94]. The particles separate along paths having the same radial coordinate, r, as those of the paths of approach. In the case of colloidal-size charged latex spheres, however, interaction forces due to double layer repulsion and attraction due to van der Waals forces come into play when sphere surfaces approach to within a distance h - 100 nm [95]. This results in asymmetric collision trajectories [96], as illustrated in the right hand panel of Figure 11. These can be analyzed and hydrodynamic theory used [95,97] to show that net interaction forces as small as 10-~3 N are detectable. In the case of latex spheres in aqueous solutions of simple electrolytes, the interaction forces have been interpreted by applying the DLVO theory of colloid stability [12,13,95] and thereby obtain values of the Hamaker constant and the retardation parameter of the van der Waals force [96]. 4.1 Theoretical Considerations The fluid mechanical problem of predicting the trajectories of two interacting, neutrally buoyant rigid spherical particles of equal size, b, suspended in a Newtonian fluid of Viscosity 77, undergoing simple shear flow has been solved [97-99], and extended to the case of unequal-sized spheres [100]. More importantly, it has been extended to the case when interaction forces, Fi, t(h), other than hydrodynamic operate at h < 100 nm [95,101]. The relative velocity of the sphere centres a distance s apart, is then given by [95]:
24
-~ = A ( s * ) G b sin20 sin2q~ + dt
C(s*)Fint(h)
(10)
3 rcb rl
and the angular motion of the doublet axis by: dO 1 dt = -4 B(s*)G sin 20 sin 2q~
d~
(11)
1
(12)
dt = -2 G[ 1 + B(s*) cos 2q~]
-u(r)
-u(r)
t
i i i i i
t X3
Repulsion
i
i
1
,, i
,, Xm 8
X~
X3
Figure 11. Left: Two-body symmetrical collision in Poiseuille flow between rigid spheres forming a transient doublet. The collision is tracked by moving the tube with a velocity-u(r), equal but opposite to that of the fluid at the midpoint of the doublet where Cartesian and polar coordinates have been constructed. Right: Projection in the X2X3-plane of the asymmetric paths, due to double layer repulsion, of the centres of two colliding 4 ktm polystyrene latex spheres in 50% aqueous glycerol with l mM KC1. At the centre is the collision sphere which cannot not be penetrated. From [96], with permission.
25 Here, 0 and r are the respective polar and azimuthal angles relative to X1 as the polar axis, as shown in Figure 11; A(s*) and C(s*) are known dimensionless functions of s* = s/b, which have been documented [95,97,99]. Equation (12 is identical with the Equation (5) describing the angular velocity of a rigid prolate ellipsoid [46] with B(s*)= (r~(s*)- 1)/(r~(s*)+ 1). Asymptotic expressions for r~(s*) have been given for large and small s* [97]. Rearrangement of Equation (10) yields the force equation:
3rtbrl ds = A(s*) 3JrbqGb 2 sin20 sin2q~ + Fint(h) C(s*) dt C(s*)
(13)
in which the term on the left represents the hydrodynamic drag force resisting approach of the particles, and the first term on the right the normal hydrodynamic force, Fn, between spheres acting along the line of their centres, being maximal for 0 = Jr/2, i.e. rotation within the X2X3-plane (Figure 11, left), at r = - ~ / 4 when the force is compressive, and at r = rt/4 where it is tensile. Equations (10)-(12) have been shown to apply in Poiseuille flow providing b/R <<1 [96]. When F~n,(h) = 0 the trajectories of approach and recession of the colliding spheres are symmetrical about the orientation q~- 0 of the doublet axis (Figure 11, left), and are defined by two constants, D and E given by the integrated form of Equation (10) [96,97]"
x l =+2s*Dfls*) and x z =+2s*fls*)[E+ g(s*)]
(14)
where g(s*) and f(s*) are integral functions of s* for which numerical values have been given [96,97]. When Fi,.(h) :r O, the paths of approach and recession are no longer symmetrical, and D and E become functions of time, increasing when F..t(h) > 0 (repulsion), and decreasing when Fi,.t(h) < 0 (attraction). 4.2 Determination of the Interaction Force In principle, Fi,~(h) can be computed from Equation (10) if one can measure the observed translational velocity, ds/dt. This turns out to be difficult since the trajectory may lie out of the median, XzX3-plane of the tube, and even for collisions in the median plane it is impossible to measure ds/dt when h < 100 nm, just when particle interaction forces begin to be important. Instead, the experimentally observed trajectories are fitted by numerically integrating Equation (10) and solving for D and E [96]. In addition, in the case of the charged latex spheres, by using expressions from the DLVO theory, it is possible to compute the Hamaker constant, A, for the latex:
F~nt(h) --
Fattr(h) +
Frep(h) ,
26
I(1+
where the van der Waals attractive force, Fattr(h ), is given by: Fattr(h ) =
12h2
Ab Fattr(h) - 12h2
p
(15a)
1 + 1.77p 2
10.98 0.434 0.0674 p
p2
I-
P3
p > 1
(15b)
and the double layer electrostatic repulsion force, Frep(h), is given by:
Frep(h) = 2~Kbe~
/ / 1 + e -rh
(16)
Here, p = 27rh/A, is the retardation parameter, ~, the London wavelength, K the dielectric constant of the suspending medium, e0 the permittivity of free space, gt0 the surface potential (usually taken to be the measured (-potential), and tr the reciprocal Debye double layer thickness. Analysis of trajectories such as those shown in Figure 11 for polystyrene latex spheres in 50% aqueous glycerol at 1 mM KC1. led to a value of A = 3x10 -21 Joule in good agreement with a value (2.7x10 -21 Joule) extrapolated from that determined in water [96]. It should be noted that Equation (10) neglects the effect of the double layer on the velocity field around the spheres. The layer introduces coupling forces and torques which should be included in the functions A(s*), B(s*) and C(s*) [101]. Fortunately, under the conditions of the experiments shown in Figure 11, fight, tcb >> 1, and electroviscous forces and torques were negligible.
4.3 Two-Body Collision and Collision Capture Frequencies 4.3.1 Collisions in the absence of interaction forces Traditionally, the frequency with which two spheres collide to form doublets in a sheared suspension containing No singlets was computed assuming, as originally done by Smoluchowski [102], that all particles move along linear trajectories until they collide with a sphere. A reference sphere of radius b is located at the origin of a Cartesian coordinate system and a simple shear field, and all sphere ccntres which pass through a collision sphere of radius 2b and cross-sectional area 4~b 2 will undergo collisions. The general case of the twobody collision frequency, J~, between a reference sphere of radius b~ and spheres of radius b z , concentration N2 per unit volume, is then given by: 4
3
Js - -~ (1 + b 1/b 2) 3N2Gb 1
(17a)
27
32
= --~ NoGb
3
(17b)
for equal-sized spheres of radius b. The classical Smoluchowski theory has been corrected using the rigorous hydrodynamic theory of the curvilinear particle collision trajectories described by Equation (10) with F~,~(h) = 0 [103]. The difficulties in defining a collision were eliminated by defining a "curvilinear collision sphere" of radius p,, around the reference sphere. Correction factors to be applied to the results of the rectilinear theory (collision sphere radius = 2b) were then calculated for the cross-sectional area of the collision sphere, the collision frequency and the average doublet lifetime as a function of the ratio p,,,/b. Curvilinear and rectilinear theories were shown to be compatible at p,,,/b = 2.38, which appears to correspond to the minimum dimensionless distance of approach of the spheres recognized experimentally by different observers as a collision [47].
4.3.2 Collisions in the presence of interaction forces When interparticle forces act between the spheres, the collision capture frequency per particle, J,, giving the number of collisions that result in permanent doublet formation, is then defined as" 32
Js = - ~ aoNo Gb3 =
8aocG zc
(18a) (18b)
where ao = JJJ is the capture efficiency and c is the volume concentration.. Capture efficiencies have also been computed using rigorous hydrodynamic theory. In this case the collision capture cross-section was found by integrating the trajectory Equations (10)-(12) forward in time by trial and error [104]. Capture efficiencies were calculated as a function of the ratios of the interaction forces Fr~p and Fa,,, to the hydrodynamic force for equal sized [104] and unequal-sized spheres [105]. In the absence of repulsive forces, for equal sized spheres, ao decreases with increasing G, as would be expected from Equation (13), since with increasing G both the hydrodynamic drag and the hydrodynamic normal force between spheres increases while the interaction force remains constant. When both repulsive and van der Waals forces act, ao first increases at low G, then decreases at moderate and high G. The values of ao depends strongly on the ratio of the sphere radii, being appreciably greater for equal sized particles, since the minimum distance of approach between spheres actually increases with decreasing radius ratio.
28 Measurements of the number of doublets formed in the Poiseuille flow of 1 ~m diameter polystyrene latex spheres at 1 mM KC1 demonstrated that ao decreased with increasing G in going from tube centre to the periphery [104]. Experiments with aqueous suspensions of 2.6 ~tm diameter polystyrene latex spheres in Poiseuille flow showed that at 10 mM KC1, many of the two-body collisions resulted in the formation of permanent doublets [106]. Some of the spheres of these doublets were in a secondary potential energy minimum at distances h between 15 and 17 nm, as shown by measurements of the period of rotation [Equation (6)]. Theory shows that the dimensionless period of rotation, TG, depends not only on h but also on whether the spheres are capable of independent rotation [97]. There is a different solution for each of these cases, and TG at a given s* is markedly greater for independently rotating spheres than for rigidly coupled spheres. By contrast, studies carried out in the presence of a cationic polyelectrolyte revealed the formation of permanent doublets many of which rotated as dumbbells of rigidly-linked spheres, being in a primary potential energy minimum, and exhibiting values of TG close to the predicted value of 15.62. At sufficiently high polymer concentration, polymer bridge formation appeared to provide the most reasonable explanation of their flow behaviour [107]. 4.4 Two Body Collisions Between Blood Cells In the circulation, aggregation of blood cells and their adhesion to the vessel wall is most often mediated by cross-bridging of biological polymer adhesion molecules (carbohydrates and proteins) through non-covalent receptor-ligand linkages. These interactions occur under flow-induced normal and shear stresses, and therefore, characterization of the physical strengths of such bonds is an important prerequisite for understanding adhesion processes such as occur in cancer metastasis, platelet thrombosis and leukocyte margination and extravasation. Modem techniques in physical chemistry, cell and molecular biology have made it possible to relate the microrheology of cell-cell and cellwall adhesion to the actual biological polymer molecules (receptor and ligand) and bonds involved in these processes. One has therefore been able to study the effect of fluid mechanical forces on intermolecular reactions and deal with the transduction of fluid mechanical forces via the cell membrane into the cell interior through the activation of second messenger systems [108-110] and the relation between the chemistry of cell motility and the forces generated by the cell [111]. Some studies, concerned with rheology at the molecular level, are now described in this and the subsequent Section.
4.4.1 Application to shear-induced aggregation of platelets The kinetics of the adenosine diphosphate (ADP)-induced aggregation of native human platelets in plasma and of washed cells in physiological salt solutions subjected to shear in small tubes have been investigated. The ADP
29 concentrations chosen were those just able to induce aggregation, so that the effects of shear and time of exposure to ADP would be at their most sensitive. The magnitude and time of exposure to shear were varied independently by using a double infusion system [112] in which the cells and agonist simultaneously entered a tiny mixing chamber and flowed through preset lengths of polyethylene tubing at mean transit times from 0.2 to 86 s and mean tube shear rates, , from 40 to 1920 s -~ (representative of those in the circulation; Table 3). The suspensions were collected into glutaraldehyde to arrest the reaction, and all particles in the volume range 1 to 105 ~tm 3 counted and sized using an aperture impedance counter. Aggregation was measured by the decrease in the number of singlets, and the number and size of aggregates. Adenosine diphosphate-induced aggregation is known to be mediated by the plasma protein fibrinogen which specifically binds to the activated form of the platelet membrane integrin glycoprotein complex, GPIIb-IIIa [113]. Most likely, aggregation occurs via cross-bridging of fibrinogen molecules between receptors on adjacent cells, (Figure 12) [114]. Successful cross-linking requires simultaneous binding of opposite ends of the fibrinogen molecule to two cells immediately after activation of the GPIIb-IIIa complexes, and before their saturation with fibrinogen, an unlikely scenario given the high plasma fibrinogen concentration (2x107 molecules available per platelet) and receptor surface density (5X104 sites per cell). A model of aggregation therefore requires either a low affinity for fibrinogen binding with the continuous breaking and making of bonds, or the time-dependent exposure of new
i
t'~, C
/
r
r
Figure 12. Schematic diagram of platelet activation by ADP causing shape change from smooth discocyte (A) to spiny spheroid (B) with pseudopods, and transformation of the membrane glycoprotein IIb-IIIa receptors (inset) into active forms (o) capable of binding fibrinogen (Fb; C) which can cross-link cells producing doublets. From [115], with permission.
30 receptors that permit cross-linking to occur during the time of collision. In platelet-rich plasma (no red cells present) prepared from blood anticoagulated with sodium citrate ([Ca ++] -- 45 ~tM), at 0.2 ~tM ADP, the presence of a shear-dependent initial lag time indicated that both a weak and a strong platelet-platelet bond is involved in aggregation at physiological shear rates [116]. As illustrated in Figure 13 the initial rate of disappearance of singlets into microaggregates was the highest at the lowest shear rates, but later in time decreased markedly. As the shear rate increased, the initial rate of aggregation decreased; nevertheless a relatively high rate of aggregation was sustained at later times. These results were interpreted using two-body collision theory even though activated platelets are not rigid spheres, but spheroechinocytes (spiny spheroids) with numerous pseudopods of appreciable length capable of significantly increasing the effective collision cross-section of the cell [117]. With platelets, the interaction force is in part repulsive, due to like surface charges (negative charges on the platelet surface glycoproteins rich in sialic acid), and attractive, due in part to van der Waals forces, but mainly due to polymer bridging by fibrinogen. Since net attractive forces appear to operate, a 0 was expected to decrease with increasing shear rate.
[
'
I
I
'
'
'
- ~
100 ~
'
335
80
--~ -~
'
'
41.9
S-1
-[
S -1
1335
S-1
60
t~
~
40
r9~
2o
0
20
40 60 80 M e a n Transit Time, s
100
Figure 13. Effect of shear rate on the ADP-induced aggregation of platelets in plasma, expressed as % singlets remaining vs mean transit time in the tube (R = 0.6 mm); [ADP] = 0.2 gM; bars are + 1 S.E.
31 Indeed, ao, computed from Equation (18b) over the first 4 seconds, was greatest at the lowest shear rate (= 0.26), but decreased markedly to 0.017 and <10 .3 at the intermediate and highest shear rates, respectively. Thus, it appears that a weak bond between cells of the microaggregates formed within the first few seconds, diminished in strength with time, and the low collision frequency at low shear rate was unable to sustain a high rate of aggregation. As shear rate increased, the initial decrease in ao led to a decrease in rate of aggregation, yet at later times the high collision frequency, aided by the arrival of a more shear stress-resistant bond sustained a high rate of aggregation. In the presence of red cells at 40% hematocrit, the initial rate of single platelet aggregation in whole blood is an order of magnitude greater than in platelet-rich plasma [118,119]. There has been a long-standing discussion on whether this is due to a chemical (release of ADP from the red cells) or a physical effect of the red cells. As shown in Figure 14, some aggregation of platelets in blood at low shear rate was observed in the absence of any added ADP, and this could be abolished by the addition of an ADP scavenger enzyme [119]. However, there remained a large effect believed to be a physical one. An obvious candidate for the increased rate of aggregation is the increased platelet diffusivity in blood, described in Section 3, which might lead to an
100
._
-
~~
9~
-
-
- _ -
-
=
- _ -
-
-- "" -- ~BloodL,H = 40%, no ADP
-
-
8o 60 40
20 0 0
10
20 30 40 M e a n Transit Time, s
5O
Figure 14. Plot as in Figure 13 showing marked effect of RBC on the ADP-induced aggregation of platelets in blood in tube flow at = 42 s -a. From [118], with permission.
32
increased two-body collision rate. However, arguments based on calculations of a Brownian translational motion two-body collision frequency [102]: JD = 16~bNoDr
(19)
show this is unlikely. Here, D r is now the effective diffusion coefficient, obtained from measurements of the radial dispersion of platelets in blood undergoing tube flow (of the order of 5• -~ m 2 s-~). Computations indicate that the two-body collision frequency would increase by 156% at = 42 s -~, but only by 19% at = 335 s ~, over that in platelet-rich plasma, too small to explain the large increase in the observed initial aggregation rates [118]. Instead, the hypothesis that an increased a0 results from an increased collision doublet lifetime, To, was tested. To this end 2.5 ~tm diameter latex spheres were suspended 40% red cell ghost suspensions in plasma as a model of platelets in blood and the measured distribution, '~'Dmeas, compared with that in plasma in the absence of multibody interactions, and with that, TDtheor , predicted from Jeffery's theory. From Equation (7) with r~ = 1.98 for a doublet of rigidlylinked, touching spheres [F~,,(h) = 0],, the doublet lifetime is: - ~ 4"97 tan-l(1 tan q~0) TDtheor-
(20)
where ~0 =-~00 is the collision angle when the spheres are first seen to make apparent contact. In the absence of other interactions, the doublet is predicted to rotate until the orientation q~ = +r the reflection of the collision angle, is reached when spheres separate. Figure 15 illustrates the erratic time course of the ~0-orientation of the axis of a collision doublet, compared to that computed from Equation (20). For 320 doublets, the m e a n TDmea/'~Dtheor : 1.61 + 1.80 (S.D.) compared to 1.00 + 0.31 for 90 doublets in plasma. Analysis showed that impedance of the rotation is the likely mechanism by which "~Dmeas > "~Dtheor" A more sophisticated analysis of shear-induced platelet aggregation uses population balance mathematics and a laminar shear coalescence equation [120,121] in which a particle density function expresses the makeup of a collection of singlets and aggregates of various sizes. The collision efficiency and an aggregate void fraction (to avoid the problem that platelets and are not spheres) are introduced and experimental data matched by computer analysis. A population balance analysis has also been used to model the shear-induced aggregation and disaggregation of activated neutrophils [122,123].
0
E F F E C T OF FORCE ON THE K I N E T I C S OF C E L L - C E L L A N D CELL-WALL ADHESION
Adhesion between cells is generally mediated by weak non-covalent molecular linkages which, in the system at rest, can nevertheless persist for a
33
90 60 30
-30
]
9
I I I I
A
,
~ 0.05
_ .
Measured Theoretical
I
I 1
-60 -90 0
, "
I 0.10
,
I 0.15
i
I
0.20
0.25
Time, s Figure 15. Time course of the ~0-orientation of a collision doublet of 2.5 gm latex spheres in a 40% ghost cell suspension (O) compared to that for an undisturbed doublet (A) formed at the same initial ~ = -~o and G, predicted to separate at the mirror image q~ = +~Oo. Dashed lines give time interval = "~Dmeas(> "~Dtheor) o f apparent contact between the spheres. From [119], with permission. long time. In contrast, the linkages quickly rupture when mechanical stress is applied to the cells. Thus, molecular attachments appear to be maintained by kinetic traps and bond rupture should be regarded as a kinetic process far from equilibrium, with apparent bond strengths expected to depend on the rate of force application. With the advent of ultrasensitive mechanical techniques such as the atomic force microscope [124,125], the biomembrane force probe [126; Section 5.3] and optical tweezers [127], the properties of single adhesive bonds at cell interfaces can be tested with nanoscale resolution. In this Section, the focus is on bonds made and broken in the flowing blood, and discussion is limited to the kinetics of bond rupture between freely suspended cells in shear flow, and to the kinetics of leukocyte rolling along, and adhesion to the vessel wall. In both cases, the experimental results have been modelled using relations for the effect of an external force on the off rate constant of a single bond.
5.1 Models of Bond Rupture under Force 5.1.1 Bell model Bell [128] was the first to treat the increased rate of bond dissociation under force in biological systems, in particular with respect to receptor-ligand bonds,
34 applying to such bonds a model of Zhurkov [129] for the force dependence of bond rupture based on empirical studies of the fracture of macroscopic solids. This model is actually an extension of transition state theory in gases [130] in which it was proposed that intermolecular forces in a gas could be treated as a one dimensional random walk in a potential energy well. The probability of escape depends on the depth, E 0, of the well and the natural frequency of the bond in vacuum, "to (-- 10 -~3 s for atoms in a solid). For a parabolic energy barrier it was shown that the escape or break-up time in the absence of an extemal force is given by: t o = "to exp (E o/kBTK)
(24)
where the exponential represents the probability that thermal fluctuations provide enough energy for the barrier to be surmounted (for the transition state to be reached). Zhurkov introduced the force dependence by postulating that the extemal force, f per bond, acts directly along the reaction coordinate x to reach the value xa at the transition state, and that the force reduces the energy barrier in a linear fashion; thus tb= To exp (E 0 - f x ~ )/kBT K = t o exp (-z'f)
(25a) (25b)
where Z = xa/kBTK. Therefore, when f reaches the value f/3 - Eo/xa, t~ = "t'0, and bond failure is almost instantaneous. By identifying tb with 1/~r the reverse, or off-rate constant, the connection to bond reaction rates is made: tr
= tCog~ exp (z'f)
(26)
where tCog~ = 1~to. The probability of bond breakage Pb in a short time interval, At, is then given by [ 131 ]" Pb = 1 -- exp (-tCoffAt) = 1 - exp [ 0te x p(x'f) At]
(27)
5.1.2 Evans model In the Bell model, all the force-driven features are lumped into one p a r a m e t e r - a length x~, often referred to as the 'range' or 'width' of the bond, that gives little insight into molecular events. An alternative form was therefore proposed [132] for forces sustained at a level nearf~: tb= "gOexp -f-
(28)
where ~:0 is again the time for rapid bond rupture and m was used to capture variations in rupture behaviour from ductile (m < 1) to brittle failure (m > 1). Evans et al. showed that any realistic technique for putting stress on bonds will
35 have a rise time or rate of loading and this, coupled with the inherent stochastic nature of receptor-ligand bonds, will lead to the peaks observed in the histograms of bond strength from mechanical probe experiments [132,133]. Hence, the strength of a single bond is governed by the rate of loading, and they have used the Brownian dynamics theory of Kramers [130] to show that the rate follows a general form given by: to# = to#~ gff) exp [AEo(f)/kJK]
(29)
which follows the required phenomenological descriptions for the force-driven kinetics [134]. These models of bond rupture under force have been used to analyze experimental data obtained on the break-up doublets of red cells, and on the roiling of leukocytes on endothelium, as described below. 5.2 Cell-cell adhesion- break-up of doublets of red cells Studies of two-body interactions between charged latex spheres described above led to similar experiments on the interactions between sphered red blood cells. Rigid sphered swollen human red blood cells (SSRC) of known antigenic type were prepared, and permanent doublets formed by cross-linking the spheres with the corresponding antibody [42]. Normal red cells were sphered and swollen in 0.2M buffered glycerol containing sodium dodecyl sulfate, fixed in 0.08% glutaraldehyde, washed and suspended in buffered media containing viscous co-solvents such as glycerol, sucrose or 40 kDa dextran. In the absence of antibody, two-body collisions between the 6.3 l.tm diameter SSRC involving transient doublet formation were found to be markedly asymmetric with an apparent repulsion, the trajectory constant E increasing and the minimum h > 50 nm [135], i.e., a distance large compared to that at which double layer interactions become important. The result was likely due to the interaction, principally steric, between the glycocalyx of the colliding cells.
5.2.1 Hydrodynamic force of break-up Instead of studying 2-body collisions, therefore, the model of doublets of red cell spheres, cross-linked by antigen-antibody linkages was used in order to measure the hydrodynamic forces required for bond rupture. The SSRC were suspended in buffered aqueous 48-56% sucrose of known viscosity containing 0.075-0.6 nM of monoclonal anti-blood group antibody. Permanent doublets of SSRC formed through two-body collisions when the suspensions were sheared at low G (< 10 s-~). Measurements of TG showed these doublets to be rigidly coupled [42]. For such a doublet in shear flow, ds/dt = 0, and if it can be induced to break up, then, from Equation (13), F~,,(h) can be computed: A(s*)
Fint(h) = - Fn = - C(s*-"'~37cbr/Gb2 sin20 sin2q~
(30)
36
However, the bonds linking the red cell spheres are also subject to hydrodynamic forces acting normal to the doublet axis, i.e. shear forces. The general method of calculating forces, torques and translational velocities of neutrally buoyant rigid spheres in a simple shear field [97], based on the matrix formulation of hydrodynamic resistances in creeping flow [136], was used to compute the normal force acting along, and the shear force, F~, acting perpendicular to the axis of the doublet [137]: [31]
F, = a,(h)rlGb 2 sinZO sin2dp
F s = O~s(h)rlGb2
sin 0 1 - sin 20 COS2~)
/
/1
(2sin20 cos2~-
1)2sin2r
+ cos20 C0S2~
2
[32] where a,(h) and a,(h) are force coefficients weakly dependent on the minimum distance of approach, h. 5.2.2 Measurement of the hydrodynamic force at break-up To measure the hydrodynamic force required to break up permanent doublets of SSRC, the suspensions were subjected to a slowly accelerating Poiseuille flow in the travelling microtube apparatus, and the particles tracked and videotaped until break-up occurred. From the videotape replay, the doublet radial position and velocity yielded the local shear stress as well as the particle orientation at break-up, enabling F, and F~ to be computed. It was estimated that, at an antibody concentration of 0.3 nM --- 15 IgM molecules were available over the estimated 2000 antigenic sites in the contact area of the SSRC (2.5 x 104/lal) available for cross-linking (between h = 20 and 45 nm [42]). Although this implies a maximum possible 30 cross-bridges, the actual number of bonds was most probably much smaller, as was confirmed by the computer simulation of a stochastic theory of bond rupture (see below). Thus, the fraction of two-body collisions resulting in permanent doublets at shear rates < 10 s-1, was estimated to be only ---1%; moreover it was unlikely that all available IgM molecules would be bound to antigen, and many of the bonded antibody molecules would not participate in cross-linking. In calculating the forces at break-up, it was first assumed that the cells separated the instant a critical force to break n bonds of strength f, F = nf, was reached. Figure 16 shows scattergrams of the normal force at break-up, F,,, as a function of antibody concentration and suspending phase viscosity. It is evident that there is considerable overlap in the range of F,, without any evidence of clustering around a set of discrete force levels corresponding to different numbers of antigen-antibody crossbridges, although the mean measured force increased significantly with antibody concentration (Figure 16).
37
250
<> 58%
56?
200
Z 50~
150
8
49
o
100
46%
56% m
-,~ozx 6
A
~
o O 0
~
5O 0 0.15
0.30 [IgM], nM
0.60
0.15 [IgA], nM
Figure 16. Scattergram of the distribution of the normal hydrodynamic force to separate doublets of SSRC cross-linked by monoclonal IgM and IgA antibody, suspended in buffered 46-58% sucrose in an accelerating Poiseuille flow. Also shown are mean values + 1 S.D. From [93], with permission. The absence of clustering of F, values suggested that bond rupture leading to doublet break-up was a stochastic process, both time and force dependent, as shown by Evans et al. in studies using the micropipette aspiration technique [132]. Thus, in the accelerating Poiseuille flow experiments a doublet might well have broken up at a separation force lower than that measured at break-up, had the particle been given enough time under a lower force.
5.2.3 Time and force dependence of break-up Accordingly, experiments were undertaken in Couette flow using a transparent counter-rotating cone and plate d e v i c e - the Rheoscope [51,93] - in which a constant shear stress could be applied virtually instantaneously, and the time dependence of break-up studied as a function of the applied force. As predicted, at a given applied force, there was a pronounced distribution in times to break up, while the average time to break up decreased with increasing applied force. Figure 17 (left panel) shows the observed temporal distribution of break-ups plotted as a function of the dimensionless number of rotations (t/T) from the onset of shear. It can be seen that, in the intermediate and high force ranges, most of the break-ups occurred within the first 2 to 3 rotations. In the lowest force range, the fraction of break-ups fluctuated between 1.5 and 4% in the first 7 rotations, after which no more doublets broke up.
38
0.25
0.25
o o
F n = 55 p N
F n <70pN
0.20
I I
0.15 - I O ..I
9.-o..
70
---i-
Fn> 140pN
,•
I I
0.20
9-.o..-
----A- F n = 1 6 7 p N
l 1 l 9 l
0.15 o
- 0.10
0.10 -" I
0 0 o
Fn=lllpN
A
-9 0.05
"~ 0.05 -
/A
\ "..,"1
0 0 ] 2 3 4 5 67 8 910 Number of Rotations
i
"A..-~ x
~' -~.~-]
9 *,,,4
ra~
0
|
I
0
1 2 3 4 5 6 7 8 9 10 Number of Rotations
I ,1
I
I |
Figure 17. Left: Fraction of doublets of SSRC, cross-linked by 75 and 150 pM IgM antibody, breaking up per rotation in Couette flow after onset of shear. Right: Computer simulation of break-up for the best fit parameters xa = 0.40 nm, to = 100 s, <no> = 4.0. From [93], with permission.
5.2.4 Locus of bond rupture: use of antigen-linked latex spheres There remained the question of the locus of bond rupture in the SSRC studies. Micropipette aspiration experiments have shown that it is possible to extract antigen under force from a normal red cell [132]. Subsequently, experiments in shear flow showed that blood group antigen extraction can occur even with fixed cells [ 138] which cast doubt on the assumption that breakup of the SSRC crossbridge is due to failure of antigen-antibody bonds. Therefore, studies in which a synthetic blood group B antigen was covalently linked to carboxyl modified polystyrene latex spheres and the spheres crosslinked by monoclonal IgM antibody were undertaken [139]. Here, break-up of doublets of microspheres should occur through the rupture of antigen-antibody bonds, and would be expected to lead to a significantly different force dependence on doublet lifetime. As with the SSRC over a comparable range of normal force, there was a distribution in times to break up. However, significantly higher forces were required to achieve the same degree of breakup for doublets of antigen-linked spheres than for SSRC.
39
5.2.5 Monte Carlo simulation of doublet break-up Using the Bell theory [128], a computer simulation of doublet break-up under shear was developed in which to and x, [Equation (25)] were varied to fit the data collected in the Poiseuille and Couette flow experiments. Since bond formation is thought to be a Poisson process, specification of the average number of bonds, , is all that is required to characterize the distribution of bond numbers used in the computation of the force per bond when fitting the experimental data. Each rotation was divided into 1000 equal time steps, and for each step Pb was computed from Equation (27) and the force per bond calculated from Equation (31) for the current instantaneous values of 0, ~0 and nb. A random number between 0 and 1 was chosen from a uniform distribution for each bond remaining. If the number drawn was less than Pb, the number of bonds was reduced by one, and the force per bond on the remaining bonds recalculated. Thus, bonds are postulated to break sequentially. The results obtained for the best fit of the Couette flow data with 75 pM IgM are shown in the fight hand panel of Figure 17, with x,= 0.40 nm, to = 100 s and - 2.5 bonds. It is evident that the general features of the real experiments are quite well reproduced. Computer simulation of the break-up of doublets of antigen-linked latex spheres indicated that differences between these and populations of SSRC were due to a change in bond character. The antigen sphere-IgM bond was less sensitive to the applied force, had a lower value of x, (0.12 nm) and required a higher value o f f , (1000 pN) for instantaneous rupture compared to that (360 pN) for the SSRC [139]. This supported the notion that antigen-antibody bonds were ruptured in the case of the antigen-linked spheres whereas antigen was able to be extracted from the membrane of the SSRC. 5.3 A Biosurface Force Probe In Section 5.1 it was pointed out that the strength of a single bond is governed by the rate of loading, a factor which could not be controlled in the break-up studies described above. Evans et al. [126], describe an ultrasensitivetunable force transducer that can measure forces over an incredible range from 0.01 pN to > 1000 pN - the strength of a covalent b o n d - and in which the rate of loading can be controlled. As illustrated in Figure 18, the transducer is a microbead probe attached to a pressurized membrane capsule (e.g. RBC or lipid vesicle). The pressure, P, is controlled by micropipette suction and sets the membrane tension Zm:
Rp Zm = P2( 1 _ Rp/Ro )
(33)
where R 0 and Rp are the radii of the membrane capsule and suction pipette, respectively. When a small force, f, is applied to the probe, the capsule is
40 -p
Tm
Azt
Y Figure 18. Schematic diagram of an ultrasensitivetunable force transducer formed by a pressurized membrane capsule. From [140], with permission. elongated by a displacement, Az,, proportional to the force. The stiffness constant, kf for the transducer ( f - kyAz,) is given by:
kf
=
27t-z"m ln(2Rp/R o) + ln(2R 0/rb)
(34)
r b being the radius of circular contact between capsule and microbead. Since stiffness is proportional to tension, the force sensitivity can be tuned in operation between 1 l.tN m -~ and 10 mN m -~ simply by changing P, and Azt is measurable down to 0.01 ~tm using optical techniques. The microbead is chemically conjugated to separate ligands, one for macroscopic attachment to the capsule surface and the other for focal bonding to receptors on a biological surface. The microbead is brought towards the surface beating the receptor, bond formation is signalled when fluctuations in height (which depend on the rigidity of the receptor interface) diminish markedly; similarly bond release shows up when the fluctuations return to the original level. The relative frequency of formation and release then quantitates on/off rates. Finally, the transducer can be retracted to load the force on the attachment until bond rupture occurs.
5.4 Adhesion and detachment of cells from surfaces: leukocytes In the last 15 years, much work has been devoted to elucidate the mechanism whereby leukocytes, in particular polymorphonuclear cells (neutrophils) and lymphocytes, are able to be arrested on the endothelium of post-capillary venules and subsequently to migrate out into the extravascular space. The process appears to occur in three stages: "rolling" of leukocytes along the vessel wall, followed by firm arrest of the cells, and finally transmigration through the vessel wall [ 141 ].
41
5.4.1 Rolling vs firm adhesion of leukocytes A series complex interactions involving fluid mechanical forces and the formation and breakage of receptor-ligand bonds are involved when circulating leukocytes adhere to the venular vessel wall [141]. Blood cells flow through post-capillary venules of diameter from 15-25 lam at mean velocities ranging from 0.3-1 mm s -~. Wall shear stresses are said to be of the order of 0.2 N m 2. Normally, leukocytes flowing in close proximity to the wall do not adhere to the endothelial cells, i.e., no bonds are formed between the leukocyte and an endothelial cell. However, during inflammation, stimulation of the endothelium and/or the leukocyte together with the associated low flow state results in the appearance of more peripherally located white cells [56,84]. Many of these cells appear to be "rolling" along the vessel wall with translational velocities from 10-40 ~m s -~ [142-145], markedly lower than predicted for rigid spheres very close to a plane wall [146]. Rolling is not a smooth process" the translational velocity varies continually and the cell is frequently arrested. Some of the rolling and arrested cells rejoin the mainstream, then flowing at much higher velocities; others become firmly adherent, likely because of activating stimuli received while in contact with the endothelium. These cells can later proceed to enter the interendothelial cell junctions and migrate out [147]. Each of the 3 steps of the cell-wall interaction process is mediated by different sets of receptor-ligand pairs (adhesion molecules)with different kinetics of bond formation and rupture. To elucidate the separate roles of the various adhesion molecules and the effect of fluid mechanical stress on adhesion, recourse was had to in vitro experiments in which leukocytes were allowed to roll on, and adhere to surfaces bearing adhesion molecules in a parallel plate flow chambers [148]. Except near the side walls of the chamber, there is a Poiseuille velocity distribution between upper and lower surfaces, and the wall shear rate Gw = 3Q/2wh 2, where w is the width and h the height of the chamber.
5.4.2 Leukocyte rolling: selectin mediated adhesion Since leukocytes, unimpeded by interactions with endothelial cells, likely flow past the vessel wall at velocities of the order of 10 venule diameters s ~, a rapid rate of bond formation is necessary to arrest such a cell. As regards rolling, which involves the continual making and breaking of bonds, the kinetics of bond formation and dissociation are clearly more important than the so called bond affinity, i.e., the equilibrium association constant K A = K'o,/n'oy, where too, (moles s ~) and 1r (s -~) are the on and off rate constants. The ability of the bonds to withstand high strain before rupture (high "tensile strength"; low value of xt~) will also be important in initial adhesion and rolling, an issue addressed in two mathematical models of the rolling process [131,149]. The initial slowing down of leukocytes in the bloodstream involves a class of cell transmembrane adhesion molecules known as selectins (L-selectin on the
42
leukocyte and E - and P-selectin on the endothelial cells). The ligand binding sites on selectins are calcium-dependent lectin-like domains, carbohydrate structures which recognize fucose containing oligosaccharide moieties known as sialyl-Lewis x and sialyl-Lewis ~ on the leukocyte or endothelial cell. In the case of neutrophils, L-selectin is concentrated on the tips of the microvillus-like projection of the ruffled cell membrane, favouring the formation of the selectin-carbohydrate bond with the endothelial cell. Attempts have been made to determine the parameters of Bell's model from experiments with rolling cells as well as receptor-linked latex spheres. Alon et al. [ 150], using neutrophils rolling along lipid bilayers containing P-selectin on the lower surface of a parallel plate flow chamber, measured the force dependence of ~;oZ for the P-selectin-carbohydrate ligand bond using the distribution of times during which cells were arrested. They found xt~ = 0.05 nm, an extremely small value, suggesting the linkage to be close to an "ideal" bond, the lifetime of which varies little with applied force [131,151 ]. The work also showed that both too, and tCog~ were fast compared to other known macromolecular interactions [150]. This may be compared with xa = 0.40 nm, obtained from computer simulation of the observed force dependence of rupture of a protein-protein bond between doublets of latex spheres [152].
5.4.3 Leukocyte adhesion mediated by integrins After a variable period of rolling, if activation of the leukocyte has occurred, bonds between the activated leukocyte 13/-integrins and their counter-receptors on the endothelial cell belonging to the immunoglobulin superfamily, can form after transient cell arrest and eventually induce firm adhesion. Integrins are heterodimeric cell-surface proteins consisting of one of several o~-subunits and one of several [3-subunits bound non-covalently. The ~2-integrins, Mac-l, LFA-1 and VLA-4, are the known leukocyte integrins. The immunoglobulin superfamily of receptors is defined by the presence of the immunoglobulin domain, which is composed of 70 amino acids arranged in a well characterized structure [141]. The immunoglobulin adhesion molecules implicated in leukocyte-endothelial cell interactions are cellular adhesion molecules ICAM-1 and VCAM-1 and mucosal addressing cell adhesion molecule, MAdCAM-1. The effect of wall shear stress on leukocyte-endothelial cell adhesion has been extensively studied. Selectin-mediated rolling followed by integrin-mediated cell adherence is highest in the range of venular wall shear stress, ~:w, between 0.1 and 0.4 N m -z, but decreases rapidly at higher Tw. By contrast, integrinmediated adhesion, in the absence of rolling, leads to cell adhesion only at Vw < 0.1 N m 2 [145,148], emphasizing the necessity of the selectins for successful cell arrest to occur. The physiological importance of the selectins is underlined by the recently described leukocyte adherence deficiency 2 (LAD-2) syndrome, characterized by impaired immune function as well as developmental anomalies traced to a defective fucose metabolism. The other leukocyte adherence
43
deficiency syndrome, LAD-l, characterized by life-threatening bacterial and fungal infections, is due to a structural defect in the leukocyte integrins, making the leukocyte unable migrate through the interendothelial cleft. 5.5 Force dependence of detachment of cells from surfaces Measuring the detachment of cells from a substrate to which they specifically bind is the most commonly used technique for evaluating the strength of adhesion. In the presence of an applied force f per bond, such as generated in shear flow over the substrate surface, the lifetime tb of a cell-surface bond is given by Equation (25) [128], and the value of f = fa for instantaneous bond rupture can be usefully identified with the "strength" of a single bond. A measure of the adhesion strength can also be inferred from the force dependence of the time course of the detachment of a population of cells. Measurements of particle adhesion and detachment have been made using stagnation point flow chambers - as in the "impinging jet" [153] and "radial flow detachment assay" [154] methods, in which a narrow confined stream of suspension exits from an orifice and impinges on a plane surface where it is observed under a microscope. One can then study the role of hydrodynamic, colloidal and other forces on deposition and detachment of particles on surfaces under well-defined fluid flow and mass transfer conditions. A stagnation point flow chamber was first used with native whole blood to study platelet and leukocyte deposition in order to test the thrombogenicity of surfaces (155,156).
5.5.1 Impinging jet technique Dabros and van de Ven [157] first obtained solutions for the flow field and mass transfer equations in the region of the stagnation point flow. Studies have since been carried out on the deposition and detachment of latex spheres in aqueous electrolytes [157], live and fixed E. coli bacteria on glass surfaces [158], and SSRC on glass covered with monoclonal antibody [138,159]. The coordinates of the stagnation point flow field are shown below in Figure 19. A jet of suspension of radius R impinges on a surface (glass or plastic coverslip coated with receptor) a distance h away, and is contained between the two surfaces flowing out through both sides. The flow field is described by: U~=~z
; Uz=-Yz 2
[35]
where U~ and U~ are the radial and normal components of the velocity field, r and z being the radial and axial distance from the stagnation point, S, and ~, defines the strength of the flow field, for which expressions have been given as a function of Re (153). The wall shear rate on the cover slip, Gw, is given by:
Gw = ~Ur/~z = 'yr;
[36]
44
f I~ \
\ \
\
\
\
\
\
\
\
\ \
\
\
\ \
\
lI
I
I S ~ \ \ \ \ " ~ \ \ \ \ \ i
h
\ \ \ \ ~
I
t . , 4 I 2R - - ~ Figure 19. Stagnation point flow chamber: A jet of suspension of radius R impinges on a surface a distance h away and is contained between the two surfaces, flowing out through both sides. Thus, Gw is zero at the stagnation point and increases linearly with radial distance. The region where y is c o n s t a n t - a pure stagnation point flow, extends out to r = 0.1h.. For greater r, corrections to the flow field can be used. The particles are observed in epi-normal or interference contrast illumination and events recorded on videotape. Suspensions of cells or derivatized microspheres beating ligand flow onto the surface, and the number of particles per unit surface area expressed as: n, = n= [1 -exp(-t/cr)]
[37]
where n is the number at t - oo and cr is given by the relation: l/or =
1/cr,~ +
l/crbt
[38]
cr~ being the characteristic escape time of the cells and crb~that of slowing down of the deposition by the presence of deposited particles [160]. Measurements of n~ with time in various regions of the surface yield values of the initial adhesion rate and the above parameters. The escape time (average lifetime of a cell on the surface) is a measure of the cell-surface bond strength, and can be compared for the different receptor-ligand systems. Detachment experiments are carried out with the surface covered with spheres, and then exposed to a stream of suspending medium of known viscosity at various flow rates. The tangential hydrodynamic force, F~, exerted on a particle of radius b adhering to a surface is given by [146]: F~ = 1.7x6mS;wb 2 = 1.7x6n:yrb 2
[39]
can be related to the surface bond strength through the force dependence of the time course of detachment, which is dependent on the surface density of ligand.
45
An example is shown in Figure 20 for the detachment of populations of SSRC from surface-bound antibody using the impinging jet system, under forces from 20 to 100 pN. The maximum number of cross-bridges per cell was estimated to be 7, and the average rupture shear force per bond to be 17 pN [138]. 100~" = r.~ =
80
= ""
60
E
40
o
r
~
50
"~
40
E
30
~
"~ 1~
20 10
0
20
40
60
80 "
100
120
pN
0 0
10
20
30 40 Time, minutes
50
60
Figure 20. Shear-induced detachment of SSRC on a glass surface having covalently-bound monoclonal IgM. Plot of the normalized surface density against time. The average tangential hydrodynamic force was 75 pN. Inset: plot of the surface density vs the average hydrodynamic force. From [138], with permission. 5.5.2 Radial flow detachment assay
Similar stagnation point flow chambers have been used to study receptormediated cell attachment and detachment in a radial-flow detachment assay [161,162]. The assay has been used to study the relationship between the detachment force per bond and the bond affinity, K a [163]. In this device, detachment was observed at large distance from the stagnation point flow, where Gw decreases with increasing r. Antigen-coated latex spheres were allowed to adhere to antibody-coated surfaces under static conditions, and then exposed to steady shear flow. Spheres detached from the surface up to a critical radial distance from the stagnation point, corresponding to a "force required to break the bonds" beyond which the force was too low to detach the particles. As expected, given the stochastic nature of bond rupture, the contour of the ring separating adherent from non-adherent spheres, was not a sharp one. The shear stress to detach 10 ~m latex spheres covalently coated with rabbit anti-IgG on a surface covalently coated with rabbit IgG corresponded to
46 a force of 3 nN [161], 30x greater than that found in the SSRC-IgM antibody impinging jet experiments [138]. However, the surface density of rabbit IgG antibody was 1-3 orders of magnitude greater and the number of bonds, rib, was thought to be very high.
6. B L O O D C E L L I N T E R A C T I O N S IN R E G I O N S OF D I S T U R B E D FLOW Fluid mechanical factors play an important role in the localization of sites of atherosclerosis, the focal deposition of platelets resulting in thrombosis, and the formation of aneurysms in the human circulation. The localization is confined mainly to regions of geometrical irregularity where vessels branch, curve and change diameter and where blood is subjected to sudden changes in velocity and/or direction. In such regions, flow is disturbed and separation of streamlines from the wall with formation of eddies are likely to occur. In this Section the flow patterns and fluid mechanical stresses at these sites, their effect on cell and cell-wall interactions are described, and their possible involvement in the genesis of the above mentioned vascular diseases are considered. 6.1 Model System- Blood cells in an annular vortex The flow behaviour and interactions of red cells and platelets were studied in the annular vortex downstream of a sudden tubular expansion of a 150 ktm into a 500 l.tm diameter glass tube, serving as a model of a region of flow separation and an arterial stenosis [ 164 - 166]. 6.1.1 R e d blood cells
As predicted by fluid mechanical theory [167], when dilute suspensions of cells were subjected to steady flow through the model stenosis, a captive annular ring vortex was formed downstream of the expansion. Figure 20 shows paths and orientations of the red cells in the median plane of the tube at inflow tube Reynolds number, Re0, based on upstream mean fluid velocity, , and diameter. During a single orbit, the measured particle paths and velocities, as well as the locations of the vortex center and reattachment point, were in good agreement with those predicted by the theory applicable to the fluid [164]. Over longer periods, however, single cells and small aggregates < 20 ktm in diameter migrated outward across the closed streamlines, describing a series of spiral orbits of continually increasing diameter until they rejoined the mainstream. In contrast, aggregates of cells > 30 ktm in diameter remained trapped within the vortex, assuming equilibrium orbits or staying at the center. In pulsatile flow (a sinusoidal oscillatory flow superimposed in parallel with the steady flow), the observed phenomena were qualitatively similar to those described in steady flow. The vortex varied periodically in size and intensity;
47
Re =
37.8
~
< U > = 0 . 2 3 m s "1
I i
Q
Figure 21: Single orbits and orientations of glutaraldehyde-hardened human RBC in the median plane of the annular vortex formed downstream of the expansion of a 151 gm into a 504 gm diameter tube at Re0 = 37.8. Arrows indicate location of the reattachment point; particle velocities were 2.4 (O), 3.6 (A), 3.0 (B), 6.9 (C), 94 (D), 0.84 (E), 0.95 (F) and 272 mm s-1 at (G). From [164], with permission. the axial location of the vortex center and reattachment point oscillated in phase with the upstream fluid velocity between maximum and minimum positions about a mean which corresponded to that in steady flow. At hematocrits from 15 to 45%, migration of cells still occurred and resulted in a lowering of the vortex hematocrit. In part, particle migration in the vortex was likely due to the dilution effect of cell-poor plasma taken into the vortex from the fluid layer adjacent to the vessel wall upstream of the expansion. The mechanism for the trapping of large aggregates in the vortex was qualitatively explained by using existing fluid mechanical theory [57] for lateral particle migration near a tube wall due to inertia of the fluid, and by the operation of a mechanical wall effect [ 168]. 6.1.2 Platelet aggregation in the vortex The flow behaviour and interactions of human platelets in the annular vortex were studied in platelet-rich plasma (PRP) and in washed platelet suspensions. The vortex provided favourable conditions for the spontaneous aggregation of normal human platelets through repeated shear-induced collisions of cells while circulating in its orbits [165]. In a given suspension, the formation and growth of platelet aggregates was observed in a narrow range of Re0. In PRP
48
anticoagulated with heparin, containing significant numbers of platelets with pseudopods (Figure 12) and microaggregates of 2-6 cells, the rate and extent of aggregation were greatest, with aggregates > 100 ~m in length forming in <1 minute in the range 4.5 < Re0 < 17. In PRP anticoagulated with sodium citrate, and in washed platelet suspensions, containing very few microaggregates, the aggregation was much reduced unless prior to flow, the platelets were activated with ADP at concentrations below those required to form aggregates in tube flow [116]. Then, the large aggregates seen in heparin-PRP reappeared. In pulsatile flow, the number and size of the aggregates in the above suspensions decreased markedly, presumably due to the continuously changing orbits during the alternate expansion and contraction of the vortex, which shortened the particle residence times, as well as to the large variation in the shear rate in each cycle, beyond the range favourable for platelet aggregation. Calculations of the two-body collision frequency in various regions of the vortex for which the sheafing rate of strain was known [165,167], showed that, at Reo = 12, the total 2-body collision frequency [= JNo/2; Equation (18)] in the vortex periphery alone would, on average, result in -- 5.4• collisions s -~. Thus, even for collision efficiencies as low as 10 -2, it can be seen that appreciable aggregation would occur within a minute of flow.
6.1.3 Wall adhesion of platelets in the vortex The effects of disturbed flow on initial platelet adhesion to the vessel wall were studied using an expansion tube whose inner wall was coated with collagen fibers (to which platelets strongly adhere), and suspensions of washed human platelets containing washed red cells at 0-50% hematocrit [166]. As illustrated in Figure 22, platelet adhesion was localized within the vortex and downstream on either side of the reattachment point with a local minimum at the reattachment point itself. Furthermore, platelet adhesion increased, and both adhesion peaks became more pronounced, as the hematocrit increased. Surprisingly though, the adhesion peak in the vortex decreased and flattened out as the Reynolds number increased. These results are inconsistent with a diffusion-controlled platelet adhesion (when the rate determining step in adhesion is the rate at which cells are brought to the vessel surface), which should show an increase in adhesion number density with increasing shear rate [169]. It appears that the particular flow pattern within the vortex is responsible for this localization. Thus, as illustrated in the upper panel of Figure 22, only those cells carried by the curved streamlines to within one particle radius of the surface (black spheres) interact with the vessel wall and adhere to it on both sides of the reattachment point, which is also a stagnation point [166,170]. It follows from this that platelet adhesion onto the vessel wall, whether it be the natural endothelium or an artificial surface, will be localized wherever there is a stagnation point (or a reattachrnent point if it is a result of flow separation) where blood cells are carried by the flow toward the vessel
49
A
9
'~
1.0
o \
0.8
-:tF
B
~
Re= _ 37.5 U= 4 3 . 0 m m sec -! !
...... L.. ,...
5 x 1 0 5 p l a t e l e t s ~ e "1
,4-,
_.e 0.6 _
-q~
o "~ 0.4 o 9 0.2 .Q E
Z
0
, Lmin. l
0
I;
2
IL I
I
OI
v -
Steady flow .......... Pulsatile flow AV=3~t f = 1.5 cycles sec -!
Lmax. I
I
i
i
i
4 6 8 Distance from expansion, mm
i
10
i
i
12
Figure 22. Flow patterns and platelet adhesion in the annular vortex. A: Streamlines in the expansion tube showing convective transport of spheres on radially directed curved paths on either side of the reattachment point (dash-dotted line). B: Plot of platelet adhesion density on the collagen-coated tube in steady and pulsatile flow (amplitude AV = 3 ~1) at the same mean Re0, with the reattachment point oscillating between L,,z, and L,,~,. From [166], with permission. wall along curved streamlines having a pronounced radial velocity component. If this mechanism operates in the circulation, a relatively higher adhesion of platelets, and hence a higher risk of thrombus formation is predictable, not only in regions of disturbed flow (adhesion peaks on either side of the reattachment point) such as downstream of aortic or venous valves, mural thrombi, and stenoses, but also in all the branching arteries at the flow divider where there is a stagnation point. 6.2 In vivo example of an expansion flow: venous valve The above described studies have been extended to natural blood vessels by Karino who developed a method for observing flow patterns in transparent arterial and venous segments from dogs, and arterial segments from humans, postmortem [171]. The results obtained in an isolated transparent dog saphenous vein containing a bileaflet valve [172] is shown in Figure 23 with the detailed flow patterns as observed along the common median plane of vein and valve. Here, there is an expansion flow with flow separation occurring at the
50
Re=42.1 D=2.03mm d =0.81 mm 0=53.3mm s-'
R
Figure 23. Streamlines in the common median plane of the valve leaflets in a 2 mm dog saphenous vein showing formation of a spiral vortex in each valve pocket. Solid lines are paths of 50 gm polystyrene (PS) tracer spheres close to, dashed lines are paths far from the median plane; R is the reattachment point. From [172], with permission. edge of the valve leaflet which, at physiologically representative flow rates, resulted in the formation of large paired vortices in each valve pocket, located symmetrically on both sides of the bisector plane of the valve leaflets. Particles continually entered the valve pockets from the mainstream, spent up to 10 s describing a series of spiral orbits of decreasing diameter as they moved away from the bisector plane, and eventually left the vortex. It was also shown that another smaller counter-rotating secondary vortex, driven by the large primary vortex, existed deep in each valve pocket (blank area of Figure 23) where venous thrombi are believed to originate [173,174], and where it was observed that cell concentrations were appreciably lower than in the primary vortex. In such stagnant regions, fluid velocities were extremely low allowing red cells to aggregate. The results suggest that in some pathological states, the valve-pocket vortices could act as automatic traps and generators of thrombi in a fashion similar to that described above in the annular vortex formed downstream from a sudden tubular expansion. 6.3 A Model System: Flow Patterns at T-Bifurcations The flow patterns and distributions of fluid velocity and shear rate in glass models of T-junctions have been described using cinemicrography and frame by frame analysis of the motions of 50-gm diameter latex spheres [175,176].
51 Branching angles varied from 30 to 150 ~, and the effect of side-to-main-tube diameter ratio, the branch-to-parent-tube flow ratio, Q~/Qo, and inflow tube Reynolds numbers, Reo, on flow separation were studied. Figure 24 shows the flow pattern obtained by in the common median plane of a 90 ~ uniform diameter T-junction when the main branch was partially occluded so that 80% of the flow left through the side branch (unlikely to occur in vivo). A large recirculation zone formed in the main tube consisting of a pair of spiral secondary flows located symmetrically on both sides of the common median plane of the T-junction (due to the sudden deceleration of fluid velocity as a portion of the flow is drawn off into the side tube). A small, secondary recirculation zone was formed in the side tube (due to the sudden change in direction of the fluid which continues to move to the outside of the 90 ~ corner, toward the flow divider). Particles entering the large recirculation zone described complex orbits; some of them rejoined the flow through the main tube, others entered the side branch in a paired, spiral secondary flow with pronounced radial components, and some of these circulated through the secondary recirculation zone. When the degree of occlusion of the main tube was gradually reduced, the large recirculation zone became smaller and eventually disappeared as the flow rate ratio was reversed, while that in the side branch grew in size. The critical Reo for the formation of the main recirculation zone was lowest at 90 ~ for all Q~/Qo, whereas for the side recirculation zone it decreased as the branching angle increased from 45 to 135 ~. With decreasing side tube diameter, the main recirculation zone became smaller and was confined to a thin layer adjacent to the tube wall, wrapped around the mainstream [176]. The effect of radius of curvature of the walls at the junction was studied by comparing the critical Reo and the sizes of the recirculation zones in the square T-junction (Figure 24; radii of curvature < 2% of tube radius) with that in a rounded T-junction (radii of curvature -~ tube radius) [175]. The recirculation zone in the side tube formed at a much lower Re0 in the square than the rounded junction, and at a given Re 0 and Q1/Qo, a larger main recirculation zone existed in the rounded junction. It appears that the formation of the side recirculation zone is largely affected by curvature of the wall at the bend opposite to the flow divider while that of the main recirculation zone is largely affected by the curvature at the flow divider.
6.4 In Vivo Example of T-Bifurcations- Aortic T-Junctions The flow patterns in transparent segments of a dog abdominal aorta containing branches of the celiac, superior-mesenteric and right and left renal arteries have been described [177] The flow patterns illustrated in Figure 25 for the aorto-celiac junction resemble those observed in the model T-junctions, but the degree of flow disturbance, even at Reo as high as 609, is much smaller. Instead of a large main standing recirculation zone as in the glass models, there
52
Q1
Qo_~_. ~_~_~,i 9
~_ ~ ~ ~ / ' ~ f ~ , ~ - - - . . ~ _
....
------------___.__~~
_~----------------~__~
4, '~'~"_..,,S~_ ~
,,.o ,o Q 2
S
R
Figure 24. Flow patterns (paths of tracer 50 l.tm PS spheres) in the common median plane of a 3 mm diameter glass T-junction, made b_y fitting and glueing 2 pieces of tubing. Flow enters at left with mean velocity U o ; 80% left through the side tube; points show sphere positions at 22 ms intervals, numbers the velocities in mm s -~. From [175], with permission.
Q2 d/D: 4.0/7.2 Uo = 2 1 5 mms-'
Q,,Qo-O,~
Cel~rC~ery / ~llEtf",tllll
/ ~l~', X~tll
/ ~::~,J!,' F':t ~ 11 ", '~! ]~ I I
/'~
,:"ikl,
~,,,,r~ '
/
/C,oo~,t mesenteric/
........
artery
............................ 1.....
Abdominal aorta
Figure 25. Plot, as in Figure 24 of the flow patterns (paths of 200 l-tm PS spheres) at the aortoceliac artery junction of a dog abdominal aorta, with a considerably smaller flow disturbance (thin-layered recirculation zone) than in Figure 24, as only 28% of flow entered the branch, curvature of wall opposite flow divider is high, that at the divider is low. From [ 177], with permission.
53
was only a pair of recirculation zones, confined to a thin layer close to the wall surrounding the mainstream. There was no side recirculation zone, no doubt due to the gentle curvature of the bend opposite the flow divider. This characteristic was shared by all the aortic T-junctions studied, as was the very sharp curvature of the bend at the flow divider. From the results described above in the glass model T-junctions, this represents the optimum condition for minimizing the size of both regions of separated flow. Nevertheless, the curved streamlines of the recirculation zone and secondary flows will bring blood cells towards the vessel wall in a zone around the flow divider and in the side branch on the outer wall.
6.5 The Human Carotid Bifurcation The human carotid bifurcation, unlike other Y-bifurcations, exhibits a marked flow disturbance associated with a large recirculation zone located in the carotid s i n u s - the vessel expansion at the entry of the internal carotid artery from the common carotid artery. Figure 26 shows the flow patterns in the carotid sinus of a transparent arterial segment prepared from a human subject post-mortem [178]. A standing recirculation zone consisting of a pair of complex spiral secondary flows, located symmetrically on both sides of the common median plane of the bifurcation existed over a wide range of Re0, and Q1/Qo(internal/common carotid). Particles were deflected at the flow divider and travelled laterally and very slowly along the wall above and below the common median plane, almost at fight angles to, and encircling the mainstream. They then changed direction, moving back along the outer wall of the internal carotid artery at the sinus, describing spiral orbits in the recirculation zone before rejoining the mainstream. Downstream from the stagnation point (R), a strong counter-rotating double helicoidal flow developed. The formation and the size of the recirculation zone were largely dependent on Q~/Qoand on Re0. The velocity profiles were strongly skewed towards the inner walls of the bifurcation, creating a high shear field along the vessel wall downstream from the flow divider. Due to the paired standing recirculation zones in the carotid sinus, the wall shear rate, and hence the wall shear stress, changes sign and becomes negative at the separation point (S); it becomes positive again downstream of the stagnation (reattachment) point (R). Thus, in the carotid sinus, there is a region where the vessel wall is stretched in opposite directions by the counter-directed wall shear stresses. The results suggest that, under physiological conditions a standing, although pulsating recirculation zone exists in the carotid sinus, thereby affecting local mass transfer and interactions of blood cells with the vessel wall which may lead to the incidence of thrombosis and atherosclerosis in this region.
54
Reo--592 U0=266 mm s-' Q~/Q. =0.613
Q2
Qo
2a3 :
i
QI
Figure 26. Flow pattern in the human carotid bifurcation showing formation of a large recirculation zone and counterrotating double helicoidal flow in the sinus region of the internal (lower) carotid artery. Solid lines are paths in or close to, and dashed lines paths far from the common median plane. Arrows at S and R denote respective locations of separation and reattachment points. From [178], with permission. 6.6 Flow patterns and atherosclerotic wall thickening It appears that local flow patterns are involved in the localization of atherosclerosis, as is illustrated by studies of the middle cerebral bifurcation of the human circle of Willis at the base of the brain [179]. In each of the five vessels studied, atherosclerotic thickening of the vessel wall was localized around the hips of the bifurcation. The detailed flow patterns revealed that a standing recirculation zone, very similar to that observed in the carotid artery bifurcation, formed along the outer wall of one or both daughter vessels (depending on the Re in the parent vessel and Q~/Q2) at the exact locations where the atherosclerotic thickening of the vessel wall occurred. Furthermore, under the normal physiological range of flow rates and flow ratios tested, there was a strong correlation between the longitudinal length of the regions of disturbed flow and that of the atherosclerotic wall thickening. Figure 27 shows the detailed flow patterns observed in steady flow in one of the bifurcations having an almost perfectly symmetric structure and spatial arrangement of the daughter vessels. As is evident from the figure, even when Q 1 / Q 2 "" 1.0, the region of disturbed flow was much longer in the right side branch where the region of atherosclerotic wall thickening was also longer than that in the left side branch where the wall thickening was confined to only a very narrow area.
55
In pulsatile flow, the complex spiral secondary flows and the recirculation zones oscillated in phase with the pulsatile flow velocity, and the locations of the stagnation and separation points moved back and forth along the vessel wall. However, the general flow patterns remained the same as those observed in steady flow.
i il
I~/
i
[i
D, = 3.09 mm Q.= 147 ml/min
tlillli !I ll|iJ!//iHtli| ~o
Figure 27. Flow patterns at a human middle cerebral artery bifurcation in the circle of Willis with flow distributed 49%:51% in the branches. The location and lengths of the secondary flows and recirculation zones at the outer walls of the bifurcation match those of the atherosclerotic wall thickening. S and P denote separation and stagnation points. From [179], with permission. 7.
CONCLUDING REMARKS
The rheology of blood, whether at the macroscopic, microscopic or submicroscopic (molecular) level is indeed a complex subject, as so graphically and forcibly suggested by G.I. Taylor, years ago. Yet, much of the work reported in this chapter has contributed significantly towards an understanding of the rheology at all 3 levels, and this in large measure through theoretical and experimental studies of the mechanics of the motions and interactions of blood cells. It should be noted though, that the subject is much larger than that given here in a chapter whose scope was limited to aspects of blood cell and blood cell-vessel wall wall interactions. Thus, some
56 recent work had unfortunately to be omitted or only briefly reported. Examples which come to mind include the motion and wall adhesion of tumour cells in relation to cancer metastasis [180], the molecular and biophysical mechanisms of shear stress-induced signal transduction in circulating and endothelial cells, and flow-induced effects on cell morphology and function [181,182], the mechanics of cell locomotion [182], and the mechanics of the motion of RBC in microcirculatory networks, including physical interactions with the endothelial cell glycocalyx in the capillaries [24,183]. These are some of the areas of investigation at the cutting edge of present day research into the biophysics and biorheology of the circulation, using the latest state of the art technology, a long way from the work of the pioneers mentioned in the first pages of the chapter. REFERENCES 0
.
3.
.
5. 6. .
8. 9.
10. 11. 12. 13. 14. 15. 16.
L.E. Bayliss, In: A. Frey-Wyssling (ed.), Deformation and Flow in Biological Systems, Interscience, New York, 1952. L.E. Bayliss, J. Physiol. (London), 179 (1965) 1. J.R. Pappenheimer, In: Handbook of Physiology, The Cardiovascular System, sect. 2, Vol. IV, American Physiological Society, Bethesda, MD, 1984. S.P. Sutera and R. Skalak, Ann. Rev. Fluid Mech., 25 (1993) 1. R. F~hraeus and T. Lindqvist, Am. J. Physiol., 96 (1931) 562. H.L. Goldsmith, G.R. Cokelet, and P. Gaehtgens, Am. J. Physiol, 89 (1989) H1005. L. Dintenfass, Angiology, 8 (1962) 333. E.A. Evans, Biophys. J., 13 (1973) 926; ibid, idem (1973) 941.. R.M. Hochmuth, In: R. Skalak and S. Chien (eds.), Handbook of Bioengineering, McGraw Hill, New York, 1987. G.V.F. Seaman, In: D. MacN. Surgenor (ed.), The Red Blood Cell, II, Academic, New York, 1975. S. Levine, M. Levine, K.A. Sharp and D.E. Brooks, Biophys. J., 42 (1983) 127. B.V. Derjaguin and L.D. Landau, Acta Physiochemica URSS, 14 (1941) 633. E.G. Verwey and Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier Scientific, Amsterdam, 1948. D.E. Brooks, R.G. Greig and J. Janzen, In: H. J. Meiselman and D.E. Brooks (eds.), Erythrocyte Mechanics and Blood Flow, Alan R. Liss, New York, 1980. D.E. Brooks, J. Colloid Interface Sci., 43 (1973) 700. S. Chien and K.-M. Jan, Microvasc. Res., 5 (1973) 155.
57 17. E.A. Evans and D. Needham, Macromolecules, 21 (1988) 1822. 18. W.R. Milnor, Hemodynamics, 2nd ed., Williams & Wilkins, Baltimore, 1989. 19. W.W. Nichols and F.M. O'Rourke (eds.), McDonald's Blood Flow in Arteries, Lea and Febiger, Philadelphia, 1990. 20. M. Anliker, M. Casty, P. Friedli, R. Kubli and H. Keller, In: N.H.C. Hwang and N.A. Nomaann (eds.), Cardiovascular Flow Dynamics and Measurements, University Park, Baltimore, 1977. 21. J.W. Pothero and A. C. Burton, Biophys. J., 1 (1961) 565. 22. U. Bagge, P.-I. Brhnemark, R. Karlsson and R. Skalak, Blood Cells, 6 (1980) 231. 23. P. Gaehtgens, C. Duehrssen and K.H. Albrecht, Blood Cells, 6 (1980) 799. 24. T.W Secomb, In: C.P. Ellington and T.J. Pedley (eds.), Biological Fluid Dynamics, Company of Biologists, Cambridge, U.K., 1995. 25. U. Bagge and P.-I. Brgmemark, Adv. Microcirc., 7 (1977) 1. 26. G.W. Schmid-Schtinbein, S. Usami, R. Skalak and S. Chien., Microvasc. Res., 19 (1980) 45. 27. T. Karino, M. Motomiya and H.L. Goldsmith, In: J.C. Stanley (ed.), Biologic and Synthetic Vascular Prostheses, Grune & Stratton, New York, 1982. 28. P.D. Stein and H.N. Sabbah, Circ. Res., 398 (1976) 58. 29. S. Chien, Science, 168 (1970) 977. 30. P.L. Easthope and D.E. Brooks, Biorheology, 17 (1980) 235. 31. F.J. Walbum and D.J. Schneck, Biorheology, 13 (1976) 201. 32. N. Casson, In: C.C. Mills (ed.), Rheology of Disperse Systems, Pergamon, New York, 1959. 33. G.W. Scott-Blair, Nature, 183 (1959) 613. 34. G.R. Cokelet, In: Biomechanics: Its Foundation and Objectives, Prentice Hall, Englewood Cliffs, NJ, 1972. 35. D. Quemada, Rheol. Acta, 16 (1978) 82. 36. D. Quemada, Rheol. Acta, 17 (1978) 632. 37. G.R. Cokelet, In: R. Skalak and S. Chien (eds.), Handbook of Bioengineering, McGraw Hill, New York, 1987. 38. S. Chien, In: D. MacN. Surgenor (ed.), The Red Blood Cell, II, Academic, New York, 1975. 39. W. Reinke, P. Gaehtgens and P.C. Johnson, Am. J. Physiol., 253 (1987) H540. 40. G.R. Cokelet and H.L. Goldsmith, Circ. Res., 68 (1991) 1. 41. E.B. Vadas, H.L. Goldsmith, and S.G. Mason, J. Colloid Interface Sci., 43 (1973) 630. 42. S.P. Tha, J. Shuster, and H.L. Goldsmith, B iophys. J., 50 (1986) 1117.
58 43. H.L. Goldsmith and J. Marlow, Proc. Roy. Soc. (London), B 181 (1972) 351. 44. H.L. Goldsmith and S.G. Mason, J. Fluid Mech., 12 (1962) 88. 45. H.L. Goldsmith and S.G. Mason, J. Colloid Sci., 17 (1962) 448. 46. G.B. Jeffery, Proc. Roy. Soc. (London), A102 (1922) 162. 47. H.L. Goldsmith and S.G. Mason, In: F.R. Eirich (ed.) Rheology: Theory and Applications, Vol. IV, Academic, New York, 1967. 48. M.M. Frojmovic, M. Newton and H.L. Goldsmith, Microvasc. Res., 11 (1976) 203. 49. E.B. Vadas, R.G. Cox, H.L. Goldsmith and S.G. Mason, J. Colloid Interface Sci., 57(1976) 308. 50. F.D. Rumscheidt and S.G. Mason, J. Colloid Sci., 6 (1961) 238. 51. T.M. Fischer, M. Strhr-Liesen and H. Schmid-Schrnbein, Science, 202 (1978) 894. 52. S.R. Keller, and R. Skalak, J. Fluid Mech., 120 (1982) 27. 53. H. Brenner, In: T.B. Drew, J.W. Hoopes and T. Vermeulen (eds.), Advances in Chemical Engineering, Vol. 6, Academic, New York, 1966. 54. L.G. Leal, Ann. Rev. Fluid Mech., 12 (1980) 435. 55. H.L. Goldsmith, Fed. Proc., 30 (1971) 1578. 56. H.L. Goldsmith and S. Spain, Microvasc. Res., 27 (1984) 204 57. R.G. Cox and S.K. Hsu, Int. J. Multiphase Flow, 3 (1977) 201. 58. S. Segr6 and A. Silberberg, J. Fluid Mech., 14 (1962) 136. 59. H.L. Goldsmith, and J.C. Marlow, J. Colloid Interface Sci., 71 (1979) 383. 60. H.L. Goldsmith, Fed. Proc., 26 (1967) 1813. 61. H.L. Goldsmith and R. Skalak, Ann. Rev. Fluid. Mech., 7 (1975) 213. 62. A. Karnis, H.L. Goldsmith, and S.G. Mason, J. Colloid Interface Sci., 22 (1966) 531. 63. J.C. Slattery, J. Fluid Mech., 19 (1964) 625. 64. R. Skalak, Biorheology, 29 (1992) 479. 65. D. Leighton and A. Acrivos, J. Fluid Mech., 181 (1987) 415. 66. R.J. Phillips, R.C. Armstrong, R.A. Brown, A.L. Graham and J.R.A. Abbott, Phys. Fluids, 4 (1992) 30. 67. P.R. Nott and J.F. Brady, J. Fluid Mech., 275 (1994) 155. 68. C.J. Koh, P. Hookham and G. Leal, J. Fluid Mech., 266 (1994) 1. 69. N. Phan-Tien, A.L. Graham, S.A. Altobelli, J.R. Abbott and L.A. Mondy, Ind. Eng. Chem. Res., 34 (1995) 3187. 70. H.L. Goldsmith, In: T.H. Spaet (ed.), Progress in Hemostasis and Thrombosis, Vol. 1, Grune & Stratton, New York, 1972. 71. R.L. Replogle, H.J. Meiselman and E.W. Merrill, Circulation, 36 (1967) 148. 72. F.P. Gauthier, H.L. Goldsmith and S.G. Mason, Biorheology, 9 (1972) 205.
59 73. E.B. Vadas, H.L. Goldsmith and S.G. Mason, Trans. Soc. Rheol., 20 (1976) 373. 74. H.L. Goldsmith and T. Karino, Ann. N.Y. Acad. Sci., 283 (1977) 241. 75. H. Brenner, Int. J. Multiphase Flow, 1 (1974) 195. 76. V.T. Turitto, A.M. Benis and E.F. Leonard, Ind. Eng. Chem. Fund., 11 (1972) 216. 77. E.F. Grabowski, L.I. Friedman and E.F. Leonard, Ind. Eng. Chem. Fund. 11 (1972) 224. 78. I.A. Feuerstein, B.M. B rophy and J.L. Brash, Trans. Am. Soc. Artif. Intern. Org., 21 (1975) 21. 79. V.T. Turitto and H. Baumgartner, Microvasc. Res., 9 (1975) 335. 80. V.T. Turitto and H.J. Weiss, Science, 207 (1980) 541. 81. P.A. Aarts, K.S. Bolhuis, R.M. Sakariassen, R.M. Heethar and J.J. Sixma, Blood, 62 (1983) 214. 82. A.L. Zydney and C.K. Colton, Physicochem. Hydrodyn., 10 (1988) 77. 83. R. Fhhraeus, Physiol. Rev., 9 (1929) 241. 84. U. Nobis, A.R. Pries and P. Gaehtgens, In: U.Bagge, G.V.R. Born and P. Gaehtgens (eds.) White Blood Cells: Morphology and Rheology as Related to Function, Martinu Nijhoff, The Hague/Boston, 1982. 85. H.L. Goldsmith and S. Spain, In: H.J. Meiselman and M. Lichtman, (eds.), White Cell Mechanics: Basic Science and Clinical Aspects, Raven, New York, 1984. 86. A. Karnis, H.L. Goldsmith and S.G. Mason, Can. J. Chem. Eng., 44 (1966) 181. 87. G.J. Tangelder, H.C. Teirlinck, D.W. Slaaf and R.S. Reneman, Am. J. Physiol., 248 (1985) H318. 88. P.A. Aarts, S.A.T. van den Broek, G.W. Prins, G.D.C. Kuiken, J.J. Sixma and R.M. Heethar, Arteriosclerosis, 8 (1988) 819. 89. A.W. Tilles and E.C. Eckstein, Microvasc. Res., 33 (1987) 211. 90. E.C. Eckstein, A.W. Tilles and F.J. Millero, Microvasc. Res., 36 (1989) 31. 91. E.C. Eckstein and F. Belgacem, Biophys. J., 60 (1990) 53. 92. F. Seitz, Phys. Rev., 73 (1948) 1513. 93. D.F.J. Tees, O. Coenen and H.L. Goldsmith, B iophys. J., 65 (1993) 1318. 94. H.L. Goldsmith and S.G. Mason, Proc. Roy. Soc. (London), A 282 (1964) 569. 95. T.G.M. van de Ven and S.G. Mason, J. Colloid Interface Sci., 57 (1976) 505. 96. K. Takamura, H.L. Goldsmith and S.G. Mason, J. Colloid Interface Sci., 82 (1981) 175. 97. P.A. Arp and S.G. Mason, J. Colloid Interface Sci., 61 (1977) 21. 98 C.Y. Lin, K.Y. Lee and N.F. Sather, J. Fluid Mech., 56 (1972) 375. 99. G.K. Batchelor and J.T. Green, J. Fluid. Mech., 56 (1972) 375.
60 100. P.M. Adler, J. Colloid Interface Sci., 84 (1981) 461. 101. T.G.M. van de Ven, Colloidal Hydrodynamics, Academic, San Diego, 1989. 102. M. von Smoluchowski, Z. Phys. Chem., 92 (1917) 129. 103. P. Arp and S.G. Mason, Can. J. Chem., 54 (1976) 3769. 104. T.G.M. van de Ven and S.G. Mason, Colloid Polym. Sci., 225 (1977) 468. 105. P.M. Adler, J. Colloid. Interface Sci., 81 (1981) 489. 106. K. Takamura, H.L. Goldsmith and S.G. Mason, J. Colloid Interface Sci., 72 (1979) 385. 107. K. Takamura, H.L. Goldsmith and S.G. Mason, J. Colloid Interface Sci., 82 (1981) 190. 108. P. Davies, Thromb. Haemost., 70 (1993) 128. 109. S.R.P. Gudi and J.A. Frangos, In: V.C. Mow, F. Guilak, R. Tran-Son-Tay and R.M. Hochmuth (eds.), Cell Mechanics and Cellular Engineering, Springer, New York, 1994. 110. D.E. Ingber, In: V.C. Mow, F. Guilak, R. Tran-Son-Tay and R.M. Hochmuth (eds.), Cell Mechanics and Cellular Engineering, Springer, New York, 1994. 111. R. Skalak, B.A. Skierczynski, S. Usarni and S. Chien, In: V.C. Mow, F. Guilak, R. Tran-Son-Tay and R.M. Hochmuth (eds.), Cell Mechanics and Cellular Engineering, Springer, New York, 1994. 112. D.N. Bell, S., Spain and H.L. Goldsmith, Biophys. J., 56 (1989) 817. 113. D.R. Phillips, J.F. Charo, L.V. Parise and L.A. Fitzgerald, Blood, 55 (1988) 831. 114. P.L. Sung, M.M. Frojmovic, T.E. O'Toole, E.F. Plow, C. Zhu, M. Ginsberg and S. Chien, Blood, 81 (1993) 419. 115. M. Frojmovic, T. Wong and Th. van de Ven, Biophys. J., 59 (1991) 815. 116. D.N. Bell, S., Spain and H.L. Goldsmith, Biophys. J., 56 (1989) 829. 117. M.M. Frojmovic, K.A. Longmire and T.G.M. van de Ven. Biophys. J., 58 (1990) 309. 118. D.N. Bell, S., Spain and H.L. Goldsmith, Thromb. Haemost., 63 (1990) 112. 119. H.L. Goldsmith, D.N. Bell, S. Braovac, A. Steinberg and F. McIntosh, Biophys. J., 69 (1584) 1995. 120. T.K. Belval and J.D. Hellums, Biophys. J., 50 (1986) 479. 121. R.L. Drake, In: G.M. Hidy and J.R. Brock (eds.), Topics in Current Aerosol Research, part 2, Pergamon, Oxford, 1972. 122. A.D. Taylor, S. Neelameghan, J.D. Hellums, C.W. Smith and S.I. Simon, Biophys. J., 71 (1996) 3488. 123. S. Neelameghan, A.D.Taylor, J.D. Hellums, M. Dembo, C.W. Smith and S.I. Simon, Biophys. J., 72 (1997) 1527. 124. E.-L. Florin, V.T. Moy and H.E. Gaub, Science, 254 (1994) 415. 125. G.U. Lee, D.A. Kidwell and R.J. Colton, Langmuir, 10 (1994) 354.
61 126. 127. 128. 129. 130. 131. 132. 133.
E. Evans, K. Ritchie and R. Merkel, B iophys. J., 68 (1995) 2580. S.C. Kuo and M.P. Sheetz, Science, 260 (1993) 232. G.I. Bell, Science, 200 (1978) 618. S.N. Zhurkov, Int. J. Fract. Mech., 1 (1965) 311. H.A. Kramers, Physica., 7 (1941) 284. D.A. Hammer and S.M. Apte, Biophys. J., 63 (1992) 35. E. Evans, D. Berk and A. Leung, Biophys. J., 59 (1991) 838. E. Evans, In: R. Lipowsky and E. Sackmann (eds.), Handbook of Physics of Biological Systems, Vol. 1, Elsevier Science, Amsterdam, 1995. 134. E. Evans and K. Ritchie, Biophys. J., 72 (1997): 1541. 135. H.L. Goldsmith, O. Lichtarge, M. Tessier-Lavigne and S. Spain, Biorheology, 18 (1981) 531. 136. H. Brenner and M.E. O'Neill, Chem. Eng. Sci., 27 (1972) 1421. 137. S. Tha and H.L. Goldsmith, Biophys. J., 50 (1987) 1109. 138. Z. Xia, H.L. Goldsmith and T.G.M. van de Ven, Biophys. J., 66 (1994) 1222. 139. D.F.J. Tees and H.L. Goldsmith, Biophys. J., 71 (1996) 1102. 140. E. Evans, R. Merkel, R. Ritchie, S. Tha and A. Zilker, In: P. Bongrand, P.M. Claesson and A.S.G. Curtis (eds.), Studying Cell Adhesion, Springer, Berlin, Heidelberg, New York, 1994. 141. T.A. Springer, Cell 76 (1994) 301. 142. D.A. Jones, C.W. Smith and L.V. Mclntire, Biomaterials, 17 (1996) 337. 143. A. Atherton and G.V.R. Born, J. Physiol. (Lond.), 233 (1973) 157. 144. G.W. Schmid-Schrnbein, R. Skalak, S.I. Simon and R.L. Engler. Ann. N.Y. Acad. Sci., 516 (1987) 348. 145. M.B. Lawrence and T.A. Springer, Cell, 65 (1991) 859. 146. A.J. Goldman, R.G. Cox and H. Brenner, Chem. Eng. Sci., 22 (1967) 22. 147. C.W. Smith, Can. J. Physiol. Pharmacol., 71 (1993) 76. 148. M.B. Lawrence, L.V. Mclntire and S.G. Eskin, Blood, 70 (1987) 1284. 149. A. Tozeren and K. Ley, Biophys. J., 63 (1992) 700. 150. R. Alon, D.A. Hammer and T.A. Springer, Nature (1995) 539. 151. M. Dembo, D.C. Torney, K. Saxman and D. Hammer, Proc. Roy. Soc. Lond. B. Biol. Sci. 234 (1994) 55. 152. D. Kwong, D.F.J. Tees and H.L. Goldsmith, Biophys. J., 71 (1996) 1115. 153. T. Dabros, T. and T.G.M. van de Ven, Colloid Polymer Sci., 261 (1983) 694. 154. C. Cozens-Roberts, D.A. Lauffenburger and J.A. Quinn, Biophys. J., 58 (1990) 107. 155. H.E. Petschek, D. Adamis and A.R. Kantrowitz, A.R, Trans. Am. Soc. Artif. Int. Organs, 14 (1968) 256. 156. E. Nyilas, W.A. Morton, D.M. Lederman, T.-H. Chiu. and R.D. Cumming, Trans. Am. Soc. Artif. Int. Organs, 21 (1975) 55.
62 157. T. Dabros and T.G.M. van de Ven, PhysicoChem. Hydrodyn., 8 (1987) 161. 158. Z. Xia, L. Woo and T.G.M. van de Ven, Biorheology, 26 (1989) 359. 159. Z. Xia, H.L. Goldsmith and T.G.M. van de Ven, Biophys. J., 65 (1993) 1073. 160. T. Dabros and T.G.M. van de Ven, J. Colloid Interface Sci., 93 (1983) 576. 161. C. Cozens-Roberts, D.A. Lauffenburger and J.A. Quinn, Biophys. J., 58 (1990) 857. 162. A. Saterback, S.C. Kuo and D.A. Lauffenburger, Biophys. J., 65 (1993) 243. 163. S.C. Kuo and D.A. Lauffenburger, Biophys. J., 65 (1993) 2191. 164. T. Karino and H.L. Goldsmith, Phil. Trans. Roy. Soc. Lond. Biol. Sci., 279 (1977) 413. 165. T. Karino and H.L. Goldsmith, Microvasc. Res., 17 (1979) 217. 166. T. Karino and H.L. Goldsmith, Microvasc. Res., 17 (1979) 238. 167. E.O. Macagno and T.K. Hung, J. Fluid Mech., 28 (1967) 43. 168. A. Kamis and S.G. Mason, J. Colloid Interface Sci., 23 (1967) 120. 169. V.T. Turitto, In: T.H. Spaet (ed.), Progress in Hemostasis and Thrombosis, Vol. 6, Grune & Stratton, New York, 1982. 170. T. Karino and H.L. Goldsmith, Biorheology, 21 (1984) 587. 171. T. Karino and M. Motomiya, Biorheology, 20 (1983) 119. 172. T. Karino and M. Motomiya, Thromb. Res., 36 (1984) 245. 173. L. Diener, J.L.E. Ericsson and F. Lund, In: T. Shimamoto and F. Numano (eds.), Atherogenesis, Excerpta Medica, Amsterdam, 1969. 174. S. Sevitt, In: J.J. Bergan and J.S.T. Yao, (eds.), Venous Problems, Year Book Medical, Chicago, 1978. 175. T. Karino, H.H.M. Kwong and H.L. Goldsmith, Biorheology, 16 (1979) 231. 176. T. Karino and H.L. Goldsmith, Biorheology, 22 (1985) 87. 177. T. Karino, M. Motomiya and H.L. Goldsmith, J. Biomech., 23 (1990) 537. 178. M. Motomiya and T. Karino, Stroke, 15 (1984) 50. 179 T. Karino, Intern. Angiology, 5 (1986) 297. 180. A.F. Chambers, I. MacDonald, E. Schmidt, S. Koop, V.L. Morris, R. Khokha and A.C. Groom, Cancer Metastasis Rev., 14 (1995) 279. 181. J.A. Frangos (ed.), Physical Forces and the Mammalian Cell, Academic, San Diego, 1993. 182. V.C. Mow, F. Guilak, R. Tran-Son-Tay and R.M. Hochmuth (eds.), Cellular Mechanics and Cellular Engineering, New York, Springer, 1994. 183. A.R. Pries, T.W. Secomb and P. Gaehtgens, Cardiovasc. Res., 32 (1996) 654.
63
S H E A R T H I C K E N I N G AND FLOW I N D U C E D S T R U C T U R E S IN FOODS AND B I O P O L Y M E R SYSTEMS E. B. Bagley" and F. R. Dintzis b aDept, of Food Science, University of Illinois, Urbana, IL; address for mail." 756 S. Columbus, Morton, IL 61550-2428 bUSDA, ARS, National Center for Agricultural Utilization Research, Peoria, IL 61604 1. INTRODUCTION The handling of materials such as foods and food components, synthetic polymers and biopolymers, and the combining and/or transformation of these materials into useful products, normally involve complex industrial processes such as mixing, flow into and through pipes, extrusion, injection molding. To predict material behavior relevant to processing and to quantify or model behavior, rheological data are vital. Experience over the years with synthetic polymers has shown that such rheological data can also relate to molecular factors such as molecular weight, molecular weight distribution, and chain branching, and to the final finished product properties (Graessley [1]; Bagley and Schreiber [2]). Useful relationships among processing history, molecular structure, molecular aggregation phenomena, and end use applications, have been developed over the years for synthetic high polymers such as polyethylene and polystyrene. Attention should be drawn to two recent books, one of which emphasizes flow-induced structure in polymer systems (Nakatani and Dadmun [3]) and the other deals with the rheo-physics of multiphase polymer systems (Sondergaard and Lyngaae-Jorgensen [4]). There is no doubt that many biopolymers and foods can be considered as multiphase polymer systems and some of the chapters in Sondergaard and Lyngaae-Jorgensen dealing with phase transitions in shear flow, on flow-induced dissolution and crystallization in polymer systems, and flow-induced phase changes in polymer blends can be especially relevant to the phenomenon of shear thickening in biopolymer and food systems.
64 The simplest rheological behavior is that shown by relatively small molecules for which the viscosity is independent of shear rate. The next level of complexity is evident in the flow behavior of many synthetic polymers, both as melts and in solution, where at low shear rate the materials show Newtonian behavior, with this Newtonian viscosity level depending on the 3.4 power of the weight average molecular weight, above a certain critical molecular weight, of the polymer (for example, see discussion by Graessley [1]). As the shear rate is increased a shear rate is reached beyond which the steady state viscosity can begin to decrease. The details of the flow curve, such as the shear rate at which this shear thinning begins, and the degree or extent of the shear thinning, depend on molecular factors such as molecular weight, molecular weight distribution, and on the amount and type of chain branching. For solutions, concentration plays a significant role in the transition from Newtonian to shear thinning behavior as illustrated in Figure 1, taken from Launay and McKenna [5].
E
10
D
12_ 10 -1
10-2
I
I
I
1
10
I
I
I r
10 -~
10 2
10 3
10 4
?(S -1 )
Figure 1. Flow curves at 25 ~ C of locust bean gum solutions of 0.25%, 0.5%, 0.8%, 1.5%, and 1.8% concentrations (A to E, respectively). The flow is Newtonian at the lower concentrations, curves A, B, and C, at the lower shear rates. (From Launay and McKenna [5], by permission).
65 In addition to relating results to molecular factors the rheological results can be used in analyzing behavior during processing. For this it is essential to have the rheological data available for the shear rate ranges encountered during processing. Note also that these non-Newtonian materials are usually viscoelastic which give rise to important processing effects such as die swell during extrusion through pipes. The die swell is given (for cylindrical extrudates) as the ratio of the extrudate diameter to the die diameter and easily can be, for polyethylene, of the order of 3 to 5! The die swell is often correlated with commercially significant properties such as product gloss or matte. In addition, the relaxation times associated with these viscoelastic materials can give rise to significant time effects such as the complex transients observed as stress overshoots. Time effects can play an important role in the shear thickening phenomenon. Food and biopolymer systems can show all of the rheological phenomena described in the above paragraph. However, whereas many uncharged synthetic polymers show little intermolecular interactions beyond the chain entanglements often associated with materials of high molecular weight, food and biopolymer systems have three additional sources of rheological complexity. First, biopolymers are often of very high molecular weight indeed. Amylopectin, the branched component of starch, can have molecular weights above a hundred million, corresponding to perhaps 600,000 or more anhydroglucose units in the molecule. In terms of degree of polymerization this is far beyond the extent of polymerization of most common synthetic polymers such as commercial polyethylenes or polystyrenes. The massive size of this starch molecule component makes it particularly susceptible to shear degradation which gives rise to a number of experimental results not normally seen in polymer systems. Second, food systems and biopolymers in general contain a variety of molecular groups such as hydrogen bonds and ionic groups, which foster interand intra-molecular interactions which can lead to a variety of complex phenomena influencing both processability and final product properties. The effect of these interacting units can be modified or amplified by additives of various kinds which again can contribute to unusual but significant effects, of value and interest both in practical and scientific terms. Finally, and this aspect of biopolymer systems has not been extensively discussed, it seems reasonable to expect that in the biosynthesis of these polymers, where the polymer is "laid down" as a substrate in biosynthesis during plant growth, the molecules may be intimately entangled, more so than would be expected for synthetic polymers. This might well result in entanglements of an especially effective kind, particularly when the molecular weight is enormous.
66 These food and biopolymer systems, in addition to showing shear thinning effects, can also, under specific circumstances, show shear thickening behavior. Shear thickening behavior seems to be relatively rare and the interpretation of apparent thickening can be complicated by a variety of effects. Thickening is often associated with shear induced structure formation, but often the details of, and the mechanism for, the structure formation are unclear. Nevertheless, the processes leading to these shear thickening effects are of great interest and are significant and need to be carefully considered in terms of processing and process design as well as in interpretation of data used for material characterization. Shear thickening occurs for what might superficially be considered comparatively simple systems such as dispersions of rigid spherical particles. Shear thickening in such systems is often considered a reflection of volume increase under shear, or dilatancy. As Tanner [6] discusses, this concept of dilatancy goes back to the analysis of the flow of concentrated suspensions of solids by Osborne Reynolds, but Tanner emphasizes that in recent years "The term dilatant has come to be used for all fluids which exhibit the property of increasing viscosity with increasing rates of shear." This is an unfortunate development since it implies a mechanism for shear thickening which may well not be correct for many, if not most, shear thickening systems. That is, shear thickening can occur without a volume change, so that this misleading usage of the term dilatant should be discontinued unless a volume change is proved. Shear thickening is thus a phenomenon associated with complicated interacting systems. These interactions may arise from specific molecular factors, such as hydrogen bonding, or physical factors such as chain entanglement or particle crowding (in particulate suspensions or dispersions). These effects can be exacerbated during flow because of the formation of shear induced structures of various types. When shear thickening occurs in systems such as suspensions of rigid spheres, where geometrical considerations come into play, the shear thickening occurs concomitantly with system volume expansion, leading to the concept of dilatant flow. These systems can also be considered to produce shear induced structures. The objective of this chapter is to review first of all rheological measurements and theory related to shear thickening effects. Then a variety of shear thickening systems will be discussed with particular emphasis on food and biopolymer related systems. Special attention will be given to the need for a careful distinction between shear thickening alone and shear thickening accompanied by dilatancy. We wish to point out an unexpected problem in using on line computer searches to review literature relevant to this chapter. Use of the search term, "biopolymer," in DIALOG searches of Chemical
67 Abstracts (American Chemical Society) did NOT result in listing of our own publications pertaining to starch, although use of the search term, "starch," did. This indexing situation was most surprising to us.
2. R H E O L O G I C A L
M E A S U R E M E N T S AND R E L A T E D T H E O R Y
In the simplest sense a shear-thickening fluid would be considered as one in which the viscosity increases with increasing shear rate. For such a fluid a plot of shear stress versus shear rate would curve upward as shown in Figure 2a. Such plots would typically be obtained using a viscometer such as a Couette bob-in-cup instrument or a cone-and-plate viscometer. (Rheological instrumentation has been reviewed in numerous publications (Dealy [7]) and there are available useful commercial instrumentation brochures from companies such as Haake, Physica, and others.) When shear thickening is observed either in plots such as viscosity versus shear rate (Figure 2b) or shear stress versus shear rate (Figure 2a) there is an implicit assumption that the b Shear thickening J
/ / /
~
Shear thickening
Newtonian
Newtonian
thinning
? Figure 2. a. versus shear b. versus shear
Shear thinning
?
Schematic representation of flow behaviors on a shear stress (z) rate (?) plot. Schematic representation of viscosity behaviors on a viscosity (q) rate (?) plot.
plotted values of shear stress, shear rate and viscosity are steady state values. This may not be the case and the appearance of shear thickening may be a result of transient effects. Such effects are particularly likely with instrumentation in which shear sweeps are performed, for example in a Couette rheometer in which the shear rate may be cycled over a certain range in a certain time. In a recent study of effects of processing on starch solution properties, shear rate sweeps from zero to 750 reciprocal seconds and back
68 down to zero were carried out over a four minute period (Dimzis and Bagley [8]). In such experiments the possibility of transient effects cannot be ignored. Such a flow history is sometimes referred to as the thixotropic loop or T loop (see Greener and Connelly [9]). Other instrumental factors can be significant in the phenomena of shear thickening. For example, a recent contribution by Chow and Zukoski [10] deals with the effect of gap size in shear thickening of a suspension. The possibility of such experimental artifacts must be kept firmly in mind in interpreting complex rheological data. Fluids for which viscosity depends not only on shear rate but also on time can be subdivided into two classes, thixotropic and rheopectic (Tanner [6]). Thixotropic fluids are generally regarded as ones in which there is a breakdown of structure by shear while rheopectic fluids are ones (Tanner [6]) in which there is formation of structure due to shear. The history of the term "thixotropy" has been reviewed briefly by Cheng and Evans [ 11 ] but they use the additional term "antithixotropic" because in their view there are three distinct types of structural changes brought about by shear: l) structural breakdown under shear with recovery at rest (thixotropy); 2) structure buildup under shear, disintegrating at rest (anti-thixotropy or negative thixotropy); 3) structure breakdown at moderate and high rates of shear with recovery accelerated at low shear rates, the recovered structure being stable at rest (rheopexy). The difficulty with the introduction of new terminology to describe particular types of behavior is that nature keeps providing us with new phenomena which do not fit into the accepted categories of the times. Thus, starch systems (as will be discussed in more detail below) can show structure buildup under shear but the structure so formed appears to be quite stable, both at rest and under different shear fields (Dintzis and Bagley [8]). This type of behavior is not covered by the three classifications proposed by Cheng and Evans. Further, when the classifications become too numerous it often happens that different workers use the terminology in different ways. A good example of such confusion was given by Cheng and Evans, who noted in their 1965 paper "At the present time thixotropy is still used by certain rheologists to denote what is more commonly known as pseudoplasticity (that is, the decrease in viscosity with increase in shear rate) without reference to time dependence." Barnes has recently reviewed shear thickening effects for nonaggregating solid particle dispersions in Newtonian liquids (Barnes [12]). He summarizes his views in the abstract of his article as follows" "The actual nature of the shear thickening will depend on the parameters of the suspended phase: phase volume, particle size and distribution, particle shape, as well as those of the
69 suspending phase (viscosity and the details of the deformation, i.e., shear or extensional flow, steady or transient, time and rate of deformation)." Barnes notes that shear thickening is defined in the British Standard Rheological Nomenclature as "the increase in viscosity with increase in shear rate, and is to be distinguished from rheopexy which is the increase of viscosity with time at constant shear rate." Barnes also makes some interesting comments on, and gives some references to, the practical implications of shear thickening.
60
.,..:::...:'.. : .........
50
(~ CO L~
(1)
(-"
,
40 ................................... W a x y m a ! z e s t a r c.=.h ! " =""
30 . . . . . . . . . . . . . . . . . . . . . . . .
'= '
-"
| ,,.. m"
:.,'" : . ~ =iP ram'.
i~i =
................................
m""
~..-':-'-'"- ;,.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,~, ~"",: "
~r
9
,
co 20 ......... :::: ................ : ................................................. "":' 1 .'A B:uffalo starch : ~..~,~,,,,~ "
-
.."
i
.,?
i
i
1 0 - ; . . . . . . . !;, . . . . . . . ~. . . . . ~ , ~ . ~ ~ ~ ' ~ - .
9
m m=
,-" ,~0~,,,~ ~
n,~'.~., vO
i.., 1oo
,
~
.
..... ! .........
.
.
.
! .........
! ......... :
.
:
:
:
:
:
i,.. 200
i.., 300
i... 400
i,,, 500
i,,, 600
i . , ~ 700 800
Shear Rate[I/s]
Figure 3. First cycle of a shear stress/stain rate response of 2.0% (wt/vol) (solid symbols) waxy maize and (open symbols) normal maize or "Buffalo" starch solutions in 0.2N KOH at 30 ~ C. The loop designated 1 is anticlockwise; loop 2 is clockwise. This normal maize starch solution did not exhibit shear thickening behavior; the waxy maize starch solution did. (From Dintzis and Bagley [8]). Rheological measurements in any event require careful analysis to understand the processes involved in the shearing of complex fluids or concentrated suspensions/dispersions, particularly when non-steady state data are obtained as in the use of the thixotropic loop experiments in which shear rate is increased and then decreased over a certain range and in a specified
70 time period. Results can be a reflection of the relaxation times of the materials being studied. For example, Professor I. Krieger emphasized in a recent meeting that linear viscoelastic materials will always yield anticlockwise loops in plots of shear rate (abscissa) versus shear stress (ordinate), and subsequently in a private communication he demonstrated this for both the Kelvin (Voigt) and Maxwell models (Krieger [13]). Greener and Connelly [9] showed that the application of a T-Loop ramping of shear rate versus time can lead, for the Wagner model, to anticlockwise hysteresis loops or even to figure 8 plots such as are shown in Figure 3. When shear thickening effects are observed it is thus important to determine whether the observations are simply a reflection of the relaxation time characteristics of the material being investigated or whether the thickening is a reflection of structure buildup during shear. When structure buildup occurs it then becomes important to know whether this structure is stable or reverts with time (with or without further shear treatment) to its initial state. Shear thickening has also stimulated work on non-equilibrium molecular dynamic (NEMD) methods as tools for investigating the origins of shear dilatancy and shear thickening phenomena (Woodcock [14]). These NEMD calculations show that shear thickening and volumetric changes can occur even in simple classical dense fluids as kinetic manifestations of a transition to athermal particle dynamics at shear rates approaching a critical characteristic response frequency. These computations are perhaps relevant to colloidal dispersions, corresponding to suspensions of particles, but the application of the methodology to polymer systems is probably not feasible at this time.
3. SUSPENSIONS Numerous examples of shear thickening going back to the original studies by Reynolds of dilatancy of sand systems could be cited. A particularly interesting example and one that might be considered to be related to food systems is given in the work by Williamson and Heckert [15] who worked with corn starch dispersions. They demonstrated the increase of apparent viscosity with increase in shearing stress using a modified Stormer viscometer and discussed results in terms of geometrical factors (starch granule packing) and related concerns including the question of starch granule swelling in water. A more recent investigation of starch systems was included in the extensive work by Hoffman dealing with discontinuous and dilatant viscosity behavior in concentrated suspensions. His studies (Hoffman [ 16,17]) included a variety of suspensions including monodisperse suspensions of polymeric materials. In later studies (Hoffman [18]) the behavior of unfractionated
71
potato starch suspended in glycerol was examined using a parallel plate rheometer with various gap widths and volume fractions at which both shear thickening (volume fractions of 0.52 and 0.56 respectively) and discontinuous flow (volume fraction of 0.60) curves were obtained (Figure 4). These volume fraction levels are in the range where one might expect shear 10 4 LL
(n
o
ro
0 o
.on >
.o
LL
(/)
t-
r
c)
03
10 3 -
10 2
-
r or)
10
........ 10 -2
TIFf',
i ........ 10 -1
I 1
........
i 10
........
I ....... 10 2
10 3
( S u s p e n d i n g Fluid Viscosity) x ( S h e a r Rate); ( d y n e s / c m 2)
Figure 4. Influence of the volume fraction of unfractionated potato starch granules on the flow behavior of potato starch granules/glycerol suspensions; torsional flow at about 23 ~ C; O Volume fraction starch (VF) = 0.52; rn VF - 0.56; V VF - 0.60. (From Hoffman [18] by permission). thickening and dilatancy due to particle crowding and ordering into layers, as discussed by Hoffman. Although not explicitly stated the viscosity data given by Hoffman appear to be steady state values. Starch particles in general, and certainly potato starch in particular, are not uniform spheres and as Hoffman discusses one needs to consider that particle size distribution and particle shape can both influence the thickening effects. As Williamson and Heckert noted, and as was no doubt also true for the work of Hoffman, the starch granules are little affected by solvent (water, glycerol) at room temperature. In many actual food processing applications, though, temperatures would be in the gelatinization range, high enough that swelling of the starch may occur, or higher, where continued exposure to high enough temperatures, pressures and shear would lead to starch granule disruption and solubilization.
72 Further studies of the flow of potato starch suspensions have been reported by Bang et al. [ 19]. They report dilatant behavior but caution that "Not all the shear thickening is associated with a volume increase, but the term dilatancy is more broadly used for the materials which increase viscosity with increase of shear rate." However, the work of Bang et al. [19], while showing shear thickening of potato starch suspensions at concentration levels of 14 to 30 wt % at temperatures in the range of 20 ~ C-40 ~ C does not show the flow discontinuity which Hoffman reports at concentration levels such as 50% or greater. Bang et al. give a theoretical explanation that involves swelling of the outer amylopectin molecules of the starch granule for the "dilatancy" in their systems. This hypothetical swelling with the extension of the outer granule amylopectin molecules into water is proposed as the mechanism for the formation of a "three-dimensional scaffold structure" which results in the shear thickening. More data on the partial swelling of the starch granules at these low temperatures (below 40 ~ C) would be needed in establishing the validity of the Bang et al. explanation for the thickening. Rheological investigations of the "cooking" of starch granules have been made (Bagley and Christianson [20]); Christianson and Bagley [21]). Figure 5 taken from the Christianson and Bagley [21] reference illustrates the type of behavior observed. The dispersion was 25% com starch in water, heated at 65 ~ C (on the lower end of the gelatinization temperature range for this starch) for various times. After heating, a sample was removed from the heating vessel and the viscosity measured in a Haake Rotovisco Couette rheometer at 60 ~ C. The shear rate was raised from the low end (around 3 reciprocal seconds) to the high shear rate range in 3 to 5 minutes depending on the upper shear rate reached (see Figure 5). The initial uncooked dispersion had a very low viscosity which could not be measured on the equipment used for the cooked samples. After 15 minutes heating at 65 ~ C the starch granules had swollen enough that the viscosity level was now well within the measurable range for the Rotovisco. As can be seen from Figure 5 the viscosity initially decreases with increasing shear rate but a region is reached just above 10 reciprocal seconds where the viscosity begins to increase with increasing shear rate. This shear thickening is interpreted as follows. After 15 minutes at 65 ~ C the effective volume fraction of the water-swollen starch granules is now high enough to have a viscosity in the range 500 to 1000 cp. However, the granules still have a relatively rigid core and act as more or less rigid particles giving rise to the shear thickening shown in the lower curve of Figure 5. After 30 minutes the particles have swollen still more so that the effective volume fraction is such that viscosities half a decade higher than the 15 minute cook are observed. The particles, however, are more deformable, being much less rigid, more
73
100,000 At 60~ after cooking at 65~ | 15 min o 45 min A 30 min 9 75 min
~e~ e~
--0~0~
10,000 ..=....
25% (db)corn starch in water
~
~o
r .....,.
"~'---A-- A.----A--A ~ A .
Q~
1,000 " E ) ~ E ) _ E).___E)._E)~ E ) ~ E)--" E)~ E ) ~ " E)..
100
1
1
10
,
I 100
1000
Shear Rate (sec-1) Figure 5. Viscosity (cp) vs shear rate (sec l) for a 25% dispersion of normal maize starch in water, measured at 60 ~ C after cooking in a Corn Industries Viscometer at 65 ~ C for 15 (Q); 30 (A); 45 (n); and 75 ( e ) rain. (From Christianson and Bagley [21].)
plasticizing water having penetrated the granules. The evidence of shear thickening is minimal. At still longer cook times, 45 and 75 minutes, the granules occupy still greater volume fractions, are now very deformable with little rigidity, and the high viscosity dispersions show typical non-Newtonian shear thinning behavior. Such shear-thinning behavior for concentrated dispersions of deformable, swollen particles has been seen earlier (Taylor and Bagley [22]). It is particularly interesting that for highly swollen and readily deformable gel particles this shear thinning behavior even continues at dispersion concentrations corresponding to volume fractions of unity, that is for dispersions of close-packed deformable particles. The question must be considered as to whether the shear thickening is a reflection of the fact that the structural relaxation time is comparable to the
74 time required to ramp the shear rate from the low initial level to the upper shear rate. Unfortunately, if the ramp time is varied to resolve this question it means that the dispersion must be kept at 60 ~ C for longer periods of time and the experimental result will be clouded by the effects of a longer cooking period. However, the shear thickening observed for measurements made around room temperature where the ratio of ramp time to relaxation times is quite different from that at 60 ~ C suggests that the shear thickening observed is not a result of time effects. More extensive investigation of the point would be desirable. Nevertheless, the observation of shear thickening in these studies certainly seems to provide information on the swelling process and the changes in the deformability of swollen starch granules during the cooking process.
4. S H E A R T H I C K E N I N G EFFECTS IN S O L U T I O N S / D I S P E R S I O N S
It is rather remarkable that moderately concentrated and dilute dispersions/solutions have been shown to exhibit flow behavior that has been termed shear-thickening. For example, such systems include: the copper salt of cetyl phenyl ether sulfonic acid, at concentrations as low as 0.02%, (Hartley [23]); isoionic DNA (Kuznetsov et al. [24]); 5% rubber in toluene (Crane and Schiffer [25]); 5% polymethacrylic acid (Eliassaf et al. [26]); and dilute systems of polyacrylamide in glycerol-water mixtures (Dupuis et al. [27]). All exhibit antithixotropic and/or dilatant behavior. In some cases, particularly where there are strongly interacting ionic, polar or hydrogen bonding groups, mechanisms are conceivable which might lead to the structure formation implied by shear thickening effects. In other cases of flexible polymer molecules the mechanism by which shear thickening could occur are not so evident. This is in contrast to the shear thickening seen in dispersions of rigid particles where one can see intuitively how geometric and crowding effects could lead to shear thickening and dilatancy and detailed mechanisms have been proposed to describe the effects quantitatively (e.g., Hoffman [16,17]). It has recently been shown that relatively dilute starch systems can also show shear thickening behavior. Starch is a particularly interesting molecule to consider in this regard. First of all, it is of practical importance having extensive use both as a food and in industrial applications such as paper coating. Second, it is a biopolymer which exists both in linear (amylose) and branched (amylopectin) forms, and hence is of particular scientific interest. Both the scientific and technological interest in starch are magnified because the molecular weights of the starch, especially the amylopectin component,
75 can be extremely large, of the order of one hundred million or more. This means that starch, in terms of the number of monomer units per polymer molecule, is much larger than most normal commercial thermoplastic polymers such as polyethylene and polystyrene. Starch is also interesting and challenging because of its solubility characteristics. Common practice in the use of starch has been to "paste" the material by heating with mechanical stirring at temperatures up to about 95 ~ C (Whistler et al. [28]). While well beyond the temperature range at which gelatinization occurs (i.e., where starch crystalline melting occurs and significant swelling due to water sorption takes place) the starch granules are not by any means fully solubilized. Amylose may be largely in solution but relatively little amylopectin can be fully solubilized, and starch granule fragments may persist even to 100 ~ C and above. The gelatinization range depends on numerous factors such as the type of starch (corn, rice, wheat, etc.), the variety (e.g., waxy corn starch consisting primarily of amylopectin) and various agronomic factors. For common starches this gelatinization range is 60o-80 ~ C. With so many hydrogen bonding sites it seems surprising that starch is not more readily soluble in water. In fact, it is necessary to heat starch in water well above 100 ~ C (to, say, 120 ~ to 140 ~ C in an autoclave, for example) to approach polymer solubilization. Heating starch, with stirring, to such temperatures produces "solutions" that are quite different in properties from dispersion/solutions that have only been "pasted" by heating to around the 95 ~ C level (Christianson et al. [29]). Alternate routes to solubilizing starch include jet cooking (in which a starch slurry is passed through a nozzle with high pressure steam, Christianson et al. [30]) and the use of solvent systems such as dimethyl sulfoxide (DMSO), water/NaOH, water/KOH and other alkaline aqueous systems. Starch is heated above 100 ~ C in aseptic food processing. This can be a continuous sterilization process for viscous liquid foods in which high temperature is applied for a short time. Dail and Steffe [31,32], in studying this aseptic processing, found that solutions of cross-linked waxy maize starch showed shear thickening effects and that this was probably due to dilatancy instead of time-dependent thickening due to on-going pasting. In investigations of the effect of thermo-mechanical processing on starch structure and size it was found that the intrinsic viscosity of starch decreased significantly as processing severity increased (Dintzis and Bagley [33]). At the same time dilatancy was evident in samples that were not severely treated (Dintzis and Bagley [8]). Since the viscosities were obtained with a shear sweep the observed thickening could have been a result of time effects as discussed by Krieger [13]. However, the thickening in these starch solutions
76
was of an unusual type, as indicated in Figure 6 taken from Dintzis et al. [34]. In a shear sweep from rest to 750 reciprocal seconds over a period of 2 minutes, and then without pause decelerating to zero shear rate in another 2 minutes, with several subsequent repeats of this cycle, results as shown in Figure 6 were obtained. The solution freshly placed in the fixture was initially shear thinning but just beyond 100 s~ shear thickening was observed. After an increase of nearly half a decade in viscosity, the solution became again shear thinning. Subsequent lowering of the shear rate gave a viscosity/shear rate plot following a power law all the way down to the lower shear rate limits of the machine. Subsequent cycling gave viscosity/shear rate curves which to 101 o
o .
.x.
Fresh c o o k
60 ~C F l o w
3 hrs. old, stored in D e w a r
f/) C
rl
0
v
>,
o,.~.
10 0
x o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
or) O O
x
i
0
x
o
x
o o o
x
x
~ x x ,,,
:"Ooc b x ~,,,, "o
>
o o
1 0 -1 10 0
. . . . . . . .
o
o c>! o 0.~. om~
, 10 ~
. . . . . . . .
i
10 2
(~
. . . . . . . .
10 3
Shear Rate (s -1) Figure 6. Viscosity/shear rate behavior in Couette flow at 60 ~ C of a 10% (wt/vol) sample of waxy maize starch autoclaved with stirring at 140 ~ C for 15 min at pH - 7.0. Because the initial low stresses in these aqueous systems are below the operating range of the sensor, the first three data points on each of the curves AB are unreliable and probably not valid. O = freshly cooked sample; X = sample stored in Dewar flask for 3 hr prior to flow measurement. (From Dintzis et al. [34]). a first approximation followed the initial downcurve BC, from approximately 750 to about 9 s ~ of Figure 6 though the viscosity level did drop slowly on
77 successive recyclings. The result was shown not to be an artifact of the instrument (see Figure 2 in Dintzis et al. [34]). These results have been extended to waxy rice and waxy barley starches and it has been established that it is the amylopectin component that causes these effects (Dintzis et al. [35]). With careful sample preparation one can detect a very attenuated form of the shear thickening and structure formation effects in regular maize starch. This very sizeable shear thickening effect, in which the viscosity increased by almost half a decade, occurred only with freshly prepared solutions. Once the higher viscosity level had been generated, the starch solution showed, on subsequent shear rate cycling, shear thinning behavior only. Structure formation during the shear thickening portion of the initial curve was suspected and phase microscopy pictures confirmed this suspicion as shown in Figure 7. The fresh, unsheared samples of waxy maize, dull waxy maize and normal maize starches showed little evidence of any structure. Shearing of the regular or normal starch solutions generated little, if any, structure. However, shear-generated "structure", as indicated by refractive index inhomogeneities in the micrographs, is quite evident in the dull waxy maize sample and predominates in the case of the waxy maize starch. The later sample appeared quite clear, water-white, but after shearing through the shear thickening region the solution developed an obvious opalescence. Results in these studies are very sensitive to sample preparation procedures. This can be illustrated with reference to time effects observed in some solutions. These effects can be seen (Figure 7 of Dintzis et al. [34]) where shear thickening was observed occurring over surprisingly long times at constant shear rate for a gently dispersed 3% waxy maize sample in 90%DMSO-H20. However, vigorous stirring changed this time behavior significantly, eliminating the shear thickening effect. Normal maize solutions generally appear to be simple shear thinning materials. A more detailed study of 3 g waxy maize starch in 100 g of 90% DMSO showed three different types of behavior. When the dispersion/solution in a flask was stirred with a stirring bar for 1 hour at 100 rpm, one initially observed an opaque dispersion of low viscosity that soon became transparent as the viscosity increased during stirring. Finally, the fluid became viscous, elastic, and water white, but with "fish eyes" (visible regions of refractive index inhomogeneity) readily apparent. This fluid, when examined using a Rotovisco R rheometer with shear rates held at 10, 50, 100 and then again 10 s1 for 20 minutes at each rate, showed no shear-thickening behavior. This starch stock fluid was subjected to further stirring and then tested again in the Rotovisco R. After sufficient stirring of the stock fluid, shear-thickening behavior was observed in the rheometer. When the stock fluid was again subjected to additional stirring, the shear-thickening effect and the "fish eyes"
78 disappeared and the solution became shear thinning.
.
.i~ ....
9,:<~,~,,,
i:~(,"':
C
,,
~
~,~ ~,
"
9~""
7"' .
,~,,;;
...
.~
E
~
. . . . ..,.,.
'
F
.
.
.
.
.
.........
. . .~
:,~.
. .
...
...
....
:
~
lOpm
~7"'~'
'
.~"
.:
..,,,,,..,,
Figure 7. Phase Contrast micrographs of sheared (in cone and plate) vs not sheared starches in 90% DMSO-H20. A, 2.0% waxy maize starch sheared; B, 2.0% waxy maize starch not sheared; C, 2.0% dull waxy maize starch sheared; D, 2.0% dull waxy maize starch not sheared; E, 2.6% normal maize starch sheared; F, 2.6% normal maize starch not sheared. Starch concentrations chosen to provide approximately equal amylopectin amounts in each fluid. (From Dintzis et al. [34]).
5. W A T E R SOLUBLE CHITOSAN DERIVATIVES Clearly, complex processes are at work in these shear-thickening starch
79 systems but other biopolymer systems can also show surprising behavior. Thus, Yalpani et al. [36] reported on the complex rheological behavior of a branched, water-soluble chitosan derivative, 1-deoxylactit-l-yl chitosan, a comb-like polymer. The complexities reported included shear thickening seen in viscosity versus shear rate plots for 1 and 2% (wt/wt) concentration levels of this chitosan derivative in water. The authors emphasized that steady state values of shear stress were obtained in their Brabender Rheotron viscometer and used in calculation of viscosities. The onset of the observed viscosity increase occurred at a lower shear rate for the 2% than for the 1% solution. Lower concentrations (0.5% and 0.1%) showed approximately Newtonian behavior over the shear rate range of 10 to 3000 s -~. Rising temperature reduces the shear thickening exhibited by this chitosan derivative. For a 1% solution an approximately five-fold viscosity increase occurs in the shear thickening region at 25 ~ C. The shear rate at which thickening occurs increases with increasing temperature over the range 25 ~ 35 ~ C. Specifically, the viscosity peak occurs at 40 s -~ at 25 ~ C and at 35 ~ C at about 350 s -1. The magnitude of the viscosity increase in the shear thickening region decreases with increasing temperature. At 43 ~ and 50 ~ C evidence of shear thickening has pretty well disappeared and the solutions are only slightly shear thinning in character, or in the authors' words " .... at 43 ~ C, essentially Newtonian flow is again observed." Yalpani et al. also report some unusual time effects for these chitosan derivative solutions, in particular showing oscillatory "shear stress-time dependency at steady shear." To illustrate, for the 2% solution, at a shear rate beyond the peak of the viscosity-shear rate plot, oscillations in shear stress at constant shear rate is seen. The corresponding viscosity variation is quite large, from 800 to 2000 Pa.s, "without any appreciable damping being apparent over the periods (few minutes) .... investigated." The phenomenon was not observed with the 1% solution.
6. SHEAR THICKENING IN PROTEIN SYSTEMS
The effect of shear on structure, phase separation, mixing of polymers and multiphase polymer systems has been noted briefly earlier and recent books serve to spotlight the current interest in these effects (Nakatani and Dadmun [3]; Sondergaard and Lyngaae-Jorgensen [4]). The importance of these effects in food systems, particularly in protein-based foods, has been emphasized by Suchkov et al. [37]. They note that: "The problem of structure-property relationships for protein solutions under shear flow conditions is of great applied importance for modelling food protein recovery and processing. There
80 are several reasons for this. Food systems are always subjected to shear field (mechanical) treatments during processing. This occurs particularly during mixing, stirring, shaking, pouring, homogenization, pumping, extruding and cutting. Normally, the larger scale and greater efficiency of food industrial processes are related to the more intense shear forces. The mechanism of shear effect on protein recovery from dilute solutions is of significance for the improvement of technologies for protein isolation from different raw materials, for scaling up and for wider use of continuous precipitation." The same authors also note that proteins together with polysaccharides are the main components responsible for the properties and quality of foods. Mechanical treatment is an important part of the processing of these components, alone and in combination, and the rheological characteristics are critical aspects of protein/polysaccharide-based foods. Clearly, structure formation during shearing, as indicated by shear thickening effects, can be critical in influencing the final physical and organoleptic properties of processed foods.
T
o
~
o
o-1
0-2
2O
E 00 ~
15
-i
/ 7 10
I
0
10
I
C* 20
I
30
...
I
40
Cpr, % wt.
Figure 8. Phase diagram of the legumin-NaCl-water system" 0, turbidity measurements; O, phase component analysis. (From Suchkov et al., private communication and [37]).
81 The work of Suchkov et al. is of significance in the food area because they examined both the rheological and phase behavior of an important protein system, specifically salt solutions of the 11S globulin protein from broad beans (legumin). Stable single phase salt solutions of this protein "can undergo a reversible transition from the single phase to the two phase state .......... either on cooling or diluting the protein solution with water." A concentrated "mesophase" and a dilute phase are formed so that in effect the water solution changes into a water-in-water emulsion. Figure 8 (Figure 3 of Suchkov et al. [37]) shows the phase diagram for 11S broad bean globulin-NaCl-water system. Viscosity measurements of a 19.6 wt % system were made at temperatures from 11 ~ C to 29 ~ C in 1.5 ~ C intervals, and the results for temperatures 14.0 ~ to 21.5 ~ are shown in Figure 9 (Figure 2 of Suchkov et al.). The authors note that above the binodal the single phase legumin/salt solutions showed Newtonian behavior while within a range of temperatures below the binodal the system exhibited nonNewtonian behavior and specifically showed shear thickening effects. It is evident from Figures 8 and 9 that the onset of apparent shear thickening does not coincide very exactly with the development of two phases. It is not until 17.0 ~ C, well below the maximum of the phase diagram plot, that the shear thickening is seen. The author's comment (V. Ya. Grinberg, personal 0,2
--
6 (14~
0,1
-5
oo
4 (17~ 3
n
E
2 (20~ 1
~" 0,02
0,01
I
0
600
......
I
!
I
1200
1800
2400
~/, S-1 Figure 9. Flow curves at different temperatures of the legumin-NaCl-water system for a protein concentration of 19.6 wt%. (From Suchkov et al., private communication and [37]).
82 communication) that the shear sweep cycle time was about 40 rain. and that each of the coexisting phases in the protein system was a Newtonian liquid suggests that the observed shear thickening was not the result of time effects. It is also interesting to observe in Figure 9 that viscosities can be lower, at the lower shear rates, for the low temperature measurements than at the higher temperatures; for example at 14.0 ~ C the viscosity below about 750 sl is lower than the Newtonian viscosities at 17 to 21.5 ~ C. Such a reversal could be indicative of a shear dissolving process. Certainly, the results are extremely interesting and potentially valuable for process design applications. Considerably more work and extensive investigation of protein and food systems is needed along lines discussed in the literature, for example, by Wolf and Horst [38] and Tolstoguzov [39].
7. H O N E Y
Scott Blair [40] in Chapter VI, The Rheology of Honey, from his wellknown book on "Foodstuffs, Their Plasticity, Fluidity and Consistency" notes that the term "dilatancy" had been expanded "of recent years" to include "any system whose viscosity increases with increasing rate of shear beyond a certain minimum value". He noted that "the system need not contain any solid particles but can be a true solution such as gum arabic in water to which a small amount of borax has been added." He then went on to reference the work of Pryce-Jones [41 ], who, using a twin-Couette rheometer, demonstrated that a large number of honeys showed the property of shear thickening. Specific honeys, such as honey from Eucalyptus ficifolia, also possessed the property of being easily drawn into long strings, the property of "Spinnbarkeit". Scott Blair further emphasized that dilatant honey systems displayed the Weissenberg effect, as evidenced by the fact that when a glass rod is immersed in honey and rotated at certain speeds the honey will climb up the rod. This certainly implies that the system is exhibiting considerable elasticity so that one evidently is dealing with viscoelastic fluids for these dilatant honeys. Honey dilatancy is also emphasized in more recent texts, for example in Sone [42]. Honeys are essentially concentrated sugar solutions. However, for the 15 dilatant honeys investigated by Pryce-Jones [43] all contained dextran (C6H1005) n with n of the order of 8000. This would mean the dextran is a relatively high molecular weight material, approximately 1.3X106 Daltons. The concentrations of dextran are relatively low, quoted by Scott Blair as 7.2% dextran for the Eucalyptus ficifolia and 6.4% for Eucalyptus eugeniodes honeys, respectively. Of course, for molecules of molecular weights over a
83 million one should perhaps regard 6 and 7% solutions as really quite concentrated. The viscosity increase for the 7.2% solution occurs over a narrow shear rate range. At shear rates of 50 reciprocal seconds the viscosity is about 85 poise, rising to 90 poise at 75 reciprocal seconds and 130 poise at shear rates around 110 s~. That the dilatancy is due to the presence of the dextran is shown by the observation (Scott Blair [40]) that the removal of the dextran by precipitation with acetone and reconstituting the honey to its original concentration yields a product which behaves like a true Newtonian fluid. Scott Blair also notes that the addition of dextran to clover or sainfoin honeys, which do not show dilatancy, stimulates "The peculiar rheological effect" of (presumably) shear thickening and rod climbing. Although, as Sone notes, such "observations indicate that dextran is the cause of dilatancy in honey" the exact mechanism is obscure. No doubt, as in the case of starch fluids discussed above, the large number of hydrogen bonding sites accessible in the dextran, as in starch, contributes to the formation of some structure under shear which leads to a shear-thickening effect. There has been very compelling rheological evidence confirming the part played by dextran in these shear thickening effects (Sabati~ and Choplin [44]; Choplin and Sabati~ [45]; Sabati6 et al. [46]). Sabati6 and Choplin used a carefully prepared dextran and found that structure formation did occur and that the viscosities of the materials before and after structure formation differed by at least a decade, the structure formation being irreversible [44]. The shear thickening and structure formation behavior of dextran systems are indeed very similar to the behavior of amylopectin solutions showing features such as anticlockwise "thixotropic loops" (Figure 6, [45]). Choplin and Sabati~ also conclude that the structure formed in dextran, irreversible provided sufficient strain is applied, may be due (in part) to the ability of the molecule to interact by hydrogen bonding. However, this would not address the problem of the reported differences among amylopectins in the magnitude of the effects. Some interesting earlier references are provided by Sabati~ et al. [46] in reviewing the rheology of native dextran solutions and subsequently in this paper the authors conclude that, in addition to hydrogen bonding effects, "chain-chain" packing that depends strongly on molecular conformation contributes to the dextran shear thickening effects. Equivalent conformational differences no doubt lie at the root of the reported differences among amylopectins [33,34,35] but it is not at all evident how these conformational differences could be more clearly illuminated.
84
8. RECENT MODELS
Dr. Craig Carriere kindly called our attention to some recent papers describing theoretical models for the development of shear thickening. There is the phenomenological model of Hess and Hess [47]. Wang examined a model treating the dynamics of coexistence between a physically cross-linked network with temporary junctions and unbound chains [48]. Hatzekiriakos et al. [49] modeled polymer macromolecules as Hookean elastic dumbbells deforming affinely during flow. While not exhaustive, these references are of fundamental interest and computational investigation of the applicability of these approaches to specific shear thickening systems is clearly warranted. It must be noted, though, that these models imply structural reversibility which is not observed in the behavior of amylopectin.
9. A C K N O W L E D G E M E N T We very much appreciated seeing a pre-publication copy of the manuscript, "Effect of shear on the phase behavior of the legumin-salt-water system. Modelling protein recovery.," by V.V Suchkov, I.A. Popello, Ya. Valerji and V.B. Tolstoguzov. We thank these authors for their kind permission to include Figures 8 and 9 in this review and for their promptness in responding to our questions. Thanks are also due to Dr. Craig Carriere for helpful discussions. REFERENCES ~
.
.
,
~
W.W. Graessley, Viscoelasticity and Flow in Polymer Melts and Concentrated Solutions, in "Physical Properties of Polymers," Am. Chem. Soc., Washington, D.C. (1984). E.B. Bagley and H.P. Schreiber, Elasticity Effects in Polymer Extrusion, in "Rheology, Theory and Applications, Vol. V," F.R. Eirich (ed.), Academic Press, N.Y. (1969). A.I. Nakatani and M.D. Dadmun, Flow-Induced Structure in Polymers, ACS Symp. Series 597, Am. Chem. Soc., Washington, D.C. (1995). K. Sondergaard and J. Lyngaae-Jorgensen, Rheo-Physics of Multiphase Polymer Systems, Technomic Pub. Co. Inc., Lancaster, Pennsylvania and Basel, Switzerland, 1995. B. Launay, and B. McKenna, Physical Properties of Foods, R. Jowitt, F. Escher, B. Hallstrom, H.F.Th. Meffert, W.E.L. Spiess and G. Vos, (eds.), Appl. Sci. Pub., Ltd., UK (1983).
85 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.
R.I. Tanner, Engineering Rheology, Clarendon Press, Oxford, 1985. J.M. Dealy, Rheometers for Molten Plastics, Van Nostrand Reinhold Co., New York, 1982. F.R. Dintzis and E.B. Bagley, J. Appl. Polym. Sci. 56 (1995) 637. J. Greener and R.W. Connelly, J. Rheol. 30(2) (1986) 285. M.K. Chow and C.F. Zukoski, J. Rheol. 39(1) (1995) 15. D.C-H. Cheng, and F. Evans, Brit. J. Appl. Physiol. 16 (1965) 1599. H.A. Barnes, J. Rheol. 33(2) (1989) 329. I.M. Krieger, Comments made at the Rheol. Symp., Pittcon '96 Conf., Chicago and in private communication (1996). L.V. Woodcock, Chem. Physics Letters, 111 (No. 4,5) (1984) 455. R.V. Williamson and W.W. Heckert, Ind. Eng. Chem. 23(6) (1931) 667. R.L. Hoffman, Trans. Soc. Rheol. 16(1) (1972) 155. R.L. Hoffman, J. Colloid and Interface Sci. 46(3) (1974) 491. R.L. Hoffman, Advances in Colloid and Interface Sci. 17 (1982) 161. J-H. Bang, E-R. Kim, S-J. Hahn and T. Ree, 1983, "Flow Mechanism Of Dilatant Systems. (1) Starch Suspensions in Water," Bulletin of Korean Chemical Sot., Vol. 4, No. 5212-217, 1983. E.B. Bagley and D.D. Christianson, J. Texture Stud. 13 (1982) 115. D.D. Christianson and E.B. Bagley, Cereal Chem., 60(2)(1983) 116. N.W. Taylor and E.B. Bagley, J. Polym. Sci., Polym. Phys. Edition 13 (1975) 1133. G.S. Hartley, Nature 142, (1938) 161. I.A. Kuznetsov, V.F. Popenko and S.M. Filippov, Biofizika 23 (1978) 539. J. Crane and D. Schiffer, J. Polym. Sci., 23 (1957) 93. J. Eliassaf, A. Silberberg and A. Katchalsky, Nature 176, (1955) 1119. D. Dupuis, F.Y. Lewandowski, P. Steiert and C. Wolff, J. NonNewtonian Fluid Mechanics 54 (1994) 11. R.L. Whistler, J.N. BeMiller and E.F. Paschall, Starch: Chemistry and Technology, 2nd ed., Academic Press, Orlando, Florida, 1984. D.D. Christianson, E.M. Casiraghi and E.B. Bagley, Carbohydr. Polym. 6 (1986) 335. D.D. Christianson, G.F. Fanta and E.B. Bagley, Carbohydr. Polym. 17 (1992) 221. R.V. Dail and J.F. Steffe, J. Food Sci., 55(6) (1990) 1660. R.V. Dail and J.F. Steffe, J. Food Sci., 55(6) (1990) 1764. F.R. Dintzis and E.B. Bagley, J. Rheol. 39(6) (1995) 1483. F.R. Dintzis, E.B. Bagley and F.C. Felker, J. Rheol. 39(6) (1995) 1399. F.R. Dintzis, M.A. Berhow, E.B. Bagley, Y.V. Wu and F.C. Felker, Cereal Chem. 73 (1996) 638.
86 36. 37. 38.
39.
40. 41. 42. 43. 44.
45. 46. 47. 48. 49.
M. Yalpani, L.D. Hall, M.A. Tung and D.E. Brooks, Nature, 302 (1983) 812. V.V. Suchkov, I.A. Popello, V.Y. Grinberg and V.B. Tolstoguzov, Food Hydrocolloids 11(2) (1997) 135. B.A. Wolf and R. Horst, "Flow Induced Dissolution/Demixing in Polymer Systems - A Predictive Scheme" in Rheo-Physics of Multiphase Polymer Systems, K. Sondergaard and J. Lyngaae-Jorgensen (eds.), Technomic Pub. Co. Inc., Lancaster, Pennsylvania, 1995. V. Tolstoguzov, Gums and Stabilizers for the Food Industry, G.O. Phillips, P.A. Williams and D.J. Wedlock (eds.), "Applications of phase separated biopolymer systems," IRL Press, Oxford 1996. J.W. Scott Blair, Foodstuffs, Their Plasticity, Fluidity and Consistency, Interscience Pub., Inc., New York (1953). J. Pryce-Jones, J. Sci. Inst., 18 (1941) 39. T. Sone, in "Consistency of Foodstuffs," D. Reidel Co,, Dordrecht, Holland, 1972. J. Pryce-Jones, Bee World 33(9) (1952) 147. J. Sabati6 and L. Choplin, "Shear Induced Structure in Dextran Solutions" in Proceedings of the Symposium on Recent Developments in Structured Continua, D. DeKee and P.N Kaloni (eds.), Longman Scientific and Technical, (1985) 165. L. Choplin and J. Sabati6, Rheol. Acta 25 (1986) 570. J. Sabati6, L. Choplin, F. Paul and P. Monsan, Rheol. Acta 25 (1986) 287. Hess, Ortwin and Hess, Siegfried, Physics A., 207 (1994) 517. Wang, Shi-Qing, Macromolecules 25 (1992) 7003. Hatzekiriakos, G. Savvas and D. Vlassopoulos, Rheol. Acta 35 (1996) 274.
87
R H E O L O G Y OF FOOD EMULSIONS C. Gallegos and J.M. Franco Departamento de Ingenieria Quimica, Universidad de Huelva, Escuela Polit~cnica Superior, La R6bida, 21819, Palos de la Ftra. (Huelva), Spain.
1.
INTRODUCTION
An emulsion may be defined as a thermodynamically unstable heterogeneous system formed by at least two liquids that are at best only slightly soluble. The internal phase is dispersed in the other in the form of small droplets, with diameters higher than 0.1 ~tm [1-3]. Food emulsions are governed by the same principles as are other emulsion systems. The only specific requirements for these emulsions are that they must posses long-term stability (several months or years) and that they must contain only ingredients which are acceptable for human consumption [4]. Most of food emulsions are oil-in-water (o/w) emulsions, e.g. mayonnaise, salad dressing, coffee cream, cream liqueurs, soft drinks, ice creams,... [5, 6]. They are typically stabilized by an adsorbed layer of protein at the oil-water interface, although many food colloids contain a mixture of macromolecular and low-molecular-weight emulsifiers [7-10]. Different types of surface active molecules may interact with each other in different ways [11], for example inducing protein displacement from the interface [12]. On the other hand, water-in-oil (w/o) emulsions, such as butter, margarine or spreads, are more dependent on the cristallinity of the oil phase, the presence of rigid surfactants on the interface and the presence of agents which increase the viscosity of the aqueous droplets [4]. In some cases, it is possible to obtain a multiple emulsion, for instance waterin-oil-in-water (w/o/w). wl/o/w2 emulsions are systems where a w~/o simple emulsion is dispersed in a second aqueous phase w2. The oil phase exists as a thin layer, stabilized by a molecular monolayer of emulsifying agents at both oil-water interfaces [13, 14]. In many occasions, this type of emulsions are presented as
88 reduced-calorie products since this particular microstructure yields a drastic reduction of fat and energy content [15]. This chapter summarises the state-of-the-art of the bulk rheology of oil-in-water food emulsions, as well as its relationship with the emulsion microstructure and its resulting stability. 2.
EMULSION STABILITY
2.1. Mechanism of emulsion destabUization. As has been previously mentioned, stability is the most important factor to be considered in emulsion technology. An emulsion is stable when there is no change in the size distribution or the spatial arrangement of droplets over the experimental time-scale. The three main mechanisms of destabilization are the flocculation, creaming and especially coalescence of oil droplets. Creaming results from the action of gravity on oil droplets, of lower density than the continuous medium, yielding a vertical concentration gradient of droplets but no change in the overall size distribution. The rupture of the emulsion may be detected by the appearance of a separated aqueous phase at the bottom. The aggregation of the droplets without breakdown of the emulsifier layer is called flocculation. Flocculation may frequently be an intermediate step to the rupture of the emulsion by coalescence, which implies the irreversible process of the droplets becoming larger yielding a flee oil phase at the top of the sample. The dominant mechanism of instability will be determined by the interparticle and hydrodynamic interactions. Whether or not aggregation or coalescence occurs, depends on the composition and surface viscoelasticity of the adsorbed surface layer [16-23] and may be predicted by the classical DLVO theory [24, 25] or, more usually, combined with the steric stabilization theory [26]. All these mechanism are interrelated among them. Thus, flocculation generally favours creaming since the aggregates of oil droplets have a settling velocity higher than that of the individual droplets or, on the contrary, even when creaming takes place without flocculation the final state is a flocculated emulsion which may favour coalescence. 2.2. Factors affecting emulsion stability. In general, emulsion stability depends on droplet size distribution, rheology of the continuous phase and interparticle interactions [27, 28], and all of them affect the bulk theology of the emulsion. As Figure 1 schematically shows, structural parameters, bulk rheology and emulsion stability are interrelated among them. In addition to this, all of them are decisively influenced by processing conditions.
89
EMULSION STABILITY
STRUCTURAL PARAMETERS
* rheology of the continuous phase RHEOLOGY
]~.
r
* n a t u r e of particles 9 droplet size distribution - deformability internal viscosity - surface viscoelasticity concentration -
-
-
* interparticle interactions
PROCESSING CONDITIONS * temperature * residence time * agitation speed * etc.
Figure 1. Relationship among microstructure, processing, rheology and stability of food emulsions.
90 Consequently, emulsion rheology can not be understood without considering the structural parameters of the emulsion. At the same time, a rheological Characterization of these materials provides useful information in order to predict the stability of emulsions. One of the most important factors affecting emulsion stability is the droplet size distribution. It is a well-known fact that fine emulsions with narrow distributions yield a higher stability than coarse emulsions with broad distributions [28, 29]. Lower droplet size favours stability against creaming since settling velocity is a function of droplet diameter. Another crucial factor influencing emulsion stability is the strength and nature of interparticle interactions. If the attractive forces exceed the repulsive electrostatic and/or steric interactions the aggregation of oil droplets occurs and coalescence may take place depending on the surface viscoelasticity of the surface layer. Nevertheless, if the primary minimum of the energy between the approaching droplets is not deep the flocculation is easily reversible and temporary [4, 27]. As has been previously mentioned, flocculation may frequently be an intermediate step to the rupture of the emulsion, either after a creaming process or after the coalescence of oil droplets. However, in concentrated emulsions an extensive flocculation process may favour emulsion stability by forming a weak gel-like particulate network [8, 30] where the continuous phase is immobilized within the interstices [31, 32]. Classical DLVO theory indicates that an increase in ionic strength should lead to instability. However, in emulsions containing polymers such as proteins, the presence of electrolytes has been shown to modify the charge stabilization among oil droplets [32, 33]. Thus, salt affects the tridimensional conformation of ionic polymers and, subsequently emulsion stability [34]. In the same direction, stabilization may be achieved by modifying pH [35]. The strength and nature of the interactions among droplets depend on the type and concentration of the emulsifiers used, which determine if the flocculation is reversible or irreversible. The extension of this aggregation process influences the bulk theology of emulsions [36, 37]. Further stabilization may arise from the presence of low concentrations of water-soluble polysaccharides, hydrocolloids, which act as thickening or structuring agents in the continuous phase [38-40]. The main function of these hydrocolloids is to reduce the mobility of oil droplets, decreasing the extension of both flocculation and creaming processes. Traditionally, this type of stabilization has been related to the existence of a finite yield stress in the thickening solution in which emulsion droplets do not cream since the gravitational lift on the droplets is less than the yield stress. Recently, it has been suggested that the
91
stabilization takes place by the formation of a particulate network being the yield stress of this structural network the decisive factor to stabilize the emulsion [41 ]. In relation to this, another important aspect to be taken into account is the thermodynamic compatibility between the hydrocolloid and the emulsifier used [5, 38, 42]. Thus, added biopolymer may have a positive or negative effect on stability depending on its concentration and the nature of the droplet-biopolymer interactions. 0
M I C R O S T R U C T U R E AND R H E O L O G Y OF FOOD EMULSIONS
3.1. Steady-state flow behaviour The flow properties of an emulsion are among its more important physical characteristics. From a technical point of view, the unit operations related to the manufacture of an emulsion (mixing, pumping, filling, etc.) require the knowledge of its flow behaviour to assess mixing efficiency, power consumption, etc. [4345]. In addition to this, knowledge of the viscous behaviour of emulsions is basic in relation to their final use, that is to understand their response to pouring or extrusion from packs, draining, etc. [45]. Much work has been carried out on the effect of several structural parameters, such as disperse phase volume fraction and particle size distribution, on the flow behaviour of o/w emulsions. Thus, many efforts have been dedicated to understand the relationship between the viscosity and the disperse phase volume fraction, ~ [37, 43, 45, 46]. Very dilute emulsions (~) < 0.01) show a Newtonian behaviour, being the relationship between the relative viscosity and 4) defined by the equation proposed by Einstein [47]: n 1 + 2,5 fir - r l c -
(1)
where r/~ is the viscosity of the continuous phase. The classical Taylor's treatment [48] takes into account the internal circulation of the dispersed fluid:
(1 + 0.4(nc / n j ) )
qrel -1+2.5~ l+(qc/qd) JOO
(2)
where rk is the continuous phase viscosity and r/a is the dispersed phase viscosity.
92 As the volume fraction increases, the relative viscosity becomes a more complex function of ~): rl r - 1 + 2,5~) + b ~)2 + cff 3 +...
(3)
where b is a coefficient that accounts for hydrodynamic interactions between the droplets [49]. This term is usually sufficient to describe the viscosity of dispersions up to ~ = 0.2. Other equations widely quoted in the literature for Newtonian emulsions are Mooney's equation [50]:
( 25~ "~ fir = exP/1 +am,J
(4)
Eilers' equation [51, 52]: ( 1.25~) ) fir-~1 + 1_ aEd~)
(5)
or Roscoe's equation [53]: -
(6)
In addition to these, equations proposed for solid suspensions have been also used to describe the influence of ~ on the relative viscosity, with the following form [54-57]:
qr = qr ($ / d~m )
(7)
where ~, is the maximum packing fraction. In suspensions, the viscosity becomes infinite for ~ = ~m, but in emulsions ~m is a not well-defined parameter due to the deformability of the droplets. As examples of this type of equation could be cited: Frankel and Acrivos
F (~/~.,),,,3 l
rl~ = c L 1 - ($ / $,,),,3 J,
c- 9/8
(8)
93
Krieger-Dougherty
F , qr = [1 - ~ m
-[nl*m
(9)
[rl] --- intrinsic viscosity
Maron-Pierce-Kitano f i r -
1- * 7-2 ~mJ
(10)
However, concentrated emulsions usually show non-Newtonian behaviour. This behaviour has been related to either droplet flocculation or the non-Newtonian behaviour of the dispersed phase [58]. Concerning the evolution of viscosity with shear rate, the general picture shows three different regions, a constant viscosity, rio, at low shear rates, a power-law decrease in viscosity, and finally a constant viscosity, rl~, at high shear rates, characteristic of the unflocculated system. Wilkinson or Carreau models have been widely used to describe this behaviour [9, 58], although depending on the oil volume fraction all these regions may not be observed [36, 59, 60]. A difference of several decades between 11o and rl~ is usually found in a flow curve of concentrated food emulsions obtained using a controlled-stress rheometer [61]. These flow curves are the result of a dramatic structural breakdown which may be attributed to both an irreversible process (droplet coalescence) and a reversible one (deflocculation, droplet alignment, droplet deformation). Consequently, the relationship between viscosity and volume fraction should include the influence of shear rate. Some authors [46] have used the above mentioned equations with non-Newtonian emulsions submitted at very high shear rates, when a high-shear-rate-limiting viscosity has been attained, due to a complete deflocculation of disperse phase droplets. In these conditions, the viscosity of concentrated emulsions can be satisfactorily described by the relation: 11oo _ exp 2 , 5 , ) qc 1- k~) where k depends on the hydrodynamic interactions among droplets.
(1 1)
94
Pal and Rhodes [62] have proposed an empirical equation to correlate results obtained with both Newtonian and non-Newtonian emulsions:
fir - fir [~ / (~)qr=lO0]
(12)
where (~b),~,.=;00is the disperse phase volume fraction at which fir- 100. The same authors [36] have developed a theoretical equation for nonNewtonian emulsions. They explain the non-Newtonian behaviour as a consequence of the formation of flocs, yielding the immobilization of an important amount of continuous phase, and, consequently, increasing the effective concentration of disperse phase. In addition to this, the effective concentration increases due to the hydration effect, that causes an association between continuous phase and emulsifier molecules. The equation results:
f i r --
1 - k o k F (3)),1-2'5'
(13)
where ko is the hydration factor, and ky is the flocculation factor, that depends on shear rate (being unity at high shear rates). Another method to describe the viscosity dependence with both shear rate and oil volume fraction has been proposed by Partal et al. [60]. This method is based on the superposition of the flow curves of emulsions with different oil volume fraction, using an empirical shift factor. The relationship between the shift factor, a(~), and the oil volume fraction is described by a Frankel-Acrivos type equation:
(14)
where 4)o is the reference oil volume fraction. The master curve shown in Figure 2 also contains the flow curves of emulsions having different emulsifier concentration. In this case the superposition was carried out by using another shift factor, which varies exponentially with the emulsifier concentration. When considering the influence of ~ on emulsion viscosity it is also necessary to know that there is a simultaneous influence of droplet size. In fact, rlrel increases as the mean droplet size decreases. Thus, for example, k in equation
95
(11) increases as droplet size decreases [63]. In addition to this, an emulsion with a broad distribution of droplet sizes will have a lower viscosity than those with a narrower distribution of particle size [29]. In this case, a model that relates the relative viscosity and ~) for emulsions containing an i-model size distribution has been proposed [46]. ~rel(~; ) = .L_Lexp i=l
ki~ i
1 -
(15)
being +i the oil volume fraction for each diameter.
100
~ 10
Carreau modd I
9
il-
l
0.1 0.01
........
' 0.1
........
~ 1
........
~
........
10
~ 100
........
~ 1000
........ 10000
a+asE+ (s-1) Figure 2. Master flow curve for emulsions containing 1-10% wt. sucrose ester and 55-85% wt. oil. Reproduced, with permission, from ref. 60. This fact has been confirmed by different authors [9, 28, 32]. Thus, Figure 3 shows that the viscosity of salad dressing emulsions containing a mixture of emulsifiers increases as mean droplet size and polydispersity decreases. This has been attributed to the fact that narrower droplet size distribution and lower mean diameter yield stronger interparticle interactions [28]. On the other hand, Partal et al. [60] have found a linear correlation between some rheological parameters (high-shear-rate-limiting viscosity and flow index) and polydispersity. However, in other cases this correlation fails, because, for instance, the emulsifier that is in
96 excess may form a gel-like continuous phase. This structuration may be favoured by increasing the droplet size, and consequently the emulsifier concentration in the continuous phase [64].
a
1 ra~
,,..a
9 ~
().1
[ ~Carreaulnodel 0.01
. . . . . . . .
I 1
0.1
] . . . . . . . .
shear
30
'
I
'
I
~ 10 rate
'
I
..........
i 100
. . . . . . . 1000
(s "l)
'
I
'
I
'
A
25 0
20
15
!
0
5
10
15 diameter
20
25
3O
(~.m)
Figure 3. Steady-state viscosity (a) and droplet size distribution (b) of salad dressing emulsions (emulsification time: r'l 3 min, O 5 min, A 10 min). Adapted, with permission, from ref. 8.
97 A large majority of food emulsions may show slip effects under steady-state shear [61, 65-68]. It has been demonstrated that multiphase systems show slip effects because of a displacement of the disperse phase away form the solid walls of the sensor system in a rheometer or the walls of pipes or tubes [69]. In the case of emulsions, this effect is even more dramatic due to the deformability of the droplets and is also enhanced when creaming process becomes important. Figure 4 shows a typical example of a flow curve showing slip phenomena. As Barnes [69] has pointed out we normally expect a single drop from a lower-shear rate Newtonian plateau through a shear thinning region towards the high-shear-rateNewtonian plateau. However, if slip occurs, an intermediate false pseudoNewtonian region is observed as Figure 4 indicates.
9
flowcurve ~ith slip effects
lip flowcuwe
"',9~ ~ log (stress)
Figure 4. Squematic flow curves obtained in a plate-plate geomet~, with smooth (slip curve) and roughened (non-slip curve) plates. As has been reported [61], the extension of slip is clearly dependent on the emulsion composition due to the existence of different interparticle interactions that influence emulsion microstructure. Thus, an emulsion containing a very soluble low-molecular-weight emulsifier shows wall slip (determined by comparison of the flow curves obtained with sensor systems having smooth and rough surfaces respectively), in contrast to an emulsion containing an only slightly soluble emulsifier which practically does not show slip effects. This difference
98
must be related to the aqueous phase behaviour. Thus, in the first case the droplets are suspended in a low viscosity continuous phase favouring their mobility, but in the other emulsion a very structured continuous phase (gel-like) takes place, apparently reducing the slip problem. However, the differences found in the steady-state flow curves carried out with different types of surfaces are much higher when the viscosity is plotted versus shear stress than those found when viscosity is plotted versus shear rate. In fact, in many cases the differences found in the viscosity-shear rate plots are not significant [61 ] and slip effects may be not taking into account if the experimental flow curves are obtained using a controlled-shear rate rheometer. This can be observed in Table I where the Carreau model parameters are compared for an emulsion measured with rough and smooth surface sensor systems respectively. Table I. Parameters of the Carreau model for a food emulsion.
TIo (Pa s)
~ (s -1)
s
P P 2 O ( - r o u g h s u f f ac-e) .......................i [ 4 i 6 "g............................1,7 ....i0 "'-'~.............................................................. 0.46
C35/4 (smooth surface) Confidence limit
3.2.
1.5 10s
0.8 10.4
0.46
+0.6 105
+1.0 10 .4
+0.02
L i n e a r viscoelasticity
The linear viscoelastic properties of food emulsions are usually confined to that range of strains below any perceivable visual movement during measurement, and though not generally applicable to large strain and stress technical applications, are nevertheless useful in assesing the microstructure and even possible the longterm stability of an emulsion [45]. The first studies on the response of concentrated emulsions in the linear region were done by creep-recovery experiments [46]. The creep behaviour of a concentrated emulsion, as a function of disperse phase fraction, often looks as despicted in Figure 5. Consequently, food emulsions behave as viscoelastic liquids, even highly concentrated emulsions such as mayonnaise [28]. This response has been modelled using different analogycal models, being the simplest combination one Maxwell element combined with one Voigt element in series, that is "one retardation process" model:
.! ( t ) -
do +
t
qo
+ J1
e(_t/~)
(16)
99
This model has been used by Kiosseoglou and Sherman [18] to describe the response of an egg-yolk film at a groundnut oil-water interface and by Matsumoto and Sherman [70] for nonionic surfactant films at an olive oil-water interface.
2.0x102
1.5x10"2
,f--, ~
l.OxlO_2
m
5.0x10-3
0.0
I
,
50
I
~
100
I
150
,
I
200
,
250
time(s) Figure 5. Creep-recovery behavior of oil-in-water emulsions stabilized with a nonionic surfactant. However most of food emulsions show a more complex rheological behaviour. Thus, Gladwell et al., [71] used a "two retardation process" Kelvin-Voigt model to fit the creep compliance response of soya oil-water emulsions emulsified with egg yolk and stabilized with xanthan gum.
J (t)= Jo + J1 e
(-t/z 1)
+ J2 e
(-t/z 2)
+ t /rio
(17)
This model has been used to describe the creep behaviour of a wide variety of emulsions [72-76]. On the contrary, oscillatory shear tests have not been extensively used for emulsion rheology [28]. As a consequence, only very little data were available in the literature until a few years ago [77], when several authors started to use this technique on different types of emulsions [37, 45, 78]. A general overview of these studies demonstrates that the frequency dependence of the storage and loss
100
moduli is dramatically dependent on the concentration of the emulsion, processing conditions and nature of the emulsifier used.
103
.......
'1
'
' ......
I
........
I
'
.......
I
.......
0121O
OO D O0 DDOODO0
102
000oO
Oo D OooO0
0
AAA &A AAA
AA
A AAAAAAAAA
A A A A A A A A9 l o l IlI I I I IO O
OO II
101
OO O O I 9 t l i i i t l I I 9 iiii
II II
00 IIII I I I I I I I I I II I I
I I I I
lO 0
10-3
........
I
10-2
,
. , ..... I
,
. , , ,,,.I
10q
10~
,
~ ......
I
10 ]
i
i
l
l
i
ii
02
~ (rad/s) Figure 6. Storage and loss moduli vs. frequency for model mayonnaise with different oil contents. I"1 75% wt., O 77.5% wt., A 80% wt., ( G ' open symbols, G " solid symbols). Adapted, with permission, from ref. 78. Commercial food emulsions, such as mayonnaise, give the kind of behaviour depicted in Figure 6, with the storage modulus higher than the loss modulus in a frequency range comprised between 10-2 and 102 rad/s, and characterized by the appearance of a minimum in G" at intermediate frequencies and a "plateau" region in G' (slope values of around 0.1). This behaviour corresponds to highly flocculated emulsions, resulting from the formation of physical entanglements among protein molecules adsorbed at the oil/water interface of the oil droplets, which leads to the formation of a structural network. On the contrary, Tadros [37] has studied the dynamic viscoelastic response of non-flocculated isoparaffinic oil-in-water emulsions in a relatively wide range of disperse phase volume fraction (~ comprised between 0.48 and 0.60), stabilized using a block copolymer (Synperonic PE). Even for the more concentrated systems, the loss modulus shows higher values than the storage modulus up to frequencies of around 10 rad/s. The terminal relaxation time increases exponentially with the disperse phase volume fraction when the latter exceeds
101
0.54. The same tendency is observed in the evolution of G" with ~. These results are explained on the basis that at ~ < 0.56, the droplet-droplet separation is probably larger than twice the adsorbed layer thickness and hence the adsorbed layers are not forced to overlap or compress. In this case, the repulsive interaction between the adsorbed layer thickness is relatively weak and the emulsion shows predominantly viscous response. However, when ~) > 0.56, the droplet-droplet separation may become smaller than twice the adsorbed layer thickness and the chains are forced to compress. This leads to strong steric repulsion and the emulsion shows predominantly elastic response. Similar behaviour has been found for vegetable oil-in-water emulsions (~) comprised between 0.7 and 0.8) stabilized by a nonionic-low molecular-weight elnu|sifier highly soluble in the aqueous phase, at very low concentration (1% wt.) [791.
lO 1
'
'
'
' ' ' " I
'
'
'
' ' ' " I
'
'
'
' ' ' " I
'
'
. . . . .
'I
'
. . . . . . .
I
'
'
'
' ' ' "
10~ ~Z
~z 10-1 GN~ (P a) 45 % 0 50 % 0 55 % 0 10-e 10-3
,
,
i
.i,,,|
.
10-2
!
I
,.,.,l
i
10q
i
i
,,,,,i
,
i
10~
i
iii.,|
i
101
,
,
3590 4345 5300
,,,,,|
,
102
,
.
iii,,
103
c0 (rad/s) Figure 7. Influence of oil concentration on the normalized viscoelasticity functions for free-starch salad dressing emulsions, l"! 45% wt., O 50% wt., A 55% wt. Reproduced, with permission, from ref. 10. However, when a high HLB sucrose ester (slightly soluble in water at low temperature) is used as the only emulsifier [80], or in addition to a macromolecular emulsifier [10], in concentrated food emulsions three characteristic regions were observed: (i) a pseudoterminal region at low
102
frequencies that shows a tendency to a crossover of both viscoelastic functions, (ii) an intermediate "plateau" region, and (iii) the beginning of the transition region at high frequencies. Figure 7 shows the evolution of the storage and loss moduli with frequency for an emulsion stabilized by a mixture of the above mentioned emulsifiers. Franco et al. [8, 10] have used an empirical model, adapted from the one given by De Rosa and Winter [81] for polymer melts, to successfully describe the three experimental regions that appear in the linear relaxation spectra of these emulsions:
-
EI/m I+)nl +
H (~)- A
for Xmin < ~ < XP
for Xp < ~ < Xmax
(18)
(19)
where ~ and ~ are the characteristic relaxation times for the onset of the plateau and pseudo-terminal regions, respectively, Xm~• and )~m~ are the reciprocal of the minimum and maximum experimental frequencies attained, m, n and c are the power-law exponents for the three different regions, and A is an empirical constant. Madiedo and Gallegos [82] have proposed a different empirical model that describes the three regions of the relaxation spectra of oil-in-water emulsions stabilized by a mixture of two sucrose esters with different HLB values: H ()~)-
a x m +13xn p
(20)
In this model, m, n and p are, respectively, the slopes of the transition, plateau and pseudo-terminal regions, ~p is the pseudo-terminal relaxation time, and the parameters a and [3 are given by the following relationships
Homm
a= (m-n)
)~o
(21)
103
1 -
n Ho
(22)
m)
Ho is the minimum value of the relaxation spectrum, which appears between the transition and the plateau regions, and ~o is the relaxation time that corresponds to this minimum. This model allows a smooth transition between the plateau and pseudo-terminal regions, in contrast to the BSW-CW equation. Studies carried out on model mayonnaise (emulsions using egg yolk as the only emulsifier) demonstrated that an increase in oil concentration (75-80% wt.) produces a wider linear viscoelasticity range and larger values of the dynamic linear viscoelasticity functions [78]. Only the transition and plateau regions of the relaxation spectra appeared. The same influence was found when the emu|sion was stabilized by a nonionic surfactant, with a high solubility in the aqueous phase, for emulsifier concentrations higher than 1% wt. [79]. In this case, a dramatic decrease in the loss tangent as oil concentration increases (60-80% wt.) was noticed in the low frequency range. This is due to the fact that a decrease in oil concentration favours the development of a pseudo-terminal region in the relaxation spectra of these emulsions, as previously mentioned. Guerrero et al. [80] have studied model emulsions stabilized by a sucrose palmitate. The tendency showed by the linear viscoelasticity functions was practically the same in the whole oil concentration range studied (60-80% wt.), although their relaxation spectra always displayed three different regions. Furthermore, the values of the loss tangent were practically independent of oil content. This fact allowed to superpose the linear viscoelasticity functions using the plateau modulus as normalization factor and a horizontal shift factor only for the less concentrated systems. The superposition obtained was good enough, although some scatter in the plateau region was shown. Similar results have been obtained by Franco et al. [10] studying salad dressing emulsions (35%-55% wt. oil) stabilized by a mixture of egg yolk and a high HLB sucrose ester. Thus, the linear viscoelasticity functions at different concentrations can be superposed using the plateau modulus as normalization factor, which increases with oil concentration. However, there is a slight, but significant increase in the slope of the plateau region as oil concentration increases, which may be related to the development of the entanglement network. On the contrary, there is no significant influence of the oil concentration on the relaxation time that defines the onset of the pseudo-terminal region of the relaxation spectra. The addition of 2% wt. starch, although favouring the emulsion stabilization at lower oil concentrations (down to 35% wt.), dampens the influence of oil content on the
104
linear viscoelasticity functions of these systems. Figure 8 shows the normalized linear viscoelasticity functions and the values of the plateau modulus as a function of oil concentration for starch-containing (2% wt.) emulsions. The BSW-CW model fails to describe the behaviour of these last emulsions. Thus, there is a sharp increase in the slope of the spectra at the highest relaxation times for the less concentrated systems (35 and 40% wt.), associated with their poorer elastic characteristics. In relation to the influence of emulsifier concentration on the linear viscoelasticity functions of concentrated food emulsions, an increase in lowmolecular-weight emulsifier concentration favours the formation of a tridimensional structural network, that is the development of the plateau region of the relaxation spectrum, yielding a significant decrease in the crossover frequency at which G ' - G". The linear viscoelasticity functions for emulsions having low emulsifier concentration cannot be superposed to the curves of the most concentrated emulsions due to the above mentioned fact, which coincides with negative values of the slope of the plateau region. These results have been found for emulsions stabilized by a nonionic surfactant as the only emulsifier [79] and for emulsions stabilized by a mixture of macromolecular and low-molecularweight emulsifiers [ 10]. It is worth pointing out that, contrary to the above-mentioned effects, Franco et al. [10] found that an increase in the macromolecular emulsifier concentration may produce a decrease in the values of the linear viscoelasticity functions. Similar results were obtained by other authors, analyzing the surface viscoelasticity of egg yolk films at the oil/water interface [18]. Although the addition of macromolecular emulsifiers has been demonstrated to favour the formation of an extensive structural network, it has been suggested that optimum viscoelasticity was achieved at monolayer saturation. At higher protein concentrations the interfacial viscoelasticity decreases because slip planes may develop at the interface [83]. The protein displacement from the interface by the low-molecular weight emulsifier may contribute to these results [84]. Although some methods have been developed to correct wall-slip on oscillatory shear measurements [85] there is no evidence of this phenomena in food emulsions, even when wall-slip caused by large velocity gradients was observed [86].
3.3. Transient flow: Non-linear viscoelasticity Different authors have studied the transient flow behaviour of concentrated food emulsions [87-90]. All of them report the appearance of a stress overshoot followed by stress decay to an equilibrium shear stress value. This behaviour has
105
been explained from two different points of view: thixotropy and non-linear viscoelasticity.
101
. . . . . . . .
n
. . . . . . . .
I
. . . . . . . .
I
. . . . . . . .
I
. . . . . . . .
t
|
i
| ||111
a
10~
o o00 t~
10_1
10-2 10-3 .
.
.
.
.
.
.
.
.
.
n
10-2
10 -1
10 0
101
10 2
10 3
c0(rad/s)
3500
-
3400 J
3300 320(I J
3100 o
Z
O
J
I
f
J
3000 I J
2900 2800 3'5
'
4'0
4'5
'
5'0
'
5'5
Oil content (% wt.) Figure 8. a) Influence of oil concentration on the normalized viscoelasticity functions for salad dressing emulsions (1"3 35% wt., O 40% wt., A 45% wt., V 50% wt., X 55% wt.). b) Evolution of the plateau modulus with oil content. Adapted, with permission, from ref. 10.
106
Figoni and Shoemaker [87] and Gallegos et al. [89] tried to describe the stress decay after the overshoot by using a kinetic model, sum of two first order kinetic functions, as follows: cy- cye - (%1 - eye1) exp (-klt) + (%2 - eye2) exp (-k2t)
(23)
The kinetic constants of this model have been related to different shear-induced processes: deflocculation and coalescence of oil droplets. However, others authors [88, 90-92] have tried to describe the above mentioned behaviour using a non-linear viscoelasticity model, the Wagner model [93], a class of strain-dependent K-BKZ-type constitutive equation [94]. Mackley et al. [95] and Madiedo et al. [96] have also used this model to predict the steady-state flow of oil-in-water food emulsions. Considering simple shear, and assuming that the memory function can be separated into time and strain-dependent components (Figure 9), the Wagner model would be reduced to the following equation: t d G ( t t' ~ - ) h (~,) y (t,t') dt' (24) -oo dt' " where G (t-t') is the linear relaxation modulus and h (y) is the damping function: cy (t) - -
h (Y) = exp ( - k 7)
(25)
being k an empirical parameter, which quantifies the level on non-linearity in the material. Nevertheless, Gallegos et al. [78, 90, 91] have also used a damping function described by the Soskey-Winter model, which fits fairly well the experimental results obtained with different types of emulsions: 1 h ( 7 ) - l + a 7b
(26)
In the previous integral equation, the relaxation modulus may be described in terms of a discrete set of Maxwell elements [91, 95]. In this case, the Wagner model results: t ~gi -(t-t')/~. cy (t) - - "f =~1~ e 1 h (Y) Y(t,t') dt' -ooi l
(27)
107
Introducing equation (25), the apparent viscosity for a shear rate, +, can be calculated as
n =
gi Xi
(28)
z
i= 1 (l+kXi~;) 2
lO 4
........
i
........
i
........
I
........
:
i 102
........
i 103
10 3
10 2
~"
101
El i D~
[] nclc~tan_,.,_~ o
~ lOo 10 -]
10 ~
........ 10 -]
i 100
........
i 101
........
time(s) Figure 9. Linear and non-linear relaxation moduli for a salad dressing emulsion, ( i G(t) from H(X), X G(t) from the Ninomiya-Ferry approximation). In other cases [78, 90] a power-law equation was used, yielding the following integral equation: cy(t) -
I m C ( t - t') - m - 1 h (y) ~,(t,t') dt' --o0
(29)
Figure 10 shows the experimental and calculated values of R(t) (ratio between the shear stress after instantaneous imposition of shear rate, ~(t), and the steadystate stress, r~(oo)) for a transient flow test carried out at 0.6 s-~ As can be observed, the fit of the model is acceptable at low shear rates. Higher shear rates produce increasing differences between experimental and calculated values.
108
Other authors [96] have used a continuous relaxation spectrum, calculated from the experimental values of the dynamic moduli, using the Tikhonov regularization method [97, 98], to predict the steady-state flow behaviour of different emulsions. In this case the Wagner model results t oo e_(t_t,)/Z cr (t) - - ~ ~ H (X) h (~,) qt(t,t') dlnkdt'
(30)
--o0 --o0
and the steady-state viscosity may be calculated from the following integral equation: 11 ( f ) -
oo X H(:X) [ d lnX -oo (1 + kXf)2
(31)
2~F ~ = 0 . 6 s -1 1.5
1.O~-
III~ X~
- Xll X--M - - X T X - - J - - X
0.5
0.0
,
0
I
25
~
I
50
~
I
75
~
I
100
~
I
125
~
I
150
time (s) Figure 10. Relationship between transient and steady-state stress values as a function of shearing time for a salad dressing emulsion, (11 experimental values, X Wagner model prediction). In addition to the rheological tests, optical observations of the sheared materials were also reported using a purpose built optical shearing cell. Wall-slip, micro-
109
domain movement, chaining and changes in droplet size distribution, were all observed under different shear conditions and depending on the nature of the emulsifier used. The authors conclude [96] that the Wagner model gives acceptable results, in a relatively wide shear rate range for vegetable protein-stabilized emulsions. The overestimation of the apparent viscosity at low shear rates is thought to be due to sample slipping between the rheometer plates. On the contrary, the model largely fails for a whole egg-stabilized commercial mayonnaise. This difference is explained on the basis that the shear rate is inducing significant changes in the droplet size distribution of mayonnaise throughout practically the entire rheological range studied. On the contrary, the vegetable protein-stabilized emulsion only shows a significant change in droplet size distribution at very high shear rates. The authors remark this fact may indicate that the applicability of the Wagner model is closely related to shear-induced structural changes in emulsion microstructure. 4. INFLUENCE RHEOLOGY
OF
PROCESSING
ON
FOOD
EMULSION
Emulsification is a complex unit operation in which many variables influence the processing and the final rheological characteristics of the product. The manufacture of emulsions is an energy-intensive and highly dynamic process, which usually requires the application of mechanical energy. The two critical steps are the consecutive disruption of droplets and their coalescence, both of which are favoured by an intense agitation. Consequently, the improvement of the emulsification process requires the measurement of the droplet size of the dispersed phase and its polydispersity, as well as knowledge of its rheological properties [28, 32]. The emulsification process may be greatly affected by the viscous and viscoelastic properties of the continuous phase at which the disperse phase is added. Thus, Gallegos et al. [64] have studied the influences that temperature, agitation speed and emulsification time cause on the droplet size distribution and viscoelasticity of vegetable oil-in-water emulsions stabilized by a well-known polyoxyethylene nonionic surfactant, which produces a low-viscosity micellar continuous phase, and manufactured using an anchor impeller. Figure 11 shows the droplet size distribution (DSD) curves as a function of the agitation speed of the anchor impeller (N). A well-pronounced maximum at droplet diameters larger than 5 ~tm and a very smooth maximum at lower sizes are noticed, although the curves move to lower sizes as N increases. Only one
110
maximum was found when the emulsion was prepared with a rotor-stator turbine (4000 rpm). These DSD curves determine the linear viscoelastic response of the emulsions. As can be observed in Figure 12, the systems prepared with the anchor impeller show higher values of the loss modulus at low frequencies, althought G' and G" curves crossover at a frequency O3c.Above this characteristic frequency, a plateau region in G' develops as agitation speed increases. Similar behaviour was also found by Tadros [37] for weakly flocculated emulsions. On the contrary, the emulsion prepared with the rotor-stator turbine only shows a plateau region, similar behaviour to that found for highly concentrated o/w emulsions stabilized by a macromolecular emulsifier [78]. This behaviour corresponds to an emulsion having much smaller sizes and narrower droplet distribution than those prepared with the anchor impeller. An increase in temperature favours the coalescence process, leading to larger droplet sizes and broader distributions. In the same way, the plateau region in G' tends to vanish and the crossover frequency increases. On the contrary, an increase in the emulsification time enhances the development of the plateau region.
l0
O.
~
~
Diameter (~m)
Figure 11. Droplet size distributions for emulsions containing NPE-PEG-10, prepared with different process conditions, (m 1oo rpm, 9 150 rpm, A 200 rpm, V 250 rpm, I"! 300 rpm, • 4000rpm). Reproduced, with permission, from ref. 64.
111
Different results are obtained if a sucrose ester nonionic surfactant, which forms a gel-like structure in the continuous phase for a wide range of concentrations and temperatures [98], is used as emulsifier. An increase in agitation speed or emulsification time also produces a decrease in droplet size and polydispersity [64]. However, an increase in the agitation speed produces a decrease in the values of the dynamic viscoelasticity functions, because the gellike structure tends to vanish and the continuous phase of the emulsion becomes less viscoelastic. In fact, the viscoelastic properties of these emulsions depend on the balance between the formation of a larger interfacial surface and the breakdown of the gel-like structure of the continuous phase during processing. 1000
100
IIIIQIll ~ o ~ O O Q o ~ ~ ~~ m o ~oooooo~ oQ
10
u
oo
oo
Oo
0
Oo
O~
o
[]
O DB []
9
9
9
mm m
~
........
o.o '
........
o11
........
........
........
oo'
......
Figure ]2. Evolution of the storage and loss modu]i with frequency, for an emulsion containing NPE-PEG-] 0 as emulsifier, prepared with different process conditions, (I-I ]00 rpm, O150 rpm, V 250 rpm, X 300 rpm, + 4000 rpm). Reproduced, with permission, from ref. 64.
In relation to the manufacture of protein-stabilized emulsions, protein denaturation by heat tends to improve emulsifying and foaming capacity by enhancing macromolecular flexibility and surface hydrophobicity [5]. A large number of researchers have studied the emulsifying capacity of whey proteins by heating the solution prior to the addition of the oil phase [84, 99-105]. This thennal denaturation favours emulsion stability and improves the rheological
112
properties, both effects being related to the formation of a gel structure. Similar results have been obtained by Gallegos et al. [64], using vegetable protein as emulsifier.
6000
,
,
,
,
,
,
,
,
,
,
,
5000
,
9
o oo
4000
9
/
o'
20OO
O go 9
[] [] O O 0 ooO
[]
[]
~" ~ 2 ~
v ~.
n []
0 0 0
0
o
0
OOO
O0
40
0 O0
20
O0 0
o
O00
0_ - - ~ B ~
0
[]
DO
[] 9
O 0
9
9 9
",p O O0
9
9
9
%
9 9
9
00000
~ O9
uNNN 9
0
100
80
m u m m n
9
l
9
o~
1000
,
9 %
,,I
3000
,
@@@ @@@@
@@@
@
.A@ .
40
? * *, *
60
I
80
,
I
100
,
I
120
,
I
140
time (min) Figure 13. Effect of the lecithin/protein molar ratio (R) on developing storage modulus (1 Hz) of whey protein emulsions. Lecithin/protein molar ratio (R): II R=I6, 9 R=6, I"1 R=4, O R=2, 9 R=0. Dashed line shows temperature programme. Adapted, with permission, from ref. 102. Dickinson [106, 107] has also studied emulsion gelation by covalent crosslinking of proteins using an enzyme. The resulting values of the storage modulus are compared to those obtained by heat treatment of the emulsion. Furthermore, data for some protein gels without emulsion droplets were also obtained. It is concluded that, whereas the enzyme treatment produces stronger protein gels than the heat treatment, the opposite is the case for emulsion gels. This is attributed to topological constraints imposed by the permanent nature of the covalent crosslinks in the enzyme-set system which restricts further reinforcement of the network structure [106]. However, the G' values for enzyme-set emulsions gels are slightly less frequency-dependent than those of equivalent heat-set emulsion gels. This is explained by the fact that enzyme-set system theology is more
113
similar to that of a classical polymer gel with "chemical" bonds, whilst the heatset system rheology is typical of a "physical" gel with breakable bonds.
10
3
'
,
,
"
.....
I
. . . . . . . .
~
. . . . . . . .
I
,
,
,- .....
I
. . . . .
A
r
102
%
model
BSW-CW
i01
,
. . . . .
10-2
10 a
,,l
,
,
......
1 0 -1
.
.
.
.
.
.
"1
{
,
,
,
,,,,,l
i0 ~
.
.
.
.
.
' " I
. . . . . . . .
,,,{
101
'
' '
'
' ' " 1
10 3
10 2
'
'
'
. . . .
'i
'
'
' '
' ' " -
b lO 3
i::1 rn
t~
io ~
:2 101 BSW-CW
10 0 0 "z 1
. . . . . . . .
,
10 -1
model
. . . . . . . .
I
. . . . . . . .
10 ~
,
101
. . . . . . . .
,
10 z
. . . . . . . .
10 s
)~ (s) Figure 14. Influence of the processing variables on the relaxation time spectra of salad dressing emulsions, a) Influence of emulsification temperature: 1"I 20~ 9 without thermal control, A 50~ b) Influence of emulsification time and agitation speed: [21 8000rpm-5min, O8000rpm-3min, A 5000rpm-5min, V 5000rpm-3min. Reproduced, with permission, from ref. 8.
114
The same author has also investigated the effect of low-molecular weight emulsifiers, added after emulsification but prior to thermal processing, on the linear viscoelasticity functions of protein-stabilized emulsions [84, 108, 109]. The results indicate that the rheology of the emulsion gel, produced by in situ heat processing, is significantly influenced by the surfactant used. Thus, Figure 13 shows the evolution of the storage modulus during the thermal processing, as a function of temperature, for different lecithin/protein molar ration, R. These results are consistent with lecithin-protein complexation at the oil-water interface and in the bulk aqueous phase. Much more complicated behaviour was found when a polyoxyethylene sorbitan monolaurate was used. In this case, the values of the storage modulus present a maximum for R = 1, which is related to a partial displacement of protein from the oil-water interface by the water-soluble surfactant [110]. The influence of processing on emulsions stabilized by a mixture of macromolecular and low-molecular-weight emulsifiers has been also studied by Franco et al. [8]. Figure 14 shows the linear relaxation spectra of emulsions, prepared in a pilot-plant colloidal mill or with a rotor-stator turbine (lab-scale), as a function of rotational speed, residence time and temperature of emulsification. As can be observed the slope of the plateau region increases with the processing variables, because of the development of an entanglement network. This is related to a decrease in mean droplet size and polydispersity of the emulsion, yielding stronger inter-droplet interactions [28]. 5.
CONCLUDING REMARKS
The rheology of food emulsions is mainly dependent on the strenght of interdroplet interactions. Dilute emulsions (i.e. milk) have a low-viscosity Newtonian behaviour. On the contrary, concentrated food emulsions show gel-like rheological characteristics. This behaviour can be attained by increasing the disperse phase volmne fraction, or by different flocculation mechanisms (heat denaturation of proteins, covalent cross-linking of proteins, etc.). However, many other structural parameters also influence the rheological (viscous and viscoelastic) response of emulsions (i.e. droplet size, polydispersity, droplet deformability, etc.). In qualitative terms, the influence of all of them on the rheology of emulsions is well understood, as well as their dependence on the emulsion processing. However, because of the lack of good model systems, i.e. with different degrees of polydispersity, the quantitative understanding of the relationship among stability/microstructure/rheology/processmg of food emulsions represents a very considerable challenge for the future.
115 REFERENCES
1. W. Clayton (ed.), Theory of emulsions and emulsification, Churchill, London, 1923. 2. P. Sherman, (ed.) Emulsion Science, Academic Press, London 1968. 3. P. Becher (ed.), Encyclopedia of emulsion technology, II. Marcel Dekker, New York 1985. 4. D.G. Dalgleish, In Emulsions and emulsion stability, SjOblom, J. (ed.), Marcel Dekker, New York 1996, 287. 5. E. Dickinson, G. Stainsby, Food Technol., 41 (1987), 75. 6. E. Dickinson, An Introduction to Food Colloids. Oxford University Press. Oxford 1992. 7. J.M. Franco, A. Guerrero, C. Gallegos, Grasas y Aceites, 46 (1995), 108. 8. J.M. Franco, A. Guerrero, C. Gallegos, Rheol. Acta, 34 (1995), 513. 9. J.M. Franco, M. Berjano, A. Guerrero, J. Mufioz, C. Gallegos, Food Hydrocoll., 9 (1995), 111. 10. J.M. Franco, M. Berjano, C. Gallegos, J. Agric. Food Chem., 45 (1997), 713. 11. D.C. Clark, P.J. Wilde, D.R. Wilson, R.C. Wunsteck, Food Hydrocoll., 6 (1992), 173. 12. E. Dickinson, In Interactions of Surfactants with Polymers and Proteins, Goddard, E.D., Ananthapadmanabhan, K.P., (eds.) CRC Press, Boca Raton 1993, 295. 13. S. Matsumoto, In Macro and microemulsions: theory and applications, Shah, D.O. (ed.) ASC Symposium Series, Washington 1985, 272. ! 4. S. Matsumoto, J Texture Stud., 17 (1986), 141. 15. B. De Cindio, D. Cacace, Int. J. Food Sci. Technol., 30 (1995), 505. 16. E.E. Uzgiris, H.P.M. Fromageot, Biopolym., 15 (1976), 257. 17. E. Dickinson, E.W. Robson, G. Stainsby, J. Chem. Soc. Faraday Trans., 79 (1983), 2939. 18. V.D. Kiosseoglou, P. Sherman, Colloid Polym. Sci., 26 (1983), 520. 19. E. Dickinson, S.E. Rolfe, D.G. Dalgleish, Food Hydrocoll., 2 (1988), 397. 20. D.G. Dalgleish, Colloids Surf., 46 (1990), 141. 21. E. Dickinson, J.A. Hunt, D.G. Dalgleish, Food Hydrocoll., 5 (1991), 403. 22. A. Kondo, K. Higashitani, J. Colloid Interface Sci., 150 (1992), 344. 23. J.L. Klemaszewski, K.P. Das, J.E. Kinsella, J. Food Sci., 57 (1992), 366. 24. B.V. Derjagum, L. Landau, Acta Physiochem., 14 (1941), 633. 25. E.J.W.Verwey, J.T.G. Overbeek, Theory of the Stability of Liophobic Colloids, Elsevier, Amsterdam 1948.
116 26. W. Heller, T.L. Pugh, J. Chem. Phys., 22 (1954), 1778. 27. D.H. Melik, H.S. Fogler, In Encyclopedia of Emulsion Technology, III, Becher, P. (ed.) Marcel Dekker, New York 1988, 3. 28. R.R. Rahalkar, In Viscoelastic Properties of Food, Rao, M.A. and Steffe J.F. (eds.), Elsevier, London 1992, 317. 29. P. Sherman, J. Pharm. Pharmacol., 16 (1964), 1. 30 E. Dickinson, Colloids Surf., 42 (1989), 191. 31 P. Sherman, J. Colloid Interface Sci., 24 (1967), 67. 32 A.R. Carrillo, J.L. Kokini, J. Food Sci., 53 (1988), 1352. 33 M. van den Yempel, J. Colloid Sci., 13 (1958), 125. 34 E. Vernon-Carter, P. Sherman, J. Dispersion Sci. Technol., 2 (1981), 399. 35 J.A. Hunt, D.G. Dalgleish, J. Agric. Food Chem., 42 (1994), 2131. 36 R. Pal., E. Rhodes, J. Rheol., 33 (1989), 1021. 37. Th.F. Tadros, In First World Congress on Emulsion, vol. 4, Paris 1993,237. 38 E. Dickinson, In Gums and Stabilisers for the Food Industry, vol. 4, Wedlock, D.J., Williams, P.A. (eds.). I R Press, Oxford 1998, 244. 39. Y. Cao, E. Dickinson, D.J. Wedlock, D.J., Food Hydrocoll., 5 (1991), 443. 40. Dickinson, In Progress and Trends in Rheology, IV, Gallegos, C. (ed.). Steinkopff, Darmstadt 1994, 227. 41. A. Parker, P.A. Gunning, K. Ng, M. Robins, Food Hydrocoll., 9 (1995), 333. 42. R.L. Scott, J. Chem. Phys., 17 (1949), 279. 43. R. Pal, Int. J. Multiphase Flow, 15 (1989), 1011. 44. R. Pal, AIChE J., 39 (1993), 1754. 45. H.A. Barnes, Colloids Surf. A, 91 (1994), 89. 46. P. Sherman, in Encyclopedia of Emulsion Technology, Becher, P. (ed.), Marcel Dekker, New York 1983, 405. 47. A. Einstein, Investigations on the Theory of the Brownian Movement, Dover, New York 1906. 48. G.I. Taylor, Proc. Royal Soc., A138 (1932), 41. 49. G.K. Batcherlor, J. Fluid Mech., 83 (1977), 97. 50. M. Mooney, J. Colloid Sci., 6 (1951), 162. 51 H. Eilers, Kolloid-Z, 97 (1941), 313. 52 H. Eilers, Kolloid-Z, 102 (1943), 154. 53 R. Roscoe, J. Appl. Phys., 3 (1952), 267. 54 S.H. Maron, P.E. Pierce, J. Colloid Sci., 11 (1956), 80. 55 I.M. Krieger, T.J. Dougherty, Trans. Soc. Rheol., 3 (1959), 137. 56 N.A. Frankel, A. Acrivos, Chem. Eng. Sci., 22 (1967), 847. 57. J.S. Chong, E.B. Christiansen, A.D. Baer, J. Appl. Polym. Sci., 15 (1971),
117
2007. 58. R. Pal, J. Rheol., 36 (1992), 1245. 59.Y. Otsubo, R.K. Prud'homme, Rheol. Acta, 33 (1994), 29. 60. P. Partal, A. Guerrero, M. Berjano, J. Mufioz, C. Gallegos, J. Texture Stud., 25 (1994), 331. 61. J.M. Franco, C. Gallegos, H.A. Barnes, J. Food Eng., submitted for publication. 62. R. Pal, E. Rhodes, J. Colloid Interface Sci., 107 (1985), 301. 63. F.L. Saunders, J. Colloid Sci., 16 (1961), 13. 64. C. Gallegos, M.C. Sfinchez, A. Guerrero, J.M. Franco, In Rheology and Fluid Mechanics of Nonlinear Materials, Siginer, D.A. and Advani, S.G. (eds.), ASME, New York 1996, 177. 65 A. Yoshimura, R.K. Prud'homme, J. Rheol., 32 (1988), 53. 66 J. Carnali, H.A. Barnes, J. Rheol., 34 (1990), 841. 67 R. Pal, Chem. Eng. Commun., 98 (1990), 211. 68 R. Pal, Chem. Eng. Sci., 52 (1997), 1177. 69 H.A. Barnes, J. Non-Newtonian Fluid. Mech., 56 (1995), 221. 70 S. Matsumoto, P. Sherman, J. Texture Stud., 12 (1981), 243. 71 N. Gladwell, R.R. Rahalkar, P. Richmond, J. Food Sci., 50 (1985), 1477. 72 H.J. Rivas, P. Sherman, J. Texture Stud., 14 (1983), 251. 73 H.J. Rivas, P. Sherman, J. Texture Stud., 14 (1983), 267. 74. V.D. Kiosseoglou, P. Sherman, J. Texture Stud., 14 (1983), 397. 75 N. Gladwell, R.R. Rahalkar, P. Richmond, Rheol. Acta, 25 (1986), 55. 76 J.M. Madiedo, J. Mufioz, M. Berjano, C. Gallegos, In Progress and Trends in Rheology IV, Gallegos C. (ed.) Steinkopff, Darmstadt 1994, 281. 77. T. van Vliet, J. Lyklema, M. van den Tempel, J. Colloid Interface Sci., 65 (1978), 505. 78. C. Gallegos, M. Berjano, L. Choplin, J. Rheol. 36 (1992), 465. 79. M.C. Sfinchez, M. Berjano, C. Gallegos, Afinidad, submitted for publication. 80. A. Guerrero, P. Partal, M. Berjano, C. Gallegos, Prog. Colloid Polym. Sci., 100 (1996), 246. 81 M.E. de Rosa, H.H. Winter, Rheol. Acta, 33 (1994), 220. 82 J.M. Madiedo, C. Gallegos, Applied Rheol. 7 (1997), 161. 83 D.E. Graham, M.C. Phillips, J. Colloid Interface Sci., 76 (1980), 240. 84 E. Dickinson, S.T. Hong, J. Agric. Food Chem., 43 (1995), 2560. 85 A. Yoshimura, R.K. Prud'homme, J. Rheol., 32 (1988), 575. 86 L. Ma, G.V. Barbosa-Cfinovas, J. Food Eng., 25 (1995) 397. 87 P.I. Figoni, C.F. Shoemaker, J. Texture Stud., 14 (1983), 431. 88 O.H. Campanella, M. Peleg, J. Rheol., 31 (1987), 439.
118
89. C. Gallegos, M. Berjano, F.P. Garcia, J. Mufioz, V. Flores, Grasas Aceites, 39 (1988), 254. 90. C. Gallegos, M. Berjano, A. Guerrero, J. Mufioz, V. Flores, J. Texture Stud., 23 (1992), 153. 91. C. Gallegos, J.M. Franco, Les Cahiers de Rheologie, 14 (1995), 107. 92. P. Partal, Ph.D. Thesis, University of Seville, Seville, 1995. 93. M.H. Wagner, Rheol. Acta, 15 (1976), 136. 94. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston 1988. 95. M.R. Mackley, R.T.J. Marshall, J.B.A.F. Smeulders, F.D. Zhao, Chem. Eng. Sci., 49 (1994), 2251. 96. J.M. Madiedo, C. Bower, M.R. Mackley, C. Gallegos, Grasas Aceites, submitted for publication. 97. J.M. Madiedo, Ph. D. Thesis, University of Seville, Seville, 1996. 98. J.M. Madiedo, J. Mufioz, C. Gallegos, In Rheology and Fluid Mechanics of Nonlinear Materials, Siginer, D.A. and Advani, S.G. (eds.), ASME, New York 1996, 151. 99. N.K.D. Kella, J.E. Kinsella, Biochem. J., 255 (1988), 113. 100 J.E. Kinsella, D.M. Whitehead, Adv. Food Nutr. Res., 33 (1989), 343. 101 R. Jost, F. Dannenberg, J. Rosset, Food Microstructure, 8 (1989), 23. 102. Y.L. Xiong, J.M. Aguilere, J.E. Kinsella, J. Food Sci., 56 (1991), 920. 103 R.A. Yost, J.E. Kinsella, J. Food Sci., 57 (1992), 892. 104. R.A. Yost, J.E. Kinsella, J. Food Sci., 58 (1993), 158. 105. D.J. McClements, F.J. Monahan, J.E. Kinsella, J. Texture Stud., 24 (1993), 411. 106. E. Dickinson, Y. Yamamoto, J. Agric. Food Chem., 44 (1966), 1371. 107. E. Dickinson, In Proceedings of 1st Intern. Symposium on Food Rheology and Structure, Zurich 1997, 50. 108. E. Dickinson, Y. Yamamoto, Food Hydrocoll., 10 (1996), 301. 109. E. Dickinson, Y. Yamamoto, J. Food Sci., 61 (1996), 811. 110. J. Chen, E. Dickinson, J. Sci. Food Agric., 62 (1993), 283.
119
WORMLIKE MICELLAR SURFACTANT SOLUTIONS: RHEOLOGICAL AND FLUID MECHANICAL ODDITIES R. Steger a a n d P.O. B r u n n b
~Rheotest Medingen GmbH, ROdertalstrafle 1, D-01458 Medingen b. Dresden, Germany bUniversitgit Erlangen-Niirnberg, Lehrstuhlfiir Strfmungsmechanik, Cauerstr. 4, D-91058 Erlangen, Germany
1. INTRODUCTION In aqueous surfactant solutions the surfactant molecules form aggregates (termed micelles) at some critical concentration cmc (= critical micelle concentration). Its exact value depends on the size of the hydrophobic part of the molecule. The hydrophilic group plays also a role, in particular whether it is ionic or nonionic. At concentrations above cmc surfactants are present in the form of micelles and monomers. Micelles are fragile dynamic objects which are constantly formed and destroyed by the addition and loss of monomers. This permanem exchange of material is in real thermodynamic equilibrium. Usually the miceUes formed above cmc are of globular shape and remain so up to rather high surfactant concentrations. Yet, some systems exist for which, even at high dilution, a second critical concentration, termed ct. (transition concentration), ct > cmc, exists at which rod- (or worm-) like micelles are being formed. This usually happens when additives like oppositly charged surfactants, organic counterions or uncharged components like aromatic hydrocarbons are added [1,2]. Thus, above c, the rod- (or worm-) like micelles are thermodynamically more stable than globular micelles. The dynamic nature of their creation and breakup implies that micellar solutions cannot be regarded as suspensions of well-def'med particles with a given size. Some of the kinetics, which govern these processes, is well understood (e.g.[3]). As soon as rod like micelles are present in the solution, viscoelastic behavior is encountered in viscometric measurements [4,5]. In addition, the solutions then
120
show the ability of drag reduction in turbulent pipe flow [2,6,7] as well as enhanced resistance in porous medium flow [8,9]. The system studied in this paper is the surfactant N-cetyl-N,N,Ntrimethylammoniumbromide (C~6TMA-Br) with sodiumsalicylate (Na-Sal) as counterion (equimolar solution). Most of the results reported will refer to a concentration of 1000 weight-ppm (___-2.3 mmol), a concentration well above ct (at least in the temperature range tested by us, 25~ _< T _<50~ [ 10].
2. RHEOLOGY
2.1. Experimental The viscometric results reported have all been obtained by the commercial Couette viscometer Haake CV 100 - RV20. It is of the CR type (constant rate), the outer cylinder being the rotating one. Details about the measuring systems employed (dimensions) have been given before [ 10]. 10 -~ I000 p p m
o 9
4 h 20 rain
T=35"(3
0
4 min
DA45
x
1 min
Surfactant
EiO000D~% o
~
i
1 0 -2 -,==4
0 r o
o
0
-,,...4
-3
10
XXXXX .~
.
v
.
.
. .v'
I ~
."
1
Shear
.
r
9 9 9
'~l
=
'
"
9
9 ~
10
Rate
x
i .v
9
l
10 z
in
9
i~
9 in|
lb a
1/s
Figure 1. Influence of the sweep time (varying between 1 minute up to 4 hours) on the apparent viscosity as a function of apparent shear rate (R / R~ = 1.078) o
121
Using the software provided by the manufacturer a sweep test reveals time dependent behavior up to a sweep time of four hours (Sweep times longer than this do not change anything). Starting essentially from the base line of the solvent viscosity, 11~, the apparent shear viscosity 11 shows sudden shear thickening at a critical apparent shear rate ~, ~, from 11 ~ 11, up to some value rl p. The apparent viscosity will then stay essentially at that plateau value 1"1p up to an apparent shear rate, termed ~,,. Increasing the shear rate fin-ther leads to shear thinning behavior (see figure 1). On a double-logarithmic plot the slope for ~ > ~,, is minus one, which in terms of shear stress x means that there exists a critical shear stress ~ ~ ( = 11p ~,~), at which rl ,,jumps" from its plateau value 1"1p to some lower value (see figure 2).
10-1 . . . . . . . . .
. .
o
Surfactant 1000 ppm T=35 C DA45
13.,
r x
4h 2 0 min 4 min 1 min
},
~ r-.,I
1 0 "2
r
r,/3
0 0 o
~
>.
0
-
~ xXXXX
IO-
- ....
I 0-3
.
.
...-.,,=
10 2
Shear
,
x
r
...... . , , . , . ,
X X ....... , - , - . . . . . ,
10-I
Stress
T in
Figure 2. The results of figure 1 as a function of shear stress
'w-
r
w: i : w T C
I
lO
I
Pa
122
It is of interest to note that while the quantities ~ , rl p and ~ are time dependent (as far as the sweep time is concerned) ~ r is not. Thus, the rheological characteristics of the solution seem to be i) time dependend behavior (which has been shown to be rheopectic [10], ii) a sudden increase of rl at some (time dependend) apparent critical shear rate ~ and iii) a sudden decrease of 11at some ( time independent ) shear stress ~ . In order to avoid a sweep time dependence all viscometric results to follow have been obtained with a sweep time of more than four hours. For completeness one should note that the sudden increase in flat some apparent critical shear rate ~ is found for many surfactant solutions in the concentration range where rod-like micelles dominate [4,5]. For that reason it is not astonishing to see that it has received considerable attention in the literature. The picture that at "~ a new phase is suddenly ,,created", the so called s_hearinduced-s_tate (SIS) is intriguing since it allows one to consider the q , ~
Surfactant 1000 ppm T=35 C
co
t3-,
a DB45 * ME4-5 A ME46
10-1 o000~
9r-',,I
t~
rl o
10
o o~o
-2 "
O
~
A
Q
9 r-,,,ll
& & A A A A
A
D 0 o
0 r r~
10
-3
kAk - dk'
**~0~
I~II
AAA A A
9r,-,I
b,. 10
-4 11 - -
It
"l-- 9 : I
I-i
n1011 ! ....
Shear
Iv '" Iv-~-I --IV -II ~11 i ~
2
- -W~ '' W-- 9 ~
W W W31~01
Rate ~ in I/s
Figure 3. Influence of the relative gap width ~con the apparent flow curve A: K=1.037, *" K=1.078; ~ K=1.176
123
jump on the basis of a phase transition (like a sol-gel phase transition). Readers interested in this subject are referred to the excellent papers of H. Hoffmann and, repectively, H. Rehage [5,11,12]. The viscometric results of our solution show a dependence of 1"1upon the"type and the size of the measuring system used. For example, by using three systems of different relative size (indicated by the fact that K:, the ratio of the outer to inner cylinder, 1<-R o /R] > 1, differs) we find that )~ increases while "iqp simultaneously decreases with decreasing 1< (figure 3). If for 1<~ 1 there is any jump (rl p > 1"1s) at all then this jump is bound to be rather small. Interesting is another, and totally unexpected, fact namely that there is also a dependence upon the actual gap width. Figure 4 shows that three systems with the same K - 1.078 furnish (rather unsystematic) results. Two Mooney Evert systems (termed ME) imply an increase of 11with increasing gap width, the double gap device (DA) contradicting that somewhat, since the gap widths (1.375mm and 1.63 ram) are between the others. We conclude from these results that the apparent flow curves depend upon the details of the measuring system used. 10-1
_
o * e
Surfactant 1000 oppm T=35
ooo
.
12., 17~
DA45 ME45 ME30
oooooo O
%'*******~o ~ Oo 0 ~ ~ oo o
10 -2-
B I,,,,-f
0
0 B
~r,,,I
~
O to r~
10
10
-3
-40-1 1
......
'i'
' '
Shear Figure 4.
'
' ' ~' 10'"
. . . . .
R a t e ~ in
"i~)
2
'
' "'
'"10
3
1//s
Influence of actual gap width AR- R - R, on the apparent flow curve o. AR=l.085 mm; , AR=l.63 mm; o- AR=1.375 and 1.63 mm (K = 1.078 in all three cases)
124
2.2. Theoretical considerations
In a concentric cylinder viscometer (Couette type) the outer cylinder (radius Ro) rotates with angular velocity f2 relative to the inner one, (radius R~, height h). The torque M, needed to hold the inner cylinder at rest, is measured. If viscometric flow prevails the shear surfaces are concentric cylinders. Thus, the shear rate ) will be a function of r, r being the distance from the axis of rotation, do (1) dr
Here o = o(r) is the angular velocity of a cylindrical surface of radius r. A torque balance reveals that the shear stress x varies like r-2, so that
with "r., t h e shear stress at the inner cylinder. Expressed in terms of measurable quantities it is given by M (3) x ~ = 27r.hRiZ
For the measuring systems employed by us the shear stress thus varies across the gap width by 7.5, 16.2 and by 38.3 % (~: =1.037, 1.078 and 1.176). Assuming the fluid to adhere at a solid surface the other measurable quantity, namely f~, is given by Ro
do
1
Rj
dr
2
(4)
"ci Ti/
r" 2
T,
It is important to note that this implies f~ - f~('t,, "t o ) = f~('r,,, K 2 ) . Principally it is possible to invert this relation (and thus to get the shear viscosity rl ) if experiments are performed with different systems (different K's) using the same shear stress at the outer cylinder, i.e. fixed x o = "r., / K : . For in that case we get from equation (4) the shear rate at the inner cylinder, ~, Of~ (5) c31ogK "tO
125
Thus, in a plot of f2 versus log~: the slope will be ~,. It is clear that such a procedure will be extremely time consuming (to obtain from experimental data the slope rather accurately will require many measurements) and we know of no attempt that this procedure has ever been tried. An alternate way is to employ a mean value theorem to equation (4) with the result 2K2 K2
~
(6)
xi
-1
where u is the appropriate mean shear stress, x o _
(7)
f2
with 9
C
_
-1
(8)
_
4n~:2hR 2
a purely geometrical quantity. Thus, measurements of M and
f2
suffice
to determine
rlcorrectly.
Unfortunately ~(or ~ ) is not known and it is necessary to a priori choose a particular value for it. Commercial instnmaents, and also the one used by us, use for the shear rate the expression [13] 2~:2 (9) Ira
K 2 -- 1
which by equation (6) corresponds to the choice u = x,. While correct for a Newtonian fluid as well as in the limit of infinitely small gap width (~: ~ 1 ) it will for f'mite gap widths in general be incorrect. Not the fluid-specific shear viscosity rl but an apparent viscosity rio, which depends on relative gap width as well, is the consequence,
126
M f2
(10)
All experimental results reported thus far are apparent ones (since all results reported in the last chapter are apparent ones we have for simplicity omitted the subscript a ~ w h i c h emphasizes this fact ~ on q and ~. When we now compare the theoretical results with experimental ones the latter results will get this subscript) so that the observed dependence upon K (see figure 3) was to be expected. Since these results point towards decrease in the height of the jump for decreasing K (see figure 3) and since ~t and ~ coincide in the limit K ~ 1, the question arises whether the jump seen in rio (see figure 3) is indeed consistent with a jump in rl at some critical rate ~,~. Such a jump of q implies a jump of rio at a critical apparent shear rate ~ a~, which by equation (4) is given by K 2 InK2 r
-
-
~gC
(11)
1( 2 --1
Since ~cShOuld be a fluid specific critical shear rate, ~ , according to this relation, should increase with ~c. Experimentally, the opposite is found (figure 3). The rheological relations presented do not admit a jump of rl~ at some critical shear stress x . This, however, is observed (see figure 2). The way out of this dilemma is to abandon (at least) one of the assumptions on which the formulas rest, which are i) the flow is visco-metric (with concentric cylinders as shearing surfaces) and ii) there is no slip. If the flow is not viscometric then none of our theoretical relations are valid. On the other hand if the flow is viscometric but slip occurs then equation (4) has to be modified,
-- ~'~]f + ( ~'] s i + ~'~'2s o )
(12)
with f~e given by equation (4) and f2~(f~,o) the angular slip velocity at the inner (outer) cylinder. The relations 4-10 would involve the rheologically relevant
127
angular velocity f21 -f2~(x,,K) rather than the angular velocity f2of the outer cylinder. To obtain it, the slip contribution has to be known. Making the usual assumption that the angular slip velocity can be written as shear stress dependent slip velocity u divided by the radius of the corresponding cylinder, i.e.
,[
f2,~ + f~,o = ~
,
2]
u,(x~) +-u,(x~~: / K )
(13)
shows that slip effects become more and more important the smaller system (small R~) is. This was to be expected and the experimental results presented in figure 4 for the two Mooney -Evart systems (ME) are in accord with this feature. That the results for the double gap device (DA) do not follow this vend could possibly be of different origin. Labeling for the DA system the radii of the cylinders (in increasing order)/~ to R,, we have (with ~c=/~ //~ = R~ / R3) ~"~-- ~"~fl (~1, K) + (~'~S1 + ~"~$2) (14 a) for the 1-2 gap, and simultaneously
~'-'~-- ~"~/3($3, K) + (~"2S3 + ~"2S4 )
(14b)
for the 3-4 gap. If no slipping takes place then f~z,- f2z:and consequently x3 - x,. It is this very fact which ~ without slip ~ allows one to cast the result for rl = rl(~ ) (for rl, = 1"1a (~t,,, K )) in the form of equation 7 (or 10) with the system constant C given by C~
(K 2 -1)
(15)
47tK2h(R~ + R32)
With slip, however, (f2~,+ t~:) will, in general, not equal (~"~S3"~~"~$4) SO that the relation between x3and x~is not known (Using the fact that f2~ cannot decrease with increasing shear stress and that u should increase with x requires
128
(f2s3 + f2s,)<__(f2s~ + f2s,) and consequemly x~ >_x,. This implies that slip effects in the 1-2 gap are larger than in the 3-4 gap. Since K =/~ //~ - R3 / R, the 1-2 gap is the smaller one so that this result was to be expected.) This being the case there is no possibility to list a formula for the shear viscosity. Using a double gap device in viscometric measurements in cases where slip might occur is an extremely poor choice. Formally using equation (10) (with C given by equation (15), as was used to obtain the corresponding results displayed in figure 4) and f'mding odd results could possibly be traced back to this feature. Since f2eaccording to equation (4) stays constant at constant shear stress x, (equation (4)) the sudden decrease of rl~ at x would require a sudden increase of (f2" + f2o ) . In other words on reaching x any increase in f2 would leave f2I unaffected but merely serve to increase the slip. Physically such a possibility cannot be discarded.
3. SLIT FLOW
To possibly shed light on some of the puzzles presented we arm to slit flow (dimensions of the channel: length l m, height: 3 mm, width 30 mm). Previous investigations, employing a modification of the customary Laser Doppler velocimetry (termed GRLDA-method [14]) such that local shear rates can directly be determined, revealed that the jump of the shear viscosity, determined via the relation 11 - - ( A p I L)y I (dul d y ) - (Ap I LI)Yl/~
(16)
where Ap / L is the applied pressure drop per unit length, u is the (axial) velocity in the x direction and y is the traverse direction measured from the midplane, is accompanied by a sudden jump of the local axial velocity fluctuations, u'. Prior to the jump the root mean square (RMS)-values are in the 2 % range, while they suddenly jump into the 10% range exactly at that point, where rlsuddenly increases, too [14]. The fact that RMS-values in the 2% range are found for low shear rates already is astonishing and most likely a reflection of the dynamic
129
nature of surfactant solutions. The sudden jump to values of about 10% does come a big surprise. From a fluid mechanical point of view this hints more in the direction of turbulent flow. Since the Reynolds number was 81 (based on 1"1+) this would be associated with some type of structural (or early) turbulence. Proceeding along that fluid mechanical line of thought one would put for the axial velocity u, u=~+u' (17) (with ~ ' - 0 ), where ~ denotes the mean. The average of the equation of motion (in the average flow direction, x) thus reads (Here we have assumed that the mean of fluctuating quantities show no systematic variations in the axial direction, i.e. ~ ~2 _ ~ - f , = 0).
0=@+
d
(~v+-'
-p~'v-')
(t8)
Here xxy - ~ + X~yis the x-y component of the extra stress tensor and v' denotes the traverse velocity fluctuation. Equation 18 implies for the average shear stress ~+ the relation
Ap -' + pu' v ' +xy - (----L--y- + xy )
_
(19)
It was shown elsewhere [14] that if isotropic tm'bulence prevails (v'= u', i.e. pu'v'=-p~ '2) Newtonian behavior ('t~,=rls(du/dy), i.e. ~'~v= 0) would quantitatively account for our results. In other words, the observed increase in q, as determined via equation 16, would be fictitious, since that increase is solely due to the (sudden) occurrence of the Reynolds shear stress 9u'v'=-p~ '2. The viscosity remains constant, 1"1-- q~. (According to equation (19) q satisfies rl~(d-a/ dy)= -(Ap / L)y if ~ < ~ and - -(Ap / L)-O~ '2 if ~ > Yc) To study the hypothesis of isotropic turbulence, v' ,~-u', Laser Doppler velocimetry (in the backward scattering mode) was employed, which allowed us not only to determine the axial velocity u but also (by rotation of the optical system by 90 ~ the traverse velocity v - v'[ 15]. Figure 5 shows a typical result. It
130
0.16 I"-"-I
C,6TMA-Sal +
-~0.14
1000
NaBr
wppm
~
25
~"JO. 12 1:3 ~
~o.~o
-~- - m -
-
"~ 0 0. 8
-
/ ,(
>,0.06 ~
RMS o f t h e -'--'--'--'--'-RMS of t h e
v-component u-com-ponent
cccco
\
r
o 0.04
-
t.,.~ 9
> 0.02 0.00
~T
P
i
I
-1.5
I
t
t
w
-1.0
w
I
~
=
i
-0.5
-I
l
0.0
I
i
w
I
~
i
0.5
I
i
I
i-w
I
=
1.0
3"
1.5
y - p o s i t i o n in flat c h a n n e l [ m m ]
Figure 5. The average velocity ~ as well as the RMS-values for the fluctuation of the u-component and, respectively, the v-component 0.030
t 0.025 :
C,eTMA-Sal + NaBr I000 wppm
-
25
/
~
RMS~
rn 0.020 -
o o o o o RMS v - c o m p o n e n t
.
,,~,,,,,~
0.015
=~
U~
0 . 0 1 0 -2_
0.005
~
_
o
0---.o
0.000
o
0 o
o
o., o.~ o.~ o.~ o.~ o~. 1.b 1.'1 1~. 1~. 1~. 1.~ 1.o y - p o s i t i o n [mini
Figure 6.
The RMS-values for, respectively, u' and v', close to the wall
131
follows from these results that Iv'l << lu'l (even close to the wall, figure 6)i.e. the Reynolds shear stress, though non zero, is too small to account for the observed increase. If the flow is indeed ,,turbulent", then it has to be a rather anisotropic type of turbulence. Interesting in this connection is the fact that the surfactant solutions investigated here do indeed show an essentially one-dimensional type of turbulence under truly turbulent conditions (high Reynolds number) e.g. [16]. Since isotropic turbulence would, in our case, imply Newtonian behavior and vice versa, we conclude that the fluids rheology is definitely non-Newtonian. Thus, the observed increase in ,, q" as given by equation 16,should mainly be due to a sudden increase in average shear stress fluctuations. Formally assuming for the shear stress an arbitrary functional dependence f to the shear rate ~, ~ - f ( ~ ) , we, get, by Taylor expanding the right hand side around ~ and then averaging
In our case up to the onset the viscosity is essentially constant ( f ( ~ ) = rl,~ ), so that one anticipates T' = 0 (as is true for a Newtonian fluid). However, if one realizes t h a t - at least d i r e c t l y - after onset the flow is not (strictly) viscometric then the notion z - f ( ~ ) is incorrect [17] and x =rl,~ +x'
with -V ~ 0
(21)
is entirely possible. Without a priori knowledge about the fluids rheology, we know nothing about ~'. For completeness we note that x'will not only depend upon the shear rate, but on other variables as well [17]. Another possibility exists which might also be responsible for the observed increase of ~-Iyl/~ at onset. This has to do with the average itself. For an average to describe the (average) situation properly it is necessary that the mean peaks around the instantaneous distribution (e.g Gaussian distribution). Performing measurements at a constant distance (0.5 ram) from the wall and changing the local shear rate ~ by varying the volume flow rate we find that the axial velocity distribution is indeed properly peaked for low shear rates (below onset) but is
132
essemially plateau like for high shear rates (figure 7). In this latter case the arithmetic mean used in LDA studies, 1 -W- , U , N .
0.25
(22)
..,
I
r
; i
;J
!
t
I
;
I I"1~
I
CTA-SAL + N a B r T25~
@.2@~
I
I
1
l i I l{
i
i
I'
I
l ~
1000 ppm
i
O Shear Rate 0.5 1Is Shear Rate 5.2 1Is
O) 0
Shear Rate 236 1Is
~0.15 "0 1:::;
<8.1@
_
O. 05
@. @@
,
9
f
i ~o= , I
@.1
Figure 7.
~
~ I =
=~
=~=~-ri
=r
:-
"
r ~--"
, -" § ~-g-:'4-I--/
Ve]ocity [era/s]
|
1@
4
@@
The probability density function for various shear rates
in which N, is the number of measurements, for which the velocity u, has been measured ( N = ~ N, ) seems highly questionable. Considering the logarithmic i
scale of the abscissa of figure 7 it becomes clear that equation (17) is not representative of the average behavior for high shear rates. These results point more in the direction of some kind of multiphase flow, for which it is (almost) equally likely to find velocity values well below fi (according to equation (17)) as well as above. This hints in the direction of a shear induced state, (SIS), in which monomers, micelles and, for the most part, larger entities
133
(SIS) are in thermodynamic equilibrium. Thus, if we assume that there is, after some critical shear rate, ~,, an additional SIS state, and that this SIS state is far more dynamic (Since turbulent flow is customarily associated with vortices of different size and since we have no evidence of such vortices in our case we refrain from calling the flow turbulent.) then the state of wormlike micelles existing up to ~c, then the viscometric results make sense. What we have to abandon totally is that after onset there is viscometric flow. Quite the contrary, for ~ > ~ c, the flow is definitely non-viscometric. It may be viscometric in the mean (even this is open to question, see figure 7), but it is fluctuations around the mean which bring in the strong (or odd) dependence upon the actual geometry in relations valid only for truely viscometric flow. Since even prior to onset the flow is not strictly viscometric (~'2/~2 ~ 2% for slit flow), we expect the critical shear rate 1'c (actually ~ ) to be geometry dependent, too.
! .00
i
~l- 8j~
CTA-SAL + NaBr 1000 p p m T25~ X - 0~ S h e a r Rate 236 1/s
>a oo 0 . 6 ~
~ I o~ i:>
' I
0
'
20 "
i O
1O
2O
30
Time
Figure 8.
4~
50
60
Is]
Intensity fluctuations for flow birefringence measurements (after onset), extinction angle X - 0~
The fact that after onset the flow cannot be viscometric (at least not in its classical terms) is strengthened further by two additional observations. According
134
to the SIS hypothesis the micelles are all lined up in the flow direction after the critical shear rate ~ h a s been reached [e.g. 11]. Flow birefringence measurements will thus produce the extinction angle %- 0 ~ . While we fred this to be true on average there are rather huge instantaneous fluctuations around this mean (see figure 8). Corresponding measurements of the instantaneous pressure difference fluctuations also hint at an extremely dynamic behavior (onedimensional) of the flow after onset (figure 9). Attempts to study (after onset) such behavior by considerations restricted to truly laminar viscometric flow seem futile.
1 .013 ~,
~0.60
CTA-Sal + NaBr 1000 p p m T - 25 ~ S h e a r R a t e 236 1 / s ~,1~1~
i [
I
i
.oo
i
i
I
i i
r ' ] " l ~ r
I
i
~ r
~ i
I
!
I l k ' i l l
3o.
Time
Figure 9.
t:
I
I
o
I
[
~ IT
i
I'1
i
r
I
I
!
~ '1 I
5o.
1 I
I'l
I
ii
'
e
[s]
Difference pressure fluctuations for the results presented in figure 8
For completeness we note that slip effects can def'mitely be discarded. In slit flow we never observed any tendency of slip, irrespective of the volume flow rate used. Figure 5 is a representative example for this fact.
135
4. SUMMARY AND CONCLUSIONS
Since the tendency for slip has to be discarded the results presented revealed the non-viscometric flow behavior (at least in its classical sense) under seemingly viscometric flow conditions. We do believe that these results reflect the dynamic behavior of micellar surfactant solutions, especially in the so called SIS state. This state is a rather dynamic one such that classical viscometric studies, though useful if performed under identical conditions, may be of little use, if an engineering attempt is made to use the data in any other system or for, respectively, upscaling. ACKNOWLEDGEMENT Partial support of this study by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. REFERENCES
.
2. 3. 4. 5. .
.
.
9. 10. 11. 12. 13.
S.Gravsholt, 1973, Pro. Intern. Congr. Surface Activity, 6, 807 A.J. Hyde, D.W.M. Johnstone, 1975, J Colloid Interf Sci 53, 349 D. Ohlendorf, W. lnterthal and H. Hoffmann, 1986, Rheol Acta 25, 468 T. Nash~ 1956, Nature 177, 948 H. Hofmann, M. L0bl, H. Rehage, I. Wunderlich, 1986, Tenside Dertergents, 22, 290 K. Schmitt, P.O. Bnmn and F. Durst, Progr. And Trends in Rheology II, 1988 Steinkopf, 249 K. Schmitt, F. Durst, P.O. Brunn, Drag reduction in fluid flows, 1989 Ellis Harwood Ltd., 205 E. Ruckenstein, P.O. Brunn and H. Holweg, 1988, Langmuir 4, 350 J. Vorwerk, P.O. Brunn, 1994, J Non Newtonian Fluid Mech 51,79 A.M. Wunderlich, P.O. Brunn, 1989, Colloid Polym Sci, 267, 627 H. Hoffmann, G. Platz, H. Rehage, W. Schorr und W. Ulbricht, 1981, Ber Bunsenges Phys Chem 85, 255 H. Rehage, I. Wunderlich und H. Hoffmann, 1986, Progr Colloid & Polymer Sci 72,51 G. Schramm, Einf0hrung in die Rheologie und Rheometrie, Gebr. HAAKE GmbH
136
14. 15. 16. 17.
A.M. Wunderlich, P.O. Brunn and F. Durst, 1989, Rheol Acta 28, 473 R. Steger, 1994, Dissertation, Universit~it Edangen-Ntirnberg H.W. Bewerdorff and D Ohlendorf, 1985, Proc 5 Symp Turb Shear Flows, (ed. J. Lumley), Ithaca, NY, 241 G. B0hme, 1981, Str0mungsmechank nicht Newtonscher Fluide, Teubner, Stuttgart
137
TIME PERIODIC FLOWS J. D u n w o o d y Department of Applied Mathematics ~ Theoretical Physics, The Queen's University, Belfast B T7 1NN, N. Ireland. 1. I N T R O D U C T I O N
Time periodic flows have a prominent position in rheological experiment and fluid characterization, since they are used extensively to determine the relaxation functions, or their spectra, which figure in the constitutive description of viscoelastic fluids such as polymer solutions and polymer melts. Invariably the amplitude of the oscillations in the viscometers designed for this purpose are small, or infinitesimal, so that linearised constitutive theories have been deemed adequate to describe the flow(cf. Ferry[I]). In addition it is standard practice to neglect the inertia of the motion generated by some external stimulus, particularly in parallel plate and cone and plate instruments. Such simplifications, if made a matter of course, are a source of error in the analysis of experimental data as stressed by Schrag[2] in the case of parallel shear flow. This is even more likely for parallel cylinder viscometers, such as those used by Oldroyd et. al.[3] and Markowitz[4], because of the nature of the fluids tested in these instruments. Of course, viscoelastic fluids and non-Newtonian fluids in general always require a nonlinear constitutive law to adequately describe their varied behaviour, particularly when subject to dynamic forces which produce strains of considerable magnitude. There is some doubt as to whether or not the standard regime of experiments used to obtain data on particular fluids is broad enough to yield a material description which is complete and suitable for all circumstances(cf. Vrentas et. al.[5]). For this reason, but also because the flows have inherent scientific interest, the review below is considered opportune.
138
In what follows attention is restricted to those time periodic flows which are defined to be viscometric according to the definition given by Truesdell & Noll[6], which is nonstandard and includes time dependent flows. Thus, elongational flows are excluded. Time periodic perturbations of steady viscometric flows are included, but only if both the steady flow and the perturbation satisfy a common, specific constraint on the velocity components which delineate a particular class of curvilineal flows. While Schrag[2] has emphasised the role of inertia, too often it is understated, and indeed it is frequently overlooked that some of the standard viscometric flows are dependent on neglible inertia for their existence. Also, the effects of nonlinearity must be a consideration when the amplitude of the oscillations is not small; but not so large as to make the existence of the flow questionable(cf. Hatzikiriakos & Dealy[7]). If the physical circumstances are right, both nonlinearity and inertia will have contributory and interacting effects on the possible motion. Throughout this article it is assumed that a fluid adheres to solid boundaries, that is to say that no-slip boundary conditions are applied. 2. V I S C O M E T R I C
FLOW
The geometry of the most commonly used viscometers (cf. Ferry [1]) is such that the flow may be considered to be one of the curvilineal flows described by Truesdell & Noll[6], in which the particle paths are conveniently expressed in terms of orthogonal curvilinear co-ordinates. For the viscometer flows these are invariably rectangular Cartesian, cylindrical polar or spherical polar co-ordinates. In general these flows may be time dependent, and in particular time periodic in the sense that the flow characteristics at a given point in the flow repeat after a given period of time. In the appropriate set of orthogonal curvilinear co-ordinates the velocity of a fluid particle at x and time t has the contravariant components with respect to base vectors ei(x): v
-
o
,
v
-
v
t)
,
-
v
1, t )
,
(1)
where x ~ - x.ei(x) and ei(x) are the duals of ei(x). Furthermore, it is assumed that the dependence on the coordinate x I is such that the ratio
Or
)
(2)
139
is independent of time, or equivalently that Ov 2
Ox ~ = f (x * )q(x 1 , t)
,
Ov 3 OX 1 -- h ( x 1 ) q ( x 1 , t )
(3)
As seen below, this is necessary for the physical components of the shear stress on the surfaces x I constant to be in time independent ratio, so that while fluctuating in magnitude they maintain their directions at each point on the surfaces x 1 constant. For these motions the displacement vector u(x, r; t) of a particle at past time r relative to its position x at present time t has the components ,,1_0,
u2 _
v 2 (x 1 , a ) d a ,
u a-
S
v2(x 1,a)da
,-oc
(4)
< r <_ t .
The relative deformation gradient is then Ft(x,r) -
6~ +
-~Txjda
(5)
ek({) | &j(x),
where { -- x + u(x, r; t)
(6)
and the base vectors
(r) are orthonormal. For these orthonormal vectors there exists a time dependent orthogonal tensor Qt(x, r) such that
(8)
a~(x) - q,(x, ~).a({), and so
,/
q~(x, ~).F,(x, ~) - v 77)(~)
(9)
~
/O00/
Equivalently, the tensor defined by (9) has the matrix of components
g(kk) (x) k
Vg-~JJ)({) ~j
+ k(~; t)
~ o o
/300
(10)
140
where k(s; t) -- --~/ f : q(x 1, t -- r ) d r , O~
-
-
- I g 1 1 1 ( g 2 2 ( f ) 2 + g33(h)2), /
,/ 11
,.)/--1
lg22f,
-
-
(11)
7--1 v/ g111g 33 h ,
and 0 < s - t - 7 < oc. The incompressibility condition requires that the Jacobians satisfy the condition g(x) - H g(,,)(x) - I I i
-
(12)
i
This condition is satisfied if (i) the coordinates are rectangular Cartesian or circular cylindrical polars (ii) the coordinates are spherical polars and the velocity component v 2 - 0. Indeed in both these cases g(ii)(x) - g(ii)(~)
(13)
for all 7-, and the ratio c~ 9 fl is then independent of time, being dependent on x only as is clear from (ii). At each point x at present time t a basis
i~: i~ - e l ( x )
,
i2 - c~62(x) + fle3(x) , i3 - -f162(x) + o / e 3 ( x )
(14)
may be defined such that the relative right Cauchy-Green strain tensor is Ct(x, t - s) - FT(x, t - s).Ft(x, t - s) = I + k(s; t ) ( N T + N) + k2(s; t ) N T . N ,
where N has the component matrix with respect to the basis ik" O 0 1 0 0 0
0/ 0 0
(16)
141
For the class of incompressible simple fluids with constitutive law expressing the stress tensor T as a non-linear tensor valued function of the strain tensor history: T + pI - S,% 0 ( C t ( t - s) - I) - S ~,=0 (k(s; t ) ) ,
(17)
it follows from the principle of material frame indifference: Q . T . Q T - - p I + S~%o ( q . c t ( t -
s ) . q T - I)
(18)
for arbitrary orthogonal Q (cf. Truesdell & Noll [6]), that T - (i3.T.i3) - T ~ o (k(s; t ) ) ( N + N T) + Sls%o (k(8; t)) N T . N + ,$2~=o (k(s; t ) ) N . N T
(19)
From this relation and (14) the physical components T
--
(20)
ei(x).T.ej(x)
of the stress tensor may then be readily obtained. In particular T<12> - o/~C~=o (]g(8; t))
,
T<13> - fl~c~_-0 (k(8; t))
(21)
with similar relations obtainable for T~23> , T < l l > - T<33> and 7'<22> T<33>. Hence, in these time dependent curvilineal flows the physical components of the rate of strain tensor
(22)
1 [Vv + (Vv) T] and stress tensor T have the property D<12> " D<13> -- T<12> " T ( 1 3 > - a - f l
(23)
2.1 T i m e P e r i o d i c Flow
By a suitable choice of Q in (18) it is readily established that the shear stress functional in (19) and (21) is odd in the history k(.; t), i.e. :r, o
t)) -
t))
(24)
142 Also, if the non-zero components of the velocity are time periodic, i.e.
271 i -- V (X 1 t) a2
v i ( x 1, t ~- - - 1
for
i -- (2, 3),
(25)
then q ( x 1, t "F- -271" ) -- q ( x 1 , t ) a2
(26)
and k ( s ; t + 2a- ) ~d
-7
-- --'7
~o ~ q ( x 1, t -~- -27r - CO
/o
r)dr (27)
q ( x 1 , t -- r ) d r - k ( s ; t)
Hence, time periodic curvilineal flows have the property that the shear stress components defined in (21) are periodic:
271 T (x 1, t -}- - - ) -- T (x 1, t) a)
for
i -- (2, 3).
(28)
In passing it is noted that SlsC~__0 (k(8; t)) and 8 2 ~ 0 (k(s; t)) are even functionals of their arguments, so that all the stress components are periodic. If the velocity is anti-periodic, v i ( x 1 t + -7r) ~d
-v i(xl
(29)
t)
and therefore periodic, then the history k(8; t -~- - - ) -- - - 7 ~d -- "7
for all 0 _ s < ~ .
/o s q(x 1, t +
/o
~d
r)dr
(30)
q ( x 1, t -- r ) d r - - k ( s ;
t)
It follows from (21) that if (29) applies then
71" T (X 1, t -~ - - ) -- - - T < l k > (X 1, t) a2
fo r
k - (2,3),
(31)
i.e. the shear stresses are anti-periodic. However the normal stress differences are not(cf. Coleman & Noll[8]).
143 It is evident that the converse of the statements that (25) implies (28) and (29) implies (31) is true if and only if
(32)
ceT<12> -~-/~T<13> - ~~ 0 (k(8; t))
is invertible. While this is to be expected in the neighbourhood of the zero history k(s; t) for all 0 ___ s < ~ , it should not be taken for granted elsewhere (el. Hunter & Slemrod [9] and Truesdell & Noll, w Thus the generation of time periodic flows of arbitrarily large amplitude by application of time periodic forces must be treated with caution. The observations of Giacomin, Hatzikiriakos et al. in a series of papers ineluding references [7,10,11 ] and also those of Durand et al. [12 ] and their introduction of 'wall slip' are also pertinent to this point. Because of the above observations, it is considered prudent to consider firstly those flows as above which give rise to strain histories such that Ik(s; t)l is moderate for all 0 _ s < co. To this end it is assumed that k(s; t) C E where
)U--
{
/0
k(s;t)"
e-P~k(s;t)ds < o o , ~ ( p ) > 0
}
,
(33)
and k(t; p) --
e-P~k(s; t)ds < e ~ , ~(p) > 0
j~0~176
(34)
is an element of a Fr(~chet space of functions analytic in a right half-plane of the complex plane. Further, it is assumed that E is a subset of the domain of the functionals in (19). All k(s; t) such that [k(t;P)l<M<~176
,~(p)>0
(35)
belong to/C, and in particular all periodic flows with transform 27r
k(t;- p) - 1 -Pfe- :~____e~fo -z- e-P~q(t - s)ds
(36)
satisfy this requirement for all p such that ~(p) > 0 through proper choice of v q ( t -
1
s) -- 2 (t rD 2) ~. In addition 27r
-
q(t + - - - s) - q ( t - s) 4, k(t + cO
27r CO
;p) - k(t; p)
(37)
144 Among the materials covered are the Coleman-Noll[8] fading memory fluids requiring that for some real/3 > 0
1 > fO c~ e -;3~ { k2(s;t) + -~ lk4(s;t)} ~ ds > fo c~ e-Z~[k(s;t)lds,
(38)
the K-BKZ class for which
7-(k(~; t)) -
~- (k(~; t), ~)d~
j~0(X)
(39)
exists on (33), and various special models such as those named for Maxwell, which form sub-species of the above (cf. Crochet, Davies & Walters [13]). 2.2 Equations of M o t i o n It is assumed that each of the stress functionals appearing in (17) and (19) is Fr~chet differentiable over its domain in some normed vector space of functions on [0, ~ ) , which intersects with (33). For example, corresponding to T~0(. ) there exists a linear functional dT~=o(.) defined by
%~o (k(~; t) + ,~xk(~; t)) - LTo (k(~; t)) e----~0 6 (40)
d T ~ o (k(s; t)). Ak(s; t) - lim
for all Ak(s; t). Formally, the relation (40) may be replaced by dT (/c(t; p)). Ak(t; p) - lim 7 (k(t; p) + eAk(t; p)) - r (k(t; p))
e----~0
(41)
6
where
T (k(t; p)) -- ~'s~ (/~-1 [k(t; p)]) , } dT (~(t; p)) -- d~/-s~ (/~-1 [k(t; p)])./~-1
(42)
are compositions of operators. Also, if
Vi (xl , t; p) _ --[c~ e -psv i(x I t -- s)ds Jo
(43)
145
it follows that
Vi =
O~ i
Ot
_~_pOi
and
~1~
(44)
0~3 = -flpk(t;p). g•_33_ v lOxl
0 ~2
Ox 1
i -- (2 3) '
'
= -apk(t; p)
(45)
In general the extra stress tensor (46)
S-T+pI has the physical components
S -
I g( ii) i g(JJ) Sj - v/g(ii)g(jj)S ij
(47)
and in terms of them the equations of momentum balance are
flv/g(jj)
=~
OvJ
Ovj
v/g(j)) 0 i
+ ~ir
v/g(~) Ov/g(~) v~v~ +
- - ~ -t- v s Ox------~ -- g(JJ)
~
1
Ov/g(JJ)ox s vJ Vs ]
v/g(jj)
S~/g ) -t- 10v/g(jj) ~<jj>
Oxi
v/g(ii)g(jj)
1
{ Ov/g(JJ)
V/'g(ii)g(jj)
OxJ
2 ~SKij> OX i
g(jj) --
OxJ
Ov/g(~) OxJ
SKii>
}
--
1 op v/g(jj) OxJ
(48) Whether or not the preceding purely kinematic considerations are empty is determined by whether or not the equations (48) for specific choice of coordinates admit appropriate solutions for the velocity components (3) with rate of shear strain histories (11). It is evidently true that the form of the constitutive equation (17) must have a bearing on this, but the question of existence is most often avoided by the convenience of equating the inertial terms making up the left hand side of (48) to zero on the grounds that they are of no consequence. Solutions to the resulting divergence equation, involving the stress tensor only, may then be sought by the inverse, semi-inverse methods well known to elasticians. However, the errors that can arise from this simplification are all too commonly overlooked(cf. Schrag[2]).
146 2.2.1. Plane Flows. For these flows the chosen coordinate system is rectangular Cartesian with the cordinates of a fluid particle (x 1 , x 2 , x 3) identified with (x, y, z), while (~-1, ~-0,
V3 -- 0 ==~
k(S; t) -- -- ~0 ~ 0V2(X,
t -- r) d r
(49)
5x
so that (48) reduce to
0S<11> ~x
0S<~2> Ox
OP = o Ox @
Ov 2
N
P-gi-
Op
o
(~0)
Oz
It follows from (50) that if a pressure p
-
(51)
a(t)y + b(t)+ S<11>
is applied the solution of equations (50) reduces to the solution of aS<12> _ a(t) -- p
Ox
~v 2
(52)
Ot
subject to ak(~; t) Ova(x, t - ~) (53) Os Ox If it is assumed that the inertial term in the right hand side of (52) is neglible and then set to zero, solutions to (52) and (53) may be sought in the separable form v~ - ~ {~y(x)},
(54)
~- r
which implies that
S<12> ----~/-s~
( ~ If( x) --~(1
-
e-'~)
Then there are solutions if and only if
])
f(x)-
V'(x),.
(55)
147
(i) a(t) = 0 = f ' ( x ) = 0 while dT~=o exists, ~ - a (t) # 0. (ii) T~--0 is linear, f'(x) is constant and a(t + -5-) The first of these conditions relate to oscillatory plane shear flow and are appropriate to gap loading(cf. Ferry[I]). However, in order that (52) yield a unique value of S<12> corresponding to each f ( x ) it is necessary and sufficient, since the frequency w must be absolutely small for neglible inertia, that lim
w--~0
~
,, (1 - e
r O.
b(M
(56)
If this condition does not hold then the possible effects of hysteresis must be considered(cf. Hunter &: Slemrod[9]). The second set of conditions relate to plane pulsatile flow, but the condition that T ~ 0 is linear is very restrictive. Among the fluid types satisfying it are the Navier- Stokes fluids and certain 'second order fluids' (cf. Truesdell & Noll[6]). Of course if the strain amplitudes are infinitesimal a linearized theory, which is compatible with both sets of conditions, is applicable. Existence of periodic solutions to (52), for a special model proposed by Slemrod[14], is assured by a theorem of Rabinowitz [15] (cf. Dunwoody[16]), if the nonlinearity in the equation is sufficiently weak. 2.2.2. Cylindrical Flows. For these flows the curvilinear coordinates (x 1, x 2, x 3) are identified with the cylindrical coordinates(r, 0, z), while in (11) - ~(~)
,
~-
~(~)
,
~-
Z(~)
,
(57)
so that the physical components of the extra stress tensor S are from (17) s<~j> - ~<~j>(~).
(58)
Hence, the equations (48) become for these flows
1 0 rot
(rS)-
S<22> r
1 0 (~'S<12>)-~- S<12>
r Or
r 10 rot
(rS<13>)
Op = Or
-pro 2
1 0p = - p r o
r O0 Op Oz = - p z
oo
oo
(59)
148 where t~ - o0 etc. It follows from these that the pressure required to support such flows is of the form
P--~<11>-t-f
~<11>--~<22>r - ~ - p ( r b )
dr + a(t)O + b(t)z.
(60)
The velocity components (t), i) are obtained as the solutions, if such exist, to the equations
OS<12> 2 pro.9 + a(t) Or + r S<12> -r ' 0S<13> 1 Or + -S<13>r - p2 + b(t) ,
(61)
where S<12> = c t ( r ) T ~ o (k(s; t)) ,
S<13) -- /~(r)~s~_-o (k(8; t)) ,
(62)
subject to
O 0 ( t - s) --r
OF
Ok(s; t) ~ Os
--
~ oz.(r)~
1
-
O i ( t - s) Or
ok(
;t)
-
(63)
c~(r) 2 +/3(r) 2
and a ( r ) , / 3 ( r ) , a(t) and b(t) are arbitrary. In most practical cases
a(t) -- 0
(64)
due to the physical requirement that the pressure be single valued in 0. If the inertial terms pro and p~ are neglible and equated to zero, and the choice b(t) - 0 (65) is made, then equations (61) are satisfied by
,
r
(66)
149
where C1 and C2 are real constants.This corresponds to the choice OZ(I')" / ~ ( f ) -
(67)
C1" rC2,
in (23), and so from (62) and (63) it follows that t)) -
-;r-
+ r2C~.
(68)
Given that the modulus of the right hand side of this relation, for all r in a finite interval, is within the range of the functional T,~=0(-) over the domain/C, then if the functional is one-to-one and invertible over/C the shear strain history k(s; t) is periodic by (27) and uniquely determined by (68). The velocity components are then determined by (63)1,2. However, only in the case that the functional is linear would the shear strain history be simple harmonic. Couette flow corresponds to the degenerate case / 3 - 0 e=> (72 - O,
(69)
Og -- 0 ~ C1 -- 0.
(70)
and sliding cylinder flow to
Even for these cases there is no known physical device which could enforce stresses of the form (66) at cylindrical boundaries. Another interesting type of flow falling within this general class is pulsatile (axial) flow in a tube or annular pipe. However, for it
b(t) 7k 0
(71)
in (61), so that the stresses (66) no longer satisfy the equations of motion. Most often practical interest has been centred on small oscillations superimposed on steady flow and inertial effects are not always neglected (cf. Barnes, Townsend & Walters [17]). Mena, Serrania & Van Ziegler [18] have performed experiments in which perturbation flows of all the above types have been superimposed on steady axial flow.
150 2.2.3 Torsional Flows.
The type of flow envisaged is that which may be caused by independently rotating about the common perpindicular axis through their centres one or both of two parallel circular plates containing a fluid between them. Again the chosen coordinates are circular cylindrical except that in this case the curvilinear coordinates (x 1, x 2, x 3) are identified with (z, 0, r). Also c~-1,
/3-0,
V3--#--O=~ k(s; t) - - r
O-w(z,t),
fo ~ O~(z,Ozt - ~) du
(72) .
so that S<12> -- "~s~
(~:(8; t)) -- S < 1 2 > ( r , Z, t)
(73)
while from (14) and (19) (74)
S<13> -- S<23> -- O.
The equations of motion then become
OF = 0
0S<11> Oz
0S<12> Oz 0S<33> S<33> -- S<22> Or + r
Oz
10p = pr iiv
(75)
r O0
019
02
Or = - p r
subject to
0k(,; t)
~r
Ot
O~(z, t Oz
~)
(76)
From the first and second of these P -- S<11> -I- g(r, t)O,
(77)
and for a single valued pressure g(r,t) --0.
(78)
151
Therefore, S<12> -
fir f (:u(z,t)dz + h(r, t)
(79)
and it is clear that (73) and (76) are incompatible with (79) in general, unless some simplifying assumptions are made. If the inertial terms in the right side of (75) are neglected then (73),
00
(76) and (79) are compatible if in (76) ~z is independent of z. In particular this is true if the velocity ~) has the time periodic, separable, linear form O - N(az + b)e *"~ (80) which corresponds to the physical condition of 'gap loading' again. The analysis for the steady flow case and references to experimental work is to be found in Truesdell & Noll[6]. The restriction that time periodic flows of this type may only occur under gap loading conditions also applies to the following type of flows. 2.2.4 Cone and Plate Flows.
The chosen coordinate system is spherical polars, and for convenience (x l, x 2, x 3) are identified with (0, r r ) i n this system. Again the velocity has only the one component which is
and v3 -
§ -
/ ~ .-j0~ 0
k ( s ; t ) - -sinO
(0, t -
~
u)
du.
(83)
Hence, as for the previous type of flows, S<13> -- S<23 = 0,
(84)
S<12> - g < 1 2 > ( 0 , t ) .
(85)
while The flows envisaged could be generated by the independent rotations of a circular plate and half-cone about the axis of the cone which is perpindicular to the plate surface, the fluid being contained between.
152
The equations of motion for these flows are from (48) cot 0 1 cop _ p r sin 20 ~2 r (S<11>-S<22>) r CO0 -2 ' 1 0S<12> cot 0 1 0/9 sin 0 r r O0 ~- 2 r S<12> --pr r sin 0 0r
1 0S<11>
r
00
+
1 {S -~- S < 2 2 >
0S<33> Or
-- 2 S < 3 3 >
}
Op Or
r
--pr sin 2 0 ~2.
(86) It is clear from (85) and (86)2 that if
0p or
= o,
(87)
which is the condition necessary for the pressure to be single valued in r there is incompatibility between (83) and (86)2 unless the inertial term is zero, which is the case for steady flow. In addition,the relations (19) and (83) imply that 0 (S<11>- S<33>)o-7
~0 ( s < ~ >
- s<~>)
so that there is compatibility between (83) and
- 0
(86)1,3 if and
(88)
only if
0 (~----~{Sis%0 (]9(8" t)) -~- $2c~ 0 (]9(8" t))} - 2 p r 2 sin 2 0r O~ ' ~= ' 00'
(89)
which cannot be satisfied for all r and 0 since O0 r O.
(90)
This incompatibility has not always been recognised(cf. MacDonald, Marsh & Ashare [19]). On equating all the inertial terms to zero in the right hand sides of the equations (86) they and (83) are compatible if and only if
c(t)
~<12> =
- 2 sln 0
0 {SiCs=O ~ (k(8", t))-~- $2 s=O c~ (k(8" t))} - O. .,
00
(91)
153
These conditions are satisfied for all fluids, but only approximately, when
2
c___0_~ 7~,
0r = - a (t) , (90
(92)
T~=o (fo a ( t - u)du) - c(t), where e is small. In particular, the time periodic form
- (aO + b)e ~ t
(93)
is possible under these stringent conditions, which again corresponds to a form of of 'gap loading'. The stresses 7'<12>, T < l l > and T<33> at the boundaries required to maintain the flow are then readily obtained from (86) 1,3 once r is prescribed. As noted by Ericksen[20], there are two distinct approximations involved in the above analysis, one dynamical in nature and the other geometrical.
3. S M A L L A M P L I T U D E
FLOWS
In these flows a time periodic viscometric flow with small amplitude of displacement is assumed superimposed either on a rest state of the fluid, or a steady viscometric flow of it of the same type. The contribution that the neglect of inertia might make to error of measurement is the basic consideration.
3.1 Gap Loading and Inertia Since inertia is not being ignored, the discussion must be restricted to either plane or cylindrical flows, and to avoid the unnecessary complications arising from the geometry of the flow the former of these two types is chosen. A comprehensive treament of 'simple shear' has been given by Schrag [2], but a revised version influenced by the desire to extend it to the nonlinear theory of viscoelastic materials has been given by Dunwoody [21] and is preferred. It is assumed that in a time time periodic plane shear flow m~x Ik(~;t)l << 1
0<s
V t.
(94)
154
Therefore, if in particular (55) applies
f(x) << 1
~u 2
I-~ I/~ << 1,
~
(95)
so that while the relative strain is required to be small there is no such restriction on the strain rate, except at low frequencies; but, this observation is general and a derivative of (11). Under these circumstances, an e ~ 1 may be chosen such that
k(s; t) - Ek(s; t)
(96)
and it then follows from (41) that
7~0 (~k(~; t)) - ~ 0 ( 0 ) . ~(~;t)+ o(~).
(97)
In the case of oscillatory plane shear flow an appropriate value for E is the A ratio ~ , where A is the amplitude of the oscillations of one of a pair of parallel plates which are a distance L apart, the other being fixed. There is then no motion when A - 0 (cf. Dunwoody [21]). Also, for a periodic flow of the form (54) with f ( x ) - V' (x) constant, a complex valued dynamic viscosity may be defined by ~ * ( ~ M ) - d~]"s~
9 [/,03-1
( 1 - e-*"~*)] ,
(98)
which has the assumed limiting values #-
lira q*(w)
(99)
~,~ "--+ 0
the Newtonian viscosity corresponding to steady shearing for which k(s; t) - - e s , and Elim ~wq*(w), (100) o.J ----~ o o
the instantaneous elasticity, or rigidity corresponding to an impulsive uniform shear for which k(s; t) - - e H ( s ) . For example, if the linear operator in (97) has the representation
d%~o(O), k(~; t)- fo ~ G(s)k(s;
t)ds
(101)
155
and G(s) the relaxation spectral decomposition
ak
-~/ak
(102)
k=l
then n
n
k=l
k=l
To facilitate estimation of the importance of the inertial forces relative to the viscoelastic forces in these motions it is convenient to introduce a set of nondimensional variables as follows:
LSc -
x,
ewLv
-
v2
co--li- t
~COIT]*(CO)lCr -- S<12>
(104)
In terms of these new variables a nondimensional form of the equation of motion (52) with a(t) - 0 is obtained which is
Oa = R* O--v-v
o~ (7
R* = PwL2
oi' --
I~*(~)1 '
--(O2]T]*(a2)l) -1 d T 2 o ( O ) 9 f0 ~ 0~(~,-0~i -
(105)
r d+.
Corresponding to the above limits on the dynamic viscosity the nondimensional parameter R* has the limits lim R* = pwL2 .o--+0 #
,
lim R* = pw2L2 , .J--+~ E
(106)
which are the equivalents of a Reynolds Number and Mach Number respectively. A separable solution to (105) of the form
v(~,b- ~ [v(~)~l, } V(O)-O, V(1)-l,
(107)
is readily obtained by solving the ordinary differential equation in V(2)"
'*-(~) v" I~*(~)1
( ~ ) - m*v(~) - o,
(lO8)
156 since from (98) d ~ o ( O ) 9f0 De - * ~ ' d ~
-
coU*(co) .
(109)
The appropriate solution is sin/3:~ sin/3 '
V(:~) -
(110)
/32 _ - ~ n * l n * ( ~ ) l
r/*(aJ)
'
and it is uniquely determined because q*(co) is by assumption strictly complex, so that cos/32 and sin/~2 have no zeros for any values of 2 7~ 0. For neglible inertial effects, equivalently 'gap loading', R* < 1 ,:~
IA~l << 1
(111)
and then
V(2)- 2
(112)
approximately. The shear stress arising from the motion (110) is
- ~
[~*(~)cos~] i-~*-~-)lsin3
(113)
'
and, if (111) holds, it is approximately q*(co) ] (l --
O'g - -
~1~
Iv*(~)l
(114) "
The error involved in using (114) in place of (113), regardless of the constraint (111), is indicated by the ratio of their amplitudes and the difference in their phase angles which are evaluated from
I~--2
~-~ I
~d
arg ~gg -- arg
s--~n~
(115)
respectively. If arg [r/*(co)] -- - r
(116)
157
then it follows from (110) that
1{
3 - - OZl -- LOZ2-- R*2"
cos
2
4
-4-~sin
2
4
"
(117)
Important limiting values of the ratio ~_x are obtained from (99) and C~l (100), viz. Od2 (i) 1, OZl 7: a2 (118) (ii)
w --+ e c ~
r ~
-~ , ,
--
- O,
OZl
which correspond to N e w t o n i a n f l u i d response and e l a s t i c s o l i d response respectively as expected of these materials. Dunwoody[21] has demonstrated through a plot of the ratio ~-~ against frequency ca using a specific Ctl K-BKZ model designed by Luo & Tanner[22] to fit data on LDPE melts, how these asymptotic values are connected by a monotone decreasing function of w. In addition, using the same model, he has demonstrated the monotone increasing behaviour of ,7.(~o) as a function of ca, which implies that R* for the specific fluid increases with w for fixed gap width L and density p. For the computation of relaxation moduli it is essential to have computed values of r/* (w) over a wide range of frequencies 0 < w _< wM < oc such that max
It/* (w) ]
<< 1 V w > WM
(119)
0 <(-O~WM
(cf. K a m a t h & Mackley[23]). This condition is in clear conflict with the condition (111) for gap loading, unless the gap width L is severely reduced at the higher frequencies. Following the example of Schrag[2], the arguments of the function in the right sides of (115) may be written as /~COS/~20Zl(1--~a2)COS{Ctl(1--~c~---~-2) 2 } a l al
(120) sin/3
sin (OZl (1 - ~2-~-)}
Then, assuming 0 < r < 2, values of the ratios in the left hand sides of (115) are plotted in Figure 1 and Figure 2 as functions of 2 for various values of ~_x in the range [0 ~ 1] and al > 0 9 O~1
158
0~1 -- 0 . 2 5
O~ 1
1 . 1.02
1.00
0
4
-
0.5
-
~
. . . . . . . . . ~ . . .~ . . . . . . .
0 99
"
"
1i
"~\"K~
0.9~
~
0.98 0.75
0.98
0.50 0.25
0.98 a 1
0.2
0.4
X
0"~'N.~K"-5 0.75
0.96 0.94 0.92
0.5 0.25
0.75
--
'
x
"
a 1 -- 1.0
1.05
0.75
/ .
0.2
0.95
0.4
.
.
.
.
.
.
X
0
0.75
0.9 0.85
' ' "
"
0.2
0.9
05
0.8
0 25
o.7
"
"
0.4
~
X
O. . . ~ I
0.5
0.25
Figure 1. Variation of [a___[ across the gap width for specific Crg
values of a l and ~a2 _ 0.25, 0.5, 0.75 in each plot. cr
al
-
-
0.25
OL1 ~
0.03 0.02 0.01
~
. . . . . . . .
-0.01
0.75 0.I
0 5
0 25
0.05
0 25
X
.6
0.8
0.5
0.75 0 5
I - 0 . 0 5
~
.
.
.
.
.
.
al--
OL1 -- 0 . 7 5 0.3
0.75
0.2
0.5
0.4
0.i
0.25
0.2
X
.6
0.8
I
1.0
0.75 0.5 0 25
. . . . . . . . . . .
_
0
.
1
~
.
6
~ .
0.8
1
-0.2
.
.
.
.
.
.
X
.6 0.8
i
Figure 2. Variation of arg (~-~_)across the gap width for the \"91
same values of a l and a___22_ 0.25, 0.5, 0.75 again in each plot. C~l
159
It is remarkable that their variations are so significant at relatively small values of a l - R*89
~~_a
+
and values of ~ ~__xin the middle of the
range [0, 1]. The above observations cannot be overemphasised. 3.2 A s y m p t o t i c a p p r o x i m a t i o n s In the case of small strain oscillating cylinder flows the constraint (94) is satisfied and the relations (96) and (97) again hold. Thus, the equations of motion (59), with/3 - 0 , reduce with errors O(e 2) to the single equation 1 0 S<12> " /" C~" ( / ' S < 1 2 > ) - t " -- - p r o (121) r
and
P - po + . / p r O 2 d r
(122)
is single valued in 0. The strain history is from (63)
t - S*)ds, _ ek(s; t). k ( s ; t ) - - / r O 0 (Os*
(123)
The flows envisaged are contained by two cylinders of radii a and b, a < b, so that the gap width is L - ( b - a). (124) Therefore, there are two fundamental lengths which must figure in the considerations of dimensional analysis, viz. the the inner radius a and the gap width L. It is evident from (41), (97) and (123) that, as in the case of plane flows, a separable solution to (121) may be sought which is time periodic. Since equation (121) is explicit in r it is appropriate to choose a as the more fundamental of these two lengths, and so corrresponding to (104) the non- dimensional set of variables
a~ -- r,
ea~av -- tO,
a~-lt-
t,
elU*(a~)la -- S<12>
(125)
is introduced for these flows. In terms of them the equation (121) is replaced, on setting v - ~ [V(~)e ~t] , (126)
160
by the ordinary differential equation (127)
,~v"(, ~) + ,~v(,~)+ ( Z ~ - 1 ) - 0 where
~ _ _~R:I,*(~)I ,*(~)
and
R;
=
pwa 2
(128)
(~)1
I,*
"
The boundary conditions which are applied may be V(1)-0,
(129)
V(I+L)-I+L,
which correspond to the standard Couette flow. Alternatively, they may be V(1 + L ) - (1 + L)~ [~e'~] (130) d (~_lv(?~)) -- fl2:~V(1) -- 0 d§ § _
_
,
where the non-dimensional parameter S defined by the relation aS=
K-w2I 27rPw2a3L
(131)
depends on K, the restoring constant of a torsion wire attached to an inner cylinder of radius a, length L and moment of inertia I. Thus the conditions (130) are representative of a popular device in which the oscillations of an inner cylinder, created by the forced oscillations of an outer cylinder through the medium of a fluid contained between the cylinders, are resisted by the torsion wire (cf. Oldroyd et. al. [3], Markovitz [4], Walters [24]). The phase constant c in (130) allows for a difference in the phase angle of the oscillations of the two cylinders, and in arriving at (130) the identity b- ~0 (132) has been employed to eliminate 0 in favour of O. The solution corresponding to the boundary conditions (129) is (1+L) V ( ~ ) -- -- 2-W-(L-) { J l ( ~ r ) Y l ( f l ) -
tL/-
Jl(~)
[ t1+ Lt]
Yl(flr)Jl(fl))
v~(~)
[ t1+ Lt]
,
(133)
161 That corresponding to the the boundary conditions (130) is more complex,but the amplitude of the velocity at the inner cylinder and the phase angle of the outer cylinder relative to it are obtained from
V(1)- ~(1 + L ) e *c [Y2(/3)J1(/3)- J2(/3)Y1( z ) ] / D , D - Sl [/3(1+].)] Y1 [/3(1+ L)] v~(~) J~(9) + ~ J1 [/3(1+ L)] Y1 [/3(1+ L)] r Jl(/~) YI(~)
(134)
and the requirement that V(1) is real. The conditions that W and D are non-zero guarantees that there are no 'normal modes' in either case. Because the arguments of the Bessel functions in (133) and (134) are complex, asymptotic expansions have been employed in their evaluation(cf. Oldroyd [25]). In practice this causes no serious error as it is most likely that the fluids tested in cylindrical viscometers will be 'short memory' fluids of low viscosity. Interest will therefore be concentrated on reasonably high frequencies. As noted by Joseph[26], high frequencies in the argument of rj* (co) correspond to small times in the arguments of the relaxation modulus G(s) in (101), from which the asymptotic formula
G(0) ~ , G'(0) (~)~ ~ o(~co2)
~*(~)~E~
(1 35)
is readily derived, assuming G(s) is sufficiently smooth at s - 0. For small enough values of co
(136) and then the Bessel functions may be replaced by their power series expansions to obtain a power series in/3 to replace (133) say (cf. Walters [27]). On the other hand, for intermediate values of w Waiters [27], in his critique of previous work by Markovitz [4], has shown how to obtain a Taylor series in powers^of ~L for V(1) by expanding its expression in (134) about its value at L - 0. The condition for strong convergence of this series is ^
19IZ, << 1 ~
R*L 2 =
pooL2
= n* << 1,
(137)
162
which is the equivalent of the 'gap loading' condition of plane flow. The condition (135) is of course even more stringent than (136). Under the conditions Ifll >> 1 r
>> 1,
R* <
(138)
the primitive asymptotic approximation to (131) 1 sin fld: ~/( 1 + ].)a 1+2 ' V(~) ,-, ~ sin fir_,
1 + 2 -- F,
(139)
may be obtained through asymptotic expansions of the expressions in (133) similar to those used by Oldroyd[25]. Aschoff & Schiimmer[28] have performed direct calculations to compute the shear stress from (133) and have concluded that there was no significant error resulting from the use of (139) in place of (133) when L , - 1 and R* > 0.01 r R* > 2.25.
(140)
It is clear from comparison of (110) and^(139), on equating flL in the latter to/3 in the former and similarly ~L -1 to d:, that the approximation (139) is not based on a 'gap loading' assumption, i.e. the influence of inertia is accounted for fully. In this respect it differs fundamentally from the Taylor expansions proposed by Waiters [27], also Markovitz [4], because (139) is derived via asymptotic expansions in inverse powers of /3, while the Taylor expansions derived by Waiters [27,] involve positive powers of/3. The approximation (139) is actually based on geometric considerations, and succeeds so well, as claimed by Aschoff & Schiimmer[28], because the Bessel functions in (133) have asymptotic expansions for large flF, -rr < arg(fl) < re and 1 _< F ___ (1 + ]_,). 3.2 P e r t u r b e d S t e a d y Flows The flows considered in this subsection are most often described as nearly viscometric flows (cf. Pipkin & Owen[29]). However, here the term viscometric flow is inclusive of the time dependent curvilineal flows described in w and indeed of all the time independent, or steady viscometric flows which in turn include the steady curvilineal flows. Thus, the terminology adopted is that of the critical reference work by Truesdell
163 & Noll[6]. Nontheless, the analysis of perturbed steady viscometric flows must be subject to consistency conditions on material functions which in their most general form have been derived by Pipkin & Owen[29]. In all cases the velocity components in the appropriate curvilinear coordinate system are assumed to have the form
?31
-
-
0,
Vi -- V(0 i ) (X 1 )-~-- V~I ) (X 1 , t) "
for
i - - (2, 3)
(141)
where again equations (2), (3) and the analysis following them in w applies. It is also assumed that the condition [v(1)l
Iv(0)l
<<1
(142)
holds. The split (141) is compatible with (3) if and only if, Ov 2
OV 3
Ox I - f ( x 1) [1 + q(x 1, t)] ,
Ox I = h(x 1) [1 + q(x l, t)] ,
(143)
where [q[ << 1,
(144)
so that in (11)
k(~; t)
-
-
(
.~, + .y
/o
q ( x ~, t - ,.)d~
)
(145)
-- ]~(0)(8; t) -~- ]~(1)(8; t). The condition (143) ensures that for all s finite
k(1)(8; t) ~ ]~(0)(8; t),
(146)
and if q(x 1, t) is time periodic then all of w applies, i.e. k(1)(s; t) and the extra stress tensor S are also time periodic. ]The flow considered by Tanner & Simm0ns[30] which is the sum of a time periodic perturbation orthogonal to a steady plane shear is not a curvilineal, or locally viscometric flow as defined by Truesdell & Noll[6].
164 For a time periodic perturbation of a steady viscometric flow the parameter e in (40) may be chosen as e -- max Iq(x I , t)l ,
(147)
so that in (19) and (21)
~Ys~ (k(0)(8; t)-t- k(1)(8; t)) -- ~s~ (k(0)(s; t)) -]- d~K20 (k(0)(8; t)). k(1)(8; t) -1- 0(6) __ r-p(1) O(6), - T(<~189 + ~ <12> +
(148) with similar asymptotic forms obtainable in an obvious way for the other components of S and in particular for the normal stress differences Sl s%0 (') and 82 ~%o('). Therefore,
S - S (0) -~-S (1) -}-O(6),
(149)
where S -
S,% o ( C t - I) -
S ~ = o ( k ( s ; t)) ,
(150)
and if the pressure p _ p(0) + p(1) _1_0(6),
(151)
where p(0) is the pressure arising from the steady flow: V p (~ -
V.
S (~ ,
(152)
then S (1) and p(1) are the perturbation stress tensor and pressure which must be substituted in (48) to balance the perturbing motion with velocity v(1) ( xl , t). It follows from the above that all of the classes of flow considered in ~2 may be treated in this way, subject to the proviso of their existence in the presence of inertia, as before. The consistency relations of Pipkin & Owen[29] relate the 13 independent components of the tensor valued linear functional in the expansion
s-
+
G 0 (st-oI0 ( - 8)- i, t153/
165
where g(s) is a general perturbation of a steady viscometric history, to the viscometric functions T -- "/'s=0(--")/8)
~1
- - S l c~
- &
~
,
e~ s=0
(154)
and their derivatives with respect to 7. Dunwoody & Joseph[31] provided their own derivation of these relations appropriate to a 'locally' two dimensional perturbation of the steady flow. Their relations involve just 9 of the components of the linear functional. Since the perturbations considered here are 'locally' one dimensional a sufficient set of consistency conditions are .
-
.
-
(155)
derived from (17), (19)and dS (o)
/
h
d7
0G (o) 07
= dS %o
(156)
t))
t)
"
07 The consistency relations quoted by Barnes et. a1.[17], attributed by them to Pipkin[32], apply to a very special material class having a multilinear functional constitutive form. Bernstein[33] has obtained similar consistency relations for the general K-BKZ fluid class by setting ~(1)(8; t ) - Ae Lwt ( 1 - e - ~ )
,
A ~< 1,
(157)
in (145), substituting the corresponding g(s) in (152) and then taking the limit a~ ~ 0, but again some of his relations are peculiar to K-BKZ fluids. Coleman & Markovitz[34] using the same method obtained relations similar to Bernstein's for a class of second order fluids approximating the fading memory fluids of Coleman & Noll[8]. The flows which have been dealt with in the literature using this type of approximate analysis fall within the classes described in w so that in the case of torsional and cone and plate flows (cf. Jones & Waiters[35]) inertial effects must be neglected. Bernstein[33] in his analysis of sinusoidal oscillations superimposed on steady plane shear flow of KBKZ fluids has taken no account of inertia. Barnes et. al.[17] have taken
166
account of inertia in their analysis of pulsatile pipe flow and also 'second order effects', but have employed a special second order theory in doing so. Phan-Thien[36] and others have followed suit. Their aim of course has been to match their results to specific observations. In all cases a single ordinary differential equation governs the spatial dependence of the time periodic perturbation, so that the analysis differs little from that required for the equation resulting from letting 7 -~ 0. Indeed, Osaki et. a1.[37] interpreted their experimental data in terms of the analysis of Markovitz[4] referred to in w However, besides the effect that 7 7~ 0 has on the perturbation functionals (154), the interaction of the perturbation with the basic flow has considerable practical interest(cf. Jones & Waiters[35]). Both of the two previous nondimensional analyses leading to conditions for the use of the 'gap loading' approximation may be modified to be applicable to the corresponding perturbed steady flows.
4. L A R G E A M P L I T U D E F L O W S The above description is applied to all those viscometric, time periodic flows which are excluded from the considerations of w because their amplitude of oscillation in strain satisfy neither (94) nor (146). For the sake of simplicity the discussion is restricted to flows oscillating about a rest state, and only plane shear and cylindrical shear flows are considered because the effects of inertia, and in particular its interaction with non-linearity in the constitutive relation for the fluid, are the main interest. Therefore, the work of Jones & Waiters[35] on cone and plate flow while noted is not detailed. Besides, their work is limited to special second order theories, which is also true of the work of Barnes et. a1.[17], Townsend[38], Davies et. al. [39] and Phan-Thien[36] on pulsatile flow in cylindrical pipes, and all are limited by the restriction that they involve time periodic perturbations which are small in an absolute sense. Here the aim is generality which is illustrated by application to a K-BKZ model proposed by Luo & Tanner[22] to describe LDPE melts.
4.1 M o d e r a t e A m p l i t u d e Expansions Considerable experimental data obtained from large amplitude oscillatory shear(LAOS) flows has quite recently been recorded by such as Hatzikiriakos & Dealy[7], Giacomin, Jeyaseelan, Samarkas & Dealy[10]. Many of the perplexities which these investigators have attempted to ad-
167
dress in interpreting their data and observations, such as whether or not there is 'wall slip', are avoided by resricting attention to oscillations in the shear with amplitudes which are moderate, but in a sense dictated by material parameters. Mathematically one is then also on safer ground vis-a-vis the question of the existence of time periodic flows using the classical assumption of 'no slip' of the fluid at solid boundaries. Of course if one neglects inertia such questions are trivially answered in the case of plane shear, but unfortunately the predictions of the theory are at odds with the observations for arbitrary LAOS. ~. 1.1. P l a n e S h e a r Flows.
For simplicity of presentation the shear stress is written in the form (42)1 but as a function of the rate of strain using the identity p k ( t ; p) - -O~f~(t; p) ,
~(t; p) --
e-P~v(t - s)ds ,
f0 X)
(158)
where a simplified notation for partial derivatives is being introduced, to obtain S < 1 2 > -- O" -- 7" ( - - p - 1 O x V ) -- g ((:;gxV;p) .
(159)
Then the equations of motion (52) and (53) governing plane shear flow with a ( t ) - 0 reduce to the single equation in terms of ~(x, t)"
p-laxG(x, t) - O2tv(x, t) -F pOtv(x, t) ,
(160)
where the identity (44) has been used. For strains which are moderate in some sense, best defined within the context of the constitutive description of an individual material, it is assumed that g ( 0 ~ ; p) has a polynomial approximation to any desired degree of accuracy, i.e. m
g (axy;p) - ~
r:0
g2r+l (axV,...,
axv;p)
<~
(161)
for some m, where the terms under the sum are formed by multilinear operators defined on /(:2,-+1. It follows from (24) and (159) that these multilinear operators must be of odd order. Such approximations may
168
be obtained by Taylor expansions involving Fr6chet derivatives or differentials of the functional T~0(. ) if it is defined on a vector space of functions intersecting /(:(cf. Liusternik & Sobolev[40]). But, there are alternative to such esoteric methods as is now shown for a specific model. It is important to obtain estimates of the error involved in such approximation, rather than appeal to asymptotics based on infinitesimals(cf. Dunwoody[21]. The use of asymptotic methods in viscoelastic fluid dynamics has had an unfortunate history, due to indiscriminate application. The model chosen to illustrate the preceding argument is that of Luo & Tanner[22], which is a specific form of the general K-BKZ model designed to fit data obtained from experiment on LDPE melts(vide. Wagner[41], Papanastasiou, Striven & Macosko[42]). For it the shear stress in a plane shear flow is
a -- --
+ k2(s;t) t ) d s '
J~0 M ( s )
M(s) - ~n --ak
(162)
n an integer. If the flow is time periodic with frequency co, so that k(s; t) - k
s; t + - -
(163)
61C,
co
and Ik(s; t)l < c~2 V s 6 [0, ec) a n d t 6 ( - c o , ec) ,
(164)
then as shown by Dunwoody[21] the formula (162) for the shear stress is approximated by O"
m
~ f (1 -
e -2~/w'~k ) - 1 ak j[o27r/w -s ~
r=0 k=l
e
/ak
k2r+l(s't) , (--a) ~
ds + R m
(165) where the remainder
Rm
max <E O<s<27r/~s
k2m+3(s;t) ctm+l
(166)
which is independent of t. The physically significant constant
n E - ~ k=l
ak
(166)
169
is the i n s t a n t a n e o u s elasticity as defined by (100). The equivalent of (165) in the form (162) and the notation of (161)is G -- --
7:
M(8)
:f::
]~2r+1(8;t)
~=o
(_a)~
ds+Rm
(167)
?n
g2r+l (Oxv,...
= ~
OxV;p)-F nm.
_
r--O
D y n a m i c viscosities r/~*(a~) of order r may be defined in terms of the multilinear operators g2r+l(') defined on the Fr&het space of functions analytic in the right half of the complex plane. They are ?];(CO) -- g2r+l ((PAr-bDO)--I, -. ., (P-~-bCd)--l;P)
(168)
which for the sample fluid have the complex values n
,
r#~(co)- (-c~)-~(2r + 1)! ~
~2r-F1
]-[2r+l
akA k
(169)
In particular, r/~)(co) -- r/* (aJ)
(170)
is the dynamic viscosity defined by (98), and it has the limits (99) and
(100). A nondimensional form of the equation (160) is obtained in terms of the nondimensional variables (104) except that the quantity e is replaced A by the ratio ~ , where A is the amplitude of the displacement of the 'upper' of two plates, a distance L apart, which contain the fluid between them. In addition to (104) p - - c@,
fj2 _ A ~ ,
k(t) - -
(A) ~
/3-10~,~,
(171)
and
1,7:(~,)I-Z--
~2,.+1 (o~v,..., o~; p) - g2,~+1 ( o ~ 2,
, 0 ~ 2"p)
(172)
170
which generalises (104)4. It is noted that the equivalence
0~ _ (@)
(p ~+ ' ~,) ** o~ ~ -
(p ~+~)'
(173)
the definition of r/; (w) in (168) and the multilinear property of the operators g2~+l (',. 9 "; p) imply that !) ((i5 + ~ ) - 1 , . . . , (/5 + t ) - i ;i5) -- arg (q*(w)) .
(174)
The nondimensional form of (160) sought is then m
0~ [Dr02~+l (0~O,..., 0e#; t5) + / ) m ] -- R* (0t2O +/30tO) ,
(175)
r--0
where R*=
pwL2 ( wA ) 2r r/r*(w)) I~;(~)1' D,.- -~ rl~)(w .
(176)
The parameter D~ has asymptotic limits as w --+ oc and w --+ 0. In particular, for the sample fluid lim D ~ -
CO--* OO
2) ~ aL 2 ,
(177)
which corresponds with the limit (106)2 on R*. The low frequency limit corresponds with (106)1 and is appropriate for 'retarded ' flow(cf. Dunwoody[21]). The set of D~, 1 _< r <_ m, where rn is arbitrary, have the norm O
sup
(D,.) ~
-/)
w,~
.
l
This positive scalar, which is assumed to be bounded above as a function of w, as it is for the sample fluid, acts as a measure of the degree of nonlinearity in (175). If and only if D << 1 is the nonlinearity of no consequence. Also, it is clear from (176) that the frequency not only has an influence on the relevance of the inertial terms represented by the
171
right side of (175), but it also contributes to the nonlinear effects caused A primarily by relatively large ~ . A nontransient solution to (175) and the boundary conditions v(0, t) - 0,
9(1, i) - ~ \/5 +
(179)
may be sought in the form of an asymptotic series" v(:c, t) -- v0(z,
t)-4- Dvl(z,
(180)
t)-t- 9 9 9 9
As is to be expected the leading term in this series is given by (107) and (110). The second and succeeding terms satisfy the boundary conditions ~(0, t ' ) - ~ ( 1 , t ) - 0
(181)
r__l,
so that they are obtained successively and independently from D1 oq~0 3 (oqs,~o ' oqs,~o, oqs,~o; 1~)
O,
-
R*
+
-
D2 0~5 (0~ ~o,.. , 0~ ~o; t3) D
D1 0~.03 (0~,91 0~,g'o 0~9o;/5) D
~
+ etc. (182) and so on, by the method of variation of parameters, or what is much the same t h i n g - though better for computation- the method of Green's functions. The unique solution of the homogeneous equations satisfying the boundary condtions (180) is zero for all 1 _< r, so that it is then apparent from the multilinear form of the operators in the right hand sides that the particular solutions have the form
f~r(X; t) -- ~
~[p'4- (2k q- 1)/,]-le(2k+l)~iXl(2k+l)(:~) k=O
}
.
(183)
The variable coefficients Xl(2k. t_1) in these finite harmonic series are found by solving, by either of the above methods, the ordinary differential equations obtained by substitution of (183) in the appropriate equation in
172
(182) and equating coefficients of e (2k+l)Li. The left hand sides in the resulting equations involve the dynamic viscosities with argument w replaced by (2k + 1)w. The details of the calculations are described by Dunwoody[21], who computed the actual shear stress occurring at the upper plate using the Luo & Tanner[22] model and two terms in the series (180) only. His numerical results obtained for A / L - 2 , w - 100 rad/sec, and various Reynolds numbers are plotted in Figure 3, where the upper profile is obtained from the linear theory. If the gap loading assumption is made, equivalently R* is set to zero, then the series (179) reduces to the single term
-
eti ) ~.
(184)
p+
This transform of the nondimensional gap loading velocity produces a shear stress, constant across the gap, ag-
Iq~(w)] ( ~ ) 5 g
+R~,
(185)
where
~g -- ~ ~-~g2rA-1 ,.=o
t5+ ~
-~- ^ ,... ~-tp- ~ '/5+ ~
^ "/3 p - ~'
.
(186)
Dunwoody[21] has computed the value of the shear stress crg again using the Luo & Tanner[22] model for the same frequency and amplitude ratio as above, but retaining four harmonics and terms O ([A/v/--dL]S). For comparison a plot from his results is shown in Figure 4,where again the upper profile is obtained from the linear theory, and it is apparent that the higher order harmonics have a significant effect. The shear softening due to nonlinearity alone is also significant. It is also observable by comparison of the shear stress profiles in Figure 3 and Figure 4 that the effect of increased inertia is to increase the shear softening at the upper plate, which bears out the general conclusion to be drawn from Figure 1. This observation can be compared with the effect that a pulsatile pressure has on the mass flux of a fluid through a pipe(cf. Barnes et. al.[17]).
173
R*
R* - 0.7
0.0
-
60000 40000
-<0;of/ o
0.010.02k~3
0.010.0~03
~/:~176176176 o R*-- 1.2
R * - - 1.0
40000
40000
\ 0.0
_
0j/OoZeS 0.010
3 --
--
-40000
3
0
~j~oooo ~L
Figure 3. Profiles of Dunwoody[21] f o r -
measured in Pascals as calculated by
A
A
- 2, w - l O O r a d / s e c , both fixed, and L R* -- 0.0, 0.7, 1.0, 1.2. The upper profile was obtained from the linear theory, and the ordinate scale varies with each plot.
75000 50000 250 ,'5/ ,
,
.
_o.o~
.
.
.
|
.
.
.
.
|
,
/ /
.
_o,o~, _o.//
.
.
.
.
,
.
.
0.01
.
.
|
.
0.02 ~ 3
ooo
000
-75000 Figure 4. Profile of ~r~L measured in Pascals as calculated by A A Dunwoody [21] for - - - 2 , w - l O O r a d / s e c and R* -- 0.0, the L upper profile again obtained from the linear theory.
174
4.1.2. Cylindrical Shear Flows. As in the case of plane shear flow either of the solutions (133) or (134) may be made the basic term of a series solution of the equations (121) for Couette flow, taking account of both inertia and nonlinearity in the constitutive equation as in w With slight variations in the argument the method to be employed is that described by Dunwoody[43], where essentially just two terms in the expansion (161) were retained, which as seen in w may not be quite sufficient for accurate quantitative analysis. Also, as discussed in w the fluids likely to be tested in oscillating cylinder instruments would lead to circumstances necessitating full account of inertia in the analysis; but, the use of asymptotic approximations such as (139) would reduce the computations considerably. As previously related, Barnes et. al. [17] have analysed pulsatile flow in circular pipes, where the flow is supported by a pressure
P - Po - t5 [1 + ee ~'t] .
(187)
For particular models they have shown via an analysis in which terms O(e 3) are n e g l e c t e d - the theory is therefore a second order one - that the mean flux of the fluid is enhanced by the presence of the oscillations in the pressure, hence by inertia also, and that the effect is O(e2). Davies et. a1.[39] highlighted the fact that there was disagreement between theory and experiment in that the theory of Barnes et. a1.[17], as well as their own based on a different model, predicted that flow enhancement should decrease with the frequency of the oscillations while experiment showed the opposite. Phan-Thien[36] used a generalised Maxwell fluid model for the same problem, and he found that the mean flux enhancement increased with frequency. 4.2 E x i s t e n c e of L A O S In order that a time periodic flow exist it must be possible to represent the velocity with the property (25) in the form of a Fourier series oo
,
,
kfk- ]fk].
(188)
k~--oo
To be specific, the assumption that a solution to the equations (52), (53) and (21) with a - 1 a n d / 3 - 0 is gross if p 7~ 0, because it is evidently
175
not true in general that the shear stress in the left hand side will separate into the form S<12> --
~ gk (f(x))e k'- -cx:,
(189)
~kwt ,
where f(z)-(...fl2(x
)
, f l 1 (x) , f) (z) , fl (z)
,..
.)
,
(190)
in general.It is therefore surprising that a perturbation analysis for LAOS on the assumption that R* << 1 has never been undertaken. It is the author's conjecture that plane, rectilineal, oscillatory shear flow of arbitrarily large amplitude satisfying the standard no slip boundary conditions does not exist, because the velocity must be two dimensional if plane shear flow is to exist at all, i.e. it must be n o n r e c t i l i n e a l . If his conjecture is correct then it would explain the complexities encountered by Hatzikiriakos & Dealy[7] and Adrian & Giacomin[44] arising from the imposition of a unidirectional oscillatory displacement of large amplitude on one plate of a parallel plate viscometer. Theories involving wallslip (cf. Graham[45]) have been put forward to account for the complicated 'quasiperiodic' experimental data. Nonrectilineal oscillatory plane shear of arbitrarily large amplitude, based on work of Antman & Guo[46] on large amplitude shearing oscillations of incompressible, nonlinearly elastic solids, is the subject of a current study by the author.
REFERENCES 1. J.D. Ferry, Viscoelastic Properties of Polymers, Wiley, New-York, 1980. 2. J.L. Schrag, Deviation of velocity gradient profiles from the gap loading and surface loading limits in dynamic simple shear experiments, Trans. Soc. Rheol., 2 (1977) 399-413. 3. J.G. Oldroyd, D.J. Strawbridge and B.A. Toms, A coaxial-cylinder elastoviscometer, Proc. Phys. Soc. B, 64 (1951) 44-57. 4. H. Markovitz, A property of Bessel functions and its application to the theory of two rheometers, J. Appl. Phys., 23 (1952) 1070-1077. 5. J.S. Vrentas, D.C. Venerus and C.M. Vrentas, Finite amplitude oscillations of viscoelastic fluids, J. Non-Newtonian Fluid Mech., 40 (1991) 1-24.
176
o
.
.
.
10.
11.
12.
13. 14.
15.
16. 17. 18.
19.
C. Truesdell and W. Noll, Non-linear Field Theories of Mechanics, Encyclopedia of Physics (Ed. S. Flfigge), Band III/3, SpringerVerlag, Berlin, 1965. S.G. Hatzikiriakos and J.M. Dealy, Wall slip of molten high density polyethylene I. Sliding plate rheometer studies, J. Rheol., 35 (1991) 497-523. B.D. Coleman and W. Noll, Recent results in the continuum theory of viscoelastic fluids, Ann. N.Y. Acad. Sci, 89 (1961) 672-714. J.K. Hunter and M. Slemrod, Viscoelastic fluid flow exhibiting hysteritic phase changes, Phys. Fluids, 26 (1983) 2345-2351. A.J. Giacomin, R.S. Jeyaseelan, T. Samarkas and J.M. Dealy, Validity of separable BKZ model for large amplitude oscillatory shear, J. Rheol., 27 (1993) 387-410. A.J. Giacomin and R.S. Jeyaseelan, A constitutive theory for polyolefins in large amplitude oscillatory shear, Polym. Eng. Sci., 35 (1995) 768-777. V. Durand, B. Vergnes, J.F. Agassant, E. Benoit and R.J. Koopmans, Experimental study and modelling of oscillating flow of high density polyethylenes, J. Rheol., 40 (1996) 383-394. M.J. Crochet, A.R. Davies and K. Walters, Numerical Simulation of Non-Newtonian Flow, Elsevier, Amsterdam, 1984. M. Slemrod, Damped conservation laws in continuum mechanics, Non-linear Analysis and Mechanics (Ed. R.J. Knops), Research Notes in mathematics, Pitman, 1978. P.H. Rabinowitz, Periodic solutions of non-linear hyperbolic partial differential equations II, Comm. Pure App. Math., XXll (1969) 15-39. J. Dunwoody, Time periodic rectilineal simple fluid flows, J. NonNewtonian Fluid Mech, 53 (1994) 83-98. H.A. Barnes, P. Townsend and K. Walters, On pulsatile flow of nonNewtonian liquids, Rheol. Acta, 10 (1971) 517-527. B. Mena, F. Serrania and A. yon Ziegler, Further developments in the use of oscillatory dies in polymer extrusion, Progress and Trends in Rheology IV (Ed. C. Gallegos), Steinkopff-Verlag Darmstadt, 1994. I.F. MacDonald, B.D. Marshe and E.A. Share, Rheological behaviour for large amplitude oscillatory motion, Chem. Eng. Sci., 24 (1969) 1615-1625.
177
20. J.L. Ericksen, The behaviour of certain visco-elastic materials in laminar shearing motions, Visco-elasticity :Phenomenological Aspects, Academic Press, 1960. 21. J. Dunwoody. The effects of inertia and finite amplitude on oscillatory plane shear flow of K-BKZ fluids such as LDPE melts, J. Non-Newtonian Fluid Mech., 65 (1996) 195-220. 22. X.-L. Luo and R.I. Tanner, Finite element simulation of long and short circular die extrusion experiments using integral models, Int. J. Numer. Methods Eng., 25 (1988) 9-22. 23. V.M. Kamath and M.R. Mackley, Redetermination of polymer relaxation moduli and memory functions using integral transforms, J. Non-Newtonian Fluid Mech, 6 (1989) 119-144. 24. K. Walters, The motion of an elastico-viscous liquid contained between coaxial cylinders. II, Quart. J. Mech. Appl. Math., 13 (1960) 444-461. 25. J.G. Oldroyd, The motion of an elastico-viscous liquid contained between coaxial cylinders. I, Quart. J. Mech. Appl. Math., 4 (1951) 271-282. 26. D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, SpringerVerlag, New York, (1990). 27. K. Walters, The motion of an elastico-viscous liquid contained between coaxial cylinders. II, Quart. J. Mech. Appl. Math., 14 (1961) 431-436. 28. D. Aschoff and P. Schiimmer, Evaluation of unsteady Couette flow measurement under the influence of inertia, J. Rheol., 37 (1993) 1237-1251. 29. A.C. Pipkin and D. Owen, Nearly viscometric flows, Phys. Fluids, 10 (1967)836-843. 30. R.I. Tanner and J.M. Simmons, Chem. Eng. Sci., 22 (1967) 10791082. 31. J. Dunwoody and D.D. Joseph, Systematic linearization for stability of shear flows of viscoelastic fluids, Arch. Rational Mech. Anal., 86 (1984) 65-84. 32. A.C. Pipkin, Modern Developments in the Mechanics of Continua, Academic Press, London, 1966. 33. B. Bernstein, Small shearing oscillations superposed on large steady shear of the BKZ fluid, Int. J. Non-linear Mech., 4 (1969) 183-200.
178
34. B.D. Coleman and H. Markovitz, Normal stress effects in second order fluids, J. App. Phys., 35(1964) 1-9. 35. T.E.R. Jones and K. Walters, The behaviour of materials under combined steady and oscillatory shear, J. Phys. A, 4 (1971) 85-99. 36. N. Phan-Thien, Pulsatile flow of polymeric fluids, J. Newtonian Fluid Mech., 4 (1978) 167-176. 37. K. Osaki, M. Tamura, M. Kurata and T. Kotaka, Complex modulus of concentrated polymer solutions in steady shear, J. Phys. Chem., 69 (1965) 4183-4191. 38. P. Townsend, Numerical solution of some unsteady flows of elasticoviscous liquids, Rheol. Acta, 12 (1973) 13-18. 39. J.M. Davies, S. Bhumiratana and R.B. Bird, Elastic and inertial effects in pulsatile flow of polymeric liquids in circular tubes, J. NonNewtonian Fluid Mech.,3 (1977) 237-259. 40. L.A. Liusternik and V.J. Sobolev, Elements of Functional Analysis, Ungar, New York, 1961. 41. M.H. Wagner, Analysis of stress-growth data for simple extension of a low- density polyethylene melt, Rheol. Acta, 15 (1976) 133-135. 42. A.C. Papanastasiou, L.E. Scriven and C.W. Macosko, Integral constitutive equation for mixed flows : viscoelastic characterization, J. Rheol., 27 (1983) 387- 410. 43. J. Dunwoody, A comparison of periodic flow of moderate amplitude in two viscometers, Developments in Non-Newtonian Flows (Eds. D.A. Siginer and S.E.Bechtel), ASME, FED-Vol. 206, AMD-Vol. 191, 1994. 44. D.W. Adrian and A.J. Giacomin, The quasi-periodic nature of a polyurethane melt in oscillatory shear, J. Rheol., 36 (1992) 12271243. 45. M.D. Graham, Wall slip and the nonlinear dynamics of large amplitude oscillatory shear flows, J. Rheol., 39 (1995) 697-712. 46. S.S. Antman and Z.-H. Guo, Large shearing oscillations of incompressible nonlinearly elastic bodies, J. Elasticity, 14 (1984) 249-262.
179
S E C O N D A R Y F L O W S IN T U B E S OF A R B I T R A R Y S H A P E Mario F. Letelier a , Dennis A. Siginer b
a Departamento de Ingenieria Mecdmica, Universidad de Santiago de Chile, Casilla 10233, Santiago, Chile o Department of Mechanical Engineering, New Jersey Institute of Technology, University Heights, Newark, NJ 07102-1982, USA
1. INTRODUCTION Conduit flow is a common occurrence in many industrial and biological systems. It is also, in some cases, a convenient approximate model for studying fluid motion through porous media, filters, tissues, and other slow fluid motion in complex solid matrices. Current technological advances in many fields require a good understanding of the dynamics of fluids other than Newtonian in conduit flow. This knowledge is necessary in order to estimate energy loss, transport properties, and many other variables of industrial interest. Viscoelastic fluids constitute an important class among non-Newtonian fluids, the study of which is rendered more difficult by several properties and phenomena exhibited by these fluids such as stress relaxation, strain recovery, die swell, normal stress differences, drag reduction, and flow enhancement [1]. The flow of non-linear viscoelastic fluids in non-circular pipes may lead to the occurrence of secondary flows, a phenomenon not well covered in the technical literature. Secondary flows have a significant influence on important industrial phenomena, such as transport, and energy loss. A large number of competing viscoelastic constitutive models exist to predict flow phenomena. Integral models seem to predict experimental data better [2]. In this chapter, the simple fluid of multiple integral-type models with fading memory is considered. Secondary flows are determined in the case of laminar
180
longitudinal flows in approximately triangular and square conduits, when the flow is driven by small-amplitude oscillatory pressure gradients. The chapter is organized as follows. The mathematical background is developed and the summary of a novel analytical method devised by the first author and co-authors for determining the velocity field of laminar Newtonian unsteady flow in non-circular pipes is presented in Section 2. This is followed by an analysis in Section 3 of the pulsating flow in circular pipes of a viscoelastic fading memory fluid of the multiple integral type. Results of these sections are combined in the next section, where a mathematical expression for the axial velocity is developed for flow in non-circular pipes driven by a pressure gradient oscillating around a non-zero mean. The chapter closes with Section 5 where analytical steps that lead to the determination of the transversal velocity field are developed. Plots that depict the main features of axial and secondary flow fields are also presented.
2. NEWTONIAN UNSTEADY FLOW IN NON-CIRCULAR PIPES Fully developed Newtonian flows in pipes are rectilinear when they are driven by a longitudinal pressure gradient, properties are kept constant, no wall-suction is considered, and the pipe is a straight cylinder of arbitrary cross-section. No secondary flows are possible under these conditions. The linearity of the equations of motion makes it possible to use a wide variety of analytical methods that yield either exact closed-form, or approximate solutions. In this section, a summary of these results is presented based on a unified approach that is applicable to an infinite spectrum of cases where the main differentiating factor is the pipe cross-section geometry.
2.1. Analysis The flow of a Newtonian, incompressible, and isoviscous fluid through straight tubes of constant-cross-section is governed by a linear form of the Navier-Stokes equations when the flow is parallel and laminar. In this case, velocity is of the form, u = (0, 0, w),
(1)
where w is the axial velocity. If z is the axial coordinate, then, the continuity equation reduces to,
181
c~ c~
- o,
(2)
and w only is a function of the coordinates attached to the cross-section, and of time. The Navier-Stokes equations become, 0 -
0t9
0 -
~x'
3t9 ~
, p
c~ 3t
-
319
+/.tV 2 w,
(3)
,~
in which Cartesian coordinates have been used. In this, P is the piezometric pressure, that is, P = p + p g ( and P, (, g, /9, fl, t represent the fluid pressure, vertically upward directed length coordinate, acceleration due to gravity, fluid density, dynamic viscosity, and time, respectively. Since in equations (2) and (3), w = (x, y, t), and P = P (z, t), physical homogeneity demands that, 319 3z
= pf(t),
(4)
where f(t) is an arbitrary function of time, usually called the "forcing function" of the flow. The velocity shown in equation (3) is subject to the no-slip condition at an arbitrarily curved boundary 3D~ and, in general, to an initial condition for t=O, that is, w (x, y
, t) --- 0, w (x, y, 0) = w (x, y).
(5)
DE
Closed-form exact solutions of equation (3) for non-circular pipes are available for a few cross-section shapes and a variety of forcing functions. Some well-known contributions are those of Mt~eller [3] and Yih [4], for concentric annular pipes, Fan and Chao [5] for rectangular pipes, and Hepworth and Rice [6] for circular and annular sector, and rectangular shapes. Most of the results obtained are for simple types of forcing functions such as start-up, oscillatory, linearly accelerated, and similarly driven flows. Analytical methods used include separation of varfables, eigenfunction expansion, and Duhamel's principle.
182
The method of eigenfunction expansion provides a convenient tool for finding new solutions for cases where cross-sectional geometry makes possible the definition of associated eigenfunctions. In general, the equation of motion can be written as, c~
_ vV 2
w =f~ +f2(t),
(6)
3t where v = ~ / 9 is the kinematic viscosity, and, where the forcing function has been split into a steady-state, constant part f,, and, a time-dependent part f2 ( t ) . The linearity of equation (6) allows one to define, (7)
w = w~(x, y) + w2(x, y, t), and to rewrite equation (6) as, vV ~ w. = f . .
C~V2h vV ~ w:, = O. dt
(8)
3Wa p v V 2 w 2 p - f 2 ( t ),
(9)
Wz = Wz h + w zp"
dt
Let E (x, y ) be an infinite set of admissible eigenfunctions (m, n = O, 1 , 2 . . . ) , in which Cartesian coordinates are used with eigenvalues a, and /3,,. Then, it follows that, -_
)2Om [( ] +
mn
~12J
Relationships shown in equations (8) and (9) lead to oo
w, = Z m-O
oo
Z~m~ .--0
, ~.. =
A m2. ( L / v )
am
+
12)
2~
Z o
AmnEmn = l ,
(10)
183
W2h = E
Z CmnEmn e2mnt "/~mn= V a m
m=O n--O ao
'
[\l 1 )
nt"
(11)
kl 2 )
'
aO
W2p= ~_. ~_,Omn(t) Emn "
(12)
m=O n=O
Here, A~. are constant coefficients determined by means of Fourier-series analysis. C,,, are constant arbitrary coefficients determined by applying an initial condition and the orthogonality of the eigenfunctions, l~ and l 2 are appropriate length scales, and the time-dependent coefficients Dmn a r e to be computed by solving equation (9)~. When equation (12) is substituted, equation (9)~becomes,
dDm, dt
-JrVF(IQC'm~2 -~-(i~n ~2] Omn = Amnf2. I\ l 1 )
(13)
\ 12 )
Only the particular solution in equation (13) applies, since the homogenous solution has been already taken into account in equation (11). An example of this kind of exact analysis is applied to the problem of a flow that decelerates under a zero pressure gradient starting from a given initial steady state velocity W(x, y) in a rectangular pipe of cross-section 2a(x-direction) and 2b(y-direction). It is assumed that the z coordinate lies along the pipe axis, so that the initial and boundary conditions are,
w(x,y,O + ] : W(x,y), w(+ a,y,t) : w(x, + b, t) : O,
(14)
further, f =0, w(x,y,t --+~o)-0. A convenient set of eigenfunctions for this problem is defined by,
Emn = cos
(2m+l)x (2m+l)y cos , m,n = 0,1,2... 2a 2b
(15)
184
The initial steady-state velocity can be obtained by solving equation (8)~, where the constant f l will be defined as fl = k for clarity's sake. This solution is shown in equation (10). Thus, W=
oo
oo
Z ~
64k
Bran"
( - - l ) m+n 2
(2re+l) (2n+l)
(2re+l) a~
(2n+l) +
b2
Since the pressure gradient driving the flow for t > 0 + is zero, the timedependent velocity obeys equation (8), whose solution is shown in equation (11). Thus, 64k w=
4 rc r
2, mn
-
~
oe
(_1)m+n
ZZ m=~ "=~(2m + l)(2n + l ) I ( 2 m +~
n~v I(2m+1)2 4
a~
+
- - 2r ant
Em, e + ( 2n+l)2]bi
(2n+l)~ 1. ___
(16)
b~
Another useful approach is Duhamel's principle. velocity is found through an integral superposition velocity which results from a suddenly imposed gradient, then, for a general forcing function f, equation (6),
-~s
Using Duhamel's principle, method. If Ws(x,y,t) is the unit step-function pressure velocity is obtained using
(17)
w(x,y,t) = I t w s (w,y,t - s) d f (s)ds + f ( O ) w s (x,y,t) o
'
"
The structure of equation (17) shows that the boundary condition is met, since that condition is first imposed on ws. Also, the initial condition follows from W,, that is, w (x, y, O) = f (O) w, (x, y, O).
185
The associated start-up velocity of the flow has to be worked out first in order to apply equation (17). This is not necessary when the eigenfunction method is used, since the transient response to any forcing is incorporated in equation (13).
2.2. Boundary perturbation approach Exact analytical methods are very restrictive as to the scope of admissible pipe shapes. A more flexible approach was developed by Letelier and Leutheusser [7], and Letelier, et al [8]. An updated version of the approach is presented here. In the following, cylindrical polar coordinates (r, O, z) will be used. A steady state expression for the axial velocity is given by, f (R 2 r 2 w~ = 4 v ) +e,w~,
(18)
where w,~ is a homogeneous solution of equation (8)2, R is a constant length, and E, is a small arbitrary parameter. For ~ -+ 0 the velocity takes the classical Poiseuille's form. The velocity should meet the boundary condition given in equation (5)1 on a closed curve, and be unique within the domain. This can be achieved if, w,~ (o, o = o), Ic, I ~ Ic,~],
(19)
Condition (19)1 makes the velocity unique at the center of the pipe, and condition (19)2 restricts the absolute value of c~ to a value equal, or less than some critical value e~c, in order that the contour be a closed curve. A convenient set of functions to this end are, Wlh
= ~ pn sinnO L r n cosnO
(20)
where n is any positive integer. For each value of n, the small parameter ~c1 has some specific physical dimension stemming from the physical homogeneity of equation (18). For the sake of convenience, a dimensionless form of equation (18) can be used by selecting R as length scale for r, f l R 2 / 4 v for Wl, and fl / 4 v R n-2 for OC'l.In dimensionless terms, equation (18) becomes, w~ = 1- r 2 + ~1rn sin n O,
(21)
186
and the pipe contour is given by (22)
1 - r 2 + ~ r n s i n n 0 = O.
A wide variety of contour shapes develop from equation (22) according to the given values of ~, and n [8]. As l e~ I increases starting from c1 =0, the circle changes continuously up to a shape that exhibits a number of sharp corners, or cusps, equal to the integer constant n, and is obtained for ]c, ]= ]~]. When ]c~ I is greater than its corresponding critical value, then equation (22) no longer describes a closed contour. For appropriate values of Cl equation (22) describes ellipses (n = 2), approximate equilateral triangles (n = 3), approximate squares (n = 4), and so forth. The critical values of ~ can be found by applying the boundary condition, and the condition that at a cusp the velocity gradient should be zero. This leads to, n-2
I~lcl= - ( ~ n-2-2 )n2
(23)
.
In general, expression (21) can be made up of several different components such as given by equation (20), that would lead to many other shapes. For steady flow, equation (21) is an exact solution of the Navier-Stokes equations. The perturbation approach described here draws on both the steady state solutions in non-circular pipes and on the available closed-form unsteady solutions of flows in circular pipes. Let w~ be the velocity distribution of an unsteady flow in a circular pipe driven by an arbitrary forcing function. Then velocity for an unsteady flow driven by such forcing function in non-circular pipes, can be conveniently expressed as, w=
( 1 - r z + e~r"
where Fj
sin n 0 )
(r, O, t) ( j
= 1,
I we -r ~
+ c lF~ + c 2F z +...
),
(24)
2 .... ) are unknown functions to be determined from
equation (6). It has been shown that, in general, w~/(1- r 2) is finite for r = 1 [8]. In equation (24) the steady state velocity is now a "shape factor" that becomes zero at the pipe contour.
187 In the following, the analysis is carried out for the case of an oscillatory flow superimposed on a steady flow. It is, thus, assumed, f = k~ + k 2 sin cot,
(25)
where k, and k2 are constants. If the reciprocal of the angular velocity co is taken as time scale, then equation (6) becomes, in dimensionless variables, f) 0w~ _ vZ Wc Ot
=
4 + A sin t , if2 - (~ v
, 2 -
4k2
.
(26)
k~
Velocity field is expressed as w~ = 1 - r z + A ( r )
e it +
A(r)
e -it ,
(27)
where A(r) is a complex unknown function and A is its conjugate. Substitution of equation (27) into equation (26) yields, V 2 A - iA - iA, 2
(28)
with A --
2, I1_o.( t"/~r)_- 11 A = a , + ia 2. 2f'2 L I o(4iff2) '
(29)
The functions al(r ) and a2(r ) can be obtained from equation (29)1 by splitting the modified Bessel function I o into its real and imaginary parts. In this form, the expression for the velocity is, w~ = 1 - r 2 + 2 (a~ cos t - a 2 sin t).
(30)
Different versions of equation (30) have been developed in the past by several authors, among them Sexl [9], and Uchida [10]. Since ~, is a small parameter
188
( [ G ] < 1 for n _> 2), a perturbation approach can be used to find the unknown functions Fj. To this end w is expanded as, W=W
(0)
"-~-oC'1 W(1)-'~ - oO? W (2) "+ ...,
(31)
and it follows from a comparison of equations (31) and (24),
w (~
c = l - r z + 2 ( a ~ c o s t - a 2 sin 0 ,
w ( ' > = ( 1 - r 2) F 1 + w~r" 1-r 2 sinnO,
(32)
w ( 2 > - ( 1 - r 2) F2 +r"F~ sin nO,
where, ~3v~ v2
V 2 w (i) = 0
"
'
i = 1,2, .. " '
An appropriate solution for
W (1)
V2 -
d2 1d + -~ + 3r 2 r Or
1 d2 r 2 ~5~2
(33) "
in equation (33) can be found by defining,
W(~)= (.4 (') eit+ A(')e-;' )sin nO+ Dr" sin nO,
(34)
where A ~1) is a complex function of r, and D is an arbitrary constant. Upon substitution of equation (34) into equation (33) it follows that, d2A(')2dr + rl dA(1) (dr
i ~ n_~) A <~) = 0.
(35)
The admissible solution of equation (35) is the modified Bessel function I, Thus,
A ~l> = C (I) I, (i~r).
(36)
189
The constant C~) is complex, requirements. Defining,
to be determined
from physical
continuity
C (l) = C(l) + iC) ') , I, (~/if2 r) = In1 (r) + In2 (r), we obtain, A (1) - - a(~) + ia~ ~) , a(~) = C(~) I~, ( r ) - C ~ ~) 1.2 ( r ) , a~ ') = C~ l) Inl (r) +
C(1) 1,2
(37)
(r).
The first order velocity is now given by, w (') = 2 (a( 'J cos cot - a z(~) sin cot) sin n 0 + D r n sin n 0.
(38)
Setting D = 1 and equating equations (38) and (32) an equation for F/is found,
0-r2)Fl =2 [(a~1) -rnpl)COS t-(a~ 1) - r n p 2 ) s i n al
P ~ - l_r 2 ,
t] sin
nO,
(39)
a2
P2-
The functions p~ and
l_r 2. P2
are continuous functions of r. Since for r = 1 the
LHS of equation (39) is zero, this condition leads to, a~') (1) - p~(1),
a 2~) ( 1 ) = P 2 (1)
(40)
which determines the constants in equation (37). In this way F, becomes a continuous function of r throughout the domain. Equation (32)3 defines the second order velocity, where, w ~2) = (1-r 2) F 2 + rn(F~ cos t-F~2 sin t) sin 2nO,
F~(r) = 2(a~) - r " p , ) F~2(r) = 2(a~ ~)- rnp2) l_r z ' l_r 2 9
(41)
190
Equation (41) may be rendered linear in trigonometric functions by using the relation, sin 2 nO
= 1- cos2n0
Physical continuity at r = 1 demands that w ~2) should be of the form,
w '2' = 2(b~ c o s t - b 2 sin t ) + 2(a( 2' c o s t - a ~ 2' sin t ) c o s 2 n 0 . The unknown functions equation (33),
B(r)
e i' + - B ( r )
e -it ,
(42)
b, (r), b2 (r) can be obtained from a 0 - free solution of
B(r) = C o I o (x]~r), C O = Co, + i Co2, I o = Iol +ilo2,
(43)
and similarly, A ~2' : C ~2' 12,
(x/-f-t-~r) = a~2) + ia~2' , C ~2) : C(2) + iC~2' ,
I2n : 12n I .qt_ //2n2 "
The four c o n s t a n t s Col , C02 , C~ 2) and C~2) can be determined by equating equations (41) and (42), and setting r = 1. In this way, the expression for F 2 is derived as, 2 F2 - 1 - r 2
2 1-r
bl
r "F~I 4
cos t
-
b2
[(a(2)+ r n F ~ ) c o s t - ( a ~ 2 ) + 2
4
Higher order terms and several values of oscillating flow. It is in this case for e, = intervals.
4
sin t +
rnF~z)sint 1 cos 2n 0. 4
(44)
can be found in a similar manner. Computations for n = 6 .O were made by Letelier et al [8], for the case of a purely noted that the pipe contour approaches a regular hexagon, 0.1481. Wall shear stress was also computed at finite time
191
This method can be applied to flows other than oscillatory, as long as it is not necessary to impose an initial condition. Such a condition cannot be satisfied by the present analysis. 3. VISCOELASTIC FLOW IN CIRCULAR PIPES
Steady, fully developed, viscoelastic pipe flows are rectilinear when the pipe is a circular cylinder [11, 12]. This is true even when the flow is unsteady, regardless the constitutive behavior. Some special results of practical interest arise from non-linear effects. One of these results is flow enhancement in periodic flows. Several authors have explored this important aspect both theoretically and experimentally [1, 2, 14 - 18]. Integral models seem to simulate experimental data better than other models. An example of integral models is Green and Rivlin's [19] simple fluid model of the multiple integral type. For incompressible fluids, the total stress T, can be expressed in terms of the strain history G, on a particle X, T + p l = Fs:o[G(X,s)] ,
s = t-r,
(45)
where p, t and r are the pressure, time, and past time, respectively. Following Green and Rivlin, the functional F is expressed as a Fr6chet series, F [ G ( X ; ~)] : e.S <') + ~2S(2) --1-oo3S (3, -+- 0(0o 4 )
(46)
in terms of the small parameter e. S ~i) is defined as the Fr6chet stress at order i. The first two terms in equation (46) are, (47)
= I o4(S)a, (s)ds, S(2) "-- I 0 ~(S)G2
oo oo
(s)ds -]- I o I o Pl
(s1, $2 )al (s1)a2 ($2)dsldS2,
(48)
where the history of the strain has already been expanded in a series in c, G(s) = o~G1(s) + c2G2 (s) + 83G 3(s) + 0(6~ 4 ).
Similarly, the velocity and piezometric pressure are written as
(49)
192 c~3
OO
u(X,t," c) = ~ c n u ( " ) ( X , 0 ; P ( X , t ; g ) n =1
zo~np(n)(x,t).
(50)
n=l
The equations of motion and boundary conditions are, Ou
p
--VP+VoS
;
V.u=O
(51)
inD,,
Dt D,
- {(r, O,z).'O < r < R,O < 0 < 2rc, - m < z < +~},
u (R ,0 ,z)
-
O, u ( O , O , z ) < ~ .
We define u = (u,v, w). The axial symmetry of the flow requires u = v = 0 and, w (-~) = -w. Siginer [2] presented a solution for pressure gradiems oscillating about a non-zero mean. First and third order velocities were found in closed form. Flow enhancement appears at the third order of the analysis. This does not imply that the enhancement effects are small. On the contrary, as predicted by the analysis, they are quite large, as the series expansion outlined above may be considered asymptotic in nature. The second order velocity is zero, as the axial velocity is odd in c. A general analytical method for circular pipes can be developed by extending the eigenfunction approach presented in the previous section to the first order problem. The equation of motion is given by, ~42 (1)
P
-
c~
(52)
I o G ( s ) V2 w(')(r, t-s) ds + p f ( t ) .
Let, w (~) = Z C, (t) Jo
, r
,
(53)
n=l
where
Jo
is
the
Bessel
function
of
first
kind
and
order
zero,
Jo(bn) =
0; n = 1,2, .... Substitution of equation (53) into equation (52) gives,
and
193
P
dC, dt
+ R 2 ~ o G(s ) Cn (t
-- S)
as
~
(54)
D n p f (t ),
with oo
ZDnJ n:,
~ (b n r
~):1.
The time-dependent coefficients C. can be obtained from equation (54) with the help of an appropriate initial condition. For oscillatory forcing around a nonzero mean, P f = Pfo + A sincot.
(55)
C n is given the form, C n : Cnl e ic~ q- Cnl e -i~ q- C ,
that when using equation (54) yields, R2 Cnl =
C=
2[(b.2r/') 2 + (pooR 2 - r/") 2] fo R2 P
(17"- p(oR 2 -ib2Dn#7'),
D. bZ" , /.t = ~o G(s)ds,
(56)
77 - ~ o G(s)e- ~<~ - ~7' - i~7",
(57)
From which we obtain,
w ~'~ = Z c~ljoR~ n=l
.
~
Dn 2
bn
R 2
(b~. u')~ + (poor ~ _
u")~ [(u"-
poor ~-) cos cot
194
+ D. b.2 r/' sin co t]} Jo (b. R ) .
(58)
The shear relaxation modulus G(s) determines the viscosity/z in equation (56) and the complex viscosity 7-/*in equation (57). The constant parameters fo and are assumed to be 0(1). Other cases can be analyzed for different forms of the forcing function, provided it remains bounded. Another analytical approach can be developed by exploiting the linearity of equation (52), through an extension of Duhamel's principle, in the form,
t df w (') : I oWsv(r,t- 4 ) - : = ( 4 ) d 4 + f(O)w~v(r,t), ag
(59)
where Ws~ is the start-up velocity for the viscoelastic fluid at O(E), and for a unit pressure gradient given by,
Wsv(r't)-2pR2 p
(1 - e -x"' ), An = ~-'~J ~3 .--, b.J, (b.) p
(60)
~0~176
4. LONGITUDINAL VISCOELASTIC FLOW IN NON-CIRCULAR PIPES
To order c, the flow velocity field in non-circular pipes is governed by,
Ow p--=r
oo G(s)V2w(r, O. t - s)ds + pf(t), V 2 - 3z 0 c,-r2
1 0
+ - /"
1
O-9F + _ /,.2
02 ~2
~
(61)
where cylindrical coordinates (r, O, z) have been selected and the superscript in w has been dropped. The condition at the boundary cO, is, wl
=0.
Following the analysis presented in Section 2, velocity is assumed to be of the form,
195
w = (R2-r z
+ F~,r n
sin n 0) [w o (r, t) + c,H,(r, O, t) + ~ H 2 ( r O , t)+O(c3)].(62)
Where wo(R 2 - r 2) is the closed form axial velocity of the flow in a circular tube. The analysis that makes it possible to find the unknown functions H i ( r , O , t ) , i = 1,2,.., follows a close parallel to its counterpart for Newtonian flow as presented in Section 2. The main steps that lead to the determination of the velocity field for pulsating flow in pipes of arbitrary cross-section are described in the following using dimensional variables. Flow is driven by the pressure gradient defined in equation (55) with constants j~, and 2 of 0(1). The base flow that is the flow in a circular pipe under the forcing given by equation (55), has already been solved in Section 3, and is described by equation (58) in exact form. Assuming c~ small, equation (62) can be rewritten as a series in ~ , w = w ~~ + clw ~'~ + e,w ~2~ + 0 (e~), w (~ = wo(R 2 - r2),
(63)
W(1)= WO rn sin n 0 + H~ (RZ-r2), w ~ = r n Hj_~ sin n 0 + H j (R2-r2).
(64)
Equations associated with equation (64) are, ~(J)
P
cTt
(j)
-~o~G(s)V2w
(r, O , t - s )
ds
;
j _> 1.
(65)
In order to take advantage of the similarity between this problem and its Newtonian counterpart presented in Section 2, the velocity at zero order in ~, will be written as, w(O) _
Pfo (R2_r 2) + 2 [a 1(r) cos cot - a 2(r) sin co t], 4/a
(66)
where a, (r) and a2(r ) a r e infinite polynomials explicitly available from equation (58). Note that several symbols used in previous sections are used in this section with equivalent meaning, but they do not necessarily have the same algebraic structure. For example, the functions a, and a 2 have the same meaning in equations (30) and (66), but their algebraic structure is not the same in both equations. Assuming w ~ has the following structure,
196 w"' = (A ~'' (r)
e i~
+ --A("
(r)
e -ia~
) sin nO + D r" sin nO,
(67)
we obtain from equation (65), d2A (~) + 1 dA (~) rt 2 dr 2 r dr - (12 + - 7 )A(1)= 0, 12 .
.
.
.
.
.
(68)
ipco
r]*'
with r/* given by equation (57). The admissible solution of equation (68) reads as (69)
A (') = C (') I, (Ar) = a(') + ia~ '), C (') = C~ ') + i C~ '). A comparison of equation (64)~ and equations (67) through (69), yields, H, = h, (r, 0 sin n O, h , -
al
P ~ - R2_r 2 ,
P2-
2 [(a( ~)- r ' p ~ ) coscot-(a~ ')- r ' p 2 ) R2_r 2
a2
sin cot],
(70)
R2_r 2.
Similar to the steps which led to equation (40) in Section 2, the required continuity of the function h~ for R = r leads to a set of algebraic equations for the constants C(~) and c"<~) The expressions for wc~) j > 1 can be worked out following the steps shown in equations (41) through (44). Computations of equation (62) have been made by Letelier and Siginer [20] for some values of the fluid, flow, and geometry parameters. The fluid constants were selected following Siginer [1], thus, "'2
G(s) =/-to - U'oo Sk-I
s/O
~
0=
al , k( r/'oo-p)
(71)
Here r/'oo is a second Newtonian viscosity at high frequencies, O, a,, 0
197
r/'~ = O, a~ = - 5 0 g / c m ,
/-to = 200 p o i s e ,
p = 0.89 g/cm 3.
Other parameters concerning the flow and the geometry are given as, co =
10
r a d/s , P f o
_
I,
2 and the geometrical constants are, R = 3cm, ~ = 0.3845
for n = 3, ~ = 0.22
for n = 4.
The selected values of n and the associated values for E1 determine through equation (22) pipe cross-section contours that are approximately triangular and square. The longitudinal flow in these pipes exhibits a very complex pattern that is very much dependent on time and on the effect of sharp corners. In Figures 1 through 6 the isovels in the cross-section of the pipes are shown for one value of the viscosity index k. The specific times at which the isovels were drawn are such that the velocity profiles undergo several inflexion points, as shown in the companion plots. At such times, local velocity extrema appear at several points, a phenomenon that tends to increase as k decreases. All plots are dimensionless. The length scale for r is R, and the velocity scale for u is PR 2/4f10. Velocity profiles refer to the larger half symmetry axis, and to the shorter half symmetry axis for all figures.
198
F i g u r e 1. I s o v e l s f o r n = 3 ;
F i g u r e 2. I s o v e l s f o r n = 4 ;
c= 0.3845
c= 0.22
;
;
k = 0.007
k = 0.007
;
;
t = 0.14257r.
t = 0.257r.
199
1
"
0.8
0.6
0.4
0.2
I
I
0.002
0.004
0 0
Figure 3. k = 0.007
I, 0.006
I " ~ 0.008
0.01
V e l o c i t y profile at shorter symmetry axis for n = 3 ; ~ = 0 . 3 8 4 5 ; t = 0.1425rc.
2 ! .
.
.
.
.
;
.....
r
~I o
...... 0
i
0.002
0.004
0.006
0.008
0.01
U
Figure 4. Velocity profile at larger sylmnetry axis for n = 3 ; e = 0.3845 ; k = 0.007 ; t = 0.1425rc.
200
1-
r 0.8
0.6
0.4
0.2
0 -0.035
-0.03
-0.025
-0.02
-0.015
I
I
-o.o~
-o.oo5
o
Figure 5. V e l o c i t y profile at s h o r t e r s y m m e t r y axis for n = 4 ; e = 0.22 ; k = 0 . 0 0 7 ; t - 0.25~.
1.4 r 1.2 1 0.8 0.60.4 0.2 o -0.05
I
-0.04
I
-0.03
-0.02
-0.01
0
U
F i g u r e 6. V e l o c i t y profile at l a r g e r s y m m e t r y for n = 4 ; t = 0.25r~.
c = 0.22
; k = 0.007
9
201
5. SECONDARY FLOWS Reported studies of secondary flows of viscoelastic fluid are scarce. Such studies have been restricted to a limited set of geometry, such as elliptical and rectangular [12]. Xue, et al [12], apply an implicit finite volume method to a modified Phan-Thien Tanner fluid in rectangular tubes when the flow is steady. Numerical computations describe the transversal flow velocity and pathlines for several sets of relevant parameters. In Xue, et al [12], also provided a review of the relevant literature. In this section the basic aspects of an analytical method for computing secondary flow in pipes of arbitrary cross-section are presented when the fluid is described by the multiple-integral simple fluid model, and the flow is unsteady. The method will be applied to a pulsating flow driven by a forcing of the form defined in equation (55). At O(d) the equations of motion read as, 8u (2) c~
u
=- VP (2) + V'S (2) ; V'u (2) - 0
I =0 0D,
in D,,
(72)
(73)
;u(O,O,z)<~.
The velocity and extra-stress are given as, U (2)
-- U (
(74)
r, O, t) e r + v ( r, O, t) eo,
S(2) - I oG(s)A(2) (t - s)ds + I oG(s) L, (t - s)ds + (75) The transversal velocity components are u and v, and e, and e o represent the radial and azimuthal unit vectors. At the second order the axial velocity component is zero. G(s) and 7"(s1,s2) represent the shear relaxation and the quadratic shear relaxation moduli, respectively. The first Rivlin-Ericksen tensor A1O) is of the form A( j~ (t-s)= 2D [u (s) (X, t-s)] ,
j =1,2,
(76)
202
and, following Siginer [2], L I (t - s ) : ~ * ' V A~ l) + A ~ ' ) V ~ * + (A~I)v~*) T ,
4 - I ~u ''~(x,
~')d~',
(77) (78)
t)~:, 4" - *ez.
The longitudinal velocity, u a) =wez is shown in equations (62) through (70). The longitudinal velocity substitution into equations (76) through (78) using equation (72), leads to the transversal momentum equations which define the velocity field (v, w), U~t --- _ P~r ~oG(s)
-r
(rW'r r
2
2u- - T v , O -
+
ds
~,o + w , o
--~ (W'r
~, r ),o- 72 w'O r
ds
~ { _l [rw, r(S,)W, (s2)] +J0JoY(S,,S2) r r
1
"~- V [w'r
PV't
~--- - -
}
(79)
($2) w'O (SI)])0- 13 W~o (SI) W'o ($2) NSI ds2,
v s 1 P,o + ~~ G(s )[V2v + 2 u,o __~T]d r o -~T
+~o~ G ( s )
1 r
+ J'O J'O~ Y (S1,
2
1
(W'r ~:'0 + W'O ~, r )'r +----~'(W,O ~'0 )'0 +-'7(W' ~'0 r~ rr $2)
-1 [w, r (s,) w, o (s 2)],r + r
1 W'r (s2)W,o (s 1) } dSl Ms2" + -U
1
~3- [W,o
(S,)W,o
+
W'O ~
~r
)
]
ds
(S~)],O (80)
203
Equations (79) and (80) can be combined and the pressure field may be eliminated using a stream function 5Udefined as,
u (2) = V x 7-'. The resulting equation has driving terms both time independent and time dependent of type e i~ and e 2i~. This means that the secondary field has a time averaged mean part ~ (r, O) and oscillatory components with frequencies co and 2oo. The time averaged field ~ is thus defined by --i Ld-" ~-'r --" I o G ( s ) { [ V
" L1]r,O - [ r V . tl]O,r}mdS ,
+ ~o IoT"(s,,s2){[V'(A, (s,) A,(s2)]r,O - [ r V .(A, (s~) A,(s2)]O,r
}rods,
(81)
where { }. refers to the mean, that is time independent terms in the expression between { }, and the operator L has the following structure, 34
1 34
L = r ~-g + r3
~5~04
~3 ~t_
2 c7r3
1 ~2
4 ~2
r cTr2 + r 3 302
2
33
1 C
r 2 drO02 + -rT ~dr"
(82)
An approximate solution for ~ is obtained by defining, ~-/m "--
(83)
H2 (r, O)~b(r, O) + ~o,
H(r, O)=
R 2 - r 2 + ~elrn
sinnO.
(84)
H is the shape factor, q~(r,O) is an unknown function, and To is an arbitrary constant. The structure of equation (83) is the simplest form that meets the required conditions,
u
lC9~ [ m [ =0, v [ cqD r 00 c~D~ cqD~
cgk' m [ = 0 , Wm =constant. ~ cqD~ c~D~
(85)
204
Next, ~bis specified as,
~b(r, 0)- ~ r ~bj (r, 0).
(86)
j=l
In this form, ~ becomes, % (r, 0) = c, (R 2- r 2)2 ~b,(r, 0) + e~ [(R z- r 2)2 ~b2(r, 0)
(87)
+ 2(R 2- r 2) r n sin n 0 ~b~(r, 0)1 + 0 (o~3),
where ~m has been expressed as a series in powers of the small parameter e,. To 0(~), the equation to be solved reads as,
0(~, ) 9L [(RZ-r 2) ~b~(r, 0)] = g,(r) sin n 0 + gz(r) cos n 0,
(88)
where the known functions gl (r) and g2(r) are involved expressions in terms of r and of the fluid constants. The solution of equation (88) can be found by replicating the method of Section 4. It is convenient to develop the functions g/as finite polynomials in r. This can be easily achieved since g, are continuous and smooth functions and the applicable range is of the order of R. Given the structure of equation (82), for any term of the form rmsin nO on the RHS of equation (87), this equation provides a non-homogeneous solution of the form Cm r .m+3 sin nO, the constant Cm being dependent on m and n. It also provides, in general, homogeneous solutions of type,
r/r
c~r~cos (fl In -R'\cos n
C2F
sin (fl In rR)
sin n 0 ) cos n 0
(89)
where c,, c 2 are arbitrary constants, and a ~ ifl are complex powers of r, with a_~O. The solution for ~ at 0 (c~), thus, reads 5u~1) = [Pl + clr~ cos(fl In R)] sin nO+ [~2 + c2 r~ cos ~ ln R)] cos n O.
(90)
205
Only the cosine term in equation (89) has been used because subsequent steps demand that the homogeneous solution should be non-zero at r = R. We note that the functions Pi(r) are known polynomials of r. The function ~1 is determined next by equating ~m') as defined in equations (90) and (87), where the constants c~ and c2are found after putting r = R. The expression for ~m2) follows from the equation at 0(e~), L 7/~m2) =g3(r)+g4(r) s i n n 0 +
gs(r)cosnO+ g6(r)
sin 2n0 + g7(r) c o s 2 n 0 ,
(91)
in which all terms in the RHS have been rendered linear in sinjnO and cosjnO, j = 1,2. The functions gt (i =3 .... 7) depend on r and on the material constants embedded in the shear relaxation and the quadratic shear relaxation moduli G(s) and y(s~,s2) which appear in equation (81). Equation (91) can be solved using a similar technique as that applied in the first order problem. The functions g, are approximated as finite polynomials of r. Computation of the homogeneous and non-homogeneous solutions of equation (91) is straightforward. All arbitrary constants are determined from the required spatial continuity for r = R, after -equating ~m~) computed from equation (91), and its equivalent expression obtained from equation (87). Material constants involved in the driving terms in equation (81) were adopted from Siginer in [2]. Plots of the streamlines for the same values of the fluid, flow and geometry parameters used in Section 4 are shown in figures 7 and 8. A very complex pattern of cells is common between the two cross-sectional shapes chosen. The main cells appear a number of times equal to twice the number of cross-sectional sides. Minor cells are present close to the corners, and to the center of the pipe, depending on the value of k. These results can be extended to other shapes by changing the values of n and/or c~. Secondary flows appear for any value of ez ~ 0. They do not exist only for c ~ - 0 , in which c a s e ~gCm ~ ~/0 =constant in the whole domain, consequently u = v = 0.
206
Figure 7. Transversal streamlines for n = 3 ; e = 0.3485 ; k = 0.007.
J
Figure 8.
Transversal streamlines for n = 4 ; e = 0.22; k = 0 . 0 0 7 .
207
ACKNOWLEDGEMENTS This work was partially supported by grant 197-0810 of the Chilean National Fund for Scientific and Technological Development. The authors acknowledge important institutional support provided by the departments of Mechanical Engineering at the University of Santiago of Chile, and the New Jersey Institute of Technology. REFERENCES 1. Siginer, D. A., J. Fluids and Structures, 6 (1992) 719. 2. Siginer, D. A., Int. J. Engng.Sci., 29 (1991) 1557. 3. Mueller, W., Z. Angew. Math. Mech., 16 (1936) 227. 4. Yih, C. S., Fluid Mechanics, McGraw-Hill, New York, NY 1969. 5. Fan, C., and Chao, B. T., Z. Angew. Math. Phys., 16 (1965) 351. 6. Hepworth, H. K., and Rice, W., J. Appl. Mech., 37 (1970) 861. 7. Letelier, M. F., and Leutheusser, H. J., J. Engng. Mech., 111 (1985) 768. 8. Letelier, M. F., Leutheusser, H. J., and M~rquez, G., J. Engng. Mech., 121 (1995) 1075. 9. Sexl, T., Z. Phys., 61 (1930) 349. 10. Uchida, S., Z. Agnew Math. Phys., 7 (1956) 413. 11. Oldroyd, J. G., Proc. R. Soc. London Ser. A, 283 (1965) 115. 12. Xue, S. C., Phan-Thien, N., and Tanner, R. I., J. Non-Newt. Fluid Mech., 59 (1995) 191. 13. Rahaman, K. D. and Ramkissoon, H., J. Non-Newt. Fluid Mech., 57 (1995) 27. 14. Barnes, H. A., Townsend, P., and Waiters, K., Rheologica Acta, 10 (1971) 517. 15. Sundstrom, D. W., and Kaufman, A., Ind. Eng. Chem. Proc. Des. Dev., 16 (1977) 320. 16. Manero, O., and Waiters, K., Rheologica Acta, 19 (1980) 277. 17. Davies, J. M., Bhumiratana, S., and Bird, R. B., J. Non-Newt. Fluid Mech., 3 (1977) 237. 18. Phan-Thien, N., J. Rheol., 25 (1981) 293. 19. Green, A. E., and Rivlin, R. S., Arch. Rat. Mech. Anal., 1 (1957) 1. 20. Letelier, M. F., and Siginer, D. A., Rheology and Fluid Mechanics of Nonlinear Materials, S. G. Advani and D. A. Siginer (eds.), 121-128, American Society of Mechanical Engineers, New York, NY 1997.
209
EFFECTS OF N O N - N E W T O N I A N FLUIDS ON CAVITATION D.H. Fruman Groupe Ph~nom~nes d'Interface Ecole Nationale Sup~rieure de TECHNIQUES A VANCEES 91761 Palaiseau Cedex- FRANCE
1. I N T R O D U C T I O N The change of phase, from liquid to vapour, occurring when the temperature is kept constant and the local pressure is decreased to become equal or less than the vapour pressure of the liquid, is called cavitation. For this definition to hold, it is necessary for the liquid to contain dispersed micro bubbles of non condensable gas, called nuclei, offering interfaces where opportunities exist for the change of phase. The liquid can be either stagnant or in movement. In the first case, the pressure change is due to the propagation of pressure waves generated by ultrasound generators, boundary vibrations, sudden valve closures, underwater explosions, etc. In the second case, the pressure reduction is associated, through the Bernoulli equation, with the increase of the liquid velocity as it occurs in the flow around hydrofoils, and by extension in pumps, turbines and propellers, in the contraction of pipes, in venturi tubes, etc. In separated flows, such as jets and wakes, cavitation occurs in the low pressure regions of the vortices developing in the free shear layers. Cavitation over solid surfaces can occur in the form of either individual travelling bubbles or sheets. The latter continuously shed bubbles which, when penetrating in a region where the pressure has increased above the vapour pressure, will collapse and rebound with an amplitude decaying with time as a result of, essentially, viscous dissipation. The consequences of cavitation are the reduction of the hydrodynamic performance of liquid machinery - pumps, turbines and propellers - and hydraulic circuits, the emission of noise, the generation of vibration and the erosion of materials. Bubble cavitation occurrence is essentially determined by the vapour pressure of the liquid. Surface tension opposes bubble growth and determines the level of "tension" to which the fluid has to be subjected before the onset of cavita-
210
tion. Also, the viscosity of the liquid, assumed to behave as a Newtonian fluid, will affect the bubble growth and contribute to the damping of the oscillations following collapse. Even in the case of such liquids, characterized by the simplest relation between strains and stresses, explaining and interpreting the inception of cavitation and the consequences of its development on engineering applications is not a simple task and numerous textbooks [1-4] have been dedicated to consider these issues. When the liquid displays a non-Newtonian behavior, as it often happens in the chemical engineering processes and when polymer additives are dissolved in Newtonian liquids, the problem becomes much more complex. It has to be recognized that the larger impetus on the research concerning cavitation in non-Newtonian fluids comes from the interest developed during the 1960's by the use of dilute solutions of high molecular weight polymers, known as being very effective turbulent drag reducers, in naval hydrodynamics and hydraulics (see the review paper by Acosta and Parkin [5] for a summary of work on cavitation in polymer solutions up to 1975). It was soon recognized that these solutions were capable of modifying strongly the flow field on a length scale characterizing the boundary layer for very small concentrations, even below one ppm. As a consequence, the experimental investigations on the effect of flowing drag reducing polymer solutions on cavitation were faced with the following question, i.e. was the behavior of individual bubbles altered because of:
i)
the modification of the surface tension, the vapour pressure and the rheological properties of the liquid phase, or, ii) the changes of the local pressures, as a result of the modifications of the flow structure and of the boundary layer. In presenting the most remarkable results we will try to single out which of these approaches is predominant. The spherical growth and collapse of a gas bubble has interested rheologists because it creates an extensional flow which can be used to estimate the elongational viscosity of polymeric liquids. Although numerous theoretical and experimental investigations have been conducted on this matter, we will only talk about them occasionally because they belong, in our opinion, to the area of characterization methods rather than to that of hydrodynamic cavitation. In this presentation we consider first the theoretical approach of the dynamic equilibrium of a single spherical bubble in an unbounded liquid. This is followed by some theoretical results concerning the effect of Newtonian and non-Newtonian properties on bubble growth and collapse. Because the conclusion of this section is that these properties have a very limited impact on the life of spherical bubbles, we move to the presentation of experimental results obtained in the somehow academic conditions of a single bubble interacting with solid walls or affected by shear flows. For these non-spherical deformations it is shown that the viscoelastic properties have a significant
211
effect. We proceed next to review results obtained in situations such as cavitation of submerged jets, on hemispherical bodies and hydrofoils, and in vortices. We show that, it is usually difficult to decide whether the flow or the bubble dynamics modifications are responsible of the observed changes on cavitation behavior. However, it appears that the former are, generally, predominant. We proceed by presenting recent results concerning the effect of polymer solutions on the cavitation occurrence in very confined spaces. Finally, we consider the important problem of cavitation erosion.
2. SINGLE SPHERICAL BUBBLE BEHAVIOR 2.1 The generalized Rayleigh-Plesset equation Let us consider a single bubble growing spherically in an unbounded liquid. In a spherical coordinate system (r, 0, ~), with the origin at the center of the bubble, the only velocity component is radial and, in order to satisfy the continuity equation, is given by, u-
R(t))
/~(t)
(1)
r
where R ( t ) is the bubble radius at time t, and [ ~ ( t ) = d R / d t . component of the momentum equation is given by,
OU OUI o~p f)Trr 2 Trr- TOO - Tr162 P -&-+UOrr - - - ~ - r + Or + r
The radial
(2)
where p is the liquid density, p is the pressure anywhere in the liquid, and Trr, TOO and 7:~r are the normal components of the deviatoric stress tensor related by,
Trr + Tr162+ TOO = Trr + 2 r162 - 0
(3)
Because of (1) and (3), expression (2) becomes, after integration between infinity and R, p
(
RR + 2
- p(R)-
Poo (t) + Trr (oo)-- Tr r ( R ) + 3
dr JR
(4)
r
where p ( R ) and poo (t) are, respectively, the pressure at the interface on the liquid side and the pressure far from the bubble. The stress Trr (cx~) can be
212
assumed to vanish far from the bubble. The pressure at the interface on the liquid side is given by,
p(R) - Pv + Pg
2~
R + Trr (R)
(5)
where pg is the non condensible gas pressure within the bubble, p v the vapour pressure and 7' the surface tension. By determining p g from the initial conditions of the nucleus of radius R0 and gas pressure PgO, expression (5) becomes,
P(R ) - Pv +
(
27'/(R~/ 3k 2~+Trr(R )
P O - P v + -~O
--R
R
(6)
where k is the polytropic index and P0 the pressure at infinity at t = 0. Substituting (6) into (4) we obtain the generalized Rayleigh-Plesset equation 3
) - - - - ( P o o ( t ) - - P v ) + ( P o - p v ) ( - R ~ ) 3k
2
.01
+3
R0
R
R
(7) --dr
R r
allowing one to compute the time evolution of the radius of the bubble as a function of p~ (t) under the assumption that the liquid is incompressible and that thermal and mass transfer effects are negligible. The terms on the left hand side represent the contribution of inertia, while, on the right, the first term is the cause of the bubble growth, the second term the contribution of the partial gas pressure, the third term the effect of surface tension and the last term that of the rheological properties of the liquid. We would like to point out here that, whatever the rheological behavior of the liquid, for cavitation to exist, the pressure far from the bubble has to be reduced to a value at least equal to a critical pressure for which an asymptotic growth phase of the bubble, at constant velocity, will exist. In the case of the static equilibrium of a bubble undergoing an isothermal expansion and ignoring the contribution of the last term in equation (7), the critical pressure is given by, 4 ?' Pc - P ~
3 Rc
(8)
213
with,
Re-Ro
E3R0/ 2 -~ --~- P O - P v + -~o
(9)
The order of magnitude of the asymptotic growth velocity when the bubble is subjected to p~ < Pv is given by,
[~g =I3Pv--PC~p
(10)
The characteristic time for a vapour bubble, having an initial radius Rch, to collapse under the effect of a pressure step is, [
~R = O. 915
Rch ~[ P VP~ -Pv
(11)
called the Rayleigh time. The velocity at the final stages of the collapse is given by,
/~c=
-3
p
--R--
(12)
where p~ > Pv. It is clear that the velocity at the interface during collapse can reach values much higher than those of the velocity for asymptotic growth. If the pressure is reduced but remains above the critical value obtained using expressions (8) and (9) and a given radius of the nucleus, it will not achieve the conditions for asymptotic growth and will oscillate before reaching its equilibrium. Therefore, depending on the nuclei size distribution and the local pressure, some bubbles will enter the asymptotic growth phase and cavitate and others will only oscillate with limited amplitude.
2.1.1 Newtonian fluid In the case of a Newtonian liquid, equation (7) reduces to,
3
~'rr d r - - 4 1 ~ - R
r
R
(13)
214
where ~t is the viscosity of the liquid. Numerical integrations of equation (13) have been conducted for numerous cases of time dependent reference pressures. As an example, Figure l a shows the radius of a bubble having a 10 ~tm initial radius as a function of time for a steep decrease of pressure, from 101 000 Pa to 100 Pa, constant pressure during 0.3 ms, and a steep increase thereafter. The initial slow growth of the bubble corresponds to the non cavitation region before the critical radius (= 4.2 • 10 -5 m) is reached. Comparison of the inviscid case (/~ = 0) to the water case (Ft = 10 -3 mPa. s) shows that the effect of the viscosity is relatively small and results in a slight decrease of the maximum radius. This decrease becomes larger if the viscosity increases to five times that of water. If the results are plotted in non dimensional form, using as scales the maximum radius and the Rayleigh time computed with the maximum radius, the results for the collapse can be hardly distinguished whatever the viscosity, Figure lb. The difference in the dimensionless time for the growth phase is associated to the modification of the Rayleigh time of each one of the bubbles. The straight discontinuous line has a slope corresponding to the asymptotic growth velocity, computed using expression (10) and pv = 2400 Pa. This velocity is larger than the one achieved by the bubbles at the end of the growth phase. This effect is essentially associated with the initial diameter of the nuclei; for smaller diameters, the asymptotic velocity is reached more rapidly, leading to bubbles whose final diameters will not be too dependent on the initial ones. For example, a tenfold decrease of the initial radius, from 10 to 1 ~tm, will, under the same pressure versus time conditions, lead to a maximum radius of the bubble only 60% smaller. 2.1.2 Non-Newtonian purely viscous fluid
By ignoring the probable contribution of elasticity, a power-law fluid has been adopted by Shima and Tsujino [6] as a model of carboxymethylcellulose (CMC) aqueous polymer solutions. The stress term in Equation (7) is now written in the form,
f S Zr__Zrdr = _4 ( 2 )n-1 m r
n
n-1
R
where m and n are the power law parameters and 77(R,/~, m, n) the apparent viscosity. They numerically investigate the collapse and rebound of bubbles of different diameters to a sudden increase of the pressure far from the bubble and do not consider the bubble growth phase. Their results do not allow one to show that power law behavior is responsible for the acceleration of the damping of the oscillations, as compared to those in water.
215
0.35 [- R, m m
0.30
p, Pa
~Pa.s
0.25 tI
/
0
//~'/
10 -3
0.20
\//
///" X -3
\~
101000
/
/~
o.15 0.10 0.05 ~ f /
lO0 ms
0
t 0.1
0 1.0
t 0.2
t 0.3
0.4
_R/Rma
1
b)
0.8 Pa.s 0 10-3 5x 10 -3
0.6 0.4
t-t(R
0.2
max )
0 - 12
I rR - 10
-8
-6
-4
-2
0
2
Figure 1 : a) Pressure and radius of the bubble for an inviscid fluid (~t = 0), water (~t = 10-3 Pa.s) and a more viscous fluid (r = 5 • 10-3 Pa.s) as a function of time, b) radius normalized with the maximum radius and time normalized with the Rayleigh time (expression (11)) for the maximum radius. The straight discontinuous line has a slope corresponding to the asymptotic growth velocity, expression (10).
216
Table 1 9 Fluid properties and main characteristics of the radius versus time evolution in Figure 2. rn
n
R ( t = 0.3 ms)
mPa.s n 0.895 10 50 450
0.35
1 0.8 0.65 0.4
mm
/~(t = 0.3 ms) m/s
tmax ms
R mm
A %o
0.3228 0.3203 0.3190 0.3192
1.053 1.047 1.041 1.037
0.3332 0.3330 0.3328 0.3328
0.1861 0.1848 0.1841 0.1847
4.24 6.14 4.48
R, m m
0.30 0.25 O.2O 0.15 0.10 0.05
r
0
I //jl / 0.3 0
I
0
0.1
0.2
0.3
0
0.1
0.2
n = 0.4 I 0.1
I 0.2
I 0.3 t, m s
Figure 2 : Radius of the bubble as a function of time resulting from the pressure evolution shown in Figure 1 and a nucleus radius of 10 ~tm f o r : a) N e w t o n i a n fluid /1 = 0 . 8 9 5 m P a . s ; b) p o w e r law f l u i d with m = 10 m P a . s 0.8 and n = 0.8 ; c) power law fluid with m = 450 m P a . s 0.4 and n = 0.4. No effect due to the rheothinning property of the fluid can be detected.
We have conducted the integration of the generalized Rayleigh-Plesset equation with the integral term of equation (14) and the pressure variation shown in Figure 1 for a Newtonian fluid of ~t = 0.895 mPa.s, and three power law fluids whose characteristics are given in Table 1. These characteristics are such that the evolution of the radius of the bubble with time is practically indistinguishable from the reference Newtonian situation, as shown in Figure 2 for two of the three power law fluids. Indeed, Table 1 shows that, as compared to the Newtonian case : the radius achieved at the end of the low pressure plateau (third column) differs, at the most, by 1.2 p e r c e n t ; the
217
interface velocity at the same time (fourth column) differs by a maximum of 1.5 percent; the total duration of the bubble life (fifth column), which ends when the interface velocity reaches the sound velocity in the liquid, differs by a maximum of 0.1 percent; and the mean bubble radius (sixth column) and the differences of the mean bubble radius relative to the maximum radius in the Newtonian fluid, A, differ by less than one percent. In spite of the considerable differences of the instantaneous apparent viscosity, 77, the behavior of the bubble remains the same as that of an equivalent Newtonian fluid and vice versa.
2.1.3 Non-Newtonian viscoelastic fluid The time history of spherical bubbles in non-Newtonian viscoelastic fluids has received considerable attention since the earlier works of Fogler and Goddard [7, 8] for a fluid model including stress accumulation with fading memory. Later, Ellis and Ting [9] have employed a second order fluid and Yang and Lawson [10], Chahine [11] and Ting [12] an Oldroyd fluid. The Fogler and Goddard results presented large elastic effects, while all other authors conclude that viscoelasticity has a very limited retardation effect (Ting, [12]) on bubble growth and collapse, provided the material constants are compatible with dilute polymer solutions properties. As stated in the paper by Kezios and Schowalter [13], when equation (7) "is solved for conditions appropriate for cavitation .... the dynamics are dominated by inertia, and the contribution of rheology to the integral on the right-hand side is of little consequence. This is fortunate, in that one need not be concerned with fluid viscosity, let alone elasticity. But it is also inconclusive, because it leaves open the question of interactions between bubbles and the fact that in a shear field one does not expect the bubbles to be spherical". These findings have been further substantiated theoretically by a recent and very well documented paper by Riskin [14]. By incorporated the polymerinduced stress calculated using a "yo-yo model" which accounts for the unravelling of the polymer molecules into equation (7) without the noncondensible gas contribution, the author computes the growth and collapse phase of a vapour bubble. He concludes that the growth of the bubble is not affected by the polymer, but the final stage of collapse is. He shows that, there is a total arrest of the collapse, with a bubble wall velocity reduced to nearly zero when the radius becomes about or less than 10% of the radius at the initiation of the collapse. However, under such situations, the velocity of the wall prior the sudden onset of the viscoelastic effects will be, because of expression (12), if (P~-pv) = 10 5 Pa and p = 10 3 k g / m 3, = 815 m/s. This is large enough to invalidate the assumption of incompressibility for both the liquid and the gas phases. Therefore, none of the theoretical approaches of spherical bubble deformation in purely viscous and viscoelastic fluids indicate the possibility of strong
218
changes on the bubble behavior. Thus, we next consider results concerning experimental single bubble behavior for a variety of conditions to establish under what circumstances the non-Newtonian behavior becomes significant. 3. S I N G L E B U B B L E 9 E X P E R I M E N T A L 3.1 Stagnant fluid The results of numerous experiments conducted to investigate the behavior of spark generated bubbles in an unbounded f l u i d - water, polyethylene oxide (PEO) and Guar Gum solutions - were presented in a paper by Ellis and Ting [9]. The authors showed that the polymer additives, even at concentrations as high as 1000 ppm of PEO, for which the aqueous solution display very marked viscoelastic effects, did not affect the behavior of the bubbles in any significant way, that they remain spherical during growth and collapse and that the duration of the collapse phase is equal to the Rayleigh time (equation (11)) computed using the maximum radius of the bubble.
A
1.2 1.0 0.8
- Rcma x _
o
-
A
~
o
RA
AAA
10
9
o~ A
I
~
0.6 - ~
AA 0 0
0.4
Q 0
0.2
I
9 9
A
17= Rc max / L
0~
77
o
o |zx I
0.5
I
I
I
1.0
1.5
2.0
o
I
t /"t" R
9 o 9 A
1.39 1.25 0.56 0.50
Fluid 250ppmPEO Water 250ppmPEO Water
Figure 3 : Behavior of spark generated bubbles in water and a polymer solution in the vicinity of a solid wall. The distance between the center of the initial bubble and the wall is L, RA is the distance between the position of the interface opposite to the wall and the center at the origin, Rc is the lateral extension of the bubble, t is the time and z is the Rayleigh time computed for Rcmax. From Chahine and Fruman [15].
219
These results were confirmed by Chahine and Fruman [15] using the same bubble production technique and distilled water and a 250 ppm solution of PEO (Polyox WSR 301) as testing fluids. They also show that the time and amplitude of the first and second rebounds were unaffected by the polymer additive. Very recently, Brujan et al. [16] produced laser-induced cavitation bubbles in solutions of CMC and polyacrylamide (PAM) with a concentration twenty times larger than the one used by Fruman and Chahine. Even for this very large concentration and for the very viscoelastic PAM solution, the bubble behavior is "little affected by the rheology of the fluid and the most important parameter for bubble oscillation is the infinite-shear viscosity". Since it was substantiated that a re-entering jet developed if the bubble collapsed close enough to a solid wall, and hence the viscoelastic properties of the polymer solution might affect the development of the perturbation initiating the re-entering jet, Chahine and Fruman [15] also investigated this configuration. Typical results are shown in Figure 3, and demonstrate that when the distance to the wall decreases as compared to the maximum lateral extension of the bubble, the polymer solution introduces a retardation effect over the initiation of the re-entering jet. Notice in this figure that the position of the interface is, at the end of the collapse, below the center of the bubble at origin. This investigation was, we believe, the first to demonstrate that polymer solutions seem to have a stabilizing effect on bubble dynamics, especially with regard to deviation from sphericity. Brujan et al. [16] did also experiments in the vicinity of a solid wall and confirmed these early results. Chahine and Morine [17] and Morine [18] conducted tests with spark generated bubbles confined between two parallel walls. This geometry offers a rather peculiar situation of collapse because the bubble, initially spherical, develops an hour-glass shape during the collapse phase and splits into two bubbles which collapse against each wall. In this situation the departure from sphericity occurs along the mid-plane between the two parallel walls. Figure 4 shows some of the results obtained with concentrations of 125 and 250 ppm of PEO. Clearly, the lengthening effect on the bubble life duration as a result of the presence of the walls is significantly reduced in the case of the polymer solutions. The most striking is that, in these fluids, the departure from sphericity of the bubbles is considerably delayed. Interestingly enough, results obtained with a 50% glycerin/water mixture, having a viscosity about ten times larger than water, demonstrate little change compared with those obtained in pure water, in agreement with the results obtained by integration of the Rayleigh-Plesset equation. Chahine [19] published results concerning the behavior of bubbles in the vicinity of a liquid-air interface and showed that, in this peculiar situation, the deformation of the bubble in the polymer solution is enhanced and made more non-spherical. This peculiarity can be associated to the fact that the free surface also deforms considerably during the growing phase, forming a jet
220
which ascends vertically up to distances of many times the maximum bubble diameter. Chahine [19] pointed out that, in the polymer solution, the jet is axisymmetric, smoother and thinner and its surface looks stretched at the beginning. "Later, the jet becomes distorted and loses its symmetry while in water the axial symmetry is conserved". 1.2 1.0
m
CD I CD m a x
a pOx.
m
mcx nyi_n -o-o
nl
m
9
[]
m A I k v
0.8
AOAA AolLX
_
O zx o A
0.6 - 0 ~
0
A
A
•
i X @
o 0.4
[]
D xe
[]
OA 0 A
0.2~
0
0
I 0.5
0
I 1.0
I 1.5 ]7
~N"~"N'~"~N'~NNN'~'N~
I
b
9 9 9 •
0.59 0.64 0.70 0.54
I 2.0
|
x
@
[] x EX
I 2.5
fluid water water water water +glycerin
m
o A []
Oi @
[] I
3.0 t/TR
fl
fluid
0.56 0.61 0.80
250 p p m 250 p p m 250 p p m
Figure 4 : Behavior of spark generated bubbles in water and a polymer solution in between two parallel walls. The center of the initial bubble is in the middle of the gap, the lateral extent, CD, of the bubble is plotted divided by its maximum value, t is the time and "t" is the Rayleigh time computed for CDmax. =CDmax/b. From Chahine and Morine [17] and Morine [18].
3.2
Shear
flows
A detailed investigation of the effect of a controlled shear flow on the deformation of laser-generated bubbles was conducted by Kezios [20] and Kezios and Schowalter [13]. Because of the shear, the bubble is elongated, as indicated in the insert of Figure 5, and as time proceeds it reaches a maximum deformation amplitude and then recedes during the collapse phase. In Figure 5, results obtained with polyacrylamide (PAM) and PEO solutions are shown.
221
100
-~
,~m
0 0
80 0
Fluid Water 500 ppm I000 ppm 500 ppm 1000 ppm
0
60
A
0 A
A O O
PAM PAM PEO PEO*
6o
40 A
20
O r7 0
100
R 3=ab 2
[]
0.5
I
I
I
1.0
1.5
t/'C R
- ~ ,~m
0 0
5
[] A [] 0
o
0
A =a-R
DOn
0
80
0
ml m
o
oOm
60
m
m o
AA
IIO A
0
0
o [] A A mll A
l
40
@ 20-
A
A
B o
0
A I
I
I
0.5
1.0
1.5
I
t / Z"R
Figure 5 : Mean bubble deformation amplitude (see insert) as a function of time divided by the Rayleigh time corresponding to the maximum equivalent spherical radius for water and polymer solutions and a shear rate, y, of 170 s-1. The asterisk refers to a low molecular weight PEO. From Kezios and Schowalter [ 13]. They demonstrate that the departure from sphericity is significantly reduced in polymer solutions, in particular in PAM solutions. Interestingly enough, the authors found that increasing the concentration beyond a critical value reverses the results and they speculated that this can be caused by the relative increase of the solution viscosity as compared to its elasticity. Needless to say that these authors have consistently checked that, whatever the polymer and the polymer
222
concentrations, the bubbles remain spherical during all the process of growth and collapse when the shear rate was absent. Ligneul [21] performed experiments with spark-generated bubbles immersed in the shear layer developed by a rotating cylinder, with velocities between 1 and 5 m/s, inside a reservoir of much larger dimensions. By comparing the behavior in water and solutions of PEO with 50 and 250 ppm concentration, he concludes that the influence of the polymer solution is to maintain sphericity during bubble collapse (but not during bubble growth). 4. C A V I T A T I O N IN A M O V I N G F L U I D
4.1 Jet cavitation The cavitation number is defined by, a - Pr - Pv 1
(15)
with P r the pressure in the reservoir where the jet is discharging and V j the bulk velocity of the jet. Depending on the way the experiments are performed, either by decreasing the pressure from a non r situation until cavitation is reached or by increasing the pressure from a r situation until the non cavitating conditions are achieved, the inception, a i, or desinent, a d , cavitation numbers are obtained respectively. Hoyt [22-25] conducted experiments with a nozzle situated 60 r below the free surface of an open reservoir. Preliminary test results [22] showed a sharp decrease of the incipient cavitation number when the concentration of the homogeneous polymer (PEO) solutions was increased up to 10 p p m ; afterwards the reduction was not as rapid. These results were later confirmed and extended by modifying the upstream turbulence level [25]. All the data is plotted in Figure 6. For these concentrations, the fluid behaves as Newtonian with an increase of the viscosity of about 10 and 25%, as compared to the solvent, for respectively 10 and 100 ppm. The surface tension of the polymer solutions was reduced very sharply, by as much as 10%, when the concentration increased from zero to about 40 ppm and reached a plateau afterwards. The tensile strength, the nuclei concentration and the air content, at saturation or very slightly below, are practically unaffected in the polymer solutions for the concentrations mentioned earlier. Because the surface tension is reduced we may expect, based on equations (8) and (9), an increase of the critical pressure and an advanced cavitation. Since all other properties are unchanged and the cavitation inception is delayed, Hoyt concludes that the causes are of an hydrodynamic origin.
223
~f
Inception cavitation index
0.4
Turb. Turb. generator level, %
) 0.3
No <2 Yes 2 Yes 2.2 Yes 2.6 Hoyt [21]
nX 0.2
0.1
Polymer concentration, ppm 0
I
I
I
I
I
I
I
I
10
20
30
40
50
60
70
80
Figure 6 : Incipient cavitation number as a function of polymer concentration for a submerged jet issued from a 6.35 mm diameter nozzle at four conditions of the upstream turbulence level. Reynolds number were in the range of 2 • 105. From Hoyt [25].
Baker et al. [26] obtained the desinent cavitation numbers, o"d, instead of o"i as in the case of Hoyt, for a constant concentration, 100 ppm, of the same polymer, a nozzle diameter of 50.8 ram, and a Reynolds of 6 x 105, larger than in the case of Hoyt's tests, for varying concentration of air in the circulating fluids. When the concentration was below saturation (<4 ppm) the polymer did not introduce any appreciable change, but, for supersaturated conditions (>4 ppm) cavitation desinence was delayed. The considerable difference in behavior between Baker's and Hoyt's results can be due to the much larger size of the nozzle, by near a factor of ten, the reduction of the jet velocity, by a factor of about three, and the choice of the desinent instead of the incipient cavitation number as a reference. Indeed, even in the case of pure water, increase of the total gas content causes a strong increase of c7d, in particular for supersaturated conditions. Hoyt and Taylor [27] conducted visualizations of cavitating events using a nozzle of only 2.92 mm diameter for saturated and undersaturated conditions. The photographs clearly demonstrated that, for cavitation numbers comparable or lower than those for which the water jets show extensive cavitation in the shear layer, the drag-reducing polymer solutions, with a concentration of only 25 ppm, drastically reduced the presence of individual cavities. Moreover, the bubbles are larger and their surface is smoother than for the conditions of limited cavitation in water jets.
224
1.4
- Desinent cavitation index
1.2 1.0
AA
e#
-
0.8 0.6
o
4
o
0 0
o ~
Jet origine
Water Water 100 ppm PEO 100 ppm PEO
Nozzle Orifice Nozzle Orifice
0
0.4 0.2
Fluid
Total gas content, ppm
0 0
I 2
I 4
I 6
I 8
I 10
I 12
I 14
Figure 7 9 Desinent cavitation number as a function of total gas content for submerged jets of water and an aqueous solution of 100 ppm of poly(ethylene oxide) issued from a 50.8 mm diameter nozzle and orifice at Reynolds number of 6 • 105. From Baker et al. [26].
Arndt [28] advanced an explanation of the Baker et al. [26] and Hoyt [22-25] results based on the difference in the hydrodynamic behavior of jets depending on the values of the Reynolds numbers. His analysis is based on the fact that, for equal Reynolds number, the order of magnitude of the strain rate in the contraction of the small nozzles, will be larger than for the large nozzle. Thus, this effect, coupled to the transitional state of the jets in the Hoyt and Taylor [27] experiments, is expected to lead to a significant viscoelastic effect on the pressure field and to justify the inhibition of cavitation. However, no explanation is given on why cavitation is delayed in the Baker et al. [26] experiments in the supersaturated regime. Already in 1974, Ting [29] correlated jet cavitation inhibition data by Hoyt [23] with the drag-reducing properties of the polymer solutions. He showed that the percent reduction of incipient cavitation and of drag reduction can be expressed in the same way; as a function of the polymer concentration and an intrinsic concentration, which is a measure of how rapidly cavitation surpression (drag reduction) increases with increasing concentration. The characteristic numerical values obtained in both situations are in very good agreement. In conclusion, jet experiments do not show conclusively that cavitation is inhibited under all flow conditions. It appears rather clearly, however, that modification of the flow structure is the main effect with little, if any, contribution from the fluid properties on inception. When cavitation is well develo-
225
ped, the cavities are larger than those existing in the solvent and their surface is smoother. These modifications can be directly related to experiments showing that polymer solutions delay the departure from sphericity of single bubbles and, as a consequence, their eventual break-up due to surface deformation and instability.
4.2 C a v i t a t i o n near blunt bodies
The effects of polymer solutions on cavitation inception on blunt bodies (ogives) was first investigated by Hoyt [30]. For axisymmetric bodies, many other papers followed; Ellis et al. [31] and Baker et al. [32], in homogeneous solutions, and van der Meulen [33, 34] and Gates and Acosta [35], for injected polymer solutions. Reitzer et al. [36] conducted experiments using a circular cylinder. As an example, Figure 8 reproduces the inception cavitation numbers obtained by Gates and Acosta [35] for a hemispherical nosed body as a function of the Reynolds number for various values of the non dimensional injection rate of polymer fluid, defined as,
G-
cQ rc D U Sts
(16)
where c is the concentration, Q the volume flow rate of the ejected solution, D the diameter of the hemisphere, U the free stream velocity and 8ts the boundary layer displacement thickness calculated at specified positions along the body surface. There is a marked reduction over the whole range of Reynolds numbers, but the relative effect seems to decrease when the Reynolds number increases. Moreover, when the concentration of the ejected fluid is increased, the polymer effects increase also but seem to reach a saturation beyond which further increase causes a reduction of the polymer efficiency. The saturation effect is exemplified in Figure 9 where the cavitation inception number has been plotted for a constant Reynolds number of 7.5 x 105 as a function of the polymer solution injection rate (up) and the dimensionless polymer solution injection rate (down). It is interesting to note that what seems to be really important is the amount of polymer present in the reference thickness of the boundary layer. Gates and Acosta [35] have plotted the maximum percent reduction in inception on the hemisphere nose as a function of the Reynolds number for their results and those previously published (Figure 10). It is very interesting to see that the effects are qualitatively comparable in spite of the differences in the way the polymer molecules are brought to the body wall (carried by the main flow or ejected at the nose). The differences, between the results of van
226
der Meulen [34] and Gates and Acosta [35], may be due to many factors and, in particular, the level of turbulence and nuclei concentration of the cavitation tunnels employed in their investigations.
0.7 - cyi ~
Hemisphere body
9
0.6
:
iBm
i ~
0.5
.
. o/.--.
0.4
0.7
o [] ii Q
A A
10-Sx Re I 8 9
K/ 0.3
/
106x G
I
I
I
I
I
3
4
5
6
7
t
I
0 0.35 1.75 2.3 7 9 14.6 23.6
(3"i NSRDC body 106• G
0.6 o O
0.5
0 0.35
(9 O
0.4
| o
0.3 3
I 3.1
I 3.2
I 3.3
3.4
I 3.5
I 3.6
I 3.7
1 0 - S x Re I I I 3.8 3.9 4
Figure 8 9 Inception cavitation numbers for a hemispherical and a NSRDC body as a function of the Reynolds number for various values of the nondimensional injection rate of polymer fluid. From Gates and Acosta [35].
227
0.65 F O'i D O
0.60
50 ppm 500 ppm
O
0.55 -0 0.50
-
|
0.45 O I
0.40
I
Q, 10-6 m3/s I
0
I
I
I
I
I
I
5
I
I
I
I
I
15
10
0.65 ~- O"i
50 ppm 500 ppm
O
0.60
OO O
0.55 0.50 0.45 |
0.40 0
I 1
I 2
I 3
I 4
I 5
I 6
I 7
106•
G
I 8
Figure 9 9 Cavitation inception number for a constant Reynolds number of 7.5• as a function of the polymer solution injection rate (up) and the dimensionless polymer solution injection rate (down). From Gates and Acosta
[35]. In an attempt to interpret these results, van der Meulen [37] and Gates and Acosta [35] visualized the boundary layer along the body surface in the region of separation. They both come to the same conclusion: the polymer additives remove "the laminar separation by stimulation of transition (Gates and Acosta [35])", causing "transition to turbulence at much lower Reynolds numbers than the pure solvent (van der Meulen [37])". This conclusion is further substantiated by the results of tests performed by these authors with a Schiebe body, which, theoretically, should not exhibit laminar boundary layer separation. They showed that, indeed, the polymer solution ejection has no effect whatsoever on either the type of cavitation or the inception index. Moreover, as signalled by Gates and Acosta [35], tripping the boundary layer transition on the hemispherical headform has been demonstrated to be more
228
effective than the polymer solutions in delaying cavitation occurrence. It seems therefore that, for these tests and for the Reynolds numbers employed, the polymers act by modifying the behavior of the boundary layer and have little, if any, effect on the individual bubbles at inception. We will see below that even if the cavitation is very much developed, these boundary layer effects are still of importance.
60 - A o ' i
,o~
O"i 50
-
A
40
o [] A
@ @
O0
30 A
0 | o
0
@
o
A
0 20-
0
0 0
0
0
@
@ 0
@
[] []
10-
[]
0 0
1
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
conc., Reference ppm 500 van der Meulen [33]
2o}
50 20 500
2o } 80
Ellis et al. [36] H o l l et al.
Gates & Acosta [34] Baker et al. [31]
10-SRe I I 9
10
Figure 10 9 Composite plot of maximum percent reduction in inception on the hemisphere nose as a function of the Reynolds number. From Gates and Acosta
[35]. Brennen [38] investigated the effects of homogeneous dilute polymer solutions on the characteristics and appearance of the interface of well developed cavities produced behind a cylinder and spheres operating at Reynolds numbers, relative to the diameter of the obstacles, comprised between 1 0 4 and 1.2 • 105. The author showed that the polymers caused a wavy instability of the wetted surface flow around the headforms at the initiation of the cavity. This instability can be related to the effect of the polymers on the transition and separation mentioned above. Bazin et al. [39] presented one of the earlier results on the effect of the injection of rather concentrated (5000 and 10000 ppm) solutions of PEO WSR 301 on the surface of a cylinder downstream of the stagnation point. They found, for unspecified Reynolds numbers, that the injection of the polymers inhibits the development of cavitation and that the noise level at inception conditions is significantly reduced by the polymer additive, especially in the high frequency band (> 25 kHz).
229
14
l O - 3 x Pac ' P a
12 10
c
= 0 ppm
c = 3 ppm m
-
(
zone
zone
one V
-
}~
o~ 0.5
10-SxRe t
0.6
0.7 I
15
0.8 I
0.9
1.0
1.1
1.2
1.3
1.4
I
I
I
I
I
I
I
I
10 9
8
7
6
5
4
3
2
(7
Figure 11 : Acoustic pressure as a function of the cavitation number and of the Reynolds number for water and a 3 ppm PEO homogeneous polymer solution flowing around a circular cylinder. From Reitzer et al. [36].
Reitzer et al. [36] have also investigated the flow around a cylinder in an open loop cavitation tunnel. A 1000 ppm PEO solution was ejected upstream of the test section with a flow rate such that the concentration of the homogeneous solution in the test section reached an homogeneous concentration of 3 ppm after mixing. Their main results are summarized in Figure 11, where the acoustic pressure is plotted as a function of the cavitation and Reynolds numbers for water and for the homogeneous polymer solution. For water, there is a region, zone I, where the flow is non cavitating, up to a Reynolds number, defined with the free stream velocity and the diameter of the cylinder, of 85 000, for which cavitation is initiated and isolated bubbles can be seen. Beyond this Reynolds number, cavitation develops and the acoustic pressure increases up to a maximum (zone II), after which it decreases over a limited range of Reynolds (zone III). After a relative minimum, the noise increases very sharply, reaches another maximum much larger than the previous one (zone IV) and then decreases as sharply as it had increased (zone V). By comparison, the polymer solution completely inhibits zone II, which is replaced by the continuation of zone I, and zone III, where the noise was decreasing in the case of water, is replaced by an increasing region that fits perfectly zone IV. The polymer has no effect whatsoever on zones IV and V where cavitation occurs respectively in the form of alternating bursts and an elongated vapour cavity. From these and other data, the authors conclude, as in other previous works, that the cavitation retardation associated with the
230
presence of the polymer molecules is more likely due to the modification of the flow field around the cylinder than an effect of the macromolecules on the dynamics of individual bubbles.
0.65 -
Inception parameter
0.60
9
0.55 -O
00
oemo o o o
O
PEO-FRA
9 A
[]
9
v~mvvV
9 Am~ mm
0.50
9
t o~ .. 9 9
9
[]
~pr Ipr V
A
m m [] A Ak A A y y y v
0.45
A
V
0 ppm 100 ppm 250 ppm 500 ppm
A
V"
9
IPr
0.40 1 0 - S x Re 0.35 0.8 0.65 -
I 1.0
I 1.2
I 1.4
I 1.6
I 1.8
I 2.0
I 2.2
Inception parameter
0.60 -
9
O0 9
i~ 9 9
9
-o e ~ 9 emo oo 9
mum
0.55
PAM-273
nnn
0 []
[][] mmmam m 9 0.50
! 2.4
At
A
mum
T
~?V V 0.45
_
k
A
0 ppm 100 ppm 300 ppm 500 ppm
V
0.40
10-5• Re
0.35 0.8
I
I
I
I
I
I
I
I
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
Figure 12 9 Inception cavitation numbers as a function of the Reynolds numbers, computed using the local azimuthal velocity and the diameter of the protrusion, for pure water and three concentrations of PEO and PAM. From Ting [40].
The experiments with cylinders can be compared to those conducted by Ting [40] in a specially designed unit allowing him to perform experiments using very small amounts of test fluids. It consisted of a circular plate, on the surface of which were two protrusions having a diameter and height of 2.9 mm, rotating inside a casing. The protrusions were observed through a transparent wall in order to determine, visually, the initiation and the extent of the
231
cavitation. The inception cavitation numbers as a function of the Reynolds numbers, computed using the local azimuthal velocity and the diameter of the protrusion, for pure water and three concentrations of PEO and PAM, are plotted in Figure 12. For the largest concentration, PAM seems to be more effective over a larger range of Reynolds numbers as a result, probably, of its larger capacity to sustain high shear rates with moderate degradation. Ting also signalled that the appearance of the polymer cavities was more transparent than in water and showed a regular and smooth wavy pattern, as beautifully demonstrated by Brennen's [38] visualizations. 4.3 Vortex cavitation It is well known that viscoelastic fluids cause very strong effects in rotating flows. In particular, while in a Newtonian (or even inviscid) fluid the rotation of a rod makes that the surface of the liquid is depressed near the rod, in a polymeric fluid the liquid will climb the rod. This is the famous Weissenberg effect. The non-Newtonian liquid develops a normal stress contribution which acts in such a way that the depression induced by the rotation of the liquid is overcome, and even exceeded. In fact, the so called "vortex inhibition phenomena" has been used as a method to estimate quantitatively and qualitatively the viscoelastic properties of liquids (Gordon and Balakrishnan [41]). It seems therefore natural to assume that the contribution of normal stresses may have a significant effect on the cavitation which may occur in confined (vortex chambers) or unconfined (tip vortices) situations. For an axisymmetric line vortex, the radial component of the momentum equation, assuming that the radial component of the velocity is zero, is given by, =
Or
R
OTrr
Trr -- TOO
Or
r
(17)
where V o is the tangential velocity, and Tii the extra stresses (i = r, 0). The pressure on the axis of the vortex is obtained by integrating equation (17), from infinity to r = 0. It is easy to see that the contribution of the rotation, whatever the velocity distribution, is to decrease the pressure on the vortex axis. In a Newtonian fluid the extra stress terms cancel. In non-Newtonian viscoelastic fluids, the experience shows, as stated above, that these terms contribute to diminish the centrifugal effect. Inge and Bark [42] conducted experiments with a specially designed elliptical wing having a maximum chord of 0.16 m and a half span of 0.238 m. They reported results obtained by ejecting concentrated polymer solution into the water stream through a multiperforated injector situated one meter upstream of the foil and for homogeneous polymer solutions with concentrations
232
between 0.01 and 12 ppm of PEO Union Carbide WSR-301. In Figure 13, the values of a modified cavitation number, taking due account of the different values of the incidence angle and of the free stream velocity, are plotted as a function of the polymer concentration in the homogeneous solutions for three conditions of cavitation development. The results show very clearly that the incipient cavitation occurrence is significantly delayed for concentrations larger than 1 ppm. The same is apparent for the other conditions of cavitation although the changes are proportionally smaller. Interestingly enough, the permanent cavity and the attached cavity situations cannot be distinguished for the largest concentration. The experiments with the polymer ejection gave qualitatively similar results.
o"n O o
O Incipient cavitation [] Appearance o f a p e r m a n e n t cavity ZX Cavity reaches the w i n g
~
0
o
[]
El
....
c, ppm 0
I 0.01
I 0.1
I 1
]
,,,
10
Figure 13 9 Values of a modified cavitation number, taking into account the effect of the different values of the incidence angle and of the free stream velocity, as a function of the polymer concentration for three conditions of cavitation development. From Inge and Bark [42]. Fruman [43] investigated the behavior of the tip vortex of a NACA 16020 cross section elliptical foil in water and in water with polymer solutions ejected from a 0.5 mm diameter tube at a distance of 20 mm upstream of the tip. Prior to the polymer tests, experiments conducted by ejecting a 50% water + glycerin mixture, with a viscosity of 10-2 Pa.s, did not show any significant contribution of the augmentation of the viscosity of an otherwise Newtonian fluid. Ejecting a 500 ppm PEO solution when a quite large vapour tube occupies the vortex core considerably modifies the appearance of the cavitation. At an ejection velocity of about half of the free stream the
233
continuous very long cavity is reduced to a very short cavity of about half of a maximum chord length and only scattered isolated bubbles are carried downstream. By doubling the ejection velocity these entrained bubbles are eliminated and only the shortened cavity remains. In terms of the incipient cavitation number, Figure 14 shows inception results obtained with an ejection of a 250 ppm PEO WSR 301 solution at an ejection velocity slightly larger than the free stream.
2.5
cYi
no ejection polymer
2.0
.....
1.5
1.0 o
0.5 5
I
I
I
7.5
10
12.5
Figure 14 9 Inception cavitation numbers as a function of the wing incidence angle during the ejection of a 250 ppm PEO WSR 301 solution from a 0.5 mm diameter tube situated 20 mm upstream of the wing tip. From Frurnan [43].
Thus, it appears that the polymer solutions have a strong effect on the tip vortex cavitation occurrence and development. However, it is not clear what is the mechanism responsible for such changes because, in these experiments, neither the tangential velocities of the vortex nor the lift of the wing have been measured. The latter is a very important factor since the tip vortex intensity is directly related to the mid-span circulation of the foil and, as shown by Wu [44] and Kowalski [45], lifting surfaces immersed in flowing homogeneous polymer solutions display a reduction of lift. In order to elucidate these aspects, experiments were conducted by ejecting the polymer solution from the tip of the wing through a small diameter orifice (Fruman et al. [46]), measuring both the tangential velocity distributions and the incipient cavitation numbers. The results show a significant reduction of the incipient cavitation number and, more interesting, a decrease of the
234
maximum tangential velocity and of the slope of the velocities in the core region when the polymer solution was ejected, Figure 15. Measurements of the lift did not show any change during polymer ejection (Aflalo [47], Fruman and Aflalo [48]) in agreement with the fact that the tangential velocities in the potential region were unchanged.
V, m/s qi, m 3/s
2
0 0 . 7 x 10 -6 2.2 x 10 -6
1
0
-1
-
U o o = 5 m/s
~=10 ~ ~2
-12 1.5
I
I
I
I
I
I
I
I
I
I
-10
-8
-6
-4
-2
2
4
6
8
10
V, m/s
1.0 0.5 !
--0.5
-1.0
U -
-1.5
-12
=5m/s 0~=5 ~
I
I
I
I
I
-10
-8
--6
-4
-2
0
I
I
I
I
I
2
4
6
8
10
Figure 15 : Vertical (tangential) velocities as a function of distance to the vortex axis for different ejection rates of a 1000 ppm PEO WSR 301 solution. Measurements conducted at 0.20 m (five maximum chords) downstream the tip. From Aflalo [47] and Fruman and Aflalo [48].
235
(P ~ - P o)P / (P ~ - P o)w
1.0
u
[] []
0.8
5 5.5 5 5.5
[] []
[]
0
0
0.6
a;
, m/s
o
5 5 10 10
[]
0.4
0
0
0.2 107x q-/ 0 0
I 0.5
I 1.0
I 1.5
I
2.0
Figure 16 9 Ratio of the pressure difference between infinity and the vortex axis for the ejection of the polymer solution and the pure water as a function of the reduced injection rate. 1000 ppm PEO solutions. From Aflalo [47]. Using these velocity distributions it was possible to integrate between a position far from the vortex, where the pressure is p=, to a position on the vortex axis, where the pressure is P0, the Vff/r term of equation (17) in the case of pure water flow, index w, and polymer ejection, index p. The ratio of the resulting pressure difference contributions are plotted in Figure 16 as a function of the non dimensional ejection flow rate, m
qi =
qi c
U~ b Cma x
The results show that the polymer ejection causes a reduction of the pressure difference as compared to the pure water situation. This reduction is largely sufficient to justify the delay on the cavitation occurrence even if the contribution of the normal stresses in equation (17) is ignored. An investigation on the effect of homogeneous polymer solution on tip vortex cavitation was conducted by Fruman and Aflalo [48]. The homogeneous solution, of about 10 ppm, was obtained by injecting, upstream of the water tunnel recirculating pump, a concentrated polymer (1500 ppm).
236
, Cavitationnumber at time t umber at t = 0
1.00 (
0.95
0.90
0.85
-
Time after beginning mixing, s 0.80 0
i
i
i
1
I
200
400
600
800
1000
At time t=O
Attime t= lOOs
At fin~ t = 9OO s
Figure 17 : Top : evolution of the cavitation number as a function of the time elapsed after the beginning of the mixing of a 1500 ppm PEO solution into the recirculating water of the cavitation tunnel. Bottom : photographs showing the elliptical foil (left) and the tip vortex cavity initiated at the tip and extending downstream for pure water (t =0) and with increasing homogeneous polymer concentrations (t =100 and 900 s). From Fruman and Aflalo [48].
237
The photographs of Figure 17 show that, in spite of the fact that the modification of the flow rate and the reference pressure caused a significant reduction of the cavitation number during the process of mixing, for an initially well developed cavity the build-up of the polymer concentration in the cavitation tunnel recirculating water was responsible for a significant decrease of the cavity length. It is interesting to note also that the polymer seems to be more effective at distances from the tip of several maximum chords and do not causes the cavity to detach from the foil.
3.0
8
2.5
iO
--
0
water polymer
@0
w~
2.0-
(9 []
0
Om
o
0
[]
1.5-
u
0 0
o
0~
o~ o
[]
1.0-
O~
~ 0.5 0.1
c
Z
I
I
I
I
I
t
I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 18 9 Desinent cavitation numbers as a function of the lift coefficient for the NACA 16020 elliptical foil in an homogeneous polymer solution. Open and closed symbols refer to two free stream velocities, 8 and 8.4 m/s. From Fruman and Aflalo [48]. The measurements of the lift of the foil showed that it was markedly reduced in the homogeneous polymer solution. The tangential velocities were also considerably reduced in both the core and the potential region. The latter is in direct agreement with the reduction of the lift and hence of the mid-span circulation. Figure 18 shows the desinent cavitation numbers as a function of the measured lift coefficients in water and homogeneous polymer solution. The effect of the polymer solution is to degrade the performance of the foil since, for the same lift coefficient, the critical cavitation number is larger. The effect of the polymer solution on the lift is largely more important than that on the inhibition of cavitation inception. Since the boundary layer of the NACA 16020 section used in these tests is very sensitive to laminar-to-turbulent layer transition (Pichon et al. [49]) it may well be that the extremely large decrease
238
of the lift coefficient is due, in part, to the already mentioned destabilizing effect of the homogeneous polymer solutions. The effect of doubling the size of the foils and increasing the free stream velocity, increasing thus the Reynolds number, on the effectiveness of the polymer ejection at the tip of the foil was investigated recently by Fruman et al. [50] with results comparable to those already reported, provided the ejection flow rate was scaled with the Reynolds number. A very detailed survey of the tangential and axial velocity distributions at different positions along the vortex path was also conducted in the very near region (less than a maximum chord) downstream of the tip. The original results, showing a decrease of the maximum tangential velocity and of the slope of the solid body rotation core, were confirmed. Again, the increase of the pressure on the vortex axis due to the velocity modification gives a decrease of the desinent cavitation number, Figure 19.
2.0
w Cp
llum
I
/
"~-- crd withoutejection
J
~
1.5
L
0,~ .. ~
_
m
I0 r
e
no ejection
cre with polymer ejection
1.0
III
I
0.5 0
I 0.2
I
I 0.4
I
I 0.6
I
I 0.8
X /C max I I
1.0
Figure 19 : Pressure coefficient on the vortex axis, computed using expression (17) ignoring the stress terms, as a function of the distance to the tip. The values of the desinent cavitation numbers for the conditions of pure water and polymer ejection flows are also indicated. The minimum of the pressure coefficients are very close to the desinent cavitation numbers. From Fruman et al. [50]. The ejection of polymer solutions at the tip of the foils to delay cavitation has been applied with success by Chahine et al. [51 ] to a five blade screw propeller of about 29 cm diameter. Because of the relatively low aspect ratio of the blades, the position of the tip vortex detachment was not as easy to establish as
239
in the case of the elliptical foils. Therefore, a preliminary investigation was conducted in order to determine the positions of ejection ports allowing the ejected fluid to be carried out into the tip vortex. Once these positions were chosen, experiments were conducted to determine the effect of the polymer concentration and solution flow rate. As an example, Figure 20 shows the cavitation number for inception as a function of the polymer concentration for the pure water conditions, for water/glycerin injection and for polymer injection. The most important result is that, depending on the position of the ejection ports, the effect of polymer injection can be unnoticeable or achieve a decrease of the incipient cavitation number of up to 35%.
17 - C a v i t a t i o n n u m b e r
water + glycerin injection blade #5, no injection blade #1 }injection blade #3
16 15 14 13 12
~ ~ o l y m e r 10
concentration, ppm
I
I
I
I
I
I
I
1000
2000
3000
4000
5000
6000
7000
Figure 20 9 Inception cavitation number as a function of the polymer concentration for the pure water conditions, for water/glycerin injection and for polymer injection. From Chahine et al. [51 ]. Inge [52] investigated the contribution of the stresses due to an Oldroyd Bfluid to the pressure within a weak vortex. The numerical computations showed that the pressure on the vortex axis was increased with only a small influence on the velocity field. This finding is not confirmed by experiments, which show that, whatever the conditions - polymer ejection or homogeneous solutions -, the azimuthal velocity profiles are modified enough to give an increase of the pressure on the vortex axis. Hoyt [53] investigated the effect of homogeneous polymer solutions of PEO on the onset of cavitation in a vortex device where the liquid was injected tangentially from a single port and evacuated through an axial pipe. The results are summarized in Table 2. Incipient cavitation numbers are defined as
240
the difference between the discharge pressure and the vapour pressure divided by the pressure difference across the device, and the air content is non dimensionalized by the value at saturation. There is a large inhibition effect for the PEO solutions, even at very low concentrations. However, a non drag reducing polymer, Carbopol, does not display an inhibition effect. Table 2 : Incipient cavitation number and air content for tests in a vortex chamber. From Hoyt [53]. Test liquid Dearated water Tap water PEO, 8.2 ppm PEO, 12 ppm PEO, 16 ppm PEO, 17.5 ppm Carbopol, 20 ppm
Incipient Cavitation Number 0.253 1.480 1.330 0.547 0.513 0.100 0.107 1.300
Air content/Air content at saturation 0.284 0.701 0.755 0.709 0.709 0.785
Bismuth [54] conducted tests in a long vortex chamber where the fluid was introduced tangentially at one end through eight rectangular slits and evacuated axially at the other end. The reference pressure was measured in a large reservoir situated downstream of the vortex chamber and used to determine the incipient cavitation number using, instead of the pressure drop across the chamber as in Hoyt [53], the kinematic head at the exit section. For a 10 ppm concentration of PEO, the cavitation onset was advanced, as compared to the pure water results, Figure 21, in complete disagreement with the earlier results by Hoyt [53]. It has to be pointed out that, because of the different definition of the cavitation number, direct comparison of these results is difficult. However, if it is accepted that the drag reducing properties of the polymer solution should increase, at equal pressure drop, the flow rate, the results by Hoyt [53] should correspond to much larger inhibition effects. On the other hand, velocity measurements made by Bismuth [54] did show that the tangential component of the velocity increased, as compared to pure water, when moving from the wall towards the center of the vortex. This increase was large enough to justify, by integration of the simplified radial momentum equation (equation (17) without the normal stress terms), the enhanced cavitation characteristic of this flow. Moreover, since the flow rate determines the ejection tangential velocity, and hence the circulation, Figure 21 can be compared to Figure 18, where the lift coefficient is also proportional to the tip
241
vortex circulation. In both cases there is augmentation of the incipient cavitation numbers as compared to pure water.
20
[]
[]
19
18
17
-
16
-
EO]
f
I
I
IQ' 10-3m3/~
15 1.1
1.0
1.2
1.3
1.4
I
I
I
I
11
12
13
14
Ejection tangential velocity, m/s
Figure 21 9 Inception cavitation number in the vortex tube as a function of the flow rate for water and a 10 ppm homogeneous polymer (PEO) solution. From Bismuth [54].
Bismuth also performed experiments by ejecting, through an orifice of 1 mm diameter situated at the end opposite to the evacuation, semi dilute solutions of PEO. In Figure 22, the ratio between the incipient cavitation number with polymer ejection and the one for pure water are plotted as a function of the ratio of flow rates, ejection/total, for different values of the reference pressure in the downstream reservoir. As shown, the polymer ejection significantly inhibits cavitation onset. This figure is very much analogous to Figure 16 and leads us to think that polymer ejection in the center of a confined vortex or of a tip vortex has the same qualitative effect. In well developed cavitation conditions, the vapour tube on the chamber axis is made to disappear over increased distances when the rates of ejection of the polymer solution increase.
242
~Rp CrRw PR
1.00
0.86 0.98 1.17
0.95 _
[]
0.90
1.23
0
1.35
o 0.85 []
0.80
qi /Q x103 0.75 0
I
I
I
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Figure 22 9 Ratio between the incipient cavitation number with polymer ejection and the one for pure water as a function of the ratio of flow rates, ejection/total, for different values of the reference pressure in the downstream reservoir. From Bismuth [54].
4.4 Cavitation in confined spaces The flow occurring between a moving and a fixed wall separated by micron size gaps is of the lubrication type and is characterized by very large pressure gradients in the flow direction and, within the usual assumptions, nearly constant pressures in the direction normal to the flow. Moreover, because the minimum pressure can be made equal or less than the vapor pressure, cavitation can locally occur. It can be theorized that the elongational flow prevailing in the vicinity of the minimum gap will promote the occurrence of a viscoelastic contribution, if polymer solutions are employed, and modify the conditions for cavitation onset and development. Ouibrahim et al. [55] have conducted experiments in a micron size space, with a minimum gap, e, comprised between 5 and 20 ~tm, confined by a rotating drum of radius, R, equal to 7 cm, and a fixed flat wall with a Newtonian fluid (water) and a nonNewtonian viscoelastic fluid (600 ppm aqueous solution of PEO WSR 301). In this peculiar situation, the product of (e/R)Re, where Re is the Reynolds number computed with the drum tangential velocity, mR, and e, are much smaller than unity as lubrication theory requires. The cavitation number, is defined as,
Prey -- Pv or- I -~- P ( ( o R ) 2
(18)
243
where Prey is a reference pressure. In the case of water at inception conditions, intermittent cavitation spots, scattered along a very thin band parallel to the cylinder axis, appear downstream of the minimum gap. If the cavitation number is decreased, the cavitation spots increase in number, become less scattered, adopt the shape of an arrowhead with the tip facing the upstream of the rotating cylinder and show bubbles escaping from the trailing edge. When the cavitation number is further reduced, a string of nearly regular cavities is formed. They have characteristic dimensions which are of the order of a few millimetres, separated by thin lateral liquid films. m
O' i
(:rip= 2514 Re -1.88 lO
•iw =
f
O o
7287 Re
-1.84
polymer water Re
5
I llll 5 6 789
10
I 2
I 3
456789
2
10
I 2
I 3
Figure 23 9 Incipient cavitation numbers as a function of the Reynolds number for water and polymer solution and e - 10 ~tm. From Ouibrahim et al. [55]. In the case of the polymer solution and of incipient conditions, intermittent tiny cavitation spots in the shape of rod-like cells scattered along a very thin band parallel to the cylinder axis, appear downstream of the position of the minimum gap. When the reference pressure is decreased below the one for cavitation inception, a nearly continuous vapour cavity, characterized by a practically straight leading edge and a more or less undulating trailing edge, develops parallel to the axis of the rotating cylinder. This continuous cavity results from the lateral coalescence of the arrowhead cavities, present only for a very small range of the reference pressure. The difference in morphology is not a consequence of the modification of the shear viscosity, since tests conducted with a water/glycerine solution, having a viscosity of 11 mPa.s, much larger than the one for the polymer solution, 2.05 mPa.s, shows
244
arrowhead cavities analogous to the ones observed in the case of water. When the cavitation number decreases, the trailing edge waviness increases while oscillations occur in the flow and axial directions. For extreme conditions the movement of the cavity trailing edge becomes chaotic. In Figure 23, the inception cavitation numbers as a function of the Reynolds number (computed using the solvent viscosity and the minimum gap) have been plotted in logarithmic coordinates for e = 10 ~tm. As shown, the initiation of the cavitation is delayed in the case of the polymer solution but the slope remains the same as that in the case of water. It can be hypothesized that the inhibition effect is associated with the elastic contribution of the normal stresses, which has the net effect of reducing the incipient cavitation number, that is to say, to increase the pressure over the Newtonian contribution. The added pressure, APe, can be estimated thus from the difference of incipient cavitation numbers at an equal Reynolds number, by, 1
APe -
2 t%
fie,
9 8 7
tYiw l.
-
(19)
APe, Pa
Pa.s ~
,-
-, --0.8236
3
-
6 5 2
4 3
2
Ou/Ox, s -1 10 10 3
I
i
I
I
i
i I tt
2
3
4
5
6789
:
105 2
10 4
Figure 24 9 Estimated elastic contribution to the pressure and elongational viscosity as a function of the estimated elongational strain rate. From Ouibrahim et al. [55].
245
In Figure 24, the values of APe have been plotted, together with an estimate of the elongational viscosity defined as, ~e = ~
APe
c)u "/ Ox
(20)
where c)u / Ox (=70 co) is the maximum elongational strain rate, as a function of the latter. The elastic pressure contribution increases with the strain rate increasing while the elongational viscosity decreases. The latter is up to four orders of magnitude larger than the shear viscosity (a Trouton ratio as high as 104). These Trouton ratios and elongational viscosities are comparable to the ones obtained for a variety of polymer solutions displaying an elastic behavior. It seems that the effect of the polymer solution in the present situation is due to the development of normal stresses in the elongational flow in the space confined between the moving and the fixed wall. 5. C A V I T A T I O N
EROSION
The inhibition or enhancement of material erosion caused by cavitation in non-Newtonian liquids has been a subject of much conflict and debate. This is not surprising since, in the case of Newtonian fluids, the mechanism of cavitation erosion is not well understood either in stagnant or flowing liquids. Indeed, erosion seems to result from the combined action of the pressure wave, generated during the near spherical phase of collapse of the bubbles, and the reentering jet generated during the terminal, highly non spherical, phase of collapse. Moreover, in flowing liquids, the intensity of erosion (rate of material removal) increases very rapidly when the free stream velocity increases. Finally, the interaction between the mechanical and crystalline properties of the eroded material and the hydrodynamics of the bubble can be quite strong. Tests in stagnant fluids have been conducted by inducing ultrasonic cavitation on metallic samples and determining the weight loss as a function of time. Ashworth and Procter [56], with copper test specimens placed at 1.3 mm below the tip of an ultrasonic probe (horn) obtained, as compared to the results in pure water, a decrease of the incubation time and an increase of the erosion rate with a 1000 ppm concentration of the PAM agent in distilled water, Figure 25. For a concentration of 100 ppm, no significant change in behavior occurs. It has to be mention that, because polymers in solution are partially degraded by extensive exposure to ultrasonic cavitation, these investigators took the precaution to have a fresh solution continuously fed into the test reservoir in order to achieve an equilibrium solution. In a glyceroldistilled water mixture, having a viscosity comparable to that of the "equilibrated" 1000 ppm PAM solution, the incubation time increased by a
246
factor of three as compared to pure water and the erosion rate decreases slightly.
W e i g h t loss, mg
J
[]
50 Fluid
40
J
water glycerol-distilled water mixture
"o
)J
100 p p m P A M 1000 p p m P A M
20 10 E r o s i o n t i m e , rain I I
0
10
20
30
40
50
60
Figure 25 9 Weight loss as a function of time for copper specimens subject to ultrasonic cavitation. From Ashworth and Procter [56].
Weight loss, mg
f = 19.5 k H z A = 38 ~tm t I = 295 K
50 ..O
J
40
Fluid
30 0
.~/
20 -
[] A 0
10
0
I
i
I
I
I
I
10
20
30
40
50
60
water 100 p p m P E O 500 p p m P E O 1000 p p m P E O
Figure 26 9 Weight loss as a function of time for aluminium specimens subject to ultrasonic cavitation. From Shima et al. [57].
247
Ten years after these early results, Shima et al. [57] conducted very similar experiments, but used PEO instead of PAM and had the aluminium test specimen attached to the vibrating rod. Their results are very much different of those of Ashworth and Procter [56]. First, no incubation time seems to be necessary either in water or polymer solution; second, a 100 ppm solution shows a behavior similar to that of water with a slight increase of the weight loss; third, for 500 and 1000 ppm solutions the weight loss increases for a short time after initiation and decreases significantly thereafter, Figure 26. When the 1000 ppm polymer solution is moderately degraded the weight loss increased in the early stages and decreased afterwards while for a fully degraded solution the results are very close to those for pure water. Again, a water/glycerol mixture decreased the material removal. Tsujino [58] pursued the research by Shima et al. [57] and investigated the effect of the peak-to-peak amplitude. He showed that when the amplitude is decreased, from 38 to 25 ~trn, water and a 1000 ppm PEO solution have exactly the same behavior during a very short duration (-10 min) after initiation and then differ, the aggressiveness being much decreased in the polymer solution. 50
Weight loss, mg
f = 19.5 kHz t I = 295 K
40 30
A= 38 l
-
20
a
m
~
~
A, vibratory amplitude
~'0
-
A Shimaet al. [57] 9 Tsujino[58]
-
/
_
/
/
/
~
' \ A = 25 gm
O Ashworth and
Procter [56]
10 I Test ti~ne, min I 0
0
10
20
30
40
s0
60
Figure 27 9 Weight loss as a function of time in water. Adapted from Ashworth and Procter [56], Shima et al. [57] and Tsujino [58]. As shown, the results obtained by Ashworth and Procter [56] on one side, and Shima et al. [57] and Tsujino [58] on the other, are markedly different in spite of the fact that the two polymers used in the tests are very effective drag reducing agents, display analogous viscoelastic effects, and have, for equal concentrations, comparable shear viscosities in solution. The major differences
248
have to do with the materials and with the position of the test specimens in the ultrasonic pressure field. However, in the case of pure water, the weight loss versus time remains nearly linear regardless of the specimen material, the position of the sample and the amplitude of the ultrasound, Figure 27. In the polymer solutions, a linear behavior has been shown to exist by Ashworth and Procter, while a power law behavior, with a power of less than unity, has been demonstrated by Shima et al. and Tsujino. In a quite recent work, Hoyt and Morgan [59] have reported results analogous to those of the Japanese researchers when a 200 ppm PAM was added to a 6% mixture of tomato puree in water. How then to interpret the changes in the case of the polymer solutions ? No explanation has been proposed as yet, although the visualisation conducted by Shima et al. and Tsujino show conclusively that the development of the cavitation bubbles in the more concentrated PEO solutions is very different from the one observed in pure water. In the latter case, a cloud of bubbles develops in an orderly fashion on the surface of the horn with a thickness increasing from the rim to the center and, away from this surface and around the axis, a column of small bubbles extending quite far away (one to two times the diameter of the horn). When the erosion progresses, the thickness of the cloud seems to increase on the surface and, in particular, around the center of the sample. In 500 and 1000 ppm polymer solutions, isolated bubbles, clouds or filaments, occupying a cylinder of at least the diameter of the horn, occur away the sample surface and on the side wall of it, in what can be said to be a disorderly fashion (as compared to water). When time, and hence erosion, progresses, bubble clouds become denser and thicker on the surface and on a column extending away from it. Therefore, it seems that, in the polymer solutions, low pressure regions, where cavitation can occur, are developed because of macro scale (of the size of the sample diameter) changes in regions which are not even supposed to be affected by the vibration, such as the side wall of the sample and the column mentioned above. For the test conditions of Shima et al. [57] and Tsu~ino [58], it seems reasonable to hypothesize that the very dense and thick cavitation, developing on the test specimen surface when erosion progresses, has a cushioning effect that reduces the damage due to bubble (cloud) collapse. For the Ashworth and Procter test geometry, this cushioning effect is reduced because of the distance between the sample and the tip of the vibratory horn and the modification of the macro scale (which is in this case of the order of magnitude of the space between the sample and the tip). Ting [60] offered a completely different explanation of the Ashworth and Procter results. He claimed that the enhanced erosion resulted from the "solidlike" behavior of the reentering jet and the additional impact stress developed in the extremely rapid deformation process undergone by the fluid. In view of the results by Shima et al. [57], Tsu~ino [58] and Hoyt and Morgan [59] this reentering jet erosion enhancement effect does not seem to be of primary
249
importance. In fact, because all of the single bubble behavior investigations indicate that the effect of viscoelasticity is to keep the bubbles spherical much longer than in the Newtonian solvent, it seems reasonable to expect, if no macro scale changes of the flow occur, a reduction of cavitation erosion. Only one paper (Shapoval and Shal'nev [61]) refers directly to erosion in a flowing PAM solution. In their tests, cavitation was generated in the wake of a cylinder attached to the wall of a rectangular section. They determined the cavitation numbers corresponding to the onset of cavitation and to the situation corresponding to a cavity having a given length. In the polymer solution onset is quite significantly delayed as compared to water. For a cavity having a length of the cavitation zone equal to the cylinder diameter, the cavitation number in the polymer solution was also reduced but proportionally less than for the onset condition. The removal of material from a sheet of rolled lead placed on the wall of the test section behind the cylinder was reduced in the case of the polymer solution when the length of the cavity was twice the diameter of the obstacle. Assuming that the pattern of the eroded regions, limited to the shear layer, reflects the structure of the flow behind the cylinder, the authors note that cavitation in the polymer solution appears to be analogous to that in water at larger values of the Reynolds number. Again, in these Russian experiments it is impossible to conclude the reasons for the apparent decrease of the erosion. This is more so considering that the region where the erosion imprint was analyzed corresponds to the shear layer in the wake and to conditions which seem to be far from those prevailing at the closure of an attached cavity where a stagnation point occurs. This latter situation was discussed by Ting [60] for travelling bubbles. 6. C O N C L U S I O N A considerable number of investigations have been conducted towards the understanding of non-Newtonian fluid effects on cavitation occurrence and its consequences. The major conclusions can be summarized as follows 9 9 for single bubbles, non-dimensional time evolution of spherical bubbles in a stagnant and unbounded domain is unaffected by the non-Newtonian properties of the fluids, - in a domain bounded by a solid or free surface and in a shear layer, the non-spherical collapse of bubbles is affected by the non-Newtonian properties which delay the departure from sphericity, 9 in moving liquids such as 9 in submerged jets, results show that cavitation can be either delayed or unaffected by the non-Newtonian fluids and that major changes seem to be associated with modifications of the structure of the free-shear layer, -
-
250
-
in flow around blunt bodies, major changes appear to be correlated with the early occurrence of laminar to turbulent transition in drag reducing polymer solutions, without any major contribution associated with bubble dynamics, in tip-vortex flow situations with homogeneous polymer solutions, the non-Newtonian effects seem to enhance, for equal foil lift coefficient, the cavitation occurrence, while, in pure water and polymer solution ejection from the wing tip, there is a systematic cavitation inhibition, - in confined vortex flow in homogeneous solutions, the results show either cavitation enhancement or inhibition, without an explanation for the causes of such conflict, - in lubricating type flows, cavitation is delayed as a result of the development of a normal stress contribution to the pressure. Erosion due to cavitation can also be either augmented or diminished depending on the way the tests are performed. In summary, besides for a few controlled flow situations, such as the ones utilized for investigating single bubble behavior, there is not a definite answer concerning the favorable or unfavorable effect of non-Newtonian fluid properties on cavitation occurrence and the consequences of its development. The major obstacle to a complete understanding of this problem lies on the fact that polymer solutions have, at least for the range of flow parameters encountered in laboratory test situations, strong effects on the boundary layers developing along streamlined bodies, on the structure of large eddies in shear layers of separated flows, on the lift of hydrofoils, etc. These large scale effects are further complicated by the behavior of the individual cavitating bubbles, essentially determined by the local pressure and the presence of solid boundaries or fluid interfaces at distances smaller than the bubble diameter; thus relatively small scales. It is our opinion that some of the many questions that remain unanswered may receive an appropriate response if test can be conducted at much larger Reynolds numbers than those achieved up to know and with very dilute polymer concentrations. However, the degradation of the polymer molecules may have then a negative impact on the results of the cavitation tests. -
ACKNOWLEDGEMENT The authors wishes to thank the support he has received over many years by the Ministry of Defence, France, to conduct research on polymer solutions effects in hydrodynamics. He is deeply indebted to Professors Jack W. Hoyt and William R. Schowalter for taking the time to read a drat version of this chapter and provide the author with many correction and wise comments.
251
NOMENCLATURE r
D e
G k L m,n P po pc Pd Pg PgO p(R) Pv p~(t)
Prez qi
Q R
Ro RA Re
Rc Rch R(t) (r, O, (p) t U
U
vj vo l~ts Y 7? r
concentration diameter of headforms minimum gap in lubrication tests non-dimensional injection rate of polymer fluid polytropic index distance between the center of a bubble and a wall power law parameters pressure pressure at infinity at t = 0 critical pressure pressure in a reservoir non-condensible gas pressure within the bubble non-condensible gas pressure at t = 0 pressure at the interface of a bubble on the liquid side vapour pressure pressure far from the bubble reference pressure non-dimensional flow rate volume flow rate of ejected solution radius of the rotating drum in lubrication tests radius of a nucleus distance between the position of the bubble interface, opposite to a wall, and the center Reynolds number critical radius initial radius for a collapsing bubble bubble radius at time t spherical coordinate system time radial velocity component free stream velocity bulk velocity of a jet tangential velocity boundary layer displacement thickness surface tension and shear rate apparent viscosity viscosity
252
P O"d O"i "t'R
Trr, TOO, T ~ CO
APe
elongational viscosity liquid density desinent cavitation number incipient cavitation number Rayleigh time normal components of the deviatoric stress tensor angular velocity of the drum in lubrication tests added pressure in lubrication tests
REFERENCES ~
.
.
.
.
6. ~
8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
R.T. Knapp, J.W. Daily and F.G. Hammit, Cavitation, McGraw-Hill, New York, 1970. D.H. Trevena, Cavitation and tension in liquids, 1987, Adam Hilger, Bristol and Philadelphia. W. Tillner et al., The avoidance of cavitation damage, Mechanical Engineering Publications Limited, London, 1993. C.E. Brennen, Cavitation and bubble dynamics, Oxford University Press, Oxford, 1995. A.J. Acosta and B.R. Parkin, J. of Ship Research, 19, 1975, pp. 193-205. A. Shima and T. Tsujino, Chemical Engineering Science, 31, 1976, pp. 863-869. H.S. Fogler and J.D. Goddard, Phys. Fluids, 13 (5), 1970, pp. 1135-1141. H.S. Fogler and J.D. Goddard, J. Appl. Phys., 42 (1), 1971, pp. 259- 263. A.T. Ellis and R.Y. Ting, NASA-SP 304, Pt. 1, 1974, pp. 403-420 (with discussions). W.J. Yang and M.L. Lawson, Journal of Applied Physics, 45 (2), 1974. G.L. Chahine, 1974, Thesis, Univ. Paris VI, 1984, Rapport de recherche ENSTA 042. R.Y. Ting, Phys. Fluids, 20 (9), 1977, 1427-1431. P.S. Kezios and W.R. Schowalter, Phys. Fluids, 29 (10), 1986, pp. 31723181. G. Riskin, J. Fluid. Mech., 218, 1990, pp. 239-263. G.L. Chahine and D.H. Fruman, Phys. Fluids, 22 (7), 1979, pp. 14061407. E.A. Brujan, C.-D. Ohl, W. Lauterborn and A. Philipp, Acustica, 82, 1996, pp. 423-430. G.L. Chahine and A.K. Morine, ASME Cavitation and Polyphase Flow Forum, 1980, pp. 7-8. A.K. Morine, 1982, Thesis, Univ. Paris VI, Rapport de recherche ENSTA 179, 1983. G.L. Chahine, Applied Scientific Research, 38, 1982, pp. 187-197.
253
20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.
P.T. Kezios, PhD dissertation, Princeton University, 1984. P. Ligneul, Phys. Fluids, 30 (7), 1987, pp. 2280-2282. J.W. Hoyt, ONR Drag Reduction Workshop, Boston, 1970. J.W. Hoyt, 16th American Towing Tank Conference, Sao Paulo, Brazil, 1971. J.W. Hoyt, ASME Cavitation and Multiphase Flow Forum, pp. 44-47, 1973. J.W. Hoyt, ASME Journal of Fluids Engineering, 98, 1976, pp. 106-112. C.B. Baker, J.W. Holl and R.E.A. Arndt, ASME Cavitation and Polyphase Flow Forum, 1976, pp. 6-8. J.W. Hoyt and J.J. Taylor, ASME Journal of Fluids Engineering, 103, 1981, pp. 14-18. R.E.A. Arndt, ASME Cavitation and Polyphase Flow Forum, 1980, pp. 9-10. R.Y. Ting, AIChE Journal, 20 (4), 1974, pp. 827-828. J.W. Hoyt, 1l th International Towing Tank Conference, Tokyo, 1966. A.T. Ellis, J.G. Waugh and R.Y. Ting, Journal of Basic Engineering, 92, 1970, pp. 459-466. C.B. Baker, R.E.A. Arndt and J.W. Holl, 1973, App. Res. Lab. Tech. Memo. 73-257, The Penn. State Univ. J.H.J. van der Meulen, ASME Cavitation and Polyphase Flow Forum, 1973, pp. 48-50. J.H.J. van der Meulen, 1976, Doctoral Thesis, Netherlands Ship Model Basin. E.M. Gates and A.J. Acosta, 12th Symposium on Naval Hydrodynamics, National Academy of Sciences, Washington, 1978. H. Reitzer, C. Gebel and O. Scrivener, J. of Non-Newtonian Fluid Mech., 18, 1985, pp. 71-79. J.H.J. van der Meulen, ASME Cavitation and Polyphase Flow Forum, 1976, pp. 4-5. C.E. Brennen, J. Fluid Mech., 44, 1970, pp. 51-63. V.A. Bazin, Y.E.N. Barabanova and A.F. Pokhil'ko, Fluid Mechanics Soviet Research, 5, 1976, pp. 79-82. R.Y. Ting, Phys. Fluids, 21 (6), 1978, pp. 898-901. R.J. Gordon and C. Balakrishnan, Nature Physical Science, 231,1971, pp. 177-178. C. Inge and G. Bark, TRITA-MECH-83-12, 1983, The Royal Institute of Technology, Sweden. D.H. Fruman, ASME Cavitation and Polyphase Flow Forum, 1984, pp. 73-76. J. Wu, Lift reduction in additive solution, Journal of Hydronautics, 3 (4), 1969, pp. 188-200. T. Kowalski, Journal of Hydronautics, 5 (1), 1971, pp. 11-14.
254
46. D.H. Fruman, D. Bismuth, S.S. Aflalo, in The influence of polymer additives on velocity and temperature fields, B. Gampert ed., SpringerVerlag, Berlin, 1985, pp. 399-4. 47. S.S. Aflalo, Thesis, Univ. Paris VI, 1987. 48. D.H. Fruman and S.S. Aflalo, ASME Journal of Fluids Engineering, 111, 1989, pp. 211-216. 49. T. Pichon, A. Pauchet, A. Astolfi, D.H. Fruman, J-Y. Billard, Journal of Ship Research, 41 (1), 1997, pp. 1-9. 50. D.H. Fruman, T. Pichon, P. Cerrutti, J. Mar. Sci. Technol., 1, 1995, pp. 13-23. 51. G.L. Chahine, G.F. Frederick and R.D. Bateman, ASME Journal of Fluids Engineering, 115, 1993, pp. 497-503. 52. C. Inge, TRITA-MECH-83-05, 1983, The Royal Institute of Technology, Sweden. 53. J.W. Hoyt, ASME Cavitation and Polyphase Flow Forum, 1978, pp. 1718. 54. D. Bismuth, Thesis, Univ. Paris VI, 1987, also, Rapport de Recherche ENSTA 215. 55 A. Ouibrahim, D.H. Fruman and R. Gaudemer, Phys. of Fluids, 8 (7), 1996, pp. 1964-1971. 56. V. Ashworth and R.P.M. Procter, Nature, 258, 1975, pp. 64-66. 57. A. Shima, T. Tsujino, H. Nanjo and N. Miura, ASME Journal of Fluids Engineering, 107, 1985, pp. 134-138. 58. T. Tsujino, Ultrasonics, 25, 1987, pp. 67-72. 59. J.W. Hoyt and J. Morgan, ASME Cavitation and Multiphase Flow Forum, 1989. 60. R.Y. Ting, Nature, 262, 1976, pp. 572-573. 61. I.F. Shapoval and K.K. Shal'nev, Sov. Phys. Dokl. 22 (11), 1977, pp. 635-637.
255
LOW-DIMENSIONAL DESCRIPTION OF VISCOELASTIC TAYLORVORTEX FLOW
Roger E. Khayat Department of Mechanical & Materials Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9
1. INTRODUCTION The influence of inertia and elasticity on the onset and stability of Taylor vortex flow (TVF) is examined for an Oldroyd-B fluid. The rigid-flee (stick in the azimuthal direction and slip along the axial direction) and rigid-rigid boundary conditions are used. Both formulations lead to essentially the same qualitative stability picture, but the former conditions allow a much simpler formulation, more easily amenable to algebraic manipulation and analysis. The resulting equations reduce to the Lorenz equations for a Newtonian fluid and small number of modes. A truncated Fourier/Chandrasekhar representation of the flow field, and the Galerkin projection, lead to a closed nonlinear dynamical system with sixteen degrees of freedom. In contrast to the six-dimensional system derived previously [Khayat, Phys. Fluids A 7 (1995) 2191 ], the present model is capable of capturing the dynamical behavior observed experimentally for a (Boger) fluid with negligible inertia. The improvement stems mainly from the proper choice and inclusion of higher-order modes that account for normal stress effects. For flow with dominant inertia, the stability picture is similar to that for a Newtonian fluid: steady TVF sets in as the Reynolds number reaches a critical value, Reo which decreases with fluid elasticity or normal stress effects, and is strongly influenced by fluid retardation. As elasticity exceeds a critical level, a subcritical bifurcation emerges at Re o similar to the one predicted by the Landau-Ginzburg's equation and the previous six-dimensional model. It is found that slip tends to be generally destabilizing. The coherence of the model is established through comparison with existing linear stability analyses, and experimental measurements and flow visualization.
256
Recent linear stability analyses and experiment indicate a dramatic departure in the stability and bifurcation of the Taylor-Couette flow of viscoelastic fluids in comparison to Newtonian fluids. The Taylor-Couette flow of Newtonian and viscoelastic fluids involve a rich sequence of fundamental flow regimes that are observed during experiment, coveting the range from laminar to turbulent motion. For a Newtonian fluid, it is observed that at a sufficiently small Reynolds number, Re, there is a unique stationary (Couette) flow, which is globally stable, except perhaps near the ends of the cylinders. When Re is raised in the range near a critical value, R e o the stationary flow loses its stability, and develops a regular cellular vortex structure in which closed ring vortices alternating in sign are wrapped round the axis of rotation. In this range of Reynolds number, the flow remains essentially time independent and axisymmetric. This is the well known Taylor vortex flow (TVF) after G. I. Taylor, who was the first to examine this flow regime both theoretically and experimentally [ 1]. At higher Re value, the stationary cellular structure loses its stability in turn to another structure with a different number of vortices [2,3]. The upper limit of the TVF regime coincides with the emergence of time-dependent flow and the breaking of axisymmetry. Simultaneously, there appears a uniformly rotating pattern, or wavy vortex flow (WVF), of tangential azimuthal traveling waves superimposed on the cellular structure of TVF [4]. The TVF-WVF transition was carefully investigated by Fenstermacher et al. [5] and Gorman & Swinney [6] through flow visualization and spectral techniques. They obtained an accurate picture of the spatio-temporal flow structure which was later theoretically examined by Rand [7]. As the WVF sets in, the amplitude of the azimuthal waves is shown in the experiments to grow continuously with Re. The velocity power spectrum indicates the appearance of a single discrete frequency component with its harmonics. As Re is further increased, a second fundamental appears, with a gradual increase in the smallscale structure. At higher Re value, there is a broadening of the base in the power spectrum, indicating the onset of aperiodicity or chaos. For highly elastic (Boger) fluids, normal stress effects appear to prohibit the onset of steady TVF. The experimental measurements of Muller et al. [8] clearly demonstrate the existence of a purely elastic overstable mode when the Deborah number exceeds a critical value for a fluid with vanishingly small inertia. Periodicity appears to be lost as the level of elasticity is further increased. The influence of inertia was later examined by Baumert & Muller [9] through flow visualization. More extensive experiments are, however, needed that could elucidate on the possible route(s) to instability and transition to turbulence. One of the main objectives of the present paper is to actually predict the conditions
257
under which transition may occur in the Taylor-Couette flow of viscoelastic fluids. Fluid turbulence is among the most challenging areas in the physical sciences, and has been the subject of intensive research over the past hundred years or so. Similarly to any flow in the transition or turbulent regime, the Taylor-Couette flow involves a continuous range of excited spatio-temporal scales. In order to assess the effect of the motion of the arbitrarily many smaller length scales, one would have to resolve in detail the motion of the small scales. This issue remains an unresolved one since, despite the great advances in storage and speed of modem computers, it will not be possible to resolve all of the continuous range of lengths scales in the transition regime. It is by now well established that loworder dynamical systems constitute an alternative to conventional numerical methods as one strives to understand the nonlinear behavior of flow [ 10]. For the Rayleigh-Benard thermal convection problem, the simplicity of the Lorenz equations [11], and the rich sequence of flow phenomena exhibited by their solution, have been the major contributing factors to their widespread use as a model for examining the onset of chaotic motion [12]. Despite the severe level of truncation in the formulation of these equations, some of the basic qualitative elements of the onset of thermal convection and the destabilization of the cellular structure have been recovered through the model. More general problems in fluid dynamics have also been treated using low-dimensional descriptions and the theory of nonlinear dynamics [13]. These methods are based on the expansion of the flow field in terms of a complete set of orthogonal functions, Fourier series or other standard basis functions, and the Galerkin projection technique whereby the initial set of partial differential equations is decomposed into an infinite set of ordinary differential equations governing the time-dependent expansion coefficients. The system is then truncated, leading to the finite-dimensional dynamical system. These methods have mainly been applied to Newtonian fluids, and the author appears to be the first to extend their application to nonNewtonian fluids [ 14-19]. While the problem of Taylor-Couette flow has been extensively investigated for Newtonian fluids, relatively little attention has been devoted to the case of viscoelastic fluids. In addition to the relatively recent interest in polymeric flows, the onset of chaos and turbulence is far less widespread in viscoelastic than in Newtonian fluids. To our knowledge, there has been no experimental evidence of the existence of (deterministic) chaos for highly elastic polymeric solutions similar to the one found for Newtonian fluids [5,6]. Recently, however, there has been a growing interest in the Taylor-Couette flow of viscoelastic fluids [20,21 ] for a recent review). Some of the early experimental work on the stability of rotating viscoelastic fluids is that of Giesekus [22,23]. Using polyacrylamide
258
solutions [23], he showed that the critical value of the Reynolds number at the onset of Taylor vortices may be smaller or higher than the Newtonian value depending on polymer concentration. At high polymer concentration, when elastic effects become dominant, the critical Reynolds number was found to decrease with the flow Deborah number, De. However, at high polymer concentration, solutions may exhibit significant shear thinning. Thus, ideally, both viscoelastic and shear thinning effects must be accounted for through a suitable constitutive model. The later experiments of Haas & Biahler [24] tend to show similar results. The experiments of Muller et al. [8] clearly demonstrate, for the Taylor-Couette flow of a Boger fluid, the existence of a purely elastic time-periodic instability at a critical rotation rate. The experiments were conducted using Laser Doppler velocimetry (LDV) measurements of the axial velocity component of a polyisobutylene-based fluid between two concentric cylinders, with the outer cylinder being at rest and the inner cylinder rotating. The results show an oscillatory flow at a vanishingly small Reynolds number (Re -~ 7x 10-3). The flow appears to undergo a transition from the purely azimuthal Couette flow to time periodic flow as the Deborah number exceeds a critical value, which is in good agreement with the value based on earlier linear stability analysis of an Oldroyd-B fluid [25]. The LDV measurements show that the oscillatory behavior is not localized but appears to be spread throughout the flow. As the Deborah number increases from the critical value, the amplitude of oscillation increases monotonically with De. The corresponding power density spectra show peaks, which are instrumentally sharp at the fundamental frequency, the growth of harmonics, and eventually subharmonics, reflecting, perhaps, the presence of period doubling or emergence of a second fundamental frequency (quasiperiodicity). The influence of inertia (shear rate) on the evolution of TVF was later examined by Baumert & Muller [9] who performed flow visualization on Boger fluids using reflective mica platelet seeding. Parallel theoretical work, for axisymmetric and non-axisymmetric flows, has also been conducted on the linear stability of viscoelastic fluids [25-27] and nonlinear calculations [28-30], with new phenomena being constantly attributed to fluid elasticity [31,32]. Unlike Newtonian fluids, which obey Newton's law of viscosity, viscoelastic fluids are not governed by a similar universal constitutive law. The dependence of the predicted flow behavior on the particular choice of a constitutive model adds another difficulty in our attempt to interpret an already complex flow situation, particularly in the transition and turbulent regimes. Larson [33] carried out a linear stability analysis in the narrow-gap limit,.using the Doi-Edwards and K-BKZ constitutive equations. He found that the dependence of the critical Reynolds number on De is generally non-monotonic, but the flow is increasingly destabilized by fluid elasticity in the higher De range
259
[33]. Numerical calculations were carried out by Northey et al. [30] for the Taylor-Couette flow of an upper-convected Maxwell (UCM) fluid neglecting inertia. The non-axisymmetric flow solution was obtained by Avgousti & Beris [28] and Avgousti et al. [29]. Elastic overstability was also proved through various linear stability analyses [25,26], showing that the base (Couette) flow can lose its stability via a Hopf bifurcation as the Deborah or Weissenberg number exceeds a critical value. Most of existing analyses are, however, limited to the linear regime. In our quest to probe the nonlinear regime, we have undertaken the current series of studies on the influence of fluid elasticity on Rayleigh-Benard thermal convection and Taylor-Couette flow. Khayat [14-17] developed a fourdimensional dynamical system for the thermal convection of strongly elastic fluids of the Oldroyd-B type. Such a system constitutes a generalization of the Lorenz equations [11] to include viscoelastic fluids. The critical Rayleigh number at the onset of the convective cellular structure was found to be the same as for Newtonian fluids. This is a direct consequence of the fact that the nonlinear terms (at least to the degree of truncation adopted) are the same as those in the Lorenz equations; no convective or upper-convective nonlinear terms survive in the constitutive equations. The conductive state thus loses its stability to the two steady convective branches C1 and C2, say, through a supercritical bifurcation similarly to Newtonian fluids. The two convective branches lose their stability in turn through a Hopf bifurcation as the Rayleigh number exceeds a value that, this time, depends strongly on fluid elasticity and retardation. It was also observed that fluid elasticity tends to precipitate the onset of chaotic motion, while fluid retardation tends to delay it. Above a critical value for the Deborah number, the flow behavior departs significantly from that of a Newtonian fluid. All three fixed branches remain unstable in the supercritical range for any value of the Rayleigh number. Thus, for D e > D e c, no steady convection can set in, and the cellular structure is always periodic in time (overstable), with the corresponding Fourier spectrum showing a sequence of period doubling as D e is increased. The existence and stability of the Hopf bifurcation corresponding to the onset of overstability were established in a later paper [15] using center manifold theory. These findings have prompted us to adopt a similar approach to the Taylor Couette flow of viscoelastic fluids in the narrow gap limit [18,19]. Our earlier work [ 18] focused on the influence of elasticity and retardation on the sfability and amplitude of the Taylor vortices with inertia dominating the flow. A truncated Fourier representation with Galerkin projection of the flow field similar to that of Kuhlmann and associates for a Newtonian fluid was proposed. Kuhlmann [33], and later Kuhlmann et al. [35] examined the
260
stationary and time-periodic TVF, in the narrow-gap limit and arbitrary gap width, respectively, with the inner cylinder rotating at constant and harmonically modulated angular velocity. The solution to the full Navier-Stokes equations was obtained by implementing a finite-difference scheme, and an approximate approach based on the Galerkin projection. Comparison of flows based on the two methods led to good agreement. In the approximate Galerkin projection [ 18], the velocity and stress components assume a truncated Fourier representation in space, with the expansion coefficients being functions of time alone. Since elastic effects were assumed to be weak, higher-order normal stress terms were neglected, thus leading to a six-dimensional nonlinear system, and reducing to Kuhlmann's three-dimensional system for a Newtonian fluid [18]. In both models, the velocity is assumed to adhere to the cylinder surfaces in the azimuthal direction, while it slips in the axial direction (rigid-free boundary conditions). The corresponding rigid-rigid boundary model leads to more realistic values of the critical Taylor number and the corresponding wave number, but to a less realistic value for the torque. Kuhlmann's three-dimensional system turned out to be equivalent to the Lorenz system with the Prandtl number equal to one. In this case, Kuhlmann's model cannot predict the destabilization of the Taylor vortices, and therefore cannot account for the onset of chaotic behavior. The situation is exactly similar for the Lorenz system when the Prandtl number is less than or equal to one [36,37]. In the previous work [18], a similar level of truncation in the Fourier representation for the velocity and stress was adopted for an Oldroyd-B fluid. Similar levels of truncation have been widely used for the Navier-Stokes and energy equations [13,38]. Examination of the influence of additional modes on flow and temperature fields [39-42] indicates that many of the gross features predicted by low level models are essentially recovered by the higher-order models. In the viscoelastic model, there are two Fourier modes in the azimuthal velocity and shear stress components that are expected to lead to nonlinear terms in the stress equations. This is in contrast to thermal convection [14-17] where the nonlinearities originate from the convective terms in the energy equation. While the stability and bifurcation picture at the onset of the cellular structure in the Rayleigh-Benard convection of viscoelastic fluids is similar to that of Newtonian fluids, the presence of nonlinearities in the stress equations of the Taylor-Couette flow of viscoelastic fluids has a drastic influence on the stability of the Taylor vortices and the transition to chaos. While the two models suffer from a similar level of truncation, the addition of viscoelastic effect, even to a very weak extent, appears to alter dramatically the stability and bifurcation picture. Most importantly, unlike the Newtonian model, elasticity tends to destabilize the TVF leading to chaotic behavior and the emergence of a Lorenz
261
type attractor. As the Deborah number increases, the onset of TVF occurs at a Reynolds number that decreases with De. Beyond a critical De value, the exchange of stability takes place via a subcritical instead of a supercritical bifurcation. Since the previous formulation [18] is not valid in the presence of dominant elastic effects, most results from existing linear analyses, finite amplitude numerical calculations, and experiments on the purely elastic overstability could not be recovered. The aim of this chapter is to investigate the onset and stability of finite amplitude TVF with elastic effects dominating over inertia. It is shown that the purely elastic overstability can only be predicted if the higher-order normal stress terms, neglected in the previous formulation [18], are properly accounted for. Particularly, the addition of the azimuthal normal stress component z00 leads to additional coupling with higher-order eigenmodes that can no longer be neglected. The resulting nonlinear dynamical system involves sixteen instead of six degrees of freedom. Although more cumbersome, and therefore less amenable to algebraic manipulations, the present expanded model is more accurate in its predictions, and leads to good agreement with existing formulations and experiments. Both the rigid-free and rigid-rigid boundary conditions are used and the resulting flows are compared. It is found that the two formulations lead essentially to similar qualitative flows. The former conditions allow a much simpler formulation. It also allows the examination of the influence of the higher-order normal stress terms when compared with the previous formulation of Khayat [18]. Additional calculations are also carried out in the absence of inertia, based on a dynamical system that accounts for yet higher order normal stresses, in an attempt to reach a quantitative agreement with the measurements of Muller et al. [8]. The chapter is organized as follows. In w the derivation of the nonlinear dynamical system is discussed. The stationary solutions and their stability are examined in w The influence of inertia for weakly and strongly elastic fluids is examined through numerical calculations, which are presented in w Discussion and concluding remarks are covered in w 2. DERIVATION OF THE NONLINEAR DYNAMICAL SYSTEM The derivation of the nonlinear dynamical system for a viscoelastic fluid is summarized in this section. The governing equations are solved by assuming an infinite discrete Fourier/Chandrasekhar representation in space for the flow field, which, upon truncation and application of the Galerkin projection, leads to the sixteen-degree-of-freedom nonlinear dynamical system (16NDS) that governs the expansion coefficients.
262
2.1. General equations and boundary conditions Consider an incompressible viscoelastic fluid of density 9, relaxation time X1 and viscosity q. In this study, only fluids that can be reasonably represented by a single relaxation time and constant viscosity are considered. The fluid is assumed confined between two infinite and concentric cylinders of inner and outer radii R i and R o , respectively, and the flow is axisymmetric. The inner cylinder is taken to be rotating at an angular velocity fl, while the outer cylinder is at rest. The conservation of mass and linear momentum equations are, respectively:
V . u = O,
Re
+ u . Vu
)
= - V p -/-Rv V 2 u - V. z"
(1)
where V is the gradient operator, t is the time, u is the velocity vector and x is the elastic part of the deviatoric stress tensor obeying the following constitutive equation [43]"
De
+
u. V r- (17u) t . z" -
1
r. Vu = - r -
(2)
Vu- ( V u ) t .
The various dimensionless groups in the problem, namely the Reynolds number, the Deborah number, D e , the solvent-to-polymer viscosity ratio, R v , and the gap-to-radius ratio, ~ (not explicitly appearing in the equations above), are given by:
Re,
Re -
dRi.O w
'
De =
2 R .O d
"
Rv
=
22 A1 - / Z 2
-
, tip
c=
d
,
(3)
Ri
where qs and rip are the solvent and polymer viscosities, respectively, and 22 (0 <_22 < 21) is the fluid retardation time. Note that in this case q = rls + qp and v= r//p. Other related dimensionless groups are the Taylor number, Ta, the Elasticity number, E, and polymer-to-solution viscosity ratio, a: Ta
cRe 2
E-
De Re"
Zip r/
1 9 Rv + l
(4)
263
The boundary conditions on the cylinder walls are prescribed as follows. Regardless of the nature of the two cylinders, the no-penetration condition must apply at any time. The no-slip conditions in the azimuthal direction are assumed to hold. There remains the conditions along the axial direction. Generally, stick (rigid) conditions should be assumed. However, the use of the less realistic fleeslip conditions simplifies greatly the formulation. Such conditions were previously used by Kuhlmann [34], and more recently by Khayat [18]. Similar unrealistic conditions have extensively been adopted in the Rayleigh-Benard thermal convection of Newtonian fluids [44] and viscoelastic fluids [45]; the two planes at which the temperature differential is imposed are assumed to be free surfaces. In the present work, the free and rigid conditions will be used. Thus, the term rigid-flee (RF) boundary conditions will be referred to if slip along the axial direction applies, and the term rigid-rigid (RR) boundary conditions will be referred to if stick is used. It will be concluded that little qualitative difference emerges between the two formulations. As mentioned above, the introduction of free conditions is customary in the case of a Newtonian fluid [34,35], and should be compared to the free-surface assumption usually adopted in the RayleighBenard convection problem [ 13,38,44,46]. In that case, both the lower and upper planes, separated by the fluid, are assumed to be free surfaces. This is an unrealistic assumption, but it makes the problem formulation much easier to handle since trigonometric rather than hyperbolic (Chandrasekhar) functions can be used in the expansion of the solution along the gap. For Taylor-Couette flow, a slip along the Z direction tends to enhance the onset of Taylor vortices given the absence of an axial friction force acting between the fluid and the cylinder walls. In the narrow gap limit, linear stability analysis for a Newtonian fluid gives the minimum value of the critical Taylor number and corresponding wave number for the onset of TVF, respectively, as Ta m = 1695 and k m = 3.12 if rigid-rigid boundaries are assumed [44], whereas for rigid-free conditions, Tacm = 654 and k m = 2.23 [34]. The neutral stability curves in both cases are qualitatively similar. This is in close analogy to the lowering threshold in the Rayleigh-Benard problem by using free-free boundary conditions [4]. While rigid-rigid boundary conditions tend to give more accurate values for the critical Taylor number and wave number, the mixed (rigid-free) boundary conditions lead to a torque value at the onset of TVF much closer to what existing formulations [34,35,47,48] predict. As to the nonlinear range, both rigid-rigid and rigid-free conditions have been used with the same level of truncation in a three-dimensional dynamical system [34]. The two models do not lead to any significant qualitative difference in behavior. Based on these
264
arguments, one may then suspect that not much further insight would be gained if rigid-rigid boundary conditions were used in the case of a viscoelastic fluid, at least at the qualitative level. This is indeed confirmed in the present study through comparisons with existing linear stability analysis results, experimental measurements and flow visualization. 2.2. Governing equations in the narrow-gap limit Consider the flow between two concentric cylinders in the limit when the radius R i / R o is very close to one. Suitable scales are sought for length, time, velocity and stress. There are, in fact, several choices possible [49-51]. One obvious choice for the length scale is d - R 0 - R i . In this case, one expects e = d / R i to emerge as the natural perturbation parameter for the problem, no matter what the remaining scales are. However, a careless choice of these scales can easily lead to the reduction of the equations to those governing plane Couette flow if all terms of O(e) and higher are neglected. Additional complexities are also expected in the case of a viscoelastic fluid since it is difficult to properly assess the magnitude of the non-Newtonian contribution of the stress components [18]. The flow induced by relatively high rate of rotation of the inner cylinder is of special interest here since it leads to the generation of large normal stresses. Following Tabeling [51], one velocity scale is chosen for the flow components along the azimuthal direction, and another for the components along the radial and axial directions [ 18]. If x, y and z are taken as the reduced coordinates in the radial, azimuthal and axial directions, then the problem
consists of finding the variables
Ui
and ~0" where i,j = x,y,z.
2.3. Elimination of the z dependence The general solution of equations (1) and (2) is carried out by seeking a periodic flow field along the cylinder axis. Thus, the velocity and stress fields are represented by infinite Fourier series in the z direction, with the expansion coefficients depending on x and t. The Fourier modes have a fundamental wavelength ~/k (in units of d) in the z direction. After elimination of the pressure in the momentum equations, the flow variables can be written as spectral sums: OO
O0
ui(x,z,t)
-
Zum(x,t)e
imkz 9
m - -oo
(5) m ~ ~oo
265
Additional relations among the Fourier coefficients are obtained by examining the physical symmetry of the flow. A symmetry allowed by equations (1)-(2), similarly to the Navier-Stokes equations [53,54], occurs when the flow is invariant under z ~ - z, and uz ~ - U z , rce ~ - r a e (~ = x, y). This amounts to having invariance in flow when the Taylor-Couette apparatus is turned upside down [54]. Thus, in addition to being axisymmetric, the velocity and stress fields must satisfy additional symmetry conditions It is not difficult to see that this symmetry is preserved under the nonlinear multiplication in both the convective and upper-convective terms of equations (1)-(2). The symmetry leads to additional simplification Upon application of the Galerkin projection method, which consists of multiplying equations (1)-(2) by the corresponding modes above and integrating with respect to z from 0 to ~k, an infinite set of coupled partial differential equations governing the expansion coefficients is obtained. If equations (1)-(2) were linear, the various modes involved would have separated in a manner similar to the case of a Newtonian fluid [44], and one would examine the firstorder z-dependent terms. In the present nonlinear context, some of the zerothorder terms are also retained in order to ensure that part of the nonlinear convective terms in the momentum equations, and the upper-convective terms in the constitutive equations do not vanish in the projection process.
2.4. The nonlinear dynamical system The most crucial step in the problem formulation is seeking an orthogonal representation form of the expansion coefficients in (5) and imposing a suitable level of truncation leading to the final nonlinear dynamical system. A judicious selection process must thus be applied for the choice of the various modes in order to ensure the physical and mathematical coherence of the final model. Not only does the approximate flow solution have to satisfy the imposed boundary conditions at the inner and outer cylinders, it must also reduce to the corresponding solution for linear flow: 1) in the limit of zero-Elasticity number for a Newtonian fluid, and 2) in the limit of vanishing inertia for a viscoelastic flow. Since the linear behavior of a Newtonian fluid is well understood [44], attention is first focused on the linear stability of a viscoelastic fluid with no inertia. In this case, the governing equations reduce to those of Larson et al. [25] in the narrow-gap limit. The discussion is limited to the case of an upper-convected Maxwell (UCM) fluid. Thus, R v is set equal to zero, and inertia effects are neglected in the momentum equations. If the nonlinear terms are neglected in the remaining
266 0
A
equations, u~and (3' become decoupled from the remaining variables. For linear stability analysis, one sets u l ( x , t ) = U ( x ) e - i a ) t , with similar expressions for the remaining variables, where co is generally complex. In this case, the stress components are explicitly expressed in terms of the velocity and velocity gradient through equations. Upon elimination of the z velocity component from the continuity equation, a fourth-order equation is obtained for U(x) as in Larson et al. [25]. In this case, the quantity A= 2 oDe2 / k ( 1 - i a)De)2 becomes the eigenvalue of the problem once the boundary conditions at the inner and outer cylinders are imposed. The solution of the resulting eigenvalue problem, with both the RR and RF conditions, can be obtained using the direct method, similarly to Larson et al. [25]. Further details are given elsewhere [56]. The eigenvalues are determined using an iteration scheme on the initial guesses and the secant method until the determinant is less than an imposed small tolerance. The eigenvalues A obtained using this method are found to be all imaginary, leading to the same dispersion relation as that of Larson et al. [25]. It is found that the critical (minimum) value of the Deborah number and corresponding wave number for the onset of the most unstable (overstable) mode are, respectively, Dec = 5.77e -1/2 and kc = 4.44, compared to the values 5.92e -1/2 and 6.7 in the case of RR boundary conditions [25]. This critical value for the onset of linear overstability, and, more generally, the marginal stability curve in the (De, k)-plane, should serve as limits (corresponding to small departure from the base flow) for the flow field predicted by the nonlinear theory, where only an approximate solution can be found. An approximate solution is first sought for the linearized problem, and is then compared with the exact solution from the direct method. For RF boundary conditions the following truncated Fourier representation is adopted: U ( x ) = U 1 sin 1rx + U 2 sin 2 erx,
x ~ [0, 1]
(6a)
which satisfies the RF conditions, whereas the solution that satisfies the RR conditions is expressed in terms of the even and odd Chandrasekhar functions denoted, respectively, by @l(X) and @2(x) and defined in [55]: U ( x ) = U 1 ~ l ( X ) + U 2 tl~2(x),
x ~ [-1/2, +1/2]
(6b)
The corresponding characteristic equation for A, for both the RF and RR conditions, may be written in the form:
267
)22/2
k6CiA2-(k 2 C2 ( k - C 3
=0,
(7)
where C1, C2 and C3 are constants involving the inner product of the orthogonal functions and their derivatives. These constants take, of course, different values depending on whether equation (6a) or (6b) is used. The exact solution, when compared with (6a) or (6b), gives an estimate of the magnitude of the error resulting from the t~ncation in solution. One should expect that the error remains of the same order when such a truncation is applied to the solution of the full nonlinear equations for small departure from the base flow. It is clear from eq. (7) that the neutral stability curves based on the approximate solutions 6a and b, those based on the exact solution using the RF boundary conditions, and the exact solution from [25] using the RR conditions exhibit similar behavior. At small wave number, the critical Deborah number for overstability to set in decreases sharply with k, reaches a minimum at k = kc, and increases again for k > k c. All curves indicate a general flattening around kc, confirming the experimentally observed wide range of wave numbers at which overstability sets in [9,20]. On the basis of these observations, and further analysis of the influence of inertia and fluid retardation (see section 3), one can now proceed and impose the type and number of modes that may adequately describe the nonlinear flow field. Although the solutions and results for both the RF and RR formulations are presented in this work, more details will be given concerning the RF problem. The RF formulation presents advantages as (i) the RF boundary conditions lead to a much simpler formulation than the RR conditions, thus allowing further algebraic manipulation and analytical investigation using various tools from the theory of nonlinear dynamical systems, (ii) the resulting equations reduce to the six-dimensional system studied earlier [ 18], thus allowing the examination of the influence of the higher-order modes, previously neglected, (iii) these equations, in turn, reduce further to the Lorenz system for a Newtonian fluid, a system extensively investigated in the past thirty years or so [ 12]. It is first observed that, if only one fundamental eigenmode was kept in expressions (6), one would obtain the trivial solution to the eigenvalue problem. This corresponds exactly to the level of approximation adopted in the previous work [18], where the flow of highly elastic fluids (De >> Re) could not be treated. It will be seen later that this amounts to neglecting important normal stress terms in the limit Re ~ O. The general solution, must then be chosen so that (i) linear behavior is recovered for small deviation from the base flow, (ii)
268
the Newtonian flow is recovered in the double limit De, Rv ~ 0, and (iii) the boundary conditions are satisfied. Only two terms tumed out to be sufficient in the expansion in x for all the velocity and stress coefficients u m ( x , t ) and rn~ij(x,t). For the RF conditions, trigonometric (Fourier) functions are used, similarly to (6a), except that the expansion coefficients are time dependent. For the RR conditions, hyperbolic (Chandrasekhar) functions are used to satisfy the stick conditions, similarly to (6b). There are thus sixteen independent modes that are retained. Higher-order terms are included in the radial and axial velocity, and the normal stress coefficients as opposed to keeping only one term in the previous six-dimensional system [18]. The truncation level adopted in the previous work [ 18] is similar to the one used by Kuhlmann [34] and Kuhlmann & Lficke [35] for a Newtonian fluid, where two terms are kept only for the velocity and stress in the y direction, and one term is retained for the remaining variables. This level of truncation does not leads to the trivial solution of the (linearized) eigenvalue problem in the limit Re ~ 0 [25]. Upon projection of the Fourier modes for the RF formulation, or the Chandrasekhar modes for the RR formulation, a set of sixteen ODEs are obtained, which govern the time dependent expansion coefficients. The two sets of equations are complicated and will be given explicitly elsewhere [56]. In the present study, each set of equations will be referred to as the sixteen-degree-offreedom nonlinear dynamical system (16NDS). The projection consists of multiplying equations (1)-(2), in the narrow-gap limit, by the appropriate mode and integrating with respect to x over the intervals [0, 1] and [- 1/2, + 1/2] for the RF and RR formulations, respectively. Upon elimination of higher-order axial and radial velocity, and normal stress coefficients, the system can be reduced to the six-dimensional system derived previously [ 18,56], which is given here for reference" (J- V + X-aRvU, l? - - U W + r U + Y - a R v V , - uv + b(Z-
)( - - 5(X
+ aU),
]" -
~ WX
- b UZ -
2 = fl(gY-
r
(8)
RvW),
- rX -
8 ( Y + a V
5(Z + a W),
),
269
where U, V, W , X, Y and Z are time dependent and are related to the expansion coefficients [18]. Note that a dot denotes differentiation with time. The parameters in system (8) are given by: 1
r = k 2 z'3Ta,
r=
2
b - 41r 2 r,
17= mb
+
1"
k 2'
~- E
,
(
)
(9)
qo- 31r2 - k 2 r
In the double limit E, R v --~ 0, that is, in the case of a Newtonian fluid, equations (8) reduce to the three-dimensional system derived by Kuhlmann [34], or to the Lorenz system with the Prandtl number equal to one [ 11 ]"
O-v-u,
12 - - U W
+ rU - V,
t~-
UV - bW.
(10)
It is important to observe that the nonlinear terms in the viscoelastic 16NDS stem from both the convective terms in the momentum equations and upperconvective terms in the constitutive equations. This is in sharp contrast to the case of Rayleigh-Benard convection of a viscoelastic fluid [14-16] where no nonlinear terms survive in the stress equations. The consequence of such nonlinear terms in stress, as we shall see, is drastic on the flow behavior, as opposed, for instance, to the cases of a convected Maxwellian fluid with less nonlinear terms, or linear Maxwellian fluid with no nonlinear terms in stress. It is at this point, perhaps, that one may begin to appreciate the difficulty with analyzing viscoelastic fluids given the marked difference in flow behavior subject to one constitutive model or the other. 3. L I N E A R S T A B I L I T Y ANALYSIS Before proceeding with the numerical solution of the 16NDS, subject to some appropriate initial conditions, it is useful to carry out local stability analysis in an attempt to unravel some of the fundamental differences between Newtonian and viscoelastic fluids. In section 2.4 the linear stability of the Couette flow was already examined in the absence of inertia for a Maxwell fluid (Re = R v = 0). Consider the influence of inertia on the stability of a viscoelastic fluid. For small values of E (or De) and Rv, one expects the behavior of the flow in phase space to be similar, in both the Newtonian and viscoelastic regimes, at least around the fixed points. As E increases, the stability picture changes, giving rise to a
270
periodic solution, or overstability, which is the result of normal stress effects that are usually not present in the case of a Newtonian fluid.
3.1. Stability of the circular (Couette) flow As in the case of a Newtonian fluid, one of the steady-state solutions of the 16NDS is the origin in phase space, which corresponds to purely circulatory motion in the azimuthal direction (base flow). As the Taylor (or Reynolds) number exceeds a critical value, T a c , additional fixed points emerge. For a weakly elastic fluid the stability picture is expected to be similar to that for a Newtonian fluid, and, in the limit of a small number of modes, to the stability of the Lorenz system [11]. One can also refer to the more complicated set of Newtonian equations resulting from taking the double limit of the 16NDS as D e , R v --> O, but the resulting stability picture is essentially the same as that based on the Lorenz equations. In the case of the Lorenz model (10), one easily deduces that at r = r c = 1, t w o additional fixed branches emerge, corresponding to the onset of (axisymmetric) Taylor vortices in opposite directions" Ws = r -
U s = G = + [ b ( r - 1)] 1 / 2 ,
1.
(11)
For the 16NDS, the critical value of r = r c (or T a c for the Taylor number) at the emergence of the two fixed branches is not as transparent as for equations (8) or (10). The value of r c tends to generally decrease as the level of elasticity increases. The value of r c corresponding to system (8) gives a reasonable estimate of this tendency [ 18]"
rc = ~
"
a
or
k2+.21,i 1 i ,
Ta c =
k )2
1 + a rE
(12)
where the term between square brackets constitutes the Newtonian contribution from the Lorenz system. In this case, T a c has a minimmn at !
k = km = ~[aE
.
.
.
.
n. 2
lr 2
(aE
+ 1) + aE
1] 1/2 1
Clearly,
in t h e c a s e
of a Newtonian fluid,
= 1 and k m = h a l 2 . Similar conclusions are reached when the formulae are cast in terms of the Reynolds number, R e , and Deborah number, D e . We shall use whatever notations are convenient as the present results will be compared rc
271
with those based on existing linear stability analyses. Comparison will be made with existing formulations that use exact solutions for the eigenvalue problem, and with the previous model that involves a lower number of eigenmodes [ 18]. Joo & Shaqfeh [27] carried out the numerical solution of the linear stability problem, using the RR boundary conditions, and examined the influence of inertia and elasticity for a UCM fluid. It is thus helpful to assess the accuracy of the present approximate method against their results. Fig. 1 displays the neutral
300
I
I
I
I
I
I
'I
De - 0.0 2.5 5.0 7.5 10.0
250
I
..... ..... .......... .....
/
200
~D
150 9f,,,,\ "('..', ,,
100
I
..-"
I"
UNSTABLE ~ "r
..'" ...... . .... ""'"""
_.."'I""
,,
, i-'"
"~ \ " ", ~ 9 "\'..... ".... 9 "\ "... " . ".-..
..--" _.~-'"
...-'"
...- ~
,,.
..-""
.i"
9...........
. ....... .'" . .......... .-'~'~"
..
. ............ . ~ . ~ . ~ " ........... ~ . . ~ . ~ ' ~
. . . - o"
50 "
'
I
I
I
I
I
1
2
3
4
5
,
,
,
7
8
I
6
Fig. 1. Influence of fluid elasticity and marginal stability curves for a UCM fluid (Rv = 0) based on the RR boundary conditions and e = 0.01. stability curves in the (Re, k)-plane for the onset of steady TVF for 0 < De <_ 15 and e = 0.1. Note that these curves are based on the linear stability analysis of the 16NDS. For all De values, the critical Reynolds number Rec = Re(k = k ) decreases roughly like 1/k2 near k = 0, reaches a minimum, and then increases roughly like k2 for large k. There is a shift in the value of km towards the fight as
272
De increases, thus reflecting the greater difficulty for axisymmetric Taylor vortices to be observed in the case of highly elastic fluids. On the other hand, this minimum becomes less localized with increasing fluid elasticity, resulting in a wider range of wave numbers at which Taylor vortices set in. However, for De > 10, the neutral curves show again a very localized minimum, with an increasingly narrower range of wave numbers for the Couette flow to lose its stability to steady TVF. For a given k value, Rec decreases generally with De roughly like 1~De. This is, perhaps, the most important result in fig. 1" fluid elasticity tends to precipitate the onset of axisymmetric Taylor vortices at any value of the wave number in the axial direction. The results of Joo & Shaqfeh (reported in [27]) display the (Re, k) neutral stability curves for the same range of the Deborah number. Their results should be regarded as exact. They show good agreement with the curves based on the current approximate method. The approximate curves tend to give slightly lower values for Re c and ktn . Neutral stability curves in the (Re, k) plane were also obtained using the RF conditions. This enabled the assessment of the influence of the boundary conditions on the stability picture. The RF marginal curves lead to essentially no qualitative disagreement with the curves in fig. 1. Both figures show a localized minimum at k -- km that increases with De, a tendency of the curves to flatten around the minimum for the more elastic fluids, a general decrease in the critical Reynolds number as fluid elasticity increases, and the narrowing range of practical k values for the onset of steady TVF. Consider now the important question whether there is any (at least qualitative) change in the stability picture due to the inclusion of higher-order modes in the 16NDS as opposed to keeping only six modes as in system (8) [18]. A direct quantitative comparison was carried out by examining the influence of fluid inertia, elasticity and retardation through Re, E and Rv, respectively. Comparison shows that the stability picture is qualitatively the same in both cases. The inclusion of higher-order modes appears to make little difference for Newtonian and weakly elastic fluids. In fact, the curves corresponding to small values of E (E = 0 and 0.01) were essentially identical. Deviation between the two sets of curves became more evident for the larger E values. This is expected, as discussed above, since higher-order modes, previously neglected in [ 18], become important as the level of fluid elasticity increases. In this case, the neutral stability curves appear to be flatter as a result of the inclusion of higher-order modes. Similar conclusions are reached as to the influence of fluid retardation on the critical Taylor number Tac for an Oldroyd-B fluid with E = 0.5. Fluid retardation
273
tends to delay the onset of Taylor vortices. Note that in the limit Rv ~ 0% one recovers the Newtonian marginal stability curve (E = 0). The influence of Rv on the loss of stability of the base flow is to be expected since retardation tends to delay the onset of instability. Comparison shows a growing deviation and flattening of the curves for the lower Rv values considered. Recall that as retardation decreases, elasticity becomes relatively more dominant, and the influence of the higher-order (normal stress) modes is expected to be stronger. It is interesting to note that the curves in fig. 1 are reminiscent of those corresponding to the onset of overstability obtained from the linear analysis of Larson et al. [25] (see also fig. III-11 by Larson [20]). It is clear in both cases that fluid retardation tends to prohibit the onset o f steady and oscillatory Taylor vortices. Although the mechanism of onset of instability is different in the two situations (given the absence of inertia in the analysis of Larson et al. [25]), the role of fluid elasticity (at least for an Oldroyd-B fluid) appears to be very similar to that of inertia. This is particularly obvious from fig. III-11 in [20], which clearly shows the destabilizing influence of both inertia and elasticity. Additional calculations and comparisons with existing results were also carried out to closely examine the influence of fluid elasticity and retardation. The results will only be commented upon here but not shown for lack of space. The behavior of the minimum value of the critical Taylor number Tacm(E,Rv)-Tac(k=km,E,Rv) as function of the Elasticity number and viscosity ratio reveals a very similar trend as established previously (fig. 3 in [18]) on the basis of model (8). It is found that Ta m drops sharply with E at small elasticity in the case of a UCM fluid, in a manner similar to that predicted by earlier linear analyses (see fig. 111-6 in [20] and fig. 8 in [57]) for the case of corotating cylinders). The decrease in Ta m is considerably attenuated at the larger E values. As to the influence of fluid retardation, there is a flattening in the (Ta m , k) curves as Rv increases. This behavior agrees with that obtained from the linear analysis of Thomas & Walters [58] in the case of a Maxwell fluid, and those of Ginn & Denn [59] for a second-order fluid corresponding to a second normal stress of zero value. In this case, the formulation based on the secondorder fluid is expected to lead to the same stability picture as that based on a Maxwell fluid with small E value. Similar agreement is reached, as to the influence of fluid elasticity on the value of the wave number km at which the critical value of the Taylor number has a minimum, upon comparison with earlier formulations [ 18].
274
3.2. Stability of the Taylor vortex flow and bifurcation The previous section (3.1) deals with the loss of stability of the (purely circulatory) Couette flow to steady or oscillatory TVF depending on whether inertia is significant or negligible. In this section, attention is focused on the stability of the TVF itself. For a Newtonian fluid, the emergence of the two nontrivial branches at r = 1 is accompanied by an exchange of stability between the origin and the two fixed branches via a supercritical bifurcation. An exactly similar situation is encountered in the Rayleigh-Benard convection of a Newtonian fluid [37], and even in the convection of a viscoelastic fluid of the Oldroyd-B type [ 18]. For the Taylor-Couette flow of a viscoelastic fluid, the two solution branches correspond to the nontrivial steady solution of the 16NDS. A closed form solution such as (11) is difficult to obtain in this case because of the nonlinear coupling in the flow field and the large number of equations involved. The solution is thus obtained numerically using the damped Newton-Raphson method. On the other hand, the steady-state solution of system (8) is possible to obtain in closed form, and is written here for reference as [ 18]: (U 2 - 5b)(U 2 - 8 ) + ( a R v + a fa+ 5 ) ( b a R v - a f a + 5 ) U 2 r
~
a ( b R v - (a- 1)U 2 + b 5(a + 5 ) Ws = Xs
U Z + (a + 5 ) r - tY baRv - a fa + tY '
= -aUs,
Ys = ( a R v - r - W s ) U s ,
Vs = Us, Z s = aRv Ws -
(13)
b
It is thus clear that the solution of the first equation in (13) leads, similarly to the Lorenz system, to two nontrivial solution branches corresponding the onset of Taylor vortex flow. The steady-state solution of the 16NDS also leads to the emergence of two nontrivial branches at r = rc(E, Rv, k), which will be referred to as C 1 and C2, that are symmetric with respect to the r, and correspond to the onset of Taylor vortices in the two opposite directions. Unlike the case of a Newtonian fluid, the pitchfork bifurcation at r = rc is not always supercritical. In fact, our previous study based on system (8) shows that the bifurcation is supercritical only for weakly elastic fluids. When the Elasticity number E exceeds a critical value, E = Esub, the bifurcation becomes subcritical. This bifurcation picture is confirmed when higher-order modes are included as we shall see next.
275
For weakly elastic flows (E < E s u b ) , the 16NDS leads to an exchange of stability similar to the case of a Newtonian fluid, which takes place at r = rc(E, Rv, k), except that the critical value r c now depends strongly on fluid elasticity and retardation. As was established above, rc is different from one, and becomes increasingly smaller as E increases. At r = r c, a supercritical bifurcation emerges. The base flow, which is stable for r < r c, loses its stability to the two steady branches C 1 and C2 as r exceeds r c. Thus, for r > r c, the solution evolves to either one of the two branches depending on the initial conditions. As r increases and reaches a critical value, the two branches lose their stability as will be seen below. This stability and bifurcation pictures remain essentially unchanged until E exceeds E s u b , when the supercritical bifurcation gives way to a subcritical bifurcation at r = r c. The influence of the Elasticity number on the bifurcating branches at r = r c is depicted in fig. 2 for a UCM fluid ( R v = 0) and k = 3. These curves correspond the RF conditions, and should be compared to the bifurcation diagrams based on system (8) as shown in fig. 5 of [ 18], and included here in the inset of fig. 2 for reference. Because of symmetry with respect to the r-axis, only one branch (C 1) is shown. The curves in the figure are obtained numerically, contrary to those based on the analytical expressions (11) and (13). There are several important aspects to be observed from fig. 2, and the inset therein, in comparison to the Taylor-Couette flow of a Newtonian fluid. It is first observed that there is a strong dependence of r c on fluid elasticity. For small E, there is a supercritical bifurcation at r = rc, with r c becoming increasingly smaller as E increases, and decreasing all the way to zero. At the critical value E = E s u b ( R v , k), which in this case ( R v = 0, k = 3) is equal to 0.018, the bifurcation changes from super- to subcritical. This corresponds to a change in concavity at (r = r c, U] = 0). In the supercritical regime, when r < r c, a small disturbance of the base flow decays exponentially according to linear theory. The nonlinear terms in the 16NDS remain small in this case. As r exceeds r c, linear theory predicts an exponential growth of a small disturbance of the base flow. This growth is, however, halted by the stabilizing nonlinear effects. Although linear theory may well describe the onset of secondary flow, it fails to give the magnitude of the steady disturbance. In the subcritical regime (E > E s u b ) , when r < r g (shown here for the curve E = 0.05), the base flow is globally asymptotically stable [44]. When r g < r < r c, t w o bifurcation solutions exist for the disturbance from the base flow. In this case, any disturbance below a certain threshold decays to the origin. Above the threshold value, the base flow is destabilized. In this range of r values, the flow is commonly defined as metastable [60]. Whether the bifurcation at rc is subcritical or supercritical,
276
i
1.8
I
I
I
I I
I
I
I
I .-
"
!
I
;
E = 0.000 0.006 ..... 0.018 ...... 0 . 0 4 0 ........... 0.050 ..... 0.100 .....
...............:.;.::.-.....................
1.6 ..., ........
1.2
:..
_
9 9
'.
....
...-.
//
1.6 0.8
.... --- .......................
1.4 0.4
1.2
o.o.
.~ . . - o. I
0 0
i
0.2
I
i
0.4
/.
..-'"
j . ../
s/.
il
1
0.8
0.6
1.2
,'"
1.4
E sub = 0 . 0 1 8
.-"
,,"
,,,./"
,'
/1"
.................................................... - ......................................................?.':" ............................................ s
0.8
~
',
I'".,,
I
<-'- ................. -.,
"-.. 9~" ~- ~
-.i
I
I
I"I ....
11....
, I
, I i ,
",.
,,.
0.2
i
0.2
, I I ,
""..
", i
~ ~
",..
/
~.
i i I
,
-
/ t
-
.~
/
)r / ............................... " / .......................... -
",
\
/
t
; ................
~.
-~
i' rg
:
-.-'.. ..................
x
0.4
/
~ "-.
~ "~. .....
..................
.'
:
...
,
,
_/-
i
~.~
"~I il
i- . . . . . . . . . . . . . . . . . . . . . . .
.......
,,.~,...
0.6
0.4
l"
_ ..........................
/
/
/
I
-
I
I
I
0.6
0.8
~ *
I
I
1
1.2
.
.4
I I rh
rc
r
Fig. 2. Bifurcation diagrams and influence of higher-order modes for a UCM fluid (Rv = 0) with k = 3 and e = 0.01. Super- and subcritical bifurcations for weakly and strongly elastic flows corresponding, respectively, to E < Esub and E > Esub. For the subcritical curves, the stable regions are indicated by breaks in the curves and are delimited by two arrows. Inset shows bifurcation diagrams based on lower-order theory eqs. (8) with Esub = 0.014. the TVF loses its stability at some critical r value r h > r c (shown here for the curve E = 0.05) Consider the stability of the steady branches C 1 and C2 as r is increased from r c. It is helpful, however, to first recall the situation for a Newtonian fluid. The discussion is focused on the Lorenz system (10) since the inclusion of higherorder mode does not alter the qualitative picture. Linearization of the Lorenz equations around the steady-state solution leads to a characteristic equation of the third degree. The conditions for the loss of stability of the toroidal TVF is of particular interest here. For a Newtonian fluid, the two nontrivial fixed points C 1
277
and C2 are always sinks for any value of r. At r = 1, a pitchfork bifurcation occurs, while the origin remains a saddle point with a one-dimensional unstable manifold. Note that this is exactly the same situation for the Lorenz equations with the Prandtl number equal to one. In this case, no Hopf bifurcation occurs since the characteristic equation does not possess a pair of purely imaginary roots [61 ]. The roots have a real part that remains negative for any value of r. Thus, for r > r c = 1, the three-dimensional Newtonian model cannot predict the destabilization of the TVF (through a Hopf bifurcation, for instance), nor can it entertain a chaotic solution. This situation remains unchanged when higher-order modes are retained in the solution. The presence of fluid elasticity in the constitutive equations appears to be a sufficient condition for the existence of a Hopf bifurcation. Indeed, at r = r h ( E , R v , k) > r e, linear stability analysis around C 1 and C2 indicates that these two branches lose their stability via a Hopf bifurcation emerging at r - r h. In fig. 2, sub
the range of r values for which the branch C 1 is stable (for E > E ) is indicated by breaks in the lines between two arrows. It is observed from the figure, that for highly elastic fluids (E > 1), the range of stability of the C1 and C2 branches decreases to zero, so that the (steady) TVF is unstable for any postcritical r value. While the existence of the Hopf bifurcation may not be difficult to establish, the investigation of its stability can be algebraically quite involved [62]. The difficulty stems from the hyperbolicity of the fixed point at r = rh, and center manifold theory must be applied in order to determine the stability of the fixed point [63]. The situation is similar for r near r o The stability picture will be thus established through the numerical solutions of the 16NDS. 4. FINITE A M P L I T U D E TVF AND C O M P A R I S O N W I T H E X P E R I M E N T Linear stability analysis, such as the one presented in the previous section, determines the flow field as it is slightly perturbed from the base flow or from the (steady) TVF. However, it fails to give the flow structure for a large disturbance. The influence of the nonlinear terms must thus be examined through the numerical solution of the 16NDS. The influence of inertia and elasticity on weakly and strongly elastic flows is examined in some detail. A weakly (strongly) elastic flow is defined as one whose Elasticity number is small (large) enough for it to undergo a supercritical (subcritical) bifurcation at the onset of Taylor vortices. Particularly, the emergence of super- and subcritical Taylor vortices, as well as the onset of chaos, are investigated by determining the flow spatio-temporal structure in phase space, through power spectra and/or time
278
signatures, whichever representation is most insightful. The present nonlinear formulation is assessed against experiments for a weakly elastic flow and a flow with negligible inertia. 4.1. Influence of inertia on weakly elastic flow
The influence of inertia on a moderately weakly elastic Oldroyd-B fluid is examined for a fixed value of E by varying the value of D e (or, equivalently, Re). This is equivalent to fixing the level of fluid elasticity and increasing the shear rate. Baumert & Muller [9] performed experiments on the flow visualization of the Taylor-Couette flow of dilute solutions of high molecular weight polyisobutylene in oligometrie polybutene. Rheologieal measurements of these solutions confirmed that they are of the Boger fluid type (highly elastic with constant viscosity). They monitored the temporal evolution of TVF over a range of shear rates using reflective mica platelet seeding. The transition to steady TVF was observed to occur with or without oscillatory flow depending on whether elasticity was dominant or not. In this section, the calculations are carried out for a weakly elastic flow of moderately small inertia (Re .~ 100), and the transient flow behavior is compared with the observations of Baumert & Muller [9]. The aim of the present comparison is to reproduce some of the experimental observations as the base flow loses its stability to TVF. Given the uncertainty in the experimental values of the relaxation time and the wave number, and the assumptions adopted in the derivation of the low-dimensional dynamical system, only a qualitative agreement can be expected. The results are also compared with the numerical calculations of Avgousti et al. [28] whenever possible. The calculations performed in this section are based on the rigid-flee boundary conditions. For a fixed level of fluid elasticity (E fixed), experiment suggests that transition from the purely azimuthal flow to steady TVF occurs once the shear rate reaches a critical value, i.e., D e = D e c (or Re = R e ) . The transition from simple sheafing to the large toroidal axisymmetrie vortices, which span the gap, is monotonic with time as D e first exceeds D e c. At higher D e value, an oscillatory secondary flow is observed prior to the formation of steady vortices. Axially migrating mica alignment bands begin to appear, but the band formation ceases with steady vortices setting in after some time. At this point, steady TVF is observed to persist for a long time. Consider the ease of a moderately weakly elastic fluid with E = 0.16. This corresponds roughly to the low-viscosity Boger fluid used in the experiment of Baumert & Muller [9], with p = 0.856 g/c m3, rls = 2.6 poise and I"1 = 3.0 poise.
279
The value of the relaxation time ~1 was found to be equal to Ls = 0.021 s or ~t-0.11 s (with the corresponding Elasticity number equal to E = 0.164 or E t = 0.94), depending on whether steady shear or transient experiments were used [9]. The value of the viscosity ratio is set equal to R v = 6.5. Figure 3 shows the time evolution of the (dimensional) axial velocity component U z at z = k/2, where it is maximum, and x = 0.5, midway through the gap. The evolution of the velocity is displayed for various values of the shear rate: 13 < D e < 14 (81.25 < R e < 87.5), above the critical shear rate, D e c ~ 13 (Re c ~ 81.25). The curves in the figure are similar to the results reported in fig. 1 of Avgousti et al. [28], and the trend is in agreement with the observations of Baumert & Muller [9] for low and medium
0.01
0.009 -
....................................................................................................
9 ..../
0.008 -
"''".... ................................................................................................................................................................................................
0.007-
0.006-
0.005 :
/,'-
: i
0.004-
....
i /
:
tI
0.003 -
De = 13.09 13.18 13.27 13.49 13.70
0.002 i
9 :;
i
0.001 J
i
ii
/
, ,'
,'"
,"
,i /
/
,'
o I 0
5;0
10b0
!
15;0
2000
25'00
..... ..... .......... .....
30'00
3500
Time (s)
Fig. 3. Monotonic time evolution of the axial velocity for different shear rates (13 < D e < 14) and small initial perturbation of the base flow (E = 0.16, R v = 6.5, k = 3.5 and e = 0.093).
280
viscosity fluids. The initial condition taken for all the curves in fig. 3 is U1 = 0.01 with the remaining variables equal to zero. In this case, the transition from the base flow to TVF occurs via a supercritical bifurcation similar to the branches E < 0.018 in fig. 2 for a UCM fluid. In the precritical range (De < Dec), the origin in phase space (base flow) is asymptotically stable to any perturbation, and the numerical integration of the 16NDS, in the vicinity of the origin, follows simply an oscillatory decay to the origin. As De (Re) exceeds Dec (Rec), an exchange of stability occurs between the origin and the steady-state branch C1 (or C2), coinciding with the emergence of steady TVF, with increasing amplitude as De (Re) increases.
100
Theory
+ + +
+ -i-
....,
[-.,
Experiment
10
100
De
Fig. 4. Time after shear initiation of appearance of steady vortices as function of the shear rate (De). The values in the figure are based on the curves in fig. 3. The inset shows the experimental measurements of Baumart & Muller [9] for a medium viscosity fluid.
281
Typically, the flow evolves monotonically from the base flow to steady TVF over a period of time that decreases with increasing shear rate as depicted from fig. 3. This appears to be in agreement with the observations of Baumert & Muller [9] who monitored the dependence of onset time on the shear rate for a fluid with medium viscosity. Figure 4 displays the time after shear initiation of appearance of steady vortices as a function of the Deborah number (shear rate). An inset showing the values from experiment is also included for comparison, and reveals good qualitative agreement with theoretical predictions. The values in the inset are based on measurements performed for a medium viscosity fluid, with E = 44. As will be seen next, at higher shear rates, however, theory predicts that the time TVF takes to set in begins to increase with De, a trend in disagreement with what experiment suggests.
0.o45[ '~ o
0.0as
.
li
.
.
.
.
.
.
.
,
1 '11 " !
i
o.o
ii
0.0254 i1!/ 7h
|
1
~
o.o2
I
0 9005
o
it i :V~i :.\ i ,,:' '.. ..-,, i !
l, i i
:,i"
:",
".,,':'.2.~,vc.':-.-~'c:-
"----:---- .....
"
......... -............
---"-
.....
1
i. i
De = 17.47
'
".,.,i
17 81 17194
5~o
lo'oo
~;oo
2000
Time (s)
Fig. 5. Oscillatory time evolution of the axial velocity for different shear rates (17 < D e < 18) and small initial perturbation of the base flow. The parameters used are the same as in fig. 3.
282
0.028
-
(a)
0.026:
18.13 J
_
0.02416.78
0.022 De = 16.07
0.02
0.0 8 .
,
.
--
,.
-
0.028
(b) 0.027 -
q7
19.14
0.026-
[hh. De = 18.77
18.89 11111#~r162
....
-
0.025 0.024-
0.08-
....
19.38
(c)
.
.
.
.
.
.
i
0.040-0.04 -0.08 - i .....
-0
2000
1 ....
4000
6
010
0
"'!
8000
I
10000
I
12000
"
I
14000
!
16000
18000
Time (s)
Fig. 6. Time evolution of the axial velocity for 16 < De < 21. The flow for each shear rate level is obtained using the flow of the previous (lower) level as initial conditions. The parameters used are the same as in fig. 3. The figure is subdivided into three segments of 18,000 s period each. Steady TVF is always reached during the first period (a) for the range 16 < De < 18.7. Oscillatory instability develops towards the end of the second period (b) for the range 18.6 < De < 19.27. Transition to periodic behavior some time during the third period (c) for the range 19.26 < De < 21.
283
The evolution of the flow loses its monotonicity as shearing exceeds a certain level. This is depicted in fig. 3 for the curves corresponding to De > 13.2, which show a local depression in the flow amplitude before steady TVF sets in. In fact, as De is further increased, an oscillatory behavior emerges before the onset of steady vortices as shown in fig. 5 for 17 < De < 18. The secondary flow gains strength initially at a much faster rate than for the lower range of shear rates (fig. 3). The amplitude of oscillation decays with time until the onset of steady of TVF, but the frequency appears to remain constant. As De increases, the overall amplitude of oscillation increases, accompanied by a phase shift and delay in the onset of steady TVF. Eventually, oscillatory behavior is sustained for longer time for higher De values; in the limit, no steady TVF is found. Before examining the onset of sustained oscillations, it is desirable to address the issue of the influence of the initial conditions on the ensuing flow behavior. Baumert & Muller [9] report that "flow behavior was not found to be particularly sensitive to the type of velocity ramp imposed" initially. They indicate that the flow behavior commencing at the start of the final velocity was indistinguishable from the case when a plateau was imposed over a certain period. This observation is presently confirmed through additional calculations carried out by taking as initial conditions the steady TVF reached before increasing the shear rate, instead of starting again from near the origin as in figs. 3 and 5. The resulting sequence of initial flow transients, after the imposition of the higher (new) shear rate, and subsequent final (steady or oscillatory) TVF are depicted in fig. 6 for the range 16 < De < 21. The figure displays the evolution of flow over a period of 54,000 s, subdivided into three equal periods of 18,000 s. The first segment is shown in fig. 6a for the range of shear rates corresponding to 16 < De < 19. Comparison between the curves in fig. 5 and those in fig. 6a indicates that the flow behavior is essentially unaffected by initial conditions. It is interesting to note the similarity in the emergence and decay of oscillatory behavior between one steady level and the other. The figure clearly shows that the time it takes steady TVF to set in increases with shear rate. This is a result of the oscillatory transition that takes a longer time for the higher shear rate. This is indeed even more evident from fig. 6b, which shows, for the range 18 < De < 20, that steady TVF does not even set in over the time period a given shear rate is applied. The oscillations tend to decay at a slower rate as De increases, and eventually start to grow with time as indicated in the last portion of fig. 6b, corresponding to De = 19.26. This growth, which is linear upon onset, does not remain unstable over time as linear stability analysis suggests. At some time, when the signal amplitude is large enough for nonlinear effects to become significant and halt the (exponential) growth.
284
The bifurcation diagrams in fig. 2 suggest that steady TVF is reached as long as r is below rh(E, Rv, k). This corresponds to the onset of a Hopf bifurcation at De - Deh when steady TVF loses its stability to periodic behavior. For the present flow in fig. 6, Deh is found to be approximately equal to 19. The existence of a Hopf bifurcation at De = Deh is easily confirmed through linear stability analysis around the nontrivial steady-state solution(s) when two eigenvalues are purely imaginary, and no other eigenvalue possesses a zero real part. The stability of the Hopf bifurcation, however, is difficult to establish for a complex system like the 16NDS. Such an analysis is algebraically involved, but a similar analysis, based on center manifold theory, was carried out previously for the thermal convection of viscoelastic fluids [15]. Numerical integration, on the other hand, shows that the Hopf bifurcation indeed exists, and is stable as will be seen below. This situation is in sharp contrast to the case of the Lorenz system (10) whereby no loss of stability of the steady TVF is predicted. In figs. 6a and b, and for the range 16 < De < 19.15, the computation has been limited to flows with final state, after transients have died out, that corresponds to a standing wave. This also corresponds to most of the range of shear rates examined in the experiment of Baumert & Muller [9] that led to steady TVF for the low and medium viscosity fluids. However, they reported the observation of coexistence of migrating bands and distorted vortices at the highest shear rate considered. It is thus desirable to numerically examine the transition from steady to oscillatory TVF. The sequence of flows displayed in fig. 6c is obtained for De above Deh subject to the initial condition corresponding to the steady TVF at De slightly below Deh. Avgousti et al. [28] reported having numerical difficulties for a UCM fluid when they took the slightly perturbed standing wave as their initial condition. Figure 6a shows the transition from steady to oscillatory TVF, that is, from the flow at De = 19.26 to that at De = 19.38. The figure also shows the transition to another oscillatory state at De = 20.56. The first transition is preceded by a destabilization of the steady TVF to a traveling wave of amplitude (exponentially) growing with time, which is reminiscent of the behavior reported in figs. 6 and 9 of Avgousti et al. [28]. The exponential growth (instability) ceases immediately upon increase of the shear rate, giving way to a regular and stable oscillatory vortex structure. This structure is sustained for as long as the shear rate is maintained at De = 19.38. Upon further increase of De, another regular oscillatory structure sets in with a larger amplitude. The modulation of the oscillation indicates the emergence of period doubling. Additional theoretical and experimental investigations are obviously needed if the flow at still high shear rate is to be explored. It is possible that the flow may not remain
285
axisymmetric, a fact that does not seem to be suggested by the experiments of Baumert & Muller [9]. 4.2. Weakly inertial flow and influence of boundary conditions The nonlinear behavior is further examined by focusing the calculations on the influence of fluid elasticity in the presence of very weak inertial effect. One of the most interesting phenomena encountered in the Taylor-Couette flow of a viscoelastic fluid is the emergence of overstability that is attributed to fluid elasticity, which is otherwise absent in the case of a Newtonian fluid. The existence of such a purely elastic overstable mode was proved through linear stability analysis [25] and numerical calculations [30] carried out in the absence of inertia. Recently, Muller et al. [8] conducted Laser Doppler velocimetry (LDV) measurements of the axial velocity component of a Boger fluid between two concentric cylinders (~ = 0.0625), with the outer cylinder being at rest, and the inner cylinder rotating at constant angular velocity. The measurements show an oscillatory flow at a vanishingly small Reynolds number. The flow appears to undergo a transition from the purely azimuthal Couette flow to time periodic flow as the Deborah number De exceeds a critical value, Dec, which is in good agreement with the value predicted by earlier linear stability analysis of an Oldroyd-B fluid [26]. The LDV measurements show that the oscillatory behavior is not localized but appears to be spread throughout the flow. As the Deborah number increases from the critical value, the amplitude of oscillation increases like (De - Dec) 1/2. The corresponding power density spectra show peaks, which are instrumentally sharp at the fundamental frequency, the growth of harmonics, and eventually subharmonics, reflecting, perhaps, the presence of a period doubling or quasiperiodic motion. A similar sequence of flow behaviors is also obtained from the finite-element calculations (with inertia neglected) of Northey et al. [30] for a UCM fluid. These authors, however, report having numerical difficulties in obtaining the solution at the higher Deborah numbers (possibly coinciding with the onset of period doubling). Their calculations are thus limited to an extremely small range of postcritical Deborah numbers. As discussed earlier, the system (8) used in our previous study [18] could not possibly reproduce any of the reported experimental results for vanishingly low-Reynolds number. This fact is easily verified if the inertia terms are dropped from system (8). In this case, the acceleration terms on the left-hand side, together with the curvature term (V) on the fight-hand side of the first equation, are neglected. It is then easy to deduce that one recovers the trivial solution or the purely azimuthal flow. Thus, the level of truncation adopted in the previous model makes the formulation inadequate
286
for the investigation of the purely elastic overstability or even a flow at very small Reynolds number. In general, however, the presence of inertia, no matter how small it may be, prohibits the base flow from losing its stability to the overstable mode. Instead, the base flow loses its stability first to steady (and not oscillatory) TVF since there is always a finite range of r values over which the branches C 1 and C2 are stable (see fig. 2). The influence of fluid elasticity is now examined for vanishingly small inertia in the postcritical range of the Deborah number, in an attempt to recover the flow sequence observed in the experiment of Muller et al. [8]. One cannot expect full quantitative agreement between theory and experiment given the level of approximation in the 16NDS. Moreover, the wave number k needs to be imposed in the present calculation. This quantity is not known from experiment; the wave number is difficult to establish under unsteady conditions of flow [9]. The comparison between theory and experiment is covered next for the RF and RR formulations. Consider the flow with negligible inertia, and set r = 10-6. Also let Rv = 3.75 and e = 0.0625 corresponding, respectively, to the viscosity ratio of the fluid and the gap-to-radius ratio used in the experiment [8]. Although the wave number is not specified in the experiment, it is set k = 4 based on wave numbers reported in other experiments on Taylor-Couette flow of viscoelastic fluids [9]. Thus, only the Deborah number will be varied. The flow is examined as De is increased from zero, that is from the Newtonian level. Referring to fig. 2, it is seen that for De = E = 0 the base flow is unconditionally stable to any perturbation since r << 1 in the present case. This situation remains practically unchanged until De reaches the theoretically predicted critical value, Dec = 29.7 or 29.3, depending on whether the RF or RR boundary conditions are used. These values are based on the purely elastic linear stability analysis. At this point, the base flow is supposed to lose its stability, but this is numerically detected at slightly higher De values. Since inertia effects are small (r = 10-6), one observes an exchange of stability between the base flow and oscillatory TVF, since no steady TVF can set in in the absence of inertia. As the Deborah number is increased beyond the critical value, the amplitude of oscillation increases. The resulting sequence of flows is identified through the Fourier spectra shown in figs. 7a and b for the calculation based on the RF and RR boundary conditions, respectively. The range of Deborah numbers studied is the same as in the experiment: 32 < De < 53, and should be compared with that reported in figs. 7-10 of Muller et al. [8], where it is noted that De = 0. 911DeM~lle~.
288
Theoretically, oscillatory TVF is expected to emerge at De = 29.3 for the RF formulation; in practice, however, it is detected at a higher Deborah number. At De = 33, the corresponding Fourier spectrum shown in fig. 7a clearly displays periodic motion after the base flow becomes unstable. This periodic behavior persists as De increases. The flow oscillates around the origin (base flow), with an amplitude that increases as De increases. The spectrum shows the fundamental frequency and six odd harmonics. Additional harmonics emerge upon further increase of the Deborah number as shown for De = 39.7. Further increase in the elasticity level (De = 41.2) leads (most apparently) to a doubling in the period as depicted from the figure. It is not easy, in fact, to establish whether one is dealing with a bifurcation into period-2 motion or quasiperiodic behavior on a toms. At De = 43.57, the power spectrum indicate clearly the presence of chaotic motion. A very similar sequence of flows is obtained on the basis of the RR formulation. The qualitative similarity is obvious from fig. 7b. Although the two formulations predict that the oscillatory TVF sets in at essentially the same critical Deborah number (29.3 vs. 29.7, for the RR and RF conditions, respectively), the emergence of higher harmonics, subharmonics and chaos occurs at higher De values for the RR formulations (fig. 7b). This confirms, once again, that stick conditions tend to delay destabilization. The sequence of flows predicted by the present model is clearly comparable to that reported by Muller et al. [8]. Experiment predicts a loss of stability of the origin at a critical Deborah equal to 32.3 while the present formulation gives Dec = 29.7. Direct quantitative comparison of the actual velocity amplitude is impossible since not all experimental parameters are available. It is possible that the experiment was in fact not conducted at fixed Reynolds number, and that the Deborah number was increased by increasing f2. Both theory and experiment predict the increase in amplitude of the velocity signal, the emergence of higher harmonics in the Fourier spectrum, and eventually that of subharmonics (corresponding to the onset of period doubling) as De increases. Figure 8 shows the bifurcation diagrams for the square of the velocity amplitude based on the present model and the measurements (inset) from [8]. In the figure and inset, numerical calculation and the experimental measurements show the onset of periodic motion as the Deborah number exceeds the critical value. Both sets of data show that the amplitude of oscillation grows like (De Dec) 1/2, in agreement with the prediction based on asymptotic analysis in the limit De --~ Dec. Although the qualitative agreement between theory and experiment is obvious, the quantitative comparison is also encouraging, especially for the RR results, given the assumptions adopted in the present formulation: level of truncation involved, axisymmetric flow, narrow-gap
289
0.14-
0.01
Experiment
0.008 0.12
0.006 O
0.004 0.1-
0.002 0 20
2;
3;
0.08 -
0.06-
0.04-
0.02-
020
2'5
!
30
35
40
45
5'o
55
De
Fig. 8. Bifurcation diagrams based on the present model (RF and RR formulations) and the square of the axial velocity component amplitude. Inset showing the experimental measurements of Muller et al. [8]. approximation, the constitutive model, and, very importantly, the uncertainty in the value of the wave number. The subject of elastic overstability is an important phenomenon in viscoelastic fluids, and remains somewhat unexplored in the nonlinear regime. 4.4. Purely elastic overstability The experimental measurements of Muller et al. [8] were conducted under conditions of vanishingly small inertia. A closer quantitative comparison between theory and experiment is obtained when a modified twenty-mode dynamical system (20NDS) is used, which accounts for even higher normal stress effects than in the 16NDS, with the Reynolds number set equal to zero [ 19]. The most influential normal stress modes are carefully selected to ensure that the
290
relevant dynamics is captured by the approximate model and solution. This is first done by referring to the results from linear stability analysis as above. Comparison between the approximate and exact solutions leads to good agreement, especially in the lower wave number range. The model takes into account more effectively the influence of normal stresses (which lead to the Weissenberg rod-climbing phenomenon), and is thus adequate for the flow of a highly elastic fluid (with negligible inertia), thus allowing direct comparison with the experiment of Muller et al. [8]. Not all needed experimental flow parameters were explicitly reported in ref. [8]. The test fluid used in the experiment has a (constant) viscosity 11 = 162 Pa s, and consists of 1000 ppm of a high molecular weight polyisobutylene, dissolved in a viscous, low molecular weight polybutene of viscosity rls = 128 Pa s, so that the solvent-to-polymer viscosity ratio Rv - 3.76. The fluid relaxation time )~ varies depending on the rheological technique used to measure it. Steady shear flow data give )~ = 3.3 s, while transient relaxation experiments lead to 10.9 s [8, 9]. The inner and outer cylinder radii were 8 and 8.5 cm, respectively, so that ~ = 0.0625. Although the inner cylinder angular velocity f~ was not explicitly given in the experiment, its value can still be inferred from the values of the experimental Deborah number, DeM, which was introduced by Muller et al. [4] 2
as: DeM -
(1 + M = De. It appears that there was only one (1 + ~2 _ 1 . Note that De ~--~0
fluid used throughout the experiment, and DeM was probably varied by varying only the inner cylinder speed, f~. Hence, from the range of DeM values reported, the corresponding value of the inner cylinder speed can be given by f~ = DeM/77.06 for ~. - 4.4 s. Muller et al. [8] reported that the highest Reynolds number, Re, reached in the experiment was of the order 7x 10-3. Indeed, if one considers the value of f~ corresponding to the highest Deborah number reported (De M = 54.5), one finds that Re = 2.86 x 10-3 (assuming the density 9 ~ 1 g/cm3). The experimental wave number, k (in units of d), at which overstability is first observed, was also not reported by Mullet et al. [8]; its measurement may have been difficult under transient conditions. Its exact value, however, is not crucial in this case since the critical Deborah number for the onset of overstability does not depend strongly on the wave number, over a wide range of practical values: k e [4, 8] for Rv = 3.76 as linear analysis suggests [20]; this is reflected by the flattening of the corresponding neutral stability curve in the (De, k) plane around the critical value Dec. The wave number will be fixed to k = 4.85 for all subsequent calculations. This is the minimum value of the wave number
291
that corresponds to De c = 32 as predicted by the linear stability analysis based of the twenty-mode model. This value is also close to wave numbers reported in other experiments on TC flow of viscoelastic fluids [20]. Thus, as in the experiment of Muller et al. [8], only the Deborah number will be varied (by varying f2) in the following calculations and results. Consider the flow as De is increased from zero, that is from the Newtonian level. Linear stability analysis indicates that the Couette flow is unconditionally stable for De < Dec = 32. In the absence of inertia, an exchange of stability takes place between the circular Couette flow and oscillatory TVF at the critical Deborah number, since no steady TVF can set in. This is in contrast to a Newtonian fluid in which case only steady TVF sets in at the critical Reynolds number. The main variable of interest is the axial velocity component, Uz(r,z,t), which is obtained from the solution of the twenty-mode nonlinear dynamical system. Here r and z are, respectively, the radial and axial coordinates. In the experiment [8], Uz was measured at the point located midway through the gap, and, probably, where it is maximum over a wavelength in the axial direction.
Thus, the amplitude of uz(r= -Ri- - + - ~Ro ,z
d ) will be monitored next. = -~-,t
The experimental critical value of the Deborah number, at which oscillatory motion was first detected, was reported to be equal to 32.3, and happens to be slightly larger than the theoretical value De c = 32 predicted by the present linear stability analysis. As De is increased beyond the critical value, the amplitude of oscillation increases, confirming the existence and the stability of the Hopf bifurcation, in agreement with experiment [8]. Calculations are carried out for the same range of Deborah numbers as in the experiment: 32 < De < 50. At De = 32.5, the velocity signature and corresponding Fourier spectrum display periodic motion after the purely circular (Couette) flow becomes unstable. The amplitude of oscillation remains relatively small (0.008 cm/s). The power spectrum indicates the presence of a dominant frequency of 0.02 Hz and a weak second harmonics. This periodic behavior persists as De increases, with the flow oscillating around the origin (Couette flow). At De = 43.57, the motion remains periodic around the origin, with an increase in amplitude to 0.052 cm/s. There is an increase in the fundamental frequency to 0.0298 Hz and the emergence of four significant even and odd harmonics. This trend persists as De is further increased with the eventual emergence of additional harmonics. Figure 9 shows the Hopf bifurcation for the square of the velocity amplitude based on the twenty-mode model 20NDS, and the measurements from [8]. Both experiment and theory suggest that the amplitude of oscillation grows like (De -
292
9
0.01
0.009
!
!
/t
Theory 0.008
-
0.007
-
/
,
,s
/' 9 O 9
0.006 O
eq
0.005
,,'"
-
Experiment
II
ii s I
0.004
i
i
it"
i
i
,/ t
0.003
-
0.002
-
0.001
-
9"
i
t
/ i r ,/
0
20
I
I
25
30
9
I
I
I
I
35
40
45
50
.....
I
55
60
De
Fig. 9. Bifurcation diagram and comparison between theory (dashed line) and experimental measurements (diamonds) of Muller et al. [8]. Dec) 1/2 in agreement with the prediction based on asymptotic analysis in the limit D e --+ D e c. Figure 10 displays the dependence of the dominant frequency
and its harmonics on the Deborah number. The frequency tends to increase with De almost linearly. Unlike the amplitude, the frequency exhibits a jump at the
critical Deborah number. This means that any initial weak velocity amplitude at the onset of oscillatory TVF has a dominant frequency that is relatively easy to detect. The agreement between the computed and measured frequencies is obvious from the figure. The apparent growing disagreement for the higher harmonics is to be expected. Any initial discrepancy at the dominant frequency level is simply amplified as it is multiplied by two for the second harmonics, by three for the third harmonics and so on. A closer quantitative agreement between theory and experiment can hardly be envisaged given, on the one hand, the uncertainty surrounding experimental
293
0.25
0.2
0.15
'
I
I
fl
~
f2 f3 f4
+ [] x
..... ...... ...........
E x p e r
T h e o r
l
y
I
I ='
-
X X
m e
X
n
.
X
....................
...............
.....................
[]o.OO...o- . . I'. r.l. . . ..oo[] o o.o.--'"~176
o.O .o~o.-OO~176 o oO~176
-
..o"
0
I
..........
.............................................................
t
b
0.05
I
-
"4"
0.1
I
........
-
"'
20
I
25
,
I
30
,,
l
35
l_
40
I
45
_.
I
I
50
55
60
De
Fig. 10. Frequency diagram and comparison between theory (dashed line) and experimental measurements (diamonds) of Muller et al. [8]. The figure shows the square of the amplitude of axial velocity at x = 88 as a function of De. conditions, and, on the other, the lack of a universal and accurate constitutive model for viscoelastic fluids. The sources of discrepancy between theory and experiment are related to limitations of both Newtonian and viscoelastic flow formulations. The lack of a theory capable of predicting the value of the axial wave number, k, constitutes a major difficulty. The prediction of the value of k remains an unresolved issue (in a given formulation, it is usually simply imposed from experimental observation). In the case of viscoelastic Taylor-Couette flow, however, the measurement of k is difficult under transient flow conditions [9]. Other parameters and variables are also difficult to obtain from the experiment of Muller et aL [9] and had to be deduced. Additional uncertainty originates from the type of constitutive model used. Although the Oldroyd-B equation predicts the behavior of constant viscosity highly elastic fluids, it does not incorporate the spectrum of relaxation times that is characteristic of real fluids. More
294
complicated constitutive equations, accounting for the nonlinear dependence of the transport coefficients on the rate-of-strain tensor, may also be examined. The present formulation accounts for nonlinearities stemming from the upperconvective terms in the constitutive equation. Another source of discrepancy can come from end effects in the Taylor-Couette apparatus that have been neglected in the present formulation. The narrow-gap approximation is also a limiting assumption. Inertia effects can also play an influential role despite the fact that the experiment was conducted at a vanishingly small Reynolds number (Re < 1O 2). In general [18], the presence of inertia, no matter how small it may be, prohibits the base flow from losing its stability to the overstable mode. Instead, the base flow loses its stability first to steady (and not oscillatory) TVF since there is always a finite range of Re values over which the branches corresponding to steady TVF are stable. 5. DISCUSSION AND CONCLUDING REMARKS A low-dimensional dynamical system approach is proposed for the simulation of the Taylor-Couette flow in the narrow-gap limit of a viscoelastic fluid of the Oldroyd-B type. Among the clear advantages of introducing such a severely truncated representation for the flow field are the lower cost involved in the computation, and, more importantly, the increased possibility of obtaining the actual flow field under almost any condition without encountering numerical instability. The rather elementary Oldroyd-B constitutive model is adopted in the study for two main reasons. First, since the aim of the study is to examine the influence of elasticity on the onset and stability of TVF, the use of the Oldroyd-B model becomes justified for a class of the so-called Boger fluids, for which the viscosity remains sensibly constant over a wide range of shear rates, with the corresponding normal stress levels nevertheless substantial. A polyacrilamide solution in a maltose syrup/water mixture typically constitutes such a fluid [63]. Second, the Oldroyd-B constitutive equation is one of the simplest viscoelastic laws that accounts for normal stress effects (leading to the Weissenberg rodclimbing phenomenon). Other more realistic phenomenological [43,64] or molecular-theory based models [65-67] are more difficult to handle, and are likely to lead to a stability picture different from the one predicted by the present analysis. For instance, the presence of shear thinning, not accounted for by the Oldroyd-B equation, will likely have a destabilizing effect since the effective Reynolds number increases as the viscosity decreases with increasing shear rate [17,33]. Other more complicated constitutive equations, accounting for the
295
nonlinear dependence of the transport coefficients on the rate-of-strain tensor, may also be examined. The present formulation accounts for nonlinearities stemming from inertia in the momentum equation, and the convective and upperconvective terms in the constitutive equation. The nonlinear dynamical system is derived by expanding the flow field in double Fourier series in space for the rigid-free (RF) boundary conditions, and Fourier/Chandrasekhar series for the rigid-rigid (RR) conditions. The set of ordinary differential equations, governing the time-dependent coefficients, is obtained after applying the Galerkin projection onto the various modes, and adopting a suitable truncation to close the hierarchy of equations. The more severe truncation level used in the previous work [ 18] led to system (8), derived by neglecting higher-order normal stress terms. These terms, however, tend to become significant for highly elastic flows or in the limit Re ~ O. The present 16NDS takes into account more effectively the influence of normal stresses; thus being more adequate for a flow in the limit of vanishingly small Reynolds number, and allowing direct comparison with the experiment of Muller et al. [8]. This also allows the assessment of the influence of the higherorder modes on the solution by comparing the solutions based on the two models in the linear and nonlinear regimes (see below). If the higher-order terms are neglected, the resulting system (8) reduces, in the double limit E, Re ~ O, to the three-dimensional system (10) derived by Kuhlmann et al. [34,35] for the Taylor-Couette flow of a Newtonian fluid, or, equivalently, to the Lorenz equations with the Prandtl number set equal to one. The presence of the higherorder modes is found to make little difference for weakly elastic and Newtonian fluids, but is significant for highly elastic fluids. Regarding the boundary conditions, rigid boundaries are assumed in the azimuthal direction, while free (slip) boundary conditions are taken along the axial direction for the RF formulation, and rigid (stick) conditions along the axial direction for the RR formulation. The RF conditions give a somewhat less realistic value for the critical Reynolds number in the case of a Newtonian fluid, but a more accurate estimate for the torque [34]. The use of one type of boundary conditions or the other does not seem to alter qualitatively the stability picture. Despite the severity of the level of truncation, both formulations yield good agreement with existing linear stability analyses and experiments in the nonlinear regime. A judicious selection process of the most influential modes is carried out to ensure that the relevant dynamics is captured by the approximate model 16NDS and solution. This is first done by referring to the results based on linear stability analysis. The exact and approximate solutions are obtained for the linear
296
problem, in the absence of inertia, using the direct method. In this case, the eigenvalue problem is relatively simple to solve since only constant coefficients are involved. Only two modes, given by (6), are retained in the approximate solution, leading to good agreement with the exact solution, especially in the lower wave number range. Comparison between the approximate solutions based on rigid-rigid and rigid-flee boundary conditions indicates that the type of boundary conditions used does not influence the overall stability picture. This confirms the earlier conclusions reached in the case of a Newtonian fluid [34]. In fact, most earlier results based on existing linear analyses are recovered (see fig.
2). It is found that the critical Taylor number, Tao at the onset of axisymmetric Taylor vortices, not only depends on the disturbance wave number, but also decreases with increasing level of elasticity. At Ta - Ta c (r = rc), two bifurcating steady-state solution branches C1 and C2 emerge, corresponding to TVF in two opposite directions. These solutions lose their stability when the Taylor number exceeds a critical value Tah (r = rh), coinciding with the birth of a Hopf bifurcation. It is found that Tah also decreases with E. Thus, at least in the absence of shear thinning, fluid elasticity appears to destabilize the base flow, precipitating the onset o f Taylor vortices, which, in turn, tend to get destabilized, leading (eventually) to chaotic motion, earlier in comparison with the ease of a Newtonian fluid. The stability and bifurcation pictures are summarized in fig. 2 for a UCM fluid. The validity of the model is further assessed in the nonlinear regime by examining the influence of higher-order modes, neglected in the previous model [ 18], on the nontrivial steady-state solution branches C 1 and C2. The additional modes stem from normal stress effects and are thus found to have most of their influence on highly elastic flows, and are rather insignificant in the high Reynolds number range or for a Newtonian fluid. Indeed, for a Newtonian or weakly elastic fluid, the inclusion of the higher-order modes does not alter significantly the results in both the linear and nonlinear (fig. 2) regimes. From the analysis and numerical results based on the present model, fluid elasticity appears to alter dramatically the flow behavior in comparison with the flow of a Newtonian fluid. One of the most striking differences in flow behavior between the two flows is the destabilization of the TVF for a viscoelastic fluid that cannot be predicted on the basis of the (axisymmetric) Newtonian equations. Thus, unlike the Newtonian model, the Oldroyd-B model predicts the onset of a Hopf bifurcation (at Re = Reh) in the postcritical range of Reynolds or Taylor number (Re > Rec). Hence, to the degree of the present truncation, the Newtonian equations predict that the TVF is always stable regardless of the
297
value of the Reynolds or Taylor number. This is somewhat expected since, for a Newtonian fluid, the destabilization of TVF coincides, in practice, with the loss of axisymmetry. This coincidence does not seem to hold for viscoelastic fluids [9]. If fluid elasticity is accounted for in the constitutive equation, a different stability picture appears to emerge. In fact, linear stability analysis of the 20NDS around the TVF leads to a characteristic equation (not presented in this chapter) that happens to admit a pair of purely imaginary roots, coinciding with the birth of a Hopf bifurcation at Re = Re h as discussed earlier. The value of Reh depends strongly on fluid elasticity, fluid retardation and wave number as illustrated in fig. 2. In an effort to examine the role of inertia on the onset of TVF and its finiteamplitude instability, numerical calculations are carried out at intermediate elasticity level (E = 0.16) by varying the Deborah (or, equivalently, the Reynolds) number. The results (shown in figs. 3-6) indicate that, for weakly elastic flows, there is an exchange of stability at De = De c (Re = Rec) between the base flow and TVF, through a supercritical bifurcation similar to that in Newtonian fluids. At some point in the postcritical range (De > Dec), the steady TVF loses its stability, in turn, as De (Re) is further increased to De = Deh (Re = Reh) when a Hopf bifurcation sets in. The influence of shear rate (De) was investigated experimentally by Baumert & Muller [9]. Their flow visualization focused essentially on the early transient behavior once a shear rate is imposed. They showed that steady TVF sets in after a period of monotonic or oscillatory vortex growth depending on whether the imposed shear rate is low or high. The results based on the present formulation lead to good qualitative agreement with experiment and the nonlinear calculations of Avgousti et al. [28]. For a highly elastic fluid, of Elasticity number greater than the critical value, Esub, the steady-state solution undergoes a subcritical bifurcation at Re = Rec (fig. 2) similar to that predicted by the Landau-Ginzburg's equation [44]. In this case, the range of stability of the TVF becomes increasingly narrower as E increases, giving way to (time) oscillatory vortex structure. Indeed, beyond a critical level of elasticity (E > 0.1 in fig. 2) and for vanishingly small Reynolds number, the base flow loses its stability to oscillatory and not to steady TVF. This prediction confirms the experimental observation of Muller et al. [8]. Unlike the previous model [18], the present system led to favorable comparison with experiment (figs. 7 and 8). The present theory predicts the sequence of periodic behaviors observed as the Deborah number is increased: (1) loss of stability of the base flow to an oscillatory flow at a critical Deborah number (Dec = 29.3 and 29.7 for the RF and RR formulations, respectively, as predicted by the model vs. 32 from experiment), (2) growth of amplitude of the velocity signature
298
like (De - Dec) 1/2, in agreement with asymptotic analysis, with the emergence of higher harmonics in the Fourier spectrum, and (3) emergence of subharmonics as De is further increased, reflecting (most likely) the bifurcation into period doubling and, eventually, chaos. Finally, an augmented twenty-dimensional dynamical system, 20NDS, is also used to describe highly elastic TVF. The model is derived without inertia effect, and higher normal stress terms are added. This approach constitutes a first systematic and accurate theoretical prediction for the purely elastic overstability observed by Muller et al. [8]. It is shown that the finite amplitude TVF can be effectively described if higher-order normal stress terms are properly accounted for. Particularly, the addition of the azimuthal normal stress component leads to additional coupling with higher-order eigenmodes that are of O(eDe). The resulting nonlinear dynamical system involves only twenty degrees of freedom. The sequence of flows predicted by the present model is comparable to that reported by Muller et al. [8]. The model predicts the sequence of periodic behaviors observed as the Deborah number is increased: (1) loss of stability of the base flow to an oscillatory flow at a critical Deborah number (Dec = 32 as predicted by the model vs. 32.3 from experiment), (2) growth of amplitude of the velocity signature like (De - Dec) 1/2, in agreement with asymptotic analysis, and (3) the emergence of higher harmonics in the Fourier spectrum as De is further increased [ 19].
REFERENCES ~
2. 3. 4. 5. .
7. 8. .
10.
G. I. Taylor. Phil. Trans. Roy. Soc. A 223 (1923) 289. T. B. Benjamin. Proc. Roy. Soc. Lond. A 239 (1978) 1. D. E. Shaeffer. Math. Proc. Camb. Phil. Soc. 87 (1980) 307. D. Coles. J. Fluid Mech. 21 (1965) 385. P. R. Fenstermacher, H. L. Swinney & J. P. Gollub. J. Fluid Mech. 94 (1979) 103. M. Gorman & H. L. Swinney. J. Fluid Mech. 117 (1982) 123. D. Rand. Arch. Ration. Mech. Anal. 79 (1982) 1. S. J. Muller, E. S. J. Shaqfeh & R. G. Larson. J. Non-Newt. Fluid Mech. 46 (1993) 315. B. Baumert & S. J. Muller. Rheol. Acta 34 (1995) 147 G. R.Sell, C. Foias and R. Temam Turbulence in Fluid Flows: A Dynamical Systems Approach (Springer-Verlag, 1993).
299
11. 12. 13.
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.
38. 39. 40.
E. N. Lorenz. J. Atmos. Sci. 20 (1963) 130. C. Sparrow. The Lorenz Equations, (Springer-Verlag, New York 1983). H. N. Shirer & R. Wells. Mathematical Structure of the Singularities at the Transitions Between Steady States in Hydrodynamic Systems (Springer-Verlag, Heidelberg 1980). R. E. Khayat. J. Non-Newt. Fluid Mech. 53 (1994) 227. R. E. Khayat. J. Non-Newt. Fluid Mech. 58 (1995) 331. R. E. Khayat. Phys. Rev. E 51 (1995) 380. R. E. Khayat. J. Non-Newt. Fluid Mech. 60 (1996) R. E. Khayat. Phys. Fluids A 7 (1995) 2191. R. E. Khayat. Phys. Rev. Letts. 78 (1997) 4918. R. G. Larson. Rheol. Acta 31 (1992) 213. E. G. S. Shaqfeh. Ann. Rev. Fluid Mech. 28 (1996) 129. H. Giesekus. Rheol. Acta 5 (1966) 39. H. Giesekus. Prog. Heat Mass Transfer 5 (1972) 187. R. Haas & K. Btihler. Rheol. Acta 28 (1989) 402. R. G. Larson, E. S. G. Shaqfeh & S. J. Muller. J. Fluid Mech. 218 (1990) 573. E. S. G. Shaqfeh, S. J. Muller & R. G. Larson. J. Fluid Mech. 235 (1992) 285. Y. L. Joo & E. S. G. Shaqfeh. Phys. Fluids A 4 (1995) 2415 M. Avgousti, B. Liu & A. N. Beris. Int. J. Num. Meth. Fluids 17 (1993) 49. M. Avgousti & A. N. Beris. J. Non-Newt. Fluid Mech. 50 (1993) 225. P. J. Northey, R. C. Armstrong & R. A. Brown. J. Non-Newt. Fluid Mech. 42 (1992) 117. A. Groisman & V. Steinberg. Phys. Rev. Letts. 77 (1996) 1480. M. K. Yi 7& C. Kim. J. Non-Newt. Fluid Mech. 72 (1997) 113. R. G. Larson. Rheol. Acta 28 (1989) 504. H. Kuhlmann. Phys. Rev. A 32 (1985) 1703. H. Kuhlmann, D. Roth & M. LOcke. Phys. Rev. A 39 (1988) 745 E. Ott. Chaos in Dynamical Systems (Cambridge University Press, Cambridge 1993) J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York 1983). G. Veronis. J. Fluid Mech. 24 (1966) 545. J. H. Curry. Commun. Math. Phys. 60 (1978) 193. H. Yahata. Prog. Theor. Phys. 59 (1978) 1755.
300 41. 42. 43. 44. 45. 46. 47. 48. 4.9. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 63. 64. 65. of 66. 67.
H. Yahata. Prog. Theor. Phys. 61 (1979) 791. H. Yahata. Prog. Theor. Phys. 59 (1978) 1755. R. B. Bird, R. C. Armstrong & O. Hassager. Dynamics of Polymeric Liquids, vol. 1, 2nd Ed. (John Wiley & Sons, New York 1987). P. G. Drazin & W. H. Reid. Hydrodynamic Stability (Cambridge University Press, Cambridge 1981). H. Harder. J. non-Newtonian Fluid Mech. 36 (1991) 67. J. B. McLaughlin & P. C. Martin. Phys. Rev. A 12 (1975) 186. A. Davey. J. Fluid Mech. 14 (1962) 336. J. T. Smart. J. Fluid Mech. 4 (1958) 1. P. Chossat & G. Iooss. The Couette-Taylor Problem. Springer-Verlag (1991). M. Nagata. J. Fluid Mech. 169 (1986) 229. P. Tabeling. J. Phys. Lett. 44 (1983) 665. D. V. Boger. J. Non-Newt. Fluid Mech. 3 (1977/78) 97. R. D. Richtmyer. NCAR Tech. Note TN-176+STR (1981). P. S. Marcus. J. Fluid Mech. 146 (1984) 65. S. Chandrasekhar. Hydrodynamic and Magnetohydrodynamic Stability, Dover (1961). R. E. Khayat. J. Fluid Mech. (in press). D. W. Beard, M. H. Davies & K. Walters. J. Fluid Mech. 24 (1966) 321. R. H. Thomas & K. Walters. J. Fluid Mech. 18 (1964) 33. R. F. Ginn & M. M. Denn. AIChE J. 15 (1969) 450. C. Normand, & Y. Pomeau. Rev. Mod. Phys. 49 (1977) 581. S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York 1990). J. E. Marsden & M. McCracken. The HopfBifurcation and Its Applications (Springer-Verlag, New York 1976). J. Carr. Applications of Center Manifold Theory (Springer Verlag, New York 1981). K. Walters. Rheometry: Industrial Applications. Research Studies Press (1980). R. I. Tanner. Engineering Rheology. Oxford University Press (983). R. B. Bird, R. C. Armstrong, C. F. Curtis & O. Hassager. Dynamics Polymeric Liquids, vol. 2, 2nd Ed. John Wiley & Sons (1987).. M. Doi & S. F. Edwards. The Theory of Polymer Dynamics. Oxford University Press (988). B. C. Eu & R. E. Khayat. Rheol. Acta 30 (1991) 204.
301
NON-NEWTONIAN MIXING WITH HELICAL IMPELLERS AND PLANETARY MIXERS
RIBBON
Philippe A. Tanguy ~ and Edmundo Brito-De La Fuente 2
1 Department of Chemical Engineering, Ecole Polytechnique Montreal P.O. Box 6079, Station Centre-ville, Montreal H3C 3A7, CANADA 2 Food Science and Biotechnology Department, Chemistry Faculty "E", National Autonomous University of Mexico, UNAM, 04510 Mexico, D.F., MEXICO
1. INTRODUCTION Mixing is a very common processing operation, which accounts for about 15 % of all unit operations in the chemical and food industries. The range of possible mixing duties is extremely wide and may involve a single phase (in which case agitation is a more appropriate wording) or several phases. Moreover, the phases may be of a different nature, solid, liquid and gas. All combinations may be found depending on the process. Table 1 summarizes some important practical mixing applications. Table 1. Typical mixing applications Kneading of Pastes Dispersion of Agglomerates Flocculation Gas-Liquid Dispersion Liquid-Liquid Dispersion Reaction Enhancement Slurrying Storage Blending
Crystallization Emulsification Flotation Heat Transfer Liquid Blending Polymerization Solids Suspension Uniformity Maintenance
302
A mechanical mixer is a relatively simple device. It comprises a vessel, an impeller mounted on a shaft and a drive assembly. In mixing process development, a number of issues must be addressed for a given mixing application, namely 9 the selection of mixing equipment (impeller, drive) and operating conditions (speed, temperature, mixing duration in batch mode) 9 the design of the vessel (shape, aspect ratio, type of bottom, position of feed and side streams) and the internals (baffles, draught tube, heat exchangers) 9 the evaluation of performance (power consumption, mixing time, circulation patterns, circulation rate, distribution of shear rate) 9 the control of composition non-uniformity (segregation). Turbulence is the main physical mechanism responsible for mixing with low viscosity fluids. The impeller power is split into the power dissipated by the circulating fluid and the power dissipated by the sheafing action of the impeller in the vicinity of the blade tip. The respective amount of these two power components depends on the impeller type. When the viscosity is high, turbulence can hardly be achieved and when it is, the power drawn by the impeller is extremely large, yielding high operating costs and a significant temperature increase in the fluid bulk. Viscous mixing is a laminar process based on the "stretching-foldingbreaking" principles described in [1 ]. Mixing is obtained by the mechanical decrease of the striation thickness by shear and/or extensional forces up to a scale where molecular diffusion may fully play its role. This is a rather "gentle" flow process, which usually requires time. As with other fluid mechanics problems, the Reynolds number is the relevant number to characterize the significance of the viscosity effects. The Reynolds number is defined as: Re
-
pND2
(1)
77 where p and 1] are the fluid density and dynamic viscosity respectively, N is the rotational speed of the impeller and D its diameter. Below Re = 10, the viscous effects predominate, yielding a quasi-creeping flow. The mixing regime is laminar. At the other end of the Reynolds number scale, the mixing regime is turbulent. Depending on the impeller, the turbulence
303
threshold is comprised between Re = 104 and Re = 3.104. The intermediate mixing region above the laminar regime and below the turbulent regime is called the transition regime. In principle, mixers can be operated in any of this regime, provided adequate power is available. In practice, for viscous fluid, as turbulence cannot be achieved in the vessel, the mixing regime is at best in the transition region and often in the laminar region. Experience shows that the mixing regime is always laminar when the fluid viscosity is above 50 Pa.s. High viscosity fluids usually exhibit non-Newtonian properties. NonNewtonian flows are very common in the process industry. Typical examples include the manufacturing of rubber, plastics, petroleum, detergents, cosmetics, pharmaceuticals, cement, food, paper pulp, paints to name a few. A wide variety of flow situations are encountered in practice (flow in transfer lines, in processing equipment). Our discussion in this chapter will focus on non-Newtonian mixing and we refer the reader to the literature [2] for the treatment of the general principles governing nonNewtonian fluid mechanics. The most common non-Newtonian behavior is shear-thinning, i.e. the viscosity decreases with an increasing shear rate. This behavior is very often encountered in polymer manufacturing or in fermentation. Yield stress is often present especially with gels and highly concentrated suspensions. The yield stress phenomenon is associated with the presence of structures in the fluid that requires a minimum amount of deformation energy to break up. The mixing of yield stress fluids (or B ingham fluids as they are often called) is really difficult due to the particularly complex fluid mechanics in the vessel. A well-mixed cavern forms in the fluid bulk around the impeller [3] due to the shearing action of the blades. Far from the impeller, the rate of deformation is not sufficiently high to break the structure and the fluid is at rest. The same phenomenon may be encountered with shear-thinning fluids although in a less severe manner [4]. Another particular behavior is thixotropy, which is manifested by a time-dependent shear-thinning. From a physical standpoint, it is believed that thixotropy is related to the presence of organized microstructures whose conformation changes under an applied stress. Examples of such fluids are chocolate and paper coating colors. Thixotropy can be described in engineering terms as substances with several yield stresses [5][6]. Likewise B ingham fluids, thixotropic fluids may be difficult to mix. Shear-thickening is the opposite behavior of shear-thinning, i.e. the viscosity increases with the shear rate. This behavior is observed with
304
colloidal suspensions at very high concentration like high solids mineral slurries and with starch. Fluids exhibiting this property must be handled very carefully. Indeed, as the power consumption is proportional to the viscosity (as it will be seen later in this chapter), if shear-thickening properties develop during the process, the impeller may stall in the vessel or, worse, the motor may pull out. Viscoelasticity may add another level of complexity in the mixing of non-Newtonian viscous materials. Like shear-thinning, viscoelasticity is encountered in polymerization and fermentation. Apart from the wellknown shaft climbing phenomenon by the fluid (known as the Weissenberg effect in rheology) viscoelasticity increases significantly the power consumption in the vessel and strongly alters the circulation patterns [7].
2. S U M M A R Y OF MIXER DESIGN PRINCIPLES The mixer performance, expressed in terms of energetic, dispersing sheafing or blending efficiency, is directly related to the impeller performance. The selection of the appropriate device requires a very careful appraisal of the mixing duty, and a systematic stepwise approach based on: 9 Identification of the operations to be carried out 9 Detailed physical characteristics of the phases (Table 2) 9 General features of the impeller (shear rate, circulation rate, pumping pattern) 9 Selection of impeller candidates 9 Final selection Table 2. Phase characteristics
Liquid
Solid
Gas
specific gravity viscosity curve yield stress temperature range wt. %
Specific gravity Particle size distribution Settling velocity Wettability- solubility wt. %
Flowrate Pressure Solubility
There are two possible ways of classifying an impeller, according to the discharge flow it produces, or according to its size with respect to the
305
vessel. In the first category, there are radial discharge impellers (flat turbine), axial discharge impellers (propellers) and tangential flow impellers (anchors). If the diameter of an impeller is significantly smaller than the diameter of the vessel, this impeller is an open impeller. On the contrary, if the impeller exerts a scraping action at the vessel wall, it is called a closeclearance impeller. We show in Figure 1 typical examples of open and close-clearance impellers.
Figure 1: Flat blade turbine (open impeller) and helical ribbon screw (closeclearance impeller) In viscous and non-Newtonian mixing, close-clearance impellers are by far the best choice due to their superior top-to-bottom pumping capacity and wall scraping action as compared with open impellers. If the medium is a paste, multiple impellers comprising high speed tools for intense shearing in the bulk (emulsification head, dispersing disk) and low speed wall scraping blades may be advantageously used. We show in Figure 2 an example of kneading equipment. Sometimes, the various impellers are mounted on a rotating carousel (planetary mixers). With this technology the kinematics of the impellers is such that all the vessel volume is swept at regular intervals. There is therefore no dead zone. There is not a unique choice of mixing technology for a given application and several impellers may likely satisfy the mixing requirements. The final selection will be based on both process operating conditions and economics. Design parameters to consider include operating
306
mode (batch or continuous), volume to be processed, mixing time, and pressure in the vessel. The design of the vessel and the internals is also critical from mechanical and economic reasons: shape, dimensions, open or closed tanks, positions of feeds and outlets, baffles, heating coils are all important technical decisions that will directly impact the overall mixing system efficiency.
Figure 2: a multiple tool kneading head In industry, the use of empirical correlations has been the standard for the design of mixing systems. Nowadays, stringent process efficiency requirements make the use of pilot experiments in association with computer modelling an essential step of the development phase. From the formulation developed in the laboratory by the chemists, the process is first developed at pilot scale (typically 200 to 500 1) and then some scale-up guidelines are used to design the industrial unit. Computer modeling can be used at both steps, for the pilot scale and for the eventual production line. The selection of the mixing scale-up criteria to be used for the design of industrial facilities is an open issue. The fundamental question to address is how a process changes when the scale is changed, the objective being obviously to maintain process similarity irrespective of the scale. There are two broad classes of similarity: flow and transport phenomena. Flow similarity may involve the geometry (shape similarity), the kinematics (speed similarity) and the dynamics (force similarity). As for
307
the transport phenomena, one may have thermal similarity (same temperature gradients), chemical similarity (same concentration gradients) and rheological similarity. In the latter case, the same distribution of deformation rates (shear and extension) is sought between the pilot process and the scaled-up installation. In practice, when the process scale is changed, the new design will be based on keeping some parameters equal. The most common scale-up parameters are: 9 mixing times (in laminar regime, the mixing time is inversely proportional to the impeller speed) 9 power per unit volume 9 impeller tip velocity 9 effective shear rate or process viscosity. An approximate similarity can be obtained if the same geometry and the same kinematics is used, while using proportionality rules. For instance, if we want to keep the impeller tip speed ND constant, if D is increased, N should be diminished proportionally. An improved similarity would consider identical energy dissipation scales. The main difficulty is the effect of walls and bottom surface whose influence decreases drastically as the vessel size is increased. This becomes a serious constraint when heat transfer is involved during the mixing.
3. NON-NEWTONIAN MIXING F L O W S From an historical perspective, non-Newtonian mixing flows in agitated vessels have been first dealt with simple models based on simple physics principles and correlations, and then with computer fluid dynamics (CFD) approach. These two approaches will be described here, the scope being restricted to close clearance impeller mixers and planetary mixers. 3.1 Flow models in a mixing vessel Several flow models have been proposed in the literature to predict power consumption with close clearance impellers. They fall in two main groups, namely flow between two coaxial cylinders and drag-based analysis As it was mentioned in Section 1, the mixing of highly viscous and rheologically complex fluids is a slow process both at the macro and micro scale. As the flow regime is laminar, the theoretical notions given in the following paragraphs are valid mainly for this regime.
308
3.1.1 Coaxialflow model The basic idea is to depict the impeller as a solid cylinder rotating inside another cylinder (the vessel). This geometry is known as the Couette geometry. Let us consider the case of a shear-thinning fluid obeying the power law model:
O~rl)-- 2(n-1)12Klrl n-'
(2)
where K is the consistency index, n the shear-thinning index and T the shear rate. It is easy to show that in the Couette geometry [8], equation (3) expresses the variation of the shear rate with the radius, namely:
[rd(Vo/r)l_[ 2~]n 1(1)2 /n
dr
(3)
r1-27n --- r7-21n
Equation (4) gives the shear rate at the inner rotating cylinder (or impeller):
I 1-(rl[r2)a/n2)1[ 2or2] nO-(r, l r 2
(4)
r22-rl 2
In these equations, f~ is the rotating speed (in rad/s), V0 the angular component of the fluid velocity, and r~ and r2 are the inner and outer radius respectively. From a mixing perspective, two cases of practical interest are included in equation (4). The first one occurs when the impeller shear rates are only contained in a small volume very close to the blade. This situation is common in laminar mixing, where the flow generated by the agitator does not reach the tank wall as if the impeller was rotating in an infinite medium. For this case, the usual procedure is to assume that rl/r2 essentially approaches zero. Then, equation (4) becomes:
7, -
4n:N 1 n
(5)
309
According to equation (5), the shear rate is linearly related to the impeller rotational speed, N, and the rheology. The other interesting case arises when r~/r2 approaches 1. This situation can be found when using very small clearances in the mixing geometry. Then, equation (4) becomes:
2 ~2 r 2 ~/
~
2
r 2
I
(6)
2
--
r1
The flow patterns around mixing impellers are fairly complex and the controlling factors of the drag on an impeller are difficult to appraise. In practice, simplified flow situations are used to estimate the viscous energy dissipation, and then deduce the drag or power requirements. In nonNewtonian mixing technology, the flow model around an impeller blade must obviously include the shearing rates or shearing stresses. This is a direct consequence of the fact that the viscosity of these fluids to an imposed stress is not constant but depends on the magnitude of the deformation rate. The knowledge of shear rates in stirred mixing tanks is central to power input calculations and equipment scale-up. Empirical and theoretical approaches have been developed in the literature to estimate shear rates. In the following paragraphs, these approaches are discussed and compared.
3.1.2 Metzner and Otto Approach In late 50's, Metzner and Otto [9] introduced the following rule to define an apparent viscosity in the mixer. If a Newtonian fluid and a nonNewtonian fluid are agitated in the laminar regime under the same operating conditions and in the same equipment (so that the power input measured is the same), then, because all variables are identical, the average viscosity is the same for both fluids. This quite important result for mixing practice was expressed by equation (7):
y'- KsN
(7)
where Ks is a constant of proportionality which has to be determined experimentally for each impeller geometry of interest. In other words, the average fluid shear rate is related only to the impeller speed.
310
It is worth noting here that equation (7) is only valid for laminar mixing, although it has been used for the transition regime. The value of Ks is typically estimated by performing duplicate power input measurements, one with a Newtonian fluid and one with a non-Newtonian fluid, in the same mixing system. Thus, if the rheological behavior of the fluid is expressed by a power-law model, then, from the experimental Newtonian power input curve, at fixed N, an apparent or effective viscosity defined as:
~a -- K(Yav )n-1
(8)
is calculated by:
770-77
(9)
where 1"1is the Newtonian viscosity, PnN and PN are the non-Newtonian and Newtonian power respectively. From the apparent viscosity calculated in equation (9), the corresponding value of )'av is estimated by equation (8). Upon repeating this procedure for pairs of (N, ~/av)values, the value of Ks is determined. It is worth nothing here that as n tends towards the limiting Newtonian case (n=l), equation (5) gives 7 = 41-IN = 12.6N or Ks = 12.6. This is very close to the universal value of Ks = 13, originally proposed in the literature [9] for several fiat-blade turbines. The Metzner-Otto correlation has been extensively used to characterize the average shear rates of non-Newtonian inelastic fluids [ 10], and it is routinely recommended as a standard procedure in mixing and unit operations textbooks. This algorithm is considered a useful approach for the estimate of hydrodynamic similarities on scale-up although it might not represent the rheological properties existing in the mixing vessel. It is a current practice to consider Ks as a constant for given impeller geometry. However, some authors have reported that this constant might depend on the power law index [11][12][13], in particular for highly shear thinning fluids. A compilation of experimental Ks values for helical ribbon (HR) and helical ribbon screw (HRS) impellers is shown in Table 3. A detailed analysis on the correlations proposed for Ks clearly shows that the effect of wall clearance might be the most important one. The value of Ks increases as the wall clearance decreases and this becomes more pronounced as the ratio r~/r2 approaches the limiting value of 1.0.
311
Table 3. Ks data available in the literature (HR and HRS impellers)
Type ,
D/d
n
Ks
i
Comments
HR
1.02- 1.12
0 . 4 - 1.0
66.06 a
Weak Ks(n) Ks(geometry)
[14]
HR
1.059
0 . 2 - 1.0
25
Ks = constant
[15]
HR
1.10- 1.11
0 . 3 5 - 1.0
27
Ks - constant
[16]
HRS
1.03
0 . 3 5 - 1.0
24.58
Weak Ks(n) Ks(geometry)
[17]
HR
1.053
0 . 5 - 1.0
36.73
Ks = constant
[18]
HR
1.056
0 . 2 7 - 1.0
30
Ks = constant
[19]
HR
1.042- 1.19
0 . 5 - 1.0
27.60 a
Ks(geometry)
[20]
HRS
1.056 - 1.118
0 . 2 6 - 1.0
30.6
Ks = constant
[21]
HR
1.11 - 1.37
0.17 -0.65
79.85 b
Strong Ks(n) Ks(geometry)
[22]
HR
1.05 - 1.163
0.35 -0.75
24.68 a
Ks(geometry)
[23]
HR
1.05- 1.33
26.80 a
Ks(geometry)
[24]
HR
1.11
17-40
Weak Ks (n)
[13]
0 . 1 8 - 1.0
Ks (geometry) HR & HRS
1.135
0 . 0 9 - 1.0
7 - 36
Ks (n) for n < 0.5 Ks (geometry)
[12]
This value is calculated for D/d = 1.10. Value estimated for n = 0.6; D/d = 1.135 and 1/d - 3.24, where 1 = length of impeller blade. Regarding the functional dependence of Ks on the fluid flow properties, for most authors, Ks may be considered as independent of the rheological properties. This is particularly true for those authors who have strictly followed the Metzner and Otto assumptions. However, for authors who developed correlations based on the Couette flow analogy principle, the resulting Ks was found to be a function of the shear-thinning level.
312
3.1.3 Extended Couette Flow Approach Bourne and Butler [25] investigated the mixing of viscous Newtonian and shear thinning fluids with helical ribbons, using flow pattern visualization and liquid velocity measurements. They proposed the existence of three main flow zones: 9 the core flow surrounding the impeller shaft 9 the region between the blades of the impeller 9 a highly sheared zone located between the tank walls and the impeller blades.
The authors used the Couette flow analogy to describe the highly sheared zone substituting the impeller by a solid cylinder. Setting as a characteristic fluid velocity the tip speed velocity ND, a particular form of the Reynolds number for power law fluids was proposed:
pN2-nD2 R e p~ =
K
(10)
Thus, the familiar expression for the dimensionless power input is given by:
Np Re pl
P
KNn+ ] D3 =Kp(n)
(11)
In the above equations, no Ks values must be known a priori to handle experimental power input data. Equations (10) and (11) have been the basis for the work of Chavan and Ulbrecht [ 17], and Rieger and Novak [ 18]. Unlike in [25], Chavan and Ulbrecht considered only the geometry of the helical ribbon impeller to define an equivalent cylinder diameter to be used in the Couette flow analogy. More recently, the use of an equivalent diameter based on the ribbon geometry and the fluid rheological properties has been suggested as improvements over previous studies [26][13]. Figure 3 is a typical representation of power input experimental data obtained with a HR impeller for a wide range of shear thinning fluids, using equations (10) and (11). Details regarding the experimental setup and geometry of the HR impeller are given elsewhere [27]. As suggested by the results of this figure, the power consumption decreases as the fluid becomes more shear-thinning (decreasing values of n) at the same Reynolds number. It is also clear that as n tends towards the limiting Newtonian value (n=l), the power data tends to the Newtonian power correlation.
313
10 8
--
i.
l
I
I
I
I
I
I I
'
'
'
'
' ' ' 1
'
'
'
'
'
' ' ' 1
'
'
'
'
'
' ' ' 1
_
_
10 5
Q
-
-
"'..
_
.%
_ _ _
_
_
10 4
_--
o
_
Z
~
"O
e o
_ _ _ _
_
10 3
__ _ _
_
[]
n = 1.0
*
n = 0.77
/x
n = 0.70
0
i
10 2
n
9
_ _ I
1 0 .4
=
0.5
(Newtonian)
"'-.O.'.-.+ :-: ~-~" -, . ' O . ~I k . ' &
3
"
I
I
I
I
'll
1
I
I
I
0 .3
'
' ' I l l
I
I
I
,
, , , , I
1 0 .2
1 0 ~
_ ..
"',II~"Lx "O
n = 0.28 I
.
"'',
I
i
"J, " .
6An
i
i
i
n
i
l
-
1 0 ~
Repl
Figure 3. HR impeller power curves Figure 4 is the plot of the Kp(n) function corresponding to the data shown in Figure 3. The Kp(n) function can be represented by a non-linear model [11][ 12], b and c being fitting parameters, namely: n-1
Kp (n) - Kp(n=l) bn-1 c
n
(12)
It should be noted that the representation of the non-Newtonian power input as shown in Figure 3 is not very convenient since at least two dimensionless parameters, n and Re, influence the power number. Extensive experimentation would thus be necessary to include the effect of other variables. Another drawback is that this type of representation is not general and unique. A unifying principle can, however, be established by comparing the Metzner and Otto approach with equation (11). Indeed, it can be shown [ 11 ] that:
_
Np
Repl - -
P KN,,+~ 9 3 "- K p (n) - Kp(n=l) Ks n-1
(13)
314
or
II-
1
1
Kp(n)
Ks-
mN--2-+]d3
180
160
-
140
-
Q 9 [] 120
-
100
-
80
-
A
n-1
(14)
gp
G e l l a n 1.5 % Xanthan 3% X a n t h a n 2 % - C M C 1% Xanthan 1.5%-CMC 1.5% Xanthan 1.75%-CMC 1.25% Xanthan 1.65%-CMC 1.35% Xanthan I%-CMC 2% CMC 2.5% CMC 1.5% Newtonian fluids Kp
-
J
'
// /
/ /
/ / / /
-
a= /
60
40
20
-
0 0.0
,
I 0.2
,
I 0.4
,
I 0.6
=
I 0.8
, 1.0
Figure 4. Variation of the power constant with the power law index
As suggested by equation (14), Ks plays the role of a shift factor which should coalesce all the non-Newtonian power input results into the Newtonian curve, yielding a unique power input (master curve). We show in Figure 5 the result master curve corresponding to the data of Figure 3. As it can be observed, the whole set of non-Newtonian power input data are superimposed on the Newtonian curve when using the experimental Kp(n) function as shown in Figure 4 and equation (14).
315
10 6
dl 10 5
- 1
Np Re = 162.55
-!
[exp. Newtonian correlation]-
U3 "=t 10 4 -
Z Q.
II
10 3
-
Z
10 2
-
10 ~ 10-3
9 9 9 9 9 []
Xanthan 3 % "~1~ -i Gellan 1.5% ' ,I~~, Xanthan 2%-CMC 1% Xanthan 1.5%-CMC 1.5% Orn Xanthan 1.75%-CMC 1.25% CMC 2.5% I
1 0 -2
Re = p
10 1
10 ~
I
I
I
I
I I
10 ~
N 2"n d 2/In Ks n'l
Figure 5. Power master curve for HR impeller It is worth noting that the functional form of equation (14) makes the Ks function very sensitive to small variations in power consumption measurements, particularly in the highly shear thinning region (0.5 < n). This is illustrated in Figure 6 for a power measurement uncertainty of 10%. It can be seen that the interpretation of the results on the actual value of Ks is delicate. With the measurement uncertainty, all conclusions are possible: Ks may be taken either as a constant (independent of the fluid rheological properties), or as a decreasing or even an increasing function of the flow index. The most significant variations of Ks are found in the highly shear thinning region, 0.1 < n < 0.5. Undoubtedly, these results shed some light on the apparent opposite findings shown in Table 3. In the opinion of the present authors, the differences found in the literature could be easily explained if the impact of experimental errors on power measurements had been fully accounted for.
316
40
''''1''''~'''1''''1''''1''''1''''1''''1''''1'''
\
35
30
-.
c=l.1
~Oo
c=l.O 25
20
/
c = 0.913 (exp.
data)
15
10
5 i 0.0
,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,I,,,,
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 6. Ks sensitivity to power consumption uncertainty
3.1.4 Drag force-based analysis This approach considers that the torque exerted on the rotating impeller is due to the drag force exerted by the fluid flow around the impeller blade. The forces acting on the blade are namely [28]: 9 the dynamic pressure due to the normal velocity at the blade (known as normal drag or form drag), 9 the tangential stresses caused by the fluid friction (skin drag) along the surface of the blade. If the impeller tip speed is taken as the characteristic velocity, the mixing power can then readily obtained with the following equation:
P - z ( 2 ~ N ) - 2~rbNFdo
(15)
where rb is the radius of the impeller blade, and Fd0 the drag force in the 9 direction. The drag force can be written in terms of the drag coefficient, Cd, as"
317
( l pVo2~
(16)
F d - C d -~
where A is a characteristic blade area and V0 is the angular component of fluid velocity. In the literature, only a small number of correlations based on the drag flow analogy has been proposed. The existing ones differ mainly in the expression used for Cd. It is worth noting that most of these correlations are limited to Newtonian fluids. For non-Newtonian fluids, one correlation is presented in Table 3 [22]. For other works on this topic, the reader is referred to [29]. In the following section, we now turn to the computer modelling approach of mixing flows. 3.2 Numerical simulations We consider the flow of a viscous fluid generated by an impeller rotating in the center of a cylindrical vessel. Using symmetry of revolution arguments, the velocity and the pressure fields can be readily obtained by resolving the standard equations of motion in the non-Galilean frame of reference of the impeller; in other words by considering a stationary impeller and a rotating vessel. In this Lagrangian viewpoint, the equations of change read as: - Vp + divT - p(v.Vv
f
divv = 0
+ co o (co o r ) + 2o9 o v )
(17)
-, (Irl)r
where v is the velocity vector, p the pressure, r the radial coordinate and ~0the angular velocity. The symbol o denotes the standard vector product. This equation is written without considering the unsteady term. This approach is valid provided the flow in the Eulerian viewpoint is periodic. For completeness, the viscosity function in equation (17) must be made explicit with a rheological model. Typical models used in mixing simulations include the Newtonian model (constant viscosity), shearthinning (power law and Carreau models) and yield stress (HerschelBulkey) fluids.
318 Power law model
O~'l)-- 2(n-1)/2
girl'-'
(2)
where K is the consistency index and the n the shear-thinning index. These parameters can be readily obtained by performing a logarithmic regression on the flow curve. Carreau model
r/~?'l) r/= + (770 r/oo)(1+ 2(1t,I ~s _
_
2
(18)
where 7/0 is the zero-shear viscosity, r/_ the high-shear viscosity, t, a fluid characteristics time and s the slope of the viscosity curve. This slope is related to the shear-thinning index of the power-law model by s = (1- n)/2. Herschel-Bulkley model
TO
r/~'l)- 21/217,--~+
2 (n-l)/2
Klrl" '
(19)
where ~:0 is the yield stress, K the consistency index and n the shearthinning index. When the viscosity of the fluid is constant, the familiar B ingham model is obtained:
TO
+ r ] rTOy])-~21,2it I
(20)
For mathematical well-posedness, the equations of change must be completed with appropriate boundary conditions. In the Lagrangian frame of reference chosen, the boundary conditions read as [30]: 9 at the vessel wall and bottom: v = Xwallo r 9 on the impeller: v=0 9 at the free surface: Vz = 0 The last boundary condition implies that the free surface in the vessel is fiat. Experience shows that this assumption is valid in viscous mixing.
319
The above equations are solved using the Galerkine finite element method. Let us recall briefly the principle of this method. We first rewrite the equations of motion using tensor notations, namely:
f div FI = f div V = 0
(21)
II = - P 5 + 271)' With help of the variational calculus, these equations may be written as a saddle-point problem, namely:
Inf
Sup ~
~t (r(v)) 2 d ~ -
~a P div v d f ~ - ~ f v d ~
(22)
The equilibrium equations of this saddle-point problem are:
a(v, lg)-b(lg, p) = ( f , ~ ) , V ~
f
~ [H~ (f~)] 3
b(v, O) - O, V O e L 2 ( a ) a (v, N) - ~a/1 grad v grad N d ~
(23)
b(v, ~) - fa ~ div v df~
where H and L are appropriate Sobolev spaces and (u, v) - ~~ uvdf~
(24)
The principle of the finite element method is the transformation of the above variational problem into a set of algebraic equations that can be readily solved with the tools of linear algebra. For this purpose, suitable approximations are used for the velocity, the pressure and the test functions. In order to guarantee stability and convergence of the numerical solution, these approximations must be carefully chosen. In three-dimensional mixing simulations, the super-linear tetrahedral element PI+-Po and the tetrahedral quadratic element P2+-P1 are state-of-the-
320
art elements [31]. Tetrahedral elements are a much better choice than quadrilateral elements for several reasons: 9 they are more flexible for unstructured grids 9 their accuracy suffers less from distortion 9 they are less costly to use that their equal-order quadrilateral counterpart. After discretization, and using the penalty-Uzawa algorithm for the treatment of incompressibility, the matrix problem to solve is of the form: A
rv
i+1
--
BT p i = F
A r = A + rB rB
(25)
pi+l _ p i _ r n v i + l
where A and B are the elliptic (viscous) matrix and the divergence matrix respectively and r is the penalty parameter. The reader is referred to [32] or an advanced textbook on finite element methods in fluid flow problems for a detailed exposure of the above principles. For very viscous fluids and pastes, a single impeller may not be sufficient to mix the media and stagnant regions may occupy most of the vessel volume. In such a case, multiple impeller kneaders and planetary mixers must be used. The fluid mechanics of mixing inside a planetary mixer is much more complex than with a single impeller. From a simulation standpoint, the complexity arises from the fact that there is no symmetry of revolution usable to simplify the flow description. The most convenient viewpoint for flow description is the classical Eulerian viewpoint of the laboratory and the problem must be dealt as fully transient. In order to tackle kneading flows, a new approach has been developed called the virtual finite element method [33]. This method is nothing but a subset of the broad category of domain imbedding finite element method. In short, the idea is to represent in the volumic finite element mesh, the impeller as kinematics constraints. For this purpose, the impeller is discretized with control points on which the kinematics is known. Then these additional constraints are implemented into the finite element discretization. Mathematical optimization methods (penalty method and Lagrange multipliers) are used to enforce these constrainnts. The final matrix problem is modified accordingly and an extra step is required to update the Lagrange multiplier iteratively.
321 The flowfield is obtained after resolution of the set of governing equations. This field adequately post-processed may yield very useful information for design purposes (power consumption, shear and extensional force profiles, mixing segregation) and process understanding (dispersion and distribution mechanism, chaotic features). We propose to illustrate some of these capabilities in the next section.
4. R E S U L T S A N D D I S C U S S I O N 4.1 Validation and Couette flow analogy The validation of the above numerical method is carried out for the mixer shown in Figure 7. The impeller is a single flight helical ribbon closeclearance impeller.
D s = 0.013 rn W=0.03
m
C = 0.0125 m
P = 0 . 1 8 5 rn
D = 0.210 m
H = 0.24 m
~I
Figure 7: Single flight helical ribbon impeller mixer We first show in Figure 8 the variation of Np vs Reg for Newtonian and shear-thinning power law fluids (CMC solution, 1.5 wt. %) at room temperature up to a Reg value of about 2. It can be seen that the agreement between the numerical predictions and the experimental data is excellent, which confirms the findings of an earlier study [30].
322
1111111
I :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: . . . . . . . . . .................. . . . . . . . . . . . . .9. . .
' ........... ..:~ . .. . ... . ...
' ....... .: . . . . . . . .
9............... .............. .~ ~ ........... ................... ..... ~~-.:---i--!"-i/ .:....... ,~~.... ........ :...,......
~. . . . . . : . . . . . ~ . . . . i . . . ~ . . x..:" ................... : ~ . . . . 1 .~. . *: . ..' . :. i .!. ' :~ . . . . . -. . . . . . . . . . . . . . ,. .~. ., .:
................. ~........ ..~.-.~
..... i--..;...,..,..--~.
: ........... :i . . . . . . . . . . .
.................. .:. .......
c. . . . . . . ,~: . . . . . .
.~ . . . . . . . . . . . ~ . . . ~ . . . . ~ . . . : . . :" . . . . 2~. . . d . , . .:~ . . . ~ .: . J . - : "
~........ --...*...+.--t---~--.,--
I ................i........~-...".....~-.4-.~--~-!--i-..........4 ......~ ........i.....~'----i'"'-'"'!"§ ..............................:.-....... . !.....i---i----i---!-i-; ...................~...........i.......~....~-----~---!---J-i-.................. ~.......... -~ ...... ~-.... "...~-'..i-*-i
: Z"
~ ......... i ....... .~ ..... ~-4 ...... .~-~--
!
9
i i ! i:~!
9
100
. .................. .......................
; ........... , ~i ....... .......
:..
................... . ......... r .... :~ " ~ ~ . - - ' ~ . " - . . _ ~ , . ..................
. : ...........
: *. . . . . . . .
: " ......
-L....~-~..,..'.~...-'-"..,.~....:-.1..~,-~ . .
tf
0.10
. . x. . . ...... .
§i ........... .......
~~........ . . . . . . . ,~ i ,..... - . . . . : . i-.--:------i--.*---,~ ........... ...-
~".'-*.'~"..'~:::::~2..~........ ~ .... .~ ..... ~....-...,-;..,. ................. ... ......... ; ...... ~....~....~__..'_...:.
:-~..~_
: ! : "'"'~"'T":
................. ~........... ~. ..... - . ~ ' ~ - - . ~ - ~
10
..............
i i ii.l~'
9 i ~": ...................
' ~ ...........
: ~ .......
~. ~ :. : ..... :'""~
~ 3.-.:., : ." :
................ ~ ........... ! ..... i ..... i--.-~ .......
i 1.00
......
num. n--1.00
[ ]
num. n=0.651 exp. n-l.00 ,exp. n~O.esJ
i---i-
i! iii i 1 10.00
Re. F i g u r e 8: P o w e r c u r v e in the l a m i n a r m i x i n g r e g i m e W e p r e s e n t in F i g u r e 9 the i s o - s u r f a c e ( s u r f a c e o f c o n s t a n t v a l u e s ) o f the p r o c e s s v i s c o s i t y for the s a m e C M C s o l u t i o n a n d a r o t a t i o n a l s p e e d o f 10 R P M . In this case, a v a l u e o f 3.37 s -~ w a s o b t a i n e d for the e f f e c t i v e rateo f - d e f o r m a t i o n y i e l d i n g a p r o c e s s v i s c o s i t y v a l u e o f 4 . 8 6 Pa.s.
F i g u r e 9: L o c a t i o n o f the p r o c e s s v i s c o s i t y
323
It can be seen that the process viscosity is located along the impeller blade and not in the gap between the ribbon and the wall, as it could be expected with the Couette flow analogy. This is to our knowledge the first time such a result is presented. The numerical prediction of the variation of Kp(n) vs the shearthinning index is compared with the experimental results in Figure 10. 150 -.-e- numedcal 1 -..a-- experimental 100 r
:Z 50
.... 0
~ ..... 0.2
i .... 0.4
:, . . . . . . . 0.6
0.8
1
Figure 10: Variation of the power constant with the power law index It can be seen that Kp(n) does not vary linearly with n. Indeed, although the slope is constant above n=0.6 as already noted in [ 17], the slope tends to decrease for higher shear-thinning fluids. The same trend is observed numerically and experimentally, although to a lesser extent in the latter case. In Figure 10, there is a discrepancy between the numerical predictions and the experimental determination of Kp for the lower values of n. The difference could be imputable to the larger numerical errors inherent to the evaluation of stiff velocity gradients in the simulation. We present in Figure 1 l a a plot of the normalized angular velocity Vz/(Vr 2 q- Vz2)1/2. It corresponds to an isometric view of the cross section plane intersecting the central shaft. Figure 1 lb shows a top view of the same field at mid-height. The spectrum goes from dark grey (small angular velocity) to light grey (high angular velocity). These two figures were obtained for the CMC solution (1.5 wt.%, room temperature) at N= 10 RPM. In Figure 11 a, the downward pumping zone appears clearly in the vicinity of the shaft, where the rotational speed is negligible.
324
//
,jJ Figure 11" Pumping zone of the HR impeller - a) side view; b) top view In Figure 11, two pumping regions can be observed" a (downward) pumping zone in the center and an (upward) pumping zone near the helical ribbon impeller. The downward pumping zone is bounded by an offcentered circular envelope which can be regarded qualitatively as the Couette equivalent cylinder.
4. 2 Mixing flows in Dual helical ribbon impellers We now consider the flow in a typical polymerization reactor provided with a dual helical ribbon impeller (Figure 12). Usually, in industry, these impellers are preferred over single helical ribbons due to their improved performance. The simulation has been carried out using the Galerkine finite element method described before and the Lagrangian frame of reference approach. The surface finite element mesh is shown on Figure 6 as well. We show in Figure 13 a typical circulation profile. The axial velocity has been plotted at two horizontal cross sections. The scale is from-0.12 m/s (pumping down velocity) to 0.13 m/s (pumping up velocity). This picture gives a fairly good representation of the circulation in the vessel. The helical ribbon blades push the fluid at the top in the vessel periphery while scraping the wall. The downward circulation region is located along the shaft, the flow being mainly horizontal at the top (pointing inward) and at the bottom (pointing outward).
325
Figure 12: Double HR impeller
Figure 13" Vertical velocity profiles
Figures 14a and 14b represent a typical dispersion pattern obtained with such an impeller. These results are obtained by tracking with time the position of particles initially clustered and injected in the bulk in the vicinity of the linking rod between the impeller and the shaft. The trajectories are computed using an adaptive 4th order Runge-Kutta method with time-variable steps. In Figure 14a, the "stretching-folding-breaking" mechanism can be clearly identified. The particles initially agglomerated gradually separate one from the next (like a widening ribbon) while turning around the shaft during their travel towards the vessel bottom. In the bottom region, they break apart and disperse in the bulk, circulating outward. Figure 14b shows a similar behavior after a longer tracking time. The dispersing-homogenizing capability of the impeller appears very clearly.,
Figure 14: Dispersion pattern- a) initial stage; b) homogenizing stage
326
Figure 15 shows the pumping pattern in the vessel from the top. This figure is another way of looking at the circulation pattern. The pumping down region (on the left) covers most of the region confined within the helical ribbons while the pumping up phenomenon is restricted to the ribbon region only. The "Couette" cylinder appears remarkably.
Figure 15: Downward (left) and upward (fight) circulation region
4.3 Mixing flows in planetary kneaders Planetary kneaders can be seen as a special type of closeclearance mixers. They are mixing systems of choice for pastes, thickenerbased fluids and for cross-linked polymers. The modeling of the kneading flow is a new application of CFD which is receiving an increasing attention. In this work, we consider the flow in a twin-blade planetary kneader from APV. The system comprises a slow plain blade (twisted paddle) and a fast hollow blade mounted off-centered on a rotating carousel (Figure 16).
Figure 16: Planetary kneader computer model
327
The carousel to plain blade speed ratio and plain blade to hollow blade speed ratio are both 0.5 by construction. The results shown hereafter were obtained with a carousel speed of 15 RPM, a plain blade speed of 30 RPM and an hollow blade speed of 60 RPM. Contrary to the previous close-clearance impellers, the kinematics of the blades is such that no convenient simplifying frame of reference can be used. The virtual finite element method is then required to tackle the flow simulation. We show in Figures 17a and 17b the typical flowfield in the kneading vessel for an industrial cross-linked polymer. In Figure 17a, a detailed flow pattern in the region of the blade is shown. Although the blade has a twisted shape similar to a screw, the circulation appears rather horizontal, an indication of a poor top-to-bottom pumping. Figure 17b shows the general circulation pattern at mid-height in the kneader. Intense shearing appears between the blades and in the clearance between the blade tips and the wall. Far from the blade, the flow is slow and almost stagnant near the wall. .
"~
.'~
-.'~
.
.
.-,
.
.
c
.
~.
.
.
,--
.
.
.
.
.
.
.
.
.
.
.
.
~
. . . .
Figure 17" Velocity profile in the kneading vessel We show in figure 18 the dispersion pattern in the vessel. This result was obtained in the same way as with the double helical ribbon impeller. The clustered particles are initially injected at mid-height in the bulk close to the region where the blades mutually interact in an intense shearing region. It can be observed that the axial dispersion is limited to the central core of the kneader, confirming the deficiency of the top-to-bottom circulation observed in Figure 17. Radial dispersion appears, however, very
328
good as shown by the very chaotic trajectory patterns that covers most of the cross-section surface of the vessel. This proves if necessary the relevance of using more than one impeller to break the flow symmetry in viscous mixing, and therefore enhance the process efficiency.
Figure 18: Dispersion pattern; left: side view; fight: top view.
5. CONCLUSION In this chapter, we have shown how the design of mixers and kneading equipment for non-Newtonian viscous fluids can be carried out using published correlations and/or numerical techniques. It is now possible to predict mixing performance during the process development phase, before building up equipment. It is worth mentioning that this field is still knowing significant developments, and it is expected that more complex problems involving multiphase systems and more complex rheology will be tackled in the near future. ACKNOWLEDGMENTS The technical contribution of Dr. F. Bertrand and Mr F. Thibault from Ecole Polytechnique is gratefully acknowledged. Thanks are also directed to DGAPA (UNAM) and NSERC for the financial support that made possible several contributions in this chapter.
329
REFERENCES ~
2. .
4. .
6. .
~
.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
Ottino J., 1989, The Kinematics of Mixing, Cambridge Univ. Press. Skelland A.H.P., 1967, Non-Newtonian Flow and Heat Transfer, Wiley. Eklund D.E. and J.E. Teirfolk, 1981, TAPPI J., 64, 63. Solomon J., T.P. Elson, A.W. Nienow and G.W. Pace, 1981, Chem. Eng. Comm., 11, 43. Barnes H.A., 1997, J. Non-Newt. Fluid Mech., 70, 1. Rauline D., P.A. Tanguy and P.J. Carreau, Mixing of Thixotropic Fluids, in preparation. Hocker H., G. Langer and U. Werner, 1981, German Chem. Eng., 4, 133. Brodkey R.S., 1967, The Phenomena of Fluid Motions, The Ohio State University Press, Columbus. Metzner A.B. and R.E. Otto, 1957, AIChE J, 3, 3. Doraiswamy D., R.K. Grenville and A.W. Etchells, Ind. Eng. Chem. Res., 1994, 33, 2253. Brito-de la Fuente E., J.C. Leuliet, L. Choplin and P.A. Tanguy, 1992, AIChE Symp. Ser., 286, 28. Brito-de la Fuente E., L. Choplin and P.A. Tanguy, 1997, Trans. IChem. E., 75, 45. Carreau P.J., R.P. Chhabra and J. Cheng, 1993, AIChE J., 39, 1421. Bourne J.R. and H. Buffer, 1969, Trans. IChem. E., 47, T263. Coyle C.K., H.E. Hirschland, B.J. Michel and J.Y. Oldshue, 1970, AIChE J., 16, 903. Hall K.R. and J.C. Godfrey, 1970, Trans. IChem. E., 48, T201. Chavan V.V. and J. Ulbrecht, 1973, Ind. Eng. Chem. Process Des. Develop., 12, 472. Rieger F. and V. Novak, 1973, Trans. IChem. E., 51,105. Nagata, S., 1975, Mixing" Principles and Applications, Kodansha and Wiley. Sawinski J., G. Havas and A. Deak, 1976, Chem. Eng. Sci., 31,507. Chowdhury R. and K.K. Tiwari, 1979, Ind. Eng. Chem. Process Des. Dev., 18, 227. Yap, C.Y., W.I. Patterson and P.J. Carreau, 1979, AIChE J., 25, 516. Kuriyama M., K. Arai and S. Saito, 1983, J. Chem. Eng. Japan, 16, 489. Shamlou P.A. and M.F. Edwards, 1985, Chem. Eng. Sci., 40, 1773.
330
25. 26.
27.
28. 29. 30. 31. 32. 33.
Bourne J.R. and H. Butler, 1969, Trans. IChem. E., 47, T11. Brito-de la Fuente E., L. Choplin, A. Tecante, and P.A. Tanguy, 1994, in Progress and Trends in Rheology IV, C. Gallegos, J. Munoz and M. Berjano (Eds.), Darmstad-Steinkopff, Germany, 272. Brito-De La Fuente E., J.A. Nava, L. Medina, G. Ascanio and P.A. Tanguy, 1996, Proc. XII International Congress on Rheology, A. AitKadi, J.M. Dealy, D.F. James and M.C. Williams (eds.), Canadian Rheology Group, 672. Patterson W.I., P.J. Carreau and C.Y. Yap, 1979, AIChE J., 25, 508. Tatterson G.B., 1991, Fluid Mixing and Gas Dispersion in Agitated Tanks, McGraw-Hill. Tanguy P.A., R. Lacroix, F. Bertrand, L. Choplin and E. B rito-De La Fuente, 1992, AIChE J., 38, 939. Bertrand F., M. Gadbois and P.A. Tanguy, 1992, Int. J. Num. Meth. Eng., 33, 1251. Tanguy P., L. Choplin and M. Fortin, 1984, Int. J. Num. Meth. Fluids, 4, 441. Bertrand F, P. A. Tanguy and F. Thibault, 1997, Int. J. Num. Meth. Fluids, 25, 719.
331
VISCOELASTIC FINITE VOLUME METHOD N . P h a n - T h i e n a n d R.I. T a n n e r
Department of Mechanical and Mechatronic Engineering The University of Sydney NSW 2006, Australia 1. I N T R O D U C T I O N
Analytic solutions to non-trivial viscoelastic flow problems are rare due to the complexities of the constitutive equations and the nonlinearities of the conservation equations. To make any progress, we have to abandon the search for the analytic solution and seek an approximate solution via a numerical method, which is either a finite difference (FDM), a finite volume (FVM), a finite element (FEM) or a boundary element method (BEM); see for example, Crochet, Davies and Walters [1], Reddy and Gartling [2], Phan-Thien and Kim [3] and Patankar [4]. Earlier numerical schemes failed to converge at a relatively low level of flow elasticity when elastic effects become comparable with viscous effects [5]. The level of the flow elasticity is characterised by either the Weissenberg number Wi (product of a characteristic fluid relaxation time and a typical shear rate), or the Deborah number De (the ratio of a characteristic fluid relaxation time to a characteristic time scale for the flow, which is usually taken to be the reciprocal of the wall shear rate in a fully developed flow region). Recent progress in FEM, such as the streamline-upwind (SU) [6], the explicitly elliptic momentum equation (EEME) [7], the elastic-viscous split stress (EVSS) formulation [8], and the adaptive viscoelastic stress splitting (AVSS) scheme [9], have been reviewed elsewhere in this book. Therefore we will be concerned with the Finite Volume Method here, which has been very popular in high-Reynolds number Newtonian flows [4], but only made its presence felt in computational viscoelastic fluid mechanics recently. Our review begins with the formulation, the implementation, and some two and three-dimensional problems that we have had recent success with, including the channel flow past a cylinder, and the three-dimensional entry flow.
332
2. F O R M U L A T I O N
2.1 Governing Equations We are concerned with a general time dependent and isothermal flow of an incompressible viscoelastic fluid where the governing equations take the form: C o n t i n u i t y e q u a t i o n (conservation of mass)
V-u--O,
(1)
M o m e n t u m e q u a t i o n s (conservation of linear momentum in absence of body forces) p ( ~ - + V- (uu)) - V . a ,
(2)
where t is the time, u is the velocity vector with components {u, v, w}, and a the Cauchy stress tensor, given by a = - p l + S,
(3)
with p being the hydrostatic pressure, 1 the unit tensor, and S the "extra" stress tensor (not necessarily traceless), which is related to kinematic quantities by an appropriate constitutive equation. C o n s t i t u t i v e equations In most studies, either a differential (Maxwell-type) or an integral constitutive equation is used to model the fluid rheology. Integral constitutive equations are less popular because of the need of particle tracking. A review of constitutive equations, covering both microstructural and continuum view points is given in Huilgol and Phan-Thien [10]. Here, we will be mainly concerned with the P T T model [10, 11], where the extra stress tensor is written as S - 2~ND + 7",
(4)
where ~N is the Newtonian-contribution viscosity, D - (Vu + Vu T)/2 is the strain rate tensor, with T denoting the transpose operation, and 7- is the polymer_contribution stress, which evolves according to the following constitutive equation g r + A v = 2r/m0D'
(5)
333
where A is the fluid relaxation time, r/m0 is the "polymer-contributed viscosity", (.v) represents the following convected derivative: v Or v - ~ + V - ( u ~ ' ) - (L - ~D)7" - v (L - ~D) T ,
(6)
and Ae g = 1 +--tr
(r),
(7)
r/m0
where L - V u T is the velocity gradient tensor, and ~, e the material parameters for the P T T model [11]. By introducing the retardation ratio defined as/3 - ~,n0/~0 with r/0 = 7/N + 7/m0 being the total viscosity, the Oldroyd fluid B [12] (when ~ and e are set to zero) is recovered with the retardation time being A2 - (1 - / 3 ) A; in addition, the Upper Convected Maxwell (UCM) model is recovered with /3=1. The Reynolds number R~ defined as R~ - p U h / ~ o , and the Weissenberg number W~, defined as We = A U / h , are used to characterise the flow and the fluid elasticity, where h and U are typical length and velocity scale in the flow, respectively. To compare numerical to experimental results, a zero shear rate relaxation time, defined by N1 ('~)
(8)
2r
is used for the experimental fluid. The Deborah number D e - A ~ based on the maximum wall shear rate "~w in the full developed downstream channel is sometimes used instead of W~. As an example, for the planar contraction flow (constant viscosity elastic fluids [13]), this is thrice We when the upstream aspect ratio of the duct is sufficiently large.
2.2 Split stresses Our numerical experimentation leads us to adopt the EVSS formulation of Rajagopalan et el. [8]. That is, the extra stress (Eqn. (4)) is re-written as:
Sij - 271odij +
Y]ij.
(9)
Substituting (9)into (3) and (5)yields
OUi
0 (
OUi~
OP OY]ik
334
(11)
gEij + A ~ i j - 2~7/0 [(1 - g)dij - A ~ij
Thus the dependent variables are now ui, p and Eij. With this coupling between the kinematics and the stresses, we find that the kinematics calculations are less sensitive to the gradual loading of elasticity via the pseudo-body force term OEik/OXk, thereby improving the stability of the calculations. 2.3 B o u n d a r y C o n d i t i o n s To the set of governing equations one must add a set of relevant boundary and initial conditions.
Slip or no-slip on a solid surface The no-slip boundary condition at a solid surface is usually adopted in most studies, where the fluid velocity assumes the velocity of the solid surface. This assumption works well for viscous fluids, but there is a large amount of experimental data suggesting that it may not be relevant for polymeric liquids in some circumstances. There are extrusion experiments with polymer melts [14-18], which suggest that wall slip may be responsible for melt fracture. In these experiments, the occurrence of the extrudate irregularities occurs above a critical wall shear stress, which is accompanied by a fluctuation in the pressure drop. A phenomenological approach to the slip boundary condition has been proposed by Pearson and Petrie [19] where the slip velocity is taken as an empirical function of the wall shear stress. A polymer network model has been proposed recently to account for the dynamic slip velocity [20]. Here, we just simply adopted the no-slip boundary condition at a solid surface, and note that real progress in this area will be made by a careful consideration of the microstructure near a solid surface. With this (Dirichlet) boundary condition, the velocity field is prescribed as u=u0,
OnSu,
where u0 is known on the boundary S~.
Traction boundary conditions Sometimes the traction vector is given on a part of the boundary, say St. This type of (Neumann) boundary conditions take the form t=er-n=t0,
on St,
where t is the traction vector. This yields the normal traction, tn
"
-
-
n - O ' - n -- --p + S- n = n - to,
o n St,
(12)
335
and the tangential tractions
t-t-nn=t0-t0-nn,
on St.
Note that we use Su to denote parts of the boundary where velocity is prescribed, and St, parts of the boundary where the traction is prescribed. These surfaces need not be singly connected, but may consist of several patches. Indeed both velocity and traction boundary conditions can be prescribed at a given location, but in different directions, as in the case of the extrusion problem. Robbins boundary conditions arise in some slipstick problems, where the slippage velocity is determined by the tangential traction (shear stress) at the wall.
Free surface boundary conditions For problems with a free surface, e.g., bubbles, extrusion, etc., the location of the free surface is not known and must be found as part of the solution procedure. Here, a kinematic constraint can be used as the condition to locate (implicitly) the free surface. In a steady flow, the kinematic constraint for a free surface is
u-n=0.
(13)
For unsteady flow, if r t) = 0 is the location of the free surface, then always remains zero on the free surface and its material derivative must also be zero there. The free-surface kinematic constraint thus takes the form ar 0--7 + u - V r - 0. (14) Since n - + V r is a normal unit vector on the free surface (the sign can be chosen so that n is the outward normal unit vector), this constraint also takes the form
1 0r IVr a t
+ u..
= 0.
(15)
In addition to this, the traction on the free surface is known from the physics of the problem. If there is no surface tension, for example, then the traction vector is zero on the free surface. For the case where the surface tension is not negligible, we recall that the surface tension 9/is postulated to be the force per unit length acting along the edges of the free surface. The equilibrium condition on an arbitrary surface element A S reads
fAS [er]. n dS + fAC "~q dl -- 0,
336
where [er] 9n - (er + - er-) 9n is the jump in the normal traction, with er + being the stress on the positive side of n, and or- the stress on the negative side of n, and q is the unit vector normal to the boundary curve AC, but tangential to the interface AS. This is the mathematical statement of zero force on AS. From Stokes' theorem, the surface integral over A C can be done by parts,
/~c7 q dl - /~sV7 d S - /As7
(V-n)n
dS,
leading to (p- - p + ) n + (S + - S - ) . n + V 7 - 7 ( V . n ) n - 0.
(16)
Note that the mean curvature of the surface is given by [21] V n .
.
.
1 ~~
1 .
.
R1
(17)
R2
Symmetry boundary conditions Very often the computational domain can be reduced with the help of symmetry, these sylmnetry boundary conditions are quite easy to state: a plane of symmetry has no normal velocity component, and no tangential stress (tangential tractions or vorticity are zero), i.e.,
u-n=O,
t-t.nn=O,
(18)
where t is the surface traction.
Inflow outflow boundary conditions At the inflow to the solution domain, we know something about the flow, usually the velocity field. For viscoelastic fluids, the stresses are also required there (in essence, these represent the information carried with the fluid from its previous deformation history). Therefore, the boundary conditions at the inlet are usually Dirichlet boundary conditions. Note that all components of the stress cannot be arbitrarily prescribed. First, the stress components should be consistent with the specified kinematics (but need not be), otherwise steep stress boundary layers will be set up. Secondly, if traction boundary conditions are also given there, then the traction component tangential to the boundary would involve the stress components alone without the pressure terms, and therefore these stress components cannot be prescribed arbitrarily. Finally, Renardy [22] has pointed out that specifying all the stress components at the inflow may lead to an over-determined mathematical problem. It may be prudent in practice to retain the time derivative terms in the (differential) constitutive equations,
337
even though we may be dealing with steady-state solutions, and integrate the equations in time through an initial stress state until a steady state solution is reached, including the stress components at the inflow. This ensures that a physical solution is obtained. At the outlet, the flow is usually well developed and arranged so that a uni-directional flow results. Outflow boundary conditions therefore usually take the form of no transverse velocity and no axial traction.
Boundary Conditions for the Pressure Boundary conditions on the traction vector arise naturally from a weighted residual method. However, in the finite volume method the equation for the pressure is usually solved separately from the velocity field, by using the continuity constraint on the linear momentum equations, and therefore boundary conditions for the pressure may be required. The correct boundary conditions for the pressure can be derived either from the traction boundary condition, or from the momentum equations. Thus, a traction boundary condition (to) will turn into a Dirichlet boundary condition for the pressure:
p=S:nn-t0.n,
on St.
(19)
Otherwise, a Neumann boundary condition for the pressure will result from the momentum equation, if the kinematics are fully prescribed, = '1
-p
+ u. Vu
+ v.
s + pb
,
on
(20)
Note that a traction boundary condition at one point on the boundary will implicitly set the pressure. If there is no traction boundary condition, then the pressure can only be determined up to an arbitrary constant. It is important to keep in mind that a traction boundary condition is not the same as the pressure boundary condition; the former is a physical quantity that we can actually impose on the fluid, the latter is a derived quantity, arising only because we are interested in solving the Poisson's equation for p in isolation. If the set of equations, continuity, momentum, are solved jointly, then p would inherit the correct boundary conditions from the boundary conditions for u and t, and there is no need to impose anything on p at all.
Initial Conditions For time-dependent flows, a set of initial conditions is required. The initial velocity prescribed must be divergence free. For viscoelastic fluids, a set
338
of initial values for the stress components is also needed. Note that the incompressibility constraint also forbids an impulsive start and stop, normal to the boundary, since
fs
u.n
dS-O.
3. F I N I T E V O L U M E M E T H O D (FVM) The finite volume method has been very popular in computational fluid dynamics dealing with Euler and Navier-Stokes flow problems. There are two main approaches in the finite volume method.
3.1 Chorin-Type Methods The first approach uses an artificial compressibility condition to satisfy the continuity equation. A pseudo-transient formulation is then adopted in the momentum equations and the steady state solution is considered as the asymptotic solution of a time-dependent problem with time-independent boundary conditions (if need be), and these are computed by a timemarching scheme. The method is due to Chorin [23] and is very commonly used in computational fluid dynamics dealing with high Reynolds number flows. Integration in time can be carried out either implicitly or explicitly. In implicit schemes, it is necessary to solve a system of equations at each time step, making the scheme very expensive for large three-dimensional problems. However, such schemes can be made unconditionally stable and relatively large time steps can be used. On the other hand, in explicit schemes, it may be necessary to solve only a simple mass matrix, or it may not be necessary to solve any system of equations at all. The main disadvantage of explicit schemes is that the time step is restricted by the Courant-Friedrichs-Levy stability condition. However, there are several ways to accelerate convergence such as residual averaging, local time-stepping and multigrid schemes. Phelan et al. [24] use a similar scheme, but with a modification to suit hyperbolic types of constitutive equations, for solving cavity-driven flow problems for the UCM model. The same method has also been used to solve the slip-stick problem of Maxwell-type constitutive equations [25].
3.2 S I M P L E R - T y p e Methods The second approach in finite volume method is much more popular and forms the basis of some commercial packages dealing with Navier-Stokes equations. The resulting algorithms are known as SIMPLE (Semi-Implicit Method for Pressure-Linked Equations), SIMPLER (SIMPLE Revised) [4],
339
SIMPLEC (SIMPLE Consistent) [26], or PISO (Pressure-Implicit with Splitting of Operators) [27], to name a few. In these schemes, the momentum equations are solved sequentially, and the pressure is corrected from the continuity, equation and the linearised momentum equations. Applications of the SIMPLER algorithm to viscoelastic flows include the 4-to-1 entry flow and the die entry problems [28-30], the non-isothermal flow past a cylinder [31], the flow past an eccentric cylinder problem [32]. The PISO method has been applied to the non-circular pipe flow of the Criminale-Ericksen-Filbey model to study the effects of the weak secondary flow on the pressure drop [33]. A variant of the SIMPLER method, termed SIMPLEST (SIMPLE with Splitting Technique) has been used recently to study secondary flows in non-circular pipe flows, and the 4-to-1 threedimensional flows [30, 34]. Formulation In the FVM, the flow domain is subdivided into a set of non-overlapping control volumes (CV). The grid nodes (storage locations for the dependent variables) are located at the centre of each of the control volumes. The discretised equations of the dependent variables are obtained by integrating the mass, momentum conservations over each control volume as follows: 1. First, all of the governing equations are cast into the form of a general transport equation: AO(I) -b-/- +
0 (Auk,I)) -
a
k
+
where (I) is the dependent variable which can be a component of a vector or a tensor and even a constant. The coefficients A, F have different meanings for each different dependent variable, and Sr is called the source term, which lumps all the terms that cannot be accommodated in the convective and diffusion terms, and is specific to the particular variable (I). 2. Next, the general transport equation is integrated over the control volume surrounding the node, namely node P, in the flow domain and the time interval St, using the divergence theorem whenever possible,
340 Here, A V is the volume of the control volume and ~ v the corresponding face area with n k being the unit outward normal to the face. 3. Finally, a proper discretisation scheme (the control volume-based interpolation functions) is adopted for the temporal and spatial approximations of the dependent variable. Generally, the first-order backward Euler implicit formula is used for temporal differences because of its simplicity and its unconditional stability for numerical calculations. As to the volume integral of the source term S~, the value at the central node of the control volume, namely S~, is assumed to prevail over the whole control volume, and it can be linearised in term of the nodal value Op as usually assumed in FVM. In this way, the final algebraic approximation equation which relates Op to its neighbouring nodal values can be written as o ap~p -- ~ anb~nb -4- Sc --[-ape~p, nb
(23)
where o_ A
ap
~-~AV,
o
-
ap -- E anb + ap -- Sp. nb
(24)
Here, the subscript p refers to the central node P, and the summation is to be taken over all of the neighbor nodes nb of the node P. An overbar means the applied values are evaluated using the known fields from the previous time (or iteration) level, and the coefficients anb are the functions of the dependent variables, and their structures depend on both the form of the control volume chosen and the approximation scheme used. It is these coefficients that determine the spatial accuracy of the final solution. In our calculations, the power law scheme proposed by Patankar [4] is employed for the formulation because it covers central difference and upwind difference, and gives an excellent approximation to the exact exponential solution for the one-dimensional convective-diffusive equation. In addition, FVM with power law scheme, instead of central or upwind difference, has better conservational properties and thus produces physically realistic solutions even for coarse meshes. In the power law scheme, the coefficients are chosen as follows: anb -- D n b f ([Pnbl) + [sign(nb)Fnb, 0~ ,
(25)
where Dnb -- (FA/ Xi)nb is the local diffusion conductance; Fnb -- (Au~A)nb the "mass" flux passing thought the corresponding face A normal to i
341 direction of the control volume, sign(rib) is +1 for upstream face and - 1 for downstream face, and f (I Pnb I) is the function of the local Peclet number defined by Pub -- Fnb/Dnb, which is given by
f ([Pub[) -- [0, (1 -- 0.11Pnbl)5],
(26)
where the symbol [a, b~ means the greater of a and b.
Constitutive equation discretisation The viscoelastic model is also in the form (21), without the diffusion term (F - 0). To ensure numerical stability a first-order upwind difference is used for spatial discretisation. Thus, the discretised constitutive equation will take the similar form to (23) (thereafter we use the symbol "1- instead of E for the elastic stress tensor Eij 2flTIodij to avoid confusing with the summation symbol E)" --
7"ij
--
ij ij ij ap Tp -- E anbTnb -~- Sc ~, nb
(27)
where the superscripts ij refer to tensor components while reserving subscripts for the grid node and the overbar for the values from the previous time (or iteration) level. The constant part of the source term, in which the stresses and deformation are approximately piecewise-constant in each control volume, takes the form (no sum in i and j)"
S:~ -
ap~'p~ + A V {2/3~0 (1 - g)dp j + )~ (1 - ~)[(dpi + d jj) ,0] CpJ + )~ [~k (1 - 5kj)+ 2/3~0d~k] [(OuJ I
(28)
\Oxk/p \Oxk]p
0 while the coefficients ai.j and and SpJ - - A V [g
ap0
take a form like (24) with F - 0, A -
)~ (1 - ~ ) [ - (dpi + dJpJ), 0]] - E sign(nb)Fnb,
A,
(29)
nb
where Fnb -- ()~uiA)nb can b e thought of being the "mass" flux passing through the corresponding face A normal to i direction of the control volume. The term Esign(nb)Fnb in 5'~J is actually the continuity constraint nb
which should approach zero when the solution converges.
342 We note that there is an extra convective term in the source term in the constitutive equation; this extra source term should be consistent with that of the convective term of the stresses. Thus, upwind difference should be applied to its discretisation. This leads to 0
-
With upwind difference, the resulting coefficients anb will be anb -- [sig~(nb)Fn~, 0].
(31)
Therefore, the sufficient condition for convergence for the adopted numerical algorithm-TDMA (typically alpj > E anb) is satisfied even for the
nb
steady-state calculation with a ~ being zero. Some work [35] used a pseudotransient constitutive equation for the steady-state calculation to ensure a resulting diagonally dominant matrix, and so obtained convergent solutions for the inertialess flow of UCM fluid (g - 1) through a 4 9 1 abrupt axi-symmetric contraction with a much higher value of the Weissenberg number (up to W i - 6.25). The pseudo-transient method for the steady-state calculation is actually equivalent to a local under relaxation: the positive inertia coefficient ap0 damps out possible oscillation during the stress iteration. In our previous work [32, 34, 36], an artificial diffusion term -g~-2~k\ ox~] with c~ being an artificial diffusion coefficient is introduced on both sides of the constitutive equation, and in discretisation, the current values are taken for Tij on the left hand side; while the known values from previous iteration level for that on the right hand side. As a result, depending on the spatial discretisation scheme used, the coefficients anb in (27) will take the form
anb- Dnbf
~
-4- [-sign(nb)Fnb, 0],
(32)
where Dub -- ((~A/6xi)nb denotes the local diffusion conductance, and function f ( I D ~ b l ) h a s the different form for different schemes. For example, when the central-difference scheme is adopted, we have \1
I/
1 05,
,
343
Thus, with the scheme adopted, the convective term of the equation can be thought of being always discretised using upwind difference, and the diffusion term is discretised using central difference, but the resulting local diffusion conductances are different for different schemes used, that is ' + [sign(nb)Fnb, O~ - anb d + anb ~ , anb -- Dub
(34)
with D'nb -- D ~ b - 0.5 IFnbl for central difference and D'nb -- Dub for upwind d to denote the contribution of the artificial difference. Here, we use anb diffusion term discretisation, and a~b the contribution of the convective term discretisation, that is:
d _ D n' b ~ anb c _ [sign(nb)Fn~, O~ anb
(35)
To ensure all of the contributions to be positive, we need !
(36)
Dnb > O.
Obviously, this is always the case when upwind difference is used. However, for the case when central difference is employed, we require that
Dub ~ 0.5 IFnbl.
(37)
Thus, we have
(
Dub--
1)
1-- ~
Dub,
(38)
with Dub -- IFnbl and ~; is an arbitrary constant satisfying ~; _> 0.5. This can be thought of being the criterion for the proper selection of the artificial diffusion coefficient c~. Similarly, the source term will contain an extra artificial diffusion term _ f z x u _ ~ko \(c~~176 dV, and central difference is applied for its discretisation. The discretised form for the extra source term will have the form:
~dj -- _ y~ DnbTnb _ij + "fp3 ~ Dub. nb nb
(39)
Thus, the final discretised constitutive equation can be cast into the form c + adnb) Tub ij + s~J 2t- ~d -"j, (ap T ad T apO_ ~ipj) ~ "" _ E (anb (40) nb where C apC :~-~ anb and a d :~--~ anb,d nb nb
(41)
344
which can be re-arranged to yield ....
(ap + a ~ - ~ipj)
+ [(a;
+ aO -
I v " _c
ij __ O~J
n~ ttnbTnb-1- ~c
+
)
-
+A,
(42)
where /~
__
~ d _.i j ~.d _--dj q_ ~adJ Y~ t.nb.nb ~ t~p'i p (a~ + a ~ - ~ J ) + a d
(43)
nb
will tend to zero when the solution converges. Therefore, by introducing an artificial diffusion term, the spatial accuracy of the solution is less than second-order, but an effective local underrelaxation factor
ap + ap0 _ ~ j OiPJ -- (a~ + a ~ - ~pJ) + ~ D'nb
(44)
is introduced for the stress calculations so that possible oscillation during iteration is alleviated. Structured and Unstructured M e s h U
W
w_ , ' ..O" / _ G e - O- - ,-s- =0- - - I ""'P [
@
E
D
Figure 1" The control volume for node P. In the S I M P L E R method and most of its invariants, a staggered mesh is invariably used, where the pressure nodes (and the nodes for the extra stress components), are positioned along x, y, and z directions first, then the control volume faces, at the centroids of which the velocity components are calculated, are placed midway between neighbouring grid points. Thus the x-direction velocity u is calculated at the faces that are normal to the x-direction. The same rule is applied to v and w. An example of a control volume for node P, with neighbouring nodes U, D, N, S, E and W is illustrated in Fig. 1.
345
Figure 2" An unstructured finite volume mesh for a flow past a cylinder in a channel. In the figure, the lower case letter refers to the interface between two neighbouring cells, e.g., e refers to the interface between the control volumes for P and for E. One ends up having four different types of control volumes, three for the velocity components, and one for the pressure. The staggered grid is designed to eliminate the checker-board pattern in the pressure field [4]. With structured mesh, a linear interpolation scheme is often used, resulting in an O(h) accuracy, where h is a typical linear dimension of the control volume. Although one ends up with four different types of control volumes (three for the three velocity components and one for the pressure), structured mesh has been very popular because of the ease in implementation; in addition, iterative solvers like the tri-diagonal Thomas algorithm (TDMA) can be readily applied along the coordinate lines defining the mesh. Structured mesh is not very suitable for complex geometry, however. To provide the flexibility in fitting complex computational domains, unstructured non-staggered triangle mesh have been developed by a number of authors, for example, Prakash and Patankar [37], Masson et al. [38], and Davidson [39]. An example for such an unstructured mesh for the two-dimensional flow past a cylinder in a channel is given in Fig. 2. To retain the basic structure of the iterative line-by-line tri-diagonal Thomas solver (TDMA), lines of nodes from the entry to the exit boundaries are introduced (Huang et al. [32]). If this is not possible, then the line is terminated in the domain. The nodal information (neighbouring ID's, etc.) is stored in a sweeping line array. Similar to a re-ordering of the nodes in the finite element method to minimise the bandwidth of the system matrix, this array stores all the information needed for the line-by-line tridiagonalmatrix algorithm, which only requires computer storage and computer time
346
of O(N), where N is the number of unknowns. The method has been used successfully on viscoelastic flows between two eccentric cylinders [32], and past a cylinder in a channel [36].
Solver In the S I M P L E R method and most of its variants, the kinematics are determined by solving the momentum equations, assuming that the pressure field in known. The pressure correction is then applied, by enforcing mass conservation [4]. For the viscoelastic flow computations, the source terms containing the extra stress OEik/OXk in the momentum equations are treated as pseudo-body forces with the known dynamics field obtained from the previous time (or iteration) level by solving the discretised constitutive equations. In each cycle of the algorithm, no system matrix needs to be solved, and the coupled discretised equations for the dependent variables are solved sequentially from an initial guess for all field variables (typically quiescent field for Newtonian computations). For the momentum equations, two T D M A sweeps are performed, and for pressure correction and stress equations, four T D M A sweeps. In the iterative procedure, the calculations of velocities and stresses are under-relaxed by a global factor of 0.85 ~ 0.5 depending on the elastic level, but no relaxation is need for the pressure calculation. The convergence criterion for terminating the calculation is that the integral residual of the discretised equations over all control volumes for any dependent variable is less than the input tolerance, of the order 10 -6 10 -s, and the relative changes in the values of flow field (typically the velocities) near the solid wall from one iteration to the next are of the order 10 -5 ~ 10 -7. For viscoelastic computations, the corresponding Newtonian result (A - 0) is used as the initial guessed field, and Wi is increased gradually by increasing A. The under-relaxation mechanism due to adding artificial diffusion is very effective in the stabilisation of the calculations at high elasticity. To speed up the convergence rate and save some C P U time, a three-dimensional block T D M A solver can be used, see [30]. The two-dimensional implementation was done with both structured and unstructured mesh. The three-dimensional version was implemented with structured mesh only, as three-dimensional automatic unstructured mesh generation is still an active area of research. Both versions have been well tested with simple flow problems where analytical solutions exist, including Couette and Poiseuille flows, and proved to be very robust. In the 2D Poiseuille flow, there seems to be no upper limiting Weissenberg
347
number, and an agreement to four significant figures with analytical results was demonstrated at We = 40 in [36]. The 3D version has been tested for the Poiseuille flow of the Oldroyd-B fluid in a straight pipe with square cross section. Numerical results show that no upper limit in the Weissenberg number is encountered, and the convergence is improved with mesh refinement. It is verified that adding the artificial diffusion has a similar function to increasing the under relaxation factor [34]. We now consider the flow past a cylinder in a channel, and the threedimensional entry flow. 4. F L O W P A S T A C Y L I N D E R
The flow past a cylinder in a channel has been well investigated, both experimentally and numerically, and therefore is a good candidate to test the finite volume method. Here, the flow past a cylinder, of radius R, placed on the centreline, or offset from the centreline by a distance e in a channel of width 2L, is considered. No-slip boundary conditions are applied at the walls. Along the centreline symmetry conditions are applied. The length of the downstream section is 9L, and that of the upstream section is 6L. All distances are normalised by cylinder radius R. Having fixed the model parameters and the problem geometry, the only parameter left to vary is the average velocity U of the fluid at the channel entry. The Deborah number (De) and Reynolds number (Re) are defined as
De = AU/R,
Re = pRU/Vo.
(45)
Dhahir and Walters [40] reported some experiments and finite element calculations (using Polyflow TM) on the effects of viscoelasticity and of the wall on the flow of non-Newtonian liquids past a cylinder confined by two plane walls, with particular attention to the drag force on the cylinder. Inertia effects were neglected in the calculations, and they found the streamlines nearly independent of the mean flow rate. In addition, they also found the streamlines to be essentially independent of the Weissenberg number, at low values of the latter. Flows past an asymmetrically confined cylinder have also been investigated and the effect of the eccentricity on the drag force has been obtained both numerically and experimentally. Baaijens et al. [41] investigated the same flow of a shear thinning solution PIB/C14 both numerically and experimentally. A number of constitutive equations were used in their FEM simulations, including the PTT, the Giesekus and the UCM models. The field variables were compared to experimental values, along the centre line
348
and over cross sections of the channel. In general, there was a good agreement between measured and computed field variables. Measured and computed birefringence was also presented in [42] and a qualitative agreement was found. The wall effects were considered by placing the cylinder off the centreline, and it was found that the elongational thickening of the viscoelastic fluid causes a significantly larger flow rate through the broader gap compared with inelastic fluids. Barakos and Mitsoulis [43] investigated viscoelastic flow past a cylinder symmetrically confined by two parallel plates using a K-BKZ integral constitutive equation, using FEM with a path tracking scheme. A good agreement between their results and those of [41] was found. Huang and Feng [44] also reported some numerical results, using PolyflowTM, for the flow of the Oldroyd-B fluid past a confined and non-confined cylinder, retaining inertial terms. In the finite volume simulation, the computational domain is divided into non-overlapping polygonal control volumes, as shown in Fig. 2, and the SIMPLER algorithm is adopted for the UCM model. First, the accuracy of the Newtonian calculations is assessed by comparing the drag coefficient defined by K=
4~0UR' where F~ is the axial force acting on the cylinder, with mesh refinement. The extensive mesh refinement studies are performed with different meshes by reducing both the polar mesh size hp and the maximum mesh size hm according to Table 1. By comparing the results within and across mesh groups, the refinement of the polar mesh size is found to play a more important role in producing a more accurate solution. In the table, the errors are calculated relative to the result of the finest mesh L4. Polar Maximum Mesh Nodes Drag Force Error Mesh Size Mesh Size M2 0.069813 0.4 1067 10.0634 0.03740 M4 0.0349065 0.2222 10.3178 3434 0.01306 M6 0.02617 0.1538 6504 10.3060 0.01419 M7 0.02416 0.1818 7711 10.2927 0.01546 N1 0.021 0.2566 2392 10.3260 0.01228 N2 0.0175 0.2566 2806 10.3313 0.01177 N3 0.00873 0.2566 4296 10.3513 9.86e-3 N4 0.00714 0.2566 10.4172 3.55e-3 4996 L1 0.021 0.127 6991 10.4306 2.27e-3 L3 0.00873 0.127 9695 10.4543 8.19e-7 L4 0.00714 0.127 0 10977 10.4543
349
Table 1" Drag force and relative error computed for a Newtonian fluid.
3000 2500
..... FEM . 0 6 s O uCVM 0.0s
!
2000 O,I
E o 1500
V
LI_•
1000
500
%
30 40 50 60 70 80 FLOWRATE (cma/s)
Figure 3: A comparison between the FVM (symbols) and the FEM results of Dhahir and Walters (solid and dotted lines). The best fit polynomial to the calculation of K is K(hp)
-
10.5245 - 6.4120hp,
showing an approximately linear convergence with decreasing mesh size. An extrapolation of this formula gives K - 10.5245 at zero mesh size, which is comparable to the FEM result of 10.5313 [45], and asymptotic result of 11.0199 [46]. For viscoelastic flows, mesh refinement is the only check of the consistency and accuracy of the numerical results. A monotonic decrease in drag force with the decreasing polar mesh size is found, with the maximum mesh size fixed at hm - 0.1818. For a given mesh, a decrease in K with increasing D e is found. The results indicate a good mesh convergence with 0 < D e _< 1. At higher values of D e , the convergence is slower. More iteration is needed for higher Deborah numbers. For Newtonian flow 350 iterations are needed for a convergent result, the number of iterations increases to 670 at D e - 0.914 and 800 at D e - 4.0225. Figure 3 shows a comparison of the FVM results (symbols) and the F E M results of Dhahir and Walters [40] (lines, taken from Fig. 19 of their paper).
350
6
....
,..-d:".'-.,
....
6
....
,..~:':..',
....
6, . . . .
' 4-
-
~:''''',
....
F
4[-
2
,"
'-
4-
-
2 +
'
o
:~.~ a
-2
. ~
Iz~.z-a o
o,.-~~. - 2 .~---
. ,
9
q
~a
-
b
4
,I,
. . . . . . . . .
-6_ -1 o
I""
12
-6
'"
"
~
"
_
1 2
-,JI
"
I.
I
. . . . . . . . . . .
6-2 lo
12
Figure 4: A comparison between the computed (FVM: solid lines, FEM: triangles and squares) and experiment (circle) results by H. Baaijens [42] along several cross sections for P T T fluid with A - 0.0431 s, p - 0.8 g/cm 3 and e - 0.39. Left: axial velocity, middle: normal stress difference and right: shear stress. The shear rate at entry is so chosen that D e - 2.31. We have chosen the same geometry, with L = 5/3R, and 2113 control volumes were used over the full flow domain. Using the UCM model, the force per unit length on the cylinder is computed at varying flow rates and two values of A (0.0 and 0.06 s). Inertia effects are neglected by assuming creeping flow. Figure 3 displays the variation of normalised force components F* - F = / % (cm2/s) with the flow rate. The Deborah number at the highest flow rate of 80 cm2/s is 0.64. It is seen that F= increases with the flow rate in both Newtonian (A - 0) and UCM (A - 0.06 s) cases. Increasing elasticity reduces the force on the cylinder. The direction of the force is along the flow direction, as it must. The drag force in both the Newtonian and UCM cases are in excellent agreement with the F E M results of Dhahir and Walters [40]. The flow of a 5% P I B / C 1 4 solution past a symmetrically confined cylinder between two plates of a distance 2L apart is simulated. Following [41], the P T T model is chosen. The density of the liquid is taken into account and fixed at 0.8 g / c m 3, the relaxation time is fixed at 0.04313 s and P T T
351
model parameter is fixed at e = 0.39 and ~ = 0 (these values were given in [41]). The computation is done on half of the flow domain and with mesh M4 (with a total nodes of 3434, closed to the total nodes of 3981 in [41]). Figure 4 shows a comparison between the measured and computed (finite volume) results at cross sections x = - 5 R , x = - 2 R , x = - 1 . 5 R , x = 1.5R and x = 2R at the highest measured Deborah number of 2.31, where the velocity has been normalised with the mean velocity U, and stresses with 7o - 3~oU/R. To provide a visual comparison with the measured data, the computed results in the half domain are either symmetrically or asymmetrically mapped to the other half of the flow domain. Due to experimental errors, the measured results are not symmetric about the centreline. Figure 4 shows a good agreement for the velocities at all cross sections. The measured and computed first normal stresses agree well at all cross sections except at x = - 2 R , especially near the walls, although a reasonable agreement is still found inside the flow domain. The computed and measured shear stresses also agree reasonably well in all cross sections except at x = - 5 R . At x = - 5 R and near the entry, the flow is still fully developed, the shear stress is linear with respect to y / R , and tends to zero at centre line. The computed and measured shear stress agree better in the left domain than in the right domain. The finite volume solutions in the previous subsection are also checked against F E M results [41], plotted in Fig. 4 as triangles for a l-mode and squares for a 4-mode P T T model (the F E M data were taken from Fig. 4.14 and Fig. 4.16 from [41]). In Fig. 4, F E M results are plotted at the cross sections Ix/RI - 1.5 for the first normal difference (middle figure) and shear stress (right figure) where significant differences are found between measured and computed results. The shear stress profiles are displayed on the right of Fig. 4. An overall agreement with F E M results is noted. In summary, the unstructured finite volume results are mesh convergent, comparable to F E M results, and agree well with available experimental data for the flow past a cylinder in a channel. 5. T H R E E - D I M E N S I O N A L
ENTRY
FLOW
In the study of the entry flow problem, two types of the cross-sections, circular and rectangular, are of particular interest. The corresponding flows are referred to as axi-symmetric contraction and planar contraction flows, respectively. A large number of experimental works and two-dimensional numerical simulations on the flow behavior of various types of viscoelastic fluids in both types of geometry have been performed. Most earlier work has been reviewed in the literature (see, for example [47-49]). Here, we
352
only briefly list some of the main observed phenomena. Z
Figure 5: Three-dimensional planar contraction geometry and graded meshes (one quarter of the domain).
| '0[
l)JJtll
3
(a) w =0.82
4
.,
1
2
3
(b) W = 1.29
4
/// JJli
3
(c) w.-- 2.75
Figure 6: Streamlines for the Boger fluids fitted using the UCM model with different elasticity under the same flow condition (Re - 2.1), (a) W~ - 0.82; (b) We - 1.92; (a) We - 2.75. The aspect ratio is one. In the axi-symmetric contraction flow for Newtonian fluids, it has been clearly shown experimentally and computationally [50-51] that the presence (or absence) of the upstream corner vortex is determined by fluid inertia only.
353
0.05
IIII II t i t
w.
--
3.40E-3
1/
0.00
/ / //
-~~;'[2
- - ~ -0.05
-
2.7
-
-
.----'m
2)I
-
2.9
1.0
(a) Wmi.
0.05
_
ItfJJ,{tf
-8.73E-3
li
0.00
-0.05 - ' - - ~ - -
"~-"~~'~
~"'-
,
i/
/
/ / /! / I
~".,'1
-2_~ 7 - - ~ - ~ / ! / /
--~.____._ ~
2.6
/I
lilltfltt/t
. - I ~ - - ~
i
i-.~ ~
.
ii
9
2.7
.,i
1.0
(b) 0.05
Wm.~n--- 2 . 1 4 E - 2 0.00
........
,
--..----'--'7"--
2.6
2.7
218
' -
2.9
3.0
(c) Figure 7: A close-up view of the meshes and flow fields in terms of streamlines near the re-entrant corner of Fig. 6. However, for viscoelastic flow, experimental work has demonstrated many unusual flow phenomena which distinguish it from the corresponding Newtonian counterpart, such as the vortex enhancement [52], the appearance of the lip vortex [53], and the flow transition from the steady-state to three-dimensional, time-periodic and aperiodic flow states as the flow rate (or the Weissenberg or the Deborah number) increases to some level [54]. The dependence of these flow phenomena on the fluid properties is highly non-linear, and it seems that fluid elasticity based on steady and dynamic
354
shear properties alone is not adequate to explain the different behavior for different viscoelastic fluids. It was argued that the extensional behavior of the fluid plays a more important role on the vortex activities for viscoelastic fluids: strong vortex enhancement is expected to occur for fluids with strain-rate thickening extensional viscosity; on the other hand, strainrate thinning extensional viscosity will reduce the strength of the vortex activity [55-56]. In contrast to the axi-symmetric contraction flow, where a strong vortex enhancement is expected for some Boger fluids, the vortex activity is weak in planar contraction flows [57-59]; only under some extreme flow conditions [53] does a lip vortex appear. While for some shear thinning viscoelastic fluids, such as polyacrylamide aqueous (PAA) solutions, vortex growth has been observed in both cases. More remarkably, at very high flow rates, a non-trivial 3D unstable and alternate flow pattern is observed
40
-
.
, -=
35 . - - - o ~ +
,
-
-
,
.
.
.
.
,
-
140 . . . .
-
r
= 0.4
We=0.83
We = 0.4
120 ~
we = o.83 ;
*~gzz~tb ~
We = 1.39 We = 1.88
We = 1.88 X
2s
. we
We = 1.39
30
"~
-
.
We=2.9
~
100
(~=0)
v
We=
2.9(~
-
We=2.9
(~=0)
r~
We=4.15 =O.1
~
20
,~,
80
Z--
60
-----o-----
We=
4.15
~
We=2.9(Exp.)
W e = 2 . 9 ( P, = 0.1
15 4O 20 o .............. 0
-' 2
......
-9
-.----~----,-+ ~---;
.
-6
-3
:
.
i
0
.
. . . . .
3
6 Z
1
2
.
.
-9
. . . .-6 ..
.
.
'
-3
0
"
. .._';;;;;;,_.~l:J
3
6 Z
(a) (b) Figure 8" Computed distributions of (a) axial velocity w; (b) the first normal stress difference N1 along the centreline. Some of the selected results of the three-dimensional simulation of the entry flow [30] are given here (see Fig. 5 for the flow geometry), to illustrate the performance of the finite volume method. First, the 3D flow patterns of the Boger fluid fitted using the UCM model are simulated under the experimental flow conditions used by Walters and Webster [57] and Walters and Rawlinson [58]. The streamlines calculated for the fluids with different elasticity are given in Fig. 6, at a Reynolds number of 2.1 and an aspect ratio of W / H - 1. Here, the Reynolds number R~ defined_ as R~ - pUh/rlo, and the Weissenberg number is defined as We - A U / h , where h and U are
355
the half height and the mean flow rate in the downstream channel, respectively. As we can seen from the figure, there is virtually no corner vortex, which is in good agreement with experimental observation. However, upon close-up inspection of the streamlines near the re-entrant corner, as shown in Fig. 7, there exists a tiny lip vortex which spreads over several meshes and increases in both size and strength with increasing fluid elasticity. AIthough the lip vortex feature for Boger fluids in planar abrupt contraction flows was observed only at very high elasticity (We at order of 100) in the flow geometry with higher contraction ratio [53], the trend of our simulation results is consistent with the experimental observation. There may be two plausible reasons for the discrepancies. First, the lip vortex may be very hard to discern in the flow visualisation due to its being small in size. Secondly, the UCM model may not have all the relevant physics in complex flows of Boger fluids. Recently, Quinzani et al. [61] measured the detailed flow fields of a wellcharacterised shear-thinning polymer solution (polyisobutylene dissolved in tetradecane (PIB/C14)) flowing in a 3.97:1 planar abrupt contraction with the Laser-Doppler velocimetry (LDV) and flow-induced birefringence (FIB). Their experimental results have been partly simulated by some 2D numerical simulations with the fluid modelled by an integral K-BKZ equation with a spectrum of four relaxation times [62], and by multi-mode PTT, Giesekus and UCM models [63]. All of the simulations showed a good general qualitative agreement with experimental data. But the agreement is still semi-quantitative, especially for the extensional stress near the entry section (up to 30 - 40% discrepancy depending on the level of elasticity). The PIB/C14 shear-thinning solution used in the experiment work has been well characterised by Quinzani [64] with several non-linear multi-mode models, and they found that the best quantitative fit to the transient extensional viscosity could be made using the P T T model with p - 0.8 g/cm 3, /~ 1, ~0 - 1.424 Pa.s, A - 0.06 s, ~ - 0 (or 0.1), ~ - 0.25 (or 0.05). A finite volume simulation has been performed with these parameters using a graded mesh. A comparison between the computed values for the centreline velocity (w), and the first normal stress difference (centreline birefringence) is shown in Fig. 8. Generally, the numerical results agree reasonably with experimental data; however, the peak values predicted for N1 are 20-40% higher than the observed values (depending on (). The discrepancies are believed to be mainly due to the inadequacy of the constitutive model. -
356
6. C O N C L U S I O N S
The results of the finite volume calculations presented here are rather encouraging for computations of complex flow with arbitrary geometry. The method is very robust, allowing converged solutions to be obtained at a high Weissenberg number. The results also compare favourably with those obtained with traditional finite element methods. In addition, the use of unstructured mesh allows us to handle more complex computational domain, much the same way as the finite element method. Coupled with "both-side" diffusion, which is basically an under-relaxation mechanism, the finite volume method appears to us a valid way of computing stable solutions at high Weissenberg number. ACKNOWLEDGEMENTS
This research is supported by then Australian Research Council (ARC). The calculations were performed using the facility of Sydney Distributed Computing Laboratory (SyDCom). REFERENCES
1. M.J. Crochet, A.R. Davies, and K. Walters, Numerical Solution of Non-Newtonian Flow, Elsevier, Amsterdam, 1984. 2. J.N. Reddy and D.K. Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, Florida, 1994. 3. N. Phan-Thien and S. Kim, Microstructure in Elastic Media: Principles and Computational Methods, Oxford University Press, New York, 1994. 4. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, New York, 1980. 5. M.A. Mendelson, P.-W. Yeh, R.A. Brown and R.C. Armstrong, J. Non-Newtonian Fluid Mech., 10 (1982) 31. 6. J.M. Marchal and M.J. Crochet, J. Non-Newtonian Fluid Mech., 26 (1987) 77. 7. R.C. King, M.R. Apelian, R.C. Armstrong and R.A. Brown, J. NonNewtonian Fluid Mech., 29 (1988) 147. 8. D. Rajagopalan, R.C. Armstrong and R.A. Brown, J. Non-Newtonian Fluid Mech., 36 (1990) 159. 9. J. Sun, N. Phan-Thien and R.I. Tanner, J. Non-Newtonian Fluid Mech., 65 (1996) 75-91.
357
10. R.R. Huilgol and N. Phan-Thien, Fluid Mechanics of Viscoelasticity: General Principles, Constitutive Modelling, Analytical and Numerical Techniques, Elsevier, Amsterdam, 1997. 11. N. Phan-Thien and R.I. Tanner, J. Non-Newtonian Fluid Mech., 2 (1977) 353. 12. R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids: Vol i: Fluid Mechanics, John Wiley and Sons, 2nd Edn, New York, 1987. 13. A.R. Davies, S.J. Lee and M.F. Webster, J. Non-Newtonian Fluid Mech., 16 (1984) 117. 14. E.B. Bagley, I.M. Cabot and D.C. West, J. Appl. Phys., 29 (1958) 109-110. 15. A.M. Kraynik and W.R. Schowalter, J. Rheol., 25 (1981) 95-114. 16. A.V. Ramamurthy, J. Rheol., 30 (1986) 337-357. 17. D.S. Kalida and M.M. Denn, J. Rheol., 31 (1987) 815-834. 18. F.J. Lim and W.R. Schowalter, J. Rheol., 33 (1989) 1359-1382. 19. J.R.A. Pearson and C.J.S. Petrie, Proc. 4th Int. Cong. Rheol., Part 3, (1965)265-282. 20. S.G. Hatzikiriakos and N. Kalogerakis, Rheol. Acta, 33 (1994) 38-47. 21. O.D. Kellogg, Foundations of Potential Theory, Dover, New York, 1953. 22. M. Renardy, ZAMM, 65 (1985) 449-451; J. Non-Newt. Fluid Mech., 36 (1990)419-425. 23. A.J. Chorin, J. Comput. Phys., 2 (1967) 12-26. 24. F.R. Phelan, Jr., M.F. Malone and H.H. Winter, J. Non-Newt. Fluid Mech., 32 (1989) 197-224. 25. H. Jin, N. Phan-Thien and R.I. Tanner, Comp. Mech., 13 (1994) 443457. 26. J.P. van Doormaal and G.D. Raithby, Num. Heat Transfer, 7 (1984) 147-163. 27. R.I. Issa, J. Comput. Phys., 62 (1985) 40-65. 28. J.Y. Yoo and Y. Na, J. Non-Newt. Fluid Mech., 39 (1991) 89-106. 29. Y. Na and J.Y. Yoo, Comp. Mech., 8 (1991) 43-55. 30. S.-C. Xue, N.Phan-Thien and R.I.Tanner, J. Non-Newt. Fluid Mech., in press (1997). 31. H.H. Hu and D.D. Joseph, J. Non-Newt. Fluid Mech., 37 (1990) 347377. 32. X.-F. Huang, N. Phan-Thien and R.I. Tanner, J. non-Newt. Fluid Mech., 64 (1996) 71-92.
358
33. B. Gervang and P.S. Larsen, J. Non-Newt. Fluid Mech., 39 (1991) 217-237. 34. S.-C. Xue, N.Phan-Thien and R.I.Tanner, J. Non-Newt. Mech., 59 (1995) 191-213. 35. G.P. Sasmal, J. Non-Newtonian Fluid Mech., 56 (1995) 15. 36. X.-F. Huang, N. Phan-Thien and R.I. Tanner, J. Non-Newt. Fluid Mech., in press (1997). 37. C. Prakash and S.V. P a t a ~ , Numer. Heat Trans., 8 (1985) 259-280. 38. C. Masson, H.J. Saabas and B.R.Baliga, Inter. J. Numer. Meth. Fluids, 18 (1994) 1-26. 39. L. Davidson, Inter. J. Numer. Meth. Fluids, 22 (1996) 265-281. 40. S.A. Dhahir and K. Walters, J. Rheol., 33 (1989) 781-804. 41. H.P.W. Baaijens, G.W.M. Peters, F.P.T. Baaijens and H.E.H. Meijer, J. Rheol., 39 (1995) 1243-1277. 42. H.P.W. Baaijens, Evaluation of Constitutive Equations for Polymer Melts and Solutions in Complex Flows, PhD Thesis, Eindhoven University of Technology, Netherlands (1994). 43. G. Barakos and E. Mitsoulis, J. Rheol., 39 (1995) 1279-1292. 44. P.Y. Huang and J. Feng, J. Non-Newt. Fluid Mech., 60 (1995) 179198. 45. A.W. Liu, D.E. Bornside, R.C. Armstrong and R.A. Brown, J. NonNewt. Fluid Mech., in press (1997). 46. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics. Kluwer, Boston (1964). 47. S.A. White, A.D. Gotsis and D.G. Baird, J. Non-Newtonian Fluid Mech., 24 (1987) 121. 48. D.V. Boger, Ann. Rev. Fluid Mech., 19 (1987) 157. 49. B. Tremblay, J. Non-Newtonian Fluid Mech., 43 (1992) 1. 50. M. Viriyayuthakorn and B. Caswell, J. Non-Newtonian Fluid Mech., 6 (1980) 245. 51. M.E. Kim, R.A. Brown and R.C. Armstrong, J. Non-Newtonian Fluid Mech., 13 (1983) 341. 52. P.J. Cable and D.V. Boger, AIChE J., 24 (1978) 869. 53. R.E. Evans and K. Walters, J. Non-Newtonian Fluid Mech., 20 (1986) 11. 54. G.H. McKinley, W.P. Raiford, R.A. Brown and R.C. Armstrong, J. Fluid Mech., 223 (1991) 411. 55. F.N. Cogswell, Polym. Eng. Sci., 12 (1972) 64.
359
56. J.L. White and A. Konodo, J. Non-Newtonian Fluid Mech., 3 (1977) 41. 57. K. Walters and M.F. Webster, Philos. Trans. R. Soc. London, A308 (1982) 199. 58. K. Walters and D.M. Rawlinson, Rheol. Acta, 21 (1982) 547. 59. D.M. Binding and K. Walters, J. Non-Newtonian Fluid Mech., 30 (1988) 233. 60. K. Chiba, T. Sakatani and K. Nakamura, J. Non-Newtonian Fluid Mech., 36 (1990) 193. 61. L.M. Quinzani, R.C. Armstrong and R.A. Brown, J. Non-Newtonian Fluid Mech., 223 (1994) 1. 62. E. Mitsoulis, J. Rheol., 37 (1993) 1029. 63. F.P.T. Baaijens, J. Non-Newtonian Fluid Mech., 48 (1993) 147. 64. L.M. Quinzani, R.C. Armstrong and R.A. Brown, J. Rheol., 39 (1995) 1201.
361
S E G R E G A T E D F O R M U L A T I O N S IN C O M P U T A T I O N A L ASPECTS OF C O M P L E X V I S C O E L A S T I C F L O W S
Jung Yul Yoo Department of Mechanical Engineering College of Engineering Seoul National University Seoul 151-742, Korea
1. INTRODUCTION The past quarter century has seen a significant advancement in numerical simulation of complex viscoelastic flows. In the early years, the breakdown of numerical algorithms even for modest values of elasticity parameter (such as Weissenberg number) which was observed regardless of the choice of constitutive equation, descretization method and iteration method had been a longstanding problem in the study of viscoelastic flows. Joseph and co-workers [13] suggested that the problem might be associated with, among others, hyperbolicity and change of type in the governing equations, by showing that coupling the constitutive equations of hyperbolic type with the momentum equations of elliptic type for incompressible flow led to a system of mixed character. In this regard, Song and Yoo [4] argued that a type dependent upwinding scheme like the one used in transonic flow calculations might be useful, and performed a FDM simulation of the flow of an upper convected Maxwell fluid through a planar 4:1 contraction using a type dependent upwinding scheme for the vorticity equation. This resulted in an enhancement of the Weissenberg number limit to a higher value than ever, but did not quite resolve the high Weissenberg number problem. More comprehensive discussions on hyperbolicity and change of type in steady viscoelastic flows can be found in the book of Joseph [5].
362
In the last decade, several coupled finite element methods have been developed for the solution of viscoelastic flow problems. They include the sub-element method of Marchal and Crochet [6], the explicitly elliptic momentum equation (EEME) method of King et al. [7], and the elastic-viscous split-stress (EVSS) method of Rajagopalan et al. [8]. These techniques enabled the researchers to develop f'mite element methods that were stable enough to obtain converged solutions at Weissenberg numbers as high as possible. It seems that the success of all these strategies resulted from applications of streamline upwinding technique to the constitutive equations of hyperbolic type. However, it is noted that these finite element methods were based on fully coupled algorithms, in which the velocity, pressure and stress were solved simultaneously. In addition, they were restricted to very simple geometries requiting modest number of elements even with supercomputers. For example, in one of the calculations reported in [6], the degrees of freedom for stresses were 14116 with a mesh containing 210 elements. In the meantime, another group of researchers have studied viscoelastic flows in terms of decoupled strategies, in which the velocity, pressure and stress are solved separately. In decoupled algorithms, the momentum equation is solved separately from the constitutive equation, which is significantly less expensive than their coupled counterparts. Decoupled technique is also a natural choice for integral viscoelastic models since it is not a simple task to devise coupled algorithms for them [9]. Furthermore, a properly constructed decoupled scheme should in general have a larger radius of convergence than a coupled scheme which often relies on Newton type iterations [10]. The numerical accuracy and computer efficiency of a decoupled algorithm depends on three aspects: (i) solution of the constitutive equation subject to a given velocity field; (ii) solution of the Navier-Stokes equation subject to a given viscoelastic stress field; (iii) the way how the two equations are decoupled. There has been a traditional difficulty associated with solving the NavierStokes equation for an incompressible flow, that is, the incompressibility constraint, or the coupling of the pressure and velocity. For an incompressible flow, the continuity equation in itself does not have an explicit reference to the pressure. Therefore, a pressure equation, which enables the segregation of the solution procedure for the pressure from that for the velocity, must be devised by some further manipulation of the governing equations. A very well-known way of devising such an equation is the SIMPLE (semi-implicit method for
363
pressure linked equations) algorithm [11] used in the finite volume methods. Another approach calls for a 'pressure correction', in which an initially guessed pressure is corrected successively at each iteration by adding the pressure correction obtained from the associated equation. This pressure correction is also common to artificial compressibility [12] and augmented Lagrangian [13, 14] techniques. In the framework of the FEM, this difficulty can be easily overcome by using integrated formulations in which the velocity and the pressure are obtained simultaneously. However, the global matrices become inevitably larger than those of the segregated formulations. Therefore, in segregated formulations, one can save more memory and CPU time since the pressure is obtained separately from the momentum equation. Some time ago, Boger [15] pointed out that, experimentally, flows in twodimensional geometries tend to become time-dependent and three-dimensional when viscoelasticity is important [16]. As was also discussed in the work of Evans and Waiters [17], concerning the lip-vortex mechanism of vortex enhancement in planar contraction flows, it is clear that any numerical simulation aimed at predicting the vortex growth in such flows should also be extended to three space dimensions and time dependent behaviour. These features of the flow have to be taken into account in the numerical simulation of such flows [18]. Recent researches pay more and more attention to computationally efficient algorithms, since the flows of recent interest are ultimately three-dimensional and time dependent [16]. Decoupled and segregated methods, which deemed insufficiently accurate when they first appeared, have been developed so that they now can produce credible results. Furthermore they remain very attractive because of significantly reduced core memory and CPU time requirement, which are crucial in transient and three-dimensional problems. The objectives of the present article are to examine the segregated formulations currently adopted in numerical simulations of viscoelastic flows and to discuss the possibilities of emerging algorithms. To avoid possible confusions in the usage of terminologies in this article, the term 'decoupled' is to be used for the decoupling of the extra stress into the viscoelastic component and the optional Newtonian component as they appear in the momentum equations (Keunings [9]), and the term 'segregatea~ is now used for the segregation of the pressure from the momentum equations. In section 2, governing equations for viscoelastic flows in conjunction with basic decoupled techniques for the extrastress are presented. In section 3, several strategies handling the hyperbolic
364
nature of the constitutive equations are discussed. The segregated formulations of the Navier-Stokes equation are treated in section 4. The possibilities of emerging segregated algorithms are considered briefly in the conclusion.
2. G O V E R N I N G EQUATIONS FOR V I S C O E L A S T I C F L O W S For an incompressible viscoelastic flow, the governing equations are the continuity, momentum and constitutive equations. Since the main concern of the present article is to take a general view of segregated algorithms in numerical simulation of viscoelastic flows, we do not intend to elaborate on the viscoelastic modelling. In this regard, we will simply adopt the 'upper-convected Maxwell model', which is one of the most widely studied differential models for viscoelastic flows. Then we are to consider the following set of equations:
V.u=O,
(1)
~gu
(2) v
r+A,~=2r/d,
(3)
where u is the velocity, p is the pressure, d is the rate of strain, lr is the extra stress, p is the density,/7 and A, are the viscosity and relaxation time, respectively, and v denotes the upper-convected derivative def'med by v
9
lr = - ~
+ u . V l r - V u r . l r - lr. V u
.
(4)
When equations (3) and (4) are taken in conjunction with the momentum equation (2) and the continuity equation (1), it is apparent that the extra-stress components have to be calculated together with the velocity components and the pressure. The concept of the decoupled method is based on splitting the extra-stress into the Newtonian part and the viscoelastic part as
r=2rld+
(5)
365
By substituting equation (5) into equations (2) and (3), the momentum and constitutive equations are now written as 81/
(6)
P-~
v
v
I'E + A,l-e = - 2 ~ r/d .
(7)
The above alternative formulation was first introduced by Perera and Waiters [19-20], Mendelson et al. [21] and Crochet and Keunings [22]. This was further modified by Rajagopalan et al. [8] to construct the elastic-viscous split stress (EVSS) formulation, which is categorized as a coupled method. In this coupled formulation of the EVSS, the convected derivative of d in the Galerkin discrete form of equation (7) involves second-order spatial derivatives of the interpolating function ~, which can be eliminated by performing v an integration by parts on the term O d, with the resulting boundary integral containing the flux of d through the boundary. In a decoupled formulation, however, this complication may be avoided by first calculating r/d at node points from the known velocity field and then consistently interpolating r/d (as part of the extra stress) in the same manner as 1:~ [10]. Now, the momentum equation turns out to be the Navier-Stokes equation with a pseudo-body-force V. ~:~,so that the solution always reduces to that of a Newtonian flow when A, vanishes. The velocity and the pressure can be obtained by solving equation (6) with the continuity equation in the given stress field, and the stress can be obtained by solving equation (7) in the given velocity field, iteratively. The main disadvantage of decoupled techniques lies in the iterative procedure because Picard schemes usually used in decoupled methods are often slow in convergence, which is not even guaranteed no matter how closely the initial estimates are chosen to the solution. However the decoupled technique generally requires less core memory than a coupled method, which turns out to be more efficient for larger computations. Another attractive feature of decoupled methods is the breakup of the problem into the solution of an elliptic Newtonian-like flow with pseudo-bodyforce, and the integration of constitutive equations using fixed flow kinematics.
366
Thus classical methods can be used to discretize the Navier-Stokes equation, while special techniques for the extra-stress computation may be developed to take into account the special mathematical nature of the constitutive equation involved [9].
3. NUMERICAL TREATMENT OF CONSTITUTIVE EQUATIONS IN DECOUPLED METHODS
3.1 Background The main difficulty in dealing with the constitutive equation (7) is that its type is hyperbolic. Even though this advection equation is linear in the stress with its structure being much simpler than the Navier-Stokes equation, it is still a challenge to computation because the advection equation has no built-in mechanism to smooth discontinuities in directions normal to the flow [23]. This problem is similar to that encountered in high Reynolds number flows of Newtonian fluids where high velocity along streamlines generate wild oscillations in the numerical solution. Before the advancement of the upwinding technique, the only way to eliminate these oscillations seemed to be to refine the mesh, such that convection was made not to be dominant on an element level. However, Keunings [24] showed in a study on the high Weissenberg number problem that the use of a very f'me mesh rather reduced significantly the range of the Weissenberg number for which the viscoelastic solutions could be obtained. It is because the local Peclet number, which represents the ratio of the convection to the diffi~ion, remains to be infinite from equation (7), no matter how we ref'me the mesh. Furthermore, it is reminded that one must calculate the stress closer to the comer singularity with a f'mer mesh in the case of the problem involving corner singularity, such as the 4:1 contraction flow problem.
3.2 SUPG (Streamline Upwind/Petrov-Galerkin) Method There have been several methods to overcome this difficulty, regardless of the coupled or decoupled method. In a finite element framework, the most popular approach for this problem is the Streamline Upwind Petrov-Galerkin (SUPG) method [25]. Some earlier upwind formulations for multi-dimensional problems often exhibited numerically false diffusion in the direction perpen-
367
dicular to the flow direction. The basic idea of the streamline upwind method is to add diffusion which acts only in the flow direction. Although the SUPG method has been highly successful in solving the Navier-Stokes equations for Newtonian flows, it was never introduced into solving the constitutive equations of the viscoelastic flows, until the first attempt was made by Marchal and Crochet [6] who adapted the SUPG method for their coupled method. Following Brooks and Hughes [25] and Marchal and Crochet [6], we define modified weighting function q~ as .
9 =T+ku~-v~P,
_
k =
luAl+
,eh
(8)
2
where q~ is the Galerkin weighting function, u~ and Uy are the components of the velocity vector u h at the center of an element, h~ and hy are the characteristic lengths of the element. The normalization of the velocity field is introduced so that the weighting function remains to be O(1) in regions even where the velocity field vanishes. Note that in the coupled formulation of Marchal and Crochet [6], the form of k in equation (8) is not used because it is not differentiable at u~ = U y - 0 , and one cannot calculate the Jacobian matrix with Newton type iterations. Instead, they adopted the following form:
%=
U hx)
(9)
2
Applying the modified weighting function to all terms in the constitutive equation, we end up with the consistent streamline upwind method:
q~+]T~-~-V~P ~ + , ~
2~,
=
.
An alternative method for the solution of the constitutive equation is the inconsistent streamline upwind method, in which the modified weighting function is applied solely to the advection term u. V~g of the constitutive equation:
368
~n 9 ~:e+~:E+2,q,r/d +
.V~'E d O = 0 .
(11)
Comparing (11) with the conventional Galerkin weighting form, it can be considered as if an extra term is artificially added to the Galerkin formulation. This method is equivalent to the artificial diffusion method commonly used for hyperbolic equations in the FDM or FVM. Note that the artificial diffusion method is second-order accurate in space, while conventional upwinding method is first-order accurate.
3.3 DG (Discontinuous Galerkin) Method It is well known that a cure for the numerical oscillations by introducing a certain amount of artificial diffusion into the usual upwinding technique only results in a loss of accuracy in the numerical solution. Discontinuous Galerkin (DG) method of Fortin and Fortin [26] was developed for a decoupled finite element approach to avoid smooth and stable but physically unrealistic solutions. This technique first introduced by Lesaint and Raviart [27] for the neutron transport equation was successfully applied to the numerical simulation of viscoelastic flows at high Weissenberg numbers. However, at first it was considered to be impossible to solve the steady flow problem by using the Picard iteration scheme. Therefore a time stepping scheme had to be employed, possibly because steady-state Picard's scheme does not provide information on the qualitative behaviour of the numerical solutions as Keunings [9] pointed out. Later, Fortin and Fortin [28] showed that the steady flow problem could be solved by applying the Generalized Minimal Residual (GMRES) algorithm of Saad and Schultz [29]. The value of the Weissenberg number thus obtainable was limited again and the extra-stress field showed an oscillatory behaviour. It was subsequently shown by Basombrio et al. [30] that the use of a linear interpolation for the extra stress tensor, rather than a quadratic as in Fortin and Fortin [26], significantly reduce the oscillations. However, it is quite unfortunate that for the DG method, the stress variable cannot be eliminated on the element level as, for instance, in the work of Baaijens [31] and that oscillation free solutions are only obtained at limited values of the Weissenberg nmnber. Baaijens [31] and Baaijens [32] combined the DG method with the Operator Splitting (OS) method [23], which was originally designed to decouple the
369
advection treatment and the rest of the procedure for a time-dependent problem. A shared feature of the OS and DG method is the use of discontinuous interpolations of the extra stress tensor. This allows the elimination of the extra stress variables on the element level, yielding an efficient algorithm for multi-mode fluids. More recently, Fortin et al. [33] and Gurnette and Fortin [34] have combined the DG method with the EVSS method, by which they could obtain good convergence and oscillation-free solutions for higher Weissenberg numbers. 3.4 Method of Characteristics Another method for handling the hyperbolic type equation in the decoupled finite element approach is the method of characteristics, which was first introduced in the numerical simulation of viscoelastic flows by Fortin and Esselaoui [35]. Hadj and Tanguy [36] also employed the weak formulation of the method of characteristics to compute the full convected derivative of the stress tensor and obtained solutions at a higher Weissenberg number with an Oldroyd-B fluid than that attained by Fortin and Esselaoui with a Maxwell fluid. In a recent work of Basombrio [37], the hyperbolic equations for the advective transport of stresses were integrated directly at each node using nonconservative strong formulation of the method of characteristics [38]. This method was also combined with the EVSS method by Kabanemi et al. [39] in which the hyperbolic nature of the constitutive equation was coped with using a weak formulation. 3.5 Other Methods in FDM and FVM In the finite difference or finite volume framework, the applicable methods for the hyperbolic type equations are the upwinding techniques and the artificial diffusion techniques. Hu and Joseph [40] used a conventional, first order upwind scheme for the constitutive equations. Choi et al. [41 ] and Yoo and Na [42] used the deferred correction method, which is an upwind corrected scheme involving an artificial diffusion term to attain second-order accuracy and unconditional stability, while Xue et al. [43], and Huang et al. [44] used an artificial diffusion method only. Sato and Richardson [45] used a finite volume approach for the constitutive equation, while a finite element approach was used for the momentum equation. They introduced the FCT (flux-corrected transport) concept into solving the constitutive equations, since numerical
370
schemes for advection should be TVD (total variation diminishing) in order to obtain stable results. Recently, Mompean and Deville [18] applied, so-called, QUICK (Quadratic Upstream Interpolation Scheme for the Convective Kinematics) to the constitutive equation, which was proposed by Leonard [46]. For a uniform grid system this scheme is of third-order accuracy, and for a nonuniform grid system it is of second-order accuracy [46]. The staggered grid system has been widely used in the FDM or FVM approaches, where the pressure and the stresses are located in the center of the control volume and the velocities are located on the faces of the control volume as shown in Figure 1. The staggered grid is first designed to eliminate the checkerboard pattern in the pressure field for the Newtonian flows of the FDM or FVM, which is quite similar to the mixed f'mite element methods satisfying the Babuska-Brezzi compatibility condition [47, 48] on the spaces for the velocity and pressure. This formulation of the viscoelastic flow calculation offers a major advantage in avoiding the stress calculations, for example, at the re-entrant corner of the 4:1 contraction flow problem, while the method correctly preserves the flow physics in its neighboring control volumes.
I
u
I L .
p,l: .
.
.
I _1
Figure 1 Staggered grid in the computational domain In recent works of Saramito [49, 50], the original finite volume element (Figure 2) was proposed for the mixed FEM which was somewhat similar to the staggered grid in the FVM. It is noticeable that the staggered formulation is relatively difficult to discretize the governing equations and it nearly cannot be used for an unstructured grid mostly because one cannot define a control volume consistently for arbitrarily shaped meshes. Therefore the non-staggered
371
grid system in which all dependent variables are located at the grid points becomes more and more popular in the Newtonian flow calculations. In the unstructured FVM approach of Huang et al. [44], they used a non-staggered formulation, which corresponds to equal-order element in the FEM. 19
0
E
)
9
()
l
0
E
[-] ~2
Ou 0
/
"r "t'22,p
Figure 2 Modified finite element
4. SEGREGATED FORMULATIONS IN THE NAVIER-STOKES EQUATION 4.1 Pressure Equation One of the complications encountered in solving the Navier-Stokes equation is that it includes the pressure gradient term, whereas there is no independent equation for the pressure. Furthermore, the continuity equation does not have a dominant variable in it. Mass conservation is a kinematic constraint on the velocity field rather than a dynamic one. In the FEM approach, this difficulty is simply avoided by using an integrated formulation in which the momentum and continuity equations are solved simultaneously. However, this formulation requires a large memory and computing time. An alternative way to overcome this difficulty is to construct a pressure equation in order that the pressure obtained from it intrinsically satisfies the continuity equation. In this segregated velocity-pressure formulation of the Navier-Stokes equation, velocities and the corresponding pressure field are computed alternately in an iterative sequence, in contrast to the integrated formulation. The most attractive aspect of the decoupled approach in the numerical analysis of viscoelastic flows is that it works very well with the segregated strategy for the Navier-Stokes equation which requires a much smaller memory and computing time.
372
This segregated formulation has been one of the most commonly used approaches in the finite difference procedure [51], since it does not solve the momentum and continuity equations simultaneously as in the integrated formulation of the FEM. For the purpose of solving them simultaneously, Song and Yoo [4], Choi et al. [41] and Sasmal [52] applied the stream function-vorticity formulation to viscoelastic flow problems. However this approach cannot be extended to three dimensional flows, and it needs assumptions on stream function and vorticity in treating the boundary conditions. Derivation of a pressure equation is based on combining the momentum and continuity equations. A pressure equation of Poisson type can be obtained by taking the divergence of the momentum equation (2): = v.
+v.
.
(]2)
Sato and Richardson [45] used this formulation for their viscoelastic flow calculation, in which MAC (Marker And Cell) method of Harlow and Welch [53] was incorporated into a finite element solution. A similar procedure to this formulation is also used in the three-dimensional FVM calculation of Mompean and Deville [ 18].
4.2 SIMPLE (Semi-Implicit Method for Pressure Linked Equations)-like Algorithm The well-known approach in the FVM for the pressure equation is the SIMPLE algorithm of Carretto et al. [54], which can be simply described as follows: (i) Predictor stage In this stage a guessed pressure field is used, usually denoted by p*, and by solving the momentum equation the velocity field u* is obtained. (ii) Corrector stage To obtain the velocity and pressure fields which satisfy both the continuity and momentum equations, the corrections u' and p' are obtained in this stage, and then u = + u p = p + p' This stage is repeated by treating the corrected pressure p and velocity u as a newly guessed pressure p* and velocity u* until a converged solution is obtained.
373
However, the SIMPLE algorithm does not converge rapidly because the velocity correction is obtained improperly. Its performance depends greatly on the size of the time step, o r - for steady flows - on the value of the under-relaxation parameter used in the momentum equations. It was found by trial and error that convergence characteristics can be improved by controlling the under-relaxation. Thus some modifications to this conventional SIMPLE algorithm were proposed. They are SIMPLER (SIMPLE Revised) of Patankar [11], PISO (Pressure-Implicit with Splitting Operators) of Issa [55] and SIMPLEC (SIMPLE Consistent) [56] which do not need under-relaxation of the pressure correction [57]. Recently, the FVM approaches based on these algorithms were adopted for viscoelastic flow computations, because of their attractive features of space and time savings, as well as their numerical stabilities. For steady flow problems, Hu and Joseph [40], Yoo and Na [42] and Huang et al. [44] used SIMPLER algorithm, and its modification is used in the work and Xue et al. [43], while Gervang and Larsen [58] used PISO algorithm and Luo [59] used classical SIMPLE algorithm. Note that in these algorithms the pressure and velocity fields obtained at the end of each time step (or iteration step for steady flow problems) do not satisfy one and the same momentum equation so that for time-dependent computations, sub-iterations are necessary at each time step, or very small time steps must be chosen, which accordingly degrades the efficiency of these algorithms.
4.3 Augmented Lagrangian Method In the frame work of the FEM, there is an alternative approach for the segregation of the pressure, that is augmented Lagrangian method [13, 14] which was derived from the variational formulation of the Stokes flow problem and turned out to be a powerful iterative method for solving it. Using the variational formulation of the Stokes equation, one can obtain the symmetric positive-definite functional equation, which is equivalent to the minimization problem. For the solution to this minimization problem, there have been many variants from the following very simple algorithm known as Uzawa algorithm [13, 14]-
P St - V ' ~ = - V p ' ,
(13)
374
p"+~ =p" - e V . u "
,
(14)
where m denotes the present iteration step and e is a step length for the iterative procedure. Hadj and Tanguy [36] and Kabanemi et al. [39] used this classical Uzawa algorithm for the momentum and continuity equations. Fortin and Fortin [26] and Gu6nette and Fortin [34] solved the Stokes flow problem by this algorithm combined with a condensation method [60] that eliminates the velocity degrees of freedom associated with the center of each element and the pressure gradients. This is achieved by using the continuity equation at the element level, thanks to the discontinuous approximations of the pressure. This method greatly reduces the number of unknowns, resulting in a very efficient Stokes solver. However, the convergence properties of this algorithm are not so good because the determination of e is not easy so that e is usually fixed with the iterative procedure. The way around this is quite simple but requires a few developments [13, 14, 23]. Fortin and Fortin [28] showed that an introduction of the preconditioned residual for the dependent variables enhance the convergence, but the computations still had to be performed in double precision. In fact, the algorithm (13)-(14) can be interpreted as a gradient algorithm applied to the minimization of the functional. With this interpretation in mind it is natural to seek the more effective iterative methods for the minimization of quadratic functionals, such as the steepest decent method, the minimum residual method or the conjugate gradient method. In these modified methods, the conjugate gradient method is especially attractive for solving quadratic problems because theoretically it converges in a finite number of iterations and moreover, it leads to quadratic convergence in the general case. These algorithms require the additional presence in memory, but this increased memory requirement will be justified if the automatic determination of the step length e leads to a very clear improvement in the speed of convergence compared with original algorithm (13)-(14) [ 13, 14, 23 ].
4 . 4 0 S (Operating Splitting) Scheme Clearly, the augmented Lagrangian method is not applicable in itself to the Navier-Stokes equation containing the non-linear advection term for the velocity. As mentioned in section 3, the Operator Splitting (OS) scheme has been used in order to decouple an advection treatment and the rest of the procedure
375
for a time-dependent problem. This method is also called the alternating direction implicit technique [13, 14, 23, 61], which separates the main two difficulties in the computation of the Navier-Stokes equation: the non-linearities in the momentum equation and the incompressibility constraint. In the conventional numerical simulation of viscoelastic flows, the non-linear term is usually neglected. In an attempt to answer some of the unresolved questions in their earlier work, Evans and Walters [17] investigated aqueous solutions of polyacrylamide and concluded that a lip-vortex mechanism may be responsible for vortex enhancement for all planar contraction ratios. These experimental studies consist of observations and measurements of the phenomena resulting from the change of the flow rate of the same viscoelastic fluid. Therefore, in order to simulate such flows numerically, the inertia must be retained in the momentum equations. Glowinski and Pironneau [23] proposed the 0-scheme based on the abovementioned OS method to treat the nonlinearity and the incompressibility of the Newtonian flows. It consists of three step time marching technique such that (i) First step gl n+O _ II n
P
Ok
_aV..g,+o +Vpn+O =fn+O ..[_~V. ,[,n (/gn .V)/gn ,
V.u "+~= 0 ,
(15) (16)
(ii) Second step un+l-O _ 11n+O P
(1 - 2 0 ) k
- f l Y . Zn+l-O "b (U n+l-O" V ) U n+l-O : fn+l-O + O~V..[n+O _ V p " + ~ ,
(17) (iii) Third step
P
U n+l
ign+l-O Ok _ o ~ V . ,[n+l q_Vpn+l = f n + l + ~ V . , [ n + l - O
_
V-/g n+l = 0 ,
(lln+l-O.v)lln+l-O ,
(18) (19)
376
where a = (1- 20)/(1-0)and/3 = 0/(1-0), and 0 can be selected by a numerical experiment. One can see that the first and third steps are the Stokes-like problems in which the nonlinear terms are eliminated by the second step. Thus the Uzawa type algorithm can be used in the first and third steps and the nonlinear term is treated in the second step by using the method mentioned in section 3, such as the upwinding technique. This scheme is of order two in time, and allows one to also compute steady solutions efficiently. Glowinski and Pironneau [23] introduced a preconditioning stage to the Uzawa type conjugated gradient method for an acceleration of the convergence in the Newtonian flow computations. In the case of the Stokes flow computation, the second step can be skipped since it needs not the advection treatment. However, when the 0scheme is combined with the constitutive equations of viscoelastic flows, the second step can not be skipped because the advection term in the constitutive equation is always involved. Recently, Saramito [49, 50] applied this 0-scheme to the calculation of the viscoelastic flows, but he neglected the inertia in the momentum equations, while Luo [62] extended it to the case of including the inertia in the momentum equations. Until now, no one used the 0-scheme for time-dependent flow problems. As Szady et al. [63] mentioned in their recent work of coupled approach, the 0-scheme can be efficiently utilized for the three-dimensional transient flow where it can save much CPU time and guarantee numerical stability.
4.5 Fractional Step Method The alternating direction method is closely related to the fractional step method [14]. Actually the alternating direction or fractional step method has been extensively used for quite a long time for solving time-dependent partial differential equations. Concentrating particularly on the Navier-Stokes equation for incompressible viscous fluids, the first works were performed by Chorin [64] and Temam [65]. However, the fractional step method for the Navier-Stokes equation had not been used extensively until successful result of its application to the three-dimensional unsteady turbulent flow calculation was reported by Kim and Moin [66], which was later used for the Direct Numerical Simulation (DNS) [67, 68]. A typical formulation of the fractional step method is as follows"
377 (i) First step
P At
+
3(un'V)un-(un-l"V)un-1)=-2
(ii) Second step V 2~bn+l = P V.u At
(21)
(iii) Third step
P
Un+l --fl _v~n+l At = '
(22)
where r is a pseudo-pressure and V~b#+~- Vp #+~+O(At2). One can see that the third step is derived from taking the divergence of equation (21) and requiting u n+~ to satisfy the continuity equation. There are many variants of the fractional step method such as a fully implicit fractional step method [69], due to a vast choice of approaches to time and space discretization, but they are all based on the principles described above. The fact that the pressure is segregated from the velocity in the fractional step method was utilized in the finite element analysis of the incompressible Navier-Stokes equation by several researchers [70-74]. Because the fractional step method does not include any approximation procedure, such as adopted in the SIMPLE algorithm based finite element methods [75-78], this approach is more accurate than the SIMPLE algorithm based approach for the same grid. Furthermore, in the fractional step method, the coefficients of the pressure equation are fixed and the inverse matrix of the pressure equation does not need to be calculated at every time step like the SIMPLE algorithm based approach [79]. Carew et al. [80] and Baloch et al. [81] used this fractional step method combined with the Taylor-Galerkin method for the advection treatment. The details of that scheme is described in the work of Hawken et al. [82]. Recently, Baloch et al. [83] were able to carry out a numerical simulation of a three-dimensional viscoelastic 40:3:3 expansion flow, using the Phan-Thien and Tanner
378
(PTT) constitutive fluid model, where they showed the streamlines in the center plane of the 3D expansion and made comments on the differences in the vortex development from the 2D simulation. Recently, Sureshkumar et al. [84] applied the fractional step method of Kim and Moin [66] to the DNS of the turbulent channel flow of a polymer solution by spectral method. The major difference between the 0-scheme with preconditioned conjugate gradient (PCG) method (15)-(19) and the fractional step method (20)-(22) is how to advance one time step. The two methods are both of order two in time and the Poisson equations for the pressure are involved respectively. However, sub-iterations for satisfying the convergence criteria of the momentum and continuity equations, that is, equations (15)-(16) and equations (18)-(19), are needed in the PCG method of the 0-scheme, while the Poisson equation is solved just one time at the second step (21) in the fractional step method. It is well-known that the Poisson equation for the pressure is the most time consuming one in the segregated formulation of the Navier-Stokes equations. In view of this, comparing with the fractional step method, the 0-scheme may be inefficient, particularly for three-dimensional problems, despite the fact that the Poisson equation is solved in the discrete pressure space whose dimension is much smaller than the dimension of the discrete velocity space in mixed FEM formulation. Note that the 0-scheme with Uzawa type PCG method cannot avoid the Babuska-Brezzi condition [13], and thus it must be discretized via the mixed formulation where the discrete pressure on a grid is defined twice coarser than the one used to discretize the velocity. On the other hand, the fractional step method can avoid the compatibility condition as shown in the recent work of Choi et al. [79] which combined the SUPG with an equal-order finite element formulation. This work was extended to the numerical analysis of a Bingham plastic [85]. However, the boundary condition treatments for the intermediate velocity ti and the pressure in the fractional step method is somewhat unclear and some further assumption is needed, while such an assumption is not needed in the 0-scheme.
5. CONCLUDING REMARKS Recently, the streamline upwind (SU) scheme for the control volume finite element method (SUCV) [86] has been proposed. When we consider that the
379
FEM in conjunction with the SU technique gives satisfactory results in analyzing viscoelastic flows, adopting SUCV into the existing algorithm based on the FVM seems to be promising. Furthermore, in the last decade the segregated finite element formulations [70-79, 82] have been steadily developed, which combine the merits of both the FEM and segregated FVM using SIMPLE-like or split methods. Therefore, future research efforts implementing these segregated FEM in the numerical simulation of viscoelastic flows are also highly recommended. It can be said that the earlier difficulties in the high Weissenberg number problem have been now overcome by several coupled formulations such as a sub-division method [6], EEME [7] and EVSS [8] methods, which can obtain the converged solutions at Weissenberg numbers as high as possible. In the meantime, another group of researchers using a decoupled and segregated formulation showed that they can also handle the high Weissenberg number problems. In particular, some of those segregated methods utilized special techniques used in coupled methods for hyperbolic type constitutive equations. The new trend of recent researches of a viscoelastic flow computation is the analysis on unsteady and three-dimensional flow including the inertia [16] , since those flows are related to the most interesting phenomena of nonNewtonian fluids such as the lip-vortex growth or the turbulent drag reduction by polymer additives. Therefore fully decoupled, segregated formulations of viscoelastic flows in the FDM, FVM and FEM seem to be more and more attractive nowadays due to their capability of handling those problems as well as their cost-effective properties.
ACKNOWLEDGEMENT I would like to express my deepest thanks to Mr. Taegee Min for his invaluable efforts in preparing this manuscript.
REFERENCES 1. D. D. Joseph, M. Renardy and J. C. Saut, Arch. Ration. Mech. Anal., 87 (3) (1985), 213. 2. J. Y. Yoo, M. Ahrens and D. D. Joseph, J. Fluid Mech., 153 (1985), 203.
380
3. J.Y. Yoo and D. D. Joseph, J. Non-Newtonian Fluid Meeh, 19 (1985), 15. 4. J.H. Song and J. Y. Yoo, J. Non-Newtonian Fluid Mech., 24 (1987), 221. 5. D. D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York, 1990. 6. J. M. Marehal and M. J. Crochet, J. Non-Newtonian Fluid Mech., 26 (1987), 77. 7. R. K. King, M. R. Apelian, R. C. Armstrong and R. A. Brown, J. NonNewtonian Fluid Mech., 29 (1988), 147. 8. D. Rajagopalan, R. C. Armstrong and R. A. Brown, J. Non-Newtonian Fluid Mech., 36 (1990), 159. 9. R. Keunings, Simulation of viscoelastic fluid flow, in: C. L. Tucker III (Ed.), Computer Modeling for Polymer Processing, Hanser Publishers, Munich, 1989. 10. X.-L. Luo and R. I. Tanner, J. Non-Newtonian Fluid Mech., 31 (1989), 143. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, 11. New York, 1980. 12. A. J. Chorin, J. Comput. Phys., 2 (1967), 12. 13. M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, North-Holland, Amsterdam, 1983. 14. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984. 15. D.V. Boger, Ann. Rev. Fluid Mech., 19 (1987), 157. 16. B. Caswell, J. Non-Newtonian Fluid Mech., 62 (1996), 99. 17. R. E. Evans and K. Waiters, J. Non-Newtonian Fluid Mech., 32 (1989), 95. 18. G. Mompean and M. Deville, J. Non-Newtonian Fluid. Mech., in press (1997), private communication. 19. M. G. N. Perera and K. Waiters, J. Non-Newtonian Fluid Mech., 2 (1977), 49. 20. M. G. N. Perera and K. Waiters, J. Non-Newtonian Fluid Mech., 2 (1977), 191. 21. M. A. Mendelson, P.-W. Yeh, R. A. Brown and R. C. Armstrong, J. NonNewtonian Fluid Mech., 10 (1982), 31.
381
22. M. J. Crochet and R. Keunings, J. Non-Newtonian Fluid Mech., 10 (1982), 339. 23. R. Glowinski and O. Pironneau, Ann. Rev. Fluid Mech., 24 (1992), 167. 24. R. Keunings, J. Non-Newtonian Fluid Mech., 20 (1986), 209. 25. A. N. Brooks and T. J. R. Hughes, Comput. Meth. Appl. Mech. Eng., 32 (1982), 199. 26. M. Fortin and A. Fortin, J. Non-Newtonian Fluid Mech., 32 (1989), 295. 27. P. Lesaint and P. A. Raviart, On a finite element method for solving the neutron transport equations, in: C. de Boor (Ed.), Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, New York, 1974. 28. A. Fortin and M. Fortin, J. Non-Newtonian Fluid Mech., 36 (1990), 277. 29. Y. Saad and M. H. Schultz, SIAM J. Sci. Stat. Comput., 7 (3) (1986), 856. 30. F. G. Basombrio. G. C. Buscaglia and E. A. Dad, J. Non-Newtonian Fluid Mech., 39 (1991), 189. 31. F. P. T. Baaijens, J. Non-Newtonian Fluid Mech., 48 (1993), 147. 32. F. P. T. Baaijens, J. Non-Newtonian Fluid Mech., 51 (1994), 141. 33. A. Fortin, A. Zine and J.-F. Agassant, J. Non-Newtonian Fluid Mech., 42 (1992), 1. 34. R. Gu6nette and M. Fortin, J. Non-Newtonian Fluid Mech., 60 (1995), 27. 35. M. Fortin and D. Esselaoui, Int. J. Numer. Methods Fluids, 7 (10) (1987), 1035. 36. M. E1 Hadj and P. A. Tanguy, J. Non-Newtonian Fluid Mech., 36 (1990), 333. 37. F. G. Basombrio, J. Non-Newtonian Fluid Mech., 39 (1991), 17. 38. J. P. Benqu6, G. Labadie and J. Ronat, Int. J. Numer. Meth. Fluids, 8 (1988), 869. 39. K. K. Kabanemi, F. Bertrand, P. A. Tanguy and A. A'it-Kadi, J. NonNewtonian Fluid Mech., 55 (1994), 283. 40. H. H. Hu and D. D. Joseph, J. Non-Newtonian Fluid Mech., 37 (1990), 347. 41. H. C. Choi, J. H. Song and J. Y. Yoo, J. Non-Newtonian Fluid Mech., 29 (1988), 347. 42. J. Y. Yoo and Y. Na, J. Non-Newtonian Fluid Mech., 39 (1991), 89. 43. S.-C. Xue, N. Phan-Thien and R. I. Tanner, J. Non-Newtonian Fluid Mech., 59 (1995), 191.
382
44. X. Huang, N. Phan-Thien, R. I. Tanner, J. Non-Newtonian Fluid Mech., 64 (1996), 71. 45. T. Sato and S. M. Richardson, J. Non-Newtonian Fluid Mech., 51 (1994), 249. 46. B. P. Leonard, Comput. Meth. Appl. Mech. Eng., 19 (1979), 59. 47. I. Babuska, Numer. Math., 16 (1971), 322. 48. F. Brezzi, Revue Franqaise d'Automatique Informatique Recherche Op6rationelle, Analyse Num6rique, 8 (R-2) (1974), 129. 49. P. Saramito, Math. Modelling Num. Anal., 35 (1994), 1. 50. P. Saramito, J. Non-Newtonian Fluid Mech., 60 (1995), 199. 51. P. J. Roache, Computational Fluid Dynamics, Hermosa, 1976. 52. G. P. Sasmal, J. Non-Newtonian Fluid Mech., 56 (1995), 15. 53. F. H. Harlow and P. H. Welch, Phys. Fluids, 8 (1965), 2182. 54. L. S. Caretto, A. D. Gosman, S. V. Patankar and D. B. Spalding, Proc. Third Int. Conf. Numer. Methods Fluid Dyn., Paris, 1972. 55. R. I. Issa, J. Comput. Phys., 62 (1986), 40. 56. J. P. van Doormal and G. D. Raithby, Num. Heat Transfer, 7 (1984), 147. 57. J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Springer, Berlin, 1996. 58. B. Gervang and P. S. Larsen, J. Non-Newtonian Fluid Mech., 39 (1991), 217. 59. X.-L. Luo, J. Non-Newtonian Fluid Mech., 64 (1996), 173. 60. M. Fortin and A. Fortin, Int. J. Num. Methods Fluids, 5 (1985), 911. 61. D. W. Peaceman and H. H. Rachford, J. Soc. Ind. Appl. Math., 3 (1955), 28. 62. X.-L. Luo, J. Non-Newtonian Fluid Mech., 63 (1996), 121. 63. M. J. Szady, T. R. Salamon, A. W. Liu, D. E. Bomside, R. C. Armstrong and R. A. Brown, J. Non-Newtonian Fluid Mech., 59 (1995), 215. 64. A. J. Chorin, Math. Comp., 22 (1968), 745. 65. R. Temam, Theory and Numerical Analysis of the Navier-Stokes Equations, North-Holland, Armsterdam, 1977. 66. J. Kim and P. Moin, J. Comput. Phys., 59 (1985), 308. 67. J. Kim, P. Moin and R. D. Moser, J. Fluid Mech., 177 (1987), 133. 68. H. Choi, P. Moin and J. Kim, J. Fluid Mech, 262 (1994), 75. 69. H. Choi, P. Moin, J. Comp. Phys., 113 (1994), 1.
383
70. J. Donea, S. Giuliani, H. Laval and L. Quartapelle, Comput. Meth. Appl. Mech. Eng., 30 (1982), 53. 71. B. Ramaswamy and T. C. Jue, Int. J. Num. Methods Fluids, 35 (1992), 671. 72. B. Ramaswamy, Comput. Fluids, 16 (1988), 349. 73. M. Kawahara and K. Ohmiya, Int. J. Num. Methods Fluids, 5 (1985), 981. 74. O. C. Zienkiewicz and J. Wu, Int. J. Num. Methods Eng., 32 (1991), 1189. 75. J. G. Rice and R. J. Schnipke, Comput. Meth. Appl. Mech. Eng., 58 (1986), 135. 76. A. C. Benim and W. Zinser, Comput. Meth. Appl. Mech. Eng., 57 (1986), 223. 77. H. G. Choi and J. Y. Yoo, Numer. Heat Transfer B, 25 (1994), 145. 78. H. G. Choi and J. Y. Yoo, Numer. Heat Transfer B, 28 (1995), 155. 79. H. G. Choi, H. Choi and J. Y. Yoo, Comput. Meth. Appl. Mech. Eng., 143 (1997), 333. 80. E. O. A. Carew, P. Townsend and M. F. Webster, J. Non-Newtonian Fluid Mech., 50 (1993), 253. 81. A. Baloch, P. Townsend and M. F. Webster, J. Non-Newtonian Fluid Mech., 54 (1994), 285. 82. D. M. Hawken, H. R. Tamaddon-Jahromi, P. Townsend and M. F. Webster, Int. J. Num. Methods Fluids, 10 (1990), 327. 83. A. Baloch, P. Townsend and M. F. Webster, J. Non-Newtonian Fluid Mech., 65 (1996), 133. 84. R. Sureshkumar, A. N. Beris and R. A. Handler, Phys. Fluids 9 (3) (1997), 743. 85. T. Min, H. G. Choi, J. Y. Yoo and H. Choi, Int. J. Heat and Mass Transfer, 40 (1997), 3689. 86. C. R. Swaminathan, V. R. Voller and S. V. Patankar, Finite Elements in Analysis and Design, 13 (1993), 169.
385
CONSTITUTIVE THEORY
EQUATIONS FROM
TRANSIENT NETWORK
C.F. Chan Man Fong and D. De Kee
Department of Chemical Engineering, Tulane University, New Orleans, LA 70118, USA
1. I N T R O D U C T I O N An understanding of the flow properties of non-Newtonian fluids is important in many areas of application. To solve flow problems, it is required to introduce a constitutive equation that relates the stress tensor to various kinematic tensors. The determination of a constitutive equation which describes adequately the rheological properties of a fluid under all flow conditions is one of the central problems in rheology. So far constitutive equations have been developed based on either continuum mechanics or on the molecular structure of the fluids. In the continuum approach, no explicit consideration is given to the molecular structure of the material. A relationship between suitable dynamic and kinematic variables is postulated. The conditions that this relationship has to satisfy were stated by Oldroyd [ 1]. A suitable dynamic variable is the extra stress tensor v and a suitable kinematic variable is the relative right Cauchy-Green tensor C t. The tensor C t relates the square of the relative distance between two arbitrary material points at two different times, one at the present time t and the other at a past time t'. The assumption that __zdepends on the history of _C t_ from time t' - - ~ to the present time t can be expressed mathematically as
1;-
t ~ t ! "-" ~
(Ct) oo
(1)
386
where _~ is an isotropic tensor-valued functional. Equation (1) is the constitutive equation of a simple fluid. The constitutive equation for a Newtonian fluid is 3C (2)
=
t' =t
where 770 is the constant viscosity. Various simplified forms of equation (1) have been proposed and are discussed in [2-5]. In the molecular approach, the molecular structure of the fluid is taken into consideration. In modeling a polymeric material, we first represent the polymer molecules by mechanical models. We then introduce a probability distribution and an evolution equation for the distribution. Finally we calculate the average of all quantities so that a relationship between the macroscopic quantities can be obtained. Three models, namely the bead-rod-spring model, the transient network model, and the reptation model, have been popular among rheologists. We shall discuss the development of the transient network model after a brief description of the other two models. 2. B E A D - S P R I N G M O D E L This model was proposed to describe dilute polymer solutions. The polymer molecule is idealized as a dumbbell consisting of two beads, each of mass m, joined by a spring. The beads are labeled 1 and 2; their vector positions are r l a n d r 2 and R = ( r 2 - r l ) is the vector joining bead 1 to bead 2. The polymer solution is modeled as dumbbells suspended in a Newtonian fluid of constant viscosity r/s. We introduce a d i s t r i b u t i o n f u n c t i o n F ( r 1 , r 2, trl, .7 2, t ) . such . that .F ( r 1,. r 2, 71 . , 7 2, .t ) dr .l d r 2 d r 1 d r 2 is the number of dumbbells in the position range velocities in the range r'i
t o t~i +
F__i t o F_.i + d r i
and the beads have
dT_i (i = 1, 2). It is customary to write F as a
product of a configuration distribution
!/'t(rl,r2
,
t)
and
a velocity
387 distribution ~ (F__'I, /~2) which is often assumed to be Maxwellian. If the solution is homogeneous, 7~ can be written as (L1, r2, t) = n ~ (_R, t)
(3)
where n is the number density of dumbbells. The conservation of the number of dumbbells leads to an equation of continuity of the form (Bird et al. [6]) 3t
- - ~_R -It ~ 1 1 V
(4)
where ~ 11denotes an average in the velocity space. The forces acting on bead i are:
(a)
the hydrodynamic f o r c e Ff h) which is assumed to be proportional to the difference between the average bead velocity and the velocity of the solution at that point. Assuming that the velocity of the solution is not affected by the other beads, _F}h~ can be written as
_F{h) = -~ {I]~i ]]-Vi}
(5)
where ~" is a constant and v i is the velocity of the solution at F_.i . (b)
the Brownian force _F~B~which is assumed to be of the form F{ B)-- - k T ( ~ 2 n W / ~ r i )
(6)
where k is Boltzmann's constant and T is the temperature.
(c)
the intramolecular force F~ ~ and, in the present model, it is the force due to the spring. The forces on the two beads are equal and opposite and a connector force F (c~ can be defined as
388
(7)
F (c)- _F~*)= -F~ *)
In the absence of external forces and neglecting inertia terms, the equations of motion for the beads are --~ {[I i l ]] --V1} -- kT (~)2nW/arl) + _F(c)- 0
(8a)
--~ {[I i 2 ]] -- V2 } -- k T ( O 2 n U d / O L 2 )
(8b)
- _F (c) -
0
On subtracting we obtain {[I ll~]] + (v 1 - v 2 ) } + 2kT (~)2n~/~)_R) + 2F(C) = 0
(9)
By considering the flow to be homogeneous, v 1 - V2 can be written as V_.I--V 2 -- - _ g ' R
(10)
where L is the velocity gradient. Combining equations (9, 10) yields ~1I ~11 = ~ _L- _R- 2kT
(a,env/a_13)-
2F (c)
(11)
Substituting equation (11) into equation (4) yields m
9
~t
(L-R_) ~ - ( 2 k T / ~ ) ( ~ / ~ _ R )
- 2F(C)~/~I }
(12)
The total stress/7 for the solution can be written as =l-=I = 1-=I__s + FI_p_
(13)
where =/-/sand Hp are respectively the contributions from the solvent and the polymer. The solvent is a Newtonian fluid and _/-/s is given by Us - Ps---- ns ~
(14)
389
Ps is the pressure associated with the solvent, 6 is the unit tensor, and
where
is the rate-of-strain tensor. The contributors to __/-/pare the connector, ~___p, (c)/7 and the beads,
--(b) l=lp . It is
shown in Bird et al. [6] that __/_/(pc)and __/_/(pb)are given by (c) lip= = - n < R F =l-=Ip(b) =
(c) )
(15a)
2 nkT 8
(15b)
where < > denotes the average over the configuration space. Combining equations (13, 14, 15a, b) yields __H -
Ps ~ - rls "~ - n < R F (c)) + 2 n k T ~
(16)
It has been shown (Bird et al., [6]) that at equilibrium n ( R F~C)>0 = n k T ~
(17)
From equations (16, 17) we deduce that __H- (Ps + n k T ) ~
+~=
(18a)
~_ = -rls T - n + n k T 8
(18b)
Equation (18b) is the Kramers expression for the stress tensor. To calculate
D
Cc)) (19)
390
where D/Dt is the substantial derivative, v denotes the upper convected derivative, and t denotes the transpose. From equations (18b, 19) we deduce that =rp (= __r+ 7/s _/79is given by V
= ln~ ~P 4
(20)
Equation (20) is the Giesekus form of the stress tensor. For a Hookean spring, we deduce from equation (18b) that =Vp is given by ~p = - n i l
+ nkT=8
(21) V
Combining equations (19, 20, 21) and noting that =~ = - ~ yields V _'lYp + ~1 "l~p= -
(22)
- n k T ~ , 1 "~
where A 1 = ~/4H is the relaxation time. The extra stress ~" is given by V
V
~-+ ~'I ~ = --nO (~r+ ~2 It)
(23)
where 770 - 77s + n k T ~ 1 is the viscosity and ~ 2 = /]" 177s ] (77 s -I- n k T X 1) is the retardation time. Equation (23) is the constitutive equation of a dilute polymer solution modeled as Hookean dumbbells suspended in a Newtonian solvent. Equation (23) was also proposed by Oldroyd [ 1] from a continuum mechanics approach and was labeled liquid B. An integral form of equation (23) is (Bird et al. [6]) [(nkT/k
,) e -(t-t'),~''] C~ 1(t')
-1
m ( t - t') _C t (t') dt'
m (t - t') = rlsS' (t - t') + (n k T / ~ 1) e-(t-t')/Z"
dt'
(24a)
(24b)
(24c)
391
where C -l(t') t is the relative Finger tensor, ~' ( t - t ' )
is the derivative with
respect to t' of the Dirac delta function , and m ( t - t') is the memory function. In this model, macromolecules are represented by single dumbbells. Figure 1 illustrates a more realistic model for a macromolecule consisting of N beads connected by ( N - 1) springs. In the Rouse model, the connectors are Hookean springs and a constitutive equation can be deduced as in the case where the macromolecules are assumed to be single dumbbells. Equation (24a) is modified to obtain (Bird et al. [6])
~_ = - r l s ' l +
nkT
e
P C_t ( t ' ) d t '
(25)
where/~/p is the relaxation time associated with dumbbell p.
_R2
Figure 1. Six beads connected by five springs. The Rouse model is widely used to interpret linear viscoelastic measurements. The dynamic viscosity 77' and the dynamic rigidity 77" are given by
392
rl'-rls
- nkT ~
~,lp/[1 + (klpf.0) 21
(26a)
P
rl" - n k T ~ ~21pf'0 1[1
+
(~ lp(.0) 2]
(26b)
P
where co is the frequency of the imposed small amplitude oscillations. The Aqp can be related to the molecular structure of the polymers (Rouse [7]). Increasing the number of terms in the series allows for a better fit of 7?' and 77". However, in steady viscometric flows the Rouse model predicts a constant viscosity whereas for most polymer solutions the viscosity decreases with increasing shear rate. Various improvements, such as replacing the Hookean spring by a non-Hookean spring, including hydrodynamic interaction and internal viscosity, have been proposed to the simple dumbbell model and these modifications are thoroughly discussed in Bird et al. [6]. In the models discussed so far, it is assumed that the polymer chains move freely in space, unhindered by the other chains. In concentrated solutions and in polymer melts, the motion of a chain is constrained by other chains and this constraint is taken into account in the next two models. 3. R E P T A T I O N M O D E L
3.1. D o i - E d w a r d s Model In the reptation theory proposed by de Gennes [8], the constraint that polymer chains cannot pass through each other is interpreted such that each chain is confined to move inside an imaginary tube, as shown in Figure 2. The chain is assumed to be longer than the tube and the slack is considered to be a defect capable of moving along the tube. As a result of such motion, part of the chain can move out of the original tube and a new tube is created. This type of motion is called reptation. Based on this concept, Doi and Edwards [9] derived a constitutive equation for polymeric liquids. Doi and Edwards [9] introduced the concept of a primitive chain which is the center line of the tube, as shown in Figure 2. The motion of the real chain is represented by the motion of the primitive chain. The real chain wriggles around the primitive chain. This motion is very rapid and an equilibrium is established after a time Teq (equilibration time) which is proportional to N 2, where N O is the degree of polymerization of the real chain. On a time scale
393
greater than Teq , this wriggling motion can be ignored and the motion of the real chain coincides with the motion of the primitive chain.
4,--,,
/
/
/
\
f
\
/
o"
I
\
,"
f
,_/
Figure 2. Polymeric chain ( ~ ) in an imaginary tube. (o 9 o): primitive chain. The primitive chain is assumed to be composed of freely jointed segments with step length a and arc length L. The stress "r of the primitive chain is (27)
~= -- - < Z F i _ R i > i
where
F i
and
g i
refer to the force and end-to-end vector of segment i
respectively. If the primitive chain is assumed to be continuous, any of its points at time t can be denoted by R_ (s, t), where s is the contour length measured from one end of the chain. The unit tangent u (s, t) to the primitive chain is given by u (s, t) = ~ _R(s, t) / Os
(28)
394
It is assumed that _F i e_i can be written as (29)
F i R i = Feq tl (s) u (s) ds
where
Feq is the
force acting on the primitive chain at equilibrium.
Combining equations (27, 28, 29) yields
X = --
< u (s) u (s) > ds -
L
where
Go is
(30)
a constant.
Suppose that the material is deformed such that a material particle which is at x at present time t, was at x' at an earlier time t'. The deformation _E is defined by
E = (Ox/~x')
(31)
Doi and Edwards [9] assumed that the unit tangent vector _u' (s) at time t' is related to u (s) at time t by (independent alignment approximation) u'(s) = (E_o u ) / I E
9u l
(32)
The length of the primitive chain L is assumed to be constant and the extra stress tensor ~ can be written as I: - - G o Q
Q -
(E 9 u) ( E - u) = iE.ul ~
(33a)
(33b) 0
where < >0 is an average over u in equilibrium. In equation (33a), the random motion of the primitive chain has not been taken into account. The random motion of the chain is responsible for the
395 relaxation process. This random motion is governed approximately by diffusion and a closed form solution can be obtained (Doi and Edwards [9]). The stress _vis given by
~ =-G0f_toog ( t - t ' ) __Q[E_ ' (t,_ t')]dt
(34a)
[8/(7~2p2"l;p)] exp (-t/'l;p)
(34b)
(t) = ~
p odd "~p = L2/(D/I;2p 2)
(34c)
where D is the constant diffusion coefficient of the random motion. Equation (34a) is the equation proposed by Doi and Edwards [9] and they have shown that it can be written as ~== -
f_ t M 1 [ t - t ' , I i ( t , t ' ) , I 2 ( t , t ' ) ] C t - l ( t ' ) + M 2 [ t - t ' , I i , I 2 ] ( _ C2t_- l) 1 dt'
(35a)
M 1 = ~(t-t')Hl[Ii(t,t'),I2(t,t')]
(35b)
M 2 = ~ (t-t')H2[II(t,t'),I2(t,t')]
(35c)
where I1 and related to Q.
12 are the invariants of ~_Ct1, H1 and H2 are functions which are
Equation (35a) is known in continuum mechanics as the K-BKZ equation (Carreau et al. [4]). Note that the memory functions M 1 and M2 are written as a product of a function of ( t - t ' ) only [/2 ( t - t ' ) ] and a function of deformation only [H 1(I1 ,I2), H2(11 ,I2)]" The function p ( t - t') can be obtained from linear viscoelasticity. Equations (25, 35 a to c) are similar if we set 7"/~= 0, H 1 = - 1 , and H 2 = 0. Various modifications have been proposed to the present model and they are discussed in [4, 5, 10].
3.2. Curtiss-Bird Model Curtiss and Bird [ 11] presented another version of the reptation motion and derived a constitutive equation based on their phase-space kinetic theory (Bird et al. [6]. The polymer chain is modeled as a Kramers chain consisting of
396
N identical beads freely linked by ( N - 1) massless, rigid rods each of length a, as shown in Figure 3.
C) r
Figure 3. Segments of a Kramers chain. The method employed to obtain a constitutive equation is similar to that used in section 2. It is assumed that the differences between the hydrodynamic forces on adjacent beads v and v + 1 can be written as _Fv(h+)1 _--(h) gV
--
-- ~
~v = ~ [ ~ - ( 1 - e )
13~V . { l l r- - v + l u v u v]
r l l - [ v- -( r - - +Lv
----V
+1
) - v t- -r
--
+Lv)]}
(36a) (36b)
m
where N is the average number of beads per chain, r v is the vector position of bead v relative to the center of mass (see Figure 3), fl, ~', and e are constants, and u v is the unit vector joining bead v to bead v + 1. If the link tension coefficient e is one and the friction tensor ~'v is isotropic,
397
equation (36a) is similar to equation (5). The parameter e determines the anisotropy of ~'v and Bird et al. [6] proposed e to be in the range 0.3 to 0.5. The Doi-Edwards model corresponds to the case e = 0. The constant 13 is also in the range 0.3 to 0.5. If the flow is assumed to be homogeneous, equation (36a) simplifies to F(h) v+l
--
F(h)
2_.V
~
--
Nlaa
~V
9 {iI -fi- V 11-[L- - 9 Uv] }
(37)
In the calculation of the Brownian force, it is necessary to evaluate II 1~1i U_'j]] and in order to do this, Curtiss and Bird [11] introduced the concept of reptation. They assumed that lI Lj L v ll = ( e : / 4 ) (Uj_ 1 4"- Uj) (Uv_ 1 + U_v)
(38)
where a is a constant independent of j and v for all j and v from 2 to N - 2. Equation (38) implies that the "average" velocity of a bead relative to the center of mass is in the direction of the two rods connected to the bead. That is to say, the motion of the bead is mainly in the direction of the backbone of the chain, which is the concept of reptation. In calculating the distribution function 7~, the mild curvature approximation is introduced. This means that the orientations of adjacent links do not vary significantly and the Kramers chain can be assumed to be continuous with continuous derivatives. Based on the assumptions stated earlier, Curtiss and Bird [11] obtained a constitutive equation which can be written as
~ = = - N n k T I f _ t o o g ( t - t') =Q (E_) f _ t dt'+ oo_ e ( t - t') - (8/)v) ~
( t - t') S_=_(E; L__)dt'1
exp [-/I;2p2(t- t ' ) / s
(39a)
(39b)
p odd
[5 ( t - t') = (16//1;2~) Z p odd
(1/P 2) exp [-/1;2p2(t - t')/)~]
(39c)
398
S= = (1/4rt) ~,L(t) 9 f
u u u o du
(39d)
where n is the density of polymer molecules, k is Boltzmann's constant, T is the temperature, and ~ is a time constant. If e = 0, equation (39a) reduces to the Doi-Edwards model with the time constant being defined differently. Other forms of the Curtiss-Bird equation are available and are given in Bird et al. [6].
4. NETWORK MODEL 4.1. Lodge Model The present model, the transient network model, is an adaptation of the network theories of rubber elasticity and was developed by Green and Tobolsky [12], Lodge [13], and Yamamoto [14] for modeling polymer melts and concentrated polymer solutions. In the network theory for solids, chemical crosslinks are described as permanent junctions and are forced to move together at all times. For liquids, which are capable of flowing, the junctions are temporary and liable to be destroyed. Equally new junctions can be formed. The polymeric material is represented by a network, as shown in Figure 4. The molecular segment that joins two successive junctions is an active segment, the chain that is connected to only one junction and has the other end free is a dangling segment, and the chain that is not attached to the network is a stray chain. As a first approximation, we neglect the contributions of the dangling and stray segments to the stress generated during a deformation. Let R_ be the end-to-end vector of an active segment and 7~ (R_, t) be the probability of finding a segment in the range R_ and R_+ dR. A balance of segments yields (Bird et al. [6])
3t
=
3R
(_R W) + c - 2
where c and 2? are the rates of creation and loss of segments respectively.
(40)
399
Segment
"~
) ~'
choin
~-Dongling Loose end
Segment
Figure 4. Polymer network. Note that in equation (40), unlike in equation (4), we have allowed for the loss and creation of segments. This loss and creation of segments is analogous to the loss and creation of tubes in the Doi-Edwards model. In their model, the loss and creation of tubes is due to the reptation of the primitive chain and in the network model, there is as yet no definite molecular mechanism for the creation and loss of segments. We need to assume empirically the form of c and 2 . If we assume that the junctions move affinely, it follows that 1~ = L o R
(41)
For simplicity we assume that all the segments are alike and the tension acting in each segment is
F (c). The
total stress/7 can be written as
I-I (t) = - n < F (c) R > where n is the number of segments.
(42)
400
To evaluate <
if(c)_R> we need
to solve for 7~ from equation (40) and this
implies that we need to postulate c and ,e. In the Lodge model, the following assumptions are made c = W0/~
(2, is a constant)
(43a)
2 = W/~
(43b)
_F(c) = H_R (H is a constant)
(43c)
where ~0 is the value of 7~ at equilibrium. The choice of the same constant ~ in equations (43a, b) ensures that at equilibrium there is no net loss or gain of segments. Substituting equations (41, 43a, b) into equation (40) yields OWOt =
~R~ (L= 9RW)_ + ~ ( W 0 - W)
(44)
Equation (44) seems to be quite different from equation (12), however if equation (44) is multiplied by R R and integrated over the R_-space, we obtain (Bird et al. [6]; Chan Man Fong and De Kee [15]) V
= I((RR>
-)
< R R > o = J _8
(45a) (45b)
where J is a constant. Combining equations (42, 43c, 45a, b) yields V
H+LH
= -nil J8
(46)
Noting that
FI = P ~5 + "c
(47a)
z_ = 0
(47b)
at equilibrium
401 V 8 - - ~
(47c)
we deduce from equation (46) that V "c + )~'c - - r l "~
(48)
where 77 (= ~, n H J) is a constant. Equation (48) is the constitutive equation of an upper convected Maxwell fluid (Carreau et al. [4]) and an integral form of equation (48) is
- -
m (t - t') C t (t') dt'
m (t - t') = (yl/~ 2) exp [-(t - t')/~,]
(49a)
(49b)
We note that in this case, as in the bead-spring model, we have been able to derive a constitutive equation without having to solve for 7~. The time constant ~, for the network model is empirical whereas the time constant ~,1 in the beadspring model is related to the structural parameters of the model (equation 22). The two models yield identical constitutive equations if the solvent viscosity is negligible. For clarity we have so far considered the case of a single time constant ~,. We can generalize from one time constant to a multitude of constants ~p. We can associate /~p with the different ages of the junctions (Lodge [16]) and to each ~p corresponds a __rp. Each __rp satisfies equation (48) and =r (= ~ __Vp) is p given by equation (49a) with m ( t - t') defined by m (t - t') - ~ (T~p/)~ 2p) exp [ - ( t - t' ) / ~ p ] P
(49c)
The Lodge model can describe many of the p h e n o m e n a associated with linear viscoelasticity (Carreau et al. [4]) but it does not predict shear thinning as observed in viscometric flows of polymeric fluids. One way of overcoming this shortcoming is to allow c and ,g to be functions of a macroscopic variable, such as an invariant of the shear rate, and this is considered next.
402 4.2. Shear Rate Dependent Models Several authors [15, 17, 22] have assumed that c and ,e are functions of the second invariant of the shear ~ defined by = # (1/2) trl~ 2
(50)
We now consider the multimodes model proposed by De Kee and Carreau [21 ]. Corresponding to the p-mode, the functions cp and ,gp are assumed to be Cp(t) = Lp [~t(t)] Wpo,
~ p(t)
- tlJp/Xp [~(t)]
(51a, b)
where ~up0and ~p are the distribution functions associated with the p-mode at equilibrium and at time t. Substituting equations (5 l a, b) into the balance equation [equation (20)] for the p-mode, multiplying the resulting expression by R R and integrating over the R_-space, we obtain
v < R R) p = Lp =8- < R R)p/~p
(52a)
Lp
(52b)
"
-
(4rt/3)
R4LpdR
Integrating equation (52a) yields < R R)p =
f~oo
{ Lp exp [
--
ftt'
dt"/1;p(t")] C t 1(t') } dt'
(53)
It is assumed that L p and "t'pcan be written as
Lp = [Tlpfp('~)]/~p,
"l;p = ~,pgp('y)
(54a,
b)
where r/p and Z p are constants and have dimension viscosity and time respectively.
403
Following the usual procedure, it is deduced that the extra stress tensor T can be written as
= -ffoo
m (t, t', 3;) Ct 1(t') dt'
(55a) '
YI pfp [~ (t')] exp
m ( t , t', 3~) - Z
2 )~p
ftt
dt"
(55b)
~,pgp [~(t")]
The functions fp and gp cannot be deduced from the model and have to be prescribed empirically. The following assumptions are made (De Kee [23]) fp = exp [-~tp (3 - 2c)] , go = exp [-~tp (c - 1)], fp = gp = 1, where c and
p - 1, 2, ..., k
(56a) (56b)
p - 1, 2 .... , k
p = k + l , k+2, ...
(56c)
tp are constants.
Equations (56a to c) imply that there are various types of junctions, the rates of loss and creation of the first k types are shear-rate dependent and the remaining types are shear-rate independent. Equations (55a, b) have been found to be adequate to describe the rheological properties of polymeric systems and the predictions of equations (55a, b) compared to experimental data will be examined later. Other forms of fp and gp, usually as rational functions of ~, have been proposed and are discussed in Carreau et al. [4].
4.3. Stress Dependent Models Kaye [24] assumed the rates of loss and formation of junctions are functions of the invariants of the stress tensor. He proposed that cp and ,gp are functions of Q1 and Q2 and they are defined by Q1 -
12lx-212~
'
Q2 = 213lx - 9 I
lx
I 2x + 2 7 I
3x
(57a, b)
404
Ilz - tr ~ ,
I2z = (1/2) [(trx=)2 - try=2] ,
I3x - det ~
(57c, d, e)
Note that I 1T, 12v, and 13 r are the invariants of __vand Q1 is positive for all '~"~
Equations (5 l a, b, 54a, b) are now replaced by Cp(t) = Lp [Ql(t), Q2(t)] W0 ,
2p(t) = Wp/1:p [Ql(t), Q2(t)]
(58a, b)
Lp - [lip gp (Q1, Q2)] [ )~p,
"l:p - ~,p gp (Q1, Q2)
(59a, b)
The constitutive equation is given by equation (55a) with the memory function defined by
m (t,t',Q1, Q2) - ~ P
I
TlPgP[QI(t')' Qz(t')] exp ~2 P
Iftt
dt" ~pgp [QI (t,,), Q2(t,,)]
tl
(6O) Kaye [24] considered one mode only (p = 1) and assumed g l to be a linear function of Q1 and independent of Q2. In a viscometric flow, this model predicts a viscosity which decreases with increasing shear rate and the ratio of the first normal stress difference to the square of the shear stress is constant. Phan-Thien and Tanner [25], Phan-Thien [26], and Tanner [27] have assumed that the rates of loss and creation of junctions are functions of p and (IRl2)p0 respectively. Since the segments are assumed to be Hookean (=v is proportional to < R R>),< IRl2)p is proportional to tr v__p. Their model is a stress dependent model. In addition they have assumed that the deformation is non-affine and their model is discussed in section 4.6. The Marrucci model [28] was proposed as a modification of the upper convected Maxwell model but it has been shown that it is derivable from the network theory (Jongschaap [29]) and is of the stress dependent type. Acierno et al. [28] have also introduced a structural parameter related to the segment density in their model and we shall discuss these models in section 7.
405
4.4. Strain Dependent Models A strain dependent model was proposed by Wagner [30]. He assumed the rate of creation Cp to be constant and the rate of loss ,s to depend on 11 -1
(= tr C_t ) and I 2 (= tr C_t ) . Further he proposed the loss of junctions to be due to the Brownian motion (a constant rate of loss) and to the deformation. Instead of equations (59a, b) we now have Lp
=
l]p/~,p
~p = ~,p + g (I1, I2)
,
(61a, b)
where g (associated with the loss of junctions due to the deformation) is independent of p (the type of junction). The constitutive equation is given by equation (55a) and m can be written as m (t,t , ,I1,I2)
2 = E [ (lqp/~p)
exp
P
= h (II,I2) Z
2 ('qp/~p)
IItt
e_(t_t, )/~,p
g
at" -[Ii(t")i I2(t")]
t
(62a)
(62b)
P
The damping function h is defined by
h (I 1' 12) -" expIftt'
dt" g (I 1' 12)
(63)
and is to be determined empirically. Based on relaxation data, Wagner [30] proposed the following form for h h = exp [-[3 4t~ I 1
+ (1--t~)
12 - 3 ]
(64)
where a and fl are positive constants. Wagner and Stephenson [31] added a further restriction on the function h. Equation (64) is valid only when the deformation is a non-decreasing function in time. If the deformation is a decreasing function of time, h is taken to be the minimum value of h over the relevant period. Physically this means that junctions which are lost over a previous increasing period of deformation are lost forever.
406
Comparing equations (55a, 62a) with equations (35 a to c), we observe the Wagner model is a special case of the Doi-Edwards model or the K-BKZ model. 4.5. Y a m a m o t o
Model
In all the transient network models considered so far, the functions c and A? depend on the macroscopic variables and it was possible to deduce a constitutive equation in a closed form. Yamamoto [14] proposed that the rates of creation and loss of segments depend on the end-to-end vector R. It is assumed that c and ,8 can be written as c -
fl (_R) q J 0 ,
~
-
f2(-R)~I j
(65a, b)
Since at equilibrium c = 2 , we deduce from equations (40, 65a, b) that 3q' ~)t
-
~)R
(R W) + f ( R ) ( W o - W) -
(66)
where f = fl - f2If we proceed as in the previous cases, that is by multiplying equation (66) by R R and integrating over the configuration space, we are led to evaluate
Non-Affine
Deformation
So far we have assumed that the junctions move affinely with the macroscopic imposed velocity. Lodge et al. [34] and Giesekus [35] have
407
expressed reservations concerning this affine motion assumption. A non-affine model was proposed by [25, 36]. Equation (41) is modified to
1~ = Lo R
(67a)
L = L - (112~) ~
(67b)
where ~ is a constant and is known as the slip parameter. Substituting equations (43a to c, 67a) into equation (40) and proceeding in the usual manner yields an analog of equation (40) which can be written as [ 15] ~=+~= = - r 1 7
(68a)
7 = (1 - ~) ~
(68b)
~2
__z"is the Gordon-Schowalter [37] derivative of =vand is given by Q
1;
--
D'~ = Dt
.-, ---t Lox-I:oL = = = =
(69a)
V
= 1: + (,~12) (~t,~ + ~ . ~)
(69b)
The Gordon-Schowalter derivative reduces to the Oldroyd upper, lower convected derivative and to the Jaumann derivative if ~ is set to 0, 2, and 1 respectively. Since ~ is a constant, we may write the simplest constitutive equation based on a transient network model with non-affine deformation as [combining equations (68a, 69a)] +
+
(,,?.
+ ,c._
=
-n
";'
(70)
Equation (70) is a special case of the Oldroyd [1] model derived from continuum considerations. Equation (70) predicts a variable viscosity and the ratio of the secondary (N2) to the primary (N]) normal stress differences in viscometric flows is given by
408
N2
_
~
N1
(71)
2
By allowing for a non-affine motion, the simplest model [equation (48)] of the network model has been improved considerably in the sense that a variable viscosity and a secondary normal stress difference can be predicted. The constitutive equations obtained on the assumption of affine deformation can be extended to non-affine deformation by replacing the Oldroyd derivative by the Gordon-Schowalter derivative if the equations are written in the differential form. If the equations are written in the integral form, the Finger -1
--~-1
tensor _Ct-1 is replaced by a modified Finger tensor C t= . The tensor C t=
is
calculated from the reduced velocity _v [= (1-~/2) v] instead of from v. The Phan-Thien-Tanner [25] model, which is a stress dependent and nonaffine deformation model, is written as = ~ ~
(72a)
_~P
P o
Y (tr ~ p) ~=p + )~ p ~= = -- TI p ~t_
(72b)
Two empirical forms of Y (tr r=p) have been proposed and they are 1 - ~ (tr ~
p) ~ p/lip
y
(73a, b) exp [-~ (tr ~p) ~ p [ ] q p ]
where e is a non-negative constant. If e - 0 , equation (72b) simplifies to equation (70). A non-zero value of e allows for a finite extensional viscosity (r/E) for all finite extensional rates (~). If equation (73a) is chosen, r/e increases monotonicly to a finite limit as ~ tends to infinity; equation (73b) predicts that r/e goes through a maximum as ~ goes to infinity (Bird et al. [6]). The extra stress z is arbitrary to the extent of an arbitrary isotropic pressure and C -t 1 can be replaced by (_Ct 1_ _S) o r ( ~ - - C -1 t ) in the constitutive equation. The same applies to C t .
409
An alternative form of equation (49a) is obtained by integrating it by parts. The result is
~= = f _ t o o G ( t - t ' ) _ E ( t , t ' ) . ~ (=t ' ) . _ E t (_t , t ' ) d t ' _
(74)
where G is the modulus and is related to m ( t - t') by G (t - t') = m (t - t')
~t'
(75)
If the junctions move non-affinely, equation (74) is written as =
G (t - t') E_ (t, t ' ) o "~ 9 E_t (t, t') dt'
(76)
where E is calculated from v instead of from v. In this section we have considered two major modifications of the Lodge model, namely variable rates of loss and creation of junctions and the nonaffine motion of the junctions. In the next section we shall examine the role of the dangling and free segments.
5. RECENT DEVELOPMENTS IN N E T W O R K THEORY 5.1. Stress Jump The stress jump is defined as the instantaneous finite change in the stress due to a sudden finite change in the rate of deformation. Liang and Mackay [38] have reported measurements of stress jump at the cessation of shear flow for solutions of xanthan gum. Not all constitutive equations can predict a stress jump. The equations as given in section 4 do not predict a stress jump. These equations are valid on a time scale greater that the equilibration time (Teq) and the rapid change in the stress due to the sudden finite change in the shear rate is ignored. The model proposed by De Kee and Carreau [21 ], discussed in section 4.2, can predict the stress jump if the two conditions they imposed are included. These two conditions are
410 M
Z Tip exp ( - t / ~ p ) p=k+ 1
= I]oo [1 - H ( t ) ]
(77a)
M
2Z
Tlp(t + ~p) exp ( - t / ~ p ) = ~tl~ [ 1 - H ( t ) ]
(77b)
p=k+ 1
where 7/00and ~1oo are the limiting infinite viscosity and primary normal stress coefficient respectively and H (t) is the Heaviside step function. Note that equations (77a, b) imply a singularity at t = 0 and are valid for ~ p - - > 0. Further, it is not possible to distinguish the various modes (p = k + 1, k + 2, ...) and there is no loss of generality in confining to one mode only (p = M). De Kee and Chan Man Fong [39] have shown that equations (77a, b) are equivalent to introducing a singularity to the memory function m in equation (55b). This equation is now written as m = mL + m s
mL=
~ p=l
l
rlpfp 2 exp ~p
(78a)
Iftt
~dt" g p p
1 [
ms - 11oo8'(t - t') - (~1oo/2) 8"(t - t")
(78b)
(78c)
where ~' and 6" are the first and second derivatives of the Dirac delta function. Within the network theory we can interpret m L as being the response of the active segments associated with a time scale greater than Teq. The function ms represents the response of the solvent molecules [see equation (24c)], the stray, and the dangling segments and is associated with a time scale less than Teq. To predict the stress jump phenomenon it is necessary to include the short term response which is the Newtonian and possibly also slightly viscoelastic responses [10]. This in turn implies that the memory function must have a singularity [40]. Other models that can predict the stress jump are discussed in [38, 41, 42].
411
5.2. Shear Thickening In recent years there has been a growing interest in complex fluids that exhibits shear thickening. Ahn and Osaki [43] have tabulated various systems that are shear thickening. They have classified the rheological flow curves into four types: thinning-thickening, thinning-thickening-thinning, thickening only, and thickening-thinning. The cause of the shear thickening is the flow induced complex formation of the material and the origin of this complex formation can be electrostatic interaction, hydrogen bonding, etc. Chan Man Fong and De Kee [15, 22] have included a Coulombic force between the junctions of the network model. They assumed that c and ,e in equation (40) are given by c = kcW 0 + k 1f2 (~/) q~
(79a)
= kc tlJ + k,e f l (~t) ~
(79b)
where k c, k l, and k~ are constants and the functions fl and f2 are prescribed later. To include the electrostatic force, they wrote F_(c) as F (c) = H R_ - (2q2/e R 3) R_
(80)
where H, q, and e are constants. Proceeding in the usual manner, they deduced a constitutive equation valid for steady flows and for a time scale greater than Teq. Their constitutive equation for a single mode can be written as
~
1; = + )l, m I; = =
(1 - ~ )
)~mkT
/n o ~
~
i n [2 (k 2 f 1 - k 1 fa) 7= + 7] = - b--E---=
(81)
where ~ m -" (kc + k ~ . f l - klf2) -1, E = ( k c + k l f 2 - k,e f l), n o and n are the number density junctions at equilibrium and at present time respectively, ~ is the slip parameter, k is Boltzmann's constant, T is the temperature, b and I are constants.
412
Equation (81) can be used to predict shear thinning followed by shear thickening. Tanaka and Edwards [44], Wang [45], and van der Brule and Hoogerbrugge [46] have attributed shear thickening to the dangling segments being trapped by the active network. This leads to an increase in the number of junctions and hence an increase in viscosity. The stray segments may also join the active network. By postulating approximately the form of c and ,8 and the probabilities of the stray and dangling segments joining the network it is possible to describe the four types of rheograms mentioned earlier. In the van der Brule and Hoogerbrugge [46] model the onset of shear thickening occurs at a constant value of shear stress independent of the model parameters. A non-Gaussian network can also contribute to shear thickening [16, 47, 48]. By considering both non-affine deformation which leads to shear thinning and a non-Gaussian network which contributes to shear thickening, Vrahopoulou and Mc Hugh [47] have been able to predict both shear thinning and shear thickening. 5.3. Slip-Link N e t w o r k In section 3.1 the reptation process is modeled as the motion of the primitive chain inside a tube. Doi and Edwards [9] have also proposed two other alternative models, namely the cage model and the slip-link model. Slip-links are small rings through which the chain can pass freely, as shown in Figure 5. Wagner and Schaeffer [49] have implanted the slip-link model in the transient network theory. The transient slip-link network is a network of active Gaussian segments between slip-links. Suppose that at equilibrium the distance between two slip-links is 2' 0 with n o monomer units. The end-to-end vector can be written as L = 2 ou
(82)
where u is a unit vector. After deformation, r is transformed to L' - 2oU'
r
' which is expressed as (83)
413
!
Figure 5. Slip-link network at equilibrium and after deformation. Due to the possibility of slip the number of monomers is no longer no and it is assumed that the number of monomers n is a function of u' ( - I _u' I). A slip function S (u') is defined as S (u') = n (u') / n o
(84)
The rates of creation and loss of junctions are assumed to be independent of deformation as in the case of the Lodge model, but due to slip there is possibility of disentanglement. A disentanglement function D (u') is introduced and is also assumed to be function of u'. The extra stress v is now given by -l_~m(t-t')g<(D/S) q
l
u'u'>dt'
(85)
r
where m ( t - t') is given by equation (49b) and g is unspecified. A mass balance of network segments with regard to slip and disentanglement leads to = 1
(86)
414
If S = D = 1 and g = 3, equation (85) reduces to equation (49a). By suitable choices of S, D, and g, Wagner and Schaeffer [49] have shown that equation (85) reduces to the Wagner model and the reptation model.
6. C O M P A R I S O N 6.1. Viscometric
WITH E X P E R I M E N T A L
DATA
Flows
To test the usefulness of constitutive equations it is necessary to compare their predictions with experimental observations in well defined flows. One such flow is the viscometric flow. Referred to the Cartesian coordinate system (x, y, z) the velocity field is given by v x - y ~ (y),
Vy = Vz = 0
(87a, b, c)
where ~ (y) is the shear rate. The viscometric flow is characterized by three material functions, the viscosity function (r/), the primary ('Cxx-'ryy) and the secondary ('ryy- Z'zz) normal stress differences. The viscosity and the normal stress coefficients gti (i = 1, 2) are defined by ~yx -- -- Yl (~t) ~
(88a)
%xx -- Xyy -- --/[/l(~t ) ~r
(88b)
Xyy -- ~zz = --/If 2 (~t) ~/2
(88C)
In equations (88a to c), the functions are defined when the equilibrium state has been reached. For viscoelastic systems it takes a finite time to attain the steady state. If a constant shear rate )'oo is applied at time t = 0, we define the growth functions 77+, I/t~, and
lit~ by
11+ (t, ~'oo) = -'l~yx (t, ~/oo) / Too
(89a)
V~ (t, Z/oo) - - [a:xx (t, ~(oo)- %yy (t, ~(oo)1 / ~2
(89b)
~g~ (t, "]'oo) -
(89c)
- [~yy
(t, Too)- "c= (t, ~'oo)] / ~/2
415
Similarly if a constant shear rate ~'0 is kept for a sufficiently long time until the steady state is established and the applied shear rate is stopped, which can be taken to be at time t = 0, the stress relaxation functions r/-, gt l, and ~2 are defined by
rl-(t,
(90a)
- -Tyx (t, ;/0) / Y0
~ (t, Yo) - - ['Cxx (t, ~'o)- '[yy (t, ~'o)] / ~/2
(90b)
~g2(t, Z/o) = - ['l:yy (t, ~/o)- 1:~z(t, ;/o)] / ~o2
(90c)
For the De Kee-Carreau model (equations 55a, 78a to c) the material functions are k r I (~t) = Z TIp exp (-tp~t) + 1]oo p=l
(91a)
k tJl 1 (~t) _-- 2 ~ Tlp)~ p e x p (-Ctp~/) + ~1oo p=l
(91b)
k
rl-(t, ~/0) - ~ Tip [exp (-tp~/0)] [exp (-t/~,p)] + 1]oo [1 - H (t)]
(91c)
p=l k
~ ( t , ~/0) = 2 ~
TIp~ p [exp (-Ctp~t0) ] [exp (-t/~,p)] + Igloo [1 - H ( t ) ]
(91d)
p=l
k Enp
11+ (t, ~/oo) = n (~'oo) + ~
t [1 - exp (-tp~/oo(2 - c))] - exp (-tp~/oo)
p=l
/
x exp {-t/)~p exp (-tp~/oo(C- 1))/1 + rloo[H(t)- 1] (91e)
+
k L {t
I1/1 (t, "Yoo) -- /[/1 (~oo) 4- Z np p=l
[1-exp(-tpyoo(2 - c))] - 2t exp (-tp~/oo)
-2)~peXp(-Ctp~/oo)l x exp l-t/)Lp exp (-tp~/oo(C- l))}l +/1/loo [ H ( t ) - l ] (91f) For this model the secondary normal stress difference is zero.
416
Figure 6 compares the predicted and measured values of 7"/and Iffl for a 6.0 mass % p o l y i s o b u t y l e n e solution in Primol 355, a pharmaceutical grade white oil with a v i s c o s i t y of about 0.15 Pa~
at 298 K. Figures 7 to 9 s h o w the
theoretical and experimental values of r/-, gt 1 , a n d gt I for various values o f shear rates for the PIB solution.
I
I0
I
I
I
!
I
3
_
iO 4
N
O n
i0 2
I0
-
3
D_
It"
I0
I
L ~ I 0
........
I
1 2
I 3
I 4
1 5
I 6
10 2 7
~' is-' ~
Figure 6. V i s c o s i t y and primary normal stress c o e f f i c i e n t for a 6 % p o l y i s o b u t y l e n e solution in Primol 355. 770 = 6 , 5 1 3 Pa-s, r/1 = 4 , 9 6 3 Pa~ 7/2 = 1 , 2 2 8 P a . s , 7"/3 = 2 6 3 P a . s , 7"/0o = 58 Pa-s, tl - 6 6 . 2 4 t 3 - 0 . 6 3 2 s, 140 Pa.s 2.
2,1 = 7 9 . 4 1 s, ~2 = 19.65 s, 9 m o d e l with tp - 10 tp+l.
23 = 4.21 s,
s, t2 - 6 . 4 5 s, c - 1.21,
gtloo =
417
08 \\\~\ 0.7
o
2'0
,o
t(s)
3b
Figure 7. Shear stress relaxation for the 6 % PIB solution of Figure 6. ~" data; 9 model predictions [equation (91c) with k - 3].
,o1
09 Q8
.-. 0.7._~_
06.
.~
0.5-
.,.,
0.4-
~'ox~/~,,,,,
~
=4.34 S-t
.
=
.
_
\\\
~"-- 0.30.20.10.0 0
\ \\\ \
----
20
t(s)
4'0
6'0
Figure 8. Normal stress relaxation for the 6 % PIB solution of Figure 6. ~" data; 9 model predictions [equation (91d) with k - 3].
418
~=o = 4 . 3 4 S -I ~,00= 1.37S - I
/ *-0.6-
I ~,0 = 0.137 s - !
"T-
T
T
0
2O
40
T
6O
t(s)
Figure 9. Normal stress growth for the 6 % PIB solution of Figure 6. ~" data; 9 model predictions [equation (91f) with k = 3]. It can be seen that the steady values of 7/and ~l are accurately predicted by the model. Accurate transient data are difficult to obtain and the discrepancy between the experimental data and theoretical predictions in Figures 7 to 9 can be attributed to experimental error associated with the lack of stiffness of the Weissenberg rheogoniometer model R-18 used for these experiments. High rigidity is required to reduce coupling effects between the sample and the instrument which should be capable of very rapid acceleration. It was mentioned in section 5.1 that Liang and Mackay [38] have reported measurements of shear stress jumps. They defined a stress jump ratio R as follows R-(t, T0) = rl-(t, T0) / [11 (~/0) - TIs]
(92)
419 Substituting equations (9 l a, 9 l c) into equation (92) yields R (0+, ~0) = 1
[1
-I- (Tloo--Yls)/Z lip p=l
exp(-tp~/0)]
(93)
Figure 10 compares equation (93) with the experimental measurements of Liang and Mackay [38]. We observe that R - ( 0 + , ~)0) is a decreasing function of ~'o in agreement with the experimental data.
1.0
~"~'~|~-.,,~,
_
0.7
.,.-'..,
0.5
0.3
--
0.2
-
II
I
0. I 0. I
0.2
I
I
I
I
I
0.5
I
2
5
I0
I
'.l
20
i
I
50
Figure 10. Predicted and measured stress jump ratio for solutions in xanthan gum in a mixture of 75 mass % fructose and 25 mass % water. 0.05 mass %
r/1 = 13.48 Pa.s, r/2 = 2.28 Pa.s, 77oo= 0.83 Pa-s, t 1 = 1 0 . 1 3 s , t 2 = 0.28 s. 9 99 0.025 mass % r/~ = 3.13 Pa.s, q2 = 0.91 Pa.s, r/= = 0.64 Paos, tl = 6.02 s, t2 = 0 . 1 6 s. 0.01 mass % r/~ = 0.61 Pa.s, 772 = 0.33 Pa.s, r/= = 0.53 Paos, tl=4.69s, t2 = 0 . 2 1 s. e , II, A : experimental data for 0.05, 0.025, and 0.01 mass % respectively. Furthermore R - ( 0 + , 70) is less than 1 and this implies that r/= > r/s. This is in accord with our interpretation that q= represents the viscosity of the solvent and the stray and dangling segments.
420
From equation (91a) it is seen that q is a decreasing function of ~ and to predict shear thickening we can use equation (81). Chan Man Fong and De Kee [ 15] have assumed that k~ f l - kl f2 - kc d~/m
(94)
where d and m are constants and ~ is the dimensionless form of ~. Combining equations (81, 87, 94) yields 1"1 = 1"10 (1 + 2~kcd~/m) ] (1 + d~/m) ~
(95a)
= 2r10 [~ + ~ (1 + 2 ~, k~ d ~/m)] ] (1 + d ~ m) -- 1 /
k c
(95b)
(1 + d ~/m)
(95c)
where 7"/0and ~ are constants. Figure 11 compares the values of q and ~l given by equations (95a, b) with experimental values of a partially hydrolyzed polyacrylamide solution containing 2 g/2 of NaC1 reported by Ait-Kadi [50].
0.5 .-,.. tO
i
--2
,,
g.
,,
\A\
/
- 0.5
- 0.2
g. --3-
-0.1 I
0.1 I0 ~
,
, I,
,,,I
I
,
, I ,,,,I
I01
I
,
,
0.05
I0 z
Figure 11. Plot of viscosity (A) and primary normal stress coefficient (A) versus shear rate (~) for a partially hydrolyzed polyacrylamide solution containing 2 g/2, of NaC1. Model parameters are: 7/0- 0.50 Pa-s, kc = 0.040 s -1, ~ = 0.016 s, d = 0.13, m - 0.95.
421
6.2. M u l t i s t e p Flows In section 6.1 we have considered the case of a constant shear rate being applied or removed at time t = 0. It is also possible to apply finite increments of shear rates and measure the corresponding stresses. Xu et al. [51, 52] have tested the De Kee-Carreau model for various multi-rate-step flows. Figure 12 shows the calculated and the measured shear stress for a PDMS sample (a viscoelastic standard material provided by Rheometrics) in a concave step flow. In this flow a constant shear rate ~ is applied at t = 0 and is maintained until the steady state is reached. The shear rate is then halved and is kept at this value (~/2) for a finite time after which it is increased to its original value (~5). When the steady state is reached, the shearing is stopped.
8
~6 '0 4
_
2
ol 0
o
1 20
o
I 40
o
o
o
?
o
I 60
I 80
I00
120
140
t(s)
Figure 12. Comparison of model predictions ~ : equations (55a to 56c) with shear stress data (o) for concave steps with different levels. Upper Curve" ~ - 0.5, 0.25, 0.5, 0.0 s-1. Middle Curve: ~ - 0.25, 0.125, 0.15, 0.0 s-1. Lower Curve: ~ - 0.125, 0.06, 0.125, 0.0 s-1. The calculation is based on the measured spectrum from dynamic data for the PDMS sample with c = 1.3 and f0 = 1.45 for all steps.
422 Figure 13 compares the theoretical and experimental values of "Cyx for the same material in a reverse shear rate flow. In this flow the fluid is subjected to a constant shear rate ~ at time t = 0 and after the steady state is attained the shear rate is reversed in direction but equal in magnitude (-~). This cycle is repeated.
10"
5 x 10 3
A o O. I
-5xlO
3
0
_
I 20
I 40
I 60
I 80
t
I00
120
I 140
(s)
Figure 13. Shear stress for reversed shear steps. Model predictions equations (55a to 56c). The model parameters are those of Figure 12. It can be seen in Figures 12 and 13 that the De Kee-Carreau model adequately predicts the response of the material. Details of calculations and experimental procedures are given in Xu et al. [52]. 7. D I S C U S S I O N In this chapter, constitutive equations have been derived in a fixed coordinate system. The constitutive equations can also be deduced in a convected (body) coordinate system and via an energy method. Lodge [ 16, 53] has discussed the merits of adopting the convected coordinate system in the formulation of constitutive equations.
423
Several constitutive equations based on structural kinetics have been shown to be related to the transient network theory. The Marrucci model mentioned in section 4.3 is one of them. The other models are discussed in De Kee [54]. These structural equations can be quite useful at describing the rheological properties of complex materials. De Kee and Chan Man Fong [55] have explored the capability of a structural equation in various flow situations. It is shown in sections 6.1 and 6.2 that the De Kee-Carreau model, a shear rate dependent model, can predict satisfactorily the rheological properties of polymeric systems. However, the shear rate dependent models have been objected to on the ground that they do not recover the linear viscoelastic behavior in small amplitude oscillatory flows. This objection can be refuted by noting that when the strain is small, the creation and loss of junctions are due to Brownian motion [6, 22, 31] and are independent of the imposed flow. In equations (56a, b), fp and gp are constants and we recover the Lodge model which reduces to linear viscoelasticity in small amplitude oscillatory flows. Ahn and Osaki [43] have examined the cases where fp and gp a r e in turn functions of strain rate, strain, chain length, and effective strain which is defined as the ratio of the primary normal stress difference to twice the shear stress. They found that the predictions obtained by assuming fp and gp t o be functions of the effective strain agreed best with their experimental data. Since both the primary normal stress difference and the shear stress are functions of the shear rate, we may conclude that fp and gp are functions of the shear rate. Hinch [56] in a computer simulation of the uncoiling of a polymer molecule in an elongational flow has found that the stress generated is strain rate dependent and not strain dependent. Further support for the strain rate dependent equation is given in Carreau et al. [4] and Macdonald and Carreau [57]. At present the model parameters of the network theory are not related to the molecular structure of the material and it is desirable to seek such a connection. Lodge et al. [34] have proposed to relate the rates of creation and loss to the molecular weight. In Figure 14 we have plotted tl and t2 of the De Kee-Carreau model against 7/0 M w for various polymer solutions. The curves are parallel straight lines implying that for all p, tp is proportional to
(770 Mw)~, where n is a constant depending on the polymer solutions. We also deduce that t2, t3 .... are multiples of tl and De Kee [23] found that the factor 0.1 fits the data (see Figure 6).
424
9
9
I 1 1800 I 0"s K)*
i iO?
_
sOe
i
, , ,,,l iO t
%Mw
Figure 14. Correlation between the parameters
tp
and the product 77o M
w .
The reptation model (section 3.1) has been improved by the des Cloizeaux [58, 59] double reptation model and he has been able to deduce that the viscosity 77 is proportional to M 3.4, where M is the molecular weight. In the double reptation model, the reptation of a stress point is considered in addition to the reptation inside the tube. A stress point is a point of entanglement of two polymers and if either of the two polymers reptates out of the stress point, the stress disappears. The stress point is the junction in the network theory and this prompts Mead [60] to state that "the evolution of molecular constitutive equations has gone full circle in the short span of 35 years." In the next circle, we need to examine the process of loss and creation of junctions and the deformation of the segments. The works of Wagner and Schaeffer [49] and des Cloizeaux [58, 59] which combine both the reptation and network models need further exploration.
425
REFERENCES
Oldroyd, J.G., Proc. Roy. Soc., A200 (1950) 523; A245 (1958) 278.
~
2.
Barnes, H.A., J.F. Hutton, and K. Walters, An Introduction to Rheology, Elsevier, New York, NY (1989). Bird, R.B., R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1 - Fluid Mech., Second Ed., Wiley, New York, NY (1987). Carreau, P.J., D. De Kee, and R.J. Chhabra, Polymer Rheology: Principles and Applications, Hanser, New York, NY (1997).
.
,
.
.
~
Larson, R.G., Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, MA (1988). Bird, R.B., C.F. Curtiss, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2 - Kinetic Theory, Second Ed., Wiley, New York, NY (1987). Rouse, P.E., J. Chem. Phys., 21 (1953) 1272.
11.
de Gennes, P.G., J. Chem. Phys., 55 (1971) 572. Doi, M. and S.F. Edwards, J. Chem. Soc. Faraday Trans. II, 74 (1978) 1789, 1802, 1818; 75 (1979) 38. Doi, M. and S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, New York, NY (1986). Curtiss, C.F. and R.B. Bird, J. Chem. Phys., 74 (1981) 2016, 2026.
12.
Green, M.S. and A.V. Tobolsky, J. Chem. Phys., 14 (1946) 80.
13.
Lodge, A.S., Trans. Faraday Soc., 52 (1956) 120.
14. 15.
Yamamoto, M., J. Phys. Soc. Japan, 11 (1956) 413; 12 (1957) 1148; 13 (1958) 1200. Chan Man Fong, C.F. and D. De Kee, Physica, A218 (1995) 56.
16.
Lodge, A.S., Rheol. Acta, 7 (1968) 379.
17.
Bogue, D.C., Ind. Eng. Chem. Fund., 5 (1966) 253.
18.
Bird, R.B. and P.J. Carreau, Chem. Eng. Sci., 23 (1968) 427.
19.
Meister, B.J., Trans. Soc. Rheol., 15 (1971) 63.
20.
Carreau, P.J., Trans. Soc. Rheol., 16 (1972) 99.
21.
De Kee, D. and P.J. Carreau, J. Non-Newtonian Fluid Mech., 6 (1979) 127.
8.
9. 10.
426
22.
Chan Man Fong, C.F. and D. De Kee, J. Non-Newtonian Fluid Mech., 57 (1995) 39.
23.
De Kee, D., Ph.D. Thesis, University of Montreal, Quebec (1977).
24.
Kaye, A., Brit. J. Appl. Phys., 17 (1966) 803.
25.
Phan-Thien, N. and R.I. Tanner, J. Non-Newtonian Fluid Mech., 2 (1977) 353.
26.
Phan-Thien, N., J. Rheol., 22 (1978) 259.
27. 28.
Tanner, R.I., J. Non-Newtonian Fluid Mech., 5 (1979) 103. Acierno, D., F.P. La Mantia, G. Marrucci, and G. Titomanlio, J. NonNewtonian Fluid Mech., 1 (1976) 125.
29.
Jongschaap, R.J.J., J. Non-Newtonian Fluid Mech., 8 (1981) 183.
30.
Wagner, M.H., Rheol. Acta, 18 (1979) 33.
31.
Wagner, M.H. and S.E. Stephenson, J. Rheol., 23 (1979) 491.
32.
Petruccione, F. and P. Biller, J. Chem. Phys., 89 (1988) 577.
33.
Herman, W. and F. Petruccione, J. Rheol., 36 (1992) 1461.
34.
Lodge, A.S., R.C. Armstrong, M.H. Wagner, and H.H. Winter, Pure & Appl. Chem., 54 (1982) 1349.
35.
Giesekus, H., Viscoelasticity and Rheology, edited by A.S. Lodge, M. Renardy, and J.A. Nohel, Academic Press, New York, NY (1985) 157.
36.
Johnson, C.W. and D. Segalman, J. Non-Newtonian Fluid Mech., 2 (1977) 255.
37.
Gordon, R.J. and W.R. Schowalter, Trans. Soc. Rheol., 16 (1972) 79.
38.
Liang, C.H. and M.E. Mackay, J. Rheol., 37 (1993) 149.
39.
De Kee, D. and C.F. Chan Man Fong, J. Non-Newtonian Fluid Mech., 62 (1996) 307.
40.
Gerhardt, L.J. and C.W. Manke, J. Rheol., 38 (1994) 1227.
41.
Manke, C.W. and M.C. Williams, J. Rheol., 36 (1992) 1261.
42.
Hua, C.C. and J.D. Schieber, J. Non-Newtonian Fluid Mech., 56 (1995) 307.
43.
Ahn, K.H. and K. Osaki, J. Non-Newtonian Fluid Mech., 55 (1994) 215.
44.
Tanaka, F. and S.F. Edwards, Macromolecules, 25 (1992) 1516.
45.
Wang, S.Q., Macromolecules, 25 (1992) 7003.
46.
van der Brule, B.H.A.A. and Hoogerbrugge, P.J., J. Non-Newtonian Fluid Mech., 60 (1995) 303.
427
47.
Vrahopoulou, E.P. and A.J. Mc Hugh, J. Rheol., 31 (1987) 371.
48.
Marrucci, G., S. Bhargava, and S.L. Cooper, Macromolecules, 26 (1993) 6483.
49.
Wagner, M.H. and J. Schaeffer, Rheol. Acta, 31 (1992) 22.
50.
Ait-Kadi, A., Ph.D. Thesis, University of Montreal, Quebec (1985).
51.
Xu, Y.Z., D. De Kee, and C.F. Chan Man Fong, J. Appl. Polym. Sci., 55 (1995) 779.
52.
Xu, Y.Z., C.F. Chan Man Fong, and D. De Kee, J. Appl. Polym. Sci., 59 (1996) 1099.
53.
Lodge, A.S., Body Tensor Fields in Continuum Mechanics, Academic Press, New York, NY (1974).
54.
De Kee, D., Recent Developments in Structured Continua, edited by D. De Kee and P.N. Kaloni, Longman Scientific and Technical, Harlow, U.K. (1986) 150.
55.
De Kee, D. and C.F. Chan Man Fong, Polym. Eng. & Sci., 34 (1994) 438; 35 (1995) 1031.
56.
Hinch, E.J., J. Non-Newtonian Fluid Mech., 54 (1994) 209.
57.
Macdonald, I.F. and P.J. Carreau, J. Rheol., 33 (1989) 367.
58.
des Cloizeaux, J., Europhys. Lett., 5 (1988) 437; 6 (1988) 475.
59.
des Cloizeaux, J., Macromolecules, 23 (1990) 4678.
60.
Mead, D.W., J. Rheol., 40 (1996) 633.
429
CONSTITUTIVE DERIVATIVES
BEHAVIOR
MODELING
AND
FRACTIONAL
Chr. Friedrich ~, H. Schiessel b'c and A. Blumen b
"Freiburg Materials' Research Center, Freiburg University, Stefan-Meier-Str. 21, 79104 Freiburg, Germany bTheoretical Polymer Physics, Freiburg University, Rheinstr. 12, 79104 Freiburg, Germany ~Materials Research Laboratory, University of California, Santa Barbara, CA 93106, USA
1. I N T R O D U C T I O N
The simplest decay behaviors are exponential, such as the dielectric relaxation associated with Debye and the mechanical relaxation named after Maxwell. Exponential decays depend on a single mode (or, equivalently, a single characteristic time). But most relaxation processes are governed by a large variety of characteristic times, see references [1-4] for reviews, and vast types of decay patterns follow, most popular being stretched exponentials (Kohlrausch-WilliamsWatts [2,5,6]) and power law behaviors. In this review we focus on the cases in which the decay function follows a power law for reasonably extended time or frequency intervals. Note that too short time or frequency windows do not allow to distinguish between different decay patterns [7]. The transition from the glassy relaxation zone to the transitional zone in the case of stress relaxation of a glassy polymer is an example for power-law relaxation. Consider polyisobutylene at the reference temperature To = 25~ In Figure l(a) we reproduce, following reference [8] its shear storage and loss moduli G' and G" as a ftmction of the frequency o~; in Figure 1(b) we display the corresponding shear relaxation modulus G and the shear creep compliance J as a function of time. As is evident by inspection, the high modulus plateau is followed by a power law that covers about four decades in time (Figure 1(b)) and frequency (Figure l(a)). Then a second plateau zone (called the entanglement
430
10
-4
10
G' .~
-5
8~
~.
O
-6 "7t~
8
~- 7
-7
o
6
-8
o
5
-9
O
i
i
i
i
-4
0
4
8
-12
log co/s-1 Figure l(a). Storage modulus G'(co) and loss modulus G"(co) for polyisobutylene as a function of frequency. The data are from Tobolsky and Catsiff [8].
~.~
l
i
i
-8
-4
0
e~ @ o
-10 4
l o g t/s
Figure l(b). Relaxation function G(t) and creep function J(t) for polyisobutylene (as in figure 1(a)) as a function of time.
5
~o 3
2 O
,--
*
G~- PB302
.ce,
1
~
-5
G'- PB304
l
1
1
I
I
I
I
I
I
-4
-3
-2
-1
0
1
2
3
4
5
log (aTCO) / rad s "l Figure 2. Storage modulus G' and loss modulus G" of unmodified (PB300) and urazole modified polybutadiene (PB302 and PB304) vs. the reduced frequency aTa~. The molecular weight of all samples is Mw - 31 kg/mol; the samples 302 and 304 correspond to the 2 mol % and to the 4 mol % modification, respectively.
431
plateau) shows up. This transition from one plateau to another, generally via a power law, is characteristic for amorphous polymers. Furthermore, power law behaviors also appear in the terminal relaxation zone of polymers. In the case of a single relaxation time one has a sharp transition from the entanglemem plateau to the flow zone, which obeys a typical liquid-like behavior, namely G'oc co2 and G"oc co. Physically or chemically cross-linked polymers [9], polymers with star-, H-, or comb-like topologies (see e.g. [10-19]) show a more general pattern, namely an intermediate power-law domain with G' oc co~ and G" oc coa , where cz and 13 lie between zero and one. In Figure 2 we contrast the behavior of neat, monodisperse polybutadiene to that obtained by attaching to its backbone active groups, which are able to form H-bonds. While for neat polybutadiene the shear storage and loss moduli obey G'oc o92 and G" oc co, the moduli of the modified polymers follow more general power laws. The reason for this behavior is that the active groups form temporary, random links between the polymers, fact which renders the relaxation process multimodal and cooperative; this leads here to power laws. Such laws are not the hallmark of polymers only; vast classes of substances, which range from inorganic glasses to proteins show such behaviors [1], and we like to recall the early works of Meinardi et al. [20-22] on power law relaxation in metals, in rocks and in glasses. In this chapter we focus on the possibility to portray such complex viscoelastic features of polymers by means of fractional calculus, a formalism which turns out to be exceedingly well-suited for this purpose. To this effect we start here by illustrating, using a simple example, how fractional calculus comes into play as a result of the superposition principle. We start with G(t), the shear relaxation modulus of a linear system. Now G(t) is defined as the response of the shear stress r(t) to a jump in the shear strata y(t)-yoO(t) where O(t) is the unit step function. We assume that G(t) obeys a power law, i.e.
G(t)
-
r ( 1 - p)
(1)
with 0 _ 13< 1. In equation (1) E and ~, are constants and I-'(x) denotes the Gamma function [23]; for convenience we chose in equation (1) the prefactors in such a way as to conform to the mare body of the Chapter. Due to the lmearity of the system, the response of the stress to a previous history of deformations y(t) is given by the superposition integral [24,25]"
432
r(t) - i dr' G(t- t') dy(t'___))
(2)
dr'
Inserting G(t) given by equation (1) into equation (2) we have t
Eft
r(t)- r ( 1 - p )
~dt' (t- t') -p dy(t') at'
(3)
Now equation (3) can be rewritten in the following compact form r ( t ) - E 2 p dP?'(t)
dtp
(4)
'
in which ~/dt ~ denotes the fractional derivative of order 13 [26,27] (see section 2 for details). Equation (4) with 0 <13 < 1 interpolates between Hooke's law describing solid behavior (13 - 0), i.e.
r(t)- E?'(t)
(5)
and Newton's law describing fluid behavior ( / 3 - 1)
dr(t)
r(t)- r/---~
(6)
Equation (4) is an example of a rheological constitutive equation (RCE) with fractional derivatives. As we proceed to show below, a whole series of complex behavior patterns (including crossover situations as discussed above) can be described by relatively simple fractional RCEs. This Chapter is organized as follows In the next section we give a brief introduction to fractional calculus. After surveying the historical development of fractional RCEs in section 3, we discuss in section 4 the representation of such RCEs by mechanical analogues. In section 5 we highlight the usefidness of the formalism by applying the fractional Maxwell and Kelvin-Voigt models to a variety of polymeric systems. In section 6 we discuss more complex fractional models. Finally, we conclude with a summary in section 7 and relegate some important but more mathematical expressions to the Appendix.
433
2. FRACTIONAL D I F F E R E N T I A T I O N AND INTEGRATION The straightforward extension of classical calculus to its fractional counterpart is most readily visualized by using a notation which unifies ordinary integration and differentiation:
d~f(t) -
f(~)(t) ,
for a = 1,2,3,...,
f(t),
for a =O,
at a
(7)
dt_~f(t_~) , for a =-1,-2,-3,...
together with the Weyl integral [27]:
d~f(t) _ 1 ~ f(t') dt ~ F ( L a ) dt'(t-t,)~+l
(8)
One has only to realize that for a = - 1 , - 2 , - 3 , . . , equation (8) is nothing but Cauchy's formula for repeated integration [26,27]; hence for these values of equations (7) and (8) are equivalent. Now, the basic idea is that equation (8) can be readily extended to all tx < 0; this defines fractional integration. The extrapolation to the positive a-range, a > 0, is obtained by first picking an integer n, n > ct, then performing a fractional integration of order ct - n, followed by an ordinary differentiation of order n, i.e. [26,27]
d~ f (t) = d__"_( d~-" f (t)) dt ~ dt" k, ~-" J
(9)
Equation (9) defines fractional differentiation. Let us note that also another version of fractional calculus, the so-called Riemann-Liouville (RL)formalism [26,27], is of widespread use in rheology. In RL the lower limit of the integrals in equations (7) and (8) is set to 0. The RL formalism is particularly suitable for studying the transient response of materials after switching an external perturbation on, say at t = 0, so that v(t)= y ( t ) - 0 for t < 0. In this spirit the RL version can be understood as being the restriction of the Weyl formalism to a special class of initial value problems. In the following we will use the Weyl formalism; the translation into the RL version is straightforward.
434
Weyl's fractional calculus rams out to be algebraically very conveniem: the composition role for differentiation and integration obeys the simple form
d ~ d~f dV+~f = dt " dt ~ dt "+~
(10)
for arbitrary IXand v [27]. Furthermore, the Fourier transform oo
{f(t)} - f(o)) = I dt f(t)exp(-icot)
(11)
.-oo
turns the operation d~/d# into a simple multiplication [26,27]
{ d~f(t) } : (ico) ~f(co) dt ~
(12)
Let us illustrate the usefulness of these properties using the above-mentioned rheological example. A quick comparision of equation (3) with the Weyl integral, equation (8), leads immediately to the fractional stress-strata relation r(t)=EAp d ~-' dy(t)
dt a-~
dt
(13)
Furthermore using the composition rule, equation (10) with r = r - 1 and v = 1, equation (13) turns into equation (4), as stated in the previous section. The behavior of fractional derivatives under Fourier transformation is especially useful in determining the dynamical response functions. Let us consider the complex shear modulus
G* ( co) -icol;dzG(z)exp(-icoz )
(14)
which describes the response of the stress to a harmonic strain excitation y(t) =Yo exp(icot). From equation (2) it follows by the change of variables z - t - t' that 7(co) = G*(co) y(co). Using the multiplication rule, equation (12), one finds, say from equation (4)
435 G*(co) - 7(0)) / ~(co) - E(ico,;t)p .
(15)
From the complex modulus G*(co) follow the storage and the loss moduli, G'(co) = Re(G*(co) ) and G"(co)= Im(G*(co) ), the complex shear compliance J*(co)=l/G*(co) as well as storage and the loss compliances, J'(co) = Re(J*(co)) and J"(co) = -Im(J*(co)). Furthermore, we also consider the shear creep compliance J(t) (the response of the strain to a stress jump r(t) = r 0 tg(t)), which is given here by
J(t)-/-'(1+ fl) As a direct consequence of the multiplication relation, equation (12), we can easily derive the harmonic response functions G*(co) and J*(co) of a given fractional RCE. On the other hand, the analytical evaluation of the step response functions, namely of G(t) and J(t), turns out to be a hard task in many cases. Nevertheless these responses can be derived explicitly for a whole series of fractional RCEs of practical importance, cf. sections 5 and 6.
3. HISTORICAL SURVEY OF RCEs WITH FRACTIONAL DERIVATIVES
To our knowledge, the mathematically sound use of fractional differentiation to describe rheological properties of materials starts with Gemant [28,29]. He modified the Maxwell model by introducing the semiderivative of the stress (i.e. a = 1 and fl = 1/2 in equation (30), vide infra) in order to portray the properties of an 'elasto-viscous' fluid under oscillatory excitations. Nutting, on the other hand, pioneered power laws such as equation (1) to depict experimental results [30,31], although at that time the relation between power laws and fractional derivatives was not clear to the materials' science community. Other examples for the use of power laws are the works by P. Kobeko, E. Kuvshinskij and G. Gurevitch [32] (an expression used by them is equivalent to the Cole-Davidson function of dielectric relaxation) and by Alexandrov and Lazurkin [33]. There are even claims that stretched exponentials turn into power laws for exponents smaller than 0.4 [34], see, however also reference [7].
436
In rheology Scott-Blair et al. [35-38] made an extensive use of fractional integrodifferentiation to depict through power-laws the creep and relaxation in wide classes of materials. Their works rendered clear the intimate relationship between power laws and fractional calculus, and also introduced fractional generalizations of Newton's and Hooke's models, in which the fractional elements (FE) of this chapter were viewed as arising from 'quasi-properties', representing non-equilibrium states. Thus, in their notation, property X obeys generally an expression of the following type: X_
d'~r
(17)
Viewing z as stress and ~/as strain, the property X is an extension of the usual definition of viscosity, for which a = f l - 1 . Bosworth [39] made first considerations concerning the use of equation (17). After these basic pioneering works, rheology experienced a renewed surge of activity on fractional calculus starting at the end of the '60ies. Thus Slonimsky [40] applied the calculus to study rheological phenomena of polymers, a materials' class of growing importance. He described by a fractional relation the force acting on a polymer segment and the displacement experienced by it; in fact the operator used by him can be represented as an infinite series of simple fractional derivatives. Then Smit and deVries [41] investigated several material functions, such as the complex shear modulus of the fractional Kelvin-Voigt model (having a fractional derivative of the strain) and compared the results with experimental data. Next, Memardi and Caputo [20-22] developed the ideas of fractional calculus ft~her; they provided expressions for several material functions for the fractional standard solid model (see below), for which the order of fractional differentiation was the same for the stress and the strain. The evaluation of the relaxation in this case was an important step towards the understanding of the basic properties of the whole class of fractional models. These results were not widely noticed, and later rederived in rheology by Friedrich [11-13] and Nonnenmacher [42-44], who also solved the general case, in which the two fractional derivatives are of different order. Memardi and Caputo used their expressions [20-22] to describe relaxation measurements of rock materials, metals and glasses. A study of the fractional versions of the Maxwell and Kelvin-Voigt models, where only the derivative of the stress was replaced by a fractional counterpart was performed by Koeller [45]. He also considered the generalization to parallel
437
and serial arrangements, and also pointed out the close connection between the Rabotnov calculus [46] and fractional integrodifferentiation. The Rabotnov calculus, developed in the USSR in the '60ies, is based on the Rabotnov-operator 9t5~(f) (also called fractional-exponential operator), and was used widely for portraying relaxation phenomena in solid mechanics [46]. Friedrich and Hazanov poined out the relation between 9t-~(f) and fractional integrodifferentiation [47]. One has namely:
~ - ~ ( f ) = ~k=l( - f l ) * - '
d k(~
-')f dtk(a-1)
and
~ft~- a ( f ) -
d~-lf dtg-
(18)
1
w h e r e 0 _ a < l andfl>_0. More recently, one of the goals of research in our field was the detailed analysis of the fractional Maxwell and Kelvin-Voigt models. It was soon clear that the analytical determination of the response functions of these models is a mathematically difficult task (see below). Bagley and Torvik [48] connected the molecular theory of viscoelasticity to the fractional calculus, and showed that some aspects of the theory are mirrored by the fractional Kelvin-Voigt model. Friedrich and Heymann [15,16] pointed out the intimate relation between the order of differentiation in a fractional Kelvin-Voigt model and the degree of conversion for the sol-gel transition of a crosslinking polydimethylsiloxane. Later, in 1986, Bagley and Torvik [49] considered the fractional standard solid model in more detail. They established that this model, which contains two fractional derivatives- one of the stress and one of the strata- is compatible with thermodynamics if the order of both derivatives is identical. Thermodynamical admissibility of fractional order models was also analyzed by Friedrich for several models [12]. In 1983, Rogers [50] formulated general RCEs containing a large number of sums or products of fractional derivatives, both of the stress and also of the strain. The analysis of these models was restricted to those containing mainly two fractional derivatives, and dealt with (the more accessible) frequency domain. Another use of fractional derivatives was put forward by VanArsdale who generalized the Rivlin-Ericksen and White-Metzner tensors by including fractional orders of differentiation [51 ]. Only in 1991 did Friedrich [11] as well as G1Ockle and Nonnenmacher [42] succeed in obtaining G(t) and J(t) for the fractional Maxwell model containing two fractional derivatives of different order. They showed that this solution can be expressed through a special class of mathematical fimctions which will be
438
discussed later in this Chapter. The derivation in references [44,52] is based on an integral fractional representation of the standard solid model, whereas Friedrich's derivation uses the differential picture. Friedrich pointed out that the differential fractional version is thermodynamically admissible for wide ranges of parameters, whereas the parameter range of the fractional integral version is very restricted [17]. In subsequent works GlOckle and Nonnenmaeher clarified the mathematical basis of fractional calculus by pointing out its relation to the socalled Fox-functions [42]. Moreover, they explained the interrelation between fractional relaxation functions and the time-temperature superposition principle [52]. We note that these models were successfully applied to the description of the viscoelastic properties of filled polymers [53]. In general, much of the very recent work is characterized by the search for the physical background underlying fractional calculus. The relation of this calculus to the classical sprmg-dashpot representation of complex viscoelastic materials is a central aspect of the works by Schiessel and Blumen [54-58], as well as by Heymans and Bauwens [59-61]. These works showed how power law relaxation follows from exemplary arrangements of springs and dashpots. Schiessel and Blumen succeeded in explaining the process of polymer cross-linking on the basis of such mechanical networks; they also investigated terminated ladder arrangements which mimic pre- and postgel behavior [56]. Such mechanical analogs also showed the way how physically reasonable fractional RCEs can be constructed, aspects discussed by Schiessel, Metzler, Blumen and Nonnenmacher [58] and by Heymans [61]. By employing a formal analogy between linear viscoelasticity and diffusion in a disordered structure Giona, Cerbelli and Roman derived a fractional equation describing relaxation phenomena in complex viscoelastic materials. This analogy leads to a power law expression, which is in agreement with the experimental results [62]. At the moment we are still far from being able to relate in a reductionistic way the empirical models to a microscopic background. Nevertheless, as will become evident in section 4 of this Chapter, the representation of fractional derivatives through sprmg-dashpot analogs is helpful in understanding a series of phenomena which underlie the fractional calculus formalism. Another line of research, widely pursued nowadays, is the phenomenologically oriented, pragmatic modeling. An example for this is the work by Stastna, Zanzotto and Ho [63] who provided a relation between the Kobeko ftmction [32] and fractional calculus and used it to model rheological data of asphalts. Stastna et al. found a general inversion scheme which generalizes Rogers' results [50]; it allows the representation of the complex shear modulus in the form:
439
.oo I
+ o 1,tI
(19)
In equation (19) ~lk and ~2k are two sets of characteristic times. The authors of reference [63] succeeded in deriving the corresponding fractional RCE. Note that for m = 0 and n = 1 one recovers the Kobeko function [24]. We close this section by noting that nowadays fractional calculus is of widespread use in describing different rheological phenomena for wide classes of materials. As examples from the polymer literature we refer to [19,53,64-67]. In the following section we analyze under which circumstances spring and dashpot arrangements lead to power law relaxations.
4. THE REPRESENTATION OF RCE WITH FRACTIONAL DERIVATIVES BY MECHANICAL MODELS
Fractional RCEs may be formally derived from ordinary RCEs by replacing the first-order derivatives (d/dt) by fractional derivatives (~/dt ~) of non-integer order (0 < 13< 1). This formal procedure can, however, not assure a priori that the resulting expressions are always physically reasonable. This aspect was pointed out, for instance, in references [11,42,54]. Thus it is useful to have a procedure at hand that automatically guarantees mechanical and thermodynamical stability. As a first step Schiessel and Blumen [54-57] and Heymans and Bauwens [60] have demonstrated that the fractional relation, equation (4), can be realized physically through hierarchical arrangements of springs and dashpots, such as ladders, trees or fractal structures (which we will discuss in this section). The idea is that (disregarding for the moment the specific structure of the hierarchical constructions, these will be detailed later) equation (4) is obeyed by a fractional element (FE) which is specified by the triple (fl, E, 3,). We symbolize the FE by a triangle, as shown it Figure 3(c), where also its classical counterparts are depicted: the spring (cf. Figure 3(a)) obeying equation (5) and the dashpot (cf. Figure 3(b)) with the stress-strain relationship equation (6). Then, as a second step, more complicated RCEs can be constructed by combining two or more FEs in serial, parallel or more complex arrangements; this was proposed by Schiessel et al. [58] and by Heymans [61]. We will make use of this method in sections 5 and 6.
440
E (a)
__Lq (c)
(b)
Figure 3. Single elements: (a) elastic, (b) viscous and (c) fractional.
Eo E1 rio
E2 Eg/
r12 1//
Figure 4. A sequential spring-dashpot realization of the fractional element. Let us now consider different realizations of FEs. In reference [54] Schiessel and B lumen proposed a ladder-like structure with springs (having spring constants E0, El, E2,...) along one of the struts and dashpots (with viscosities r/0, ql, q2,-.-) on the rtmgs of the ladder (cf. Figure 4). As shown in reference [54] the complex modulus G*(co) admits a continued fraction expression, namely
G * ( c o ) - Eo (it~ -1~~ (it~ -~'~ ~(it~ -~ ~..., 1+ 1+ 1+ 1+
(20)
441 where we use a standard notation for continued fractions, [a/(b + ) ] f - a/(b + f ) , cf. reference (23). Choosing in equation (20) E o = E 1 - . . = E and r/0 - rh - . . . - r/it can be shown (by comparing terminating approximations of the continued fraction with the binomial series) that the complex modulus of the infinite arrangement is given by G*(co) - E (4(ic~ + 1)1/2 - 1 2(io92) -1
(21)
where we set )~-rl/E. For a)2 <<1 equation (21) reduces to the form G*(co) "~ E(ico2) ~/2 . Therefore having the same spring constants and viscosities for the whole arrangement one gets a complex modulus with 13= 1/2, and thus the ladder model with equal springs and dashpots leads to a FE with the parameters (1/2, E, A). To extend the ladder model to arbitrary values of 13 with 0 < 13< 1, a suitable distribution for the spring constants and viscosities has to be chosen; thus an algebraic k-dependence of the material constants of the form
E k oc Tlk oCk 1-213
(22)
leads to a FE specified by (fl, E,A) [541.
r
(5 2n
G :n+lI
....
6
' r n
(
Figure 5. Recursive construction of the iterated mechanical networks of Heymans and Bauwens [60].
442
Heyrnans and Bauwens [60] and Schiessel and Blumen [56] demonstrated that values of 13 with 13r 1/2 can also be attained by arranging equal springs and dashpots into a network with a more complex structure. Thus reference [60] focuses on a class of iterated networks which are generalizations of a fractal tree model; each FE (/3,E,2) can be represented by such an arrangement to any desired accuracy. We show in Figure 5 the recursive construction of the model: One starts from an element having the complex modulus G~; this element gets then represented by four elements, two of which have G~, the two others G2 and G 3 as moduli (of. Figure 5 with n = 1 ) so that the ensuing structure still has G~ as complex modulus. From Figure 5 it is easily seen that the relation between the complex moduli of the elements has to be G,, = ~G~,, Gz,+~. In principle the iteration proceeds for each element indefinitely. By this the complex modulus G~ of the whole arrangement is c,
-
- (c4G,
G6c
) ''4 -
..
(23)
At each level n of decomposition one replaces some of the elements G, (k = 2",...,2 "+~- 1 ) by either purely elastic elements, i.e. springs with spring constant E, or purely viscous elements, i.e. dashpots with viscosity rl. For these elements one has to replace Gk in equation (23) by E or icor/ and stop their further decomposition. Let iV, denote the number of springs at the nth level of decomposition and M, the corresponding number of dashpots. From equation (23) it follows: c,
- E
--
(24)
with /3- ~ " M,,/2" and 2 = ~7/E. Thus one can reach any preset, arbitrary 13, 0 < 13 < 1 to any desired accuracy by a sufficiently fine decomposition. As a special case, by taking G 2 = E and G3 = i corl we obtain a binary tree. This network, which we depict in Figure 6, is again a FE with the parameters (1/2,E,2). The advantage of the "binary decomposition" of elastic and viscous elements is its simple analytical tractability; in many cases, however, it leads to complicated networks whose properties are difficult to grasp intuitively. Thus it is also
443
desirable to work with networks whose structural properties (such as the connectivity) and the exponent 13are related in a more obvious fashion.
I
',
',
1 I
', I
1 1 I I
1 I
Figure 6. A special case of the iterated networks shown in Figure 5" binary spring-dashpot tree. In reference [56] Schiessel and Blumen showed that for given fractal networks this relation is straightforward when the springs and dashpots are suitably chosen. The construction starts by connecting each site r~ of the given network to neighboring nodes rj by equal springs (with spring constant E) and linking each r~ to the ground via a site-dependent dashpot (with viscosity r/~ - z(r~)r/, where z(r i) is the coordination number of node r~ ). Furthermore, the nodes' motion is perpendicular to the ground. To give an example we show in Figure 7 a section of the infinite mechanical network constructed from the Sierpinski gasket. The analogy to random walks can now be seen by comparing the stresses acting on node r i (whose displacement is ~,~) r/~ ~'~ (t) - E ~-"~ [7 j(t) - 7 ~(t)]
(25)
with the master equation
dt'(r ,t)
(26)
444
which governs the probability P(r~,t) of having a random walker at site r~. In Eqs. (25) and (26) the sums run over all nearest neighbors rj of r~. The transition probabilities
w o.
in Eq. (26) obey z(rj)w o
- w -
constant. Now one can identify
formally r/, y,(t) with P(r~,t). Furthermore, the probability for a random walker to return to the origin at time t follows an algebraic form: P(t)
oc t -d' '2
(27)
where d~ is the spectral dimension of the network [68]. A power law behavior of the complex modulus, Eq. (15), follows, with ~ 3 - 1 - d ~ / 2 (cf. Ref. [56] for details). The ladder model, Figure 2, is the special case of a one-dimensional lattice with d s - 1 and thus 13- 1/2; the Sierpinski gasket in 2d (cf. Figure 7) has d,. - 2 ln(3)/ln(5) and hence fl - 1 -ln(3)/ln(5) "- 0.317.
Figure 7. Section of the infinite spring-dashpot arrangement based on the Sierpinki gasket. An interpretation of the sol-gel transition in terms of such mechanical networks can be fotmd in Ref. [56}.
445
5. FRACTIONAL M A X W E L L AND KELVIN-VOIGT MODELS Figure 8(a) shows the standard Maxwell model, in which a spring and a dashpot are arranged in series [24,25]. Now this model can be generalized by replacing these elements by the FEs (a, E 1, 3,1) and (/3, E 2, 22), as shown in Figure 8(b). Due to the sequential construction the stress v(t) is the same for both elements. Their respective stress-strain relations are 7 , ( t ) - E~'2-~ ~ d - ~ z-(t) tit_---------~ ,
(28)
Y2(t)-E21~ d-#z-( t)dt_~
where both expressions follow from the composition rule, Eq. (10). Due to the sequential construction of the fractional Maxwell model (FMM), we have 7(t) - 71(t)+ 72 (t), from which it follows that r(t) + E13;~ d ~-p v(t) d~ E2/~f122 dt a-fl __ E1/~al
dt~r(t)
(29)
Let us assume without loss of generality that ~z > ]3. Eq. (29) can be further simplified by setting 2-(E~A~/Ez2~) v('~-~) and E - E~(A1/A) ~ . This leads to z'(t) + A ~
d~-#z-(t) dt.-P
=
EA a d~y(t) dt ~
(30)
which is the RCE of the FMM [11,43,58].
E q
(a)
,E1,21)
(b)
Figure 8. The (a) ordinary and (b) the fractional Maxwell model
446
When one arranges the spring and the dashpot in parallel one obtains the standard Kelvin-Voigt model [24,25], depicted in Figure 9(a). Its generalization, which has two FEs, is shown in Figure 9(b). Because of the parallel arrangement, the stresses acting on the two elements are. additive. Following a procedure similar to that for the generalized Maxwell model (vide supra), one finds for the RCE of the generalized Kelvm-Voigt model (FKVM) [58]
r(t)
-
E,t ~ d~y(t) dt ~
+mr p
dPy(t) dtp
(31)
where we have used the same parameters X and E as for the FMM.
I
(a)
I
I
(b)
I
Figure 9. (a) The Kelvin-Voigt model and (b) its fractional generalization. The basic material functions of the FMM and the FKVM were derived elsewere [58]; we will restrict ourselves to their succinct display and to the discussion of their basic properties. Table 1 presents for both models their fundamental response functions. In Table 1 E p(z) denotes the generalized Mittag-Leffier function of argument z, whose definition and basic properties are given in the Appendix. This function is of great importance for fractional RCEs, since it occurs in many material functions in the time domain. The harmonic response functions, equations (36) to (39), follow simply from the RCEs, equations (30) and (31), by means of the multiplication rule, equation (12). Due to the sequential (parallel) construction of the FEs in the FMM (FKVM), the response J(t) (G(t)) is simply the sum of the corresponding responses of its structural parts, which obey equation (16) (equation (1)).
447
Table 1 Material functions of the fractional Maxwell and Kelvin-Voigt models FMM
c(t)
E(t/'~')-flEa-fl,l-fl(-(t//],)a-fl) (32) E-1 (_~_)a .~-1
J(t)
G*(co) J*(co)
FKVM
V(1 + a)
E
(~)e
F(1 a)
F(1 - fl)
(34)
(35)
+ V(1 + fl~--~
(i c~
1 + (ia)2) ~-~
E-' (ico2) -'~ + E < (ico2) -p
(36) (38)
E(icoZ) '~ + E(ico2) p
E-1 1+
(37) (39)
Using the asymptotic expansions of the Mittag-Leffier function and focussing on the behavior of the dynamic moduli and of the dynamic compliances it becomes clear that both the FMM and the FKVM describe two power law regions which intersect each other at the characteristic time t = 2 . In the case of the relaxation ftmction G(t) the FMM describes first a flat decrease (of slope - f l ) followed by a steeper decrease (of slope - a ), whereas for the FKVM the flat decrease follows the steeper decrease. The compliance of the FMM shows a flat increase with slope 13 followed by a steeper increase with slope a; for the FKVM the situation is reversed. Thus it follows that in the time domain the FMM can be used to describe relaxation functions which start from a plateau region and decrease according to power laws, whereas the FKVM is able to describe the relaxation via a power law to a plateau. The situation is correspondingly reversed in the frequency domain. Note, however, that a so-called S-shaped transition between two plateaus which are connected by a power law cannot be described by means of FMM or of FKVM. An example for the use of FMM is presented in Figure 10, in which the dynamic moduli of polyisobutylene are redrawn from Figure l(a). These are compared to the results of FMM with the parameters given in Table 2. Note the
448
10r
/
[]
~-~ ~ 8
FMM
~
~.
6
41 -3
, 0
3
6
9
log co / s-1 Figure 10. Description of the data of Figure 1(a) by the fractional Maxwell model (solid lines) and the fractional Kelvin-Voigt model (dashed lines). Table 2 Material parameters used in Figure 10 log E
log L
c~
13
FMM
9.09
-8.36 0.583
0.593
FKVM
5.75
-2.96 0.057
0.885
good agreement between the experimental data and the analytical expressions for FMM. On the other hand FKVM is unable to describe the observed features even qualitatively. This is especially true for the G"(o~) data where FKVM can not follow the decline found experimentally in the range of very low frequencies. In Figure 11 (which is redrawn from Figure 2) we provide another example which stresses the possibilities offered by FMM to depict relaxation data. Whereas Figure 1 is concerned with polymers in the glassy and transitional zones, here we deal with relaxation from the plateau towards the terminal relaxation zone. Figure 11 shows the storage moduli, both of two modified polybutadienes (PB302 and PB304) and also of the neat polymer, on which the modifications were performed (PB300) [64]. Note that the FMM with the parameters given in Table 3 describes in a quantitative way the observed relaxation- the focus lies
449
here on the power law behavior of the modified PBs at very low frequencies. We note furthermore that both the storage and the loss module of the neat polymer melt are quantitatively depicted by this model.
r
FMM
o Q~
~b" J"
o * o
O
0
-5
I
I
-4
-3
....
G'- PB300 G" - PB300 G' - PB302 G'- PB304
I
I
I
I
I
I
l
-2
-1
0
1
2
3
4
log (aTO) / rad S"1 Figure 11. Description of the data of Figure 2 by the fractional Maxwell model (solid lines). Table 3 Material parameters used in Figure 11 . . . . . .
logE
log)~
~
13
PB300
5.52
-1.87
0.882
0.994
PB302
5.60
-0.344
0.553
0.632
PB304
5.48
0.720
0.478
0.590
The same holds true for polystyrene, as shown in the following example, taken from reference [65]. The situation is depicted in Figure 12; as is obvious, the storage moduli of polystyrene filled with different amounts of silicagel (N20) follow a power law in an intermediate region. Here the slope of the low frequency
450
3
k3 e~o o 2
PS 100 matrix
-2
~
Model 0.5 and 5 vol. % 1 and 4 vol. % 2vol. % 3 vol. %
I
I
I
I
-1
0
1
2
3
1og (aTo3) / rad s- 1
G'(a~co)
Figure 12. Storage modulus for aerosil-filled polystyrene. The solid lines give the description through the fractional Maxwell model. The aerosil concentration increases from below. Table 4 Material parameters used in Figure 12 log E
log ;~
c~
13
log G~
PS100
4.97
-0.877
0.812
1.000
-oo
0.5 vol. %
5.07
-1.06
0.808
0.988
1.54
1 vol. %
5.17
-1.08
0.749
0.907
2.62
2vol. %
5.26
-1.12
0.718
0.867
3.09
3 vol. %
5.40
-1.03
0.649
0.784
3.99
4 vol. %
5.47
-1.15
0.597
0.725
4.15
5vol. %
5.49
-0.814
0.613
0.768
4.85
451
plateau is zero; consequently the system can be modeled by adding a Hookean spring (of strength Ge) in parallel to the two FEs. The agreement between this model and the experimental data is very good. The Ulm group [44,53] used a similar model (the integral version of the FMM) to describe the complex moduli of filled polymers. On the other hand, the example of Figure 12 demonstrates the need to go to models having more than two FEs in order to describe more complex behavior patterns, such as S-shaped transitions. Such extensions are the focus of the next section.
6. MORE C O M P L E X MODELS 6.1 Models based on combinations of 3 fractional elements
Here we present more complex models, which are combinations of three FEs. We dispense with considering the fully parallel or the fully linear combination of FEs. We view as much more interesting the parallel arrangement of the FMM with a third FE or the sequential combination of the fractional Kelvin-Voigt model with a third FE. We call these models the fractional Zener model (FZM, also fractional standard solid model) and the fractional Poynting-Thomson model
E3
q
I (a)
I "
(b)
- I
Figure 13. The (a) ordinary and (b) fractional Zener model.
452 (FPTM), respectively. We stop to note that Heymans [61] has also analyzed more complex models, which are composed of FEs and additional conventional elements. The constitutive equations of FZM and of FPTM (these being our basic, simplest models containing 3 FEs) can be derived along the lines of the previous section. We start with the analysis of the FZM. The mechanical representation of the FZM is displayed in Figure 13; it consists of an FMM (with the same parameters as in Figure 8) in parallel to an FE specified by (z, E3,A3). The stresses on the left and right branches of the arrangement, x L and XR, obey the following stressstrain-relationships: v L(t) + 2 ~-p d~-/~r L(t) = E2 ~ d ~y(t) dt~-P
(40)
dt ~
of. equation (30) and, equation (4),
E32~ d'y(t)
rn(t)-
(41)
dt r
Both stresses add, which leads to the RCE of the FZM: z(t) + 2,~-~ d~-Pr(t) = E2 ~ d~7"(t) + Eo 2r d r y ( t ) + Eo 2y+~-~ dr+"-PY(t) dt~-P
dt ~
dt r
(42)
dtr+"-P
Here we set E o - E 3(23/2) r The FPTM consists of a serial arrangement of the FKVM (with the same parameters as in Figure 9) and an FE (y, E3,//'3) (cf. Figure 14). The RCE of this arrangement can be derived by noting that the deformations of the Kelvin-Voigt element and of the FE simply add, which leads to:
v(t) +,,
E
Eo
,,~a-y
d ~-r v(t) dt ~-r
+
E Ap_r d ~-r r(t) Eo
dt p-y
where we again set E o - E3(23/2,) y
d ~y(t)
= E2,~ ~
dt ~
+
EA p d~y(t) dt p
(43)
453
( y , E3,23)
q ( a , El,21)
(a)
E2,~)
(b)
Figure 14. The (a) Poynting-Thomson model and (b) its fractional generalization.
Table 5 Material functions of the fractional Zener and the Pointing-Thomson models FPTM
FZM
(v) go 2C;-2~am~vl(Gat) G(t)
G F ~ ( t ) + G~(t)
(44)
(45) E
+~GvE(t), E+E o
2
for 3' - [3
"-'l ,1FKVM J(t)
+ ~Ej o~ ( t ) , E+E o
(46)
JFKVM(t) + Jm(t)
(47)
for r -
G*(co)
G*Mu(co) + E o (ico2) r
(48)
(ic~162 + (ic~ E- 1-[- E;l(i(.0,~,)~z_.y --[-E;l(i(_.O/],)fl_y
(49)
J*(co)
(~)-~'-~+(~'~)-~ E + Eo(i~:k) + Eo(i~X) ~-~
(50)
J ~9v M (co) + Eo ~(ico,r -~
(51 )
454
We again summarize (now in Table 5) the relaxation material functions of the two models. For the detailed derivation of the expressions given in the Table we refer the reader to [58]. Here we note only that the dynamic moduli as well as the dynamic compliances follow directly from the corresponding RCEs by means of the multiplication rule, equation (12). Moreover, the relaxation modulus (creep compliance) of the FZM (FPTM) is simply the sum of the moduli (compliances) of the corresponding substructures. On the other hand, the creep compliance (relaxation modulus) of the FZM (FPTM) is only known analytically for the special cases y -cx or ~/-13, cf. [58]; in Table 5 the case ~/-13 is presented. The constants
C,
=(E/(E-J-Eo))
and
C2
of Table 5 are
C,-(Eo/(E+Eo)) v(~-~) and C z
1/(cr-fl) "
The FZM can describe S-shaped transitions from one plateau to another via a power law. For instance, it can be shown [58] that G(t) displays three regimes for0_<7_>E0: t -~
for t<<)~
(52)
G(t) oc t -~ for )~ << t << )~, t -Y where
for )~ <
)~, ~(E/Eo)V(~-~)~.
In section 5 we have already presented an example
where the FZM fits experimental data very well for the special case ~/- 0 (cf. Figure 12). 6.2 Other models
Another approach which extends the use of FE-based models is pragmatic. The previous sections dealt with the exact solutions (relaxation functions) of models created by combining an increasing number of FEs in parallel and/or in series. The approach which we discuss here is the direct empirical modification (based on experimental findings) of the basic relaxation functions of the FE models. We now provide examples taken from the literature for such modifications.
6.2.1 Modification of the relaxation function As already mentioned (see Figure 2)the G'(o3) and G"(o3) of monodisperse linear polymers show a sharp transition from a G' oc o) 2, G" oc co behavior to a
455
G'oc co~, G"oc co-~ behavior. In the example of Figure 2 13 "~ 1/4 and a is positive and close to zero, hence G' displays a quasi-plateau. In the corresponding relaxation experiment the stress decays then first according to a power law with slope - a , which is followed by an exponential relaxation at longer times. Figure 15 shows the relaxation function G(t) of a modified polybutadiene [64] where the appearance of an intermediate power-law domain is clearly evident. In Figure 15, due to experimental limitations, the plateau can not be seen directly but it can be infered from its dynamic modulus (especially from the range aTco > 10 ), displayed in Figure 16. Now, fluid behavior is associated with the termination of all relaxation processes, say at the time )~m, and thus )~m is a natural cut-off. This can be achieved phenomenologically by multiplying the corresponding decay function with an exponential. As an example, starting from GF~(t ) (cf. equation (32)) in reference [18] the following 6-parameter relaxation function was put forward
c(o-
~
Co
exp(-t/~m)
.
(53)
In equation (53) Go is a constant, 2 and 2~ are time constants (with 2 m >> 2) and E~,~_~+~ denotes the generalized Mittag-Leffier function (c~ and 13 are related to the parameters aft ~ and f l ~ of equation (32)via a = a ~ - f l F ~ and fl = a f t ) . From the asymptotic expansion of the Mittag-Leffier function for fl < 1 it follows that for t << 9~ (plateau regime) G(t) oc t ~-~ and for )~ << t << )~m (intermediate regime) G ( t ) o c t -p . Moreover, thermodynamic stability requires that the power law exponents obey 0 < a _ A m and hence induces a fluid-type behavior. A detailed discussion of this model is given in references [10] and [18]. We only provide here the main formulae. Thus the dynamic moduli associated with equation (53) are
G*( o)-Go
+ Z/ m)
(54)
I nt- (lO)/], "Jr"/],//],m) Ot
and
J*(co)
(55)
-
G O io)/~ (i0)/9~ "Jr"2/~l~m) fl-1 "
456
Especially the dynamic moduli turn out to be particularly useful, since they lead to equations which are easy to use for fits to the data. Figure 16 shows exemplarily both the data and the obtained fits. For the PB202 sample we obtain f l - 0.578 and log/~ m = 2.48. This is in good agreement with the parameters /3'- 0.556 and log2~ - 2 . 2 4 which are determined by fitting equation (53) to the relaxation function of the sample displayed in Figure 15. In Figure 15 the glassy short-time behavior (with the plateau modulus Go ) is not in the experimentally accessible range. In such a case the data may be fitted by a simpler relaxation function, which one obtains from equation (53) by letting 2 ~ 0. This leads to (56)
G(t) - Go1. (t/2~) -p exp(-t/2m) r ~ _ . .
with Go~- rlo/2m,
where
q0 is the zero-shear viscosity,
rio-JoG(t) dt 9
Equation (56) is an empirical relaxation function of widespread use; thus its dynamic moduli were often used in fitting data (cf. reference [24]).
,~ I
PB 202
10s . ~ .
04
.~
Tref= 30 ~
_
10 3
102 10l
10 ~
101
10 2
10 3
10 4
t/s Figure 15. Relaxation function G(t) of a modified polybutadiene (PB 202, Mw = 21 kg/mol). The solid line is equation (56) with log G0l = 4.68 Pa, log km = 2.24 s and 13= 0.556.
457
10 6
10 5
r
10 4
~
10 3
Tref= 30 ~
9
PB200 PB202 PB204
o 10 2
Model
d
101 1@3 10-2 10-1 ,
, ,,,,,,1
,
,
,,i,,,I
,
, ,lll,ll
,L I III
10 0
I'll
101
I
i iiilul
!
10 2
i lllllll
,
10 3
I
I IIIll
10 4
a T O / r a d s -1
Figure 16. Storage modulus Gt(aT(O) for unmodified (PB200)and modified (PB202 and PB204) polybutadiene. The solid lines represent the fit of the generalized fractional Maxwell model according to equation (54) to the data. As another example we present in Figure 17 the dynamic moduli of a partly cross-linked polybutadiene, PB 18, in the pregel stadium [9,69]. PB 18 is a polymer with a branched structure; it is gratifying to note that also in this case fractional relaxation provides a very good description of the experimental observations. Let us now turn to the RCE associated with the relaxation function, equation (53). Starting from equation (54) it can be shown by means of the multiplication rule, equation (12), that the RCE has the following form [10]:
x(t) + Ue -'/x~ d~ [et/X~'c(t)] -G~ e-'/x" dt ~-~d~-[e'/x~ l dy(t) " dt ~
(57)
Equation (57) contains an explicit time dependence that can be removed using the product rule for fractional differential operators (cf. equation (5.5.2) of [26]). In this way, the RCE, equation (57) takes the form [10]"
458
oc
oo
r(t) + w ~ Z ( ~ ) I I , " ,=o
d~'rCt----~) " dt~' -
w i t h a, - c t - i, bj - f l - j
G O w s~
(58)
)t b' dbJT(t----~)
Z(Pf) ;=o
m
dt b,
and w - 2/2 m .
A RCE with a structure similar to equation (57) was given in reference [63]. A mechanical model composed of spring, dashpots and fractional elements is not k n o w n f o r this R C E .
10 6
_
10 5 -
10 4
-
o
G'
t~
G t~
10 3 -
10 2
Model
-
_
10 ]
101
I
I
I
,
I I till
10 2
i
i
~ ,Ittl
j
,
,
, I,l,l
10 3
10 4
j
~
J LIHl
10 5
aTO) / r a d s -1 Figure 17. Storage G'(arco ) and loss modulus G"(arCO) for polybutadiene (PB18), for details see reference [69]. The solid lines represent the fit of the generalized fractional Maxwell model according to equation (54) to the data.
459
I
I
I
I
I
I 0000(
QFIDI3D
[
/ 9
9 _
O
PB-M2
o~
2-
o
o
G'
n
G II
I
o
O
oo
l
I
-4
-3
O I O
-2
1 I
I
I
I,
-!
0
]
2
log (aT0)) /rad
3
s -1
Figure 18. Storage G'(arCO) and loss modulus G"(arco ) for an unmodified (PB) and a sidechain modified polybutadiene (PB-M2), for details see reference [70]. The solid lines represent the fit according to equation (59) to the data.
6.2.2 Modification of G*(m) and of associated materialfunctions Another modification of polybutadienes leads to yet another behavior in the vicinity of the terminal region; instead of approaching the flow region after leaving the power law zone, a second power law is observed. We present in Figure 18 this behavior, which was the research object of reference [70]. Figure 18 shows the dynamic moduli of a polybutadiene which carries sidechains in a comb-like manner; the sidechains are mesogens, and are attached to the backbone via a flexible spacer. The polymer is in the isotropic state, but its branched molecular architecture and the interactions between the mesogens lead to a dynamic behavior quite different from that of linear chains, as can be seen from Figure 18. Here the necessary modification to the material functions starts from the complex modulus G*(m). The two power law regions at the lowest frequencies
460
are depicted by the FMM; multiplying G'z,~ (co) with an additional relaxation term leads to: G*(co)- E
(ic~ 1 + (/(_O/~,1)a-fl
1 1 + (i(__O,~,2)~"
(59)
Now the other relaxation functions of the model can be calculated following the procedures outlined above. The relaxation modulus G(t) and the creep compliance J(t) turn out to be complex expressions, which can be found in reference [70]. Figure 18 shows that equation (58) reproduces successfully the data, as indicated by the fitting curve. We close by giving the RCE belonging to equation (58) [70]: d ~r(t) dt ~
d ~r(t) d a" r(t) + Xr2 ~ + 2~2r2 = E2~ at
dPr(t)
(60 /
Hence the mechanical representation of this model is a combination of 4 FEs in series; these are of the form (x, E, Z,), with x - 13, 13- ~, 13- 7 and 13- a - 7.
7. CONCLUSIONS AND O U T L O O K In this chapter we have shown how fractional calculus allows a physically sound generalization of classical models from the linear theory of viscoelasticity (see also the review by A. I. Leonov in this book). From a mathematical point of view, the operation of fractional mtegrodifferentiation is well defmed and can be easily handled in Fourier or Laplace space. Viewed technically, integrodifferentiation allows, for instance, to interpolate smoothly between Hooke's and Newton's laws. Moreover, we have introduced the simplest form of a fractional rheological model, which we call fractional element (FE), and we have provided for it several mechanical analogs, namely arrangements made out of springs and dashpots. In these (infinite) networks the order of fractional integration or differentiation can be adjusted in several ways; say, by varying the material constants of the springs and dashpots involved, or by changing the structure of the arrangement. Furthermore, we have shown how parallel or serial combinations of FEs lead to more complex models; in particular we have studied extensions of the Maxwell,
461
the Kelvin-Voigt, the Zener and the Poynting-Thomson models. The relaxation patterns of these models can be used to fit the experimental results for large classes of materials. Particularly noticeable candidates are polymeric materials which display ramified structures (such as cross-linked polymers) or whose dynamics is characterized by cooperativity (i.e. glasses). Using fractional elements one can tailor viscoelastic models with given properties, while keeping the number of parameters involved relatively low. The representation of generalized viscoelastic models by fractional analogues also allows a deeper insight into the physics behind fractional stress-strain relations. Nevertheless, we are still far from a reductionistic understanding of the fractional relaxation laws. Here, as well as in related areas, we expect much additional work in the coming years.
ACKNOWLEDGEMENT The support of the DFG through SFB 428 and of the Fonds der Chemischen Industrie is gratefully acknowledged.
APPENDIX Here we display some mathematical relations which are helpful in understanding the physical models of the Chapter. We start with the generalized Mittag-Leffier function which occurs, for instance, in the relaxation function of the fractional Maxwell model (equation(32)) or in the creep compliance of the fractional Kelvin-Voigt model (equation (35)). The generalized Mittag-Leffier function E,,~(z) is defined by [71] z k
, with la > 0 and v > 0.
The special case v - 1 yields the usual Mittag-Leffler function
(61)
E~,(z).
On the
basis of this definition, the generalized Mittag-Leffier functions for some special cases, in which la and v equal 0.5, 1 or 2, follow readily"
462
Eo.5,1(z ~
- e ~ erfc(_z ~ - - e ~ erfc(-z ~
1- 2 z
z El,l ( Z 1 ) - e z
(62) E1,2(zl) - l [e: - l] z
E2,1 (z 2 ) - c h ( z )
E2,2 (z2) _ l__sh(z) . z
All generalized Mittag-Leffier functions increase monotonically for z > 0. To obtain monotonically decreasing functions one goes to the domain of negative z. In the parameter range /~, v ~(0,1] the following asymptotic expansions for z >> 1 are of interest [26] la
z -2
for l a ~ l
(63)
V(1 1
~: F ( v - l a ) z
E,.~(-z)
-1
for v > la.
Consider now the relaxation function G(t) of the FMM, equation (32). Here the generalized Mittag-Leffier function has the parameters l a - a - 13 and v - 1-13. From definition (61) and the asymptotic expansion (63) one obtains for the two power-law regimes: G ( t ) ~: t -~
{
t -~
G ( t ) ~:
t
(a
for
t <<
for
t >> ~,.
(64)
1)
In Figure 19 we show the relaxation function of the FMM, equation (32), for r - 1 and 13- 0.5. For the sake of comparison two other relaxation functions are added, which are different from the double power-law relaxation in the long time range.
463
i
i
i
i
i
o-
i
i
i
" ~176176
~,-2
~-4
i
/\\
O -6 -8 -
(a) x "0"5 e -x /
/
-
(b) x "0"5 e"xO5 -10
i
-4 -3
i
i
-2 -1
~
,
~
,
i
0
1
2
3
4
5
log x Figure 19. Dimensionless relaxation function G ( t ) / G o vs dimensionless time t / 2 for three models corresponding to different asymptotic behaviors (a) exponential, (b) stretched exponential and (c) power-law.
,~,
0
0
"~ -2 tzr 0
-4
-6
-
F(0.5) x -05 E0.5,0.5(-x 05) -8
I
I
I
I
I
-5 -4 -3 -2 -1 0
I
I
I
I
1
2
3
4
5
log x Figure 20. Comparison of a Mittag-Leffier function, lhs. of equation (65), with one of its Pad6 approximants, rhs. of equation (65).
464 The use of the generalized Mittag-Leffier function in numerical calculations is hampered due to the sometimes slow convergence of the series in equation (61). This can be taken care of by using a functional approximation; it turns out that Pad6 approximants allow to evaluate the generalized Mittag-Leffier functions almost exactly and without much effort. The procedure is described in [13]; in the lowest order approximation it yields exemplarily for cz = 1 and 13= 0.5:
r(0.s) Eo.,,o.,(-x
l+2x
(65)
The result is presented in Figure 20; note that deviations from the exact result are small and are restricted to the transition region between the power-law domains.
REFERENCES
1. R. Richert and A. Blumen (eds.), Disorder Effects on Relaxational Processes: Glasses, Polymers, Proteins, Springer, Berlin, 1994. 2. A. Blumen, in: Th. Dorfm011er and G. Williams (eds.), Molecular Dynamics and Relaxation Phenomena in Glasses, Springer, Berlin, 1987, p. 1. 3. J. Klafter, R. J. Rubm and M. F. Shlesmger (eds.), Transport and Relaxation in Random Materials, World Scientific, Singapore, 1986. 4. K. L. Ngai and G. B. Wright, Relaxation in Complex Systems, Naval Research Lab., Springfield, VA, 1984. 5. G. Williams and D. C. Watts, Trans. Faraday Soc., 66 (1970), 80; G. Williams, Adv. Polym. Sci., 33 (1979), 59. 6. F. Kohlrausch, Pogg Ann. Physik, 29 (1863), 337. 7. J. Friedrich and A. Blumen, Phys. Rev. B, 32 (1985), 1434. 8. A.V. Tobolsky and E. Catsiff, Journal of Polymer Science, 19 (1956), 111. 9. M. Mours and H.H. Wimer, Chapter in this book. 10. Chr. Friedrich and H. Braun H., Colloid Polym. Sci., 272 (1994), 1536. 11. Chr. Friedrich, Rheol. Acta, 30 (1991), 151. 12. Chr. Friedrich, in: J. Casas-Vazquez and D. Jou (eds.), Rheological modelling: Thermodynamical and Statistical Approaches, Lecture Notes in Physics No. 381, Springer, Berlin, 1991, p. 321. 13. Chr. Friedrich and H. Braun, Rheol. Acta, 31 (1992), 309. 14. Chr. Friedrich and S. Hazanov, in: Advances in Structured and Heterogeneous Continua, Allerton Press Inc., New York, 1994, p. 173.
465 15. Chr. Friedrich and L. Heymann, J. Rheol., 32 (1988), 235. 16. Chr. Friedrich, L. Heymann and H.-R. Berger, Rheol. Acta, 28 (1989), 535. 17. Chr. Friedrich, J. Non-Newt. Fluid Mech., 46 (1993), 307. 18. Chr. Friedrich, Phil. Mag. Letters, 66 (1992), 287. 19. Chr. Friedrich, H. Braun and J. Weese, Polym. Eng. Sci., 35 (1995), 1661. 20. M. Caputo and F. Meinardi, Riv. Nuovo Cimento (Ser. 2), 1 (1971), 161. 21. F. Meinardi and E. Bonetti, Progress and Trends in Rheology II, Suppl. Rheol. Acta, 26 (1988), 64. 22. F. Meinardi, Fractional relaxation in anelastic solids, J. All. & Comp., 211/212 (1994), 534. 23. M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, New York, 1972. 24. N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior, Springer, Berlin, 1989. 25. I.M. Ward, Mechanical Properties of Solid Polymers, Wiley, Chichester, 1983. 26. K.B. Oldham and J. Spaniel The Fractional Calculus, Academic, New York, 1974. 27. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. 28. A. Gemant, Physics, 7 (1936), 311. 29. A. Gemant, Phil. Mag. 25 (1938), 540. 30. P.G. Nutting, J. of the Franklin Institute, 191 (1921), 679. 31. P. G. Nutting, Proc. Amer. Soc. Test. Mater., 21 (1921), 1162. 32. P. Kobeko, E. Kuvshinskij and G. Gurevitch, Techn. Phys USSR, 4 (1937), 622. 33. A. P. Alexandrov and Yu. S. Lazurkin, J. Tech. Fiz., 9 (1939), 1250; 1261. 34. A. LeMehaute, L. Picard and L. Fruchter, Phil. Mag. B, 52 (1985), 1071. 35. G.W. Scott-Blair and F.M.V. Coppen, Amer. J. Psychol, 56 (1943), 234. 36. J.E. Caffyn and G.W. Scott-Blair, Nature, 155 (1945), 171. 37. G.W. Scott-Blair, B.C. Veinoglou and J.E. Caffyn, Proc. Roy. Soc. Ser. A, 189 (1947), 69. 38. G.W. Scott-Blair and J.E. Caffyn, Phil. Mag., 40 (1949), 80. 39 R.C.L. Bosworth, Nature, 157 (1946), 447. 40 G.L. Slonimsky, Journal of Polymer Science: Part C, 16 (1967), 1667. 41 W. Smit and H. de Vries, Rheol. Acta, 9 (1970), 525. 42 W.G. G10ckle and T.F. Nonnenmacher, Macromolecules, 24 (1991), 6426. 43 T.F. Nonnenmacher, m: J. Casas-Vfizquez and D. Jou (eds.), Lecture Notes in Physics No. 381, Springer, Berlin, 1991, p. 309.
466
44. T.F. Nonnenmacher, W.G. G1ockle, Phil. Mag. Letters, 64 (1991), 89. 45. R.C. Koeller, J. Appl. Mech., 51 (1984), 299. 46. Y.N. Rabotnov, Elements of Hereditary Solid Mechanics, Mir Publishers, Moscow, 1980. 47. Chr. Friedrich and S. Hazanov, in: D.A. SiNner and Y.G. Yanovsky (eds.), Advances in Structured and Heterogeneous Continua, Allerton Press, New York, 1994. 48 R.L. Bagley, J. Rheol., 27 (1983), 201. 49. R.L. Bagley and P.J. Torvik, J. Rheol., 30 (1986), 133. 50 L. Rogers, J. Rheol., 27 (1983), 351. 51 W.E. VanArsdale, J. Rheol., 29 (1985), 851. 52 W.G. G1Ockle, T.F. Nonnenmacher, Rheol. Acta, 33 (1994), 337. 53 R. Metzler, W. Schick, H.-G. Kilian and T.F. Nonnenmacher, J. Chem. Phys., 103 (1995), 7180. 54. H. Schiessel and A. Blumen, J. Phys. A, 26 (1993), 5057. 55. H. Schiessel, P. Alemany and A. Blumen, Progr. Colloid Polym. Sci., 96 (1994), 16. 56. H. Schiessel and A. Blumen, Macromolecules, 28 (1995), 4013. 57. H. Schiessel and A. Blumen, Fractals, 3 (1995), 483. 58. H. Schiessel, R. Metzler, A. Blumen and T.F. Nonnenmacher, J. Phys. A, 28 (1995), 6567. 59 J.-C. Bauwens, Colloid Polyrn. Sci. 270 (1992), 537. 60 N. Heymans and J.-C. Bauwens, Rheol. Acta, 33 (1994), 210. 61 N. Heymans, Rheol. Acta, 35 (1996), 508. 62 M. Giona, S. Cerbelli and H.E. Roman, Physica A, 191 (1992), 449. 63 J. Stastna, L. Zanzotto and K. Ho, Rheol. Acta 33 (1994), 344. 64 M. Odenwald, H.-F. Eicke and Chr. Friedrich, Colloid Polym. Sci., 274 (1996), 568; M. Odenwald, Diploma-Thesis, Freiburg 1993. 65. Chr. Friedrich and H. Dehno, m: Progress and Trends in Rheology IV, Steinkopffverlag, Darmstadt, 1994, p. 45. 66. L.I. Palade, V. Vemey and P. Attane, Rheol. Acta, 35 (1996), 265. 67. S. Hellinckx, Colloid Polym. Sci., 275 (1997), 116. 68. S. Havlin and A. Btmde, in: A. Bunde and S. Havlm (eds.), Fractals and Disordered Systems, Springer, Berlin, 1991, p. 97. 69. M. Mours and H.H. Winter, Macromolecules, 29 (1996), 7221. 70. Chr. Friedrich, Acta Polymerica, 46 (1995), 385. 71. A. Erd61yi (ed.), Bateman Manuscript Project, Higher Transcendental Functions, Vol. III, Mc.Graw-Hill, New York, 1955.
467
T H E K I N E T I C T H E O R Y OF D I L U T E S O L U T I O N S OF FLEXIBLE
POLYMERS:
HYDRODYNAMIC
INTERACTION
J. R a v i P r a k a s h a
~Department of Chemical Engineering, Indian Institute of Technology, Madras, India, 600 036 1. I N T R O D U C T I O N
The rheological properties of dilute polymer solutions are commonly used in industry for characterising the dissolved polymer in terms of its molecular weight, its mean molecular size, its chain architecture, the relaxation time spectrum, the translational diffusion coefficient and so on. There is therefore considerable effort world wide on developing molecular theories that relate the microscopic structure of the polymer and its interactions with the solvent to the observed macroscopic behavior. In this chapter, recent theoretical progress that has been made in the development of a coherent conceptual framework for modelling the rheological properties of dilute polymer solutions is reviewed. A polymer solute molecule dissolved in a dilute Newtonian solvent is typically represented in molecular theories by a coarse-grained mechanical model, while the relatively rapidly varying motions of solvent molecules surrounding the polymer molecule are replaced by a random force field acting on the mechanical model. The replacement of the complex polymer molecule with a coarse-grained mechanical model is justified by the belief that such models capture those large scale properties of the polymer molecule, such as its stretching and orientation by the solvent flow field, that are considered to be responsible for the solution's macroscopic behavior. An example of a coarse-grained model frequently used to represent a flexible polymer molecule is the bead-spring chain, which is a linear chain of identical beads connected by elastic springs. Progress in the development of molecular theories for dilute polymer solutions has essentially involved the succesive introduction, at the molecular level, of various physical phenomena that are considered to be responsible
468
for the macroscopic properties of the polymer solution. For instance, the simplest theory based on a bead-spring model assumes that the solvent influences the motion of the beads by exerting a drag force and a Brownian force. Since this theory fails to predict a large number of the observed features of polymer solutions, more advanced theories have been developed which incorporate additional microscopic phenomena. Thus, theories have been developed which (i) include the phenomenon of 'hydrodynamic interaction' between the beads, (ii) try to account for the finite extensibility of the polymer molecule, (iii) attempt to ensure that two parts of the polymer chain do not occupy the same place at the same time, (iv) consider the internal friction experienced when two parts of a polymer chain close to each other in space move apart, and so on. The aim of this chapter is to present the unified framework within which these microscopic phenomena may be treated, and to focus in particular on recent advances in the treatment of the effect of hydrodynamic interaction. To a large extent, the notation that is used here is the same as that in the treatise Dynamics of Polymeric Liquids by Bird and co-authors [2]. 2. T R A N S P O R T
PROPERTIES
OF D I L U T E S O L U T I O N S
2.1. D i l u t e s o l u t i o n s A solution is considered dilute if the polymer chains are isolated from each other and have negligible interactions with each other. In this regime of concentration the polymer solution's properties are determined by the nature of the interaction between the segments of a single polymer chain with each other, and by the nature of the interaction between the segments and the surrounding solvent molecules. As the concentration of polymers is increased, a new threshold is reached where the polymer molecules begin to interpenetrate and interact with each other. This threshold is reached at a surprisingly low concentration, and heralds the inception of the semidilute regime, where the polymer solution's properties have been found to be significantly different. Beyond the semi-dilute regime lie concentrated solutions and melts. In this chapter we are concerned exclusively with the behavior of dilute solutions. A discussion of the threshold concentration at which the semi-dilute regime in initiated is helpful in introducing several concepts that are used frequently in the description of polymer solutions.
469
A polymer molecule surrounded by solvent molecules undergoes thermal motion. A measure of the average size of the polymer molecule is the root mean square distance between the two ends of the polymer chain, typically denoted by R. This size is routinely measured with the help of scattering experiments, and is found to increase with the molecular weight of the polymer chain with a scaling law, R ~ M ~, where M is the molecular weight, and u is the scaling exponent which depends on the nature of the polymer-solvent interaction. In good solvents, solutesolvent interactions are favoured relative to solute-solute interactions. As a consequence the polymer chain swells and its size is found to scale with an exponent u = 3/5. On the other hand, in poor solvents, the situation is one in which solute-solute interactions are preferred. There exists a particular temperature, called the theta temperature, at which the scaling exponent u changes dramatically from 3/5 to 1/2. At this temperature, the urge to expand caused by two parts of the chain being unable to occupy the same location (leading to the presence of an excluded volume), is just balanced by the repulsion of the solute molecules by the solvent molecules. Polymer chains in a solution can be imagined to begin to interact with each other when the solution volume is filled with closely packed spheres representing the average size of the molecule. This occurs when np R 3 ~ 1, where np in the number of chains per unit volume. Since np - pp N A / M , where pp is the polymer mass density and NA is Avagadro's number, it follows that polymer density at overlap, p~, scales with molecular weight as, pp ,-~ M 1-3~. Polymer molecules typically have molecular weights between 104 and 106 gm/mol. As a result, it is clear that the polymer solution can be considered dilute only at very low polymer densities. Since experimental measurements are difficult at such low concentrations, the usual practice is to extrapolate results of experiments carried out at decreasing concentrations to the limit of zero concentration. For instance, in the case of dilute polymer solutions it is conventional to report the intrinsic viscosity, which is defined by, [r/] -
lim ~p pp~O pp rls
(1)
where Up is the polymer contribution to the solution viscosity, and r/s is the solvent viscosity.
470
2.2. H o m o g e n e o u s flows Complex flow situations typically encountered in polymer processing frequently involve a combination of shearing and extensional deformations. The response of the polymer solution to these two modes of deformation is very different. Consequently, predicting the rheological properties of the solution under both shear and extensional deformation is considered to be very important in order to properly characterise the solutions behavior. Rather than considering flows where both these modes of deformation are simultaneously present, it is common in polymer kinetic theory to analyse simpler flow situations called homogeneous flows, where they may be treated separately. A flow is called homogeneous, if the rate of strain tensor, "~ - ( V v ) ( t ) + (~7v)t(t), where v is the solution velocity field, is independent of position. In other words, the solution velocity field v in homogeneous flows, can always be represented as v = v0 + ~(t) -r, where v0 is a constant vector, ~(t) - Vv(t) is a traceless tensor for incompressible fluids, and r is the position vector with respect to a laboratory fixed frame of reference. While there is no spatial variation in the rate of strain tensor in homogeneous flows, there is no restriction with regard to its variation in time. Therefore, the response of dilute solutions to transient shear and extensional flows is also used to probe its character as an alternative means of characterisation independent of the steady state material functions. Two homogeneous flows, steady simple shear flow and small amplitude oscillatory shear flow, that are frequently used to validate the predictions of molecular theories which incorporate hydrodynamic interaction, are described briefly below. A comprehensive discussion of material functions in various flow situations can be found in the book by Bird et al. [1].
2.3. Simple shear flows The rheological properties of a dilute polymer solution can be obtained once the stress tensor, 7", is known. The stress tensor is considered to be given by the sum of two contributions, 7" - 7"s + ~-P, where 7-8 is the contribution from the solvent, and T"p is the polymer contribution. Since the solvent is assumed to be Newtonian, the solvent stress (using a compressive definition for the stress tensor [1]) is given by, ~.s _ _ r/8 ~/. The nature of the polymer contribution "/'P in simple shear flows is discussed below.
471
Simple shear flows are described by a velocity field,
vz-O
(2)
where the velocity gradient ~yx can be a function of time. From considerations of symmetry, one can show that the most general form that the polymer contribution to the stress tensor can have in simple shear flows is [1], %
%
0
o
o
rPzz
(3)
where the matrix of components in a Cartesian coordinate system is displayed. The form of the stress tensor implies that only three independent combinations can be measured for an incompressible fluid. All simple shear flows are consequently characterised by three material functions.
2.3.1. Steady simple shear flows Steady simple shear flows are described by a constant shear rate, ~ I')yxl- The tensor ~ is consequently given by the following matrix representation in the laboratory-fixed coordinate system, ~-x/
(~176/ 0 0
0 0
0 0
(4)
The three independent material functions used to characterize such flows are the viscosity, rip, and the first and second normal stress difference coefficients, ~1 and ~2, respectively. These functions are defined by the following relations, <
_
_
-
(5)
where TPy,~'Px,7~Pyare the components of the polymer contribution to the stress tensor r p. At low shear rates, the viscosity and the first normal stress coefficient are observed to have constant values, rip,0 and ~1,0, termed the zero shear rate viscosity and the zero shear rate first normal stress coefficient, respectively. At these shear rates the fluid is consequently Newtonian in its behavior. At higher shear rates, most dilute polymer solutions show shear thinning behavior. The viscosity and the first normal stress coefficient decrease
472
with increasing shear rate, and exhibit a pronounced power law region. At very high shear rates, the viscosity has been observed to level off and approach a constant value, r/p,~, called the infinite shear rate visosity. A high shear rate limiting value has not been observed for the first normal stress coefficient. The second normal stress coefficient is much smaller in magnitude than the first normal stress coefficient, however its sign has not been conclusively established experimentally. Note that the normal stress differences are zero for a Newtonian fluid. The existence of non-zero normal stress differences is an indication that the fluid is viscoelastic. Experiments with very high molecular weight systems seem to suggest that polymer solutions can also shear thicken. It has been observed that the viscosity passes through a minimum with increasing shear rate, and then increases until a plateau region before shear thinning again [17]. It is appropriate here to note that shear flow material functions are usually displayed in terms of the reduced variables, ~?p/~?p,O,~1/~1,o and ~2/~1, versus a non-dimensional shear rate fl, which is defined by fl = )~ph/, where, Ap = [r/J0M rl~/NA kB T, is a characteristic relaxation time. The subscript 0 on the square bracket indicates that this quantity is evaluated in the limit of vanishing shear rate, kB is Boltzmann's constant and T is the absolute temperature. For dilute solutions one can show that, [?7]/[?']]0 - - ?~p/?~p,O and ~ - ?~p,O ;y/np kB T.
2.3.2. Small amplitude oscillatory shear flow A transient experiment that is used very often to characterise polymer solutions is small amplitude oscillatory shear flow. The upper plate in a simple shear experiment is made to undergo sinusoidal oscillations in the plane of flow with frequency w. For oscillatory flow between narrow slits, the shear rate at any position in the fluid is given by [1], ~x(t) - % coswt, where % is the amplitude. The tensor ~(t) is consequently given by,
0 1 0) ~(t)-%coswt
0
0
0
0 0 0
(6)
Since the polymer contribution to the shear stress in oscillatory shear flow, ~-~, p undergoes a phase shift with respect to the shear strain and the strain rate, it is customary to represent its dependence on time through
473
the relation [1], 7:.yvx -- - r / ' ( w ) a / o
cos wt-
~7"(w)"9o s i n w t
(7)
where r / a n d ~" are the material functions characterising oscillatory shear flow. It is common to represent them in a combined form as the complex viscosity, y* = rf - i r/". Two material functions which are entirely equivalent to r]/ and r]" and which are often used to display experimental data, are the storage modulus G ' - w~"/(nkBT) and the loss modulus G " - wrf/(nkBT). Note that the term involving G' in equation(7) is in phase with the strain while that involving G" is in phase with the strain rate. For an elastic material, G" - 0, while for a Newtonian fluid, G' - 0. Thus, G' and G" are measures of the extent of the fluid's viscoelasticity. In flow situations which have a small displacement gradient, termed the linear viscoelastic flow regime, the stress tensor in polymeric fluids is described by the linear constitutive relation, -
-
c(t
-
#(t,
(8)
where G(t) is the relaxation modulus. When the amplitude "~0 is very small, oscillatory shear flow is a linear viscoelastic flow and consequently can also be described in terms of a relaxation modulus G(t). Indeed, expressions for the real and imaginary parts of the complex viscosity can be found from the expression, - f0
(9)
Experimental plots of log G' and log G" versus nondimensional frequency show three distinct power law regimes. The regime of interest is the intermediate regime [17], where for dilute solutions of high molecular weight polymers in good or theta solvents, both G' and G" have been observed to scale with frequency as w 2/3. It is appropriate to note here that the zero shear rate viscosity ~p,0 and the zero shear rate first normal stress difference ~1,0, which are linear viscoelastic properties, can be obtained from the complex viscosity in the limit of vanishing frequency, lira r/',,(w) ; ~p,0 - ~a-+0
91,0 -
lira
~a-+0
2 rf(w) 03
(10)
474
2.4. Scaling w i t h m o l e c u l a r w e i g h t We have already discussed the scaling of the root mean square end-to-end distance of a polymer molecule with its molecular weight. In this section we discuss the scaling of the zero shear rate intrinsic viscosity It/]0, and the translational diffusion coefficient D, with the molecular weight, M. As we shall see later, these have proven to be vitally important as experimental benchmarks in attempts to improve predictions of molecular theories. It has been found that the relationship between [y]0 and M can be expressed by the formula, [~]o-KM
a
(11)
where, a is called the M a r k - H o u w i n k exponent, and the prefactor K depends on the polymer-solvent system. The value of the parameter a lies between 0.5 and 0.8, with the lower limit corresponding to theta conditions, and the upper limit to a good solvent with a very high molecular weight polymer solute. Measured intrinsic viscosities are routinely used to determine the molecular weight of samples once the constants K and a are known for a particular polymer-solvent pair. The translational diffusion coefficient D for a flexible polymer in a dilute solution can be measured by dynamic light scattering methods, and is found to scale with molecular weight as [2], D-~ M -"
(12)
where the exponent # lies in the range 0.49 to 0.6. Most theta solutions have values of # close to the lower limit. On the other hand, there is wide variety in the value of # reported for good solvents. It appears that the upper limit is attained only for very large molecular weight polymers and the intermediate values, corresponding to a cross over region, are more typical of real polymers with moderate molecular weights. 2.5. U n i v e r s a l b e h a v i o r It is appropriate at this point to discuss the most important aspect of the behavior of polymer solutions (as far as the theoretical modelling of these solutions is concerned) that is revealed by the various experimental observations. When the experimental data for high molecular weight systems is plotted in terms of appropriately normalized coordinates, the most noticeable feature is the exhibition of u n i v e r s a l behavior. By this it is
475
meant that curves for different values of a parameter, such as the molecular weight, the temperature, or even for different type8 of monomer8 can be 8uperposed onto a 8ingle curve. For example, when the reduced intrinsic viscosity, [u]/[n]0 is plotted as a function of the reduced shear rate/ , the curves for polystyrene in different type8 of good solvent8 at various temperatures collapse onto a 8ingle curve [1]. There is, however, an important point that must be noted. While polymer8 dissolved in both theta solvents and good solvents show universal behavior, the universal behavior is different in the two cases. An example of this i8 the observed sca]ing behavior of various quantities with molecular weight. The scaling is universal within the context of a particular type of 801vent. The term universality class is used to describe the set of systems that exhibit common universal behavior [40]. Thus theta and good 80lvents be]ong to different universality classes. The existence of universality classes is very significant for the theoretical description of polymer solutions. Any attempt made at modelling a polymer so]ution's properties might expect that a proper description must incorporate the chemical structure of the polymer into the model, since this determines its microscopic behavior. Thus a detailed consideration of bonds, sidegroups, etc. may be envisaged. However, the universal behavior that i8 revealed by experiments suggests that macroscopic properties of the polymer solution are determined by a few large scale properties of the polymer molecule. Structural details may be ignored since at length scales in the order of nanometer8, different polymer molecule8 become equivalent to each other, and behave in the same manner. A8 a result, polymer 80lutions that differ from each other with regard to the chemical structure or molecular weight of the polymer mo]ecu]es that are dissolved in it, the temperature, and so on, still behave similarly as ]ong as a few parameter8 that describe molecular features are the same. This universal behavior justifies the introduction of crude mechanical models, such as the bead-spring chain, to represent real polymer molecules. On the other hand, it is interesting to note that in many cases, the predictions of these models are not universal. It turns out that apart from a basic length and time scale, there occur other parameters that need to be prescribed, for example, the number of beads N in the chain, the strength of hydrodynamic interaction h*, the finite spring extensibility parameter
476
b, and so on. It is perhaps not incorrect to state that any molecular theory that is developed must ultimately verify that universal predictions of transport properties are indeed obtained. The universal predictions of kinetic theory models with hydrodynamic interaction are discussed later on in this chapter. 3. B E A D - S P R I N G
CHAIN MODELS
The development of a kinetic theory for dilute solutions has been approached in two different ways. One of them is an intuitive approach in the configuration space of a single molecule, with a particular mechanical model chosen to represent the macromolecule, such as a freely rotating bead-rod chain or a freely jointed bead-spring chain [13,36,45]. The other approach is to develop a formal theory in the phase space of the entire solution, with the polymer molecule represented by a general mechanical model that may have internal constraints, such as constant bond lengths and angles [16,5,2]. The results of the former method are completely contained within the latter method, and several ad hoc assumptions made in the intuitive treatment are clarified and placed in proper context by the development of the rigorous phase space theory. Kinetic theories developed for flexible macromolecules in dilute solutions have generally pursued the intuitive approach, with the bead-spring model proving to be the most popular. This is because the lack of internal constaints in the model makes the formulation of the theory simpler. Recently, Curtiss and Bird [4], acknowledging the 'notational and mathematical' complexity of the rigorous phase space theory for general mechanical models, have summarised the results of phase space theory for the special case of bead-spring models with arbitrary connectivity, ie. for linear chains, rings, stars, combs and branched chains. In this section, since we are primarily concerned with reviewing recent developments in theories for flexible macromolecules, we describe the development of kinetic theories in the configuration space of a single molecule. However, readers who wish the understand the origin of the ad hoc expressions used for the Brownian forces and the hydrodynamic force, and the formal development of expressions for the momentum and mass flux, are urged to read the article by Curtiss and Bird [4]. The general diffusion equation that governs the time evolution of the
477
distribution of configurations of a bead-spring chain subject to various nonlinear effects, and the microscopic origin of the polymer contribution to the stress tensor are discussed in this section. The simplest bead-spring chain model, the Rouse model is also discussed. We begin, however, by describing the equilibrium statistical mechanical arguments that justify the representation of a polymer molecule with a bead-spring chain model, and we discuss the equilibrium configurations of such a model.
3.1. Equilibrium configurations When a flexible polymer chain in a quiescent dilute solution is considered at a lowered resolution, ie. at a coarse-grained level, it would appear like a strand of highly coiled spaghetti, and the extent of its coiling would depend on its degree of flexibility. A quantity used to characterise a chain's flexibility is the orientational correlation function, whose value Kor (A~), is a measure of the correlation in the direction of the chain at two different points on the chain which are separated by a distance Ag along the length of the chain. At sufficiently large distances Ag, it is expected that the correlations vanish. However, it is possible to define a persistence length gps, such that for Ag > gps, orientational correlations are negligible [40]. The existence of a persistence length suggests that as far as the global properties of a flexible polymer chain are concerned, such as the distribution function for the end-to-end distance of the chain, the continuous chain could be replaced by a freely jointed chain made up of rigid links connected together at joints that are completely flexible, whose linear segments are each longer than the persistence length ~ps, and whose contour length is the same as that of the continuous chain. The freely jointed chain undergoing thermal motion is clearly analogous to a random-walk in space, with each random step in the walk representing a link in the chain assuming a random orientation. Thus all the statistical properties of a random-walk are, by analogy, also the statistical properties of the freely jointed chain. The equivalence of a polymer chain with a random-walk lies at the heart of a number of fundamental results in polymer physics.
3.1.1. Distribution ]unctions and averages In polymer kinetic theory, the freely jointed chain is assumed to have beads at the junction points betwen the links, and is referred to as the freely
478
jointed bead-rod chain [2]. The introduction of the beads is to account for the mass of the polymer molecule a n d the viscous drag experienced by the polymer molecule. While in reality the mass and drag are distributed continuously along the length of the chain, the model assumes that the total mass and drag may be distributed over a finite number of discrete beads. For a general chain model consisting of N beads, which have position vectors r~, u = 1, 2 , . . . , N, in a laboratory fixed coordinate system, the Hamiltonian is given by, 7{ --/C + r (rl, r 2 , . . . , r N )
(13)
where K: is the kinetic energy of the system and r is the potential energy. r depends on the location of all the particles. The center of mass r~ of the chain, and its velocity i% are given by 1
N
Er~ rc = N ~=1
;
1 N /'~= N E=/ ' ~ ~-1
(14)
where/-~ = dr~/dt. The location of a bead with respect to the center of mass is specified by the vector R~ = r~ - re. If Q1, Q2, ... Qd denote the generalised internal coordinates required to specify the configuration of the chain, then the kinetic energy of the chain in terms of the velocity of the center of mass and the generalised velocities Q ~ - dQ~/dt, is given by [2],
I c _ m N .2 1 2 rc + -2 V E g~t Qs Qt t
(15)
where the indices s and t vary from 1 to d, m is the mass of a bead, and g~t is the metric matrix, defined by, g~t - m E~ (OR~/OQ~). (OR~/OQt). In terms of the momentum of the center of mass, Pc = m N/%, and the generalised momenta P~, defined by, P~ - (OIC]OQ~), the kinetic energy has the form [2], 1
2
1
K: - 2m N p~ + 2 V y] G~t P~ Pt t
(16)
where, G st are the components of the matrix inverse to the metric matrix, F~t G~t gtu - 5s~, and 5~u is the Kronecker delta.
479 The probability, ~eqdrcdQ dp~dP, that an N-bead chain model has a configuration in the range dr~ dQ about r~, Q and momenta in the range dp~ dP about p~, P is given by, Peq (r~, Q, pc, P ) - Z -1 e -u/k"T
(17)
where Z is the partition function, defined by,
Z
I/f/
e -u/ksT drc dQ dp~ dP
(18)
The abbreviations, Q and dQ have been used to denote Q1, Q 2 , . . . , Qd and dQ1 dQ2 ... dQd, respectively, and a similar notation has been used for the momenta. The configurational distribution function for a general N-bead chain, Ceq ( Q ) d Q , which gives the probability that the internal configuration is in the range dQ about Q, is obtained by integrating Peq over all the momenta and over the coordinates of the center of mass, %, ( Q ) - Z -~ / / /
e -n/kBT drc dp~ dR
(19)
For an N-bead chain whose potential energy does not depend on the location of the center of mass, the following result is obtained by carrying out the integrations over Pc and P [2],
~)eq ( Q ) -
~/g(Q) e-r $ ~/-g(Q)e-r
dQ
(20)
where, g(Q) - det(gst) - 1/det(G,t). An expression that is slightly different from the random-walk distribution is obtained on evaluating the right hand side of equation (20) for a freely jointed bead-rod chain. Note that the random-walk distribution is obtained by assuming that each link in the chain is oriented independently of all the other links, and that all orientations of the link are equally likely. On the other hand, equation (20) suggests that the probability for the links in a freely jointed chain being perpendicular to each other, for a given solid angle, is slightly larger than the probability of being in the same direction. Inspite of this result, the configurational distribution function for a freely jointed bead-rod chain is almost always assumed to be given by the random-walk distribution [2]. Here afterwards in this chapter, we shall refer to a freely jointed bead-rod chain whose configurational distribution
480 function is assumed to be given by the random-walk distribution, as an ideal chain. For future reference, note that the random-walk distribution is given by,
( 1 ) N-'I N-1 Ceq (01,.--, 0N-l, (~1,--., (~N-1) -- ~ H sin Oi
(21)
i=l
where Oi and r are the polar angles for the ith link in the chain [2]. Since the polymer chain explores many states in the duration of an observation quantities observed on macroscopic length and time scales are averages of functions of the configurations and momenta of the polymer chain. A few definitions of averages are now introduced that are used frequently subsequently in the chapter. The average value of a function X (r~, Q, pc, P ), defined in the phase space of a polymer molecule is given by,
(X)e. -- f f f f
x ~:)eqdrc dQ dp~ dR
(22)
We often encounter quantities X that depend only on the internal configurations of the polymer chain and not on the center of mass coordinates or momenta. In addition, if the potential energy of the chain does not depend on the location of the center of mass, then it is straight forward to see that the equilibrium average of X is given by,
- f x r 3.1.2. The end-to-end vector
(23)
The end-to-end vector r of a general bead-rod chain can be found by summing the vectors that represent each link in the chain,
N-1 r -
y~ a ui
(24)
i=1
where a is the length of a rod, and ui is a unit vector in the direction of the ith link of the chain. Note that the components of the unit vectors ui, i = 1, 2 , . . . , N - 1, can be expressed in terms of the generalised coordinates
Q [2] The probability Peq(r)dr, that the end-to-end vector of a general beadrod chain is in the range dr about r can be found by suitably contracting the configurational distribution function Ceq ( Q ) [2],
Peq(r)- f 6 ( r -
Y~ a u i )r
i
(Q)dQ
(25)
481
where 5(.) represents a Dirac delta function. With Ceq ( Q ) given by the random-walk distribution (21), it can be shown that for large values of N and r - Irl < 0.5Na, the probability distribution for the end-to-end vector is a G aussian distribution, 3
3/2
1)a') ox. (.
_3r 2
1)a')
The distribution function for the end-to-end vector of an ideal chain with a large number of beads N is therefore given by the Gaussian distribution (26/. The mean square end-to-end distance, / r2 )eq, for an ideal chain can then be shown to be, (r2)eq - ( N - 1) a 2. This is the well known result that the root mean square of the end-to-end distance of a random-walk increases as the square root of the number of steps. In the context of the polymer chain, since the number of beads in the chain is directly proportional to the molecular weight, this result implies that R ~ M ~ We have seen earlier that this is exactly the scaling observed in theta solvents. Thus one can conclude that a polymer chain in a theta solvent behaves like an ideal chain.
3.1.3. The bead-springchain Consider an isothermal system consisting of a bead-rod chain with a constant end-to-end vector r, suspended in a bath of solvent molecules at temperature T. The partition function of such a constrained system can be found by contracting the partition function in the constraint-free case,
Z (r) - f f f f
5(r- ~ aui) e-~/kBTdrcdQdpcdP
(27)
i
For an N-bead chain whose potential energy does not depend on the location of the center of mass, the integrations over rc, Pc and P can be carried out to give,
Z(r)- C/5(r-
~
aui)r
(28)
i
Comparing this equation with the equation for the end-to-end vector (25), one can conclude that, Z (r) -- C Peq (r)
(29)
482
In other words, the partition function of a general bead-rod chain (except for a multiplicative factor independent of r) is given by Peq (r). This result is essential to establish the motivation for the introduction of the beadspring chain model. At constant temperature, the change in free energy accompanying a change in the end-to-end vector r of a bead-rod chain, by an infinitesimal amount dr, is equal to the work done in the process, ie., dA- F-dr, where F is the force required for the extension. The Helmholtz free energy of a general bead-rod chain with fixed end-to-end vector r can be found from equation (29), A(r) - - k B T In Z ( r ) -
A o - kBT In Peq (r)
(30)
where A0 is a constant independent of r. For an ideal chain, it follows from equations (26) and (30), that a change in the end-to-end vector by dr, leads to a change in the free energy dA, given by,
3kBT
dA(r) - (N - 1)a 2 r - d r Equation (31) implies that there is a ( 3 k B T / ( N - 1)a 2) r, which resists any thermore, this tension is proportional implies that the ideal chain acts like a stant H given by,
H-
3kBT 1)a2
(N-
(31)
tension F in the ideal chain, F attempt at chain extension. Furto the end-to-end vector r. This Hookean spring, with a spring con-
(32)
The equivalence of the behavior of an ideal chain to that of a Hookean spring is responsible for the introduction of the bead-spring chain model. Since long enough sub-chains within the ideal chain also have normally distributed end-to-end vectors, the entire ideal chain may be replaced by beads connected to each other by springs. Note that each bead in a beadspring chain represents the mass of a sub-chain of the ideal chain, while the spring imitates the behavior of the end-to-end vector of the sub-chain. The change in the Helmholtz free energy of an ideal chain due to a change in the end-to-end vector is purely due to entropic considerations. The internal energy, which has only the kinetic energy contribution, does not depend on the end-to-end vector. Increasing the end-to-end vector of
483
the chain decreases the number of allowed configurations, and this change is resisted by the chain. The entropic origin of the resistance is responsible for the use of the phrase entropic spring to describe the springs of the bead-spring chain model. The potential energy S, of a bead-spring chain due to the presence of Hookean springs is the sum of the potential energies of all the springs in the chain. For a bead-spring chain with N beads, this is given by, N-1
1H E qi" qi
(33)
where Qi = r i + l - ri is the bead connector vector between the beads i and i + 1. The configurational distribution function for a Hookean bead-spring chain may be found from equation (20) by substituting r - S, with the Cartesian components of the connector vectors chosen as the generalised coordinates Qs. The number of generalised coordinates is consequently, d - 3 N - 3, reflecting the lack of any constraints in the model. Since g(Q) is a constant independent of Q for the bead-spring chain model [2], one can show that, H Ceq ( Q 1 , . . . ,
Q N - 1 ) - rI. )
27~kBT
3/2
exp ( - H
Qj)
(34)
It is clear from equation (34) that the equilibrium distribution function for each connector vector in the bead-spring chain is a Gaussian distribution, and these distributions are independent of each other. From the property of Gaussian distributions, it follows that the vector connecting any two beads in a bead-spring chain at equilibrium also obeys a Gaussian distribution. The Hookean bead-spring chain model has the unrealistic feature that the magnitude of the end-to-end vector has no upper bound and can infact extend to infinity. On the other hand, the real polymer molecule has a finite fully extended length. This deficiency of the bead-spring chain model is not serious at equilibrium, but becomes important in strong flows where the polymer molecule is highly extended. Improved models seek to correct this deficiency by modifying the force law between the beads of the chain such that the chain stiffens as its extension increases. An example of such a nonlinear spring force law that is very commonly used in polymer literature is the finitely extensible nonlinear elastic (FENE) force law [2].
484
3.1.4. Excluded volume The universal behavior of polymers dissolved in theta solvents can be explained by recognising that all high molecular weight polymers dissolved in theta solvents behave like ideal chains. However, a polymer chain cannot be identical to an ideal chain since unlike the ideal chain, two parts of a polymer chain cannot occupy the same location at the same time. In the very special case of a theta solvent, the excluded volume force is just balanced by the repulsion of the solvent molcules. In the more commonly occuring case of good solvents, the excluded volume interaction acts between any two parts of the chain that are close to each other in space, irrespective of their distance from each other along the chain length, and leads to a swelling of the chain. This is a long range interaction, and as a result, it seriously alters the macroscopic properties of the chain. Indeed there is a qualitative difference, and this difference cannot be treated as a small perturbation from the behavior of an ideal chain [40]. Curiously enough however, all swollen chains behave similarly to each other, and modelling this universal behavior was historically one of the challenges of polymer physics [40,44,7,8,6]. Here, we very briefly mention the manner in which the problem is formulated in the case of bead-spring chains. The presence of excluded volume causes the polymer chain to swell. However, the swelling ceases when the entropic retractive force balances the excluded volume force. The retractive force arises due to the decreasing number of conformational states available to the polymer chain due to chain expansion. This picture of the microscopic phenomenon is captured by writing the potential energy of the bead-spring chain as a sum of the spring potential energy and the potential energy due to excluded volume interactions. The excluded volume potential energy is found by summing the interaction energy over all pairs of beads # and u, E - (1/2) E.,~=~ N E (r~ - r~), where E ( r ~ - r~) is a short-range function u u lly t
kr
E
-
-
k.T
- r.);
b
ing t h e exclua
a vol-
u m e parameter with dimensions of volume. The total potential energy of
a Hookean bead-spring chain with s is consequently, 1
N-1
r - -~ H ~ i--1
1
excluded volume interactions
N
Qi . Qi + -~v kBT Z ~,~=1 tt#u
5(r~-ru)
(35)
485
The equilibrium configurational distribution function of a polymer chain in the presence of Hookean springs and excluded volume can be found by substituting equation (35) into equation (20), and all average properties of the chain can be found by using equation (23). Solutions to these equations in the limit of long chains have been found by using a number of approximate schemes since an exact treatment is impossible. The most accurate scheme involves the use of field theoretic and renormalisation group methods [6]. The universal scaling of a number of equilibrium properties of dilute polymer solutions with good solvents are correctly predicted by this theory. For instance, the end-to-end distance is predicted to scale with molecular weight as, R ~ M ~ The spring potential in equation (35) has been derived by considering the Helmholtz free energy of an ideal chain, ie. under theta conditions. It seems reasonable to expect that a more accurate derivation of the retractive force in the chain due to entropic considerations would require the treatment of a polymer chain in a good solvent. This would lead to a nonHookean force law between the beads [7,29]. Such non-Hookean force laws have so far not been treated in non-equilibrium theories for dilute polymer solutions with good solvents. 3.2. N o n - e q u i l i b r i u m c o n f i g u r a t i o n s
Unlike in the case of equilibrium solutions it is not possible to derive the phase space distribution function for non-equilibrium solutions from very general arguments. As we shall see here it is only possible to derive a partial differential equation that governs the evolution of the configurational distribution function by considering the conservation of probability in phase space, and the equation of motion for the particular model chosen. The arguments relevent to a bead-spring chain are developed below. 3.2.1. D i s t r i b u t i o n f u n c t i o n s and averages
The phase space of a bead-spring chain with N beads can be chosen to be given by the 6N - 6 components of the bead position coordinates, and the bead velocities such that, ~O ( r l , . . . , rN, r l , . . . , rg, t ) d r 1 . . , drN d~l . . . drN
is the probability that the bead-spring chain has an instantaneous configuration in the range d r l , . . . , d r g about r l , . . . , r g , and the beads in the chain have velocities in the range d / h , . . . , drN about/'1,...,/~g.
486
The configurational distribution function ~, can be found by integrating /) over all the bead velocities, ( r l , . . . , rN, t ) -- f . . .f :P d f l . . . diCN The distribution of internal configurations r
(36) is given by,
r ( Q 1 , . . . , QN-1, t) - f ~I,'(r~, Q 1 , - . . , QN-1, t ) d r c
(37)
where, ~' - ~, as a result of the Jacobian relation for the configurational vectors [2], [ 0 ( r l , . . . , r N ) / 0 ( r c , Q 1 , . . . , QN-1)I -- 1. Note that the normalisation condition f r dQ1 dQ2 ... d Q N - 1 = 1 is satisfied by ~b. When the configurations of the bead-spring chain do not depend on the location of the center of mass, as in the case of homogeneous flows with no concentration gradients, ( l / V ) r = q~, where V is the volume of the solution. The velocity-space distribution function E is defined by, 7) (rl,...,rN, r l , - . - , I ' N , t ) -- ~ (38) Note that E satisfies the normalisation condition f . . . f E die1.., di'N = 1. Under certain circumstances that are discussed later, it is common to assume that the velocity-space distribution function is Maxwellian about the mass-average solution velocity, ---- ArM exp [
1
2 k B T [Trt(rl -- v) 2 -+- . . . nt- m ( r N
--
v)2]]
(39)
where ArM is the normalisation constant for the Maxwellian distribution. Making this assumption implies that one expects the time scales involved in equilibration processes in momentum space to be much smaller than the time scales governing relaxation processes in configuration space. Averages of quantities which are functions of the bead positions and bead velocities are defined analogously to the those in the previous section, namely, (X) - f...f
X 7) d r 1 . . , drN d ~ l . . , drN
(40)
is the the phase space average of X, while the velocity-space average is,
[[x ]] - f f z
dl'l
. . . dr'N
(41)
For quantities X that depend only on the internal configurations of the polymer chain and not on the center of mass coordinates or bead velocities, (X) - f X r d Q l d Q 2 . . , d Q N - 1
(42)
487
3.2.2. The equation of motion The equation of motion for a bead in a bead-spring chain is derived by considering the forces acting on it. The total force F~, on bead # is, F~ - Ei F(~), where the F(~), i - 1, 2,..., are the various intramolecular and solvent forces acting on the bead. The fundamental difference among the various molecular theories developed so far for the description of dilute polymer solutions lies in the kinds of forces F(~) that are assumed to be acting on the beads of the chain. In almost all these theories, the accelaration of the beads due to the force F~ is neglected. A bead-spring chain model incorporating bead inertia has shown that the neglect of bead inertia is justified in most practical situations [37]. The equation of motion is consequently obtained by setting F~ - 0. Here, we consider the following force balance on each bead #, F(h)+F(b)+F(r
(i~) -- 0
(#
1, 2 , . . . , N )
(43)
where, F(h) is the hydrodynamic drag force, F (b) is the Brownian force, F(r is the intramolecular force due to the potential energy of the chain, -# and -p(iv) is the force due to the presence of internal viscosity. These are the various forces that have been considered so far in the literature, which are believed to play a crucial role in determining the polymer solution's transport properties. The nature of each of these forces is discussed in greater detail below. Note that, as is common in most theories, external forces acting on the bead have been neglected. However, their inclusion is reasonably straight forward [2]. The hydrodynamic drag force F(h) is the force of resistance offered by the solvent to the motion of the bead #. It is assumed to be proportional to the difference between the velocity-averaged bead velocity [~i,u]] and the local velocity of the solution,
F.(h) = _r [
_ (v. + v,)]'
(44)
where ~ is bead friction coefficient. Note that for spherical beads with radius a, in a solvent with viscosity ~/,, the bead friction coefficient ~ is given by the Stokes expression: ~ = 67rr/sa. The velocity-average of the bead velocity is not carried out with the Maxwellian distribution since this is just the mass-average solution velocity. However, it turns out that an explicit evaluation of the velocity-average is unnecessary for the development of
488
the theory. Note that the velocity of the solution at bead # has two components, the imposed flow field vu - v0 + ~(t) 9r~, and the perturbation of the flow field v~t due to the motion of the other beads of the chain. This perturbation is called hydrodynamic interaction, and its incorporation in molecular theories has proved to be of utmost importance in the prediction of transport properties. The presence of hydrodynamic interaction couples the motion of one bead in the chain to all the other beads, regardless of the distance between the beads along the length of the chain. In this sense, hydrodynamic interaction is a long range phenomena. The perturbation to the flow field v~(r) at a point r due to the presence of a point force F ( r ~) at the point r ~, can be found by solving the linearised Navier-Stokes equation [1,8], v'(r) - ~ ( r -
r ' ) . F(r')
(45)
where ~ ( r ) , called the Oseen-Burgers tensor, is the Green's function of the linearised Navier-Stokes equation, n(r)-
rr ) 1 (1 + 87rq,r r-2
(46)
The effect of hydrodynamic interaction is taken into account in polymer kinetic theory by treating the beads in the bead-spring chain as point particles. As a result, in response to the hydrodynamic drag force acting on each bead, each bead exerts an equal and opposite force on the solvent at the point that defines its location. The disturbance to the velocity at the bead v is the sum of the disturbances caused by all the other beads in the chain, v" - - ~u ~ u ( r ~ - ru). F (h), where, ~ - ~v~ is given by, ~u~--
{ 87rrlsru~ 1 (1+ ru~ru~) r2 , ]zy
0
ru~--ru-r~,
for##v
(47)
for # - - u
The Brownian force F(~b), on a bead # is the result of the irregular collisions between the solvent molecules and the bead. Instead of representing the Brownian force by a randomly varying force, it is common in polymer kinetic theory to use an averaged Brownian force, F (b) --
-kBT ( 0 0r~ In
)
(48)
489
As mentioned earlier, the origin of this expression can be understood within the framework of the complete phase space theory [2,4]. Note that the Maxwellian distribution has been used to derive equation (48). The total potential energy r of the bead-spring chain is the sum of the potential energy S of the elastic springs, and the potential energy E due to the presence of excluded volume interactions between the beads. The force F(r on a bead # due to the intramolecular potential energy r is given by, F(~)_
0r
(49)
In addition to the various forces discussed above, the internal viscosity force F (i~) has received considerable attention in literature [3,38,43] though it appears not to have widespread acceptance. Various physical reasons have been cited as being responsible for the internal viscosity force. For instance, the hindrance to internal rotations due to the presence of energy barriers, the friction between two monomers on a chain that are close together in space and have a non-zero relative velocity, and so on. The simplest models for the internal viscosity force assume that it acts only between neighbouring beads in a bead-spring chain, and depends on the average relative velocities of these beads. Thus, for a bead # that is not at the chain ends,
(-
F ,(iv) = 99 ( r . + l - - r # ) ( r . + l l 2
rt~)).[~#+1__~#]]
[r.+l -- r. -- qO((r.- r._l)(r. [2rg_l)). [ [ / . _
1"#-1]]
(50)
I r u - ru_ 1 where ~p is the internal viscosity coefficient. A scaling theory for a more general model that accounts for internal friction between arbitrary pairs of monomers has also been developed [35]. The equation of motion for bead v can consequently be written as, - ~ [ ~-i-~]]- v0 - tr
A- ~ a u , . F (h) ] - kBT
0 ln~
#
Since F (h) - k , T (0 In ~ / 0 r u ) ranged to give, [[~]] - vo + tr
F(r
0r~
-4- F (~) + F (/') - 0(51)
--uF(i~), equation (51) can be rear-
1 0 ln~ + ~ Z "~,~" (--kB T~ u 0r~
+
+
(iv) )
(52)
490
where -),,~ is the dimensionless diffusion tensor [2], ~,~
5u~ 1 + ( f t ~
-
(53)
By manipulating equation (52), it is possible to rewrite the equation of motion in terms of the velocities of the center of mass r~ and the beadconnector vectors Qk, ~-/'c]] - v0 + ~ . r~
1 + 0Qk0r+ r"k ) N ( u,u,kE Bku "/t,u " (kBT OoQkln____~
1 [[Oj]] - ~" Q j - -~ Z Ajk" ( kBT 0 In 9
0r
f(iv)
(54) (55)
k
where, Bku is defined by, -Bk~ -- 5k+1,~- 5k~, the internal viscosity force, fk(i~), in the direction of the connector vector Qk is, Q'Q* I I
-
(56)
]]
and the tensor -Ajb which accounts for the presence of hydrodynamic interaction is defined by,
A.jk -- ~ B--j~,"ft,v-Bku -- Ajkl + ~(~j,k
+
~r~j+l,k+l --
~"~j,k+l - -
~-'~j+l,k)
(57)
L,, it
Here, Ajk is the Rouse matrix,
Ajk--
2 --1
for I J - k l - 0, forlj-kl-1,
(58)
0 otherwise In order to obtain the diffusion equation for a dilute solution of beadspring chains, the equation of motion derived here must be combined with an equation of continuity.
3.2.3. The diffusion equation The equation of Continuity or 'probability conservation', which states that a bead-spring chain that disappears from one configuration must appear in another, has the form [2],
O~ Ot
0
(59)
491
The independence of 9 from the location of the center of mass for homogeneous flows, and the result tr ~ - 0, for an incompressible fluid, can be shown to imply that the equation of continuity can be written in terms of internal coordinates alone as [2],
0r 0 0~ - - E o q / [ [ Qj] r
(60)
J
The general diffusion equation which governs the time evolution of the instantaneous configurational distribution function r in the presence of hydrodynamic interaction, arbitrary spring and excluded volume forces, and an internal viscosity force given by equation (56), is obtained by substituting the equation of motion for ~-(~j~J from equation (55) into equation (60). It has the form, 0r
:
-
j
0 0qj
1
qJ -
0r [0q
Ek
~(i~)
1)
0 kBT ~ OQj Aik" 0r -
+
~
j,k
(61)
OQk
Equations such as (61) are also referred to as Fokker-Planck or Smoluchowski equations in the literature. The diffusion equation (61) is the most fundamental equation of the kinetic theory of dilute polymer solutions since a knowledge of r for a flow field specified by to, would make it possible to evaluate averages of various configuration dependent quantities and thereby permit comparison of theoretical predictions with experimental observations. The diffusion equation can be used to derive the time evolution equation of the average of any arbitrary configuration dependent quantity, X( Q~, . . . , QN-~ ), by multiplying the left and right hand sides of equation (61) by X and integrating both sides over all possible configurations,
d (X) = E ( ~ ' Q j " dt j -
kBT E ( A j k " 0 l n r 9 OX ~ j,k OQk OQj
OX). OQj 0r
r
_ 1E
0X
]'OQj)
(62)
Except for a situation where nearly all the important microscopic phenomena are neglected, the diffusion equation (61) is unfortunately in gem
492
eral analytically insoluble. There have been very few attempts to directly solve diffusion equations with the help of a numerical solution procedure [9,10]. In this context it is worth bearing in mind that what are usually required are averages of configuration dependent quantities. However, in general even averages cannot be obtained exactly by solving equation (62). As a result, it is common in most molecular theories to obtain the averages by means of various approximations. In order to examine the validity of these approximations it is vitally important to compare the approximate predictions of transport properties with the exact predictions of the models. One of the ways by which exact numerical results may be obtained is by adopting a numerical procedure based on the mathematical equivalence of diffusion equations in polymer configuration space and stochastic differential equations for the polymer configuration [29]. Instead of numerically solving the analytically intractable difusion equation for the distribution function, stochastic trajectories can be generated by Brownian dynamics simulations based on a numerical integration of the appropriate stochastic differential equation. Averages calculated from stochastic trajectories (obtained as a solution of the stochastic differential equations), are identical to the averages calculated from distribution functions (obtained as a solution of the diffusion equations). It has now become fairly common for any new approximate molecular theory of a microscopic phenomenon to establish the accuracy of the results with the help of Brownian dynamics simulations. In this chapter, while results of such simulations are cited, details of the develop: ment of the appropriate stochastic differential equations are not discussed. A comprehensive introduction to the development of stochastic differential equations which are equivalent to given diffusion equations for the probability density in configuration space, can be found in the treatise by Ottinger [29]. 3.2.~. The stress tensor The expression for the stress tensor in a dilute polymer solution was originally obtained by the use of simple physical arguments which considered the various mechanisms that contributed to the flux of momentum across an oriented surface in the fluid [2]. The major mechanisms considered were the transport of momentum by beads crossing the surface, and the tension in the springs that straddle the surface. These physical arguments help to
493
provide an intuitive understanding of the origin of the different terms in the stress tensor expression. On the other hand, such arguments are difficult to pursue in the presence of complicated microscopic phenomena, and there is uncertainity about the completeness of the final expression. An alternative approach to the derivation of the expression for the stress tensor has been to use more fundamental arguments that consider the complete phase space of the polymeric fluid [2,41. A very general expression for the polymer contribution to the stress tensor, derived by adopting the complete phase space approach, for models without constraints such as the bead-spring chain model, in the presence of hydrodynamic interaction and an arbitrary intramolecular potential force, is the modified Kramers expression [21,
~-P - n p E ((r~ - rc)F (r
(63)
+ ( N - 1)npknT 1
/J
When rewritten in terms of the internal coordinates of a bead-spring chain, equation (63) assumes a form called the Kramers expression,
or
7"p - -np ~ ( Q j ~ j >
(64)
+ (N - 1)npkuT1
3
It is important to note that the presence of internal viscosity has not been taken into account in the phase space theories used to derive the modified Kramers expression (63). When examined from the standpoint of thermodynamic considerations, the proper form of the stress tensor in the presence of internal viscosity appears to be the Gicsekus expression rather than the Kramers expression [39]. Since predictions of models with internal viscosity are not considered in this chapter, the Giesekus expression is not discussed here. In order to evaluate the stress tensor, for various choices of the potential energy r it turns out that it is usually necessary to evaluate the second moments of the bead connector vectors,
dt( Q/Qk) -- ~ " ( QjQk >+ ( QjQk >. ~t +
0r
+ f(~v)]. A.mk) + ( X j m ' [ ~ 0r
2kgT --
1
( Ajk >- -~ ~m {( Qj
+ f2s) ]" qk> }
(65)
494
The second moment equation (65), which is an ordinary differential equation, is in general not a closed equation for ( QjQk }, since it invoves higher order moments on the right hand side. Within the context of the molecular theory developed thus far, it is clear that the prediction of the rheological properties of dilute polymer solutions with a bead-spring chain model usually requires the solution of the second moment equation (65). To date however, there are no solutions to the general second moment equation (65) which simultaneously incorporates the microscopic phenomena of hydrodynamic interaction, excluded volume, non-linear spring forces and internal viscosity. Attempts have so far been restricted to treating a smaller set of combinations of these phenomenon. The simplest molecular theory, based on a bead-spring chain model, for the prediction of the transport properties of dilute polymer solutions is the Rouse model. The Rouse model neglects all the microscopic phenomenon listed above, and consequently fails to predict many of the observed features of dilute solution behavior. In a certain sense, however, it provides the framework and motivation for all further improvements in the molecular theory. The Rouse model and its predictions are introduced below, while improvements in the treatment of hydrodynamic interactions alone are discussed subsequently. 3.3. T h e R o u s e m o d e l
The Rouse model assumes that the springs of the bead-spring chain are governed by a Hookean spring force law. The only solvent-polymer interactions treated are that of hydrodynamic drag and Brownian bombardment. The diffusion equation (61) with the effects of hydrodynamic interaction, excluded volume and internal viscosity neglected, and with a Hookean spring force law, has the form,
_ 0 H kBT 0 0r (66) 0 r _ _ ~ oqj" (~" Qj - --( ~-" Ajk Qk ) r + - - ~ ~ Ajk i)Qj'Oqk Ot
j
k
j,k
The diffusion equation (66) has an analytical solution since it is linear in the bead-connector vectors. It is satisfied by a Gaussian distribution,
1 ~ Qj. (er_l)jk " Qk] (Q1,..., QN-1) - Af(t) e x p [ - ~j,k
(67)
495 where Af(t) is the normalisation factor, and the tensor O)k which uniquely characterises the Gaussian distribution is identical to the second moment, o ) k -
(68)
Note that the tensors O)k are not symmetric, but satisfy the relation ~k -- Jkj- (Further information on linear diffusion equations and Gaussian distributions can be obtained in the extended discussion in Appendix A of [22]). Since the intramolecular potential in the Rouse model is only due to the presence of Hookean springs, it is straight forward to see that the Kramers expression for the stress tensor r p, is given by, v p -- - n p H ~ ajj + ( N -
(69)
1)npkBT 1
J
The tensors ~ j are obtained by solving the second moment equation (65), which becomes a closed equation for the second moments when the Rouse assumptions are made. It has the form, d t ~ k -- ~ " ajk -- erjk . ~ t _ 2 k u T A j k 1 - - -
H
~-~[ a j m A m k -b A i m ermk
] (70)
Note that the solution of equation (70) also leads to the complete specification of the Gaussian configurational distribution function r A H o o k e a n d u m b b e l l model, which is the simplest example of a beadspring chain, is obtained by setting N = 2. It is often used for preliminary calculations since its simplicity makes it possible to obtain analytical solutions where numerical solutions are unavoidable for longer chains. For such a model, substituting for ~rll in terms of r p from equation (69) into equation (70), leads to following equation for the polymer contribution to the stress tensor, ~'P + A s rP(1) -- - - n p k B T A H ~/
(71)
where ~'~1) -- d r P / d t - ~ . ~'P - ~'P . ~ i , is the convected time derivative [2] of ~'P, and AH -- (4/4H) is a time constant. Equation (71) indicates that a Hookean dumbbell model with the Rouse assumptions leads to a convected Jeffreys model or Oldroyd-B model as the constitutive equation for a dilute polymer solution. This is perhaps the simplest coarse-grained microscopic model capable of reproducing some of the macroscopic rhcological propertics of dilute polymer solutions.
496
In the case of a bead-spring chain with N > 2, it is possible to obtain a similar insight into the nature of the stress tensor by introducing n o r m a l coordinates. These coordinates help to decouple the connector vectors Q 1 , . . . , QN-1, which are coupled to each other because of the Rouse matrix. The connector vectors are mapped to a new set of normal coordinates, , , QN-1 with the transformation, Qj - ~
1-IjkQk!
where,
I-Ijk
are
(I-l-1)jk
--
1-Ikj ,
(72) the elements of an orthogonal matrix with the property such that,
~ 1]mjYImk -- ~jk
(73)
m
The orthogonal matrix Iljk , which will henceforth be referred to as the Rouse orthogonal matrix, diagonalises the Rouse matrix Ajk, j,k
(74)
IIjiAjkIIkl - at&t
where, the Rouse eigenvalues at are given by at - 4sin2(17r/2N). The elements of the Rouse orthogonal matrix are given by the expression, 2
(75)
The diffusion equation in terms of these normal coordinates, admits a solution for the configurational distribution function of the form [2] N-1
r (Q~,..-, Q~v-~) -
1-I Ok (Q~)
(76)
k=l
As a consequence, the diffusion equation becomes uncoupled and can be simplified to ( N - 1) diffusion equations, one for each of the Ck (Q~). Since the Q~ are independent of each other, all the covariances (Q~Q~) with j ~ k are zero, and only the ( i - 1) variances dj - (Q~Q~) are non-zero. Evolution equations for the variances ~ can then be derived from these uncoupled diffusion equations with the help of a procedure similar to that used for the derivation of equation (65). The stress tensor is given in terms of dj by the expression, TP--Z
WjP J
(77)
497 where, T'jP -
- n p H ~ + npkBT 1
(78)
On substituting for dj in terms of v P j from equation (78) into the evolution equation for dj, one obtains,
TjP -~- ~j TjP (1) - -
-npkBT.~j
(79)
;',[
-
where, the relaxation times Aj are given by )~j - ( r aj). Consequently, each of the "rjP satisfy an equation identical to equation (71) for the polymer contribution to the stress tensor in a Hookean dumbbell model. The Rouse model, therefore, leads to a constitutive equation that is a multimode generalization of the convected Jeffreys or Oldroyd B model. It is clear from above discussion that the process of transforming to normal coordinates enables one to derive a closed form expression for the stress tensor, and to gain the insight that the Rouse chain with N beads has N independent relaxation times which describe the different relaxation processes in the chain, from the entire chain to successively smaller sub-chains. It is straight forward to show that for large N, the longest relaxation times )~j scale with chain length as N 2. A few important transport property predictions which show the limitations of the Rouse model are considered briefly below. It is worth noting that since the Rouse model does not include the effect of excluded volume, its predictions are restricted to dilute solutions of polymers in theta solvents. This restriction is infact applicable to all the models of hydrodynamic interaction treated here. In steady simple shear flow, with ~(t) given by equation (4), the three independent material functions that characterise such flows are [2], rip -- npkBT ~ /~j ; J
t~l --
2npkBT ~ .kj ; J
q22 - 0
(80)
It is clear that the Rouse model accounts for the presence of viscoelasticity through the prediction of a nonzero first normal stress difference in simple shear flow. However, it does not predict the nonvanishing of the second normal stress difference, and the shear rate dependence of the viscometric functions. From the definition of intrinsic viscosity (1) and the fact that pp ~ N np, it follows from equation (80) that for the Rouse model, It/]0 ~ N. This is
498
at variance with the experimental results discussed earlier, and displayed in equation (11). It is also straight forward to see that the Rouse model predicts that the characteristic relaxation time scales as the square of the chain length, Ap ~ N 2. In small amplitude oscillatory shear, ~;(t) is given by equation (6), and expressions for the material functions G ~ and G" in terms of the relaxation times )~j can be easily obtained [2]. In the intermediate frequency range, where as discussed earlier, experimental results indicate that both G ~ and G " scale as w2/3, the Rouse model predicts a scaling w 1/2 [17]. The translational diffusion coefficient D for a bead-spring chain at equilibrium can be obtained by finding the average friction coefficient Z for the entire chain in a quiescent solution, and subsequently using the NernstEinstein equation, D - k n T Z -1 [2]. It can be shown that for the Rouse model Z = ( N, ie. the total friction coefficient of the chain is a sum of the individual bead friction coefficients. As a result, the Rouse model predicts that the diffusion coefficient scales as the inverse of the molecular weight. This is not observed in dilute solutions. Instead experiments indicate the scaling depicted in equation (12). The serious shortcomings of the Rouse model highlighted above have been the motivation for the development of more refined molecular theories. The scope of this chapter is restricted to reviewing recent advances in the treatment of hydrodynamic interaction. 4. H Y D R O D Y N A M I C
INTERACTION
Hydrodynamic interaction, as pointed out earlier, is a long range interaction between the beads which arises because of the solvent's capacity to propagate one bead's motion to another through perturbations in its velocity field. It was first introduced into framework of polymer kinetic theory by Kirkwood and Riseman [14]. As we have seen in the development of the general diffusion equation above, it is reasonably straight forward to include hydrodynamic interaction into the framework of the molecular theory. However, it renders the resultant equations analytically intractable and as a result, various attempts have been made in order to solve them approximately. In this section, we review the various approximation schemes introduced over the years. The primary test of an approximate model is of course
499
its capacity to predict experimental observations. The accuracy of the approximation however, can only be assessed by checking the proximity of the approximate results to the exact numerical results obtained by Brownian dynamics simulations. Finally, the usefulness of an approximation depends on its computational intensity. The individual features and deficiencies of the different approximations will be examined in the light of these observations. In the presence of hydrodynamic interaction, and with excluded volume and internal viscosity neglected, a bead-spring chain with Hookean springs has a configurational distribution function r that must satisfy the following simplified form of the diffusion equation (61), 0r
0
H
kBT
Ot - -~j9 OQj " (~" Qj - --~ y~ Ayk "Qk)r + k
r
0 0r (81) Y~ OQj "Ajk " OQk
j,k
while the second moment equation (65) assumes the form, d
2k.r
dt (QJQk> -
-
(X k>
~" (QjQk> + (QjQk) . ~t + ......r H -~-~ [(QjQm-Amk> + ]
(82)
Equation (82) is not a closed equation for the second moments since it involves more complicated moments on the right hand side. This is the central problem of all molecular theories which attempt to predict the rheological properties of dilute polymer solutions and that incorporate hydrodynamic interaction. The different approximate treatments of hydrodynamic interaction, which are discussed roughly chronologically below, basically reduce to finding a suitable closure approximation for the second moment equation. 4.1. T h e Z i m m model The Z i m m model was the first attempt at improving the Rouse model by introducing the effect of hydrodynamic interaction in a preaveraged or equilibrium-averaged form. The preaveraging approximation has been very frequently used in polymer literature since its introduction by Kirkwood and Riseman [14]. The approximation consists of evaluating the average of the hydrodynamic tensor with the equilibrium distribution function (34), and replacing the hydrodynamic interaction tensor Ajk, wherever it occurs
500
in the governing equations, with its equilibrium average Ajk. (Note that the incorporation of the effect of hydrodynamic interaction does not alter the equilibrium distribution function, which is still given by (34) for beadspring chains with Hookean springs.) The matrix Ajk is called the modified Rouse matrix, and is given by, ---
2
-
+
h*(v/i j
ki
1 ~/iJ - k-
_ 1[
1 ) v/Ij - k + 11
(83)
where, h* -- a v / ( H / z r k B T ) is the hydrodynamic interaction parameter. The hydrodynamic interaction parameter is approximately equal to the ratio of the bead radius to the equilibrium root mean square length of a single spring of the bead-spring chain. This implies that h* < 0.5, since the beads cannot overlap. Typical values used for h* are in the range 0.1 < h* < 0.3 [22]. By including the hydrodynamic interaction in an averaged form, the diffusion equation remains linear in the connector vectors, and consequently is satisfied by a Gaussian distribution (67) as in the Rouse case. However, the covariance tensors ~k are now governed by the set of differential equations (70) with the Rouse matrix Ajk replaced with the modified Rouse matrix Ajk. Note that this modified second moment equation is also a closed set of equations for the second moments. As in the Rouse case, it is possible to simplify the solution of the Zimm model by carrying out a diagonalisation procedure. This is achieved by mapping the connector vectors to normal coordinates, as in (72), but in this case the Zimm orthogonal matrix l-ljk, which diagonalises the modified Rouse matrix, N
j,k
IIjiAjkIIkl - al~il
(84)
must be found numerically for N > 4. Here, al are the so called Zimm eigenvalues. The result of this procedure is to render the diffusion equation solvable by the method of separation of variables. Thus, as in the Rouse case, only the ( N - 1) transformed coordinate variances dj are nonzero, and differential equations governing these variances can be derived by manipulating the uncoupled diffusion equations. The diagonalisation procedure enables the polymer contribution to the stress tensor ~'P in the Zimm model to be expressed as a sum of partial
501
stresses ~'jP as in equation (77), but the "FjP n o w satisfy equation (79) with the 'Rouse' relaxation times Aj replaced with 'Zimm' relaxation times Aj. The Zimm relaxation times are defined by )~j - ( ~ / 2 H ~j). From the discussion above, it is clear that the Zimm model differs from the Rouse model only in the spectrum of relaxation times. As we shall see shortly, this leads to a significant improvement in the prediction of linear viscoelastic properties and the scaling of transport properties with molecular weight in theta solvents. The Zimm model therefore establishes unequivocally the importance of the microscopic phenomenon of hydrodynamic interaction. On the other hand, it does not lead to any improvement in the prediction of nonlinear properties, and consequently subsequent treatments of hydrodynamic interaction have concentrated on improving this aspect of the Zimm model. By considering the long chain limit of the Zimm model, ie., N --+ c~, it is possible to discuss the universal properties predicted by the model. The various power law dependences of transport properties on molecular weight, characterised by universal exponents, and universal ratios formed from the prefactors of these dependences can be obtained. These predictions are ideal for comparison with experimental data on high molecular weight polymer solutions since they are parameter free. We shall discuss some universal exponents predicted by the Zimm model below, while universal ratios are discussed later in the chapter. As mentioned above, the first noticeable change upon the introduction of hydrodynamic interaction is the change in the relaxation spectrum. In the long chain limit, the longest relaxation times/~j scale with chain length as N 3/2 [29], whereas we had found earlier that the chain length dependence of the longest relaxation times in the Rouse model was N 2. In steady simple shear flow, the Zimm model like the Rouse model, fails to predict the experimentally observed occurance of non-zero second normal stress differences and the experimentally observed shear rate dependence of the viscometric functions. It does however lead to an improved prediction of the scaling of the zero shear rate intrinsic viscosity with molecular weight, [~]0 ~ N 1/2. This prediction is in agreement with experimental results for the Mark-Houwink exponent in theta solvents (see equation (11)). As with the longest relaxation times, the characteristic relaxation time Ap ~ N 3/2.
502
In small amplitude oscillatory shear, the Zimm model predicts that the material functions G ~ and G " scale with frequency as w 2/3 in the intermediate frequency range. This is in exceedingly good agreement with experimental results [2,17]. The translational diffusion coefficient D for chainlike molecules at equilibrium, with preaveraged hydrodynamic interaction, was originally obtained by Kirkwood [14]. Subsequently, several workers obtained a correction to the Kirkwood diffusion coefficient for the Zimm model [21]. The exact results differ by less than 2% from the Kirkwood value for all values of the chain length and h*. Interestingly, three different approaches to obtaining the diffusion coefficient, namely, the Nernst-Einstein equation, the calculation of the mean-square displacement caused by Brownian forces, and the study of the time evolution of concentration gradients, lead to identical expressions for the diffusion coefficient [21]. In the limit of very long chains, it can be shown that D ~ N -1/2. The Zimm model therefore gives the correct dependence of translational diffusivity on molecular weight in theta solvents. The Zimm result for the translational diffusivity has been traditionally interpreted to mean that the polymer coil in a theta solvent behaves like a rigid sphere, with radius equal to the root mean square end-to-end distance. This follows from the fact that the diffusion coefficient for a rigid sphere scales as the inverse of the radius of the sphere, and in a theta solvent, ( r2)eq scales with chain length as N. The solvent inside the coil is believed to be dragged along with the coil, and the inner most beads of the beadspring chain are considered to be shielded from the velocity field due to the presence of hydrodynamic interaction [44,17]. This intuitive notion has been used to point out the difference between the Zimm and the Rouse model, where all the N beads of the polymer chain are considered to be exposed to the applied velocity field. Recently, by explicitly calculating the velocity field inside a polymer coil in the Zimm model, C)ttinger [30] has shown that the solvent motion inside a polymer coil is different from that of a rigid sphere throughout the polymer coil, and that shielding from the velocity field occurs only to a certain extent.
4.2.
The consistent averaging approximation
The first predictions of shear thinning were obtained when hydrodynamic interaction was treated in a more precise manner than that of preaverag-
503
ing the hydrodynamic interaction tensor. In order to make the diffusion equation (81) linear in the connector vectors, as pointed out earlier, it is necessary to average the hydrodynamic interaction tensor. However, it is not necessary to preaverage the hydrodynamic interaction tensor with the equilibrium distribution. On the other hand, the average can be carried out with the non-equilibrium distribution function (67). The linearity of the diffusion equation ensures that its solution is a Gaussian distribution. Ottinger [20,22] suggested that the hydrodynamic interaction tensor occuring in the diffusion equation be replaced with its non-equilibrium average. Since it is necessary to know the averaged hydrodynamic interaction tensor in order to find the non-equilibrium distribution function, both the averaged hydrodynamic interaction tensor and the non-equilibrium distribution function must be obtained in a self-consistent manner. Several years ago, Fixman [11] introduced an iterative scheme (beginning with the equilibrium distribution function), for refining the distribution function with which to carry out the average of the hydrodynamic interaction tensor. The self-consistent scheme of Ottinger is recovered if the iterative procedure is repeated an infinite number of times. However, Fixman carried out the iteration only upto one order higher than the preaveraging stage. The average of the hydrodynamic interaction tensor evaluated with the Gaussian distribution (67)is an ( N - 1) x ( N - 1) matrix with tensor components, Ajk, defined by, o.
--
[ H(&j,k)
Ajk -- Ajk I + x/~h* ~/IJ 5' k[
+
H(&j+l,k+l)
x/'lJ- kl
_
v/l)- k+ iI
I
(8s)
where the tensors &,~ are given by, 1 "
O'l~V- ~ v
--^
0"•'1"
--
H
II~- l/] ]gBr
max(#w)-I Z ~k
j,k=min(#w)
(86)
and the function of the second moments, H(er) is,
3 H(er) - 2(2rr)3/2
f
1 (l_kk
1 e x p ( - k - o ' . k)
(87)
504
Note that the convention H(~jj)/0 - 0 has been adopted in equation (85) above. The self-consistent closure approximation therefore consists of replacing the hydrodynamic interaction tensor Ayk in equation (82) with its nonequilibrium average A---jk. As in the earlier approximations, this leads to a system of ( N - 1) 2 coupled ordinary differential equations for the components of the covariance matrix a)k. Their solution permits the evaluation of the stress tensor through the Kramers expression (69), and as a consequence all the relevant material functions. Viscometric functions in steady simple shear flows were obtained by Ottinger [20] for chains with N ~ 25 beads, while material functions in start-up of steady shear flow, cessation of steady shear flow, and stress relaxation after step-strain were obtained by Wedgewood and Ottinger [41] for chains with N < 15 beads. The latter authors also include consistentlyaveraged FENE springs in their model. Shear rate dependent viscometric functions, and a nonzero positive second normal stress difference are predicted by the self-consistent averaging approximation; a marked improvement over the predictions of the Zimm model. Both the reduced viscosity and the reduced first normal stress difference initially decrease with increasing shear rate. However, for long enough chains, they begin to rise monotonically at higher values of the reduced shear rate ~. This rise is a consequence of the weakening of hydrodynamic interaction in strong flows due to an increase in the separation between the beads of the chain. With increasing shear rate, the material functions tend to the shear rate independent Rouse values, which (for long enough chains), are higher than the zero shear rate consistently-averaged values. The prediction of shear thickening behavior is not in agreement with the shear thinning that is commonly observed experimentally. However, as mentioned earlier, some experiments with very high molecular weight systems seem to suggest the existence of shear thinning followed by shear thickening followed again by shear thinning as the shear rate is increased. While only shear thickening at high shear rates is predicted with Hookean springs, the inclusion of consistently-averaged FENE springs in the model leads to predictions which are qualitatively in agreement with these observations, with the FENE force becoming responsible for the shear thinning at very high shear rates [41,15].
505
The means of examining the accuracy of various approximate treatments of hydrodynamic interaction was established when the problem was solved exactly with the help of Brownian dynamics simulations with full hydrodynamic interaction included [46,48]. These simulations reveal that while the predictions of the shear rate dependence of the viscosity and first normal stress difference by the self-consistent averaging procedure are in qualitative agreement with the Brownian dynamics simulations, they do not agree quantitatively. Further, in contrast to the consistent-averaging prediction, at low shear rates, a negative value for the second normal stress difference is obtained. As noted earlier, the sign of the second normal stress difference has not been conclusively established [3]. The computational intensity of the consistent-averaging approximation leads to an upper bound on the length of chain that can be examined. As a result, it is not possible to discuss the universal shear rate dependence of the viscometric functions predicted by it. On the other hand, it is possible to come to certain general conclusions regarding the nature of the stress tensor in the long chain limit, and to predict the zero shear rate limit of certain universal ratios [20]. Thus, it is possible to show the important result that the polymer contribution to the stress tensor depends only on a length scale and a time scale, and not on the strength of the hydrodynamic interaction parameter h*. In the long chain limit, h* can be absorbed into the basic time constant, and it does not occur in any of the non-dimensional ratios. Indeed this is also true of the finite extensibility parameter b, which can also be shown to have no influence on the long chain rheological properties [22]. The long chain limit of the consistent-averaging approximation is therefore a parameter free model. It is possible to obtain an explicit representation of the modified Kramers matrix for infinitely long chains by introducing continuous variables in place of discrete indices [20]. This enables the analytical calculation of various universal ratios predicted by the consistent-averaging approximation. These predictions are discussed later in this chapter. However, two results are worth highlighting here. Firstly, it can be shown explicitly that the leading order corrections to the large N limit of the various universal ratios are of order (1/v/N), and secondly, there is a special value of h* = 0.2424..., at which the leading order corrections are of order ( l / N ) . These results have proven to be very useful for subsequent numerical explo-
506 ration of the long chain limit in more accurate models of the hydrodynamic interaction. Short chains with consistently-averaged hydrodynamic interaction, as noted earlier, do not show shear thickening behavior; this aspect is revealed only with increasing chain length. Furthermore, it is not clear with the kind of chain lengths that can be examined, whether the minimum in the viscosity and first normal stress curves continue to exist in the long chain limit [20]. The examination of long chain behavior is therefore important since aspects of polymer solution behavior might be revealed that are otherwise hidden when only short chains are considered. The introduction of the decoupling approximation by Magda, Larson and Mackay [18] and Kishbaugh and McHugh [15] made the examination of the shear rate dependence of long chains feasible. The dccoupling approximation retains the accuracy of the self-consistent averaging procedure, but is much more computationally efficient. 4.3. T h e decoupling a p p r o x i m a t i o n The decoupling approximation introduced by Magda et al. [18] and Kishbaugh and McHugh [15] (who use FENE springs in place of the Hookean springs of Magda et al.) consists of extending the 'diagonalise and decouple' procedure of the Rouse and Zimm theories to the case of the self consistently averaged theory. They first transform the connector vectors Qj to a new set of coordinates Q~ using the timc-invariant Rouse orthogonal matrix Iljk. (Kishbaugh and McHugh also use the Zimm orthogonal matrix). The same orthogonal matrix Iljk is then assumed to diagonalise the matrix of tensor components -Ajk. While the process of diagonalisation was exact in the Rouse and Zimm theories, it is an approximation in the case of the decoupling approximation. It implies that even in the self consistently averaged theory the diffusion equation can be solved by the method of separation of variables, and only the ( N - 1 ) transformed coordinate variances ~ are non-zero. The differential equations governing these variances can be derived from the uncoupled diffusion equations and solved numerically. The appropriate material functions are then obtained using the Kramers expression in terms of the transformed coordinates, namely, equations (77)and (78). The decrease in the number of differential equations to be solved, from ( i - 1) 2 for the covariances ~jk to ( N - 1) for the variances dj, is suggested
507
by Kishbaugh and McHugh as the reason for the great reduction in computational time achieved by the decoupling approximation. Prak.~sh and Ottinger [33] discuss the reasons why this argument is incomplete, and point out the inconsistencies in the decoupling procedure. Furthermore, since the results are only as accurate as the consistent averaging approximation, the decoupling approximation is not superior to the consistent averaging method. However, these papers are important since the means by which a reduction in computational intensity may be achieved, without any significant sacrifice in accuracy, was first proposed in them. Further, the persistence of the minimum in the viscosity and first normal stress curves even for very long chains, and the necessity of including FENE springs in order to generate predictions in qualitative agreement with experimental observations in high molecular weight systems, is clearly elucidated in these papers. 4.4. The Gaussian approximation The closure problem for the second moment equation is solved in the preaveraging assumption of Zimm, and in the self consistent averaging method of Ottinger, by replacing the tensor Ajk with an average. As a result, fluctuations in the hydrodynamic interaction are neglected. The Gaussian approximation [24,46,42,48] makes no assumption with regard to the hydrodynamic interaction, but assumes that the solution of the diffusion equation (81) may be approximated by a Gaussian distribution (67). Since all the complicated averages on the right hand side of the second moment equation (82) can be reduced to functions of the second moment with the help of the Gaussian distribution, this approximation makes it a closed equation for the second moments. The evolution equation for the covariances O)k is given by, ..
2kBT A-jk
d -
H H E ( kBT m,l,p
9
9
H
~ [O'jm"--Arnk+ Ajm" O'mk]
+
(88)
where, the ( N - 1) 2 x ( N - 1) 2 matrix with fourth rank tensor components, Flp,jk, is defined by,
3v~ h* [ O(j, l, p, k) K(~j,k) + O(j + 1, l, p, k + 1) g(~+l,k+,) = 4 [ lJ- kl
508
O(j,l,p,k+ 1) K(~,k+l) v/Ij - k - 1]3
O(j+ 1,1,p,k)K(~+l,k)] -
~ i J - k + 1]3
]
(89)
while the function K(er) is defined by the equation, U(~r)
-
- 2 f dk k-~lk _(1 (27r)3/2
kk ~-) k exp(-~l k.~r. k)
(90)
The function 0(j, l, p, k) is unity if 1 and p lie between j and k, and zero otherwise,
O(j,l p,k) - {1 i f j ~ _ l , p < k '
or
k ~_ l,p < j
(91)
0 otherwise
The convention g ( ~ j j ) / 0 - 0, has been adopted in equation (89). Both the hydrodynamic interaction functions H(~r) and g(~r) can be evaluated analytically in terms of elliptic integrals. The properties of these functions are discussed in great detail in the papers by Ottinger and coworkers [22, 24,25,48,47]. All the approximations discussed earlier (with the exception of the decoupling approximation) can be derived by a process of succesive simplification of the explicit results for the Gaussian approximation given above. The equations that govern the self consistently averaged theory can be obtained by dropping the last term in equation (88), which accounts for the presence of fluctuations. Replacing H(~r) by 1 in these truncated equations leads to the governing equations of the Zimm model, while setting h* - 0 leads to the Rouse model. Material functions predicted by the Gaussian approximation in any arbitrary homogeneous flow may be obtained by solving the system of ( N - 1) 2 coupled ordinary differential equations for the components of the covariance matrix O)k (88). Small amplitude oscillatory shear flows and steady shear flow in the limit of zero shear rate have been examined by Ottinger [24] for chains with N ~_ 30 beads, while Zylka [48] has obtained the material functions in steady shear flow for chains with N _~ 15 beads and compared his results with those of Brownian dynamics simulations (the comparison was made for chains with N - 12 beads). The curves predicted by the Gaussian approximation for the storage and loss modulus, G ~and G", as a function of the frequency w, are nearly indistinguishable from the predictions of the Zimm theory, suggesting that the oo
509 Zimm approximation is quite adequate for the prediction of linear viscoelastic properties. There is, however, a significant difference in the prediction of the relaxation spectrum. While the Zimm model predicts a set of ( N - 1) relaxation times with equal relaxation weights, the Gaussian approximation predicts a much larger set of relaxation times than the number of springs in the chain, with relaxation weights that are different and dependent on the strength of the hydrodynamic interaction [28]. These results indicate that entirely different relaxation spectrum lead to similar curves for G ~ and G", and calls into question the common practice of obtaining the relaxation spectrum from experimentally measured curves for G' and G" (see also the discussion in [34]). The zero shear rate viscosity and first normal stress difference predicted by the Gaussian approximation are found to be smaller than the Zimm predictions for all chain lengths. By extrapolating finite chain length results to the infinite chain limit, Ottinger has shown that this reduction is by a factor of 72% - 73%, independent of the strength of the hydrodynamic interaction parameter. Other universal ratios predicted by the Gaussian approximation in the limit of zero shear rate are discussed later in the chapter. A comparison of the predicted shear rate dependence of material functions in simple shear flow with the results of Brownian dynamics simulations reveals that of all the approximate treatments of hydrodynamic interaction introduced so far, the Gaussian approximation is the most accurate [48]. Indeed, at low shear rates, the negative second normal stress difference predicted by the Gaussian approximation is in accordance with the simulations results. Inspite of the accuracy of the Gaussian approximation, its main drawback is its computational intensity, which renders it difficult to examine chains with large values of N. Apart from the need to examine long chains for the reason cited earlier, it also necessary to do so in order to obtain the universal predictions of the model. A recently introduced approximation which enables the evaluation of universal viscometric functions in shear flow is discussed in the section below. Before doing so, however, we first discuss the significant difference that a refined treatment of hydrodynamic interaction makes to the prediction of translational diffusivity in dilute polymer solutions. ~
510
The correct prediction of the scaling of the diffusion coefficient with molecular weight upon introduction of pre-averaged hydrodynamic interaction in the Zimm model demonstrates the significant influence that hydrodynamic interaction has on the translational diffusivity of the macromolecule. While the pre-averaging assumption appears adequate at equilibrium, it predicts a shear rate independent scalar diffusivity even in the presence of a flow field. On the other hand, both the improved treatments of hydrodynamic interaction, namely, consistent averaging and the Gaussian approximation, reveal that the translational diffusivity of a Hookean dumbbell in a flowing homogeneous solution is described by an anisotropic diffusion tensor which is flow rate dependent [23,26,27]. Indeed, unlike in the Zimm case, the three different approaches mentioned earlier for calculating the translational diffusivity do not lead to identical expressions for the diffusion tensor [23,26]. Insight into the origin of the anisotropic and flow rate dependent behavior of the translational diffusivity is obtained when the link between the polymer diffusivity and the shape of the polymer molecule in flow [12] is explored [31,32]. It is found that the solvent flow field alters the distribution of mass about the centre of the dumbbell. As a consequence, the dumbbell experiences an average friction that is anisotropic and flow rate dependent. The discussion of the influence of improved treatments of hydrodynamic interaction on the translational diffusivity has so far been confined to the Hookean dumbbell model. This is because the concept of the center of resistance, which is very useful for simplifying calculations for bead-spring chains in the Zimm case, cannot be employed in these improved treatments [23]. 4.5. T h e twofold n o r m a l a p p r o x i m a t i o n The twofold normal approximation borrows ideas from the decoupling approximation of Magda et al. [18] and Kishbaugh and McHugh [15] in order to reduce the computational intensity of the Gaussian approximation. As in the case of the Gaussian approximation, and unlike in the case of the consistent-averaging and decoupling approximations where it is neglected, fluctuations in the hydrodynamic interaction are included. In a sense, the twofold normal approximation is to the Gaussian approximation, what the decoupling approximation is to the consistent-averaging approximation. The computational efficiency of the decoupling approximation is due both to the reduction in the set of differential equations that must be solved in
511
order to obtain the stress tensor, and to the procedure that is adopted to solve them [33]. These aspects are also responsible for the computational efficiency of the twofold normal approximation. However, the derivation of the reduced set of equations in the twofold normal approximation is significantly different from the scheme adopted in the decoupling approximation; it is more straight forward, and avoids the inconsistencies that are present in the decoupling approximation. Essentially the twofold normal approximation, (a) assumes that the configurational distribution function r is Gaussian, (b) uses the Rouse or the Zimm orthogonal matrix 1-Ijk to map Qj to 'normal' coordinates Q~, and (c) assumes that the covariance matrix ~k is diagonalised by the same orthogonal matrix, ie. Ej, k 1-Ijp~k H k q - (Q~Q~q) - dpSm. This leads to the following equations for the ( N - 1) variances dj, d
2kBT Aj - H [dj . Aj + Aj . dj] H H
A] k
where, Aj = Ajj are the diagonal tensor components of the matrix Ajk, Ajk-
and the matrix
Ajk --
(93)
~ 1-Ilj Alp l-Ipk 1,p
~
l, m, n, p
~jk is given by,
1-Iljl]pk rip,ran l-Imj link
(94)
In equations (93) and (94), the tensors Ajk and ~rr given by equations (85) and (89), respectively. However, the argument of the hydrodynamic interaction functions is now given by, 1 ^
a'tw
--
H
Itt -- v ! kBT
max(#,v)- 1
E
j,k=min(#,v)
E Him nkm o'~ m
(95)
The decoupling approximation is recovered from the twofold normal approximation when the last term in equation (92), which accounts for fluctuations in hydrodynamic interaction, is dropped. Thus the two different routes for finding governing equations for the quantities ~j lead to the same result. However, Prakash and Ottinger [33] have shown that this is ..
512
in some sense a fortuitous result, and indeed the key assumption made in the decoupling approximation regarding the diagonalisation of Ajk is not tenable. The Zimm model in terms of normal modes may be obtained from equation (92) by dropping the last term, and substituting Ajk in place of A--jk. Of course the Zimm orthogonal matrix must be used to carry out the diagonalisation in equation (93). The diagonalised Zimm model reduces to the diagonalised Rouse model upon using the Rouse orthogonal matrix and on setting h* - 0. The evolution equations (92) have been solved to obtain the zero shear rate properties for chains with N ~_ 150, when the Zimm orthogonal matrix is used for the purpose of diagonalisation, and for chains with N _ 400, when the Rouse orthogonal matrix is used. Viscometric functions at finite shear rates in simple shear flows have been obtained for chains with N ~_ 100 [33]. The results are very close to those of the Gaussian approximation; this implies that they must also lie close to the results of exact Brownian dynamics simulations. The reasons for the reduction in computational intensity of the twofold normal approximation are discussed in some detail in [33]. The most important consequence of introducing the twofold normal approximation is that rheological data accumulated for chains with as many as 100 beads can be extrapolated to the limit N -+ c~, and as a result, universal predictions may be obtained.
4.6. Universal properties in theta solvents One of the most important goals of examining the influence of hydrodynamic interactions on polymer dynamics in dilute solutions is the calculation of universal ratios and master curves. These properties do not depend on the mechanical model used to represent the polymer molecule. Consequently, they reflect the most general consequence of the way in which hydrodynamic interaction has been treated in the theory. They are also the best means to compare theoretical predictions with experimental observations since they are parameter free. There appear to be two routes by which the universal predictions of models with hydrodynamic interaction have been obtained so far, namely, by extrapolating finite chain length results to the limit of infinite chain length where the model predictions become parameter free, and by using renormalisation group theory methods. In the former method, there are two essential requirements. The first is
513
that rheological data for finite chains must be generated for large enough values of N so as to be able to extrapolate reliably, ie. with small enough error, to the limit N --4 oc. The second is that some knowledge of the leading order corrections to the infinite chain length limit must be obtained in order to carry out the extrapolation in an efficient manner. It is clear from the discussion of the various approximate treatments of hydrodynamic interaction above that it is possible to obtain universal ratios in the zero shear rate limit in all the cases. Four universal ratios that are frequently used to represent the rheological behavior of dilute polymer solutions in the limit of zero shear rate are [29],
U,p~ Ur
-
_
nkBTAI~TP'~ nkBT~l,o 2 rl~,0
U,R Ur
- lim
77P'~
n-~On~Ts(nTrR3g/3) ~2,0 ~,0
(96)
where, )~1 is the longest relaxation time, and R 9 is the root-mean-square radius of gyration at equilibrium. With regard to the leading order corrections to these ratios, it has been possible to obtain them explicitly only in the consistently-averaged case [20]. In both the Gaussian approximation and the twofold normal approximation it is assumed that the leading order corrections are of the same order, and extrapolation is carried out numerically by plotting the data as a function of (1/v/N). Because of their computational intensity, it is not possible to to obtain the universal shear rate dependence of the viscometric functions predicted by the consistentaveraging and Gaussian approximations. However, it is possible to obtain these master curves with the twofold normal approximation. Table 1 presents the prediction of the universal ratios (96) by the various approximate treatments. Miyaki et al. [19] have experimentally obtained a value of UoR -- 1.49 (6) for polystyrene in cyclohexane at the theta temperature. Figure 1 displays the viscometric functions predicted by the two fold normal approximation. The coincidence of the curves for the different values of h* indicate the parameter free nature of these results. Divergence of the curves at high shear rates implies that the data accumulated for chains with N _< 100 is insufficient to carry out an accurate extrapolation at these shear rates. The incorporation of the effect of hydrodynamic interaction into kinetic theory clearly leads to the prediction of shear thickening at high shear rates even in the long chain limit.
514
Table 1 Universal ratios in the limit of zero shear rate. The exact Zimm values and the Gaussian approximation (GA) values for U,TR and Ur are reproduced from [29], the exact consistent-averaging values from [20], and the renormalisation group (RG) results from [25]. The twofold normal approximation values with the Zimm orthogonal matrix (TNZ) and the remaining GA values are reproduced from [33]. Numbers in parentheses indicate the uncertainity in the last figure.
Zimm CA GA RG TNZ
2.39 2.39 1.835 (1) 1.835 (1)
1.66425 1.66425 1.213 (3) 1.377 1.210 (2)
0.413865 0.413865 0.560 (3) 0.6096 0.5615 (3)
0.0 0.010628 -0.0226 (5) -0.0130 -0.0232 (1)
In both table 1 and figure 1, the results of renormalisation group calculations (RG) are also presented [25,47]. As mentioned earlier, the renormalisation group theory approach is an alternative procedure for obtaining universal results. It is essentially a method for refining the results of a loworder perturbative treatment of hydrodynamic interaction by introducing higher order effects so as to remove the ambiguous definition of the bead size. All the infinitely many interactions for long chains are brought in through the idea of self-similarity. It is a very useful procedure by which a low-order perturbation result, which can account for only a few interactions, is turned into something meaningful. However, systematic results can only be obtained near four dimensions, and one cannot estimate the errors in three dimensions reliably. The Gaussian and twofold normal approximations on the other hand are non-perturbative in nature, and are essentially 'uncontrolled' approximations with an infinite number of higher order terms. It is clear from the figures that the two methods lead to significantly different results at moderate to high shear rates. A minimum in the viscosity and first normal stress difference curves is not predicted by the renormalisation group calculation, while the twofold normal approximation predicts a
515
.0
. . . . . . . .
3.5 3.0 q w
o .......
|
. . . . . . . .
!
0
~
,
..
. . . . .
,
. . . . . .
.-.,
9
9
. . . . h*=0.]5
h =0.15 h* = 0.25
J
RG
16 ~o
o
12
.......
h* = 0.25 RG
2.5 2.0
8
1.5
4
1.0[ . . . . . . . . . . . . . . . . . . . . 10 -1
10 o
101 0.005
"~
j . . . . . .
9 t -, " - r
,~.
0 l---^ :=~-'-~: ........... 10 -1 10 o 101
.........
. O 0 O
" " ~ ~ ' " " ~
~
-0.005 9"~ 9r
-0.010 -0.015 -0.020 ~
o ......
h* = 0.25 RG
-0.025 . . . . . . . . . . . . . . 1(,-' " i0 ~ 10'
p Figure 1. Universal viscometric functions in theta solvents. Reproduced from [33].
small decrease from the zero shear rate value before the monotonic increase at higher shear rates. The good comparison with the results of Brownian dynamics simulations for short chains indicates that the twofold normal approximation is likely to be more accurate than the renormalisation group calculations.
5. C O N C L U S I O N S
This chapter discusses the development of a unified basis for the treatment of non-linear microscopic phenomena in molecular theories of dilute polymer solutions and reviews the recent advances in the treatment of hydrodynamic interaction. In particular, the successive refinements which ultimately lead to the prediction of universal viscometric functions in theta solvents have been highlighted.
516
REFERENCES ~
.
.
.
5. .
.
.
.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Fluid Mechanics, 2nd edn., John Wiley, 1987. R. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory, 2nd edn., John Wiley, 1987. R. B. Bird and H. C. (3ttinger, Annu. Rev. Phys. Chem., 43, (1992) 371-406. C. F. Curtiss and R. B. Bird, Adv. Polymer Sci., 125, (1996), 1-101. C. F. Curtiss, R. B. Bird and O. Hassager, Adv. Chem. Phys., 35 (1976) 31-117. J. des Cloizeaux and G. Jannink, Polymers in Solution, Their Modelling and Structure, Oxford Science Publishers, 1990. P-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, 1979. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986 X. J. Fan, J. Non-Newtonian Fluid Mech., 17 (1985) 125-144. X. J. Fan, J. Chem. Phys., 85 (1986) 6237-6238. M. Fixman, J. Chem. Phys., 45 (1966) 785-792, 793-803. D. A. Hoagland and R. K. Prud'homme, J. Non-Newtonian Fluid Mech., 27 (1988) 223-243. J. G. Kirkwood, Macromolecules, Gordon and Breach, New York, 1967 J. G. Kirkwood and J. Riseman, J. Chem. Phys., 16 (1948) 565-573. A. J. Kishbaugh and A. J. McHugh, J. Non-Newtonian Fluid Mech., 34 (1990) 181-206. H. A. Kramers, Physica, 11 (1944) 1-19. R. G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, 1988. J. J. Magda, R. G. Larson and M. E. Mackay, J. Chem. Phys., 89 (1988) 2504-2513. Y. Miyaki, Y. Einaga, H. Fujita and M. Fukuda, Macromolecules, 13 (1980) 588. H. C. Ottinger, J. Chem. Phys., 86 (1987) 3731-3749. H. C. Ottinger, J. Chem. Phys., 87 (1987) 3156-3165. H. C. Ottinger, J. Non-Newtonian Fluid Mech., 26 (1987) 207-246.
517
23. H. C. ()ttinger, J. Chem. Phys., 87 (1987) 6185-6190. 24. H. C. Ottinger, J. Chem. Phys., 90 (1989) 463-473. 25. H. C. Ottinger and Y. Rabin, J. Non-Newtonian Fluid Mech., 33 (1989) 53-93. 26. H. C. Ottinger, AIChE J., 35 (1989) 279-285. 27. H. C. Ottinger, Coll. Polym. Sci., 267 (1989) 1-8. 28. H. C. Ottinger and W. Zylka, J. Rheol., 36 (1992) 885-910. 29. H. C. Ottinger, Stochastic Processes in Polymeric Fluids, SpringerVerlag, 1996. 30. H. C. Ottinger, Rheologica Acta, 35 (1996) 134-138. 31. J. R. Prakash and R. A. Mashelkar, J.Chem.Phys., 95 (1991) 37433748. 32. J. R. Prakash and R. A. Mashelkar, J. Rheol., 36 (1992) 789-805. 33. J. R. Prakash and H. C. Ottinger, J. Non-Newtonian Fluid Mech., 71 (1997) 245-272. 34. J. R. Prakash, in: M. J. Adams, R. A. Mashelkar, J. R. A. Pearson and A. R. Rennie (Eds.), Dynamics of Complex Fluids, Imperial College Press-The Royal Society, in press, (1997). 35. Y. Rabin and H. C. Ottinger, Euro-Phys. Lett., 13 (1990) 423-428. 36. P. E. Rouse, J. Chem. Phys., 21 (1953) 1272-1280. 37. J. D. Schieber and H. C. Ottinger, J. Chem. Phys., 89 (1988) 69726981. 38. J. D. Schieber, J. Rheol., 37 (1993) 1003-1027. 39. J. D. Schieber and H. C. Ottinger, J. Rheol., 38 (1994) 1909-1924. 40. G. Strobl, The Physics of Polymers, Springer-Verlag, 1996. 41. L. E. Wedgewood and H. C. Ottinger, J. Non-Newtonian Fluid Mech., 27 (1988)245-264. 42. L. E. Wedgewood, J. Non-Newtonian Fluid Mech., 31 (1989) 127-142. 43. L. E. Wedgewood, Rheol. Acta, 32 (1993) 405-417. 44. H. Yamakawa, Modern Theory of Polymer Solutions, Harper ~r Row, New York, 1971. 45. B. H. Zimm, J. Chem. Phys., 24 (1956) 269-281. 46. W. Zylka and H. C. Ottinger, J. Chem. Phys., 90 (1989) 474-480. 47. W. Zylka and H. C. ()ttinger, Macromolecules, 24 (1991) 484-494. 48. W. Zylk~, J. Chem. Phys., 94 (1991) 4628-4636. **
~
o~
**
~176
.o
519
CONSTITUTIVE EQUATIONS FOR VISCOELASTIC LIQUIDS" FORMULATION, ANALYSIS AND COMPARISON WITH DATA A. I. Leonov Department of Polymer Engineering, The University of Akron, Akron, OH 44325 - 0301, USA
1. INTRODUCTION Viscoelastic liquids are capable of accumulating large recoverable strains in flow. This puts the media in an intermediate position between liquids and solids and makes their rheological behavior very complicated. Though the particular properties of elastic liquids have found many useful applications in polymer processing and modern technology, no deep understanding of the nature of viscoelasticity has been reached. Since no fundamental relation is believed to have yet been discovered, at least more than ten popular constitutive equations (CEs) are in competition at present without any clue to a preferable type. For several decades, there have been many attempts to derive CEs for polymer fluids from the viewpoints of mechanics, mathematics and physics. The first approach, which was adopted by many specialists in mechanics, was purely rheological. It postulated nonlinear and quasi-linear relations between the observable variables, the stress tensor cr and the strain rate tensor e. Oldroyd [ 1,2] pioneered the method and also revealed important principles of invariance. The concept was further developed by scientists such as Rivlin, Green and Ericksen and their numerous successors [3]. Later, it was recognized that many rheological equations derived from different approaches are associated with the equations proposed by Oldroyd [ 1,2]. A great many rheological equations, both of differential and integral types have been proposed, and they were able to describe some properties of viscoelastic liquids. The lack of thermodynamic analysis is the main disadvantage of this approach. Hence, such important phenomena as dynamic birefringence, non-isothermal flow, and diffusion
520 cannot be considered simultaneously with rheological constitutive equations. Also, the viscoelastic constitutive equations are often non-evolutionary. The second approach was developed mainly by such scientists as Noll, Coleman, Truesdell, and their colleagues [4-7]. It is the rational mechanics which searched for the most general form of constitutive relations between kinematicand dynamic variables. The basic system involving the constitutive and thermodynamic equations was constructed using strict mathematics, with the CEs having to satisfy the general principles of causality, material objectivity and local action. In this way, the properties of all viscoelastic liquids can be described by a set of hereditary functionals with "fading memory", whose invariance and thermodynamic consistency were perfectly revealed. Unfortunately, there is no unique way to specify the memory functionals and hence predictions are not possible. The third approach is purely physical and explains the behavior of polymeric liquids in terms of the intra- and inter-molecular dynamics. At the beginning, this approach was used to study the behavior of dilute polymer solutions by Kargin, S lonimsky, Kirkwood, Riseman, Rouse, Zimm, Bueche and others (see [8] for a review]). For a long time, concentrated polymer solutions and melts have also been considered as "temporary networks" of entangled chains. Green and Yobolsky [9], Lodge [10], and Yamamoto [1 1] developed semiphenomenological theories that extend the theory for rubber elasticity [12]. This idea was enhanced by the rise of the "reptation" theories, which are due to the work of de Gennes [8], Edwards [13], and Doi and Edwards [14]. In the reptation approach, the creation and decay of the molecular entanglements are studied by the statistical description of a polymer molecule moving along its own axis within a "tube" created by surrounding molecules, and then the motion of the molecule is averaged over high frequency transverse Brownian motion. Another reptational or statistical approach can be found in the works by Curtiss, Bird et al. [ 15-18]. Pokrovsky and Vo|kov [ 19,20] proposed a non-reptational approach based on a generalized Langevin equation for a single macromo|ecule moving in macromolecular environment. A more fundamental approach to the linear viscoelastic properties of polymers has been recently developed by Schweizer [21,22]. This microscopic theory of the dynamics of polymeric liquids called the "mode-mode coupling" (MMC) approach, omits the specific assumptions of reptation theories and formulates the statistical properties of polymers in the most systematic and fundamental way. In contrast to the rational mechanics, the statistical methods have generated a number of specific CEs, and have been generally successful for explaining viscoelastic behavior of polymer liquids in the region of linear or weakly nonlinear deformations. Apart
521
from its poor description of nonlinear data, the physical approaches are also not free from empiricism in formulations and arbitrariness in overcoming mathematical difficulties. Additionally, recent mathematical analyses and numerical simulations revealed examples of unphysical and unstable behavior of the CEs derived by these methods. The fourth method of deriving CEs has been introduced by the author [23,24]. It investigated nonlinear viscoelastic phenomena by the methods of the quasilinear irreversible thermodynamics with the use of a "recoverable strain tensor" as hidden parameter. By this approach, a class of Maxwell-type differential CEs has been proposed under strict stability constraints which are based mainly on thermodynamics. Later, similar approach was adopted by Dashner and Vanarsdale [25] to formulate the general class of CEs almost equivalent to the author's. Almost all CEs proposed in the literature have a limited ability to describe start-up, steady state and relaxation phenomena of polymer fluids in standard rheometric flows, within a relatively narrow region of strain rates usually employed in the tests [26]. However, there are two frustrating problems: (i) none of viscoelastic CE proposed could describe the whole set of available data with one specified set of parameters, (ii) in real modem processing the values of Deborah number, De, may be at least two orders of magnitude higher than those in usual rheological tests, and in that flow region almost all CEs exhibit various instabilities, the reason for which still remains unclear. There are a lot of speculations in the literature on how the instabilities in CEs are related to those observed in flows of polymer fluids There are also contrary opinions about the physical sense of non-evolutionary behavior of viscoelastic constitutive equations (see, e.g. Refs. [27,28] with contrary views on the subject). The temptation to relate the instability in CEs to real flow instability was perhaps caused by the widely spread perception that no rheological constitutive model is globally stable. A few books [ 18,26,29-31 ] and a lot of papers on the polymer rheology do not answer the question as to which CE should be chosen to solve fluid mechanical problems of polymer processing in the usual case of large De number flows, where the nonlinear effects of elasticity are important. Therefore, general studies in this field should be aimed on searching for not only descriptive but also reliable CEs. Thus in our opinion, the principles of choice of CEs should also include, besides the descriptive ability of viscoelastic CEs, such fundamental thermodynamic properties of these as formulation of dissipation and free energy, along with stability constraints.
522 To study these fundamental properties, even for the relatively simple CEs, one has to employ a general formalism within the framework of which it is feasible to find some general stability constraints. Several formal approaches to the viscoelastic CEs have been proposed relatively recently in the literature. One of them is the local quasi-linear approach of non-equilibrium thermodynamics developed by the author [23,24,32]. As mentioned, this approach has resulted in derivation of a class of Maxwell-like CEs, with some thermodynamic constraints. Almost 10 years later, the Poisson-bracket variational approach has also been established. It was first introduced by Grmela [33,34] and then elaborated by Beris and Edwards [35]. This approach, extending the Hamiltonian formalism in classical mechanics to the case of continuum mechanics, also employs the dissipative functional. It was proved [36] that both the above approaches result in the same "canonical" formulation of general Maxwell-like CEs. Another two general formal schemes, consistent with the irreversible thermodynamics, have been developed by Jongschaap et a1.[37,38] and by Kwon and Shen [39]. The first one employs a matrix representation in the phenomenological relations between the thermodynamic forces and fluxes, with additional relations between external and internal (hidden) thermodynamic variables. The second one uses the notion of evolution of the temporary network structure. Regardless of all the differences in their detailed schemes, their equivalence to the general author's approach is evident. Another objective that should be kept in mind when proposing the CEs for polymeric liquids, is solving geometrically complicated flow problems under high deformation rate, as related to industrial processing. Complexity of flow problems for polymeric liquids is enhanced by the effects of fading memory which do not exist in viscous liquids or elastic solids. Hence, viscoelastic polymer fluids demonstrate many unique rheological effects, such as nonlinear evolution of stress under steady deformation, which cannot be seen in the other fluids. Because of this, even geometrically simple problems which may be solved analytically for viscous fluids, have often to be treated numerically in the viscoelastic case. This necessitates the elaboration of certain criteria for the selection of CEs for practical applications. The following principles of choice have been suggested [40,41 ] for the selection of CEs: (i) Stability. However well an unstable CE can describe rheometric tests, it is impossible to use it in modeling of polymer processing, since the Deborah numbers there may be at least two order of magnitude higher and the flow much more complicated. Extrapolation of the majority of CEs to the region of high Deborah numbers and 3D flows usually results in several types of instabilities in numerical flow simulations. These instabilities reflect the mathematical
523
structure of the proposed CEs. In the most cases, they are not related to the physical instabilities observed in the flows of polymeric fluids, or poor numerical algorithms, but rather to violations of some fundamental principles. (ii) Descriptive ability and flexibility. It is now well recognized that polymer melts with similar linear viscoelastic spectra can show qualitatively different nonlinear behavior. For the proper description of various flows, this requires some functions of the kinematic variables, and the associated nonlinear parameters in the CE (which vanish in the linear limit) to be specified within the stability constraints. Once these functional forms and parameters are specified for a particular polymer, the CE must simultaneously describe the entire set of available experimental data fairly accurately. (iii) Computational economy. The proposed CE should allow for numerical calculations in complex flows with as little computational effort as possible. For example, despite the good descriptive ability, it is rather cumbersome to work with models in which the elastic potential is specified in terms of the principal values of a strain measure, and it is usually conceived that working with CEs of differential type is preferable for numerical calculation than with integral ones. (iv) Extensibility. Real polymer processing is confronted with a variety of complications such as compressibility, non-isothermality, wall slip, phase transitions and separations, chemical effects (degradation, curing), etc. In principle, the CE of choice should be amenable to extension in order to accommodate these phenomena. Among the four principles listed above, the first two can be regarded as the most fundamental properties which the CEs should possess.
2.
FORMULATION EQUATIONS
OF
VISCOELASTIC
CONSTITUTIVE
There are two types of viscoelastic constitutive equations which are only in use today for practical applications. These are of differential (Maxwell-like) and single integral types. Therefore we discuss below only the formulation of these two types of CEs, since only they have been tested up to present.
2.1. Formulation of general Maxwell like constitutive equations [36] 2.1.1. Thermodynamics and general single-mode viscoelastic approach We will employ in this section the general approach of irreversible thermodynamics discussed in various books (see, e.g. [42,43]). The fundamental hypothesis of"local equilibrium" underlying the method assumes that even in a
524 non-equilibrium process, in any arbitrarily small macroscopic particle of a medium, it is feasible to operate with the same common thermodynamic functions, depending on the same variables, as in true thermodynamic equilibrium. It is also possible to involve in the analysis some non-equilibrium parameters which vanish in equilibrium; the less these are involved the more economic is the description. This assumes that the system under study is in a sense close to a thermodynamic equilibrium. For viscoelastic liquids, the true thermodynamic equilibrium is at the rest state. The same holds for elastic solids, but for the liquids there additiona|ly exists an incomplete thermodynamic equilibrium in the stressed state. Though strong flows of elastic liquids can be far away from true thermodynamic equilibrium, we can still assume that they are close to the state of incomplete equilibrium which is characteristic for stressed elastic solids. In fact, this fundamental hypothesis underlies all the theoretical treatments of elastic liquids, the molecular approaches, Poissonbracket and matrix formalisms included, and reflects our intuitive view of shear elasticity in liquids. We assume that in the simple case under study, the state variables of elastic liquids are" the temperature T and a hidden variable, symmetric second-rank non-dimensional tensor =c. We assume additionally that the tensor ___eis positive definite, which to some extent, have been justified independently [36,44]. In the following, we employ the Eulerian formulation of the constitutive equations and, without loss of generality, use a Cartesian coordinate system. The physical sense of tensor c can vary, however. In our own studies [23,24] the tensor =ewas treated as the Finger elastic (recoverable) strain C, 1, which may be measured independently. When related to molecular approaches, the tensor c is proportional to the averaged diadic built up by the end-to-end vector of a part of a macromolecule between two entanglements. In these approaches, the configuration tensor is a typical internal variable and the problem of how to measure it directly is questionable. Being mostly interested in studies of isothermal flows of isotropic elastic liquids, we can now introduce as a proper thermodynamic potential, the free energy density per mass unit, F = F( T, It, 12, 13). Here /1 - tre,
12 -
1/2(I12-tr__e2),
13 - detc,
(1)
are the basic invariants of the tensor g. Then the "thermodynamic stress tensor" 0% associated with the free energy F, as in the quasi-equilibrium elastic case [45], is introduced as follows" (o~ = 2pg=.c3F/Oc = 2p[F,c_- + F2(I,c=- s + Ffl,6=] (Fj = OF/OIj) (2)
525
Here 9 is the density and fi is the unit tensor. The only physical reason to operate with the equation (2) is the explicit assumption [23,24] that polymeric liquids always possess an "elastic limit", a quasi-equilibrium situation achieved on very rapid (instantaneous) deformations, where the temporary entanglements in macromolecules act like cross-links in cross-linked rubbers. Thus, in this limit, when g --+ _Cl the use of equation (2) can be justified. It should also be noted that no assumption of incompressibility has been made. We now briefly discuss the dissipative effects [23,24] which are associated with flows of elastic liquids. By using the laws of mass and energy conservation as well as the momentum balance, and separating the total entropy variation in the flux and origin ("entropy production"), one can easily obtain the ClausiusDuhem expression for the entropy production Ps in the case under study: TPs = -~I.VT + tr(=o-.e) - pdF/dt [ T
(3)
Here ~/is the thermal flux, __o-isthe actual stress tensor, g is the strain-rate tensor, and the symbol d/dt means the time derivative in the frame of reference associated with a moving particle of liquid. The first term on the right-hand side of equation (3) reflects non-isothermal effects and could be considered as independent of the other two, which are related to mechanical dissipation. According to the Second Law, P~ should be positive for all non-equilibrium processes and vanish at equilibrium. Considering further the isothermal behavior of elastic liquids. Using equation (2), we can rewrite the dissipative terms in (3) as follows" D = TPs[T = tr(cr-e) - tr(_O-e'e-l"1/2d__e/dt),
(4)
where D is the mechanical dissipation. The next step in developing the non-equilibrium thermodynamic approach is to represent the dissipation as a characteristic bilinear form D = ~ k " Yk where Xk are thermodynamic "forces" and Yk are conjugated thermodynamic "fluxes". As thermodynamic forces, we can naturally take the actual, o-, and equilibrium, =O-e, stresses. Then the thermodynamic flux conjugated to the actual stress =o-is certainly the strain-rate tensor g. Another thermodynamic flux, ee, conjugated to the quantity ere is a tensor which is related but is by no means equal to 1
-c 2 =
-I
.de/dt.
This is because the latter does not even satisfy the frame-
=
invariance conditions. This situation demonstrates an ambiguity in definition of
526 the second thermodynamic flux, e~. Mathematically, it associates with arbitrariness in finding a tensor from a scalar product of two tensors. Thus an additional physical assumption is needed to define the unknown quantity e~. This has been proposed in [23] as follows: not only thermodynamic force, cr~ but also conjugated to this, the thermodynamic flux ee, have to be defined as in the case o f quasi-equilibrium elastic solids. This assumption seems to match reasonably well the local equilibrium hypothesis. It also results in the following o
procedure" to find the solution, e~, from the kinematic relation c=c.__e+e.__c,valid for elastic solids (or for the total continuum) and hold the expression for the o
non-equilibrium case. Here c is the co-rotational or Jaumann time derivative of the tensor c. It is exactly the same procedure that we have used in the definition of the equilibrium stress tensor ere. In so doing we can define the quantity e, as the solution of the equation" o
f = C.ee + ee'_C_,
(5)
which has been obtained in an explicit form [24], too awkward to be reproduced here. It is easy to see from equation (5) that the term pdF/dt I T in equation (3) is reduced to tr(o-.e~), which, in turn, can be represented in the form of the second term in the dissipation equality (4). It should also be noted that this derivation, being general enough, was confirmed by an independent specific analysis of kinematics of viscoelastic deformations [24,46,47] when ___e= C~l. If the nonequilibrium stress _O-pand strain rate ep are defined as follows, _O-p- g - o-~,
__ep- __e-ee
the dissipation inequality due to equation (5) is rewritten in the form: D - TPs[T - tr(_O-p.e) + tr(o-~.ep).
(6)
Equation (6) is now represented as the typical bilinear form D = f~(k'Yk discussed above, where C~pand ee are independent thermodynamic forces, and ___ep and ee are independent thermodynamic fluxes. Two independent sources of mechanical dissipation are now clearly seen from equation (6); both of them being positive defined and vanishing in the equilibrium. They are" (i) the power produced by the irreversible stress _O-pon the total strain rate =e (the first term in (6)), and (ii) the power produced by the reversible stress o-~ on the irreversible
527
strain rate ep (the second term in (6)). Using quasi-linear scheme of irreversible thermodynamics, these were connected in [23,24,46] by phenomenological relations with kinetic coefficients represented as some rank four tensors depending on the internal parameter, tensor g. In presenting these relations, the Onsager symmetry of kinetic coefficients, proved for quasi-linear case in [43], has also been used. The structure of the kinetic rank-four tensor had also been completely revealed [24]. An interested reader can find the details in papers [24,46,47]. This strict, straightforward and complete approach can be compared with somewhat controversial matrix approach [43,44].
2.1.2. Maxwell models with quasi-equilibrium stress. We will consider from now on only the Maxwell liquids for which the actual stress tensor is equal to the thermodynamic stress, i.e. =o-= fie- (=O-p= 0). In this case, the first term in equation (6) vanishes. Then the quasi-linear phenomenological relation between the thermodynamic force, _fie and the generalized thermodynamic flux, gp, is of the form" ep,ij = m0kl(T,c)O'e,kl(Y,c).
(7)
Here M is a rank-four mobility tensor, an isotropic tensor function depending on the configuration tensor =e; the stress tensor _o-~now having the form shown in equation (2). Substituting (7) into the dissipative equality (6) with O-p = 0 represents the latter in the quadratic form of the stress tensor, D = M~st-cr~-=o-~.~t. This in turn results in the fact that the mobility tensor has to be positive definite and has the following symmetry properties" it is symmetric in the first two and second two indices and in transposition of these indices. Also, substituting expression (7) for ge into equation (4) yields the evolution equation for the configuration tensor c" V
c+~c)/0(T) = O,
~c)/O(T) = 2C-ep(T,c).
(~j(T,c)/O(T)
=
2CikMkist(T,c)O's,(_~._C))
(8) V
Here c is the upper convected time derivative and ~ c ) is a non-dimensional isotropic tensor function of tensor c defined as in equation (8) and related to the dissipative processes in an elastic liquid. Additionally, in the rest state when c--+ =_8(I1--+I2---~3; I3---~1), we have" =d(T,c)---~=0. We assume that this limit transition is regular, meaning that there is a limit to the linear Maxwell viscoelastic equation when the intensity of strain rate is very low. In the incompressible case with
528
linear dependencies Cre(=e)and =d(=c), equations (2) and (8) are easily reduced to the upper convected Maxwell model. Multiplying the first eqation (8) from the right and from the left by g-~ reduces the latter to the a "dual form'" + s
= 0,
(b = __e-')
(8a)
where b is the lower convected time derivative of the tensor b. Also, in the incompressible ease with linear dependencies of ~ and ~ ) , eqations (2) and (8) are reduced to the usual form of the lower convected Maxwell model. The difference between these is mainly in the dependence of stress on elastic strain" =o-= G=e corresponds to the case of rubber elasticity, whereas =o-= -Gb can describe the elasticity of crystals or metals. Equations (1),(2) and (8) give rise to the energy relation:
r
+ D - tr(Cre.e),
D - tr(g~- Cre.gb)/(20(T)),
(9)
where D is the mechanical dissipation. If the new kinetic rank-four tensor L(e=) is introduced as follows" Lijkl = CimCjnMnmkl
(10)
the dissipative term ~___e)in equation (8) is rewritten in the form"
~j( e=)/O(T) = LoglpOF/c3ckl
(11)
Substituting equation (11) into (8) represents the latter in the form which coincides with the evolution equation obtained by Beris and Edwards (see equation (2.13) in the second part of Ref.[35]). The tensor L has the structure [35] similar to that known for the tensor M. We now analyze the compressibility condition in the general Maxwell model described by equations (2) and (8). Multiplying equation (8) scalarly by (2=e)1 gives:
1/2d/dt(lnl3) + trep - tr__e; trep - 1/2tr[__c-l.=d(_c_)/0(T)], (12) where the function =d(c) was defined in equation (8). There is also the mass conservation equation which can be written in the form"
529
d/dt(lnp/po) = -tre.
(13)
Here/90 is the density in the rest state. Then, combining equations (12) and (13) yields:
d/dt[ln(,O/po. ~-3 ) + trep = O.
(14)
Equation (14) shows that P---~Po when c--+_6, since in this limit, trep--+0. It also shows [23,24] that if
trep = 0,
#/9o = 1/-4~-3,
(15)
exactly as in the equilibrium limit of elastic solids. Thus, the evolution equation (8) for tensor ___calso describes the law of mass conservation in compressible elastic liquid only if the condition tr__ep = 0 is satisfied. When trep ~ 0, the density variations are not described anymore by the configuration tensor c but satisfy the kinetic equation (13) or the equation of mass conservation (12). It means that, in this case, the density is not a state (thermodynamic) variable and an attempt to improve the situation by making the assumption that F = F(T,p,e=) contradicts the Murnaghan's definition (2) of the thermodynamic stress. This was the only reason why in our publications [23,24,46] we also employed the condition trep = 0. A simple constitutive equation for the compressible part of stress has been discussed in [23,24]. It includes the equilibrium pressure and also KelvinVoight modeling of compressible effects. This approach has been recently extended for non-isothermal linear compressibility phenomena with a bulk relaxation spectrum when introducing a set of scalar independent hidden parameters {~k} [48]. Consider now the incompressible case when p =/9o - const. When tr__ep :/: 0, equations (8) still hold, whereas equations (13) and (14) are reduced to: 1/2d/dt(ln/3) + tr__ep- O; tr__e= O.
(16)
Surprisingly enough, in this case, the free energy remains the same, i.e.
F=F(T,II,I2,I3), because the tensor =e is not directly related to the density. By introducing the modified free energy / ? - F - p represented by
lnI 3, the actual stress =0-can be
530 __o-=-p=~+ ere.
(17)
Here the isotropic term 'p'(a Lagrange multiplier) in equation (17) is introduced to satisfy the condition of incompressibility tre = 0; the thermodynamic tensor ere being still defined by equation (2). This means that the tensor cr~ is not defined with the accuracy of an isotropic term as in the equilibrium case, and isotropic pressure is not an equilibrium one. These features result in the fact that, even in the incompressible case, the expression (6) for the dissipation is now not invariant under the transformation, _o-e--~o-~+p__6. When the constraint trep = 0 is used, equation (15) results in the incompressibility condition [23,24],/3 = detc = 1. In this case, the modified elastic potential is introduced as" W = W(T, II,I2)-f~(I 3- 1), where W = pie. The ^
magnitude ~ here is the isotropic pressure. It is used in differentiation in equation (2) as the Lagrange multiplier, allowing us to consider all cij variables as independent. Then equation (2) results in the Finger formula for the stress: - -p6_+ 2 Wle- 2 W2c -l.
(18)
Here Wk= OWlOIk and p = ~-/;'2/2. Also in this case,/2 = trc l and the formulae for free energy and the stress tensor are exactly the same as for elastic solids. Also, isotropic pressure and the expression for dissipation (6) are invariant under the transformation: _O-e--)Cr~+ p=6. We finally stress that as shown in this section, any attempt to extend the configuration tensor approach to the compressible case is thermodynamically inconsistent. Since the Maxwell type models involving configuration tensor c were derived from the evolution equation for distribution function, it simply means that the later equation should be somehow modified. It is unknown at present what should be done to rectify the situation.
2.1.3. Maxwell models with non-equilibrium and non-potential stress tensor[36]. The above thermodynamic derivation of Maxwell-like constitutive equations was further extended [36] to include formally into consideration the incompressible Gordon/Schowalter [49], Johnson/Segalman [50] and PhanThien/Tanner [51,52] models. For this reason, instead of the natural, quasiequilibrium relation (5), the following non-equilibrium evolution equation for the tensor c was proposed"
531 o
c = ~(__C'ee+ ee'__c)
(5a)
Here ~ is a numerical parameter (-1 <~< 1, %~0). The case %= 0, related to the corotational derivative, can be also easily considered but we shall omit this. Then following the same procedure, described in the derivation of equation (8), we can finally obtain [36]: o
__o -= -p__6+ __ere• Crex= Cre(g)/~,
_c- ~(e.e + =e.___c)+ &(T,g)/O(T) = 0,
(19)
D = tr(Crex-&). In equations (19), written here in so-called canonical form, the formula for the extra stress tensor Crex is given due to some thermodynamic reasons [26]. Note that when ~,~+1, equations (19) have no perfect elastic limit and therefore they are non-equilibrium. One can also ignore the thermodynamic origin in the stress formulation ere(C) and consider formally the second formula in equation (19) as a general nonpotential stress-elastic strain relation O-~x= Crex(C,~). It can be formally done even in the case ~,=+1, where there is a limit to elastic solid behavior, ~(T,_e_)--+=0. However, in this case, it is always possible to consider such a loading-unloading procedure that will create work from nothing [53]. It means that this approach inevitably leads to the theoretical eternal motion machine and therefore is thermodynamically forbidden.
2.1.4. Examples of single mode Maxwell-type constitutive equations. We now illustrate how the particular Maxwell-type CEs can be obtained from the above general equations by specifying the terms ~ and ere in equations (19) and (2), or F in equation (1). When ~b~= g-_6,
(2p/G)F = I~-3,
ere = G__e,
(20)
where p and G are the shear modulus and density, equations (19) represent an interpolated Maxwell or the Gordon/Schowalter [49] - Johnson/Segalman [50] model. When ~=+1 or ~=0, it is respectively called the upper/lower convected or co-rotational Maxwell model. When
532
0 = 0o(T)f(I1),
g~ = g-=b,
(2p/G)F = I~-3,
_O-e=G___c,
(21 )
it becomes the general Phan-Thien/Tanner model [51,52], where f(I1) was proposed as a linear or exponential, increasing function of I1,. Again, when ~=1, we particularly call it an upper convected Phan-Thien/Tanner model. When 0 = Oo(T)f(IIe), IIe=tr(__e2), ~= 1, ~=__c-__6",(2,o/G)F=Ii-3, O-e=Gg,
(22)
where f is a decreasing function of lie, it reduces to the White-Metzner CE [54], which does not belong to the general class of quasi-linear CEs. The FENE model (see, e.g., Section 8.5.3 in Ref. [26]) can be written as: Y_,=1, 0==K(I~)c-=d, (2p/G)F=(Ic3) 1nK(I,), o-~=GK(I~)c, K(I, ) = (Ic- 3 )/([c-ll ),
(23)
where Ic=Rcz and Re is the ultimate finite dumbbell length. When ~=1, =dr=otc2+(1-2ot)c-(1-c~)__d(0
__o-~=Gs
(24)
it represents the Giesekus model [55], with a numerical parameter or. The case ~=1, =d~=B(I~)(c-__6), (2p/G)F=(3/~)lnB(I1), B(I1)--1+~(I,-3)/3
_O-e=G__c/B(I1), (25)
corresponds to the canonical representation [36] of the Larson differential model [56]. The general class of the author's incompressible CEs [23,24,47] is represented as:
~=1, r l /2c[b~(c-I~ 5/3 )-b2(c~-12~_/3)], O-e=2Wlc-2W2c-~
(26)
Here W(I~,Iz) (=pF) is the strain energy function for incompressible case. The positive functions bi(Ii,I2) should have a proper linear viscoelastic limit and their positive definiteness suffices the positive definiteness of the dissipation. The convexity constraints,
533 Fl>0,
F11Fzz>F12 z, (Fi=c3FIc3Ii,Fij=c3Filc3Ij)
F2>0,
(27)
imposed on the general form of potential F were also suggested [23,24,47]. The important implication of inequalities (27) and the proper use of this class of CEs are discussed in detail in [40]. In the simple case of bl=b2=l, with the neoHookean potential for F, it reduces to the simplest "Leonov model" which includes no nonlinear parameters.
2.1.5. Multi-mode Maxwell models. All the above single mode viscoelastic constitutive equations of Maxwell-type are usually extended to the multi-mode case. We briefly discuss below only models with potential stresses in the common incompressible case. The typical N-mode extension is as follows" N
F - y'Fk(T, ck), k=l
N
cy - ~ r ---ex
-=
(T,c ) --- ex,k
=k
(28)
~
Here Fk, Crex,kand Ck are the free energy, extra stress tensor and configuration tensor, respectively, in "k"th relaxation mode. Additionally, the general form of evolution equation presented in equation (18) holds for every configuration tensor s In the limit of very low Deborah number, this approach shows the linear viscoelastic behavior, with a discrete spectrum of relaxation times { Ok}. It means that every nonlinear relaxation mode is generated by the corresponding linear viscoelastic mode. The generalization to the multi-mode case can be justified only if the various relaxation modes are well separated, i.e. 01>>02>>...>>ON
(29)
In this case, it is reasonable to assume that the various relaxation modes act independently. All the known experimental datatestify in favor of inequalities (29). Additionally, the guess-independent Pade-Laplace method (see e.g. [57]) reveals the effective discrete linear relaxation spectrum in accord with (29).
2.2. Formulation of nonlinear single integral constitutive equations From a wide class of viscoelastic CEs of the integral type, only the single integral ones have been experimentally tested. In the common incompressible case, its general form is represented as [58]"
534 !
o%•= ~ [qg,(I,,I2,t- x)C- qg2(II,I2,t-
"r.)C-l]dx
(30)
--00
Here _O-ex is the extra stress tensor, C is the Finger total strain tensor for incompressible media, whose time evolution is described as follows" v
C-dC/dt-C.Vv-(Vv_) _ _
_._
_~_
r -C-O; ~
___
=
C[
=8 I =
T
(31)
- -
Here I l= trC, I2 = trC -1, and ~, and q32 are generally independent functions. We can also introduce many other measures of deformations. One of them, the Hencky measure, H = (l/2)lnC, will be used below. Experiments show that the simplified time-strain separable version of equation (31),
6k (Ii,/2,t-x) = m(t-x)q)k(I1,/2)
(k=l,2)
(32)
can be introduced. Here re(t-x)- dG(t-x)/dx, and G(t) is the relaxation modulus. Equations (30) and (31) are not the only single integral form of CEs. When, for example, the mixed convected time derivative is used, the CE can also be represented in an integral form (see, e.g., Ref [26]). Also, Kaye [57] and Bernstein et al. [60] proposed the potential form of equations (30) and (31):
(O, =(2p/G)OP/DI~,
~p2=(2,o/G)0~'/012;
(33)
or in the time-strain separable case"
q)~=(2p/a)3F/Ol~,
tp2=(2p/a)OF/OI2.
(34)
Here G is the elastic Hookean modulus. Equations (33) or (34) constitute the KBKZ class of single integral CEs. The potential F in equations (33) denotes the thermodynamic free energy with relaxation effects taken into account. For the time-strain separable viscoelastic CEs with potential F, the basic functionals such as the free energy W, the extra stress tensor o-e and the dissipation D are of the form [61 ]: ^
535
(35)
W = (p/G) iF ( I, ,I 2 )m(t - z)dx -oo
t
(36)
o-~ = (2p/G) IC. c3F/~C(I,,I 2 ) m ( t - , ) d , --O0
t
D - tr(crce)-dW/dt = Co~G) IF (I~,I 2 ) [ d m ( t - x) / dxld~
(37)
--oo
A comparison of equations (2) and (8) with equations (35) and (36) clearly shows that CEs of the differential type where the dissipation and free energy are generally independent, are more flexible for rheological modeling than CEs of the integral type where those quantities are roughly proportional. We now describe some particular CEs of separable single integral type by specifying q~, and q~2 in equation (30) or the potential F for the K-BKZ type in equations (34). Wagner et al. [62] proposed their first specification as: q~l=fexp(-nl 4I - 3 )+(1-J)exp(-n2 ~/I - 3 ),
I=flll+(1- /2;
r (38)
where f, nl, /
q 2:
h(I~,/2) = l/(ot-,/~), z = (/1-3)(I2-3). (39)
Here again, a and, , are positive fitting parameters. We further refer to the specification (38) as the Wagner model I, and (39) as the Wagner model II. Luo and Tanner [64] proposed the modification of the CE originally presented by Papanastasiou et al. [65] in the form: qo,= oth(Ii,I2),
qo2= ~:oth(IiJ2);
h(IiJ2) = [ot-3+flll+(1-tS')I2]1,
(40)
where c~, /3 and ~: are fitting parameters and K>0. When K=0, this model becomes identical to the one proposed by Papanastasiou et al. [65]. One can easily see that no thermodynamic potential relation exists (q012 ~ q~21) in all of the above three CEs (38)-(40).
536
The Lodge model [10], which is reduced to the integral presentation of UCM[1], q01=l,
q)2=0,
(41)
is a particular case of the separable K-BKZ class (34) when the neo-Hookean potential is used. In order to describe better the viscometric data, Larson and Monroe [66] suggested the following 4-parametric form ofF for the K-BKZ class: (2p/G)F = (3/2ct)1 n[ 1+~(I-3)/3], I = (1-fl)ll + ~ / 1 + 21312 - 1, Ot=Ko+K2tanq[K1A3/(1+A2)], A=I2-I1
(42)
Here Ko, K1, ~:2 and fl are numerical fitting constants. Another potential form was derived by Currie [67] as a close approximation of the Doi-Edwards reptation model [ 14]. It is written as (2p/G)F = (5/2)1 n[(J- 1)/7],
J = ll + 2x/I2+ 1 3 / 4 ,
(43)
and contains no nonlinear fitting parameters. A linear combination of simple potential forms, t G2 e x p ( - t 2p/~=. G! exp( -~- ). In[1 + ~( I , - 3)] + 21302 -~2 )" In[1 + 13(12 - 3)] + 2or0 -
G3 exp( - t 20--7 O3 )" ( I , - 3)
(44)
was also introduced by Yen and McIntire [68] as a partially time-strain separable version of the general potential presented by Zapas [69].Here the relation between the potential /~ and CE is shown in equations (33), Gi's and 0,'s are moduli and relaxation times, respectively, and a and fl are nonlinear fitting parameters. It should also be noted that the thermodynamic potentials proposed in the theory of rubber elasticity can in principle be applied in the case of K-BKZ class of integral CEs as well as for CEs of the differential type. In many cases, they are presented as a function of principal values Ci of the Finger tensor rather
537
than invariants of the total Finger deformation tensor. Below are several examples of these potentials.
(i) Ogden potential [70]: 2PF= ~ G,, (C.,,~ 1 2 +C~",~2 + C ~,,~2 3 _ 3)
(45)
Here Gtn are numerical parameters which can be negative or positive, and Gn are shear moduli. The potential (45) becomes identical to the Mooney potential [71] if it contains only two modes corresponding to a 1=2 and cx2= -2.
(ii) Valanis/Landel potential [72]: 3
2pF= i~[aff-Q, (lnff-C , - 1 ) + 131n.fQ-~]
(46)
where c~ and 13 are parameters with dimensionality of modulus.
(iii) The BSTpotential [73 ] 2pF-- A I ~ + B _
_
n
o
m IE,
I E - 1 (C~,,/2 , + C ,~2 2 + C ,3/ 2 ) . ~
n
n
(47)
n
Here A and B are the parameters with the dimensionality of modulus, whereas n and m are numerical parameters. As mentioned, the non-potential viscoelastic CEs proposed in the literature, are unphysical, since when applying very fast deformations, it is possible to create a perpetual motion machine from a hypothetical material subordinated to this type of CE [53]. Nevertheless, we will analyze below this type of CEs too, considering it as mathematical abstraction for the superficial data curve fitting. To make the following analyses more efficient, we now introduce a unified set of notations for both differential and single integral types of CEs which employ only upper convected time derivatives in the evolution equations. The lower convected time derivative (~=-1) in equation (19), can be equivalently rewritten in the form of upper one (~=1) [36], as shown by equation (8a). We introduce a modified pressure term defined as:
538
for differential CEs P
t
+ jq)212m(t- 1:)d1:
for integral CEs
(48)
--cO
Then using the Cayley-Hamilton identity and the invariance of rheological variables (say, the extra stress o-~) under arbitrary addition of isotropic terms in the case of incompressibility, we represents both the classes of CEs as follows: ____o-=-p'=d+ O-e,
O-~= 2,~__.SF/SR, for differential CEs
o
=e
[_~m(t_,)E(c)dx
for integral CEs
(49)
E = 2p[q01c + q02(Ilc - c 2) + q0113~ Here 8/8g is in general the partial Frechet derivative with respect to c, and with the definition for q)i in equations (2) and (30). In this notation, c becomes the total Finger strain tensor __Cin the case of integral CEs, and thus for the CEs of integral type and the author's CEs, I3=1, q)3=0 automatically. Even though the set (40) is written for hyper-viscoelastic equations, the non-potential viscoelastic formulations can also be included into consideration. The (elastic or total) strain tensor c is the solution of the following evolution problem: t el,__o = =8
v
for differential CEs (50)
= = =
=
[ Ct=t,
= C] t = t t = =8,
~(c) - =0 __ =
for integral CEs
Here ~bis the dissipative term which vanishes for integral CEs, and from now on we consider only upper convected time derivatives even for differential models.
3. STABILITY OF VISCOELASTIC CONSTITUTIVE EQUATIONS
3.1. Background In numerical simulations of viscoelastic flows, degradation of the numerical solution or lack of convergence of computational schemes has been frequently
539
observed for large or even modest values of Deborah numbers. It is thought that the main reason for this numerical malady is an improper choice of a CE which possess some bad mathematical properties (e.g., see Ref. [74], p.314). Therefore the analysis of such properties of various viscoelastic CEs as their stability and boundedness seems to be a pre-requisite for any successful numerical modeling. It will be shown in this Section that these properties are in turn, originated in our capability to incorporate basic thermodynamics laws in formulation of the CEs. We discuss in this section recent results of stability analyses for isothermal formulations of CEs for viscoelastic liquids. To begin with, we briefly describe the basic aspects of stability analysis in general, and two types of instabilities related to the formulation of viscoelastic CEs in particular. The general purpose of the stability analyses is to find the conditions of boundedness for a solution of a set of functional equations subordinate to some additional (usually initial and boundary) conditions. In our case, the total set of equations consists of momentum balance and continuity equations coupled with viscoelastic CEs. No-slip boundary conditions are usually applied at the rigid surfaces and the common set of kinematic and dynamic boundary conditions is used at the unknown free surfaces. It is assumed a "basic" solution of the set (generally, 3D and time dependent) to exist, subordinate to initial and boundary conditions. The solution is said to be stable in the Lyapunov sense, if "small" disturbances imposed on the solution at time to, which satisfy the homogeneous boundary conditions, remain to be small at any time t (to
540 The purpose of the stability analyses discussed in the following sections is not to describe the real physical instabilities observed in flows of viscoelastic liquids, but rather to reveal some constraints which should be imposed on the formulation of CEs to prevent them from the occurrence of unphysical instabilities. Therefore, we call the stability considered in the subsequent analyses not the stability of viscoelastic flows but the stability of CEs. These are related to the formulation of CEs. The formulation of nonlinear viscoelastic CEs is not as straightforward as it appears at first sight. For many specifications of CEs in the above two classes, numerical modeling of high Deborah number flows has displayed heavy unphysical instabilities, regardless of the "true physics" employed in their derivation. Two types of instabilities related to the formulation of CEs were observed for large or even modest values of Deborah numbers, and have been analyzed in the literature. These are the Hadamard and dissipative instabilities. The case of Hadamard stability is the most understood. We define the complete set of equations for viscoelastic liquids as Hadamard stable (or evolutionary, or well-posed) when the solution of the boundary-value Cauchy problems for the set at any time provides the complete initial conditions for determining the solution at subsequent instants in time [75]. Thus, the Hadamard stability allows one to continue the solutions in the positive direction of the time axis. When this is impossible, very quick blow-up instability occurs, with extremely short wave disturbances, which results in progressive failure in numerical calculations: the finer the mesh, the worse will be the degradation of the results [76]. In many cases, one can treat the Hadamard instability as a blow-up type increase in the amplitude of initially infinitesimal waves of disturbances as the wavelength tends to zero. In viscoelastic liquids, this type of instability can be associated with a nonlinear rapid response of CEs. Hence, this type of instability depends on such quasi-equilibrium properties of the CEs as the type of differential operator in the evolution equation for differential models and the elastic potential in the hyper-viscoelastic case. Rutkevich [77] initiated the study of Hadamard stability for viscoelastic CEs. Later, Godunov [78] analyzed some aspects of the stability in more detail. Some significant results were obtained relatively recently by Dupret and Marchal [75] and Joseph and co-workers (see, e.g., Joseph's monograph [76]). These studies which have analyzed the Hadamard instabilities in particular flows for particular specifications of differential viscoelastic CEs, can be summarized as follows. Joseph and co-workers [79,80] and independently Dupret and Marchal [75] proved that the interpolated Maxwell model, which involves the mixed time derivative in the evolution equation, is Hadamard unstable, except for the
541 cases of upper and lower convected time derivatives. The ill-posedness of the Gordon/Schowalter [49] or Johnson/Segalman [50], and the original PhanThien/Tanner [51,52] CEs is subject to this type of instability. Using the general method of characteristics, Dupret and Marchal [75] also showed that the WhiteMetzner model [54] is nonevolutionary, which was again justified by Verdier and Joseph [79], who employed a perturbation method in their analysis. It was found that the dependence of the relaxation time on the invariant of strain rate tensor is the cause of Hadamard instability in the White-Metzner model. Analyzing the White-Metzner model in elongational flow, Verdier and Joseph [81] also noticed a type of dissipative instability that occurs whenever the extensional strain rate exceeds the half of the reciprocal relaxation time. The method of characteristics is the most general mathematical tool for the Hadamard stability analysis. This method is, however, sometimes complicated and cumbersome. Fortunately, in the most interesting cases, it can be simplified by using the "frozen coefficient" method (see, e.g. [76]), when analyzing only extremely short and high frequency wave disturbances propagating with a finite speed. Those cases are related to all the CEs of the quasi-linear differential type as well as the time-strain separable single integral CEs. Then, the linear stability analysis of the problem can be studied locally, without considering boundary conditions. Although following Kreiss' examples [82], this local stability condition is neither necessary nor sufficient for the overall stability in the general case of nonlinear partial differential equations, for quasi-linear differential (and time-strain separable single integral) equations, the local stability analysis with the method of frozen coefficients can be employed without loss of generality [83]. The studies of Hadamard stability have also a long history in the theory of nonlinear elasticity, where many general results were obtained and understood in great detail. Among the many general stability conditions suggested, the simplest is known as the Baker-Ericksen inequality [84]. Its physical sense is that the maximal principal stress always occurs in the direction of the maximal principal strain. In the case of hyperelastic solids, the thermodynamic stability criteria called "GCN + conditions" were also established long ago (see, e.g., Section 52 in Ref. [4]). Their physical sense is the convexity of the elastic potential with respect to the Hencky strain measure. This has been also known as a condition for strong monotonicity of stress with respect to strain. In nonlinear elasticity, the Hadamard stability criteria of field equations correspond to the conditions of strong ellipticity which coincide with the stability requirements known for dynamic problems. The weaker conditions of ordinary ellipticity deliver the stability constraints for static states [85].
542 The GCN + condition is closely associated with the condition of strong ellipticity. The lack of symmetry in the representation of second-rank variables causes a more restrictive condition for the strong ellipticity than for the GCN + condition [4]. Therefore, such inequalities as the Baker-Ericksen and the GCN + may be treated as necessary conditions for the strong ellipticity or the Hadamard stability. For the important case of isotropic incompressible hyperelastic solids, Zee and Sternberg [85] have recently found the necessary and sufficient conditions for the strong ellipticity (Hadamard stability) of CEs in a form of algebraic inequalities imposed on the first and second derivatives of the elastic potential. For the compressible case, these results have been obtained earlier [86]. For viscoelastic CEs, general results on global Hadamard stability, i.e. stability for any type of flow and for any Deborah number, were recently obtained for both general classes of quasilinear Maxwell-like CEs [36] and time-strain separable single integral CEs [87]. It was found later that these algebraic criteria for Hadamard stability were confused with the necessary and sufficient conditions for thermodynamic stability, that is, GCN +. These convexity conditions for thermodynamic potential in the hyperelastic case impose weaker constraints on CEs than the criteria for Hadamard stability. The complete results on the Hadamard instabilities for both the classes of viscoelastic CEs under study were then found and published in paper [88] for incompressible viscoelastic CEs. For compressible case, the complete criteria for Hadamard stability of viscoelastic CEs were recently found by Kwon [89]. In contrast to the Hadamard instability, another important type of instability, of dissipative type, results from the formulations of the non-equilibrium (dissipative) terms in CEs. This type of instability can occur even if the dissipation is positive definite. The studies of dissipative instabilities, initiated in paper [36] for general Maxwell-like CEs, were motivated by the fact that the upper convected Maxwell model which is globally Hadamard stable, displays the unbounded growth of stress in simple extension when the elongation rate exceeds the half of the reciprocal relaxation time. For any regular flow with a given history, a sufficient condition for the dissipative stability, close to the necessary one, was proposed in [36] for CEs of the differential type. Then the necessary and sufficient condition for single integral CEs was found in paper [90]. It was also noticed in those papers that in many viscoelastic flows, neither strain nor stress histories are known, but rather a complex mixture of both. Several patterns of pathological behavior as predicted by some popular specifications of Maxwell-like CEs, related to dissipative instability, were exposed in 1D flows [91 ]. It was found that it is necessary for the dissipative
543
stability that both the steady flow curves in simple shear and in simple elongation have to be monotonously and unboundedly increasing. The combined Hadamard and dissipative stabilities have also been analyzed in paper [88] for incompressible viscoelastic CEs. The results of this analysis will be discussed at the end of this Section.
3.2 Hadamard stability criteria for viscoelastic constitutive equations We now briefly outline the mathematical procedure [88] to obtain the conditions for global Hadamard stability for quasi-linear differential and timestrain separable single integral CEs in incompressible isothermal flow of viscoelastic liquid. It has been proved several times [36,75,79,80] that the evolution equation (19) with the mixed convected time derivatives (~=+1) is Hadamard unstable. Therefore, we consider below only the case of upper convected time derivative in the evolution equation (19) of the differential type and single integral CEs of the form (36). Then the total set of equations in this stability problem consists of viscoelastic CEs (48)-(50) complemented by the equations of momentum balance and incompressibility: pdv_/dt =-Vp +V-o,
V.v=O.
(51)
We assume that the set has a solution {__c,v,p} which satisfies some proper initial and boundary conditions, and impose on the solution extremely short and high frequency, infinitesimal waves of disturbances" {8g, 8g, 8v, 8p} = ~{ ~, g, v, p }.exp[i(k.x-mt)/g 21
(52)
Here c is a small amplitude parameter (leJ << 1), c, v, p and p are (generally complex) amplitudes of the corresponding disturbances, k is a wave vector, and co is the frequency. Using just the local linear stability analysis, we can easily find the following "dispersion relation", i.e. the dependence of the frequency co on the wave vector k and the parameters of the basic flow: for differential models
1 ~,)2~2 _ I B~j'n~V~kjVmk~ 2
[_~m(t - t 1)Bij,~mdt 1 9v i k j V m k n kjvj -0.
for integral models
(53)
544
Here ~ = co- k.v is the frequency of oscillations, with Doppler's shift on the basic velocity field v taken into account. The last equation in (53) means that due to the incompressibility condition in (51), the wave vector of disturbances k is orthogonal to the amplitude of velocity dis.turbance v. The fourth rank tensor Bijmn is defined in the principal axes of tensor __cwith principal values ck as: Bijnm- (~,n~n + ~ln~,n )Gij + ~j~nnLim, Gij = (Ci q- Cj)[(q)l q- q)2(I1 - C i - Cj)],
(no sum!)
(54)
Lij = 4q)nfifj + 2q~12CiCj(211 - Ci- Cj) - 2q)21(CiCj-1+ ci'lfj) Formulae (54) show that the tensor Bjjmn depends linearly on the constitutive functions q0k and their second derivatives q0kj with respect to the basic invariants Ik, independently of whether the approach is potential or not. It is now seen that due to equation (52) and the definition of ~ , the requirement for the stability is that the left-hand side of equation (53) should be positive. Therefore, the necessary and sufficient condition for global Hadamard stability is reduced to the following: 1 ~-~2V2
2
- BijnmvikjVmk n > 0
(55)
For the integral type constitutive equations, the condition (55) provides the stability during relaxation, which is included in the global stability requirement. On the other hand, the convexity of potential F, or the GCN + condition, can be represented as follows"
B ~jmn[30]3,,m > 0,
-02 F B ijmn -- Oh ~j0h,m
= 4Cmq
0c qn
eip
0C pj
Here hij is the Hencky strain measure, h__ = (1/2) I n c . To guarantee the thermodynamic stability, the inequality in (56) should be satisfied for any arbitrary symmetric tensor ,fl,j with the condition of incompressibility, trl~ = 0. In the potential (hyper-viscoelastic) case, the identity, Bijnm = B ijmn, holds. The comparison between the inequalities (55) and (56) shows that due to the symmetry of the tensor fl,j, the condition (56) is included in the inequality (55), i.e. the condition (56) imposes weaker stability constraints than the inequality
545
(55). It means that the conditions for the Hadamard stability are stronger than those of GCN +. Employing the algebraic procedure which has been used in hyper-elasticity [85], one can finally obtain the necessary and sufficient conditions for the global Hadamard stability as the set of algebraic constraints imposed on the functions q~k: (i) lai > O,
~i = ((Pl + (p2Ci)~k/CjCk
(i~j~k),
(ii) r q- 21-ti> O, q i =(Ii-Ci)(qOl+q02Ci)+2(Ii2-212-Ci 2-
(iii) [
,+21ai +
21
q>O,
s. )[q)llW(q)12+q)21)ci-l-q)22Ci 2] Ci
(57)
j + 2 ~ j ] 2 > g k - 2 ~ t k (iCj~k).
The additional constraint in (i), the positive definiteness of the tensor c, holds by definition for the integral CEs. It was also proved for the Maxwell-like CEs of differential type [36,102], however, only for the flow situations with a given history. The above approach to the global Hadamard stability has been recently extended by Kwon [93] on the compressible case. The new quality which occurs there is the possibility of longitudinal wave propagation. In the incompressible case, the speed of the longitudinal wave approaches infinity, whereas the speed of the transverse wave is finite. Hence, perturbation of basic solutions by the longitudinal wave was not considered in this stability analysis. The result was that the wave vector is always orthogonal to the vector of the main velocity field. However, in the compressible case, the speeds of both waves have finite values. Thus for stability, the initially infinitesimal amplitude of disturbing waves of either type (or a mixed type) should remain small all the time. It means that the conditions of Hadamard stability in compressible case are more rigid than those for incompressible one. It was demonstrated [89] on the simple example of Mooney-Rivlin potential with additional term dependent on density. Two sufficient conditions for the incompressible case have also been proposed: (1) The author's condition (27): the thermodynamic potential F for the author's class of viscoelastic CEs is a monotonously increasing convex function of invariants I1, and I2.
546 (2) Renardy's condition" the thermodynamic potential F for the K-BKZ class of CEs is a monotonously increasing convex function of ~ and Although Renardy's condition has been proved only for the K-BKZ class, it also holds for the Maxwell-like CEs with upper convected time derivatives [88]. Since the author's condition (1) is stronger than Renardy's (2), it also guarantees the global Hadamard stability for the K-BKZ class. The above sufficient conditions are much more easier to employ than the necessary and sufficient conditions for Hadamard stability (57). Therefore they are very useful for a brief evaluation of the stability for new formulations of CEs. For the compressible CEs, one sufficient condition for the global stability has also been suggested [89], but it is too complicated to use.
3.3 Dissipative stability criteria for viscoelastic constitutive equations As mentioned, there can be another source of instability originated from specification of dissipative terms in viscoelastic CEs. For viscoelastic CEs of differential type, this instability may happen due to an improper formulation of the dissipative term ~ (or ~b when ~ = 1) in equations (19), even for the Hadamard stable CEs with positively definite dissipation. For single integral CEs, the instability results from fading memory effects in equations (30) and (36). Although the global criteria for dissipative stability of viscoelastic CEs are far from being complete (if it is in general possible), we discuss in this Section two specific criteria that have been proven. In the case of compressible flow, no theorem on dissipative stability is known yet, but the following theorems are presumably valid also for the compressible CEs.
3.3.1 Criterion I of dissipative stability Theorem 1.1 (the case of CEs of the differential type [36]). Consider the set of upper convected Maxwell-like CEs (8) with the positive dissipation D = D(T, Ii, 12, 13) defined in equation (9). Let the free energy F be a non-decreasing smooth function of three invariants Ik. If for any positive number E, the asymptotic inequality
O > E'll ell
when Ilcll oo
(11 11- (trc2)m)
(58)
holds, then in any regular flow, the configuration tensor =e and the stress tensor ere are limited.
547
Theorem 1.2 (the case of single integral CEs [90]). In any regular flow, the functionals of free energy (35) and dissipation (37) are bounded, if (and only if) the thermodynamically or Hadamard stable potential function F(H1,H2,H3), expressed in terms of principal Hencky strains Hk, increases more slowly than exponentially. In the theorem 1.2, principal values of Hencky strain tensor and Finger tensor for the total deformation are related as" Hi =
(1/2)lnCi,
or
=/1= (1/2)lnC,
trH = 0.
(59)
Detailed proofs and definitions are given in the papers [36,90]. While the theorem 1.1 has been proved for differential CEs as a sufficient condition close to the necessary one, theorem 1.2 provides the necessary and sufficient condition for boundedness of single integral CEs. The above theorems were motivated by the fact that the globally Hadamard stable upper convected Maxwell model displays the unbounded growth of stress in simple extension when the elongation rate exceeds the half of the reciprocal relaxation time. As the consequences of the above theorems, (i) the upper convected Maxwell model which violates Criterion I, and (ii) the K-BKZ class with a potential F represented as an increasing rational polynomial function of basic invariants Iu, are dissipative unstable. Therefore, the Mooney and the neoHookean potentials as well as the potentials for the K-BKZ class of CEs which are subordinate to Renardy's sufficient evolution criterion also violate Criterion I of dissipative stability. Since the satisfaction of only Criterion I cannot prevent the viscoelastic CEs from severe dissipative instability, an additional criterion for dissipative stability has been introduced.
3.3.2 Criterion II of dissipative stability [88] For the stability of Maxwell-like and time-strain separable single integral CEs, it is necessary that both the steady flow curves in simple shear and in simple elongation have to be monotonously and unboundedly increasing with respect to the strain rate. It has been demonstrated [91] that the violation of Criterion II results in "blow-up" instability or even negative principal values of tensor __c in simple shear. Therefore the subordination to the combined criterion "I+II" was assumed in [88] to be presumably sufficient for the dissipative stability of both the differential Maxwell-like and the time-strain separable single integral CEs, at least in simple flows.
548 3.4. Application to viscoelastic CEs. Discussion Both the Hadamard and dissipative types of instability for such two broad classes of viscoelastic CEs have been discussed in this Section. These are the quasi-linear differential and factorable single integral models with instantaneous elasticity, which are the only ones in practical use today. The problem of global Hadamard stability for these two classes of CEs seems to be completely resolved in the isothermal, incompressible and compressible cases. This problem was reduced to that well known in the nonlinear elasticity, where the complete set of necessary and sufficient conditions of stability was formulated in algebraic form. It has been demonstrated that the proposed analysis of Hadamard stability for the two classes of viscoelastic liquids is reduced to the analysis of strong ellipticity. The physical sense of this is very evident: the studies of Hadamard stability involve very rapid disturbances which create only elastic response in viscoelastic liquids. In the case of dissipative stability, the global analysis is far from being completed, if it is generally possible. However, two distinct patterns of dissipative instability have been revealed, which are related to (i) the boundedness of stress, free energy and dissipation in a start-up flow problem under a given strain history (Criterion I), and (ii) the monotonously and unboundedly increasing steady flow curves in simple shear and simple elongation (Criterion II). Furthermore, it was assumed that the subordination of CEs to the combined criterion "I + II" is presumably sufficient for the dissipative stability in the simple flows. There is a tough problem as to how to distinguish the unstable behaviors caused by poor modeling of CEs and the observed physical instabilities which those equations should also describe. However, the long history of various branches of continuum mechanics and physics teaches us that the occurrence of either Hadamard instability or/and ill-posedness in ID situations without such important physical reasons as phase transitions, etc., is a distinct sign of inappropriateness in the CEs. Thus we can treat the instabilities demonstrated in this section as being associated not with the real instabilities observed in flows of polymer melts, but rather with the improper modeling of various terms in CEs. In numerical simulations of complex flows with unstable CEs, when the flow rate becomes high enough, the occurrence of various types of unphysical instabilities is inevitable. Even in the range of moderate Deborah numbers, the existence of singular points in flow geometry such as the comer singularity in die entrance region, is sufficient to spoil the entire numerical procedure.
549 All the results of the stability analyses found in various studies for popular viscoelastic CEs, are summarized in Table 1. An interested reader can find the details of calculations in references also provided in the Table 1. It is noteworthy that CEs derived from molecular approaches such as the Larson and the Currie models, exhibit the most unstable behavior. Surprisingly enough, none (to the authors' knowledge) of the time-strain separable single integral models are evolutionary. Appendix A represents the explanation of the reasons for that given by Simhambhatla [94]. He analyzed the time-strain separability concept for CEs and concluded that the Hadamard unstable CEs of time-strain separable type cannot properly describe the experimental data of stress relaxation after step-wise loading. The instabilities revealed in Ref. [94] exactly correspond to the results reviewed in this Section. It is astonishing that many CEs become Hadamard unstable even in viscometric flows. For the CEs of differential type, only three stable specifications exist. These are the FENE, the upper convected Phan-Thien-Tanner models, and the author class of CE's (8), (26) under convexity constraints (27). However, the FENE and the upper convected Phan-Thien-Tanner models predict zero value for the second normal stress difference in simple shear flow, which contradicts the experimental evidence for polymer melts and concentrated polymer solutions. It should be noted that all the necessary and sufficient conditions obtained for single-mode CEs become, strictly speaking, only sufficient for the multi-modal approach. Even though the necessity is not proved, it is thought that due to the inequalities (29), i.e. well separateness in the relaxation times, the exact conditions for Hadamard stability exposed above for a single mode CE, will be closed to necessary for multi-mode approach. It is also evident that the threshold of instability would only be delayed to some higher Deborah number region in the multi-modal approach, if any single-mode is unstable. For some viscoelastic CEs, regularization of ill-posedness may be achieved. E.g., it is well-known that adding a small Newtonian term to the stress stabilizes Hadamard unstable CEs. However, for complex flow simulations, this may not be enough to suppress numerical instability, and when the Newtonian term becomes larger, the description of the CE will deviate from the experimental data. In the case of Hadamard stable but dissipative unstable CEs which violate the Criterion II, one can also propose the more fundamental procedure of stabilization by changing the elastic potential. For example, the Giesekus model with or
550 Table 1 shows that the combined stability criteria impose very tough constraints on viscoelastic CEs. Therefore, the serious question arises as to whether there exists a CE or a class of CEs which can properly describe all the available rheometric data for concentrated polymer solutions and melts, when satisfying all the stability constraints. We will describe such a class in the next Section. Table I Stability of viscoelastic constitutive equations Type of instability Model (Eq. #) Type of CE Dissipative Upper convected Quasilinear unstable(Criterion I) Maxwell (20) (~=1) differential Hadamard unstable Interpolated Quasilinear Maxwell(20) differential (~,1) (Johnson/Segalman Gordon/Schowalter) Hadamard unstable General Phan-Thien/ Quasilinear (~.l) Tanner (21) differential Hadamard stable; Upper convected Quasilinear dissipative stability Phan-Thien-Tanner differential depends on (21) (~=1) dissipative term Nonlinear differential Hadamard unstable, White-Metzner(22) dissipative unstable (Criterion I) Globally Hadamard FENE(23) Quasilinear and dissipative stable differential Dissipative unstable Giesekus(24) Quasilinear (Criterion II) differential Dissipative unstable Simplest Leonov(26) Quasilinear (Criterion II) (bl, = b 2 = 1) differential Globally Hadamard Leonov class (26) Quasilinear dissipative stable under stability and differential constraints (27) Hadamard unstable Larson (25) Quasilinear differential dissipative unstable (Criterion II) single Hadamard unstable Wagner I (38) Separable integral single Hadamard unstable Wagner II (39) Separable integral single Hadamard unstable Papanastasiou (40) Separable (K=0) integral
References e.g.[36] [36,75,76,80]
[36,75,76,80] [88]
[75,81] [88] [91] [91] [88] [88] [91] [88] [88] [90]
551 Luo-Tanner (40) Lodge(41) K-BKZclassunder Renardy's condition Larson-Monroe potential (42)
Separable single integral Separable single integral K - B K Z Separable K-BKZ Separable K-BKZ
Currie potential (43)
Separable K-BKZ
Yen-Mclntire (44)
Quasi-separable K-BKZ
Hadamardunstable
[88]
Dissipative unstable [90] (CriterionI) Dissipative unstable [90] (Criterion 1) Hadamardunstable [88] dissipative unstable (Criterion II) Hadamard unstable dissipative unstable (Criterion II) Dissipative unstable (Criterion 1)
[90] [88] [95] [90]
4. MODELING OF POLYMER FLUIDS WITH STABLE CONSTITUTIVE EQUATIONS Following mostly the paper [40], we demonstrate in this Section the class of differential CEs [23,24,36] which is able to consistently describe simple flow data for such basic polymers as HDPE II, PS I, PIB P-20, PIB L-80 and LDPE Melt I/ IUPAC A/ IUPAC X, while complying with the global isothermal stability constraints. For simplicity, only one or two nonlinear parameters additional to the discretized linear viscoelastic spectra are introduced for the description of data. Instead of the simple "Leonov model" which uses only the parameters of the discretized linear viscoelastic spectrum, we employ in the following sections a highly nonlinear specification of the general class of Maxwell-type CEs proposed by the author [23,24,36]. This specification subordinated to the convexity conditions (27) guaranties the both Hadamard and dissipative stabilities. Comparison of the descriptive ability of other models with experimental data is not attempted in this Section. 4.1. Selection o f a descriptive subclass from the author's CEs For the sake of simplicity, only the constitutive modeling of viscoelastic liquids with incompressible Maxwell-type CEs is considered below, with the equations shown for a single relaxation mode. The model equations are given by equations (2), (8), (26) and (27).
552
We consider firstly the modeling of dissipative terms in evolution equations (8) and (26). The following simple forms of equation (26) have been considered
[40]: i) b~ = b(I~,/2,T)/20(T), b2 = 0; ii) b~ = 0,
b2 = b(I~,/z,T)/20(T);
(60)
iii) b~ = b2 = b(I~,Iz,T)/20(T). Here, 0 = 0(T) is the relaxation time in the linear Maxwell limit. The specification i) with the Neo-Hookean elastic potential, results in a decreasing branch of the flow curve in simple shear for b = 1. This poor quality, resulting in dissipative instabilities, can be rectified by specifying more sophisticated functional dependencies either for b(I~,I2,T), or the elastic potential. However, this form was rejected due to the inconvenience for practical modeling. With the specification ii), various simple flows can be described quite accurately for several polymers. The only problem with ii) is the weak maxima of Nl predicted during start up shear flow, in comparison with experimental observations. No way of rectifying this malady was found and we therefore reject this specification also. The form iii) for which the evolution equation (8), (26) can be written as: v
20(T)c +b(I, ,12,T) [c2 + c.(I2-I,)/3 - __6]= __0
(61)
is the only one discussed and tested in the literature. It allows for plane deformations in simple shear and endows the resulting equations with the proper quality and flexibility for modeling a wide variety of polymers. Detailed comparisons with experimental data have been made [40] with this representation. In order to relate the extra stress tensor to the elastic Finger tensor during the deformation history, a functional form for the elastic potential W(I1,/z,T) = poF must be provided. Here, Po is the density, and F is the specific Helmholtz free energy. The following fairly general elastic potential:
3G(T)
W(II,I2,T) - 2(n+ 1) {( 1-[3)[(I~/3)n+1-1] + (1-13)[(I,/3)n +'-1 ],
(62)
553
has been suggested in [40]. Here G(T) is linear Hookean elastic modulus, and 13 and n are numerical parameters. With the evolution equation (61) and potential relation (62), the constitutive equations are Hadamard and dissipative stable for 0_< 13 <1 and n >0 (see the sufficient condition O for Hadamard stability and criterion 'I +II' for dissipative stability) except when the criteria for "fluidity loss" are met (see below). Equation (62) yields the Mooney potential for n = 0, and the neo-Hookean potential for n = 13= 0. The extra stress tensor can then be written due to equation (18) as: flex = (1-13) (I1/3) n __C-l] (12/3) n C"1.
(63)
In the multi-mode approach employed below the same values of nonlinear constants for each mode are used. This gives a few-parametric description of the data. Also, the conditions of Hadamard and dissipative stability used for every mode will obviously satisfy the stability criteria for the complex model. For modeling isothermal experimental data, we need specifications for the function b(1t,12), as well as a simple reduction of the potential W(It, I2) suggested in equation (62). For practical modeling, b(I1,/2) can be thought of as a deformation-history dependent scaling factor for the linear relaxation times. The simplest choice is to let b = 1, which is known as the standard "Leonov model". Henceforth, the CE with b=l and the neo-Hookean potential (n = 13= 0) is further referred as the "simple model". While this simple choice assures the proper linear viscoelastic limit, and can also be expected to describe weak nonlinearities, it may not suffice for the description of highly nonlinear phenomena. For instance, some polymers (e.g., LDPE with a high degree of branching) show great hardening in simple elongation, while others (e.g., HDPE, PS) do not. In any simple flow, if b(Ii,/2) is chosen to decrease gradually (e.g., using weak power law) with an increase in the magnitude of 11 and 12, there will be hardening relative to the simple model. A rapid fall in b(Ii,/2) (e.g., exponential) will cause the steady state components of c to be double-valued up to a critical value of the strain rate, with one stable branch. Beyond this critical value, there is no steady-state solution and the components of c increase unboundedly as in an elastic solid. This is the concept of "fluidity loss" analyzed in detail by the author [23,32]. On the other hand, an increasing b(Ii,/2) will cause relative softening. The more rapidly b increases, the more gradual will be the variation of the steady-state value of c_with an increase in the strain rate. The choice of b(I~,/2) is not as difficult as it appears. The recommended procedure is to first perform some preliminary calculations for various flows
554
with the simple model. Then, if there is disagreement with experimental data, an appropriate functional form of b function required to bring the calculations into qualitative agreement with the data, can be systematically determined. The reason this procedure is straightforward is that for this class of equations, the effects in various flows are well separated in the sense that there is some measure of flexibility in modeling their qualitative behavior independently. This is a direct consequence of the fact that whether the simple deformations are steady or non-steady, the following relationships hold for the invariants of the elastic Finger tensor c: I~ = 12
in simple shear and planar elongation
11 > 12 in simple elongation, and
(64)
11 < I2 in equi-biaxial extension.
These relations also hold for the invariants of the total Finger deformation tensor C. This is a remarkable feature of the evolution equation (61). The experience [40] in modeling the viscoelastic behavior of several polymers (LDPE, HDPE, PS, low and high molecular weight PIB) with the function b(I~,/2) showed that simple power law or exponential functions of the invariants with one or two adjustable parameters are sufficient for an accurate quantitative description of all the available data. However, unlike the elastic potential (62), no any unified form for b(I1,/2) has been found which could be used for the description of data for all polymer melts. Because of the flexibility which the modeling of the dissipative term permits, one can operate with fairly simple forms of the general elastic potential (62), such as the neo-Hookean and Mooney potentials for the description of the usual rheometrical data. However, in simple shear, with this approach, cl2 generally reaches a limit value of unity at high Deborah numbers leading to a saturation of the shear stress and therefore to dissipative instability. One remedy is to extend the discrete relaxation time spectrum at the small relaxation time end in order to effect stability until the region where physical instabilities appear. A simpler approach is to work with the existing discrete relaxation modes obtained from the usual linear experiments, and to allow the parameter n in the potential relation (62), to be a small positive number (e.g., n = 0.1). In this way, we can preserve all the predictions in the rheometric regimes with simple potentials and also effect the non-saturation of the stress at very high Deborah numbers. Still, for concrete recommendations for quantitatively modeling high
555
Deborah number flows, more data in the region of incipient physical instabilities would be welcome.
4.2. Component equationsfor simple flows For convenience, the equations and initial conditions for simple flows for a single Maxwell mode are presented below. These equations and formulae should be employed in a multi-mode approach for all the predictive calculations in comparison with data.
4.2.1. Simple shear The evolution equations take the form" 20dCll /dt + b(I)(c~ + c 212- 1) - 4,~0c 12 (65)
20dc12 / dt + b(I)Cl2 (Cll +
C22 ) --
2~C22
Cl.C.., ..-l+c
-I-1
+ c 1 + C 1 22 9
212 9~
I
1
-I
2
The system of stresses is: cy12 - G(I / 3) n C12 ;
N 1 -
G(I / 3) n ( e l l
_ C2 2 ) ;
(66)
N 2 - G(I/3)n[(1 -- ]3)(C22 -- 1)+ ]3(1-- C11)]
Here, ~, is the shear rate, O"12 is the shear stress, Nl and N2 are the first and second normal stress differences, respectively.
Startupflow The initial conditions for startup flow are: cijl t=o- 8ij.
(67)
The steady-state solution of equations (65) is of the form" Cll -- - ~ Z / 4 Z + 1;
r
Z ( I ) - (I / 2)(I - 1) 2 - 1.
- ~
/ 4 z + 1;
Ol). - ~ / z ~ - 1 / ( z + 1); (68)
556 Here, I can be obtained by solving the implicit equation" Z ( I ) - 41 + (20~ / b(I)) 2 .
(69)
Stress relaxation Here, ~, = 0. For stress relaxation following cessation of shear flow, the initial conditions for equations (65) are:
c,j[ t_-o- cij]v,t ,
(70)
where ts is the shearing time prior to cessation of flow. For the stress relaxation following the imposition of a step strain 7o, the initial conditions for equations (65), are: ell
]t = O * - 1 + 7 2. o,
]
c22 t-o*
-1;
c 12 ]t=o* -70
(71)
9
Creep and recovery For creep, let a shear stress ,:3-~ be applied at time t = 0. Let the shear strain at time t = +0 after the jump be %. Then the initial conditions for the kinematic variables are given by equation (71). The value of ~,o is obtained by substituting for C12 from equation (71) into equation (66) and setting O"12 - - CY~ For the multi-modal case, ~ ~2 is the total stress defined as the sum of the sub-stresses in the various modes. Then the shear rate, ~; is calculated by setting dcrlz/dt = 0 in equation (66) and substituting for the time derivatives of the kinetic variables from equation (65). In the multi-modal case, r~2is the total shear stress. Evidently, d 7 / dt - ~
(7] t=0" - 70).
(72)
For N modes, the set of 2N+ 1 differential equations, two per mode in equation (65) and one in equation (72), can then be solved by using e.g., a Runge-Kutta scheme with an automatically adapting step size. The conditions for recovery following unloading can be similarly derived.
4.2.2. Simple elongation The evolution equation is:
X-~dX/dt+b(I~,I2)(X 2-X-~)/(6~.0)-~;;
I~-~2+2X-~
12
_ ~-2 +2~.
557 (73) where ~ is the elongation rate. The elongation stress is: G"E
=
G [ ( 1 - 13)(1, / 3) n (X 2 - X-') + 13(I2 / 3)(X - X-2 )].
(74)
Startup flow The initial condition for startup flow is: Lit_ o - 1.
(75)
The steady-state solution for X is obtained from the implicit equation: (76)
b(I, ,I 2 )(~2 _ X-' )(1 - k-') = 6/~0.
Stress relaxation Here, ~=0. For stress relaxation following cessation of elongation flow, the initial conditions for equations (73) is:
~l t=0-- ~l+,t
,
(77)
where te is the time of extension before cessation of flow. For stress relaxation following the imposition of a step Hencky strain eo, the initial condition is: )~l t=o" - exp(eo)"
(78)
The procedure for creep and recovery calculations is analogous to that for simple shear.
4.2.3. Planar extension The evolution equation is: ~-' d~ / dt + b(I)(X 2 - ~-2) / (40(T)) = ~p;
I, - 12 - I - )~2 + )C2 + 1.
(79)
Here ~p is the planar extension rate. The system of planar extension stresses is
558
o p, - 0,, - ~22 - G(I / 3)" ( 1 - 213)(X2 - )v-2) 0 p, - 0 22 - 0 33 - G(I / 3)" [13)v2 + ( 1 - 13))v2 - 1].
(8o)
Startup flow The initial condition for startup flow is" )q t - o - 1 .
(81)
The steady-state solution for 9~ is obtained from the equation: b(I)(L 2 - X-2) - 4~p0.
(82)
Stress relaxation Here, /~p= 0. For stress relaxation following cessation of planar extension flow, the initial condition for equation (79) is:
~1t=0-- ~l tp,~p,
(83)
where tB is duration of biaxial extension before cessation of flow. The initial condition for stress relaxation following the imposition of a step biaxial Henckey strain eBo may be written as: ~1 t=O -- exp(e Po ).
(84)
Creep and recovery calculation can be performed as for the other simple flows.
4.2.4. Equi-biaxial extension The evolution equation is: L-'dX / dt + b(I,,I2)(X 2 - ~-,)(~2 + 1) / ( 1 2 0 ) - ~B; I~ - 2X 2 + X -~"
I2 - 2X-2 + X~
~
(85)
Here, ~ B is the biaxial extension rate. The biaxial stress is" ~B - G[(1 - 13)(1, / 3) n (X: -- ~-4 ) ..1_ ]~(i 2 / 3) n (~4 __ X-2 )].
(86)
559 Startup flow The initial condition for startup flow is"
)q t=o- 1.
(87)
The steady-state solution for L is obtained from the equation: b(I,,I2)(~, 2 - X-4)(X2 + 1 ) - 12/~,0.
(88)
Stress relaxation Here, ~B= 0. For stress relaxation following cessation of biaxial flow, the initial condition for equation (85) is"
~] t=0-- )L]g,,tB,
(89)
where tB is duration of biaxial extension before cessation of flow. The initial condition for stress relaxation following the imposition of a step biaxial Henckey strain eBo may be written as" XIt-o - exp(~:Bo)"
(90)
Creep and recovery calculation can be performed as for the other simple flows.
4.3. On the comparison with experimental data High density polyethylene HDPE-II, polystyrene PS-I, polyisobutylene (PIB) P-20, a relatively high molecular weight PIB, Exxon Vistanex L-80, and low density polyethylene LDPE Melt-I have been chosen in paper [40] to compare the predictions of the above constitutive equations with data. A numerous amount of data and calculations involved in the comparison demonstrated generally a great success in our modeling. The interested reader can find a lot of useful details in Ref.[40]. For four first polymers in the tested group, the specification of the function b(Ii,I2) was uniform and proposed as follows: b(h) = exp[m(/j3 - 1)].
(91)
560 Eq.(91) means that the four first tested polymers demonstrate softening behavior at high strain rates. For describing the data for LDPE Melt I with the author's class of CEs, some preliminary calculations were first performed with the simple model, as suggested in the section 3.1 [40]. With this approach, the description of shear flows was quite accurate. However, the biaxial extension damping function was overpredicted, while the hardening effects in extension flow were underpredicted. To rectify the observed discrepancies with the simple model, the relations (60) suggest that b(11,I2) should be a decreasing function of (IJlz) (see section 3.1 for the physical sense of the function b). A simple choice was made as: b(11,I2) = (I2/11) '~ .
(92)
The parameter 'm' was chosen to be 1.4, for properly describing the extension stress growth data. It was a hope [40] that with formula (93) for b(I1,I2) all the available experimental data could be described reasonably well. Indeed, the calculations according to this choice of b(I1,I2) could describe properly almost all the data for the Melt I but they failed to describe the planar elongation tests [107]. The reason for this was that in the planar elongation, as in the simple shear, 11 = 12. Thus the hypothetical rheological behavior in the planar elongation, as predicted by formula (92), is softening. However, this prediction contradicts the hardening phenomena in planar elongation, observed experimentally [ 107]. Therefore in Appendix B, a new, more physically related attempt is presented to describe the whole set of data for LDPE Melt I.
5. CONCLUSIONS The behavior of two common classes of viscoelastic constitutive equations (CEs) for polymer melts and concentrated polymer solutions was discussed. These are general Maxwell-like and single integral CEs with instantaneous elasticity. The formulation of both classes of CEs was analyzed. The Maxwell-like CEs usually employ some hidden tensor variables with different physical senses. Therefore, in spite of the generality in formulation, their evolution equations and stress relations have different features, depending on the theoretical approach used. Some artifacts related to formulation of the CEs were also exposed. Such an important effect as compressibility was discussed.
561
General results on stability for both classes of CEs were demonstrated, which included stability analyses of both the Hadamard and dissipative types. Results of the stability analyses were applied to popular CEs. The descriptive capability of a class of Maxwell-like CEs was demonstrated, whose formulation satisfies all the stability constraints. It should be noted that the data [98] for equi-biaxial extension were obtained with using lubricated squeezing technique. Recent publications [113,114] reported that experiments with this technique can involve undesirable side effects, such as distortion of sample shape [113] or uncontrolled thinning of lubricant layer [114]. Therefore there is still a need for independent equibiaxial extension data for polymers. REFERENCES ~
2. 3. o
.
6. 7. 8. ,
10. 11. 12. 13. 14. 15. 16. 17. 18.
Odroyd, Proc. Roy. Soc., A200 (1950) 1063. Oldroyd, Proc. Roy. Soc., A245 (1958) 278. Rivlin, Research Frontiers in Fluid Dynamics, 144, Interscience, New York, 1965. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer, New York ,1992. Coleman, Arch. Rat. Mech. Anal., 17 (1964) 1. Coleman, Arch. Rat. Mech. Anal., 17 (1964) 230. Coleman and V.J. Mizel, Arch. Rat. Mech. Anal., 29 (1968) 105. de Gennes, Scaling Concepts in Polymer Physics, Cornell University, Ithaka, New York, 1979. Green and A.V. Tobolsky, J. Chem. Phys., 14 (1946) 80. Lodge, Trans. Faraday Soc., 52 (1956) 120. Yamamoto, J. Phys. Soc. Japan, 11 (1956) 413; 12 (1957) 1148; 13 (1958) 1200. Treloar, Physics of Rubber Elasticity, 3rd ed., Oxford University Press, Oxford, 1975. Edwards, Proc. Phys. Soc., 92 (1967) 9. Doi and S.F. Edwards, J. Chem. Soc., Faraday Trans. II, 74 (1978) 560, 918, 1818; 75 (1978) 38. Curtiss and R.B. Bird, J. Chem. Phys., 74 (1981) 2016. Bird and J.R. Deaguiar, J. Non-Newton. Fluid Mech., 13 (1983) 149, 161. Curtiss, R.B. Bird and O. Hassager, Adv. Chem. Phys., 35 (1976) 31. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. I & II, 2nd Ed., John Wiley and Sons, New York, 1987.
562 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
Pokrovsky and V.S. Volkov, Visokomolekulamie Soed. (Russian), A20 (1978) 2700. Volkov, Int. J. Polym. Mater., 9 (1982) 115. Schweizer, J. Chem. Phys., 91 (1989) 5802, 5822. Schweizer, J. Non-Cryst. Solids, 131-133 (1991) 643. Leonov, Rheol. Acta, 15 (1976) 85. Leonov, J. Non-Newton. Fluid Mech., 25 (1987) 1. Dashner and W.E. Vanarsdale, J. Non-Newton. Fluid Mech., 8 (1981) 59, 69. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworth, Boston, 1988. Hunter and M. Slemrod, Phys. Fluids, 26 (1983) 2345. Ramamurthy, J. Rheol., 30 (1986) 337. Tanner, Engineering Rheology, Clarendon Press, Oxford, 1992. Janeschitz-Kriegl, Polymer Melt Rheology and Flow Birefringence, Springer, New York, 1983. Pearson, Mechanics of Polymer Processing (Part I: Polymers, Mechanics and Rheology), Elsevier, New York, 1985. Leonov and A.N. Prokunin, Nonlinear Phenomena in Flows of Viscoelastic Polymer Fluids, Chapman & Hall, London, 1994. Grmela, Physica, D21 (1986) 179. Grmela, Phys. Lett., A 130 (1988) 81. Beris and B.J. Edwards, J. Rheol., 34 (1990) 55,503. Leonov, J. Non-Newton. Fluid Mech., 42 (1992) 323. Jongschaap, Physica, A94 (1978) 531. Jongschaap, K.H. de Haas and C.A.J. Damen, J. Rheol., 38 (1994) 769. Kwon and S. Shen, Rheol. Acta, 23 (1984) 217; 24 (1985) 175. Simhambhatla and A.I. Leonov, Rheol. Acta, 34 (1995) 259. Leonov, Polym. Internat., 36 (1995) 187. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, NorthHolland, Amsterdam, 1962. Gyarmati, Non-Equilibrium Thermodynamics. Field Theory and Variational Principles, Springer, New York, 1970. Hulsen, J. Non-Newton Fluid Mech., 38 (1990) 93. Murnaghan, Amer. J. Math., 59 (1937) 235. Leonov, Ann. N.-Y. Acad. Sci., 410 (1983) 23. Stickforth, Rheol. Acta, 25 (1986) 447. Leonov, On the constitutive equations for non-isothermal bulk relaxation, accepted for publication in Macromolecules, 1997.
563
49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80.
Gordon and W. R. Schowalter, Trans. Soc. Rheol., 16 (1972) 79. Johnson, Jr. and D. Segalman, J. Non-Newton. Fluid Mech., 2 (1977) 255. Phan Thien and R. I. Tanner, J. Non-Newton. Fluid Mech., 2 (1977) 353. Phan Thien, J. Rheol., 22 (1978) 259. Larson and K. Monroe, Rheol. Acta, 23 (1984) 10. White and A. B. Metzner, J. Appl Polym. Sci., 7 (1963) 1867. Giesekus, Rheol. Acta, 21 (1982) 366. Larson, J. Rheol., 28 (1984) 545. Simhambhatla and A.I. Leonov, Rheol. Acta, 32 (1993) 259. Rivlin and K.N. Sawyers, Ann. Rev. Fluid Mech., 8 (1971) 17. Kaye, College of Aeronautics, Cranford, U. K., Note No. 134 (1962). Bernstein, E. A. Kearsley and L. J. Zapas, Trans. Soc. Rheol., 7 (1963)391. Kwon and A.I. Leonov, Rheol. Acta, 33 (1994) 398. Wagner, T. Raible and J. Meissner, Rheol. Acta, 18 (1979) 427. Wagner and A. Demarmels, J. Rheol., 34 (1990) 943. Luo and R.I. Tanner, Int. J. Num. Meth. Eng., 25 (1988) 9. Papanastasiou, L. E. Scriven and C. W. Macosko, J. Rheol., 27 (1983) 387. Larson and K. Monroe, Rheol. Acta, 26 (1987) 208. Currie, in G. Astarita, G. Marrucci and L. Nicolais, "Rheology", Vol.1, Plenum, New York (1980). Yen and L.V. McIntire, Trans. Soc. Rheol., 16 (1972) 711. Zapas, J. Res. Natl. Bur. Std., 70A (1966) 525. Ogden, Proc. Roy. Soc., A326 (1972) 565. Mooney, J. Appl. Phys., 11 (1940) 582. Valanis and R.F. Landel, J. Appl. Phys., 38 (1967) 2997. Blatz, S.C. Sharda and N.W. Tschoegl, Trans Soc. Rheol., 18 (1974) 145. Crochet, A.R. Davies and K. Walters, Numerical Simulation of NonNewtonian Flow, Elsevier, Amsterdam, 1984. Dupret and J.M. Marchal, J. Non-Newton. Fluid Mech., 20 (1986) 143. Joseph, Fluid Mechanics of Viscoelastic Liquids, Springer, New York, 1990. Rutkevich, J. Appl. Math. Mech., 33 (1969) 30, 573; 34 (1970) 35. Godunov, Elements of Continuum Mechanics, Nauka, Moscow, 1978. Joseph, M. Renardy and J.C. Saut, Arch. Rat. Mech. Anal., 87 (1985) 213. Joseph and J.C. Saut, J. Non-Newton. Fluid Mech., 20 (1986) 117.
564 81. 82.
83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95.
96. 97. 98. 99. 100. 101. 102. 103.
104. 105. 106. 107. 108. 109. 110.
Verdier and D.D. Joseph, J. Non-Newton. Fluid Mech., 31 (1989) 325. Kreiss, Numerical Methods for Solving Time-Dependent Problems for Partial Differential Equations, Les presses de l'Universite de Montreal, Montreal, 1978. Strang, J. Diff. Eq., 2 (1966) 107. Baker and J.L. Ericksen, J. Wash. Acad. Sci., 44 (1954) 33. Zee and E. Sternberg, Arch. Rat. Mech. Anal., 83 (1983) 53. Knowles and E. Sternberg, Arch. Rat. Mech. Anal., 63 (1977) 321. Kwon and A.I. Leonov, J. Non-Newton. Fluid Mech., 47 (1993) 77. Kwon and A.I. Leonov, J. Non-Newton. Fluid Mech., 58 (1995) 25. Kwon, J. Non-Newton. Fluid Mech., 65 (1996) 151. Kwon and A. I. Leonov, Rheol. Acta, 33 (1994) 398. Kwon and A. I. Leonov, J. Rheol., 36 (1992) 1515. Hulsen, J. Non-Newton. Fluid Mech., 38 (1990) 93. Renardy, Arch. Rat. Mech. Anal., 88 (1985) 83. Simhambhatla, The Rheological Modeling of Simple Flows of Unfilled and Filled Polymers, Ph.D. Dissertation, the University of Akron, 1994. Kwon, Studies of Viscoelastic Constitutive Equations and Some Flow Effects for Concentrated Polymeric Fluids, Ph.D. Dissertation, The University of Akron, 1994. Einaga, K. Osaki, M.Kurata, S. Kimura, and M. Tamura, Polym. J., 2 (1971)550. Laun, Rheol. Acta, 17 (1978) 1. Khan, R.K. Prud'homme and R.G. Larson, Rheol. Acta, 26 (1987) 144. Soskey and H.H. Winter, J. Rheol., 28 (1984) 625. Takahashi, K. Taku, and T. Masuda, J. Soc. Rheol., Japan, 18 (1990)18. Laun, J. Rheol., 30 (1986) 459. Leonov and A.N. Prokunin, Rheol. Acta, 19 (1983) 137. Laun, Stress and recoverable strains of stretched polymer melts and their prediction by means of a single integral constitutive equation. In" Rheology, vol.2, Plenum Press, New York (1980). Munstedt and H.M. Laun, Rheol. Acta, 20 (1981) 211. Vinogradov and A.Ya. Malkin, J. Polym. Sci. A-2, 2 (1964) 2357; 4 (1966) 135. Leonov and A.N. Prokunin, Rheol. Acta, 19 (1980)393. Laun and H. Schuch, J. Rheol., 33 (1989) 119. Wagner, J. Non-Newton. Fluid Mech., 4 (1978) 39. Laun, Rheol. Acta, 21 (1982) 464. Wagner and H.M. Laun, Rheol. Acta, 17 (1978) 138.
565 111. Giacomin, R.S. Jeyaseelan, T. Samurkas and J.M. Dealy, J. Rheol., 37 (1993) 811. 112. Meissner, Trans. Soc. Rheol., 16 (1972) 405. 113. Takahashi, T. Isaki, T. Takigava, and T. Masuda, J. Rheol., 37 (1993) 827. 114. Kompani, D.C. Venerus, and B. Bernstein, Development and evaluation of lubricated squeezing flow technique. In: Proc. XIIth Int. Congr. on Rheology, A. Ait-Kadi, J.M. Dealy, D.F. James, and M.C. Williams, Eds. August 18-23, 1996, Quebec City, Canada, p.754.
Appendix A On the invalidity of strain-time separability at quick time scales (M. Simhambhatla [94]) Strain-time separability, i.e. the factorability of the material response to nonlinear step strains into time and strain dependent parts, has been widely used as a convenient basis for the specification of viscoelastic CEs. The proponents of this assumption claim justification based on experimental observations. However, it is shown here that the same experiments for polymer melts and solutions require that the principle of strain-time separability be violated at small times following the application of step shear strains, in order to guarantee the Hadamard stability. According to strain-time separability, the stress response, to a step shear strain applied from the rest state is: o(t,y) : yh(y)G(t),
(A 1)
where h(7) is the shear damping function, and G(t) is the linear (Maxwellian) relaxation function. This means that the stress relaxation curves (log(o(t)) vs. log(t)) for various applied step strains will be parallel. This observation of seemingly parallel stress relaxation curves has been reported for several polymer melts and solutions (e.g., see [96], [97]). Consider strain-time separable CEs with a perfect elastic limit. For these equations, the step stress cl in response to a step strain 7, will be N
a(7 ) - 7h(7 ) ~ G, i=l
(A2)
566
Here, N is the number of Maxwellian modes, and Gi the linear relaxation modulus for the 'i'th mode. Now consider the experimentally determined dependence Th vs. T for LDPE Melt I [98] shown in Figure A1. The maximum in this dependence implies that 6 vs. T should also have a maximum (equation (A2)). Such maxima for Th vs. T appear for all the experimental data that we have come across (e.g., see [96-100]). It is however, easy to see that the decreasing branch of the dependence 8 vs. T, is unstable in the Hadamard sense. In order to clarify this, consider a Cartesian coordinate system (x,y) with the 'x' axis parallel to the direction of the shear displacements 'u', and the 'y' axis normal to the shear planes. The equation of motion for simple shear can then be written as" p0v/0t = 3cr/o~
(A3)
With v - Ou/c~, and 3' = Ou/0y, equation (A3) can be rewritten in the elastic limit as; /~U/C~ 2= M(]t)c32u/o~ 2
(A4)
Here 'u' is the displacement in the 'x' direction, and M(T) - c36/0y. Let uo be a basic solution satisfying equation (A4). Let us impose a small disturbance on the basic solution, so that (A5)
u - Uo + ~ fi exp[i(ky-mt)/a21
where ~ is the small amplitude parameter, 'k' the wave number, and co the frequency of disturbance. Substituting for 'u' from formula (A5) into equation (A4) and taking into account only the lowest order terms in g, yields" (1)2 =
M(To)k2/p.
( M(y o ) - d r ~
03, Ir,, )
(A6)
If M(y o) > 0, m is real and the disturbances in equation (A5) will not grow with time. However, if M(To) < 0, we have: co - +i~/(]m(y o)lk
=/p.
(A7)
567
v _L--
2.0
r - - .........
1.5
J~
I
......
",~
...........
I
. . . . . . .
[
.........
1" ....
MeltI "
0
1.0
o5 [ o
-----
Experirnenfal h(7)
=
Poinfs
,57exp(--.31T)+.43exp(--.10-67) L ...........
5
i0
I
....
~ 5.. Shear
t, . . . . . . . . . . . . . . . . . . .
20
Stroin,
L
25
................. J
........................................
50
55
-y
Figure A 1. Experimentally observed maximum in the plot 7h(7)vs. 7 for LDPE Melt I [96]. When the imaginary part of o is negative, formula (A5) indicates that even infinitesimal high frequency disturbances will grow in amplitude rapidly and unboundedly in time. Because there are high frequency disturbances in the spectrum of natural noise, the decreasing branch of the dependence t~ vs. 7 will result in severe instabilities during experimental measurements. However, since the experiments have been conducted for large step shear strains, and no instabilities have been reported, it is evident that the principle of strain-time separability should be violated at small times following the imposition of these large shear strains. This violation should be manifested in the form of an upturn in the dependence cy(t) vs. t at small times following the imposition of a step strain 7 when moving along the time axis in the negative direction, in order to have a monotonic increase in the dependence ~ vs. the step strain Y. Interestingly, this is preciselywhat was observed by Einaga et al. [96] (see Figure 2) and Takahashi et al. [100]. In experiments where yh(7) vs. 7 has a maximum, but no loss of parallelism is seen in the curves ty(t) vs. t , it
568
is presumably because data could not been obtained at very short times following the imposition of the step strains (assuming no wall slip).
........
I ..........
~ ........
~-
10 4
t0 3 0,,,. v o p..
to2
v
r~ "10 (3)
,J~ r
101
i 011
.................................
1
I
1 0 `=
..............
. . . I ...... .. ................
10 2
i
10 s
.. . . . . . .
10 4
Figure A2. Illustration of violation of strain-time separability at short times for 20% PS in Arochlor [96]. It should be noted that the deviations from the master curve of vertically shifted stress relaxation curves are consistent with the requirement of Hadamard stability. Two conclusions can be drawn. The first is that strain-time separability holds only in an approximate sense, if we are prepared to neglect the material response to quick disturbances. The second is that any strain-time separable CE is certainly unstable in the Hadamard sense in the limit of very rapid disturbances, due to the universally observed maximum in the dependence ~,h(~/) vs. ~,in those time regimes. This indicates that we cannot simultaneously describe the experimental data in the rheometrical regimes, and have the stability at high Deborah numbers for any strain-time separable CE. It is evident that the above instabilities have no physical basis, but are simply rooted in the improper extrapolation of strain-time separability to quick time scales. High Deborah number flow simulations using these CEs should therefore be avoided.
569
Appendix B On rheological description of LDPE Melt I by stable CEs M. Simhambhatla and A.I. Leonov According to Exxon material data, LDPE Melt I has molecular weight Mw=460000 and polydispersity ratio Mw/Mn=22. Linear viscoelastic spectrum for this material at 150~ is shown in Table B 1. Table B 1 Linear__v_i.sc0e!ast_.!c_s_pectrum o f,L Dp E Melt i .(.15oOc)[9:7] i
1
2
3
0i, sec
10 3
102
101
G,, Pa
1.00xl0 ~
1.80x102
1.89x103
4 10 ~ 9.80x103
5
6
7
8
101
10 .2
10 .3
10-4
2.67x104
5.86x104
9.48x104
1.29X!0!
In order to describe all available data for the LDPE Melt I we will use the simple non-Hookean potential and two-parametric expression for the dissipative term b" b(I, ) - exp[-13(I, - 3)] +
sinh[v(I, - 3)1 vO, - 3 )
-1.
(0<13 < 1, v > O)
(B1)
Here [3 and v are some numerical fitting constants. Formula (B 1) is in fact a modification of a similar expression proposed in paper [102] to describe simultaneously the hardening phenomena in the simple extension of LDPE Melt I with following softening. The first exponential term in (B1) describes the hardening phenomena in polymer melts due to orientation of macromolecules. The simple molecular model and explanations are given in paper [23] and book [32]. The second term in formula (B 1) reflects the softening phenomena in the flow of high oriented melts as described by the Eyring formula in the activated rate processes. As shown in paper [102], the softening is attributed to the thermo-mechanical degradation with a decrease in the molecular mass during extension flow of an LDPE melt. This was revealed by intrinsic viscosity measurements performed on the specimens left over from the experiments. This points to irreversible effects which are outside the scope of a purely rheological description. Interestingly, Wagner had to make a "structural irreversibility"
570
assumption [108] to properly describe recovery following extension flow for LDPE Melt I. To describe simultaneously the flow data for LDPE Melt I, the parameters 13 and v in (B 1) were chosen after a fitting procedure as" [3 - 0.15,
v - 0.03.
(B2)
With the parameters shown in (B2), formula (B1) initially show only hardening and only then, after decay of the first term, it describes "irreversible softening". It should be also noted that in simple shearing, with typical data for shear rates available in rotating instruments, the value b in formula (B 1) is all the time near unity. This fact preserves the description of simple shear data demonstrated in paper [40]. Nevertheless, we manifest in the following also some important comparisons between our calculations and data for simple shearing too. Figure B 1 indicates a good description of the steady shear viscosity and first normal stress difference over a wide range of shear rates. Descriptions of the transient stress growth during shearing experiments are depicted in Figure B2. The discrepancy for N~ at short times is probably due to the instrumental problems for these measurements [97]. 7
I
7
10
~__...~_v
10 6
I
13..
.
~
u
l
~
l
l
10
Melfl o Temp. 150 C -
~eference
o
10
6 ~..,.
o >:, -~
.,..~
10
5
10 5 ~j_~-~7~,, ,--~7_~~^,-, ~
10
c-,-
121
4
104
O
~ "I3 Q
0
m 5=
103
c~
02
10 _
9
10
3
I
u~ ,-,
t~ ol
f-] O
C~ 10
1
.,.
10-4-
I
I
I
I
1
I
I
10-.3
10-2
10 -1
10 ~
101
10 2
10 `3
Reduced
Shear
10 10 4
O ~.~.
R a t e , 7G T (s 1)
Figure B 1 .Steady shear viscosity and the first normal stress coefficient of LDPE Melt I at reference shear rate of l s -1 [97] at various temperatures (symbols).
571
a
10 5
n
c~
i
Melt l Reference
i 0
Temp.
150
C
**t:~:~O
~
z,~
~
~
r"
t..
~
-I
9
o3
.-vaT -
1s
10 2
I0-'
10 0
10 1
10 2
Shear Strain, 7
Figure B2. Transient stress growth normal stress coefficient for LDPE Melt I [97,109]. Various symbols correspond to different temperatures. We now consider shear creep and recovery. The coincidence with experimental data is good for both the shear strain and N1, for the creep condition, as shown in Figure B3. Strain recovery and normal stress decay following unloading after the creep experiment in Figure B3 are shown in Figure B4. Here, the predictions are accurate for strain recovery but N1 is underpredicted. Large amplitude oscillatory shear data, at low to moderate frequencies are indicative of dissipation during cyclical nonlinear deformation. Figure B5 shows a good agreement with the data obtained by Giacomin et al. [ 111 ] for the batch-labeled IUPAC X. We also demonstrate the capability of the model to describe the extension experiments. Figure B6 demonstrates an excellent agreement between calculations and data [112] for extension stress growth, with the use of formulae (B1) and (B2) for simple elongation. Figure B7 shows the comparison between our calculations and data [107] for the stress growth coefficient in planar extension.
572
10
I
10
I
Melt l Temp. 150
Reference
10
o C
4 o "~
12
-
10
6
_
105
_
10 4
Pa
d
z
N1
o-0
10
~" U, 121
lc" (/3
10 0
_
10 -~
10 -2
10
u
i
i
i
10 -1
100
101
102
Reduced
10 3
2
10 5
time, f / a T (s)
Figure B3. Shear strain and first normal stress difference in creep under constant shear stress for LDPE Melt I [110]. Various symbols correspond to experiments at various temperatures. 102
10
.
.
.
105
.
I
I
I
I
1
104 -
N,
t
z
> 0
~ r ~~
10
0
_
10 3
,~ -la Q
r o_ El k_ (z)
10
-1
10 -2 10 -2
Reference Temp. 150
I 10 -1
I
I
I
100
101
102
Reduced
o
C
102
101 103
time, f / a T (s)
Figure B4. Shear strain recovery and first normal stress difference after unloading following the experiment in Figure B3 [110]. Various symbols correspond to various temperatures.
573
80
I
60
-
40
_
I
1
I
I
I
1
70 = 5.0
Frequency = 1Hz
O~
-~
2o
e
0
(o)
r~
-20 o~
-40
IUPAC
X
o
150
-60 I
-80
-40
I
-30
I
-20
-10
C
I
I
I
I
0
10
20
50
40
-t
Shear
100
75 5O v
I
7o
I
=
I
Rate
(s
I
)
I
I
I
10.0 OQ
Frequenc
=
25
f/l I1)
~
0
_
(b)
-25
-
&)
rn
IUPAC X
-50
-
o
150
-75 I
-100
-80
-60
I
-40
I
-20
C
_
I
I
I
I
0
20
40
60
80
-1
Shear
R a t e (s
)
Figure B5. Large amplitude oscillatory shearing of LDPE IUPAC X [ 111 ].
574
10
6I
I
t
I
I
0.I
1.0
I
0.01
I 0 13_
1
..---2 .+r 10 v
+#
f
Melt I (I50 ~
10
I
I
I
I
I
10 -I
100
101
102
103
10 4
Time (s) Figure B6. Tensile stress growth coefficient vs. time for LDPE Melt I [112]. Finally, we also attempted to describe with the use of formulae (B1) and (B2), the data [98] for equi-biaxial elongation for LDPE Melt I. The result of comparison between our calculations and data is shown in Figure B8. It is evident that the calculated curve predicts more hardening in the rheological behavior of the Melt I as compared to the data. The possible reason for that was discussed in the Section 5 of the paper's main text.
575
106
I
0.05 s
-1
0.01 s 0 0
-1
0~ E~ 13_
10
5
.kp
+o_
Melt I
10
(125 ~C)
4
t
100
..
I
101
102
103
Time (s)
Figure B7. Planar tensile stress growth coefficient vs. time for LDPE Melt I [107]. 1.0
07
c
0 "~ c"
0
C: 0
=
0
0.5
x ~
l.a. 12I
~
~o3 IUPAC A (150 ~ 0
.1
,_
'
0.1
0.3
0.5
9
0.7
I
1.0
Strain
Figure B8. Equi-biaxial damping function for LDPE Melt I [98].
3.0
577
SCALING APPROACH IN SOLVING PROBLEMS OF COMPLEX VISCOELASTIC FLOWS WITH MULTI-MODE CONSTITUTIVE EQUATIONS OF DIFFERENTIAL TYPE A. I. Leonov*, J. Padovan** *Department of Polymer Engineering **Department of Mechanical Engineering The University of Akron, Akron, OH 44325 - 0301, USA 1. INTRODUCTION There are two types of constitutive models for nonlinear viscoelastic phenomena in polymer fluids in use today. One is of the differential and the other of the single integral type. The differential models are descriptively more rich and flexible. This is because they are able to independently model both the dissipative and elastic, non-dissipative terms in constitutive equations (CEs). However, they usually operate with a set of independent Maxwell modes generated from discrete points of the linear relaxation spectrum. Since each independent mode is generally described by a set of six partial nonlinear differential equations, this approach tends to be computationally intractable when solving 3D unsteady problems. Significant efforts were spent through the years to make specific formulations of the CEs. The interested reader can find many specifications of C E s o f both differential and integral types in recent books [1,2]. This includes comparisons with experimental data and useful discussions. Several general approaches to the derivation of single mode Maxwell-like CEs of the differential type were recently developed. The first, Poisson bracket formalism, which uses variational techniques, was introduced by Grmela [3-5] and extensively used by Beris and Edwards [6,7] for purposes of creating a general unified approach to Maxwell type CEs. These authors employed a variational derivation of CEs with the configuration tensor c treated as a hidden variable. In Refs. [7,8], this problem
578
was treated locally by means of non-equilibrium thermodynamics, and the same results as Beris and Edwards were obtained for Maxwell type CEs. Moreover, in Ref.[8] a "canonical" formulation of general Maxwell type CEs was proposed and used for the general stability analysis of CEs (see, e.g. [10]). In the multimode approach, the set of hidden tensor variables {__el,__c2, ..., cN }is usually employed. The ability of a class of multi-mode CEs (see, e.g. [2]) to describe consistently simple polymer flows was recently demonstrated [11] when satisfying all the stability constraints. For the sake of simplicity, functionally mode independent modeling of dissipative and elastic terms was employed and successfully used in paper [11 ]. The main goal of this chapter is to demonstrate that for multi-mode CEs, modal independence immediately results in a "scaling theorem" which in cases with known flow histories, reduces the solution of any 3D unsteady and nonisothermal problem to the computations for a single Maxwell mode. Although it is easy to extend the scaling theorem for the compressible case, for simplicity, only the incompressible version is considered. This is to avoid discussions [8] of compressible formulations for many viscoelastic CEs lumped together in the unified approach. The chapter is organized as follows. In the next Section 2 we briefly discuss the general unified formulation [8] for multi-mode approaches. Then in Section 3, some similarity assumptions are made which make it possible to prove the scaling theorem. In Section 4, two examples illustrate the use of the scaling approach for reducing computations for complex viscoelastic flows. Some concluding remarks are presented in Section 5 of the paper. 2. F O R M U L A T I O N
The general unified isotropic formulation proposed in Ref.[8] for N independent Maxwell modes with the set of hidden variables {Cn}, consists of: (i) evolution equations for each n-th mode, On(T)[c~tc n nt- ( v . V ) c n - Cn'(D q- o.1 C_n- ~n(Cn'e + e'c__ja)] q- ~n(~n,Cn) -- 0 (-1 < ~ n <
1)
(1)
whose most important feature is that the velocity field v is the same for each mode; (ii) the n-th modal extra stress tensor formulation: e
O'n -- Gn(T)sn(~n, Cn )
(2)
579
and iii) the expression for the total stress tensor: c~ -
-p~_ + o e
=
_
--
e '
(5" --
e =
(3)
'Y'~ (5" --n n
Here 't' is time, v is the velocity vector, N and e are the vorticity and strain rate tensors, ~ is the stress tensor, p is the isotropic pressure, fi is the unit tensor, 0n(T)and Gn(T) are the relaxation time and elastic (Hooke's) modulus for n-th mode, ~,n is a numerical parameter for each mode, and T is temperature. The elastic terms in the CEs include the evolution operators within the square brackets in equation (1) and the formulation of the stress tensor with the aid of equations (2) and (3). Here _Sn (~,,, _C_n)are some isotropic tensor functions of tensors Cn for each n-th mode. The dissipative terms in the CEs are represented by the set of isotropic tensor functions ~n(~,, ~ ) of tensors c for each n-th mode. If the values of parameters n are equal to either -1, or 0, or +1, the evolution operator in equation (1) converts to the lower convected, corotational, or upper convected tensor time derivative, respectively. It is also generally assumed that the above isotropic tensor functions _S_n(r _C_n) and ~. (~n, --C_n) satisfy the dissipative inequality and provide the CEs with a regular limit to the linear viscoelastic case. The momentum balance and the continuity equations are of the form: plat x + (x._V)x] = Vp +
e + pgk,
V.v =0
(4) (5)
Here p is the density and g is the gravity acceleration directed along the unit vector k. Finally, there is also the heat equation, which using the entropy viscoelasticity assumption (e.g., see [2]) takes the form" Cvp[c3tT + (v.V)T] = V.K-VT + J.tr(~.e).
(6)
Here C~ is the heat capacity under constant volume, J is the thermal heat equivalent of power, and K is the thermal conductivity which is generally represented as a second rank tensor. In the case of high-elastic cross-linked
580 rubbers, the stress induced anisotropy of thermal conductivity was well documented [12-14]. For polymeric liquids, it was assumed that K = K(T, =el, c2,..., ___CN)[ 15]. To the authors' knowledge, the flow induced anisotropy in heat conductivity has never been tested experimentally, seemingly because of evident experimental difficulties. Using equations (1)-(6), we consider the general 3D and unsteady problem for a flow of visco-elastic liquid in a domain gl c 9t 3, confined in the boundary Og~= c3g/1 w c3t~2, with the initial and boundary conditions" t = 0,
x e~:
v(0,x)=V(x),
_C_n(0,x)=fi, T(0,x)=To(x); (7)
x ec3gll"
v=U(T,x),
T=T~(t,x);
x (Kc~"~2:
cy=S(t,x),
T=T2(t,x)
Here To, T~ and T2 are known temperature fields, V and U are known solenoidal vector fields, and S(t,x) is a known boundary stress. We assume below that a solution of the problem (1)-(7) exists, and will make simplifying assumptions to the above formulations.
3. SIMILARITY ASSUMPTIONS AND SCALING THEOREM We now make three similarity assumptions which reduce the set of CEs (1) (3) and Eq.(6) to a more simplified form.
Assumption 1. Thermo-rheological simplicity" 0n(T) = Otn0(T),
Gn(T) = [BnG(T).
(8)
Here an and 13n are numerical factors, and 0(T) and G(T) are some characteristic relaxation time and modulus, taken e.g. either from the first relaxation mode, or from a dominant mode, or considered as mean values averaged over the linear relaxation spectrum. The assumption (8) usually holds for polymer fluids in a restricted time-temperature region.
Assumpion 2. Mechanical mode independence: gn:
8,n= 8,;
Sn(~,Cn)= S(~,Cn);
~(8,,_C_n) = ~(~,_C__n)
(9)
Thus instead of 2N independent functions _Snand ~ , equation (9)just assumes that there are only two independent non-dimensional isotropic tensor functions
581
s ( ~ , c ) a n d ~ ( ~ , c ) . It requires in particular, that the parameters appearing in these
functions be the same for any n-th mode (e.g., see [ 11 ]). Assumption
3. Thermal mode independence:
K(T,c_I,e:,...,_C_N)= R~--"K ~( T ,
C n )~ n / ~ n ,
R=(~-~ 13n /C/,n )-1
n
(10)
n
The foregoing decomposition provides for mode independent thermal effects. This formulation yields two physically valid limiting cases, namely: (i) the isotropic heat conductivity, K = n:o(T)6__,
(11)
(ii) the anisotropic heat conductivity for solids with finite elasticity, K=K(T,C-').
(12)
Here C -~ is the Finger strain tensor. It should be noted that w h e n , r the evolution equations (1) have the elastic limit: _C_n---~C1. This may occur either for intense viscoelastic flows, or for a process of solidification (0(T)--+~). We now prove the following Scaling theorem.
Assume that the flow history (i.e. the velocity field v(t,x)) is known. Let the assumptions 1-3 with equations (9)-(11) hold. Assume that for a certain value n-k, a solution { 7~,_fi,~,_~} has been obtained. It depends on coordinate x and a "stretched" time ~, and satisfies the s i n g l e m o d e set of equations: O, ~=+ (fi . V ) . 6=- 8 . ff3 + ff3 . 6 - ~ ( 6 . 8 + 8 . 8) + ~ ( ~ , ~ ) / O(ffZ) = O
(13)
6 - 1 / 2[_V__fi + (V____fi)T ],
(14)
8 - -~
O3 -- 1 / 2[V__fi - (V___fi)T 1,
+ G(T)s(_~),
ao[0,T + (_ft. V__)_fi]- V_V_.6 + bgpk,
(15) (16)
582
V__._fi- 0,
(17)
bC vp[a, T + (_ft. __V)T]= R E . [K(~', a)- VT] + J. G(T)- tr[~(a) .~1,
(18)
with the initial and boundary conditions" -
o,
X e~
x n:
__a(O,x)- _9(x),
_a(O,x)- 5,
1"
~-
O(~,X),
T-
"]-'1(~,X) ;
X E O ~ 2"
a-
S(l:,X),
T-
T2('l:,X)
T-
To
(x) (19)
Let the stretched time x, parameters 'a' and 'b', and functions ~, IA, and ~ in equations (19), be defined as follows" z-t/ak,
2 a = 1/~--'~[3nan,
b _ l / Z ] 3 n,
n
]Tj (a:,X) - Tj (t / ak , x )
n ^
V ( x ) - a k ___V(x),
]71, "I'2'
lJ(z, x_)- a k U ( t / a k , x ),
=S(x,x_)= b S ( t / a k ,_x)
(20)
Then the multi-mode solution of equations (1)-(6) for the initial boundary problem (7) is found as follows: v(t, x) - __fi(x,_x) / a k,
T(t, x) = T(T, x),
c (t,x)-~,('ta k / a n ' x), =
=n
e
=
p(t, x ) - ~(z, x) / b,
=G(T)~--,13nS( c ) = =n
(a:-t/ak)
(21)
n
Preliminary remarks. 1. There are two important features of the above CEs: (i) as mentioned, the velocity field is mode independent and given, and (ii) due to the scaling assumptions 1-3, all the equations (13)-(18) are also independent of the mode number ' n'. 2. The following scaling properties are exposed in equations (20) and (21): (i) the time in each n-th Maxwell mode is scaled by the relaxation ratio an (= 0n/0) as: ~n= t/an; (ii) the velocity in each n-th mode is scaled by the ratio an as: _fin(1;n,X)--anV(t,x); and (iii) the non-dimensional n-th modal extra stress tensor is scaled by the modulus ratio, [~n (=Gn/G). 3. It should be noted that the velocity field, v(t,x) in nonlinear multi-mode CEs cannot be found from the known solution of the problem (13)-(19) for a single
583 Maxwell mode. This can be easily proved for steady simple shearing or the simple elongation case. Proof of the scaling theorem. Using the above features, it is easy to show that the functions defined in formulae (21) through the single mode solution of the initial boundary problem (13)-(19), satisfy equations (1)-(6) and initial and boundary conditions (7) for the multi-mode approach. Firstly, the evolution equation (1) for n=k immediately follows from equations (13) and (14) after substituting there z = t/ak, and _fi(t,x) and ~(z,x) from equation (21). Because value 'k' was taken arbitrarily, equation (1) follows from equations (13) and (14) for arbitrary 'n'. Secondly, introducing into equation (16) the expressions: x = t/ak, __fi(t,x) and 15(~,x) from equation (21), as well as the formulae for 'a' and 'b' from equation (20), yields"
--[Otv+(v'g)v]~Y'~a~ ~ n n -
1
Z~n ~ +g -
n
gP [o(r)s(c k)]-t- ~-'f~ k
(22)
n
This equation holds for any (and all) values of 'k'. Then the momentum balance equation (4) can be retrieved for the multi-mode approach by multiplying equation (22) by 13kand summing over all 'k'. Thirdly, introducing into equation (18) the respective variables from (20) and (21) yields: CvakP ~[0tT
+ (v-V)T] = RV. [K(T,c k ). VT] + akG(T)J 9tr[s(c k ).e]
(23)
n
Multiplying equation (23) by 13k /ak taking the sum over all 'k', and using equation (10), yields the expression (6) for heat equation in the multi-mode approach. To finalize the proof, the multi-modal version of the continuity equation (5), as well as the initial and boundary conditions (7) must be retrieved. In particular, equations (5) and (7) follow directly from equations (17) and (19) when employing equations (20) and (21). Since in the reality neither flow, nor stress fields are known, the requirement of the theorem about the known velocity field seems at first sight quite artificial.
584
Yet, the situation with known flow field commonly happens when using any iterative numerical scheme. This will be clarified in the following Section.
4. EXAMPLES In this Section, we illustrate the scaling approach and a possible computational economy it gives for calculations of complex flows.
4.1. Start-up simple shear flows. As an easy example, we consider first an non-inertial and isothermal solution of start up problem for simple shearing. In this case, the loading of polymer liquid begins at time instant t = 0 initiating from the rest state, when a constant shear rate ~, is suddenly imposed. In typical computations, the value of ~ is confined in an interval [3; min, "Y max] where ~, min < < "Y max. Here ~ min usually belongs to the region of linear viscoelasticity, and ~, max is attributed to the highest available experimental data. We now make a choice of values { ~ k} as: ~t o -- ~ min < "~ 1 < ~ 2 < . . -< ~t n = ~t max. For any ~, from the interval, the solution of the problem for extra stress tensor in an 'i'-th mode is represented due to the scaling theorem as:
(24)
erie(t, ~ ) - Gis(t/Oi, ~' Oi).
Here s is the non-dimensional mode-independent tensor function of two variables, and 0i the relaxation time. In multi-mode approach, we assume the ordering: 01 > 02 > ... > ON where N is the number of modes. It is wise to start making calculations with the mode with highest relaxation time 0~. Then making 'n' calculations of the initial problem for the single 1-th mode with the values ~, k ( k = 0, 1, 2,...,n) in the whole time interval t > 0, we can obtain 'n' functions" ~le( t, ]t k) = Gls(t/01, ~' k01)
(k = 0, 1, 2 , . . . , n)
(25)
To calculate the extra stress tensor in any, say j-th mode, we do not need to solve once again the initial problem, but simply use the scaling approach: _%~(t, ~, ) = (Gj/G~)~]~(t0~/0j, ~ 0j/0~) - Gjs(t/aj01, aj ~ 01) ( j = 1, 2, ..., N).
(26)
585
Here aj = 0j/01. Note that since in this example, aj< 1, we need only to extrapolate the values of or1e, already computed, for different values of ~ which are not equal to q?k. Thus, to calculate the extra stress tensor using the scaling approach we just need to make 'n' calculations for a single (say, first) mode for all chosen values of shear rates r k and then restore the values of extra stress tensor by interpolation. It seems that (especially) for homogeneous simple shearing, there is no computational advantage, as compared to the conventional procedure. Indeed, using scaling one needs to compute in this case, n+l coupled ordinary differential equations for the first mode and also perform a time consuming interpolation procedure. If Ks is the amount of operations required for solving the start up problem for a single model with a single value of ~, k and Kj is the amount of the interpolation operations, the total amount of operations, K, using scaling is: Ks~ = (n~c+l)Ks + Ki. On the other hand, the conventional method of computations is performed with the following amount of operations: Kcon NKsn~o.. Here n~c and ncon are the numbers of shear rates ~ k we need to perform the complete set of computations. In order to compare the numbers of operations when using these methods, we note that the values of n~c and n~on should be generally quite different from each other. The value noo, can be taken arbitrarily and related usually to the compared data. To make the interpolation valid, the value of ns~ however, should be twice-three time more than the amount of modes, N, i.e. ns~ = (2-3)N. Thus the efficiency of scaling over conventional method is estimated by the ratio: KsJK~on ~ [(2-3)Ks + Ki/N]/(Ksn~on).
(27)
Since approximately Ki ~ N, there is no expected computational economy when using scaling in simple shearing. The same will happen for other simple 1D flows. The reason for this is that the computations of ordinary differential equations are very cheap today.
4.2. Scaling in computations of complex flows For the simplicity, we consider here only isothermal and steady, but generally 3D complex flows, with non-slip boundary conditions. A steady contraction flow in transition region between two non-symmetrically connected tubes of geometrically arbitrary cross sections serves as a good example of such flows. The level of velocity field v(x) is characterized here by the value of flow rate Q
586
commonly defined as integral of v(x)over any cross-sectional area. The values of flow rate Q are convenient to mark the values of Deborah(Weissenberg) numbers related to any real flow field. This is because the scalar linear functional Q{v(x)} is a single valued, however in general, non-monotonous. We will use below the steady viscoelastic CEs (1)-(3) under assumptions (8) and (9), along with the momentum balance equation (4) (ignoring inertia and gravity force) and continuity equation (5). In order not to miss some possible flow bifurcations, the flow field should be gradually increased from a lowest to a highest possible values. These values are correspondingly characterized by the values of flow rate, Qmin and Qmax. Here the value Qrmn commonly corresponds to the linear viscous flow regime. The value of Qmax corresponds to expected high Deborah (Weissenberg) number, and is usually limited by the occurrence of some numerical instabilities. Thus within the interval [Qmin,Qmax] we can generate the set {Qi} (Qi < Qi+l, i = 0, 1, 2,..., I), so that Qo=Qmi, and Qi=Qmax. In interesting cases when there are some regions of non-monotony in the functional dependence Q{v(_x)} a special treatment should be taken. This is outside the scope of our simple example. To solve the problem, for any value Qi from the interval, we employ the iterative algorithm [16] which has been successfully used for solving steady contraction flows in capillaries. Applying this algorithm to the above set of equations yields" m .co m-1 +co m-1 9Cnmi- r m-1 +e m-1 .c m ) + ( V im-1 ._V)=Cm,i - c=n,i ---i ---i = , = , .e. =l =i =n,i m
~(r
(28)
(n = 1 , 2 , . . . , N )
i ) / O n -- _0'
N o'im _ --pim ~ + e , m l
+ qa (e m i - eim-1 ),
13"ie,m
m ); - Z Gn-S ( ~ ' c-n,i
(29)
n=l
~-Pi
m
-- g ' o .
e,m
+ l"la
~7 2
(V m m-1 i - vi ),
V ' V im -- 0.
(m > 0)
(30)
Here equation (28) describes the evolution equation for n-th mode, equation (29) is the formulation of the stress tensor in the N-mode model, with the artificial Newtonian term (with 'viscosity' rla) included, and equation (30) describes the momentum balance and continuity equations. In equations (28)(30), the subscript 'i' indicates the flow with the flow rate Qi and the superscript 'm', the number of iteration.
587 The above iterative scheme works as follows. For a given 'i', the "initial guess" for the virtual velocity field is obtained as" v ~
oo
oo
Xi vi_l,. Here vi_ 1, is
the solution of the problem with the flow rate Qi-1 and )v~ is a numerical parameter ()vi > 1). Since the vector field v ~ is still solenoidal, there is the relation: Qi = )vi Qi-1. Thus to guarantee that given flow rate values Qi-1, and Qi should be close enough, one should take the value of )vi just slightly above 1. Using the value _,v~, the strain rate and vorticity tensors for the initial guess, =,e.~ and mi~ are easily calculated. Then equations (28) should be solved with some proper (usually given upstream) conditions to find the space distribution of values for N six-component tensors c 1 ., and concomitantly, the extra stress =n,l
tensor, ~i e'~. The last step in the first iteration is the determining of the first iteration for the velocity field, vl, and the pressure p l at the first iteration. These are found as the solution of the linear non-homogeneous Stokes problem (30) under constraint Qi = const. Then one should repeat the iterative procedure till it converges, if it possible. Obviously, for any value 'i', the artificial viscous terms in equations (29) and (30) vanish at the convergence, m --+ oo. The described iterative scheme works well [ 16], if the CEs have some good stability properties. It allows to find effectively the velocity and stress distributions for very high level of Deborah numbers (flow rates Qmax). It is evident that the scaling approach can be applied to the computations at any step of iterative procedure, since the velocity field here has been found from the calculations on the previous step. To solve equations (28)-(30), some numerical schemes employing various discretization (e.g. pure FEM, or combined FEM and upstream) methods can be applied. These are not the subject of discussion in this paper. Obviously in this case, the highest computation burden is the numerical solution of the evolution equations (28) for the N tensors =n,i c m on any iterative step ' m ' with the given flow rate Qi. This is because for any given values 'n', 'm' and 'i', equations (28) is generally the set of six coupled nonlinear partial differential equations. Once again, it is wise to start solving equations (28) with the highest relaxation mode, n=l. Then using the scaling approach, one can find the distributed values of
tensors Cn, im and extra stress tensor iterative step 'm', as follows"
=13ie'm
for any level of flow rate Qi and the
588
m
m
__Cn,i - Cl,i
r
{a n -vm-1 }
N _ Z ans(r n=l
(an
--
0
n
/ 0 9 n1,
1,2,
"",
N)
mi ) Cn,
(31)
Here we need to compute equation (28) only once for n=l and then using an from interpolation procedure, obtain the values of the functionals =m cl,i {anvm-1} a search of already computed values. This is because in equations (31), an < an-1 (On < 0n-l). Since the flow rate Q is a linear functional of velocity v, the respective values of flow rates corresponding to the velocity a~y_iv-can be found as Qni = anQi. Since an
amount of operations for retrieving the distributions Cn,i (x). Finally, let Kvp be the amount of numerical operations for solution of the Stokes problem (30). Then neglecting the amount of operations needed for tabulations of extra stress tensor, the total amount of operations for computing the problem using scaling is: Ksc ~ [Ks + Kret'(N-1)] KvpM'I.
(32)
Here M is the amount of 'm' iterations within a computational precision, I is the total number of intervals where Qi= const, and N is the amount of relaxation modes (typically, N~3-10). We now can employ in equation (32) the inequality" Kret'(N-1) < < Ks,
(33)
which holds for complex flows. Inequality (33) simply displays the fact that in complicated computations, the number of operations for retrieval of already
589
calculated data is considerable less than that for straightforward computations m
of the distributions Cn,i (x). On the other hand, using the conventional calculations, the amount of computations, K~on, can be approximately found as" Kcon~ Ks.N.KvpM.I.
(34)
Then comparing formulae (32) and (34) in the cases when inequality(33) holds, one can find that Ks/'Kcon ~, 1 / N .
(35)
Equation (35) shows that for calculations of complex flows with multi-mode viscoelastic CEs, the scaling approach can provide with a significant computational economy as compared to the straightforward computations. The key element here is the inequality (33) which holds only for complex flows. Finally, it can be demonstrated that almost the same computational economy can be achieved with the scaling approach for solving non-steady problems for complex viscoelastic flows. Here, the scaling approach can give more economy when using more stable implicit methods. 5. CONCLUSIONS
1. We have demonstrated that employing the similarity assumptions 1-3 in general formulations of Maxwell type multi-mode CEs (1)-(3) allows for scaling velocity and time in each Maxwell mode, by the mode's relaxation time, and for scaling stress by the mode's modulus. 2. We proved that with known velocity field, these scaling properties reduce the solution of a general non-isothermal initial boundary problem for multi-mode Maxwell type CEs to solving a similar problem for a single Maxwell CEs. 3. We demonstrated a computational advantage of scaling approach for solving problems of complex viscoelastic flows. Though today's computational tools do not allow to obtain the above single mode solution in all its generality, there is no doubts that this will be possible in near future. 4. As shown, the scaling approach might have its primary advantage as problem's complexity grows, i.e. when dealing with 2D/3D complex geometries, multi-mode models and unsteady problems.
590 5. We have proven the scaling theorem for very broad formulations of multimode CEs of Maxwell type. However, the very existence and stability of solutions of the CEs will highly depend on such fundamental properties as the thermodynamic consistency and stability of CEs.
ACKNOWLEDGMENTS
This work was supported in part by the NSF Grant (No CTS-932-0037). The authors would also like to thank Sunil Acharia for valuable discussions and Jay Jeong for help.
REFERENCES
1. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworth, Boston, 1968. 2. A.I. Leonov and A.N. Prokunin, Nonlinear Phenomena in Flows of Viscoelastic Polymer Fluids, Chapman & Hall, New York, 1994. 3. M. Grmela, Phys. Lett., A 111 (1985) 36. 4. M. Grmela, Physica, D 21 (1986) 179. 5. M. Grmela and P.J. Carreau, J. Non-Newton. Fluid Mech., 23 (1987) 271. 6. A.N. Beris and B.J. Edwards, J. Rheol., 34 (1990) 55. 7. A.N. Beris and B.J. Edwards, J. Rheol., 34 (1990) 503. 8. A.I. Leonov, J. Non-Newtonian Fluid Mech., 42 (1992) 323. 9. R.J.J. Jongschaap, K.H. de Haas and C.A.J. Damen, J. Rheol., 38 (1994) 769. 10. Y. Kwon and A.I. Leonov, J. Non-Newton. Fluid Mech., 58 (1995) 259. 11. M. Simhambhatla and A.I. Leonov, Rheol. Acta, 34 (1995) 259. 12. H. Tautz, Experim. Techn. Phys., 7 (1959)1. 13. K.H. Hellwege, I. Henning, and W. Knappe, Kolloid-Z. Polym., 188 (1963)121. 14. L.N. Novichenok and Z.P. Shulman, Thermophysical Properties of Polymers (Russian), Nauka i Tekhnika, Minsk, 1971. 15. A.I. Leonov, Rheol. Acta, 15 (1976) 85. 16. R.K. Upadya and A.I. Isayev, Rheol. Acta, 25 (1986) 80
591
A T H E O R Y OF F L O W IN S M E C T I C LI Q U I D CRYSTALS F. M . L e s l i e
Mathematics Department, University of Strathclyde, Livingstone Tower, Richmond Street, Glasgow G1 1XH, Scotland 1. I N T R O D U C T I O N The continuum theory for nematic liquid crystals proposed by Ericksen [1] and Leslie [2] has significantly improved our understanding of flow phenomena in these anisotropic liquids. Initially it explained flow alignment and non-Newtonian flow effects exhibited by these liquids, and thereafter contributed greatly to our insight into a variety of instabilities that can occur in these fascinating fluids under the application of magnetic and electric fields, thermal gradients, and flow [3]. Also the realisation that changes in alignment of the local anisotropic axis can induce flow, which in turn influences the changing alignment, had serious implications for applications using these materials in display devices [4-6], this leading to a greater appreciation of the theory. More recently potential for fast switching display devices has led to an increasing interest in smectic liquid crystals, and particularly smectic C liquid crystals. These liquid crystals display translational order as well as orientational order, and clearly their intrinsic layering presents new problems in terms of the formulation of an appropriate mathematical model. Motivated largely by applications in display devices, Leslie, Stewart and Nakagawa [7] have recently proposed a theory for smectic liquid crystals, which may prove useful for the modelling of such applications. Essentially it invokes two simplifications in order to reduce the mathematical complexity, assuming that the layer spacing in these materials remains constant, and also that the tilt of the alignment of the anisotropic axis with respect to the layer normal also remains fixed. For many situations these assumptions appear reasonable, but clearly there are others where they are too restrictive. The assumption of fixed layer spacing largely rests on the notion that variations in layer thickness require considerable energy, and consequently it seems reasonable when the
592
smectic is not under undue strain. Also the constant tilt presumably excludes certain thermal and pre-transitional effects. However, the theory has already proved useful in a practical context [8,9], and in this article we attempt to summarise its predictions relevant to rheology. For readers not familiar with general properties of liquid crystals, the books by de Gennes and Prost [10] and Chandrasekhar [11] give full accounts of physical properties, while that by Collings [12] provides a fairly gentle introduction. 2. S M E C T I C
THEORY
This section presents a summary of the equations proposed by Leslie, Stewart and Nakagawa [7] for smectic C liquid crystals. The theory is somewhat restricted in that it assumes that the smectic layer spacing remains constant, although the layers may bend and deform, and also that the tilt of alignment with respect to the layer normal remains unchanged. However, the theory differs from its predecessors in that it does allow for finite bend and deformation of the layers, and for non-linear displacements and flow. Like the theory for nematics it rests on the concepts of classical mechanics. With the above assumptions one can readily describe the layered configurations by employing two orthogonal unit vectors, one the unit layer normal a, and, following de Gennes [10], the second unit vector c is perpendicular to the layer normal indicating the direction of tilt of the alignment. Clearly this second vector is tangential to the smectic layer. Hence the two vectors or directors are subject to a.a-c.c--1
,
a.c-0.
(1)
Also, in the absence of any singularities or defects in the layering, as both Oseen [13] and de Gennes [10] argue one must also impose the constraint curl a = 0.
(2)
A more familiar constraint arises from the customary assumption of incompressibility, and so the velocity vector v is subject to d i v v - O,
(3)
and of course the density p is constant. The theory essentially rests on the balance laws of classical continuum mechanics, the balances of linear and angular momentum. In Cartesian tensor notation the balance of linear momentum is given by
593
(4)
pvi -- pFi "Jr-tij,j,
F denoting the body force per unit mass, t the stress tensor, and the superposed dot the material time derivative. However, the balance of angular momentum takes the less familiar form
pK~ + eijktkj + l~j,j = 0
(5)
including terms generally omitted, K the external body moment per unit mass and l the couple stress tensor, but an inertial term associated with intrinsic local angular momentum is omitted since it is generally considered negligible. Since thermal effects are ignored in the problems discussed, it is not necessary to give an equation representing a balance of energy. Here we employ Cartesian tensor notation whenever necessary, and therefore a repeated index is subject to the summation convention, a comma preceding an index represents a partial derivative with respect to the corresponding spatial coordinate, and 5ij and eijk denote the Kronecker delta and alternator respectively. The static version of the theory rests on the assumption of a local stored energy function that depends upon the two directors and their gradients, being quadratic in the latter. This energy takes the form 2w
-
+
+ 2K a . cjaj. ck
-1- 2K~eipqCpaqai,kCk "~ K~(ci,i) 2 -t- K~ci,jci,j -t- K ~ c i , j c j c i , k c k
(6)
-t- 2K~ci,jcjci,kak + 2K~eipqCpaqCi,~ak + 2K~Cci,icjaj,kck + 2K~Cci,iaj,j, the coefficients constants. The static equations can now be obtained by appeal to a virtual work formulation [7,14], and one finds that
tij - -pSij Jc flpepjkak,i
OW OW -- ~ C k , i cOak,j ak,i OCk,j /
OW
~ij -- ~papSij -- ~ i a j -[- eipq (ap \ Oaq,j
ow ) + Cp
+ tij,
(7)
OCq,j
where the pressure p and the vector fl arise from the constraints (3) and (2), respectively. The tensor t denotes the dynamic viscous stress, but the corresponding term in the couple stress is omitted since it can be shown to be zero by essentially a thermodynamic argument requiring the
594
rate of viscous dissipation always be positive. The viscous stress t is the sum of a symmetric part
tij -- ~oDij + ~lapaqDpqaiaj + ~2(Dikakaj + Djkakai) + ~3CpCqnpqciCj + ~4(DikCkCj + DjkCkCi) + ,~~n~(~icj
+ ~j~i) + ~1 ( A ~ j + Aja~)+ ~(C~cj + Cj~,)
+ )~3cpAp(aicj + a j c i ) + al(Dikakcj + n j k a k c i + n i k c k a j
(8)
+ Djkckai) + ~2[2apcqDpqaiaj + apaqDpq(aicj + ajci)]
+ ~[2~.c~D.~cicj + c.c~D.~(aicj + ajci)] + ~l(C.aj + C j ~ ) + r2(Aicj + Ajci) + 2r3cpApaiaj + 2T4cpApcicj,
and a skew-symmetric part ti~ - )~l(Djkakai - Dikakaj) + )~2(Djkckci - Dikckcj)
+ ~ c , ~ D . ~ ( ~ i ~ j - ~jci) + ~ ( A j a ~ - Ai~j) + ~ ( C j c ~ - C~cj) + )~6cpAp(aicj - a j c i ) +
V l ( D j k a k C i - DikakCj)
(9)
+ ~ ( D j k c ~ , - Di~c~aj) + ~a.a~D,~(~,c~ - ~jc.) + ~c.c~D.~(~icj - ~jc,) + ~(Ajci - Aicj + Cj~i - C,~j). where 2Dij - vi,j + vj,i , Ai - ai - Wijaj 2Wij -- vi,j - vj,i
, Ci - ci - Wijcj,
(10)
and the coefficients are constants. The above theory is that for chiral smectic C liquid crystals, or what are also called ferroelectric smectic liquid crystals, being invariant to only proper orthogonal transformations of the coordinate axes. Put differently such materials can distinguish between right-handed and left-handedness. However, should one restrict the theory to non-chiral materials, those invariant to transformations including the full orthogonal group, one must choose g~ = g~ =0.
(11)
As Carlsson, Stewart and Leslie [15] discuss, there are grounds for setting the former coefficient zero even in chiral theory, and therefore the distinction between theory for chiral and non-chiral materials rests es-
595
sentially on one elastic term that gives rise to twisted configurations in which the c director rotates about the layer normal, much as the director in cholesterics or chiral nematics rotates about a given axis to form helical configurations. It is of interest to note that the viscous stress is identical for both chiral and non-chiral smectics. An inspection shows that the above equations are invariant to the simultaneous change of sign in the two directors a and c, this being considered the symmetry appropriate to smectic C liquid crystals. However, if the material symmetry allows the independent change of sign of the two directors, then the constitutive equations simplify, the elastic coefficients K~, g ~ ~ and g ~ ~ in equation (6) and also the ~ and r terms in the equations (8) and (9) having to be set equal to zero. It is possible to re-arrange the above equations in forms more convenient for calculations that follow. To this end the intrinsic body moment in equation (5) can readily be expressed as "iss -- eijk(ajg~ak + cjgk) ~c , eijktkj
(12)
wherein
g~ = --2()~lDikak + )~3CiCpaqDpq -F )~4Ai -1- A6CiCpAp -]- r2Dikck "[- T3CiapaqDpq + T4CiCpCqDpq --~ T5Ci) , ~tc = --2()~2DikCk 4- )~5Ci -F T1Dikak + 7"5Ai).
Similarly in cases of interest the external body moment can be written
p g i - eijk(ajG~ + cjG~).
(14)
As a consequence some manipulation allows one to rewrite the balance of angular moment (5) as two equations ,j
(ow)
,j
coal -]- Ga -F {ta -}- 7ai + tzci -]- eijkflk,j -- 0
OW
(15) +
+ Z +
+
- o,
where the scalars V, # and r are arbitrary multipliers stemming from the constraints (1). Also, with the aid of equations (115) the balance of linear moment (4) can be expressed as
pi~i = pFi + a~,ak,i + G~ck,i - ~,i + (~ak,i + ~ c k , i + tij,j,
(16)
596
where
#=p+W.
(17)
The forms (15) and (16) are more convenient for our purposes in the following sections. Also, employing the forms (12) allows one express the viscous dissipation inequality as t i8j D i j
__
(18)
*~C C g a A i - gi i >_ O,
which leads to some restrictions upon the viscous coefficients. To conclude this section we attempt to clarify the physical role of the vector/3 arising from the constraint (2). If we consider planar layers subject to flow or external fields, equations (7) show that they are subject to moments ~.i --- ~.ijaj -" ~ p a p a i -- ~i q- eipq ap Oaq,j -~- Cp-OCq,j"
aj.
(19)
Thus, if the layer normal coincides with the z-axis, one has =
+
...,
4
-
+
...,
-
0 +
...
(20)
simply giving the/3 contributions explicitly. Hence, if the layers are constrained to be planar by for example confining parallel plates, this vector provides a means by which the couple stress transmits the necessary torques to maintain the parallel layering. 3. S I M P L E S H E A R F L O W Given the layered structure always present in smectic liquid crystals it is not clear as to how this class of liquid crystal will respond to flow in general, there being little by way of clear experimental evidence to guide us. Here, therefore, we choose to consider the simplest options, two cases in which the layered structure appears to be compatible with the imposed flow. In our first example our results appear to be consistent with our premise, but in our second the question does remain somewhat open. 3.1 P l a n a r layers p a r a l l e l to t h e b o u n d i n g p l a t e s Consider a smectic C liquid crystal confined between two parallel plates with the smectic layers everywhere parallel to the plates, one of which is at rest but the other moves with velocity V along a straight line in its own plane. With Cartesian axes such that the z-axis is normal to the plates and the x-axis parallel to the imposed motion, it is natural to examine solutions of the equations of the previous section in which
597
a-(0,0,1)
, c-
(cosr162
, v-
(u(z),v(z),O),
(21)
this choice clearly consistent with the constraints (1), (2) and (3). As Gill and Leslie [16] discuss, with this choice equations (16) quickly yield
[~z =
(r/~ + r/2 cos 2 r
+ ~/2 sin r
tyz - (r/1 + r/2 sin 2 r
C v ' - c~,
(22)
+ r/2 sin r cos Cu' - c2,
where the prime denotes differentiation with respect to z, Cl and c2 are constants, and the viscosities r/1 and r/o are given by 2~71 - / z o + ~2 - 2A1 + A4 , 27/2 - ].t4 -[- ~5 -[- 2A2 - 2A3 -4- A5 -4- A6, (23) this assuming that the flow stems solely from the relative motion of the plates, there being no imposed pressure gradients, or external body forces or moments, apart from gravity which is readily absorbed into the pressure p. Consideration of special cases of the above equations quickly leads one to the conclusion that T]I and 771-~-772are both positive. The equations for angular momentum (15) lead only to K~r
+ (T 1 -- T5)(U' sin r
v' cos C) - 0,
(24)
this assuming a choice of the vector fl of the form
(25)
-- (]31(Z) , /~2(Z) , 0),
being more than reasonable for the problem under consideration. Before proceeding further, it is of interest to look for solutions representing uniform flow alignment of the c-director in shear, and clearly we have two such options r
or 7r , v ' - 0 .
(26)
Also, consideration of small perturbations leads one to select
r
if (1"5 --7"l)Ut
> 0,
(/)--71"
if
(7"1 -- 7"5)U' > 0,
(27)
essentially rejecting the alternatives on grounds of stability. Thus one can have uniform flow alignment tilted forward in the direction of shear or backwards, depending upon whether 75 is greater or less than I"1 [16,17]. The former case is consistent with one's intuition, and influences the material symmetry chosen, namely that our equations should be invariant to the simultaneous change of sign of the two directors. If one were to opt for invariance to the change of sign of a or e independently, no flow alignment would be predicted for this the simplest of shear flows involving smectic C liquid crystals.
598
Returning to the general formulation one quickly obtains from equations (22) ~I(T]I -~- ~2)U t - - Cl(~l -~- r/2 sin 2 r
c2r/2 sin r 1 6 2
,11(,/1 + r/2)v' -- c2(r/1 + y2cos 2 r
clr/2 sinr
(28) r
giving the velocity gradients as functions of the alignment. Also elimination of the velocity gradients from equation (24) leads to g~r
+ (rl - T5)(Cl sine- C2 COSr
-- 0,
(29)
which readily integrates to give K~r '2 + 2(T5 -- rl)(Cl cosr + c2 sin r
-- c3,
(30)
where c3 is a constant. The boundary conditions to be satisfied in general are u(d) -
V,
u(-d)
-
v(d) -
v(-d)
0, r
-
- r
, r
- r
(31)
the gapwidth being 2d with the origin chosen midway between the plates, and r and r two prescribed angles. It readily follows from equations (28) that 771(?']1 "~- ?~2)V -- c 1 f__ (Z]l -~- ?']2 J- d
0
C2 f__ (771 -~- 7]2
si"2r
- c2 r12
r
Cl ?']2
J- d
F sin r cos
Cdz ,
d
sin r
(32) Cdz ,
d
providing two equations for the constants Cl and c2. If the choice of angles r and r is such that r
- -r
(33)
and the angle r is an odd function, then the above implies that c2 is zero, and 771(~]1 -~- ~72)V -- 2Cl
(Z]l -~ Z]2 sin 2 r
(34)
so that Cl is clearly positive. While we know rather little concerning likely values for the various material parameters, they appear in the above equations simply as three combinations, the two viscosities rh and 7/2 and the group (T5 -rl)/K~r/1. Consequently a numerical integration of the equations is relatively straightforward. Gill and Leslie [16] give details for two cases,
599
with the angle r either symmetric or asymmetric. For the former the angle r tends to either zero or r in the centre of the cell, dependent upon whether T5 is greater or less than T~. For asymmetric solutions the results are more interesting with the twist in the centre of the cell unwinding as the shear rate increases, the flow causing the twist to concentrate near the plates with a growing region of uniform alignment in the middle of the cell. This result appears to be consistent with the early experiment by Pieranski, Guyon and Keller [18]. Note that for this problem the equations are equally valid for chiral or non-chiral materials, the difference between the two, if any, being in the initial equilibrium configuration. 3.2 P l a n a r layers p e r p e n d i c u l a r to t h e b o u n d i n g p l a t e s Our second example considers what is commonly called the bookshelf geometry with the layers planar but normal to the plates, and a shear imposed parallel to the layers would appear to be compatible with the smectic layering, as Carlsson, Leslie and Clark [17] discuss. In this case we examine solutions referred to Cartesian axes with the z-axis perpendicular to the plates of the form a-(0,1,0)
, c-
(cosr
,sine(z)), v-
(u(z),
v(z),
0),
(35)
this again satisfying the constraints (1), (2) and (3). Here, however, we must confine attention to non-chiral materials, since chiral materials would require the inclusion of a y dependence in the angle r and presumably also in the flow components, which would lead to an analysis beyond the scope of an initial investigation of the simpler options. As Gill and Leslie [16] discuss, the equations of linear momentum (16) with the above choice yield
t'~z - #1 (r
+ #2(r
t' z =
+
cosr
cl,
(36)
osr
where again Cl and c2 are constants, and there are no imposed pressure gradients or fields. The viscosity functions in the above equations are given by 2#1(r
= ~Uo+ ~ 4 ~- )k5 "4- 2A2 cos 2r + 2#3 sin 2 r cos 2 r
2~t2(r
-- tel -4- I"1 -4- "r2 -4- T5 -4- 2(n3 -4- 7"4)sin 2 r
2~t3(r
--/Zo 4-/z2 A- 2A1 + A4 + (~t4 -~- ~5 -- 2A2 + 2A3 d- A5 -I- )~6) sin 2 r
(37)
500
Here also one can solve the equations (36) to obtain the velocity gradients in terms of the angle r and the imposed shear stresses. The equations of angular momentum (15) reduce to f ( r 1 6 2 -4 21 dr(C) de r
_ (~5 + ~2 cos 2r
- (T1 -~- 7"5)v' c o s r
0,
(38)
where f(r
- g ~ + g~ cos 2 r + g~ sin 2 r
(39)
but this entails a choice of the vector/3 and the scalar 7 of the form t3-(~(z)y
, 0 , ~(z)y)
, 7=~(z)-y~(z),
(40)
the prime again denoting differentiation with respect to z. There are two respects in which this solution differs from the former, both giving rise to grounds for not pursuing it in any great detail at least for the present. One is that the material parameters appear in greater numbers, and our lack of knowledge of their relative magnitudes therefore inhibits progress. The second is that the surprising dependence of the vector/3 on the y coordinate means that the resulting torques can become large, and therefore the material may seek to relax to a less strained state, say by forming domain structures. Some guidance from related experimental studies would not go amiss in this respect. To close this section we consider the alignment dictated by shear flow in this geometry. Here the options are r162
rico
where r cos2r
, v'-0,
(41)
is the acute angle defined by - -~5/~2
, provided ~5 <_ [~21,
(42)
otherwise uniform alignment is not possible in shear flow in this geometry. Note that the viscosity )~5 is clearly positive as a consequence of the inequality (18), as Carlsson, Leslie and Clark [17] discuss. Of the solutions (41) we select on grounds of stability r162162
,
if)~2u'<0 , r162162
,
if)~2u'>0,(43)
as Gill and Leslie [16] and Carlsson, Leslie and Clark [17] show. For such uniform alignment without transverse flow it is of interest to note from equations (36) and (37) that the transverse shear stress is in general non-zero, this an aspect perhaps more readily amenable to experimental observation.
601
4. F L O W I N S T A B I L I T I E S Continuum theory for nematic liquid crystals quickly gained credibility through its success in describing satisfactorily a number of instabilities that occur in these materials, when subject to external influences including flow. Here our aim is to show that similar flow induced instabilities are likely in smectic liquid crystals, being predicted by the present theory, and naturally one hopes to stimulate relevant experimental studies. In this section it suffices to give a simple illustration discussed by Leslie and Blake [19]. We return to the geometry of our first example in the previous section and thus consider smectic C layers confined between parallel plates, the layers parallel to the plates, and subjected to simple shear flow. Here, therefore, with the same choice of Cartesian axes, the z-axis normal to the plates and the x-axis parallel to the imposed flow, we again examine solutions of the form (21), and find as before that the governing equations are + (T1 -- rs)(u' sin r
g~r
v' cos C) - 0,
(r/~ + y2 cos 2 r
+ 7/2v' sin r cos r
c~,
(711 -~- 712 sin 2 r
+ ~/2u' sin r cos r - c2,
(44)
where v/1 and v/2 are again given by equations (23). Choosing the origin midway between the plates, a distance 2d apart, the relevant boundary conditions are u(d) - Y
,
u(-d)
= v(-4-d) - O ,
r162
(45)
where V is the velocity of the moving plate, and r is either zero or r. Straightforwardly the equations (44) have simple solutions subject to conditions (45) in which r162
,
t~z -
u-
Y(z + d)/2d
(~71 + r12)l//2d
,
,
(46)
v-O,
tyz - O.
As remarked above the solution anticipated is r
0
,
if r~ > T~ ,
r
,
if 7"5 < T1,
(47)
being unstable otherwise. Consider small perturbations to the solution (46) in which r
r
+ r
,
u - Y ( z + d ) / 2 d + ~t(z)
,
v - ~(z),
(48)
602
and one obtains the following equations for these perturbations g~$" + (~ - ~)~os r
~') - O,
(49)
(~1 + ~ ) ~ ' - a , ~1~' + ,7~V;~/2a - b, a and b denoting perturbations to the shear stresses. conditions for the perturbations are
The boundary
~(~d) = ~(+d) = r - O. (50) In view of the above the second of equations (49) at once implies that ~2 is zero, and elimination of ~3 between the remaining equations yields ~,, + V(7/a + rl2)(rl - r5)cos ~bo [ ~ _ 2bd/V(rll + r/2)] - 0.
(51)
2K~rl~ d If the product (rl - T5)cos Co is positive, the above equation has a solution subject to conditions (50) of the form -
2bd cos wz Y (7/1 + r/2 )(T1 - T5) COSr V(~l + ~ 2 ) ( 1 - coswd ) ' w 2 = , 2K~rlld
-
(52)
and the boundary conditions for ~ require that the last of equations (49) yield
712V / _ ~ Cdz - 2bd,
(53)
-~J d which in turn leads to
r/2 tan wd + rllWd - O,
(54)
and which straightforwardly yields positive values for w. However, if the product (rl - T 5 ) c o s r is negative, one finds that the relevant solution is 2bd cosh w z Y (r/1 + r/2)(r5 - T1) cos r Y(yl -~- Y2)(1 - coshwd ) ' w2 2K~rlld , (55) -
-
and equation (53) now yields r/2 tanh wd + rll wd - O.
(56)
As remarked earlier r/1 + 7/2 is necessarily positive, and consequently the above equation has no non-trivial solutions. As a consequence our analysis predicts an instability with respect to the particular perturbations considered for the unstable solutions at a critical velocity V~ given by
603 Vc - 2g~rll~2c/d(rll + r/2)(7-1 - 75)cos r
(57)
where ~ is the smallest positive root of ~1 x + v/2 tan x -- 0.
(58)
While other more general perturbations may lead to a lower threshold, it is perhaps premature to attempt further more complex analyses until experimental evidence becomes available, and also our knowledge of surface anchoring at a smectic-solid interface improves. With this latter aspect in mind, Blake and Leslie [20] discuss the above problem when a magnetic field is applied, this allowing consideration of the two limiting cases of strong anchoring and no anchoring of the c-director. 5. B A C K F L O W
EFFECTS
Given that the original motivation for the theory presented in section 2 is primarily to model behaviour in smectic devices, it is of more than passing interest to examine flow induced by switching of alignment in smectic cells, particularly so in view of experience of such effects in nematics [4-6]. Not surprisingly perhaps we turn first to the simplest geometry in which the layers are parallel to the bounding plates, with the alignment initially uniform due to strong anchoring at the surfaces. Application of a magnetic field parallel to the plates, but perpendicular to the initial alignment rotates the c director around the layer normal, the distortion symmetric about the plane midway between the plates. Below we present an analysis of the relaxation of this alignment when the field is removed. With a choice of Cartesian axes so that the z-axis is again normal to the plates with the origin midway between them, and the x-axis now coincident with the direction of the initial alignment dictated by the strong anchoring, it is natural to consider solutions of the form a
-
-
(0, 0, 1) , r - (cos r
t),
t ) , sin r
t), 0 ) ,
(59)
t), 0),
clearly consistent with the constraints (1), ( 2 ) a n d (3). With this choice equations (15) simply reduce to K~~z2 - 2)~5-~ + (7"1 - T5)(sin r
cos COz) -- 0,
plus expression for fl, 7, # and T, and equations (16) yield
(60)
604
0 [
C~U P Ot -
0---~ (rh + 7?2 cos 2 r
Ou
+ r/2 sin r cos r
Ov
+ ( ~ - ~-~)si~ r
Ov
0 [
P-~ - ~
Ov
~U t(r/~ + r/2 sin 2 r O--~z+ r/2 sin r cos r Oz
(61)
+ ( n - ~) r162 and an expression for the pressure p, the viscosities 771 and r/2 again given by equations (23). From the above equations it is at once evident that one must include both flow components in order to avoid an over-determinate system. The boundary and initial conditions are ~(+d) - ~(+d) - r177 u ( z , 0) - ~ ( z , 0) - 0 ,
r
- 0, 0) - r
the cell gap again 2d, and r
(62) ,
Izl < d,
a known even function of z.
Introducing the notation a - (7"5 - n ) sin r
, b - (7"1 - r5) cos r
m-r/l+r/2cos
, n - 7 ? l + r / 2 s i n 2r
2r
(63) , r-7?2sinCicosr
where r is a constant angle associated with the initial alignment, Leslie and Blake [19] replace the above equations by
02r
0r
Ou
Ov
g ~ -~z ~ - 2 ~ ~ -8-i - ~ -8-;z - b -8; - O ,
cgu
P~
Ov p-~
c92u
c92v
- m~z~ + ~ - ~
02u
02v
02r + a OzO----i '
(64)
02r
-- r -~z 2 q- n -~z 2 + b cgz cgt ,
whose solution presumably gives at least a good approximation to the initial relaxation. With the removal of a strong field, it would be appropriate to select r to be r / 2 . By introducing dimensionless variables given below, one finds that the inertial terms in the last two of equations (64) are negligible, and therefore it is reasonable to set p equal to zero in the above. As a consequence the last two equations quickly yield (mn-
c92u = ( r b - a n ) 02r ~)-~ ~ OzOt
'
(ran - r 2) c92v - (ra - bin) 02r (65) ~ o~Ot '
605 and hence with the first
02r
~162 (66) K~ Oz 3 = (OzOt' where a ( r b - a n ) + b ( r a - bin) _ 2)~5 - (75 - T1)2 - 2~ + . (67) m n - ?.2 rl1 By appeal to the viscous dissipation inequality one can show that the above parameter ~ is positive. Introducing non-dimensional variables as follows
K~ k-2,~5,
du
dv , ~--~-,
~-/~
z
kt r-d-- 7,
i=~,
(68)
the first of equations (64) and equations (65) and (66) can be rewritten 02r /)('2
0r OT
a 0~ 2A5 0~
b 0~ 2A5 0~
02~i
(ran - r~) ~ - ~ - ( r b - an) ~
-
(~
~" 0~
02r Or
- bin) ~ 1 6 2
(69)
O(Or'
03r 02r ( , 0<#<1. O~3 = # O ( O T ' # - - 2 A S Note that the dimensionless time r in the above is not to be confused with the r that occurs in equations (15). The above equation.s have solutions satisfying the boundary conditions (62) of the form (70) r r162 -q~ /" u U(r q~ /" v-V(r __
v
-
__
--
v
~
--q2"r/l~
where r
-- A(cos q~ - cosq) , U(~) - Aq(an - r~br)i (sin q~ - ~ sin q), #(mn J
v(r
(71)
- Aq(bm ~ ( m ~ - - ~at) ) (sin q~ - ~ sin q)
A an arbitrary constant and q satisfying (1 - , ) t ~ n q - q.
(72)
Superposition of the solutions corresponding to the infinity of roots q~ of this last equation leads to
606 oo
q~-- ~
An(cOsqn~ - cosqn)e -q~r/tt
n~l
an
~
bT"
oo
~t = t , ( m n _ r2 ) n=l
q ~ A n (sin q ~ - ~ sin q ~ ) e - q ~ / "
(73)
,
bm - ar oo 2 ~/# = I , ( m n - r 2) y ~ q,A~(sin q~( - ( sin q n ) e - q " 9 n=l
Finally the initial condition on the alignment (62) requires that oo
A n ( c o s q ~ ( . - cosq~) - r
,
(74)
n--1
and noting that the necessary orthogonal function is cos q m ~ , it follows that A~ =
2qn
/.1
q~ - sin q~ cos q~ ~. r
cos q ~ ( d ( .
(75)
The above solution resembles rather closely that given by Clark and Leslie [6] for a nematic, despite the fact that two flow components are induced in a smectic. Blake and Leslie [21] discuss this solution in greater detail and show that as in a nematic the induced flow can give rise to a 'kickback' effect, before the alignment finally relaxes to its initial configuration. They also briefly consider the more complex bookshelf geometry, but do not pursue this given the greater number of unknown parameters that appear in the relevant equations. Carlsson, Clark and Zou [22] and Zou, Clark and Carlsson [23] discuss the effects of coupling with flow upon the reorientation of alignment in a smectic liquid crystal due to the application of an electric field, showing that flow can play a significant role in such transients. However, their analyses neglect flow transverse to the smectic layers on the grounds that it is small. Blake, Leslie and Towler [24] give similar results including this transverse flow, while Barratt and Duffy [25] provide a linear analysis of such effects. 6. S M A L L P E R T U R B A T I O N S
OF A UNIFORM
SMECTIC
First attempts at a theory for smectics were confined to describing small perturbations of uniformly aligned samples in planar layers. The Orsay Liquid Crystal Group [26] proposed an elastic energy for smectic C
607
liquid crystals, considering only such small perturbations, and Carlsson, Stewart and Leslie [27] show that the general, non-linear energy given in section 2 reduces to exactly the Orsay energy when restricted to such small disturbances. Also, Martin, Parodi and Persham [28] developed a corresponding linear, dynamic theory, but by its very nature incapable of describing most of the effects discussed in earlier sections. However, this theory presumably prompted Galerne, Martinand, Durand and Veyssie [29] to study light-scattering by a smectic C liquid crystal, and below we use their experimental results to obtain estimates for certain combinations of parameters in the theory employed in this chapter. Consider a smectic C liquid crystal at rest and uniformly aligned with the directors a and c given by a--a
~
,
c-c
~
(76)
a ~ and c ~ being two constant orthonormal vectors, and examine small periodic perturbations of the form a-
a~
,
c -
c ~ + E5
,
v-El)
,
E -- E e x p i ( w t - q . x ) ,
(77)
where fi, 1~ and ~ are constant vectors, E is a small constant, and w and q are a constant frequency and wave vector. Since a and c are orthonormal, then one must have a~
c~
-
a~
+ e~
-
O,
(78)
but the further constraints (2) and (3)lead to qxfi--O
, q.l~-O,
(79)
the former implying that f i - c~q,
(80)
a an arbitrary scalar. Consistent with the above one naturally assumes that p-
E~
,
7-
E~/
,
p-Eft,
r--
E~-
,
fl-
Efl,
(81)
where/3, ~,/2, # and f} are constant scalars and a vector. Leslie and Gill [30] give the general forms of the equations (15) and (16) consistent with the above small perturbations. Here we wish to discuss only one special case, which assumes that the layers are undisturbed so that = 0.
(82)
608
In this event, choosing Cartesian axes so that a ~ -(0,1,0)
(83)
, c ~ -(0,0,1),
it follows at once from equations (78) that
(84)
- (c, 0, 0). Also, if we choose
(85)
I ~ - (b,O,O) , q - ( O , q s i n O , q c o s O ) , one finds from equations (16) that 15 is necessarily zero and that
( 2 + ~ + ~)b + 2 ~ c -
o,
(86)
where r / - [(#o + #2 - 2A1 + A4)sin e 0 + (#o + #4 - 2A2 + As)cos 2 0 + 2 ( ~ x - 7-1 - 7-2 + r s ) s i n 0 c o s 0lq 2, v -
[(r5 - T1) sin 0 + (As - A2) c o s
(87)
O]q.
Also, equations (15) simply yield
(2+~+~ + ~)c + i~b - 0,
(88)
where a - [g~ sin 2 0 + (K~ + g ~ ) cos 2 0 + 2g~ sin 0 cos0]q 2,
(89)
plus expressions for/3 and -~, ft and #. From equations (86) and (88) one readily obtains the equation 2 i p w + *1 iu
2wv, 2i~5w + a
- o,
(90)
or equivalently P)~5w 2 + ()~5q - u 2 + p a ) w + a 7 1 - 0
,
w -- 2iw.
(91)
In general the elastic coefficients occurring in the energy (6) are small compared with viscous terms, and therefore one can assume that p a < < )~5~- u 2.
(92)
As a consequence the two roots of equation (91) are to a good approximation given by w+ - - ' 7 a / ( ~ 5 ' 7 -
v2)
, ~'f - - ( ~ 5 ' 7 -
u2)/p~5,
(93)
609
the 's' and ' f ' subscripts denoting slow and fast modes, respectively, the former detected by light-scattering experiments. Thus we predict a decay time rd given by
74 = f?(q, O)/a(q, 0),
(94)
where ~ / - 2(X5~/-
v2)/rl.
(95)
The coefficient ~5 is positive as remarked earlier, but appeal to the viscous dissipation inequality (18) also shows that r/and f/are both positive. Galerne and coworkers [29] find that their data for the decay time observed in their experiments fits an expression analogous to our result (94), their expression for the elastic factor taking the form
B(O) = (B1 cos 2 0 -b B3 sin 2 0 + 2B13 sin 0 cos O)q2,
(96)
the coefficients those occurring in the Orsay energy [26]. However, as Leslie, Stewart, Carlsson and Nakagawa [31] show, the above expression is identical to that given by equation (89). Galerne and colleagues find also that their data is consistent with a viscosity function given by 1
1 = -- +
cos2(0
-- 0 1 )
(97)
7g, #a and Vg are viscous constants, and 01 and 02 constant angles. While the two viscosity functions appear to differ, they are in fact equivalent, as one finds by setting ~o -[" ~4 -- 2X2 q- )15 -- ttg c o s 2 02 -[- vg q- 7g c ~
01,
#o + it2 - 2A1 + A4 -- #o sin2 02 + v 9 + 7g sin2 01, ~1 - rl - r2 + r5 -- ttg sin 02 cos 02 + 7g sin 01 cos 01, 2)15 ---- ")'g ,
V/'2()15 -- )12) = -l-TgCOS01
,
(98)
~/r2(T 5 -- 71) -- + ~ ' g S i n 0 1 .
Employing the experimental values for 7g, #o and Vg and the angles 01 and 02 one obtains values for groups of viscous coefficients in the theory discussed in this chapter. Given that traditional viscometric measurements may prove inappropriate for smectics given their layered structure, it is necessary to devise suitable experiments employing small samples in which the orientation of the layers and alignment is well monitored. Gill and Leslie [32] provide the necessary theory for one further such experiment, involving the
610
reflection and refraction of a shear wave at a smectic-solid interface. However, there is some way to go in terms of determining the parameters in the above theory, but this may change with its increasing involvement in device modelling. REFERENCES .
2. 3. 4. 5. 6. 7. Q
0
10. 11. 12. 13. 14.
15. 16. 17.
J.L. Ericksen, Trans. Soc. Rheol., 5 (1961), 23. F.M. Leslie, Arch. Rat. Mech. Anal., 28 (1968), 265. F.M. Leslie, Adv. Liq. Cryst., 4 (1979), 1. C.Z. Van Doorn, J. Appl. Phys., 46 (1975), 3738. D.W. Berreman, J. Appl. Phys., 46 (1975), 3746. M.G. Clark and F.M. Leslie, Proc. Roy. Soc. A, 361 (1978), 463. F.M. Leslie, I.W. Stewart and M. Nakagawa, Mol. Cryst. Liq. Cryst., 198 (1991), 443. C.V. Brown, P.E. Dunn and J.C. Jones, Europ. J. Appl. Math., 8 (1997) 281. J.C. Jones and C.V. Brown, Proc. 16th Int. Displays Conference (1996), 151. P.G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd Ed. Oxford University Press, (1993). S. Chandrasekhar, Liquid Crystals, 2nd Ed. Cambridge University Press, (1992). P.J. Collings, Liquid Crystals: Nature's Delicate Phase of Matter, Princeton University Press, (1990). C.W. Oseen, Trans. Faraday Soc., 29 (1933), 883. F.M. Leslie, Contemporary Research in the Mechanics and Mathematics of Materials, Eds. R.C. Batra and M.F. Beatty, CIMNE, Barcelona (1996), 226. T. Carlsson, I.W. Stewart and F.M. Leslie, J. Phys. A, 25 (1992), 2371. S.P.A. Gill and F.M. Leslie, Liq. Cryst., 14 (1993), 1905. T. Carlsson, F.M. Leslie and N.A. Clark, Phys. Rev. E, 51 (1995), 4509.
611
18. P. Pieranski, E. Guyon and P. Keller, J. Phys. (Paris), 36 (1975), 1005. 19. F.M. Leslie and G.I. Blake, Mol. Cryst. Liq. Cryst., 262 (1995), 403. 20. G.I. Blake and F.M. Leslie, Meccanica, 31 (1996), 611. 21. G.I. Blake and F.M. Leslie, (pending). 22. T. Carlsson, N.A. Clark and Z. Zou, Liq. Cryst., 15 (1993), 461. 23. Z. Zou, N.A. Clark and T. Carlsson, Phys. Rev. E, 49 (1994), 3021. 24. G.I. Blake, F.M. Leslie and M.J. Towler, Europ. J. Appl. Math., 8 (1997), 263. 25. P.J. Barratt and B.R. Duffy, Liq. Cryst., 21 (1996), 865. 26. Orsay Group on Liquid Crystals, Solid St. Commun., 9 (1971), 653. 27. T. Carlsson, I.W. Stewart and F.M. Leslie, Liq. Cryst., 9 (1991), 661. 28. P.C. Martin, O. Parodi and P.S. Pershan, Phys. Rev. A, 6 (1972), 2401. 29. Y. Galerne, J.L. Martinand, G. Durand and M. Veyssie, Phys. Rev. Lett., 29 (1972), 562. 30. F.M. Leslie and S.P.A. Gill, Ferroelectrics, 148 (1993), 11. 31. F.M. Leslie, I.W. Stewart, T. Carlsson and M. Nakagawa, Continuum Mech. Thermodyn., 3 (1991), 237. 32. S.P.A. Gill and F.M. Leslie, J. Mech. Phys. Solids, 40 (1992), 1485.
613
EXTENSIONAL FLOWS Christopher J S Petrie Department of Engineering Mathematics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK.
1. I N T R O D U C T I O N The simple, practical, definition of extensional flow is a flow in which the velocity vector can be expressed in the form u = elx ,
v = e2y ,
w = e3z
(1)
in a fixed rectangular Cartesian coordinate system, (x, y, z), where (u, v, w) are the components of the velocity vector and el, e2 and e3 are the principal rates of strain, (which may be functions of time). For an incompressible fluid, the sum of the principal rates of strain is zero. Uniaxial extension is obtained when el = k ;
e2 = e3 = - 8 9
(2)
For many fluids, including all Newtonian fluids, there is no more difference between shear and extension than there is in linear elasticity. Indeed, Trouton [1] introduced what we now call the elongational viscosity (his "coefficient of viscous traction") because he wished to obtain the conventional viscosity (the shear viscosity) of some highly viscous fluids. The elongational viscosity is the ratio of stress to rate of strain in the steady uniaxial extension of a uniform cylinder of material. For an incompressible Newtonian liquid Trouton proved theoretically (and demonstrated experimentally) that this is three times the shear viscosity, r/o, r/E = 37/0,
(3)
just as the Young's modulus is three times the shear modulus for an incompressible Hookean solid. We find that the behaviour of polymeric liquids and of suspensions of long slender particles (fibre suspensions) is markedly different in shear and extension. The ratio of elongational viscosity to shear viscosity, which we call the Trouton
614
ratio, is no longer three, or even close to three. We associate this behaviour with the existence of some sort of structure within the fluid which can become markedly anisotropic during flow, for example when fibres or extended polymer molecules become aligned parallel to one another and to the flow direction. This aspect of the behaviour of the fluids mentioned is highly significant and, especially in the case of dilute polymer solutions, can give rise to Trouton ratios of 100 or 1000 and hence to stresses which are two or three orders of magnitude larger than in shear.
1.1. Definitions The Society of Rheology, in an effort to keep some sort of order in the use of terms by rheologists, has provided an unambiguous definition of "tensile viscosity" [2] - the quantity we have referred to as elongational viscosity, also known as uniaxial extensional viscosity. Tensile viscosity is defined as follows: A material is subjected to homogeneous simple extension, i.e. to a flow which is spatially uniform, with constant rate of strain, ~, in the x~-direction and - 89 in every direction perpendicular to the x 1-axis. The ratio of "net tensile stress", O'E ~ fill --flEE, to rate of strain is monitored as a function of time and the "tensile viscosity" is defined as
(4) This definition, of course, says nothing about methods of, or even the feasibility of, experimental realization. The notation r/T is sometimes used, instead of r/E, in honour of Trouton, and the elongational viscosity may be called the Trouton viscosity. If surface tension is significant, we need to be clear that the "net tensile stress" a E in our definition, Equation (4), is the measured stress corrected for surface tension according to Applied force O"E =
Coefficient of surface tension --
Area
Radius
.
(5)
We also need to be clear that the use of O E ~ O l l - - a 2 2 i s , in any case, valid only for incompressible fluids. This matter is discussed further elsewhere [3,4]. For simplicity, our discussion of other extensional flows, Equations (8), (10) and (11) below, refers to incompressible fluids in the absence of surface tension. As well as uniaxial extension, we may consider equibiaxial extension, el = e2 = ~ ;
e3 = - 2 ~
(6)
and planar extension, el = ~ ;
e2 = 0 ;
e3 = - + .
(7)
615
The equibiaxial extensional viscosity [2] is defined by r/B(~. ) = O'11 -- 0"33
(8)
and for a Newtonian fluid ~Tn = 6770.
(9)
Equibiaxial extension is kinematically the reverse of uniaxial extension, but significantly different in terms of the effect of the flow in tending to align long molecules or fibres. The idea that the functions r/E(~) and r/B(~) may be regarded as the same function, for positive and negative values of the rate of strain in the direction of the axis of symmetry, is not a particularly helpful one since there is no reason at all for supposing that the values of this function over the two ranges of values of its argument are in any way connected (except by continuity in a mathematical sense, which corresponds to Newtonian behaviour in the limiting case of small rate of strain). Planar extension is sometimes referred to as "pure shear" but it must be clearly understood that it is qualitatively different from simple shear, being irrotational (relative to the usual fixed axes). A particular point of interest for planar extension is that there are two extensional viscosities, the planar extensional viscosity ,p(g)
=
O'll -- 0"33
(10)
,
which refers to the tensile stress required to stretch the material in the x 1direction and a second "cross-viscosity" r](0)(~ -) = O"22 -- O"33
(11)
which refers to the tensile stress required to prevent deformation in the neutral direction (the x2-direction). The theoretical relations for a Newtonian fluid, with shear viscosity r/0, are Tip - -
477o ;
r/(~ = 2770.
(12)
The notation 7/(2~ is a simple example of the general notation [2,5-7] for general extensional flows. If el = ~ is the largest (positive) rate of strain, then we may define m such that e2 = m~ and then el = ~ ;
e2 = m~ ;
e3 = - ( 1 + m)~
(13)
for an incompressible fluid. The parameter m, which we take to be independent of time, describes the geometry of the extensional flow, with m = 1 for equibiaxial extension, m = 0 for planar extension, m = - 89 for uniaxial extension and
616
< m < 1 in general. The two extensional viscosities for this general flow
2 -are
~m)(~.)
= fill
-- 0"33
(14)
and Tl(m)(~.) = 0"22 - - 0"33 .
(15)
These are equal to one another for equibiaxial extension and the second is zero for uniaxial extension while for all other extensional flows we have the two physical quantities which we may try to measure. As well as for the three standard cases, m = - ~1, m = 0 and m = 1, experiments have been carded out by Demarmels and Meissner [8] for m = +89 This flow has been referred to as "ellipsoidal extension" but may best be visualized as an unequal biaxial extension, with stretching at rates ~ and 1~ in two perpendicular directions. 2. T H E E X P E R I M E N T A L E V I D E N C E Experiments for determining extensional viscosity are difficult to carry out, and even where there is a consensus about results it is important to remember that this consensus is the fruit of much painstaking experiment. For polymer melts, it has been possible for some time to obtain reliable values for low and moderate rates of strain; the review by Meissner [9] is a useful source of information. Different polymers show different behaviour in the variation of Trouton ratio with rate of strain. As has been remarked above, the value of the Trouton ratio can be large, but for polymer melts it does not reach such extremely high values as for polymer solutions. The situation is much less clear for polymer solutions, as has been summarized by James and Waiters [10]. The experiment which corresponds to the definition of extensional viscosity has not been easy to perform for these mobile (low viscosity) fluids - in fact it had been judged impractical until the work of Matta [11] and Sridhar [12-14]. It is, however, generally accepted that the Trouton ratio for polymer solutions is large. Even with the considerable uncertainty surrounding interpretation, we believe that estimates of 100 or 1000 have genuine physical significance, whether or not the measured quantity is truly an extensional viscosity. Suspensions of long fibres generally behave in a similar manner to polymer solutions. In all cases, at least for fluids which are unoriented in their natural state (i.e. when at rest and relaxed), the limiting values of extensional viscosity for small rate of strain agree with the Newtonian predictions. There is no clear evidence
617 of what happens at high rates of strain, where experiments are difficult, but what evidence there is supports the theoretical conclusion [7] that uniaxial and planar extensional viscosities should be the same. This theoretical study shows two classes of asymptotic behaviour at large rate of strain in planar and uniaxial extension. When we use constitutive equations appropriate to polymer solutions or fibre suspensions, there is an "upper Newtonian" r6gime with a constant extensional viscosity corresponding to fully extended and fully aligned molecules or fibres. With the constitutive equations that were developed for polymer melts a decreasing extensional viscosity was found. The stress must grow with rate of strain, but can do so at a rate less than linear (e.g. (rE ~ In ~ for the Phan-ThienTanner model) [7]. This decrease sometimes follows a maximum in extensional viscosity at some intermediate rate of strain. It is extremely important to distinguish clearly between measurements which do give the extensional viscosity functions directly, i.e. experiments in which steady spatially homogeneous extensional flow is attained (to a reasonable approximation), and measurements which give a stress growth function or transient extensional viscosity (or something even less well-defined). This is not to say that the experiments in which the ideal extensional flow is realised are necessarily the best experiments. If we want measurements that will allow us to predict fluid flow and stress levels in geometrically complicated unsteady flows, we may well find that some sort of transient extensional viscosity is more appropriate. For fundamental understanding of fluid deformation and flow, however, the true extensional viscosity remains an important physical quantity which we should like to measure accurately and reliably.
2.1. Extensional viscosities of polymer melts The most extensively studied material, historically, is low-density polyethylene (LDPE). The most reliable experiments, of Meissner and co-workers (past and present) [15,16], show the elongational stress growth function (commonly called the transient extensional viscosity) for experiments at constant rate of strain. This function, for uniaxial extension, is compared with 3 times the analogous function in shear at low rates of strain (this is regarded as the linear viscoelastic response). Typical behaviour of LDPE is to show the linear viscoelastic response at very low rates of strain (such as 0.001 s -1) and a greater stress at moderate rates of strain (typically 0.01 s -1 up to 1 s -1). This behaviour is sometimes referred to as "strain hardening" or "deformation hardening" (although "strain rate hardening" might be a more appropriate term). This behaviour is also reported [17] for HDPE, which does show "strain hardening", although to a smaller extent. For other geometrical forms of deformation, there are Most work until recently has been on polyisobutylene be investigated at room temperature. PIB behaves in a in uniaxial extension. In biaxial extension the response
fewer results [5,8,18]. (PIB), because it can similar way to LDPE is closer to the linear
618
viscoelastic response (multiplied by the expected factor of 6) at moderate rates of strain. In planar extension the first stress growth function (corresponding to ~Tp) lies between the uniaxial and biaxial responses (all suitably scaled), while the second stress growth function falls below the linear response after a fairly short time. Improved versions of the extensional rheometers have allowed different deformation histories (extension at constant stress or force as well as at constant rate of strain) and measurement have been made on polymethylmethacrylate [19], polystyrene [19] and polybutadiene [20]. Another recent development of considerable importance is the new multiaxial elongational rheometer from Meissner [21] which has been used for extensional flow investigations on polymer melts and blends at elevated temperatures [22].
2.2. Extensional viscosities of polymer solutions The history of extensional flow measurements for polymer solutions is full of apparently contradictory results [4,10]. The major source of this is the use of different experimental techniques, none of which, until recently, are even claimed to reproduce steady uniform uniaxial elongation. Among the techniques that have been used are (a) fibre spinning and open syphon (Fano) flow, (b) contraction flow (into an abrupt contraction or through an orifice), (c) converging flow (through a smoothly tapering channel), (d) stagnation-point flow (e.g. use of opposed jets), (e) falling weight and controlled extension. and each of these has its own advantages and drawbacks. James and Waiters [10] have collected results for one polymer solution and show, in their Figure 2.1, values claimed for the extensional viscosity over a range of three decades (10-10,000 Pa.s) for rates of strain in the range 1-100 s -1, with experimental results scattered over the whole of this region. The explanation of this is not that the experiments are all meaningless, but that, since we are dealing with a viscoelastic material, the differing flow histories and the fact that steady flow is not attained made considerable differences to results. The various collaborative testing programmes reviewed in [4] (one of which produced the results in [10] mentioned above) have made it clear that differences in the material being tested do not explain these differences. In short, the reported quantities are not "extensional viscosities". While the variety of experiments and corresponding predictions may not be diminished in the near future, recent work does suggest that reliable extensional viscosities (in the sense that experiments are reproducible in different laboratories) may be obtained for polymer solutions.
619
2.3. Other experimental measurements The measurement of extensional viscosities and stress growth functions by no means exhausts the possibilities for useful experiments on the extensional flow behaviour of polymeric liquids. Meissner [9] has consistently urged that recoverable strain should be measured as well as total strain, and this has been done for a number of polymer melts. Recent measurements on polymer solutions [23-25] have demonstrated interesting behaviour in stress relaxation at the end of a period of steady uniaxial extension. Attempts have also been made to investigate further the fundamental difference between shear and extension by looking at flows in which the history of the principal extension ratio is the same for a shear flow and an extensional flow [26,27]. Experiments were carried out by Ztille et al. [26] both with ~ increasing exponentially and with ~ constant, where ~ is the principal extension ratio. In each case both shear and uniaxial extension were used. The results for LDPE showed that at large strains and large rates of strain the rheological behaviour is determined primarily by the history of the principal extension ratio, ~(t), and not by whether the flow is an extensional flow or a shear flow. This means that they were able to show deformation hardening and deformation thinning behaviour in extensional flow of LDPE, by choosing in the first case elongation at constant rate (~ = ~f = ~ exp(~t)) and in the second case, elongation with ~ constant. Similarly in shear the apparent viscosity (measured at constant shear rate) is a decreasing function of shear rate while the corresponding quantity for "exponential shear" increases with shear rate. Samurkas et al. [27] point out that to conclude from this that the material is deformation hardening in exponential shear may be misleading. Their point is that under exponential shear a Newtonian liquid will show a similar behaviour. Indeed Ztille et al. [26] show (in their Figure 4) that the measured stresses are generally lower than the predictions for exponential shear of linear viscoelasticity. Samurkas et al. [27] compare exponential shear with planar extension, and present their result in term of the damping function for the Wagner model (see below). Their conclusion is that the damping function obtained from steady shear flow measurements is better than that obtained from planar extension in predicting behaviour in exponential shear. This tends to the opposite conclusion to that reported above [26]. However neither set of results is conclusive, and direct comparison is difficult because of the different ways in which the data are presented. 3. BASIC P R O P E R T I E S OF SOME C O N S T I T U T I V E EQUATIONS IN EXTENSION The extensional viscosity (in uniaxial extension), as well as the planar and equibiaxial extensional viscosities have been tabulated for a number of simple constitutive equations [3]. We reproduce some of these results below (Table 1)
620
and discuss two classes of constitutive equation which have become popular in recent years. These are the Wagner integral model [28] (often used for polymer melts) and the FENE dumbbell models [29] for polymer solutions. Larson [30] gives useful discussion of constitutive equations which is both more thorough and more up-to-date than [3]. Many of the results in [3] are for special cases of the Oldroyd eight-constant model, T + ,~1 '~ - / Z l (T- D + D . T) + vl tr(T- D) I + I~o tr(T)D i" = 27?0 [D + )~2
] - 2#2D" D + y2 tr(D. D) I] ,
(16)
in which the notation T denotes the corotational derivative of the extra-stress tensor, T, and D is the rate of strain tensor. The generalized convected Jeffreys model
is one useful example, with A = 0 giving the generalized convected Maxwell model. A generalized convected derivative D
V
T =T+{(T.
D + D . T) = T -
a ( T - D + D - T)
(18)
is sometimes defined, with a = 1 - ( being used in [3] and elsewhere, and Equation (16) could use the generalized convected derivative if a = #l/A1 = //2/,~2- The parameter ( takes values in the range 0 <_ ( _< 2 and correspondingly 1 >_ a >_ - 1 . The upper convected derivative corresponds to ( = 0 (i.e. a - 1) while ( - 1 (or a - 0 ) gives us the corotational derivative and ( = 2 (or a = - 1 ) gives the lower convected derivative. In terms of molecular (or micro-structural) models, only the upper and lower convected derivatives are obtained from so-called affine deformation models. In these models, hypothetical micro-structural elements in the fluid, (such as, for example, either network junctions or dumbbells) move affinely with the fluid. The parameter ( can be interpreted in terms of a non-affine deformation of elements in the fluid, i.e. a slip between micro-structural elements and the surrounding fluid. This leads to some predictions in shear flow which are regarded as undesirable and in extreme cases are contrary to physical intuition (e.g. negative stresses in response to a sudden large shear deformation). There are other consequences in extensional flows, such as non-existence of solutions to flows problems for some values of the parameter (, e.g. ( > 0.5 [3], but the whole range of non-affine convected derivatives are by no means ruled out
621
Model
tiE
Newtonian UCM LCM GCM Corotational
~B
3770 67/0 3r/ 60 (1+)~)(1-2,~) (1+4a~)(1-2)~) 3rl 67/ (1+2,~0(1- a0 (l+2aO(1-4aO 3r/ 67/ (l+a,~)(1-2a)~O (l+4a,~)(1-2aa~)
~p
r](0)
4r/o 47/ (1+2,~)(1-2,~) 4O _ 0+2aO(1-2aO 40 (l+2a)~)(1-2a,~)
27/0 2q (1+2)~) 27/ (1-2a~) 27/ (l+2aag)
Oldroyd B
3r/ 1-Ag(I+2M) 3r/(l+,~)(l_2M)
67/ ~_ l+2Ag(1-4ag) o~'](1+4)i~)(1-2~)
4r/ 4r/(1-4aAg 2) (1+2~)(1-2~)
27/ 27/ I+2Ag (l+2,~g)
Oldroyd A
l+Ag(1-2)~) 37/(1-~)(1+2~)
1-2Ag(l+4~) 6r/ (1-4~)(1+2~)
4r/(1-4~Ag 2) (1+2~)(1-2~)
,-, z r / ~1-2Ag
Table 1. Extensional viscosities for some simple constitutive equations. The abbreviations in Table 1 for rheological model names are " U C M " for the upper convected Maxwell model, "LCM" for the lower convected Maxwell model, "GCM" for the generalized convected Maxwell model. The corotational result applies to Maxwell and Jeffreys models. The Oldroyd fluids B and A are exactly equivalent to upper and lower convected Jeffreys models, respectively. by this behaviour (unless one adopts a very strict position on the mathematical requirements for a constitutive equation [31-33]). The material functions for the eight-constant Oldroyd model are 1 -/z2~" + (/Zlfl2 - 3T2)6"2 r/z = 37/0 1 -- #1~ + (#21 -- 371)~"2 1 + 2/z2~ + 4(/Zl/Z2 - 37-2)c2
r/a = 67/0 f + 2/Zl~ + 4(~u2
37-1)~2
1 + 47-2~2 r/p = 47/o 1 + 47-1~2
(20) (21)
1 - 2(#1 - ~ 2 ) c + 47-2c 2 + 8(,tt271 - ~ 1 7 2 ) ~'3
r/(2~ = 2r/o
(19)
(22)
1 + 4 T1 ~-2
where we have defined parameters 7-1 and 7-2: 7"1 = (/tO/2)(2/z1 -- 3Vl) -- ~1(/Zl -- 171) ,
(23)
622
7-2 = (#0/2)(2#2 - 3y2) - / t l ( t t 2
-/,'2)
.
(24)
These are related to the parameters al = r~ + A2 and a2 = 7-2+ A1A2 which occur in the viscometric functions 1 + cr2k2 O(k) = ~o 1 + 0"1k 2 '
(25)
~t~l(k ) ---- 2 A l r / ( k )
(26)
- 2A2r/o,
~2(k) = -(A1 - / t l ) r / ( k ) + (A2-/t2)0o
(27)
for shear rate k. One interesting aspect of this is that all the parameter combinations which appear in the expressions for the extensional viscosities also appear in the expressions for the viscometric functions so that, in principle at least, the extensional flow behaviour of the eight-constant Oldroyd fluid can be predicted from complete data on shear flow. In fact, data for shear flow allow the calculation of the characteristic times A1 and A2, which do not appear in the extensional viscosity formulae, so that in a sense, for this model, the viscometric functions contain more information than the extensional viscosity functions. Note that the actual calculation of the parameters from viscometric data is not a trivial task and, for example, to obtain the parameters ttl and #2 which are essential for extensional viscosities one must have very good data on the second normal stress function, ~2(k).
3.1. Wagner and Kaye-BKZ constitutive equations The Wagner equation started out as a form of the Kaye-BKZ equation in which the nonlinear and time-dependent parts of the material behaviour are assumed to be independent in the sense that the kernel of the constitutive equation may be factorized. It is, at present, probably the most successful compromise between simplicity and generality in a constitutive equation for the quantitative description of the rheological behaviour of polymer melts. The model may be written
T =
M ( t - t')h(I~, I2)Ct~(t ') dr'
(28)
Oo
in which M ( t - t') is the memory function, c~-l(t ') the relative Finger deformation tensor and h(I1,12) the damping function, which introduces non-linearity of dependence on deformation through the invariants I1 and 12 of the Finger tensor. The product h(I1, I2)Ctl(t ') may be thought of as a non-linear measure of deformation [30,34]. If h = 1 and M ( s ) = ( G / A ) e x p ( - s / A ) we recover the integral equivalent of the upper convected Maxwell model. A sum of exponentials for M ( s ) gives the usual discrete relaxation spectrum. Equation (28) is not, in fact,
623
a Kaye-BKZ equation unless the damping function is independent of the second invariant, I2 - see below, Equation (35). In a general extensional flow, Equation (13), we may define the relative Hencky (or logarithmic) strain, e, and the relative extension ratio, g: e = +(t - t ' ) ,
g = e ~ = e eCt-t'~
and then we have
ctl(t ')=
00 )
0 0
0
(29)
(30)
(-2(re+l)
with invariants I1 = g2 + g2m + g-2(rn+l) ,
12 = g-2 + ( - 2 m + ~2(m+l) .
(31)
The damping function is chosen to fit data from shear or extensional viscometry (or preferably both). Early attempts used functions like h = e x p ( - n ~ ) for a shear strain of ~ = k ( t - t') (and we could write ~ = vql - 3, noting that I1 = 12 in simple shear). In order to fit data from different flows (shear and extension), the introduction of an invariant K = ~//311 +(1 -/3)I2 - 3
(32)
has been successful [35] and better fits to data have been obtained using a sum of exponentials h(I1,12) = f e - n l K + (1 - f)e -n2K .
(33)
Wagner and Laun [36] found that values f = 0.57, n l = 0.310 and n 2 = 0.106 gave a good fit to shear data for the LDPE Melt known as "IUPAC A", and /3 = 0.032 gave a good fit for uniaxial extension also. Another form of damping function which is popular is due to Papanastasiou [37] 1 h(I1,12) = 1 + a K 2 "
(34)
Larson [30] notes that neither of these functions, Equations (33) and (34), fits data for biaxial extension well and he proposes a model which is strictly of the Kaye-BKZ form, with a potential function, U, and the Cauchy deformation tensor, Ct(t~), T = ft__~ M ( t -
t') 2
0U(I1, I2) 0U(I1, I2)ct(~,)] d r ' . 0-]-i C t l ( t ') - 2 012
(35)
624
The potential function used here is U(Ii,12)=~In
1+
(I-3)
(36)
with both c~ and I depending on the invariants I1 and 12: a = CO+ C2 tan -1
[ Cl (I2 -- I1)3 ] 1 +(12--I1)2 ,
(37)
I = (1 - 3)11 + X/1 + 2fli2 - 1
(38)
and, for IUPAC A, co = 0.20, cl - 0.05, c2 - 0.121 and 3 = 0.1. This does a better job of fitting all the data than the Wagner equation without the Cauchy deformation tensor, Equation (28); Equation (28) also has the defect of predicting a zero second normal stress difference in simple shear. Similarly, Demarmels found [8,38] that the Wagner equation does not give a consistently good fit to data from shear and several different extensional flows for PIB and that a better fit can be obtained with a model which introduces the Cauchy deformation tensor, Ct(tt), as well as the Finger tensor. This gives us an equation of the form proposed by Rivlin and Sawyers, which we write here T = f t_ o M ( t
- t ' ) [ h l ( I i , I 2 ) C t l ( t t ) + h2(Ii,I2)Ct(tt)]
dt t
(39)
and Wagner and Demarmels [38] propose a form for the two damping functions, for one particular PIB melt, hl(I1, I2) = (1 +/3)h,
1
h2(II,I2) =/3h,
h = 1 + a'~*lv'k~- 3)(12 - 3)
(40)
with constant values of the parameters a = 0.11 and 3 = - 0 . 2 7 for the particular PIB melt studied in [8]. The parameter/3 in this model, which gives the ratio of second to first normal stress differences in simple shear, is important in general extensional flows but unimportant in uniaxial extension. This choice of functions h l and h2 is not obtainable from a potential, so Equations (39) and (40) do not give a Kaye-BKZ equation. A variation in use of the Kaye-BKZ model [39] seeks to express the dependence of the kernel on the invariants I1 and 12 through dependence on the principal stretches (or their squares, which are the eigenvalues of the Finger tensor). This is motivated by the success of some strain-energy functions for rubbers, but has not been found to be successful so far. It has not proved possible to pick a simple dependence on the principal stretches which gives a good fit to shear and extensional data with the same values of parameters.
625
One final, well-known, defect of these models has been considered by Wagner [40]. This is the fact that even the best of the models with a damping function or non-linear deformation measure has difficulty in fitting data obtained in flows where the fluid experiences flow reversal. The simple example of this is one step strain followed by a second step strain in the opposite direction. The problem can be explained in terms of a temporary network model in terms of irreversible loss of entanglements. This suggests a damping functional as a replacement for the damping function. The damping functional proposed is the smallest value over the time interval of the conventional damping function. For a motion with a "nondecreasing deformation", defined as a deformation for which the damping function is a nonincreasing function of time, the damping function is correct. If the motion involves a "decreasing deformation", a smaller damping factor is used. This approach has some success in predicting recovery (elastic recoil) after uniaxial extension [40].
3.2. FENE constitutive equations The FENE (finitely extensible nonlinear elastic) dumbbell model is found to be useful for polymer solutions, though there are many questions, both about the theoretical foundations of the model and about quantitative agreement with data on polymer solutions. There are a number of variants on the model, which we shall discuss briefly. First we outline the basic features of the model, while avoiding a formal derivation of the equations; for details see, for example, [29,30]. The simplest FENE dumbbell model represents a polymer molecule in solution as an isolated dumbbell (two beads connected by a spring) whose motion is governed by a balance between a spring force tending to contract or coil the molecule and hydrodynamic drag on the dumbbell ends due to the solvent, which tends to stretch the molecule and align it with the flow. A Hookean spring has the disadvantage of allowing the molecule to extend indefinitely, so a non-linear spring law, commonly the Warner spring law,
HR Fa = 1 - ( R 2 / L 2) '
(41)
is used. Here R is the end-to-end vector for the dumbbell (the molecule), H is the spring constant, R = IR] is the end-to-end length and L is the maximum permitted end-to-end length for the dumbbell, so that R < L. The hydrodynamic drag force on a bead is given by a drag coefficient (d multiplying the velocity of the bead relative to the solvent. This gives us the relaxation time, A = ( d / ( 4 H ) , which is a characteristic time for an individual dumbbell to come to equilibrium under the competing action of the drag and spring forces. There is a second characteristic time, 0 = L 2 ( d / ( 1 2 k T ) which is associated with the balance between hydrodynamic drag and Brownian diffusion and a modulus,
626
G = n k T , (as in rubber elasticity, where n is the number of dumbbells per unit volume). We define the "FENE parameter", b = 30/A = n H L 2 / G
(42)
and can also introduce a characteristic molecular dimension a by
a 2 = 3L2/b = 3 k T / H .
(43)
This gives the equilibrium length of a dumbbell (when the solution is at rest), possibly multiplied by a factor like b/(b + 5), depending on the precise details of the FENE model. These ideas are used to obtain a configuration tensor, A = (RR) and the polymer contribution to the extra-stress tensor. The mean, ( ) , is an ensemble average involving the distribution function, g,(R), and the trick is to obtain equations without having to calculate if,. This is most usually done by making the Peterlin approximation, which involves pre-averaging the end-to-end length, so that Fd =
HR 1 -- ( ( R 2 ) / L 2)
(44)
instead of using the true ensemble average of Equation (41). This leads to the FENE-P model, with configuration evolution equation v XA +
1
L2 A = ~ I 1 - ( R 2 / L 2) (b + 2)
(45)
and the extra-stress T=
G(b/L2) A 1 -- ( R 2 / L 2)
Gb I + 2r/~D (b + 2)
(46)
in which r/~ is the solvent viscosity. The Chilcott-Rallison (or FENE-CR) model [41,42] is
v AA + T=
1 (L2/b) AI 1 - ( R 2 / L 2) 1 - ( R 2 / L 2) '
(47)
G(b/L2) [A - (L2/b)I] + 2~/,D. 1 - ( R 2 / L 2)
(48)
This model has qualitatively the same behaviour in extensional flows as the FENE-P model, as far as is known from investigations to date [4]. It has the property of a constant viscosity in shear, unlike the FENE-P and FENE models
627
which are shear-thinning [41], which is both a simplification and is desirable for modelling the behaviour of Boger fluids. One shortcoming of all the FENE models discussed above, as far as fitting data on polymer solutions is concerned, is that they have only a single relaxation time. A recent discussion of multimode models (i.e. bead-spring chains) by Wedgewood, Ostrov and Bird [43] points out that a straightforward generalization of the simple dumbbell leads to a complicated set of coupled nonlinear differential equations. They point out that some earlier attempts to analyse this contain serious errors and go on to propose a further approximation, the FENEPM model. The model consists of a set of N - 1 nonlinear springs joining N beads and the end-to-end vector for the i-th spring is denoted by Ri. The FENE-PM force law is taken to be Fi =
HRi 1 --(iV -- I) -I ~N~'((R2)/L2)
(49)
and the M in FENE-PM stands for the mean value taken in the denominator of Equation (49). The M is also used to denote "multimode", but it is necessar3, to remember that the FENE-PM model is not just a multimode FENE-P model, for the reasons of complexity to which we have alluded. Even so, the model obviously remains more difficult to use than the single mode FENE-P and FENE-CR models and the questions to be settled are whether the extra effort is adequately rewarded and whether the approximation introduces any undesirable side-effects. As examples of unwanted side-effects, the fact that the FENE-P model leads to equations for steady extension with multiple solutions may be instanced. This is a comparatively minor matter, which can be resolved [4] by analysis of the full equations, as is discussed below. A more interesting matter is the demonstration by Keunings [44] that the dumbbells in the FENE-P model do not actually behave as finitely extensible dumbbells. A simulation shows that a noticeable fraction of the dumbbells exceed the supposed maximum length L. This arises from the fact that the approximation leads to a distribution function for end-to-end lengths which is Gaussian (with an infinite tail), while for a true FENE model the distribution must have a cut-off at L and hence must be nonGaussian. A comparison between the FENE-P and FENE model predictions (using a stochastic simulation) [44] shows that the rheological effect of the Peterlin pre-averaging approximation is seen in a much more rapid increase in the tensile stress during the start-up of an extensional flow. The stress growth is smoother and somewhat slower for the tree FENE model. The steady state stresses are the same. None of the FENE models yields explicit formulae for the extensional viscosities (unless one wishes to write down formally the algebraic solution to a
628
cubic equation). It is therefore less easy to make simple statements about their properties. In the limit of small extension rate, the Newtonian ratios between the extensional viscosities and the shear viscosity are recovered. In uniaxial extension the viscosity curve is S-shaped and at large extension rate, an "upper Newtonian" rrgime is obtained [45], tiE = 3q~ + 6 G ~ = 3rl~ + 2b~Tp
(50)
where r/p = AG is the polymer contribution to the viscosity. As far as the FENEPM model is concemed, the model can be expected to improve the quantitative fit to data. No surprises in the qualitative behaviour have come to light so far. 4. S O M E M A T H E M A T I C A L ASPECTS OF E X T E N S I O N A L F L O W Once we have a constitutive equation and a particular flow to analyse, whether exactly or approximately, we can address mathematical questions of existence and uniqueness of solutions. These are by no means trivial or unnecessary exercises for the nonlinear systems with which we are faced, and there are particular problems in analysing boundary-value problems, even for simple onedimensional systems (and ordinary differential equations). In this section we discuss one existence problem in detail, recognizing that rheologically it is rather simple, as an illustration of the sort of problem that may have to be faced. It is an open question as to whether the limitations that are uncovered are seen as defects in the rheological model used, defects in the fluid dynamical approximations or merely as a warning that predictions obtained with the model must always be treated with a modicum of scepticism. After this, we shall discuss an example of lack of uniqueness of solutions to the steady-state equations. This has been mentioned above, and is a likely consequence of non-linearities of the sort we are introducing. The stability of such simple solutions, which is described by the full, time-dependent, dynamical equations, can be used to make a choice between the possibilities. In some cases, such as the example below, we can in fact go further than this and make some very general claims about the global behaviour of the system, which rule out some physically unrealistic solutions even though they do, formally, satisfy the steady-state equations.
4.1. Existence of solutions to a boundary-value problem Steady fibre spinning of a convected Maxwell model can be posed as a boundary-value problem which only has solutions for a limited range of values of one boundary condition. We consider the steady axisymmetric extensional flow given by Equation (2) with the spatially varying rate of strain k = U ' ( X ) in which U is the velocity at distance X from the start of the fibre (as it emerges from a spinnerette or die). The fibre take-up is at X = Ls where the velocity is
629
set to be U~, while the initial velocity is Uo at the spinnerette. If the volumetric flow rate, Q, is given, we can relate the cross-sectional area, A ( X ) to the velocity since we make the basic assumption that, to a first approximation, all quantities are uniform across the fibre. In the absence of gravity, inertia, surface tension and air drag, the five equations governing this flow for an isothermal incompressible upper convected Maxwell model, Equation (17) with ( = 0 and A = 0, are mass conservation, equilibrium of forces axially and radially and axial and radial components of the constitutive equation:
U(X)A(X) = Q,
(51)
{Txx(X) - P(X)} A(X) = F,
(52) (53) (54)
TRR(X) -- P ( X ) = O, T x x ( X ) + A {U(X)TJcx(X) - 2 U ' ( X ) T x x C u and
= 2r/U'(X)
TRR(X) + A {U(X)T~RR(X)+ U'(X)TRR(X)} = - r l U ' ( X )
(55)
in which F is the constant force at any cross-section and P ( X ) is the constitutively undetermined hydrostatic pressure (which is, in effect, calculated from the radial force balance, Equation (53)). Equations (51)-(53) can be combined to give
Txx(X)-
(56)
TRR(X)= F U ( X ) / Q
and so we have a second-order differential-algebraic system, Equations (54)-(56), for the velocity and two extra-stress components, with three conditions, Txx(O) =
To,
U(O) = Uo ,
(57)
U(L~) = U1,
two of which are needed for the differential equations and the third to determine the force, F, which has to be applied to maintain the specified take-up velocity. The extra-stress value, To, is all that is required from the flow history, corresponding to the internal structure of the fluid at X = 0, i.e. to that which has been determined by the flow of the material upstream of the spinnerette. This formulation shows the nature of the problem as a two-point boundaryvalue problem, and the question we can address is whether there are solutions for any draw ratio, DR = U1/U0, and how the answer to this may be affected by the value of To, i.e. by the flow history of the material. In dimensionless form, with u
u , Uo
x
x=-;--,
~s
c~=
~Uo L~
t,=
,1Q AUoF
,
7-o=
ToQ FUo
(58)
630 the system of equations may be reduced to c~2u" = (1 + 2c~u')(1 -
om')o~u'/u - 3it (cru'/u) 2
(59)
with # undetermined, so that the three conditions 1 u(0) = 1 , u'(0) = go = 3 a ( # + 70) - 2c~ ' u(1) = may be satisfied. It can easily be shown that, if 90>0.
DR
(60)
DR > 1, it is necessary that
What we can prove is that the value, u(1), of the solution of Equation (59) at the take-up, x = 1, is bounded above by a function of the Deborah number, a, whatever the values of To and #. This means that there is an upper limit to the draw ratio, DR, for any given Deborah number, regardless of the flow history and regardless of how great a take-up force is applied. The proof involves considering the comparison equation cr2v" = (1 + 2crv')(1 - a,v')crv'/v which can be integrated, with initial values v(0) = 1, v~(0) = 90 to give cry'= (v 3 + K)/(v 3 - 2 K ) where K = (cr9o - 1)/(2c~9o + 1) and then x =
ce
fv y3
_
2K dy.
y3+K
Equation (64) gives v implicitly as a function of
y3 9~(v, K ) =
2K
flv y3 + K dy .
(61) (62) (63) (64)
x/o~ and K ; we may write (65)
and the solution to 9~(v, K ) = x/o: is written v(x; a, 90) = ~ ( x / a , K ) . (66) We note that the condition ago > 0 implies that - 1 < K < 0.5 and can establish that dv/dK > 0 under this condition. Hence, for all admissible values of 90, v(1; c~, go) < ~(1/a,, 0.5) (67) and we can calculate the fight-hand side of Inequality (67) numerically. For example, if (x = 1/19, ~,(19,0.5)= 20.6328. Finally a comparison theorem applied to Equations (59) and (61) shows that DR = u(1; c~,#,go) < v(1; c~,go) < g,(1/~,0.5) (68) which gives us an upper bound on DR for any chosen c~, irrespective of the values of # and go.
631
4.2. Multiple steady state solutions We consider uniform uniaxial extension, Equation (2), which starts at time t = 0. We will need to specify an initial configuration in order to obtain a specific solution to the evolution equations for the configuration tensor (and hence the stress). We consider the FENE-P model, Equations (45) and (46) and define the dimensionless configuration variables y=
A~ - A22 L2 ,
tr A All + 2A22 z = L2 = L2 .
(69)
The first of these, V, may be interpreted as describing the degree of alignment of the dumbbells with the direction of elongation while the second, z, gives the mean end-to-end length of the dumbbells in the flow (i.e. the extent to which the dumbbells are fully stretched). It is obvious that 0 < z < 1
(70)
and also fairly easy to establish that
-z/2
<_ y <_ z
(71)
(see [4,45] where an argument suggested by 0ttinger is set out). If y = z the dumbbells are fully aligned with the flow, while if y = - z / 2 the dumbbells are all aligned perpendicular to the flow. The result of substituting the flow field given by Equation (2) in Equation (45) and expressing the result in terms of the variables defined in Equation (69) is the pair of evolution equations A~=sz+
(1) s
1-z
y
(72)
m,
~ = 2sy
+c (73) 1-z in which s = ~k is the dimensionless rate of strain (which may be regarded as a Deborah number) and c = 3/(b + 2) is a constant depending on the finite extensibility parameter, b. For large values of b, c ~ a 2 / L 2, the equilibrium mean square end-to-end length of the dumbbells expressed as a fraction of their maximum length (and this is small). We may examine the steady-state solutions of this pair of equations and discover that there are three, of which only one satisfies Inequalities (70) and (71). It is relatively straightforward, if tedious, show that the unphysical steady-state solutions are unstable. A more important question is whether, if y and z have initial values which satisfy Inequalities (70) and (71), i.e. are physically reasonable, the subsequent solution, (y(t), z(t)), remains physically reasonable. We answer the question by considering the behaviour of solutions represented by curves in the phase-plane, (z(t), y(t)). We shall prove that solutions which start
632
in the triangle 0 < z < 1, - z / 2 all subsequent time, t > 0.
< y < z, for t = 0, remain in that triangle for
It is easy to see, from Equation (73), that ~. < 0 for z ~ 1 - , so that z cannot reach the line z = 1 provided that z(0) < 1. Next we consider the angle between the solution curve as it crosses the line y = z and the normal the that line which points into the region V < z. If we denote this normal by the vector n, we have n = ( 1 , - 1 ) . The tangent to the solution curve (in the direction of increasing t) is parallel to the vector v, given by v = (~,, #). If the angle between these two vectors is between - 7 r / 2 and 7r/2, i.e. if the scalar product n . v is positive, then the solution curves all cross the line y = z from the region y > z to the region Y < z. When Y = z, ~, = y + c so the scalar product is n. v = (#+c)
- (#) = c > 0
(74)
and therefore solutions cannot leave the physically sensible region across the line y = z. The same argument may be applied to the line y = - z / 2 , (1/2, 1). Using this value of y gives il = s z - s z / 2 + z/2(1 - z) iz = - s z - z / ( 1 - z) + c and hence the scalar product n . v = ~ / 2 + fl = c / 2
> 0
with normal n = (75) (76) (77)
which shows that solution curves cross the line V = - z / 2 from the region y < - z / 2 into the region y > - z / 2 , as required. We may note, further, that for s > 0, i.e. for uniaxial extension (rather than for uniaxial compression or its equivalent, equibiaxial extension), # > 0 on y = 0 for 0 < z < 1 so that in extension alignment parallel to the flow is favoured (in the sense that if V(0) > 0 we have y(t) > 0 for all t > 0 while if y(0) < 0 we may expect v(t) > 0 at some subsequent time). Similarly if s < 0 alignment perpendicular to the flow is favoured, as we would expect in uniaxial compression. The same result may be proved for the FENE-CR model, Equations (47) and (48). The differences from the FENE-P model are only in the coefficient of the isotropic tensor, I, and the effect on the configuration evolution equations is that c is replaced by c/(1 - z) in Equation (73). Equation (72) remains unaltered and the arguments used above hold for the direction in which solution curves cross the lines Y = z and Y = - z / 2 . The behaviour as z ~ 1 is the same, provided that c < 1 and it is easy to see that this will be the case, since r > 1 would require b <_ 1 and the model only makes sense for b > 3 (otherwise a 2 > L2). Hence the result that solutions which start off in the physically sensible region remain there for all time is established for this model too.
633
5. SOME OUTSTANDING P R O B L E M S IN EXTENSIONAL F L O W
There are many unsolved problems in extensional flow, and so plenty of work for rheologists to do. The choice of the best constitutive equation is still an open question- to which the safest answer is often that it depends on the purpose for which the constitutive equation is required. Good quantitative agreement with extensional flow data, and data on linear viscoelastic and viscometric properties, may usually be obtained only at the price of introducing a complicated equation which may present too great a challenge in the computational simulation of complex flows. There is, of course, the question of whether the need to choose a constitutive equation may be avoided by computational methods involving direct simulation of behaviour at a molecular or micro-structural level. My view on this is that we still have a need for constitutive equations, as an aid to our understanding, even if not for our use of computers to solve problems. A useful perspective on the importance of extensional flows may be gained from the list of key challenges in polymer processing given by Kurtz [46]. Of the five key challenges listed, two ("Blown film bubble stability" and "Draw resonance") quite clearly involve extensional flow. The other three ("Screw wear", "Sharkskin melt fracture" and "Scale up problems") may involve extensional flow as a more or less important feature. Some comments on this and other issues have recently appeared [47].
5.1. Interpretation of experimental results Two pieces of work may usefully be mentioned here. The first seemed at first sight to be a good idea, but probably will not stand up to scrutiny. That is the idea of a three-dimensional plot of transient extensional viscosity as a function of strain and time [48-50]. While this appears promising for limited data, its theoretical foundations are questionable [51 ] and the effort involved in preparing three-dimensional plots, let alone using them, does not seem to be justified by any increase in our understanding of extensional flows, or by any practical application of the three-dimensional plots. Certainly it is true that a transient extensional viscosity for a viscoelastic material should never be regarded as a function of instantaneous rate of strain alone and the emphasis of that point is a useful outcome of this discussion. It is doubtful whether, in general, one more parameter can carry all the information about the state of the material (as influenced by its flow history) that is needed to give a well-determined value for the transient extensional viscosity. The second set of results is more convincing, showing how results from the "Rheotens" apparatus (in effect a melt spinning device) may be presented in a set of "mastercurves" and "super-mastercurves" [52,53]. This presents a challenge to the theorist to explain why such behaviour is observed - does it tell us something about the family of experiments or about the appropriate constitutive
634
equation which we have not yet appreciated? A final comment on experimental methods is perhaps worth making. While the analysis of complex flows, such as converging flow or flow into a contraction, does not give a reliable extensional viscosity in the sense of a fundamental material property, such flows have a large component of extensional flow. Hence if the Trouton ratio of a fluid is large, the stresses in the fluid will be predominantly those associated with the extensional part of the rate of strain. A more practical point is that, if a consistent analysis can be made, measurements on such flows should give data which can be used for reliable predictions of stresses for similar flows in industrial processes. The test of a fluid property derived from converging flow is then not so much "Does it give a true extensional viscosity?" as "Does its use give reliable predictions in related practical flows?"
5.2. Stability of extensional flows This topic has considerable relevance to polymer processing, where the prediction of flow instability gives an understanding of limitations to production rates for artefacts made from polymeric materials. There are a variety of instabilities and failures in extensional flow which the author has discussed elsewhere [54]. One recurrent theme is the need for clear distinction between different instabilities such as "draw resonance" and the classical instability of a filament due to capillarity. Similar clarity of thought and of description is needed when filament rupture is considered [55]. Instabilities in extrusion ("melt fracture", "sharkskin") are not immediately associated with extensional flow, but there are strong links according to some of the mechanisms for these two distinct flow defects [46,56]. The topic of approximations for nearly extensional flows [3,4,57] has several connections both with stability and with the analysis of industrial processes like fibre spinning and film manufacture, both by tubular film blowing and by film casting. It is therefore not merely a topic of interest to theorists as a matter of scientific curiosity, but a topic which could shed much light on our analysis of a number of industrial processes.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
F.T. Trouton, Proc. Roy. Soc., A77 (1906) 426-440. J.M. Dealy, J. Rheol., 38 (1994) 179-191. C.J.S. Petrie, Elongational Flows, Pitman, London, 1979. C.J.S. Petrie, Rheol. Acta, 34 (1995) 12-26. J. Meissner, S.E. Stephenson, A. Demarmels and P. Portmann, J. NonNewtonian Fluid Mech., 11 (1982) 221-237. C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 14 (1984)189-202. C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 34 (1990) 37-62.
635
A. Demarmels and J. Meissner, Coll. Polym. Sci., 264 (1986) 829-846. 9. J. Meissner, Ann. Revs. Fluid Mech., 17 (1985) 45-64. 10. D.F. James and K. Waiters, A critical appraisal of available methods for the measurement of extensional properties of mobile systems, in Techniques of Rheological Measurement, Ed A.A. Collyer, Elsevier, New York, pp.33-53, 1994. 11. J.E. Matta and R.P. Tytus, J. Non-Newtonian Fluid Mech., 35 (1990) 215229. 12. T. S ridhar, Recent progress in the measurement of extensional properties of polymer solutions, in IUTAM Symposium on Rheology and ComputationAbstracts, Ed J.D. Atkinson, N. Phan-Thien and R.I. Tanner, University of Sydney, Australia, pp.29-30, 1997. 13. T. Sridhar, V. Tirtaatmadja, D.A. Nguyen DA and R.K. Gupta, J. NonNewtonian Fluid Mech., 40 (1991) 271-280. 14. V. Tirtaatmadja and T. Sridhar, J. Rheol., 37 (1993) 1081-1102. 15. J. Meissner, Trans. Soc. Rheol., 16 (1972) 405-420. 16. H.M. Laun and H. Mtinstedt, Rheol. Acta, 17 (1978) 415-425. 17. J.J. Linster and J. Meissner, Polymer Bulletin, 16 (1986) 187-194. 18. A. Demarmels and J. Meissner, Rheol. Acta, 24 (1985) 253-259. 19. J.J. Linster and J. Meissner, Makromol. Chem., 190 (1989) 599-611. 20. L. Berger and J. Meissner, Rheol. Acta, 31 (1992) 63-74. 21. J. Meissner and J. Hostettler, Rheol. Acta, 33 (1994) 1-21. 22. H. Gramespacher and J. Meissner, J. Rheol., 41 (1997) 27-44. 23. N.V. Orr and T. Sridhar, J. Non-Newtonian Fluid Mech., 67 (1996) 77-103. 24. S.H. Spiegelberg and G.H. McKinley, J. Non-Newtonian Fluid Mech., 67 (1996) 49-76. 25. J. van Nieuwkoop and M.M.O. Muller von Czemicki, J. Non-Newtonian Fluid Mech., 67 (1996) 105-123. 26. B. Ziille, J.J. Linster, J. Meissner and H.P. Htirlimann, J. Rheol., 31 (1987) 583-598. 27. T. Samurkas, R.G. Larson and J.M. Dealy, J. Rheol., 33 (1989) 559-578. 28. M.H. Wagner, J. Non-Newtonian Fluid Mech., 4 (1978) 39-55. 29. R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory, (Second Edition), Wiley, New York, 1987. 30. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, 1988. 31. Y. Kwon and A.I. Leonov, J. Non-Newtonian Fluid Mech., 58 (1995) 25-46. 32. M. Simhambhatla and A.I. Leonov, Rheol. Acta, 34 (1995) 259-273. 33. A.I. Leonov, Simple constitutive equations for viscoelastic liquids: formulations, analysis and comparison with data, in this volume, 1998. 34. C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 5 (1979) 147-176. 35. M.H. Wagner, T. Raible and J. Meissner, Rheol. Acta, 18 (1979) 427-428. o
636
36. M.H. Wagner and H.M. Laun, Rheol. Acta, 17 (1978) 138-148. 37. A.C. Papanastasiou, L.E. Scriven and C.W. Macosko, J. Rheol., 27 (1983) 387-410. 38. M.H. Wagner and A. Demarmels, J. Rheol., 34 (1990) 943-958. 39. K. Feigl, H.C. 0ttinger and J. Meissner, Rheol. Acta, 32 (1993) 438-446. 40. M.H. Wagner and S.E. Stephenson, J. Rheol., 23 (1979) 489-504. 41. M.D. Chilcott and J.M. Rallison, J. Non-Newtonian Fluid Mech., 29 (1988) 381-432. 42. J.M. Rallison and E.J. Hinch, J. Non-Newtonian Fluid Mech., 29 (1988) 37-55. 43. L.E. Wedgewood, D.N. Ostrov and R.B. Bird, J. Non-Newtonian Fluid Mech., 40 (1991) 119-139. 44. R. Keunings, J. Non-Newtonian Fluid Mech., 68 (1997) 85-100. 45. C.JoS. Petrie, J. Non-Newtonian Fluid Mech., 54 (1994) 251-267. 46. S.J. Kurtz, Some key challenges in polymer processing technology, in Recent Advances in Non-Newtonian Flows, AMD-Vol 153/PED-Vol 14I, ASME, New York, pp. 1-13, 1992. 47. C.J.S. Petrie, Recent ideas in extensional rheology, in Polymer Processing Society Europe/Africa Regional Meeting - Extended Abstracts, Ed J. Becker, Chalmers University of Technology, Gothenburg, Sweden, KN 4:2, 1997. 48. J. Ferguson and N.E. Hudson, European Polym. J., 29 (1993) 141-147. 49. J. Ferguson and N.E. Hudson, J. Non-Newtonian Fluid Mech., 52 (1994) 121-135. 50. J. Ferguson, N.E. Hudson and M.A. Odriozola, J. Non-Newtonian Fluid Mech., 68 (1997) 241-257. 51. C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 70 (1997) 205-218. 52. M.H. Wagner, V. Schulze and A. G6ttfert, Polym. Eng. Sci., 36 (1996) 925-935. 53. M.H. Wagner, B. Collignon and J. Verbeke, Rheol. Acta, 35 (1996) 117-126. 54. C.J.S. Petrie, Prog. Trends Rheol., II (1988) 9-14. 55. A.Ya. Malkin and C.J.S. Petrie, J. Rheol., 41 (1997) 1-25. 56. C.J.S. Petrie and M.M. Denn, AIChE J., 22 (1976) 209-236. 57. C.J.S. Petrie, Predominantly extensional flows, in 2nd Pacific Rim Conference on Rheology - Abstracts, Ed C. Tiu, P.H.T. Uhlerr, Y.L. Yeow and R.J. Binnington, University of Melboume, Australia, pp.225-226, 1997.
637
MECHANICS
OF ELECTRORHEOLOGICAL
MATERIALS
K.R. Rajagopal Department of Mechanical Engineering Texas A &M University College Station, Texas 77802 1. INTRODUCTION Electrical and magnetic fields can significantly change the response characteristics of many materials and electrorheology is the name given to the branch of mechanics that is concerned with the flow of materials that are primarily affected by the action of electrical fields. Usually, electrorheological materials are dielectrics or semi-conductors in a non-conducting fluid, though recently Ferroelectrics have also been used. Winslow's study (see [1 ]) of non aqueous silica suspensions under the action of electrical fields seems to have been the first systematic analysis in electrorheology, though the effect of an electrical field on the viscosity of pure liquids was studied much earlier by Konig [2], Quinke [3] and Duff [4]. The work of Winslow has been followed by a great deal of work in the field, and much of this effort has been directed in fashioning such materials with a view towards producing a better fluid in virtue of the potential applications for such materials in shock absorbers, exercise equipment, valves, actuators and the like. However, for a variety of reasons such applications have not met the perceived potential for such materials. Initial attempts at manufacturing electrorheological materials were hampered by a lack of understanding of the role of water in such suspensions. Other stumbling blocks that have prevented the development of technological devices are the limited operational temperature range, the abrasive properties of the suspension that erode the devices that they flow through, the attrition of the particles, the stability of the suspension and the enormous voltage requirements that are necessary to produce the changes that are required. Much progress has been made to overcome these limitations. Polymer based particles that mitigate the problem have been developed
638
(see Bloodworth [5]), and stabilizers have been found that increase the structural stability of the suspension. Also, great strides have been made recently in decreasing the voltage requirement. A detailed account of the material science aspects of electrorheological fluids is discussed in detail in the review article by Zukoski [6], and an assessment of the technical applications of ER materials can be found in Krieger and Collins [7]. The reader is also referred to Deinega and Vinogradov [8] for a review of electrorheological materials. Here, we shall be primarily concerned with the mathematical modelling of electrorheological materials. Electrorheological fluids can be modeled starting at a microscopic level or within the context of continuum mechanics in a homogenized sense. Here, we shall restrict ourselves to continuum models; but even within the context of continuum mechanics thee are several ways to modeling electrorheological fluids. One approach is to treat the electrorheological fluid as a homogenized single constituent (see Atkin, Shi and Bullogh [9], Rajagopal and Wineman [10], Wineman and Rajagopal [11 ]). Another is to model it as a mixture (see Atkin and Craine [12], Bowen [13], Truesdell [14]) of a particulate medium and a fluid, each being treated as a single continuum (see Yalamanchili, Rajagopal and Wineman [ 15]) allowing for interactions between the two constituents. Here, we shall restrict ourselves to modeling the electrorheological suspension as a single continuum. Much of the modeling of the flows of electrorheological fluids have been restricted to one-dimension, though there has been some work on three dimensional models. Numerous three dimensional models can collapse to the same one dimensional model and at the present moment there is not a sufficient body of experimental evidence in general three dimensional flows which can be used to validate and select any one of these models as the one that is best suited. In view of this, we shall restrict our discussion to a reasonably general class of models. Experimental evidence suggests that electrorheological fluids thicken significantly on the application of the electrical field, respond in a Bingham like fashion with the yield depending on the applied field, develop normal stress differences and stress relax (see Gamota and Filisko [ 16], Yen and Achom [17], Gamota, Wineman and Filisko [18], Jordan and Shaw [ 19], Jordan, Shaw and McLeish [20]). The response of electrorheological materials is also significantly affected by the thermal conditions (Conrad, Sprecher, Choi and Chen [21 ], Jordan and Shaw [22]). Here, we shall not consider thermal effects but restrict ourselves to an isothermal analysis.
639 The fiber like structures that are formed on the application of the electrical field suggest that such fluids ought to be modeled as anisotropic fluids. This leads to an additional level of complexity which should be introduced after a better understanding of such materials is achieved. However, this aspect of the modelling is crucial and has to be reckoned with if one is to capture the behavior of electrorheological fluids. For instance, the perceived viscoelasticity of the fluid that is characterized by a time constant for the response could be due to the response time associated with the alignment of the particles on the application of the field. The electrorheological response of liquid crystals have also been studied (Carlsson and Skarp [23], Yang and Shine [24]) where the material is anisotropic even before the application of the electric field. We shall not consider these issues here, suffice it is to say that the new framework that has been developed recently that allows for variations in the synunetry of the body with various configurations that are natural to the body can be used successfully to model such materials (see Rajagopal [25]). Such an approach has been used to model crystal plasticity (Rajagopal and Srinivasa [26]), twinning and solid to solid phase transition (Rajagopal and Srinivasa [27], [28]), multi-network theory for polymers (Rajagopal and Wineman [29]) and anisotropic viscoelastic fluids (Rajagopal and Srinivasa [30]). 2. K I N E M A T I C S AND BALANCE LAWS Let f2 denote the reference configuration of a body B. By the motion of a body we mean a one-to-one mapping ~ that assigns to each point Xc f] a point x belonging to a three dimensional Euclidean space, at each instant of time t, i.e., x=x(X,O.
(2.1)
The image of f] under X, denoted as ~"~t, is the configuration occupied by the body at time t. We shall assume that X is sufficiently smooth to render all the following operations meaningful. The velocity v and acceleration a are defined through 1Various properties ~ associated with a material point at different instants of time can be defined through d~=~(X,t)=~(x,O. We denote
r ax
d~ _ 0~ a~ _ O~,v~= 0__~and grad dt a t ' a t at ox
Also, div ~ denotes tr [ grad ~] and Div ~ denotes tr [V ~].
640
O~ dx ,
(2.2)
a - 0 2 X - ~d 2.x
(2.3)
v-
Ot dt
and
Ot 2
dt 2
The deformation gradient F and the velocity gradient L are given respectively through
F: OX:Vx,
OX
(2.4)
and L - d_~v: grad v. dx
(2.5)
The symmetric and skew part of L are denoted by D and W, respectively, i.e.,
D:~(L +Lr),W:I(L-L r).
(2.6)
We shall keep our kinematical definitions and the documentation of the basic equations to a minimum while ensuring that the treatment be self-contained. A complete and proper flame-work for the study of electrorheological fluids would require the laws of electromagnetism in addition to the usual laws of thermomechanics. There are many ways of expressing the equations of electromagnetics and we shall use the Minkowskian formulation. A detailed discussion of the basic laws of field dependant materials within the context of continuum mechanics can be found in Truesdell and Toupin [31] and Eringen and Maugin [32]. We shall use the dipole-current loop model (see Pao [33]).
641
The conservation of mass is given by 2
0p +div (pv)=0,
(2.7)
Ot
where p is the density. The balance of linear momentum takes the form
(2.s)
divTr+pf+f~=P dt"
where T is the Cauchy stress 3, f is the external mechanical body force, and fo the electromagnetic force density given by
f e : : q E + l j x B +10P xB +l div[(PxB)~v] +[grad B] r M+[grad E]P; (2.9) c c Ot C qo is the electric charge density, E is the electric field, J the conduction current, B the magnetic flux, P the electric polarization, and M is defined through
(2.10)
M=M +lvxP c
where M is the magnetic polarization. The balance of angular momentum takes the form
div(x• T) +xx pf+Q = x x p
dv
dt'
(2.11)
where ~o is the electromagnetic angular momentum density given through
2
A documentation of the governing equations can be found in Rajagopal and Ruzicka [34]
3While the phenomenological continuum models assume that the stress is symmetric, some models based on particle dynamics lead to expressions for the stress that are not symmetric. We shall assume here that the stress is symmetric. A discussion of the asymmetry of stresses of electrorheological materials can be found in Rosensweig [35]
642
~e" =xxf e +Px~+MxB,
(2.12)
where g" is the electromotive force intensity given through
g'=E+lvxB.
(2.13)
c
The balance of energy takes the form
1 2) pd(e+-~lvl
+div q -div(Tv) +Of-V+pr+We,
(2.14)
where e is the specific internal energy, q the heat flux vector, r the radiant heating and w~ is the energy production density given by
where J is given through J - J - q~v.
(2.16)
Even in classical continuum thermomechanics, the specific formulation of the second law of thermodynamics is an object of much contention. Thus, the exact form of the second law in electromagnetics is far from settled. Here, we record the second law in the form of the Clausius-Duhem inequality, though alternate interpretations of the second law are possible:
p drl r dt +div (O)+p -~>_0,
(2.17)
where 11 is the entropy and 0 the absolute temperature. The Clausius-Duhem inequality places restrictions on the allowable forms of the constitutive expressions.
The above balance laws are the usual laws of thermomechanics modified to account
643
for the effects of the electrical and magnetic fields. We have to augment the above equations with Maxwell's equations in the Minkowskian form. The conservation of electric charge is
J=0.
Oqe +div
(2.18)
0t Gauss' law is given by (2.19)
div I)e=qe , where D~ is the electric displacement field given through
(2.20)
De=P+E. Faraday's law is given by
1 0B curl E +-=0.
(2.21)
c Ot
The conservation of magnetic flux is given by div B=0,
(2.22)
and Ampere's law takes the form
curl H-
1 ODe c
Ot
1
+•
(2.23)
c
where H is defined through
H: =B-M.
(2.24)
644 The system of equations (2.18) - (2.24) can be manipulated to obtain other equations that can prove more amenable to use. We shall not get into this here but refer the reader to [36] for a discussion of the same. While the equations of thermomechanics are invariant under Galilean transformations, Maxwell's equations are invariant under Lorentz transformations, and Galilean transformations are not uniform approximations of Lorentz transformation (see [37]). In general, to solve flow problems involving electrorheological fluids, it would be daunting to use the full system of equations (2.7) - (2.24) except in the simplest of problems. Thus, we need to simplify the system of equations. A gross simplification is to ignore Maxwell's equations, as well as effects due to the fields in the thermomechanical equations as field variables but treat the electric field as a parameter. Much, though not all, of the modeling in electrorheology is in this spirit. We shall discuss such an approach in some detail later. A less drastic simplification is that based on the fact that the fluid is nonconducting and that we are dealing with a dielectric, i.e., J=O,
(2.25)
and M=O.
(2.26)
It follows that (see Rajagopal and Ruzicka [34])
d9 +9 div v=0, dt
(2.27)
dv div T+pf+f = p ~ dt
(2.28)
~
de -kA0 =T.L + dP "g'+(P-g0div v,
dt
(2.29)
645
T+O 2
1 "L+
>_0,
0
(2.30)
div(E +P) =qe,
(2.31)
1 aB
curl E + - - ~ = 0
(2.32)
c at
divB=0,
(2.33)
curl B+lcurl(v• c
dqe ~+qe dt
div v=0.
1a
1
= c - ~ (E +P) ---qeV'c
(2.34)
(2.35)
In the above equations we have assumed that the heat flux is given through Fourier's law and ~ is the specific Helmholtz potential. The above equations can be further simplified on the basis of dimensional arguments for problems of interest in electrorheology. A detailed treatment of the same can be found in [36]. In general we will have to solve the coupled partial differential equations (2.27)(2.35) which is tantamount to fourteen partial differential equations and one constraint inequality for the appropriate variables, a most arduous task even under idealized conditions. Since in most problems of practical relevance, the gaps are exceedingly small, it may be reasonable to assume the electric field to be a constant (of course, this assumption could be totally inappropriate as the electric field could vary tremendously within the short gap). Such an assumption is often made in electrorheology and the electric field then plays the role of a parameter in the problem. We shall further simplify the analysis by restricting ourselves to isothermal processes and essentially ignoring the thermal variables. Even such a simplified situation serves to highlight certain interesting features concerning the flows of electrorheological fluids.
646
It is possible that there are some electrorheological fluids that undergo only isochoric motions when the electric field E is held a constant, however the density changing with the electric field. This situation is similar to fluids which can undergo isochoric flows in isothermal processes while their densities can change with temperature. Such an assumption is the starting point for the celebrated Oberbeck-Boussinesq approximations in fluid mechanics. The fact that the density can change with the electric field, but is a constant in all processes in which the electric field is a constant can be expressed through det F =f(E). A detail discussion of the consequence of the above constraint can be found in Rajagopal and Ruzicka [36]. 3. S I M P L I F I E D M O D E L S B A S E D O N T H E E L E C T R I C A PARAMETER: DIFFERENTIAL TYPE MODELS
F I E L D AS
We shall restrict our analysis to an incompressible electrorheological fluid. Further, we shall assume that the Cauchy stress is given through
T - -pl § f(D,e),
(3.1)
where -pl is the indeterminate part of the stress due to the constraint of incompressibility and D is the stretching tensor defined through (2.6). We shall assume that the material is isotropic. The response of materials of the form (3.1) has been studied in some detail by Rajagopal and Wineman [29]. It follows from flame-indifference and isotropy that f must satisfy f(QDQr QE)=Qf(D,E)Q r
VQr
(3.2)
where O is the orthogonal group. It follows from standard representation theorems that T can be expressed as (see Spencer [38]):
647 T= -pl +aIE~)E+a2D+a3D2+a4(DE~)E+E~)DE) +as(D2E|
(3.3)
+E|
where a~, I=l, ...,5 are scalar functions that depend on the following invariants Ii:tr (E@E),/2 =trD2,/3 :tr(DE| (3.4) I4=tr(D 2E| On the other hand if we require that (3.2) holds for proper orthogonal transformations O +, i.e., invariance under rigid body motions, then T has the representation T : -pl +&IE~E +tx2D+~3D2 +t~4(DE| +6cs(D2E|
+E|
+E|
+&6(MD +DM r) +&7(MD2+D2M 7,)
+&8(DMD2+D2M rD) +&9(E(~)MA+MA|
+&lo(E|
(3.5)
+MB|
where A, B and M are defined through A=DE,
B--DE
and
M=cE,
(3.6)
where e is the alternator tensor and the &; , i=l,... 10 are functions of the invariants I 1=tr(E |
=trD 2,I3: tr(D E |
trD3, (3.7)
I s =tr(D2E@E),/6 =tr[D2M TA|
rD]I 1.
Invariance under O or O § are both assumptions, whichever choice is made. We note that even in the simpler case corresponding to invariance under the full orthogonal group, where temperature effects are ignored, and the electric field is treated as a parameter, there are five arbitrary material functions that appear in the representation, and in general it would be difficult to devise a reasonable experimental protocol to
648 determine these material functions. In order to illustrate some interesting interactions between the electric field and the deformation, we shall simplify the model further. However, we shall see even in this simplified model, the presence of normal stress differences can be induced in simple shear flows due to the applied electric field. Also, the fluid can thicken, i.e., the viscosity can increase considerably with the electric field as is to be expected if the model is to describe the response of real electrorheological fluids. 3.2 Simple Shear Flow
Here, we shall essentially outline the analysis of Wineman and Rajagopal [29] to illustrate that the model (3.3) can describe normal stress differences, thickening and other behavior characteristic of electrorheological fluids. Let us consider the simple shear flow of an electrorheological fluid modeled by (3.3) with an electric field applied transverse to the direction of flow, i.e., v =u(y)i,
E=E2j +E3k
(3.8)
A straightforward calculation yields (see Rajagopal and Wineman [29])
T11_ _p+_~
~2
T22--p+ot2Ef+-~IX4 ~2 +_~ y2E2,2 T33= -p +o~2E2, +~
T12:(-~ T
(3.9) 2)Y,
~5
TI3 =--~-YE2E3, T23=o~2E2E3+~.~6]t2E2E3. where
649 Y:u'0')
(3.10)
First, suppose that E3=0. We notice that the shear stress T12 can be expressed as T12: [itt(u
2)]y,
(3.1 1)
where the generalized viscosity ~t is dependent on both the shear rate and the electric field. Thus, for appropriate forms of ~t, the model can describe the increase in viscosity that is observed in electrorheological fluids due to the application of an electric field. Depending on the nature of the fluid, it can shear thin or shear thicken and this can either enhance or ameliorate the thickening due to the electric field. It is also worth noting that in general the normal stresses Tll , T22 and T33 would be different and thus normal stress differences that are characteristics of non-Newtonian fluids are also induced by the electric field. We note that
r, 1
-TY
2,
tX4- 2_~2E32, T11-T33=TY
T22-T33=o 2(E2-E2)
(3.12)
y2 + Ix6
and thus in general the normal stress differences are distinct. We also notice that there is a contribution to the normal stress differences due to the sheafing as well as the electrical field. Moreover, we recognize that there is a coupling effect between the mechanical and electrical fields that arises in the normal stress differences Tll-T22 and T22- T33. Next, we shall illustrate the effect of the electrical field in a simple flow. 3.3 Flow Between Parallel Plates
Consider an electrorheological fluid modeled by (3) flowing between two infinite
650
parallel plates along the x-axis due to an applied pressure gradient with the electric field applied along the y-axis. It follows from the balance of linear momenttLm, in the absence of body force fields,
To,=-Cy,
(3.13)
where C . . . .Op-constant > 0 . For the problem under consideration substituting
Ox
(3.13) into (3.11 ) leads to
[g(u',E2)lu'=Cy,
(3.14)
and the above equation is solved subject to the boundary conditions
u(-h)=0,
u(h)=0.
(3.15)
It has been observed that in the presence of an electric field the fluid flows only after a critical value is reached for the shear stress, i.e., it exhibits "Bingham type" behavior. Of course, it is possible that the fluid flows ever so slowly even below the "yield stress" in that its flow is imperceptible within the time frame of the observation. To simplify the problem, we shall assume a Bingham type behavior. Thus, we shall consider a representation for Txy of the form
Oo(E)+v(E)v, T=
0,
y=0
with the assumption that Oo(0)=0.
(3.16)
651
We could assume more complicated responses wherein la depends on both y and E, i.e. ~t= la(y,E). For a fixed value of E, the maximum value of the magnitude of the shear stress occurs at y=• and its value is Ch. If Ch < Oo(E), then the maximum value of the stress is less than the yield stress and there will be no flow. Let C be such that Ch > Oo(E). Then, there is some y* such that Cy *= Oo(E). It then follows that - Oo(E)=~t(E)u '- -Ay,
y* <_y
u '-0,
(3.17)
y * <_y<_y
Oo(E)+~(E)u '- -Ay
-h<_y-
It immediately follows from (3.17) and the no-slip boundary condition (3.15) that l
C
2
u(Y)= t(E)[~(h -Y
2)
-Oo(E)(h-y)] ,
uO') =constant 1
c
u(Y) =g ~ [ ~ ( h
y*_
2-y)z-oo(E)(h +y)],
(3.18)
-h_
We notice that in the absence of the electric field, Oo(E)=0 and thus y*(E)=0, and we obtain the classical Poiseuille solution u0,)_ _~(h 2_y 2),-h <_y<_h.
(3.19)
Other boundary value problems within the context of this theory can be found in [29]. It is also shown there that a Mooney-Rabinowitch type of relationship can be established for such models. Models of the class discussed above do not exhibit viscoelasticity, and it has been observed that certain electrorheological fluids exhibit viscoelastic response (Xu and Liang [39], Thurston and Gaertner [40], McLeish, Gordon and Shaw [41] Vinogradov et.al [42]). In the next section we discuss models that can describe the viscoelastic behavior of electrorheological fluids.
652
4. Simplified Models Based On The Electric Field As A Parameter: Integral Models We shall now discuss a class of models that have a fluid like response in a certain regime and a solid like response otherwise. We shall yet treat the electric field as a parameter and we shall use a transition function A(T(t), E(t)) which delineates the regimes of response" (i) A(T(t), E(t))=0 determines the transition (boundary) between solid-like and fluid-like response; (ii) A(T(t), E(t)) < 0 the material behaves like a solid; (iii) A(T(t), E(t)) >0 the material behaves like a fluid. We shall not get into a detailed discussion of such models here, the interested reader can find a detailed treatment in [11 ]. In the solid-like regime, the stress has the form (4.1)
to
T~olid(t)=.~'-[Fto(s),E(t)] s=t o
where F (s) denotes the deformation gradient at time s with respect to the configuration at time to when the transition function is met and the material starts to exhibit solid-like behavior. Let us suppose that the material when it is solid-like is isotropic. It follows from standard arguments the T (t) has the form t
Tsolid(t)- ~ -
[Bto(t),Ct(s);E(t)] '
(4.2)
s-t o
where
Bto(t):Fto(t) Ftro(t), and Ct(s):Frt(s)Ft(s).
Next suppose in the time interval [tl, t2] the material exhibits fluid-like behavior.
653
Then, t
Tfluid(t) =~'-[~-t(s),tl;E(t)] '
(4.3)
tI
and from standard arguments it follows that t
Tfluid(t) =,~ [Ct(s),t, ;E(t)].
(4.4)
tI
It is possible to define response functionals which can have a more general structure (see Wineman and Rajagopal [11 ]): co
T(t)= G [ F ( t - s ) , E ( t - s ) ] ,
(4.5)
s---O
which in the absence of the history of the electric field reduces to the definition of a simple material (see Noll [43], Truesdell and Noll [44]). The above models are too general to be of use. Thus, we shall turn our attention to a specific model which can describe the viscoelastic behavior of electrorheological materials. Consider the following representations for the solid-like and fluid-like part of the response (see Wineman and Rajagopal[ 11]):
654
Tsolid- -pl +[~IE|
+[~6(132Et~)E+E|
+[~213+[~3]~2+[~4D+[~5(l~E|
+E|
[~7(DE@E +E|
t
+f Ll(t-slCt(slds 0 t
t
+f f L2(t-Sl't-s2)[Ct(Sl )1~ t(S2 ) +(~ t(s2)C_,t(Sl)]dSldS2 0 0 t
+f L3(t-s)[C t(s)E|
+E|
t(s)Elds
0 t t
+f f L4(t-s,,t-s2)[Ct(S1){~ t(s2)EI~E + E ~ t(S1)(~ t(s2)E]dsldS2 0 0 t
+f Ls(t-s)[f3(tl(~ ,(s)+(~ ,(s)B(tllds 0 t
+f L6(t-s)[B(t)C t(s)E@E +E@]3(t){~ t(s)E]ds, 0
and for the time t > tl when the response is fluid-like
(4.6)
655 t
Tnuid = -pl +y~E|
+f Ml(t-s)Ct(s)ds t1
t
f
+: :M2(t-Sl,t-s2)[~'t(Sl)C-~t(s2)+f-,,(~2)C-,t(Sl)]dSldS2 tl t1 t +f M3(t-s)[Ct(s)E| +E|
(4.7)
t1 t
t
+ff t1
t(s2)E~)E+E~)(~t(s,)Ct(s2)]Edsids2,
tI
where I~:B t (t)-I and Ct(s):Ct(s)-I . The scalars 13i,I= 1,--7, and Yi, I=1, 2 are constants. Also, Li, i--1, - - 6 and M., i-1, --4 satisfy certain continuity and monotonicity properties. The model outlined above is too general to be useful and it is unlikely that an experimental protocol can be devised where all the arbitrary material functions can be determined. The purpose of documenting the model is to merely show general, principles of continuum mechanics can be used to develop general enough models can describe the response characteristics of electrorheological fluids. Of course, the above model could be further simplified with the multiple integral terms ignored which would then lead to models similar to those used in viscoelasticity to describe the time dependant behavior of such materials. For instance, there has been considerable interest in the response of electrorheological materials when subject to oscillatory shears (see McLeish, Jordan and Shaw [42], Gamota and Filisko [16], Otsubo, Sekine and Katayama [45]). Wineman and Rajagopal [11] use the above model to study small simple shear and oscillatory shear in electrorheological fluids. In the case of the oscillatory shear problem the stress oscillates with the frequency to of the oscillatory shear, and G~ and G~ the storage and loss modulii of linear viscoelasticity depend on the electric field and the phase angle depends on both the frequency to and the electric field E. They are able to correlate and explain phenomena observed in the oscillatory response of electrorheological fluids by Gamota, Wineman and Filisko [16], and Gamota and Filisko [46], [47].
656
General three dimensional continnum models that can describe the response of electrorheological materials such as thickening, stress-relaxation, normal stress differences, yield etc., have been presented. The simple shear flow of such fluids has been analysed and even in such a simple flow it is found that the models are capable of describing the characteristics exhibited by real electrorheological materials such as thickening due to the applied electric field, normal stress differences and stress relaxation.
657
BIBLIOGRAPHY
1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11.
12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
W.M. Winslow, 1949, J. Applied Physics, 20:1137. W. Konig, 1985, Ann. Phys., 25:618. G. Quinke, 1897, Ann. Phys., 62: 1. A.W. Duff, 1896, Phys. Rev., 4: 23. R. Bloodworth, 1995, Electrorheological Fluids Based on Polyurethane Dispersions, Bayer Corporation Report. C.F. Zukoski, 1993, Material properties and the electrorheological response, Annual Reviews in Material Science, 23: 45. I. Krieger and E.A. Collins, 1992, Electrorheological fluids, A Research Needs Assessment, Washington, DC: US Dept. Energy, Office Energy Res. Program Analysis. Y.F.Deinega and G.V. Vinogradov, 1984, Rheol. Acta, 23: 636. R.J. Atkin, X. Shi and W.A. Bullogh, 1991, Journal of Rheology, 35" 1441. K.R. Rajagopal, R.C. Yalamanchili and A.S. Wineman, 1994, International Journal of Engineering Science, 32" 481. A.S. Wineman and K.R. Rajagopal, 1995, On Constitutive Equations for Electrorheological Materials, Continuum Mechanics and Thermodynamics, 7: 1. R.J. Atkin and R.E. Crame, 1976, Q.J. Mech. Appl. Mathematics, 17: 209. R.M. Bowen, 1976, Theory of Mixtures in Continuum Physics Vol. 3., ed. A.C. Eringen, Academic Press. C. Truesdell, 1984, Rational Thermodynamics, Springer-Verlag. K.R. Rajagopal, R.C. Yalamanchili and A.S. Wineman, 1994, International Journal of Engineering Science, 32" 481. D.R. Gamota, A.S. Wineman and F.E. Filisko, 1991, Journal of Rheology, 35- 399. W.S. Yen and P.J. Achorn, 1991, Journal of Rheology, 35: 1375. T.C. Halsey, J.E. Martin and D. Adolf, 1992, Phys. Rev. Letters, 68" 1519. T.C. Jordan and M.T. Shaw, 1991, Electrorheology, MRS Bulletin, 38. T.C.Jordan, M.T. Shaw, and T.C.B. McLeish, 1992, Journal of Rheology, 36: 441. H. Conrad, A.F. Srecher, Y. Choi and Y. Chen, 1991, Journal of Rheology, 35: 1393. T.C. Jordan, M.T. Shaw, and T.C.B. McLeish, 1992, Journal of Rheology, 36: 441. T.Carlsson, K. Skarp, 1981,Mol. Cryst. Liq. Cryst., 78: 157. I. Yang and D. Shine, 1992,Journal of Rheology, 36: 1079.
658
25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.
K.R. Rajagopal, 1995, Constitutive Relations and Material Modeling, Univ. Of Pittsburgh Report. K. R. Rajagopal and A. Srinivasa, Mechanics of the inelastic behavior of Materials, Parts I and II, to appear in International Journal of Plasticity. K.R. Rajagopal and A. Srinivasa, 1995, International Journal of Plasticity, Vol. II: 653. K.R. Rajagopal and A. Srinivasa, 1997, International Journal of Plasticity, Vol. 13: 1. K.R. Rajagopal and A. S. Wineman, 1990, Archives of Mechanics, Vol. 42: 53. K.R. Rajagopal and A. Srinivasa, Viscoelastic Anisotropic Fluids, in preparation. C. Truesdell and R. Toupin, 1960, The Classical Field Theories of Mechanics, Handbuch der Physik, Vol. 3, Springer-Verlag, Berlin. A.C. Eringen and G. Maugin, 1989, Electrodynamics of Continua, Vols. I and II, Springer, New York. Y.H. Pao, 1978, Electromagnetic Forces in Deformable Continua, Mechanics today (New York) (S. Nemat-Nasser, ed.), Pergamon Press, 209. K.R. Rajagopal and M. Ruzicka, 1996, Mechanics Research Communications, 23: 401-407. E. Rosensweig, 1992, Magnetic Fluids, Cambridge University Press. K.R. Rajagopal and M. Ruzicka, Mathematical Modeling of Electrorheological Materials, submitted for publication. J.M. Levy-Leblond, 1977, Rivista del Nuovo Cimento, 7:187-214. A.J.M. Spencer, 1995, Theory of Invariants, Continuum Physics (New York) (A.C. Eringen, ed.), Vol. 3, Academic Press. Y.Z. Xu and R.F. Liang, 1991, Journal ofRheology, Vol. 35:1355. G. B. Thurston and E.B. Gaetner, 1991, Journal of Rheology, Vol. 3: 1327. T.C.B. McLeish, T.C. Jordan and M.T. Shaw, 1991, Journal of Rheology, 35: 427. G. V. Vinogradov, P. Shulman, Yu. G. Yanovskii, U.V. Barancheeka, E.V. Korobko and I.V. Bukovich, 1986, Inzh.-Fiz, Zh., 50: 605. W. Noll, 1972, Arch. Rational Mechanics and Analysis, Vol. 48: 1. C. Tmesdell and W. Noll, 1992, The Non-Linear Field Theories of Mechanics, Springer-Verlag, Berlin. Y. Otsubo, M. Sekine and S. Katayama, 1992, Journal of Rheology, 36: 479. D.R. Gamota and F.E. Filisko, 1991,Journal of Rheology, 35: 399. D.R. Gamota and F.E. Filisko, 1991, Journal of Rheology, 35: 399.
659
CONSTITUTIVE EQUATIONS FOR ELECTRORHEOLOGICAL FLUIDS BASED ON MOLECULAR DYNAMICS R. Tao
Department of Physics, Southern Illinois University at Carbondale, Carbondale, IL62901, USA 1. INTRODUCTION Electrorheological (ER) fluids are a class of materials whose theological characteristics are controllable through the application of an electric field. A typical ER fluid consists of colloidal dispersions of dielectric particles in a liquid of low dielectric constant. When an electric field is applied, the effective viscosity of the ER fluid increases dramatically. If the field exceeds a critical value, the ER fluid turns into a solid whose shear stress continues to increase as the field is further strengthened [1-5]. The above novel properties are the result of a structure change in ER fluids under an electric field. It is known that upon application of an electric field the dielectric particles in ER fluids form chains spanning between the electrodes. The chain will aggregate to form thick columns. The body-centered tetragonal (bct) lattice was predicted theoretically as the ground state structure for an ER fluid and has been verified experimentally [4,7]. The microstructure of ER fluids is a fundamental issue. The viscosity increase and solidification of ER fluids are all related to this microstructure. The mechanical and physical properties of ER fluids also strongly depend on the induced structure [8]. Recently, the ER effect has also been used to produce new composite materials [9,10]. Therefore, information about the microstructure of ER fluids under various conditions is very important for the growing number of applications. In this paper, we will start from the first physics principle to derive constitute equations for ER fluids based on molecular dynamics. Then we will apply computer simulations to investigate the structure of ER fluids. Our ER system is confined between two electrodes located at z = 0 and z = L respectively, upon which a voltage may be applied. The system consists of spherical dielectric particles of diameter a and dielectric constant ep suspended in a nonconducting liquid. The liquid has a dielectric constant of e / a n d viscosity r/. In an electric field, each particle has an in-
660
duced dipole moment of p - o~ef(a/2)3Eto~ where a - ( e p - e l ) / ( e p + 2el) and Eto~ is the local field. The formation of stru&ure is driven by dipolar interactions, viscous drag forces, and Brownian motions. Molecular simulations on this model have shown that the ER fluid can readily form the bct lattice under a strong electric field [11]. Several other computer simulations on a similar model have also been reported [12-16]. In our extensive simulations, we will examine the detail of the induced ER structure under various conditions. Our results indicate that ER fluids under an electric field may develop into five different structures under different conditions. In a weak electric field, ER fluids can move from a liquid state to a nematic liquid crystal state which only has ordering in the field direction, but no ordering in other directions [17]. This ordering indicates chain formation along the field direction. In both liquid and liquid crystal states, the columnar particle density remains uniform. When the electric field is strong and the thermal fluctuation is weak or moderate, ER fluids develop into a body-centered tetragonal (bct) lattice. In a very strong field without or with very weak thermal fluctuations, ER fluids may develop into a polycrystalline structure consisting of many small bct lattice grains. If both the electric field and thermal fluctuation are strong, ER fluids develop into a glass-like structure in which the particles are aggregated together to form thick columns, but the structure has no appreciable ordering. In all of the last three structures, the columnar particle density peaks in a small region, indicating a thick-column structure. Our simulation also provides information about the solidification and the chain formation time in ER fluids. The chain formation time is much shorter than the solidification time. Both of them depend mainly on the ratio of the Reynolds number to the Mason number, especially in the over-damped case. Here the Reynolds number is the ratio of the inertial force to the viscous force while the Mason number is the ratio of the viscous force to the dipolar force. However, when the viscosity is not too strong, the ratio of the Brownian force to the dipolar force also plays a role in these two time scales. 2. CONSTITUTIVE EQUATIONS We use the Langevin Equation to describe the motion of particle i,
d2ri _ dri m--~- - F i - gTrarl-~+Ri(t ).
(1)
Here Fi includes all electric forces on particle i, 37rorlvi is the Stokes' drag force, and Ri(t) is a Brownian force. Fi is given by the following
661
expression . _.-~.[fiJ
Fi --
+fidrep ]+fi self +fi wall
jr
(2)
where fij is the force acting on particle i by particle j and all of j's images, firfp is a short-range repulsive force to prevent particles i and j from overlapping, f~lf is the force on particle i due to all its own images, and f~an is a short-range repulsive force to prevent particle i from penetrating the two electrodes. The dipolar force exerted by particle j on particle i is given by 3/) 2
(3)
(efr~j) [e~(1-3 cos 20ij)-eo sin 20ij]
where rij - r i - rj and Oij is the angle between the z direction and the joint line of the two dipoles. When a dipole is placed inside a capacitor at r i - (xi, yi, zi), an infinite number of images are produced at (xi, yi,-zi) and (xi, yi, 2Lk 4-zi)for k - -4-1, +2,--.. The force that the j t h particle and its images exert on the ith particle is given by
fij,,
_
p2 oo 4s 3r 3 ST~fli j STrZ~' STrZj efL----4 y~ (xi -- Xj) f(1 ( L ) cos( L .....) cos( L ) Pij ,
s=l
f ij,y
_
p2
~
ef L4 ~
s=l
p2
fij~z
4
8371_3
(Yi - Yj)1(1(
sTrPij,)
L "cos(
Pij
co
ef L~ y ~ 4s3 7r3I(o( STcPij ~-"
sTrzi
8~zi~
"L "
cos(
8~zj,)
L "
(4)
87FZj
" L )Sin( L,, )cos( n )
s--1
where Pii - X/(xi - Xd)2 + (Yi yj)2 and Ko and Kt are modified Bessel functions. The force on a particle due to its own images is in the z-direction and given by
f self3p2[1 i,z
8E f
~176 -- 7Z -}- y ~
s=l
1
1
]
(Zi -- 8 L ) 4 - (zi q- 8 L ) 4
(5) "
To introduce hard spheres and hard walls into the simulation, we use a
662
short-range repulsion between two particles ,~p 3p2e, e x p [ _ l O O ( r o / a _ l ) ] fij ~- e_/a 4
(6)
and the short range repulsion between a particle and the electrodes as fwall
3p2ez -- e - ~ { e x p [ - l O O ( z i / a - O . 5 ) ] - e x p [ - l O O ( ( L - z i l / a - 0 . 5 ) ) } .
(7)
The random force Ri(t) has a white-noise distribution,
(Ri.~) - O.
(Ri..(O)Ri.~(t)) - 67rkBTa~6.a6(t)
(8)
where kB is Boltzmann's constant and T is the temperature. We introduce a subinterval T which is shorter than the time steps used in the integration of Eq.(1) but much longer than the molecular collision time 1 [.t+rp. [18]. The average of Ri(t) over T, Ri,~(t, T) -- -~a, - . , , ( t ' ) d t ' has a Gaussian distribution
W(R,..(t..))
-
1
(9)
where ~t - V / 6 r k B T a ~ / ' r . For a time step 6t > T, we can divide it into many subintervals of duration T, in which all quantities except Ri(t) can be treated as constants. Then, for any smooth function r X , = ft ~+6~r has a probability distribution
W(X,)
- (Trq)-'/2 e x p ( - X 2 / q )
where q 12~vkBTa~ (~)d~ and is independent of T. Although the value of T is not very uniquely defined, Eq.(10) implies that our results do not depend on a specific choice of T. The intrinsic time scale in Eq.(1) is to - m / ( 3 r a ~ ) . We re-scale the variables, t - tot*, F i - FoF* where F o - 3p2/(ela4), R i - ~R*(t), and ri - ar* in Eq.(1). The scaling produces a dimensionless constitutive equation ri''* + ri ' * - A(F* + BR*)
(11)
where A - Foto/(3rrla ) and B - gt/Fo. It is interesting to note that A is the ratio of the Reynolds number to the Mason number. The
663
Reynolds number R is the ratio of the inertial force m v 2 / a t o the viscous force 37raCy where v is the speed of dielectric particles. Hence R = mv/(37rrla 2) = vto/a. The Mason number M n is the ratio of the viscous force to the dipolar force, M n = 3raTlv/Fo. Then, it is clear that A - R / M n . In our problem, dielectric particles have negligible speed before the electric field is applied or after the solid structure is formed. During the process, the particles have steady or typical speed, either. As a result, there is no typical Reynolds number or Mason number in our problem. However, their ratio, A, is independent of the particle speed and hence a good parameter for our simulation. We also note that parameter B is the ratio of the Brownian force to the dipolar force. In addition, 1 / ( A B 2) - (1.5T/to))~ where A = ( p 2 / E l a 3 ) / k u T , a crucial parameter [19] in study of ER system in equilibrium process [17]. However, we must remember that our electric-field induced solidification is a non-equilibrium process, during which the temperature is not uniform throughout the system. 3. MOLECULAR DYNAMICS SIMULATIONS
In our simulation we use 122 particles confined in a space of dimensions L~ - Ly - 5a and L~ - 14a. This corresponds to a volume fraction of r - 0.183. There are periodic boundary conditions in the x and y directions. We probe the structure at each step using the following three order parameters, N
1 pj - [ ~ ~
(12)
exp(ibj- ri)[
i=1
where the bj are the reciprocal lattice vectors of the bct lattice bl-
(27cIo)(2etzlv~-ez), b 2 - (2~lo')(2etylV/-6-ez), b3
- 47re~/a. (13)
Of the three unit vectors, ez is along the field direction and e~,ey~ are along the intrinsic axes of the bct lattice. In the calculation of the order parameters, we must rotate the coordinate system around the z axis to find the intrinsic axes of the structure which maximizes pip2. The order parameter P3 characterizes the structure along the z direction while pl and p2 characterize structure in the x-y plane. In the simulation, we assume that an electric field is applied at t - 0 instantaneously. Then we apply an adaptive step-size control RungeKutta method to integrate the equation of motion. It is clear from
664 Eq.(11) that the final structure depends on the two parameters A and B. We examine the dynamic process by monitoring the particles' positions, velocities, and the structure's order parameters. In most cases, after application of the field, the particles quickly move to form chains, then the chains aggregate to form a thick structure. Afterwards, the particles usually fluctuate slightly around their positions in the structure and the order parameters of the system change very little. Hence we are able to determine the solidification time and analyze the final structure. The other interesting quantity is the chain formation time which is shorter than the solidification time. As seen from equations (6) and (7), collisions may interrupt the structure formation if some particle's position change in the integration is too big. Therefore, we specify a value 5r~. If the largest position change among all the particles' motion during one time step 5t is greater than 5r~, the next time step is reduced. Otherwise, 5t will be increased. Our method speeds up the integration and effectively preVents possible problems from the collisions because we control the position change by selecting a proper 5r~. Since the time step changes at every step with this method, we must employ Eq.(10) to handle the Brownian force. There are some special situations which need to be discussed. In the over-damped case, the viscous force becomes so strong that we have i~* - -i~*+A(F*+BR*) ~ 0.
(14)
The over-damping condition requires IA(F*+BR*)I << 1.
(15)
The constitutive equation (11) can then be simplified to i~ - A(F* + BR*).
(16)
Now, setting t* - (/A, we have
d r * / d ( - F*+BR*
(17)
which is independent of A. Therefore, in the over-damped case the final structure is independent of A, but the solidification time is inversely proportional to A. From Eq.(15), A << 1 / ( F ~ , + B) is required to see the over-damped case. The maximum force F ~ , comes when there are collisions between the particles or collisions between the particles and the
665
electrodes. In our simulation, we have found that when A _< 10 -a, the final structure has its three order parameters independent of A and the solidification time is proportional to I / A , indicating that the system is in the over-damped situation. The effect of the random force B R* in Eq.(11) needs special attention. Since R,*. is a Gaussian deviate, we should compare the magnitude of F/* with B. When the particles are randomly distributed, a typical value of F/* in Eq.(11) is ,,~ n 4/a where n is the particle density. In our simulation n - 0.3486 which gives an estimation of 0.2 for F/* at a random distribution. When two particles get very close, their dipolar interaction force has a typical value ,,~ 1. However, when the particles aggregate to form a final structure, the joint forces on every particle are vanishingly small. Therefore, it is easy to understand from E q . ( l l ) that if B << 0.2, B R ~ has little effect on the early dynamic process, but may have some effect on the final structure since the joint forces on each particle are small. If B >> 1, B R ~ becomes the leading force in Eq.(11). 1.0
(a)
05 f , , , , , . , , , , . ~ ~
y- ry
structure
0.8 04
0.6
BCT
13:1
Lattice
07
0.4
2" x~.~~ass-like
0.2
BCT
~
Lattice
,.<
structure ~
----
: 7 - / ' 7 7 " / 7 - . / 7 ./-7- 7 " 2 7 7" 7
0.0
I . . . . . . . .
A o.o
I'
........
0.2
I .........
0.4
I .........
0.6
O~ Glaas-like
O~
structure
Nematic liquid crystal Liquid
O(
,
!
,
l.
,
!
,
!
,
I "''''''~"
0.8
1.0
A 0.0
0.2
0.4
0.6
03
1.0
Fig.1. (a) Different structures under various conditions. In the shaded area, ER fluids may develop into a poly-crystalline structure. The boundary between the liquid and the nematic liquid crystal has A B 2 ,,., 0.25 (~ ,,o 6.7). The boundary between the bet lattice and liquid crystal state has A B 2 ,,~ 10 .2 (A ,,o 167). (b) The same results presented with axes A and ~. The poly-crystalline structure and glass-like structure are nonequilibrium products. 4. STRUCTURE OF ER FLUIDS
If the ER system can always evolve into its equilibrium state after
666
an electric field is applied, the final structure may only depend on the parameter A - p2/(e.la3ksT ). However, in the actual dynamic process, the ER system may not be able to reach the ground state after it is trapped in a local energy minimum state. The viscous force enhances such a possibility by a quick reduction of the particles' kinetic energy.
z(~)
z(~)
14.00
.~(:
14.00
!
,, ,7
0 /
it ~
g.s3
7 O0
7.00 (
467
4.67
(
2 ~3
0 00" 3.
2. ]3
..~2. '1"92 6i
2.42
x(, v
0.9t
1.42
~
Fig.2. The initial state has the particles randomly distributed in space.
O. O0 4
~.413 ,9
d
0.56
Fig.3. A good body-centered tetragonal lattice is formed with Pt, P2 ~ 0.8 and ps >__0.9.
Varying the two parameters A and B in the simulation can be realized by changing the applied electric field and temperature in the experiment. We have clearly seen five different structures of E R fluids in a wide range of A and B" liquid, nematic liquid crystal, glass-like structure, poly-crystalline structure, and bct lattice. Figure 1 depicts the final structures the system evolves over a wide range of A and B. In order to make a comparison, we derive all these final structures in our simulation from an initial randomly distributed state (Fig.2). In Fig.la, we present the result with axes A and B. In Fig.lb, we present the same result with axes A and )~. The boundary between the liquid and the nematic
667
liquid crystal and the boundary between the bct lattice structure and the nematic liquid crystal structure seem to have A = 6.7 and A = 167 respectively. Although this is expected for an equilibrium statistical physics, we should note that the final structure in our simulation may not be an equilibrium state. Therefore, Fig.1 may differ from a conventional phase diagram. 4.1 BCT Lattice and Poly-crystalline Structure As seen from Fig.l, the ordered state, a bct lattice, has been obtained in quite a wide range of A and B. In Fig. 3, we plot a typical bct lattice structure with order parameters p], p~ ~ 0.8 and p3 >_ 0.9. The projection of tile three-dimensional structure onto the x-y plane shows a centered square lattice (Fig.4), a typical characterization of the bct lattice [4]; the marked square also has its side ~ lv/~.5.ha; and the characterization of these chains is also correct for the bct lattice [4].
"1 .--/
~ 3
.'7~
/
"
/.
~ 2 9
....
0
l
I ' ' ' ' 1 ' - ' ' " 1
1
2
....
3
I ....
4
I
5
x (~) Fig.4. Projection of the bct lattice on the x-y plane. The marked square has its side ,-~ x/1--5a. We also note that there is a small region inside this ordered region where the system may likely develop into a poly-crystalline structure (Fig.5). This small region has a very small B and a big A (___ 0.1). Examination of this structure reveals that the system has thick columns consisting of several bct lattice grains. However, these grains do not form a single crystal. There is some mismatch, mainly caused by their rotation around the z axis by slightly different angles. In Fig.6, we plot a part of a thick column of poly-crystalline structure which clearly shows a twist of bct lattice grains. Since these rotations do not affect P3 very much but reduces p] and p~, P3 remains ,-, 0.9 while Pl and p2 are reduced to ,-, 0.5. The poly-crystalline structure is a produ~:t of fast solidification.
668
Because of very small B in this region, the E R system may not be able to relax into a good crystal. It is thus easy to understand that in this region the final structure is somehow sensitive to the initial random state. The computer simulation confirms the conclusion, too: in this region from some r a n d o m initial state the E R system may develop into a good bct lattice, while from some other random initial state the system ends up in a poly-crystalline structure. To further understand the issue, we have paid special attention to the situation of B = 0 which can be realized at zero temperature. If A <__ 10 -a, the system is over-damped. The final structure has the three order parameters independent of A: pl and p2 around 0.62, and P3 around 0.89.
ZCa) z(,,)
14.oo
14.00 11.67 11.57
9.33 9.,]3 7.00
7.00 4.57 4.57
2.33 2.33 0.00 4.17 60
0.00
2.77
+
4.
Fig.5. ture.
f~ A poly-crystalline struc-
.9G
i I. 0 o
~ .L.o,
Fig.6. A thick column of polycrystalline structure which clearly sllows a twist of bct lattice grains. (:7.
If A is between 10 -3 and 0.1 with B - 0, the final structure is improved with Pl and p2 around 0 . 8 - 0.9 and pa around 0 . 9 0 - 0.93, a rather good bct lattice. The order parameters of the final structure in
669
this region depends on A. As A continues to increase, the three order parameters are fluctuating in the poly-crystalline region. For example, when A > _ 0.1 and B - 0, the final structure developed from the initial state in Fig.1 has three order parameters around 0.5 while P3 is close to 0.6. However, the final structure at A _> 0.1 and B - 0 is now very sensitive to the initial random state. For example, after changing the initial state at A - 0.1 and B = 0, we have ended up with a good bct lattice" pl and p2 are close to 0.84 and p3 is close to 0.92. This also implies that the ER system at B = 0 can be easily trapped in a local energy minimum. The final structures derived at a moderate B is not sensitive to the initial random state because the ER system can get out from a local minimum energy state and develop into the global energy minimum state with the help of the thermal fluctuations. z 14.00
14.0
11.67
11.6
9.J3
9.3
7.0
7.00
4.6
4.67
2.3
2.33 0.0
3.62 48 ~~
,r e
0.00 4. o.83
Fig.7. In a nematic liquid crystal state, the particles do not aggregate together. D(x, y) is quite uniform over most of the region.
Fig.8. In a nematic liquid crystal state, the system has ordering in the z direction but almost no ordering in the x-y directions.
We have compared the final structures in the poly-crystalline region
670
when we fix A and increase B. For example, at A = 1.0 and B = 0, the final structure derived from the initial state in Fig. 1 has Pl and p2 around 0.5-0.6 and pa abot/t 0.83. When we increase B, although we continue to start the system from the same initial state, the final structure improves. For example at A = 1.0 and B = 0.1, the final structure has Pl and p2 around 0.8 and P3 around 0.87. This implies that a moderate B can help the system relax into a global energy minimum state. z
(~,)
14.00
14.00 11.67 11.67
9. ,1,1
9.,13
7.00
1.00
4.67 4.67
2. ,I,1
2.33
o.o
,1.92
2.30 .x,
0.00 !
,68 ] ....
z.~l
1.15
x(~) Fig.lO. In a glass-like structure In a glass-like structure, tile ER fluids form thick columns in a small columbut t h e t h r e e order p a r a m e t e r s ar~ nar region, a main difference bealmost the same as that of tile ne tween a glass-like structure and matic liquid crystal state. liquid. 4.2 Nematic Liquid Crystal and Glass-like Structure W h e n we increase B, equivalent to raising the t e m p e r a t u r e , we come from the region of the bct lattice to a region of final s t r u c t u r e with significant P3, but small pl and p2. Typically, pl and p2 are around 0.3 while P3 >_ 0.6. This implies t h a t the system has ordering in the z direction with weak or no ordering in the x-y directions. From the definition of a particle density Fig.9.
D(x, y) is peaked
671
N
D(r) - E
5(r-rj)
(18)
j=l
where rj is the position of the j t h particle center, we define a columnar density, L
D(x, y) - / D(r)dz.
(19)
0
After analyzing the columnar density of these structures, we find that within this region, there are two slightly different structures. When A is relatively small and B is relatively strong, D(x, y) is quite uniform, as in Fig.7. This implies that in these structures, the particles do not aggregate together to form thick columns though there is some ordering in the field direction. The system remains in a liquid state, but similar to a nematic liquid crystal structure (Fig.8). When A is relatively strong and B is relatively small, the ER fluids form thick columns. As indicated by D(x, y) in Fig.9, the particles are concentrated in a small region, a main difference distinguishing this structure from a nematic liquid crystal structure. The three order parameters of this structure are not too much different from that of a nematic liquid crystal structure. Although there is some ordering in the fielddirection, there is no significant lateral ordering. Therefore, this is a glass-like structure (Fig.10). In this region, the strong electric field forces the particles to aggregate to form thick columns, but the thermal fluctuations prevent the system from forming a crystalline structure. 4.3 Liquid A further increase of B leads to a region which has the final structure in a liquid state (Fig.l 1). All three order parameters are very small for these structures. The particles are randomly and quite uniformly distributed in the space, as seen from D(x,y) in Fig.12. In this region, the random Brownian force is too strong to prevent formation of any ordered structures.
4.4 Non-equilibrium Process and Boundaries. Our simulation shows a dynamic process. The poly-crystalline structure is a product of non-equilibrium processes. The difference between the glass-like structure and liquid crystal is only in the columnar density, not in the ordering: In a glass-like structure, the particles aggregate together while they do not in a nematic liquid crystal (see Fig.7 and
672 Fig.9). Therefore, poly-crystalline and glass-like structures may not be closely related to the equilibrium state. On the other hand, both the boundaries between the liquid and nematic liquid crystal and the bct lattice and liquid crystal structure seem to be related to the equilibrium state. Although they are not exact, these two boundaries both have A B 2 roughly as a constant. Since the parameter A is proportional to 1/(AB2), these two boundaries are roughly along the lines of a constant A [17]. The boundary between the liquid and the nematic liquid crystal has A B 2 close to 0.25, corresponding to A ,.- 6.7, The boundary between the bct lattice and liquid crystal state has A B 2 close to 10 -2, corresponding to ,.- 167. z (,)
DCx,v)
14.00
14.00
11.57
11.67 [~
9.33
7.00
9.,33 I li~ ~ 7.O0 f-~ II "
4.67
4.57 ~t II
2..13 0.00
4. .f" (~j
4.20 45
0.75
0.0~
~"~
Fig.ll. In a liquid state, three order parameters are vanishingly small and the particles are randomly distributed in the space.
Fig.12. In the liquid state, D(x, y) is quite uniform over the whole region.
5. RESPONSE TIME OF ER FLUIDS
ER fluids are marked for their fast response to an electric field. A number of experiments established that a typical response time of ER fluids is of the order of milliseconds. This response time is Usually defined as the time needed for ER fluids to have a significant viscosity increase
673
immediately after an electric field is applied. In our simulation, we define the solidification time as the time interval between the application of an electric field and the establishment of a final structure [20]. It is clear that our solidification time should be longer than the response time since ER fluids deliver a significant increase of viscosity before they reach their final structure. However, these two time scales are closely related and our solidification time has clear physical meaning and is important for applications as well. The relationship between the solidification time and the parameters A and B is in Fig 13. We note that at a fixed B, the solidification time is getting longer as A gets smaller. In the over-damped case, Eq.(17) indicates that the solidification time is inversely proportional to ' A. Our simulation verifies this conclusion,
t~o,id 60/A.
(20)
-
This relationship holds up to A ,,~ 10 -2. In real time, for example, at A - 10 -3, this solidification time is of the order of a second. As the value of A increases, the viscosity reduces. When the system is not overdamped, the solidification time further decreases as A increases, but this reduction is slower than 1/A. For example, as A increases from 10 -2 to 10 -1 , the solidification time only slightly reduces. 109
lOa
o
B=I B=0.1 B=O.01
o
k: o'3
"
107
E 0
m
.+J .p..i
10e
= 10s o~.,4
104
103
.,,l
1 0-e
!
t
i |ll|ll
1 0-s
i
i
| tJl|ll
__l
1 0-4
!
Ill|l|l
1 0-3
I
!
* |nlJll
10-2
I
I
I |Ill|
1 0-f
A
Fig.13. The relationship between the solidification time and the parameters A and B.
674
At a fixed A, the solidification time increases with B. This is due to high fluctuations of the Brownian motion. However, the effect of B is significant only when B is large enough. For example, if B < 10 -2, the solidification time is almost unaffected by B. If B > 10 -2, the thermal fluctuations delay the solidification process. For example, at A = 10 -~ and B = 10 -2, the formation of a bct lattice structure takes about 1.738 • 105t0 while at A = 10 -3 and B = 10 -1 the solidification of a similar bct lattice takes 2.81 x 105t0. In our simulation, we also determine the chain formation time by examining the order parameter p3. From Fig.14, it is clear that the chain formation is much faster than the formation of a final structure. This again implies that the particles in ER fluids form chains first, then chains aggregate together to form thick columns. Typically, the chain formation time is about one third of the solidification time or shorter. In the overdamping case, the chain formation time is also proportional to 1/A. We also notice that in real time, the chain formation time is of the order of milliseconds, the same order as the response time found in engineering applications.
'~ I%
lOe
%,.
I-
\\
o
B=I
o ,
s=o.1 s=o.ol
lO7
=
1 oe "~
lOS
104
10a 1 o-e
10-S
10-4
A
10-a
10-2
10-I
Fig.14. The relationship between the chain formation time and the parameters A and B. 6. DISCUSSIONS In this section, we want to compare our simulation results based
675
on the constitutive equations with experiments. For a real E R system, such as dielectric particles in petroleum oil, e / ~ 2, % >> 1, ~ ~ 0.2 poise, a ~ 10#m, and the mass density of the particle p ,.., 3 g / c m 3. We estimate to ~ 8.33 x 10 -7 s. If we choose the subinterval 7 = 0.4t0, then as E0 varies from 0 to 4 K V / m m at T=300 K, A changes from 0 to 10 -2 and B reduces from c~ to 10 -3. When A - 10 -2 and B - 10 -2, for example, our simulation finds the chain formation time is about four milliseconds while the bct lattice and the solidification time is less than one second. In the experiment, the chain formation takes milliseconds to complete, but the formation of bct lattice is slower than that in our computer simulation
[7].
As the particle size becomes big, the inertial time to and A increase. For example, if the above ER fluid has everything the same except a ,.~ 100#m instead of 10#m, then we have A ,,~ 1 at E - 4kV/mm. Hence, from Fig.l, we notice that ER fluids with large particles are easy to develop into a poly-crystalline structure in the non-equilibrium process. This interesting result is useful in production of composite materials by the ER effect [9,10]. Our results at B = 0 are interesting enough to warrant some experimental investigation. The fact that the final structure at B = 0 is sensitive to the initial state indicates that the Brownian force plays an important role in driving the ER system from a'local energy-minimum state into a global energy-minimum state. On the other hand, if B is too strong, the thermal fluctuations prevent the system from forming a good bct lattice. Therefore, an experimental determination of this range of B will be very interesting. This goal may be achieved by examination of ER fluids at cryogenic temperatures. Our simulation also reveals that the response time defined in ER engineering applications is related to the chain formation time. We have also found a relationship between the solidification time and the viscosity, temperature, and electric field. It will be very interesting to see if this relationship holds in experiments. A CKN OWLED G E M ENTS
This research is supported by a grant from National Science Foundation DMR-9622525. REFERENCES
1. Electrorheological Fluids, edited by R. Tao and G.D. Roy (World Scientific Publishing Comp., Singapore, 1994). 2. L. C. Davis, J. Appl. Phys., 72 (1992), 1334; 73 (1993), 6so.
676
3. H. Block and J.P. Kelly, US Patent No. 4,687,589 (1987). 4. F.E. Filisko and W.E. Armstrong, US Patent No. 4,744,914 (1988). 5. R. Tao and J. M. Sun, Phys. Rev. Lett., 67 (1991), 398; Phys. Rev. A, 44 (1991), R6181. 6. J. E. Martin, J. Odinek, and T. C. Halsey, Phys. Rev. Lett., 69 (1992), 1524. 7. R. Tao, J.T. Woestman, and N.K. Jaggi, Appl. Phys. Lett., 55 (1989), 1844. 8. T. J. Chen, R. N. Zitter, and R. Tao, Phys. Rev. Lett., 68 (1992), 2555. 9. G. L. Gulley and R. Tao, Phys. Rev. E, 48 (1993), 2744. 10. X. Wu. X. Zhang, R. Tao, and R. P. Reitz, Bull. of Amer. Phys. Soc., 41 (1996), N.1, 191. 11. C. A. Randal, C. P. Bowen, T. R. Shrout, G. L. Messing, and R. E. Newnham, in ref. 1, p516. 12. R. Tao and Q. Jiang, Phys. Rev. Lett., 73 (1994), 205. 13. D. J. Klingenberg, F. van Swol, and C. F. Zukoski, J. Chem. Phys. 91 (1989), 7888; 94 (1991), 6170. 14. N. K. Jaggi, J. Stat. Phys. 64 (1991), 1093; W. Toor, J. of Colloid and Interface Science 156 (1993), 335. 15. K. C. Hass, Phys. Rev. E, 47 (1993), 3362. 16. R. T. Bonnecaze and J. F. Brady, J. of Chem. Phys.,96 (1992), 2183. 17. H. X. Guo, Z. H. Mai, and H. H. Tian, Phys. Rev. E, 53 (1996), 3823. 18. R. Tao, Phys. Rev. E, 47 (1993), 423. 19. For example, see, S. Chandrasekhar, Rev. Mod. Phys. 15 (1943), 1; R. Reif, Fundamental of Statistical and Thermal Physics (McGrawHill, New York, 1965), 560-562. 20. P. M. Adraini and A. P. Gast, Phys. Fluids, 31 (1988), 2757. 21. H. See and M. Doi, J. of Phys. Soc. of Japan, 60 (1991), N.8, 2278.
677
ELECTRO-MAGNETO-HYDRODYNAMICS AND SOLIDIFICATION
G. S. Dulikravich
Aerospace Engineering Department, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
1. INTRODUCTION Fluid flow influenced by electric and magnetic fields has classically been divided into two separate fields of study" electro-hydrodynamics (EHD) studying fluid flows containing electric charges under the influence of an electric field and no magnetic field, and magneto-hydrodynamics (MHD) studying fluid flows containing no free electric charges under the influence of a magnetic field and no electric field. Traditionally, this division was necessary to reduce the extreme complexity of the coupled system of Navier-Stokes, Maxwell's and constitutive equations describing combined electro-magnetohydrodynamic flows. Recent advances in numerical techniques and computing technology, as well as fully rigorous theoretical treatments, have made analysis of combined electro-magneto-hydrodynamic flows well within reach. A survey of electro-magnetics and the theory describing combined electro-magnetohydrodynamic (EMHD) flows is presented with an emphasis on describing the intricacies of the mathematical models and the corresponding boundary conditions for fluid flows involving linear polarization and linear magnetization. This survey concludes with a presentation of EHD and MHD flow models involving solidification. NOMENCLATURE b - electric charge mobility coefficient, kg A s2 B_B_ = magnetic flux density vector, kg A 1 s2
678
(
d = Vv_ + Vv
= a v e r a g e rate o f d e f o r m a t i o n t e n s o r , s -~
Do
= e l e c t r i c c h a r g e d i f f u s i o n c o e f f i c i e n t , m 2 sl
D=eo_E+P
= e l e c t r i c d i s p l a c e m e n t field v e c t o r , A s m -2
e : c~T + (v._v)/2
= total e n e r g y p e r u n i t m a s s , m 2 s ~
E
= e l e c t r i c f i e l d v e c t o r , k g rn s 3 A -~, or V m ~ = e l e c t r o m o t i v e i n t e n s i t y v e c t o r , k g m s 3 A -1
E=E+vxB
f g
= mechanical body force vector per unit mass, m s2 = a c c e l e r a t i o n d u e to g r a v i t y , rn s -2 = h e a t s o u r c e or s i n k p e r u n i t m a s s , m 2 s -3
h
a_=B_/ J= J,
to-M + Jd
= m a g n e t i c f i e l d i n t e n s i t y v e c t o r , A m -1 = electric current density vector, A m 2 = electric conduction current vector, A
m 2
J d = V_qo
= e l e c t r i c drift c u r r e n t v e c t o r , A m -2
M
= total m a g n e t i z a t i o n v e c t o r p e r u n i t v o l u m e , A m ~
M=M+vxP
= m a g n e t o m o t i v e i n t e n s i t y v e c t o r p e r u n i t v o l u m e , A m -1
P
= p r e s s u r e , k g m -~ s -2
P qo
= total p o l a r i z a t i o n v e c t o r p e r u n i t v o l u m e , A s m -2 = total or f r e e e l e c t r i c c h a r g e p e r u n i t v o l u m e , A s m -3
q
= h e a t flux v e c t o r , k g s -3
S
= e n t r o p y p e r u n i t m a s s , m 2 kg-' K" s -2
T
= absolute temperature, K
V
= fluid v e l o c i t y v e c t o r , m s -~
GREEK SYMBOLS = v o l u m e t r i c t h e r m a l e x p a n s i o n c o e f f i c i e n t , K -1 E
= C h o r i n ' s ( 1 9 6 7 ) artificial c o m p r e s s i b i l i t y c o e f f i c i e n t = d i e l e c t r i c c o n s t a n t ( e l e c t r i c p e r m i t t i v i t y ) , k g -~ m -3 s 4 A 2
ro = 8 . 8 5 4 x 10 -12
= v a c u u m e l e c t r i c p e r m i t t i v i t y , k g ~ m 3 s4 A 2
8r = S / F , o
= relative electric permittivity
K
= thermal conductivity coefficient, kg m s3 K l = e l e c t r i c c o n d u c t i v i t y c o e f f i c i e n t , k g -1 m -3 s 3 A 2
P
= fluid d e n s i t y , k g m -3
_x_v = 2 g v d + g v E I ( V 9v ) - N e w t o n i a n v i s c o u s stress t e n s o r , k g m -1 s -2 x
EM
= e l e c t r o m a g n e t i c stress t e n s o r , k g m -1 s -2
679
v
EM
x__=x__ + x__
= stress tensor (viscous plus electromagnetic), kg m -~ s -2
~t
= magnetic permeability coefficient, kg m A 2 s -2
~to = 4~ x 10 - 7
=
ktr
relative magnetic permeability = shear coefficient of viscosity, kg m ~ s -~ = second coefficient of viscosity, kg m ~ s -~
= ~t [ k t o
~tv ~tv2 E = er _ 1 M = [t r _
1
= x__~'d
magnetic permeability of vacuum, kg m A -2 s 2
=
= electric susceptibility = magnetic susceptibility = electric potential, V = viscous dissipation function, kg
m ~ s -3
2. BACKGROUND
The scientific field of study that analyzes the ability of electro-magnetic fields to influence fluid flow-field and heat transfer has been investigated for decades. The equations that are most often used to model this phenomena consist of the system of Navier-Stokes equations for fluid motion coupled with Maxwell's equations of electro-magnetics augmented with the material constitutive relations. The field studying these flows is often called electromagneto-dynamics of fluids [ 1], electro-magneto-fluid dynamics (EMFD) [2-5], electro-magneto-hydrodynamics [6], magneto-gas-dynamics and plasma dynamics [7], or the electro-dynamics of continua [8-10]. The full system of governing equations has, until recently, been far too difficult to solve because Navier-Stokes system becomes very complex when modeling flows involving turbulence, chemical reactions, multiple phases, non-Newtonian effects, etc. When coupled with Maxwell's equations, the complexity of the combined EMHD system is raised by orders of magnitude. To reduce this complexity, the analytical modeling has traditionally been divided [11] into flows influenced only by externally applied electric fields acting upon electrically charged particles in the fluid, and flows influenced only by externally applied magnetic fields without electric charges in the fluid. The former are called ElectroHydrodynamic (EHD) flows [12] and the latter Magneto-Hydrodynamic (MHD) flows [13]. More recently, rigorous continuum mechanics treatments of EHD [14] and unified EMHD flows [9,10] have been developed. These continuum mechanics approaches are limited to non-relativistic, quasi-static or relatively low frequency phenomenon [ 15-17].
680 This chapter should provide an introductory survey of the background theory to allow implementation of numerical analysis of unified EMHD flows and of classical MHD and EHD flows with addition of liquid/solid phase change. An overview of electro-magnetic theory with concentrated effort placed on descriptions of the electric and magnetic fields and electric charges and currents will be made to provide a physical understanding of the field-material interactions causing polarization and magnetization effects. The system of equations governing the unified EMHD theory and the corresponding boundary conditions will be presented together with its fully conservative form that is ready for numerical discretization.
3. POLARIZATION AND GAUSS' LAW Charge polarization is created when electric charges of opposite signs are separated by a distance. Although many references define several sources of polarization [ 18], there are essentially two main sources of polarization: natural and induced [13]. Natural polarization arises from natural dipoles and charged particles. An example of a natural dipole is a water molecule which has a geometry such that the centers of positive charges and negative charges do not coincide. Since the molecules are allowed to move freely and orient randomly, water will not have polarization on a continuum level. Now consider the fluid water as it is frozen with an applied electric field. An induced polarization will be created by the electric field by inducing an initial charge separation in neutral particles [19], by causing greater charge separation within the molecules, and by causing molecular alignment with the applied electric field in case of natural dipoles [19]. Once locked in the ice crystal structure, the water molecules will no longer be able to change their position or orientation. Consequently, even after the electric field is removed, the ice will still have polarization on a continuum level since the polarization caused by the electric field aligning the water molecules was literally frozen into the ice. From this example it may seem that there is no reason, when dealing with fluids, to consider natural polarization. This, however, would be an erroneous assumption. Though the natural polarization may show no continuum effects without the presence of an electric field, in an electric field the total polarization, _P, combines both the induced polarization due to the electric field and the natural polarization of the molecules which are now aligned by the electric field [ 13, p.22].
681
If polarization is assumed to be a linear function of the steady or relatively low frequency electric field, then it can be defined as P__- GoXE(E__4- v x B)-- gp(E q- v x B ) = GpE
(1)
The electric displacement vector then becomes [19, p.164] [9, p.178]. __D-- 8oE_E_+ P__= 8o(1 + xE)E + CoXE~ x B -- go8r___E+ 8p V X B - ~___+ 8p V X B
(2)
where the material property, ZE, is the dielectric susceptibility. It is typically obtained experimentally [20, p.86] and could be a function of frequency. Electric charges come in two types: free and bound. Free charges arise from electrons in the outer or free atomic shells and from ions. Bound charges are those arising from the molecular geometry and displacement of atomic inner electron shells [13, p.21]. Gauss's law for a linearly polarizable medium then becomes [ 13, p.22]
V.D_=qo
(3)
or
V. (ao___+ P_)= V .(~_ + epVX B_B_)= qo
(4)
At this point it is important to note that qo multiplied with the charged particle drift velocity, v d , creates the convection or drift electric current, Jd [ 13, p.67], while polarization current, J_p, is defined as the variation of the total polarization with respect to time [ 19, p. 121 and p. 147].
4. M A G N E T I Z A T I O N AND A M P E R E - M A X W E L L ' S L A W If the material in question may be considered linear, that is, if the magnetization is a function of one material property and the strength and direction of the applied magnetic field, then the magnetization is defined as [9, p.178] [19, p.164] [20, p.92-96] [21, p.371-377] ~M
Po ~1+
~M)
682 In addition to the electric currents arising from magnetization and direct charge motion, other phenomenological currents have been observed and must be taken into account when defining the total current, J [9, p.162-163]. Introducing the effects of magnetization and polarization and rearranging constants, the Ampere-Maxwell's law of electrodynamics may be rewritten as [ 13, p.30]
VxB__--po VxM-k-Jd +~p -k-
(6)
Magnetization and magnetic field vectors are often combined to form the magnetic field strength vector, H__,defined as H - B _ M
(7)
~to The total current, J, is defined as the sum of the apparent magnetization current, V x M, charge drift current, Jd, and phenomenological polarization currents, J, [13, p.26] since the contribution to the magnetization current by intrinsic magnetization is zero. The Ampere-Maxwell's law for polarizable, magetizable media can therefore be written as [ 19, p.132] 0D -
VxI-I=-j
(8)
0t Detailed descriptions of these equations can be found in any number of texts [19,20,21].
5.
A MODEL OF UNIFIED ELECTRO-MAGNETO-GASDYNAMICS (EMGD)
The full system of equations governing unified EMGD flows consists of the Maxwell's equations governing electro-magnetism, the Navier-Stokes equations governing compressible fluid flow, and constitutive equations describing material behavior. Assuming a single-phase fluid and only one type of charged particles in the fluid, this set has a minimum of 12 partial differential equations that contains 13 unknowns: p, qo, T, p, and the three vector components of _v, E__, and B, respectively. The thirteenth equation is the equation of state for the
683
fluid. The foundations of the electro-magneto-gasdynamic (EMGD) theory were formulated by Eringen and Maugin [9,10] and are based on continuum mechanics [22-25]. The rigor with which the constitutive, force, and energy terms were derived leads to a model more complete and robust than any of those found in classical literature [8,7,1,11-13,18-21 ]. Dulikravich and Jing [6,26] have shown that a compact vector form of the unified EMGD system can be written as a combination of the Maxwell's electro-magnetic subsystem and the Navier-Stokes fluid flow subsystem. The Maxwell's subsystem (consisting of seven PDE's) is composed of Ampere-Maxwell's law for polarizable and magnetizable medium 8D -
-
Ot
VxH=-J
(9)
that can also be written as OE 8t
Vx--=--l+ eo eo
-
(10)
Faraday's law 0B -
~VxE=0
(11)
8t and conservation of electric charges 8qo ~+ 8t
V._J_ = 0
(12)
that is a combination of Gauss' law V.DD_ = qo
(13)
and the Ampere-Maxwell's law. Conservation of magnetic flux V.B =0
is also a part of the Maxwell's subsystem, but is not solved for explicitly.
(14)
684
The second part of the unified EMGD is the viscous, compressible flow Navier-Stokes subsystem consisting of five PDE's and an equation of state of a perfect gas. It is composed of conservation of mass equation OP + V.(p_v)- 0 0t
(15)
and a conservation of linear momentum (including electromagnetic effects) 0p_v
+V (v_gv+pI-~) 9
_
--
V (v(P 9
__
B/)
X
--
V ((B_B_M II+(___ P)I) 9
9
=
"
-"
P
S
V
(16)
0t
Here, I is the identity (unity) tensor and S v is a vector of source terms. following dyadic identifies were used in equation (16)
The
(va). M_= v. ((B. M _ ) I ) - ( ) . B_
(17)
(vE). e = v. ((E. e)i)-(ve)._E
(18)
Conservation of energy equation is also a part of the Navier-Stokes subsystem
0(Pe----~)-t- V- (pev + (pI-__x)-v)+ V. ~ 1 - p h - 9 E . ~ Ot
-
-
-
+ Iql. D B - J c .1:: = 0 Dt
--
Dt
(19)
-
It can be replaced by the entropy generation equation [9,2,4]
9
Ds _ oh +_______~@V. _ 9DtT
-T5
J-
Dt
-+&.E
-- Dt T
(20)
The viscous stress tensor for a non-linear fluid is given as v
1; - 21avd + ).tv2I(V. _v) + oqd 2
(21)
685 In the case of a media with non-linear physical properties, the unified EMGD formulation for the electric conduction current and the heat flux can be expressed as [9, p. 161-162].
Ic -- O'1~q- G2~'--~ -t- ~3 d2 "E + G4VT 4- ~5 d" VT + 0"6d2" VT + o7E x B (22)
+ cY8(tl-(E x B ) - (d- E) x B__)+ cY9VT xB + o'lo(d_. (VW x B ) - ( d " VT) x BB)+ (Yll(B 9E)B__+ o'12(B_-VT)B__ q = K1E -I'-K2d" ~ q- K3d2 "E q- K4VT + tcsd-VT + K6d2. VT + K:vE_x B
(23)
+ ~a(a=. (g • B_)- (a_. E) • B_)+,,:9VT • B_ + Klo(d" (VT x B ) - ( d - V T ) x B)+ Kll(B'E)B + KI2(B 9VT)B
The electro-magneto-thermal stress tensor for a non-linear fluid can be expressed as [9, p. 177-178]
I;
EM
= ~2~ (~)E + ot3B (~)B + o~4VT (~)VT + or5(_E| d. E)s + 0~,6 (_E~)d 2 -~__.1+o~7(VT | d-VT)s+ ot8@T |
2- VT~
-ko~9(_d.'W-W-d)q-O,,loW.d-W -+-Otll(d=2 . W - W . d
- w 2 . . w),- 0,3( |
T/s +
2)
E | E-W/s
+ O~15(___.~) ~ W" F)S -t- O~16~" ~(~)W2 9E_~ + oqv~W-(E| VT- VT | + o~,18d-(E (~ V T - VT |
(24)
s
0~,18(E 1~)VT - VT |
where ~____W=Wij=gijkBk, while the subscript s indicates symmetrization. Expressions for total polarization, P, and magnetization, M, of non-linear media can be modeled with expressions of similar complexity [9, p.175]. In these formulas, c~i, cE and Ki are the physical properties of the media. Most of these coefficients are still unknown although their exploitation can offer potentially significant benefits in applications involving interacting electric, magnetic, thermal, and stress fields. This theory is valid for the frequencies of the electric and the magnetic fields that are less than approximately 1 kHz and for fluid speeds considerably less than the speed of light [14-17]. For higher frequencies, certain physical properties become functions of the frequencies. For higher speeds, relativistic effects will have to be taken into account.
686
6. CONSERVATIVE FORMS OF E L E C T R O - M A G N E T O HYDRODYNAMIC (EMHD) SYSTEM A necessary condition that an iterative numerical solution of the EMGD system will converge to the exact solution of the analytical EMGD system as the computational grid is infinitely refined, requires that the EMGD system must be rewritten in a fully conservative (divergence-free) form. This is especially needed if strong gradients of dependent variables are expected to exist in the solution domain. The fully conservative forms can then be used directly in the finite difference, finite volume, or finite element discretization of the EMGD system and its iterative integration process. In the following derivations, it will be assumed that the fluid is incompressible, homocompositional, that it has linear polarization and linear magnetization properties, and that the frequencies of the applied electric and magnetic fields are less than approximately 1000 Hz for this mathematical model to be realistic. These are the only assumptions to be used in this model which will be referred to as a unified electro-magneto-hydrodynamics (EMHD). A fully conservative EMHD system in a vector operator form is given as [6]
OE_E_
Vx
8B 8t Oqo
-H-
=
8o
Ot
(25)
S E -
+ VxE = 0
(26)
+v.J_
= 0
(27)
V.v
= 0
(28)
ot
Ov - + v . l v v + - (1p i 8t
-8e - + - V1 & p
P
)) - 1 v . I v ( l , • B)+(B.M)I+(_EE.p)!]=S v P
.
.
.
.
.
-=
(29)
-
.(pe_v+ ( p I - x=)-_v + _q ) = S e
For simplicity of notation we can define the following terms as [6,26]
(30)
687 1
F=/.to (r
1
M) =--~t
(31)
1 =-
(32)
ep =8oZ E --e--e o
(33)
=
A=
1
Co(1+ Z E)
(1 p +
zEI+Pe p B . B
(34)
8 = pf_ + qo_E + J_• a + (v_). P + (va). M + v . (v(_P • ~_))- v . (~_v + p i - a)(35)
_Dt=V•
IB /
(36)
- -M_M_ - J _ = V x H - J _
go
Pt = A D t +A(VxE_)xx 80 + p(1 + Z E
Dt +
(37) x
A[Rt 9 - P_P_x (V x E__)]x B
If we now assume that the fluid which is subjected to applied electric and magnetic fields is of Newtonian type and if we allow only for linear polarization (equation 1) and linear magnetization (equation 5), the constitutive relations for the electric conduction current and for the heat flux vector become [9, p.173-174]
lie ~ ~I(E~ -b X x B)-4- t34VT -t- o'7(]E -+-v x B__)x B -!-(Y9VT • B q- 0"1,(B 9(~E + Ir • B__))B-b o12(B- VT)B
(38)
688 _q- K1(_E + v x B_)+ ~:4VT + K:7(_E+ _v x B_) x B (39)
+ K:9VT x B_B+ KI, (B_-(__~+ v x B))B + KI2(B- VT)B_
Then, the EMHD source terms can be given in a compact vector form [6,26] as
S_E =
1 (j_.+ Pt )
(40)
~o s_V _ [ + _ h o E + l 9
(V x _ ) x ___-(VM). B_B- (VP__)-E+ (J + P--t)x B_B_]
S e --h +!(EwvxB__.). [(v-v)e+ J_c + ~t]--!~Z M B. ((v-V)B-V x F~) 9
(41)
(42)
9
Notice that these source terms have been formulated in such a way as not to contain explicit time derivatives [6,26].
6.1 Fully conservative Cartesian form of the EMHD system The EMHD system of equations (equations 25-30) can now be written in a general conservative form in terms of (x,y,z) orthogonal coordinate system as cqQ ~+ &
c3E
0F 3G +~+~=S 0x 0y 0z
~
(43)
Here, the solution vector of unknown quantities is given as
{
Q=
E x,
Ey, E z,
B x,
By, B z,
qo, p Vx Vy v z e
}"
(44)
where the asterisk symbol designates transpose of a vector. The vector of source terms (those terms that do not contain divergence operator) is given as
- { x S , Sy, S z, 0, 0, 0, 0, 0,Sx,v Sy,v Sz,vse} S=
(45)
689 In equation (44), Chorin's [27] artificial compressibility coefficient, 13, was used to create the unsteady term in the mass conservation since physical unsteady term does not exist in the mass conservation for incompressible fluids. By combining equations (5), (7), (1), and (31), the Cartesian components of the magnetic field intensity vector can be defined [6,26] as H x = gB x + t~pVy(E z + vxBy - VyB x ) - 8pV z (Ey + vzB x - vxB z) Hy = ~By + t3pVz (E x + vyB z - VzBy) - 13pVx (E z + vxBy - vyB x )
(46)
Hz = g B z + gpvx(Ey + vzB x - v x B z ) - gpvy(Ex + vyB z - vzBy) The flux vectors in equation (43) can then be defined as
Hz/eo
/~o
-Hy/eo 0
-Ez
-Ex
Ey
Jy ~=Vy
Jx
E= Vx
2 l
Vx
Evvx-txy+
EP
[V~y--}-~~)--'lTyy-- N~BP-VyN~yB)
VxVy--?
Ev,vzt z+vz y t (47)
(48)
690
Hy/eo
-nx/~o 0
0 Jz G=,Vz 1
VzVx-~ (~xz+ Vx~ ~)
~z +!(p-~zz-~.~-Vz~z ~) p-
(49)
eVz§ Here, we have written components of (__Px B_B_) as NxPB= PyBz _ PzBy
(50)
PB
Ny =PzBx - P x B z N PB = PxBy - PyBx
In addition, we have defined the terms NEPBM-- EX Px + EyPy + EzP z + B x ( Bx - H x ) + By ( B y _ Hy ) + B z ( Bz - H z)
go
0T
(51)
go
(52)
Nap = BxP x + ByPy + BzP z
0T
go
0T
N ~ - ax ~ + ar ~ + az--~z
(53)
691 NV~ _ v x ( - p + Xxx)+ Vy'l~xy+ Vz'ISxz
v,~ Ny - v x'l;xy + Vy ( - p + "l;yy) + Vz'l;yz
(54)
NV~ - VxXxz + Vy'l;yz + Vz ( - P + Zzz)
Components of the electric current vector, J, were defined as
(~1 ~p
~ 0"7 PB ~Z 0T +~N x +cy9(B zB ) ~XX {~p ~-~Z y
Jx=vxqo+~Px+~4
+
O'11 NBpB x + O'lzNBTBx ~;p ~1
= Vyqo + - - e y ep +
+0" 4
OT c~
+--N
Sp
Y
+(I 9
~Z B x - ~ - B z )
{3"11 - NBpBy + o'12NBTBy E;p
O"1 Jz - Vzqo + ~ P z
+ 0"4
{~p
OT
+
G7
~ZZ ~p
PB OT OT Nz + (I9 By B x)
(~-
~-
O'11.NBpB z + (I12NBTBz 8p
(55)
and heat flux vector c o m p o n e n t s were defined as
qx = ~~(~px + K4 ~~
_ KI
~
1(7 PB + K9( ~ ,Bz - o~F B ) + ~KI~ +~Nx N B p B x + K:12NBTBx 8p -~C~Z Y 8p
1(7
PB
K:7
Pa
{ly -8pPy 1" K4 ~0y' 1 " ~8pN y -
Clz - }c--LPz+ K:4
~p
0T
~Z
+~Nz
8p
0T
0T
NBpBy + K12NBTBy (56) + K9(-~-ZBx - ~ - Bz)+ 1(1! 8p
~
+ 1(9(
aT
By - - ~ - B x ) +
Kll
- N B p B z + K12NBTB z
8p
692 7. CHARACTERISTIC-BASED I N F L O W AND O U T F L O W BOUNDARY CONDITIONS For most boundary value problems of electro-magneto dynamics, jump conditions are exclusively used [9,28] to formulate solid wall boundary conditions where a discontinuity occurs. At the inflow and outflow boundaries where no surface or line discontinuities exist, an alternative approach based on conservation law for continuous surfaces or lines become necessary. Characteristic boundary condition formulation [29,30], which starts from a characteristic form of the EMHD system, will be sketched here since it leads to non-reflecting boundary condition formulation [31-36,26]. To find the characteristic boundary conditions, it is first necessary to determine analytical expressions for all eigenvalues of the characteristic system. The most common approach is to use one of the symbolic programming languages software (LISP, MACSIMA) in order to determine analytical expressions for each eigenvalue. Since these software packages cannot be used for systems that have more than five coupled partial differential equations, in the case of a complete EMHD system which has twelve coupled partial differential equations, it is impossible to find the eigenvalues using available symbolic programming software. Consequently, we will use an alternative approach in which we will divide the unified EMHD system into a Maxwell's subsystem and the Navier-Stokes subsystem [33]. Each of these two subsystems will then be analyzed separately by finding the analytical expressions for its eigenvalues by hand. 7.1 C h a r a c t e r i s t i c - b a s e d b o u n d a r y conditions for M a x w e l l ' s s u b s y s t e m
For example, characteristic treatment of the Maxwell's subsystem can be formulated by rewriting the fully conservative Maxwell's subsystem
at
+~
Ox
+
Oy
+~
SEM
az
(57)
in a non-conservative (characteristic) form as
a0EM+ AEM 0EM + BEM 0t
-
0x
0y
+ CEM --
0EM -- SEM 0z
(58)
693
In order to perform characteristic analysis for Maxwell's subsystem, care must be exercised to ensure that all the terms appearing in the fluxes EEM, FEM,GEM are expressed as functions of the primitive variables
-
{
QEM--Ex,
E z,
Ey,
B x,
B z, qo
By,
}*
(59)
For illustration, the flux vector EEM can be extracted from equation (47) as 0 Hz/eo -Hy/8 o (60)
0
EEM --
-E z Ey Jx
For fluids with linear polarization and magnetization, Hz and Hy are the same as in equations (46), while Jx is given in equation (55). The flux vector Jacobian matrix A___EMis obtained as
A EM -- 0EEM = -OQEM
0
0
0
0
0
0
0
azl
a22
0
a24
a25
a26
0
a31 0
0 0
a33 0
a34 0
a35 0
a36 0
0 0
0
0
-1
0
0
0
0
0
1
0
0
0
0
0
_a71
a72
a73
a74
a75
a76
Vx_
(61)
where the coefficients are a21 = --~EVy
a31 =--~Ev z
(62)
a22 = zEv x
a33 = ~Ev x
(63)
694 a24 = ~Ev x Vz
a34 = --~Ev x Vy
(64)
a25 - ~EVyVz
a36
= -Z E
(65)
a26-
-
+Vy
VyVz
a35 = - ~ + Z ~go
ktgo a71 = 0-1 + ~llB2x
+Vz
a72 = cY7Bz + o l l B x B y
a74 = t37(VyBy + VzBz) + Oll(ExB x +
- 0 " 1v z -
O"7(Pz + epvxB
(67) (68)
a73 = -(YTBy + O'llBxB z
a75 =
(66)
NBp )
+ O'I2(NBT +B x
)--0"9 0T +~11Ey B +
0"7 ~p
0T vy
a76 =O,Vy + - s - ( P y - g p v x B z ) + 0 9 ~ +
/9"I'
(69)
Bx 0I'
(70)
0T 0z
(71)
Ol,EzBx +Ol2B x
Matrices BEM and CEM may be obtained in the same fashion as equation (61). After tedious algebraic manipulations [26], the vector of eigenvalues of the flux vector Jacobian matrix A s s is found as
-
{
k~M = 0, Zk, k~, 0, Z~,, k~,, Vx
}"
(72)
This means that the eigenvalues ~1 "- ~4 - - 0 , while ~'7 -- Vx" The remaining four eigenvalues can be obtained from the fourth order algebraic equation ~4 "t- (/,EM ~3 + VEM~,2 + ~EM ~ + 6EM -" 0
(73)
where the coefficients in the fourth order characteristic polynomial are ~EM ------a22 -- a33
(74)
695
VEM "- a22a33 - a26 + a35
(75)
~EM -- a26a33 - a22a35
(76)
~EM -- a25a36 -- a35a26
(77)
The four eigenvalues are the analytical roots given as +
10
~E=--4 "
~'E=--~
+
19
fil
1 (II/EM-I-nEM~
(78)
"i-~O2EMI--~(II/EM'F~-)EMo)
(79)
EMI+-vi-602MI---2
~/'1
EMI--
1
~:~ - -~-r
i
~1
1
- -~(v~,,, - a~.,,,o)
(80)
jm~ 0 2 M 2 -- -~1 (ll/EM -- ~"~EMo)
(81)
+ igc, G
~'B -- -- ~ (I)EM2 --
1
Here, different terms are defined as 2 _ 4 V E M + 4~EM = O~EM + (I)EMo (I)EMI -- (gEM + ~/ (I'EM
(82)
2 (g EM - 4 v EM + 4~4/EM = (X,EM --OEM o
(83)
(I) EM2 = ~ EM --
(I)EMo = 4Ot,2M --4VEM + 4q/EM
(84)
~"~EMo = ((~EMVEM -- 2~EM)/OEMo
(85)
3
ZEM =
3(aEMYEM_ 48EM) _ VEM 2
(86)
(87)
696 [
3 YEM = VEM (4~EM -- (XEM~tEM) I VEM
xl4VEM6EM-- ~tEM 2 -- (XEM~EM) 2
6
2
27
(88)
For illustrative purposes, the following are the eigenvalues in the case of onedimensional EMHD flow where Vy - v , - 0 and a22 = a33 and a25 = 0. Hence
~-~-
~E--~B--~
;[E IE2 / 1 x Vx+ x Vx+4 ~;o~to(1 +
" X
Vx--
2)]
X M ) -- ~EVx
xE2Vx +4
--Z Vx eo~to(1 +
(89)
(90)
)
1 Since /~o~-~o equals the speed of light in vacuum, it seems that for most
practical applications the incoming and the outgoing electromagnetic waves will not be influenced by the fluid except in the situations where the fluid is very highly ionized or when the fluid moves with a speed comparable to the speed of light. In the case of a pure electro-magnetics without any fluid motion, polarization, magnetization, or electric charges (v = __P= M =qo =0), these eigenvalues reduce to the eigenvalues of Maxwell's equations for electromagnetic fields in vacuum [35] 1 -
0, 4
1 ' -
1 S'
0, 4 '
1 -
}*
(91)
After introducing the similarity transformation matrix S__EMof the flux vector Jacobian matrix ~ r M ' the eigenmatrix ~EM corresponding to ~EM becomes ~"
+
~EM diag[ 0, ~E, LE,0, L+B,L-B, v x ] -
-
(92)
where )~E, )~E, )~a, )~a are given by equations (78-81). For locally one-dimensional problems, wave propagation direction is well defined. For multi-dimensional problems, there is no unique direction of
697
propagation, because the flux vector Jacobian matrices AEM , BEM , CEM cannot be simultaneously diagonalized. Therefore, characteristic boundary condition analysis allows that only one of these matrices (relating to only one coordinate direction) can be diagonalized at a time. In the case that the x-coordinate is in the main flow direction, premultiplying the equation (58) with the inverse of the similarity matrix, SEI~, gives
SE1M~QEM
~-t-
Ot
'~ -1 ~QEM ~EMSEM ~-t-
=
c~x
-I ~' SEMHEM -- 0
(93)
Here, vector HEM is given as
H r u = BEta 0QEra + CwM aQEM SEM --~ = aZ
(94)
For the hyperbolic system, time dependent boundary conditions could be derived based on the principle that outgoing waves are described by characteristic equations, while the incoming waves may often be specified by a non-reflecting boundary condition [31,32,36]. Following this approach, the characteristic and non-reflecting boundary conditions at the inlet boundary x = a and at the outlet boundary x = b can be given by the i-th equation of the system -1
(93). Here, the left eigenvector Si,EMis the i-th row of S ~
S-1 C3QEM I=i,EM -O~ + Li,EM + Si,EMHEM) x=a,b - 0
(95)
where L~,EM- 0 for incoming waves, while for outgoing waves (96) ,
63x
698
7.2 Characteristic-based boundary conditions for Navier-Stokes subsystem Similar derivations can be used to determine analytical expressions for the eigenvalues and the non-reflecting boundary conditions of the Navier-Stokes subsystem of the unified EMHD as shown by Dulikravich and Jing [26]. Characteristic treatment of the Navier-Stokes subsystem of the unified EMHD system can be performed by converting its conservative form 63QNs + aENs - -b aFN S -1 63GNs = SNS
at
ax
as
az
(97)
into its non-conservative (characteristic) form g31~NS t- ANS 691~NS I- BNS C3QN----------~s + CNS g31~N---------~s = SNS & 0x 0y -0z
(98)
where the solution vector of unknowns is given as
-
{
QNS= P/~,
Vx, Vy, Vz,
e
}*
(99)
From equation (47) it can be seen that flux vector ENs becomes
Vx 2 Vx +
/
P ENS=
~,~1
p % __I'~BM Vx~x / P
VxVy VxVz
P PB p 'l~xy VyNx ~xz
ev_~ P
Vz x
N_~ '~ P
|
/
(100)
!|
J
Terms related to d,d_ 2 and VT will not be considered in the evaluation of coefficients of the flux vector Jacobian matrix ANS since they are associated with first derivatives of velocity, v, or temperature, T. The flux vector Jacobian matrix ANS = c3ENs/0QNs then becomes
699
0
1
0
0
0
~[p
a22
a23
a24
0
0
a32
a33
a34
0
0
a42
a43
a44
0
_Vx[5/9
a52
a53
a54
Vx_
ANS -
(101)
The coefficients in this matrix are given in detail by Dulikravich and Jing [26]. Eigenvalue vector of the flux vector Jacobian matrix ANs is
-
{
~,NS = Vx, Uu, Z+v, Uw, Ue
}"
(102)
which can be written as a diagonal eigenvalue matrix ~'Ns== diag[Vx' Z+~' Uv' Uw' U~]
(103)
The eigenvalues Uu, X~, Uw, Ue are obtained analytically by solving a fourth order characteristic polynomial (similar to equation 73) where
~ VNS
= - ( a ~ +a3~ +a,4)
-
a22a33 + a22a44 + a33a44 - a34a43 -- a24a42 -- a23a32
-
(104)
(105)
)'NS -- a34a43a22 - a22a33a44 - a24a32a43 + a24a33a42 + a23a32a44 -
a23aa4a42 + (a33 + a44)13 P
~NS -- (a34a43
_
a33a44
)_~
(106)
(107)
9
so that the four eigenvalues are
~+-"
4
NS1 -I-
ONS1 ---2 "(ll/NS q- ~'~NS~)
(108)
700
~/NS q- ~'-~NSo)
(109)
~+w -- -- 4 (I) Ns2 q" i--~(I)Ns2 -- 2 (V NS -- ~'~NSo)
(110)
_lt9 ~1 2 )~+e- 4 NS2-- ~ ~Ns2 --2(~I/NS- ~'-~NSo)
(111)
--'4"(I)Ns1 --
1
NSI --
~/1
2
1
with the coefficients given by equations of the type similar to equations (82-88). Characteristic waves defined by the Navier-Stokes equations in the EMHD system have a great dependency on both fluid dynamics and electro-magnetodynamics, in particular, the electro-magnetic properties of the media and electro-magnetic field quantities. When electric and magnetic fields are absent, these eigenvalues reduce to the well-known eigenvalues of a classical NavierStokes system for Newtonian, incompressible flows. These eigenvalues are {Vx,V~VxV x + c, Vx - c }. Here, the equivalent local speed of sound is defined as C - - 4 V 2 + (~/p) X
"
Following Thompson's approach [30,31], non-reflecting boundary conditions for the Navier-Stokes subsystem are hence formulated as follows. The characteristic form of Navier-Stokes subsystem influenced by the electromagnetic effects is possible to write as
SN s r
&
~ aQNs + ~NsSNIs ~ + = 0x
-1 ~ S__NsHNs=0
(112)
where the i-th equation is S_ 1 ~QN_______SS =,,NS & + NSSN s OQNs & + Si,NsHNs _ 0
(1 13)
and the new source vector is HNs - BNS aQN-----~s+ CNS 0QN------As- SNs Oy = Oz
(114)
701
Here, the left eigenvector S--i,NS "1 is the i-th row of S~ls
--1--/I
Si.NS O0NS + Li,NS + Si,NsHNs Ot
-- 0
(115)
x=a,b
where Li,NS --0 for incoming waves, while for outgoing waves
,NS
0X
(116)
Practical implementation of Thompson-type [31-33,36,26] non-reflecting boundary conditions deserves further comments. The essence of his approach is that one-dimensional characteristic analysis can be performed by considering the transverse terms as a constant source term. In order to provide well-posed non-reflecting boundary conditions in multi-dimensional cases, substantial modifications may be required to take into account the transverse terms at the boundaries [37,38]. It should be emphasized that physically there are cases where flow information propagates back from the outside of the domain into the inside through the boundaries by the incoming waves [39]. This fact makes it possible that building a perfectly non-reflecting (absorbing) boundary condition [40] might lead to an ill-posed problem. Under these circumstances, corrections may be needed to make them partially non-reflecting.
7.3 Numerical integration of EMHD system It is often highly desirable to have a time-accurate unsteady solution to the governing EMHD equations. One numerical integration algorithm that could be used is an advanced form of the dual time-stepping technique, also called an iterative-implicit technique, originally developed by Jameson [41 ]. To create an instantaneous picture of the solution of the entire EMHD system at a given physical time, equation (43) must be driven to zero in its entirety, not, as is commonly done in time-marching techniques by driving only the physical time-dependent term to zero. To this end, a pseudo-time derivative is added to the EMHD system (equation 43) which can be rewritten as c3Q 0E 0F 0G -00 + ~ + ~ + ~ + ~ _ S c3z 0 t 0x 0y 0z
(117)
702
or as
aQ=
_
aQ &
(118)
where ~ is a composite of the spatial and source terms and is called the residual. Thus, given a physical time step the governing equations are time marched in pseudo time, x. Upon convergence, the fight-hand side of equation (118) becomes zero and the solution at the desired physical time level, t, is obtained. Note that the pseudo-time dependent variable vector, 0 , does not have to be the same as the physical time dependent variable vector, r An additional concern of great importance is that the system of equations develops zero terms in the pseudo-time dependent variable vector, 0 , for incompressible fluids, fluids without electric charges, or systems in which the electric and magnetic fields are non-interacting. This poses significant problems for time-marching numerical solutions. This problem may be alleviated, however, by proper selection of pseudo-time dependent variable vector, 0 , and through the use of matrix preconditioning. By premultiplying I~ with a properly selected matrix, it is possible to directly control the system eigenvalues. This prevents development of zeros in the pseudo-time dependent variable vector, 0 , and vastly improves iterative convergence rates over a wide variety of flow regimes (low and high Mach and Reynolds number combinations). The preconditioning matrix, F'(0), for the EMHD system could be based on one developed by Merkle and Choi [42] for the Navier-Stokes system. The preconditioned EMHD system may be written as
r, aQ_ aQ ax -&
=
(119)
Equation (119) can be transformed to a body-conforming non-orthogonal curvilinear time-dependent (~, 1-1, ~; t) coordinate system. A high order of accuracy is desired to properly resolve unsteady motions. A finite difference scheme using fourth order accurate spatial differencing and second order accurate physical time differencing could be used while the solution is advanced in pseudo-time using a four-stage Runge-Kutta scheme which is second order accurate for non-linear problems. Fourth order accuracy should be selected for
703
the spatial derivatives based on extensive research completed by Carpenter et al. [30] which found that a Runge-Kutta advanced fourth order accurate scheme provided the best convergence and stability of higher order schemes at reasonable computational cost. Second order accurate differencing in physical time could be selected based on stability and convergence studies performed by Melson et al. [43] who found that for a Runge-Kutta advanced dual timestepping scheme second order backward differencing provided the most stable physical time discretization while providing excellent resolution. The new physical time step could be treated implicitly in pseudo-time, while all old physical time steps and spatial derivatives could be treated explicitly. This is unlike Jameson's early method [41] that treats both the physical time and the spatial derivative explicitly and causes a restriction on the maximum physical time step allowed. The discretized preconditioned system may be written as 0 ~ =(~" F'F-I+ a i -~
(120) A'I~ "~/~ i _
-
~ z- ~- r i m + l ,
i-1
_4, m2,,t (~ n-,-1= 0 4
i=O
(121)
(122)
where m=1,2,3,.., represents the physical time step, n=1,2,3,.., represents the pseudo-time step, and i=1,2,3,4 is the Runge-Kutta stage number. Also, F = 0 0 / / ~ and c~i are the Runge-Kutta coefficients. Note that the physical time-dependent term on the right hand side of equation (121) is held constant for all four Runge-Kutta stages.
8. S U B M O D E L S
OF EMHD
Until now, the numerical solutions of the unsteady three-dimensional EMHD flows that have been reported in the open literature [34-36] did not account for polarization or magnetization effects and did not involve charge density transport equation. The reason is that the complete unified EMHD system is very large having extremely complicated source terms and two extremely
704 different time scales for the electro-magnetic fields and the flow-field. Consequently, a number of simplified versions of the EMHD system have been traditionally used in practical applications. These submodels can be grouped in two general categories: EHD models and MHD models [ 11-13,44]. From the unified EMHD model, it can be seen that the electromagnetic field is not the only cause of electric current and that the temperature gradient is not the only source of heat conduction as is commonly assumed. The electric field, magnetic field, heat conduction, and deformation (strain) may couple to produce charge motion and heat transfer. These couplings are called phenomenological cross effects and may be placed in four general categories: 1) thermoelectric, 2) galvanomagnetic, 3) thermomagnetic, and 4) second order effects [9, p.161163]. These categories are based on the source of the effect and each will be described in turn, as will be a comparison between classical EHD and MHD models and the unified EMHD theory. The comparison concentrates on similarities and differences between electro-magnetic force and electric current and heat conduction terms in the EHD, MHD, and EMHD models. The inadequacies of simple superpositioning of classical simplified models to fully describe the unified EMHD flows are also noted. Couplings between the temperature gradient and the electric field cause thermoelectric effects so that a temperature gradient in the material produces an electric current (Thompson effect), while applied electric field produces heat conduction in the material (Peltier effect). These two effects together are known as the Seebeck effect and form the basis for thermocouples. Also note that the or, term in the electric conduction current (equation 22) and the K4 term in the heat conduction (equation 23) are the ohmic charge conduction and Fourier heat transfer, respectively. When the electric and magnetic fields are simultaneously applied but are not parallel, electric current (Hall effect) and heat conduction (Ettingshausen effect) perpendicular to the plane containing the electric and the magnetic fields are induced in the media. These effects are termed galvanomagnetic [9, p. 161-163]. When the temperature gradient and the magnetic field are simultaneously applied but are not parallel, electric current (Nernst effect) and heat conduction (Righi-LeDuc effect) perpendicular to the plane containing the temperature gradient and the magnetic field are induced in the material. These effects are termed thermomagnetic. It should be noticed (equation 22) that the interaction of the average rate of deformation tensor and the electric field can also create the electric current, while the interaction of the material deformation tensor and the electric field can create the temperature gradient (equation 23). These piezo-electric and piezomagnetic effects can further be enhanced if the material is non-isotropic.
705
8.1 Classical e l e c t r o - h y d r o d y n a m i c s ( E H D )
As mentioned previously, EHD flows are those in which magnetic effects may be neglected and charged particles are present, while only a quasi-static electric field is applied so that the magnetic field, both applied and induced, may be neglected [ 11 ]. One of the implied assumptions is that the flows are at non-relativistic speeds, although in astrophysical flows this assumption cannot be made [1]. Atten and Moreau [44] present a detailed coverage of classical EHD modeling and discuss the relative importance of terms in the force and electric current through stability analysis. With these assumptions, the Maxwell' s system reduces to [ 11 ] V . D - V.(~E_)- qo
(123)
c3q~ ~ - V . , J - 0 &
(124)
With classical EHD assumptions, the electro-magnetic force in the unified EMHD theory reduces to" (125)
f_.v_,M_ q~E + (V_E)-P = qo_E+ (VE). e,pE
This is not the form of the electro-magnetic force usually seen in classical EHD formulations [11]. Through the use of thermodynamics and the material constitutive equation of state, the electric force per unit volume in EHD is most often used in the following equivalent forms [10, p.505-507][8, p.59-63] f EM
:
qo_E- E . E Ve + - V
-
--2-
-
2
2
E-E p -
p=const
(126) T=const
constl
(127)
The three terms in the equation are the electrophoretic, dielectrophoretic and electrostrictive terms, respectively. The electrophoretic force or Coulomb force is caused by the electric field acting on free charges in the fluid. It is an irrotational force except when charge gradients are present [45].
706
The dielectrophoretic force is also a translational force, but is caused by polarization of the fluid and particles in the fluid. The dielectrophoretic force will occur where high gradients of electric permittivity are present. This condition will be true in high temperature gradient flows, multi-constituent flows, particulate flows [ 18] or any time the electric field must pass through two contacting media of different permittivities [46]. Grassi and DiMarco [47] treat the dielectrophoretic force as it applies to bubbly flows and heat transfer. Poulter and Allen [45] note that the dielectrophoretic force produces greatest circulation when the dielectric permittivity is inhomogeneous and non-parallel with the applied electric field. The last force, the electrostrictive force, is a distortive force (as opposed to the previous translational forces) associated with fluid compression and shear. The electrostrictive force is usually smaller than the -phoretic forces. It is present in high pressure gradient flows, compressible flows, and flows with a non-uniform applied electric field. Pohl [18] describes this phenomenon in greater detail. Classical EHD modeling derives directly from the unified EMHD theory. Thus, the electric current density, using EHD assumptions, reduces to J = qo_V+ OlE + o-47T
(128)
However, this is not the form seen in classical EHD models [ 11 ] which typically define the conduction electric current as only the first term of equation (22). However, more advanced classical EHD models define the current as [9, p.562] J- = qo_v + J_c = qo_v + qob E_- DoVqo
(129)
The last two equations imply that the temperature gradient is directly related to the electric charge gradient. This may be shown to be true based on the Einstein-Fokker relationships, derived from studies of Brownian motion [25, p.264-273], which relate any concentration gradient to a charge mobility and a diffusion. Newman [48] also provides a detailed discussion of the concepts of diffusion and mobility. The electric charge diffusion term is often neglected where only limited amount of free charges are available [49]. By introducing classical EHD assumptions in the unified EMHD theory, the equation (23) for heat flux reduces to CI = KI__E+ K4VT
(130)
707
The classical EHD models neglect the contribution to heat transfer from the electric field so that equation (130) reduces to Fourier's law of heat conduction. Cl = -~:VT
(131)
Although classical EHD modeling seems to neglects heat transfer induced by the electric field and electric current, Joule heating effect ( - I , . _ E term from EMHD equation 19) is usually included in the EHD computations [50,51 ]. 8.2 Classical m a g n e t o - h y d r o d y n a m i c s (MHD) The classical modeling of MHD assumes non-relativistic and quasimagnetostatic conditions. It implies that electric current comes primarily from conductive means and that there are no free electric charges in the fluid [11 ]. With these assumptions Maxwell's system becomes
V.B_.= 0 VxE= -
(132) 0B -0t
(133)
Vxl-l= /
(134)
V-J -0
(135)
The modifications to the Navier-Stokes relations come from the electromagnetic force on the fluid from which all induced electric field terms have been neglected. Using the MHD assumptions, the electro-magnetic force per unit volume in the unified EMHD theory becomes [ 11 ] [EM = j x B_B+_ ( V B ) . M___
(136)
The second term, source of dimagnetophoretic and magnetostrictive forces, is typically neglected in classical MHD [10, p.508]. Thus, the electro-magnetic force per unit volume in the classical MHD is modeled as [ 11 ] frM = j x B
(137)
708
By making MHD assumptions, the conduction current in the EMHD can be expressed with equation (38). However, classical MHD theory usually defines the electric conduction current as [ 10, p.510] Jc = cYlE + o4VT = cYlE + O'l(V x B) + cY4VT
(138)
Here, (5'4 is the Seebeck coefficient [9, p.174] which in some classical MHD formulations is not used [11 ]. Clearly, the classical MHD formulations neglect a significant number of physical effects [52,53]. Similarly, in classical MHD modeling, Joule heating is often included in the energy relation, but the heat transfer constitutive relation remains the same as in equation (131). In comparison, the unified EMHD model for the heat flux with classical MHD assumptions can be expressed with equation (39). It could be concluded that classical EHD models include many important effects and correspond to the unified EMHD theory well, while classical MHD formulations need improvements in the force, current, and heat transfer terms. As in classical EHD modeling, it is important to be aware of the fact that many force, current and heat transfer terms can be written in several different forms, each of which is equivalent. It is, therefore, important to recognize the potential danger of simply adding terms from different EHD and MHD models.
9. S O L I D I F I C A T I O N W I T H E L E C T R O - M A G N E T I C FIELDS During solidification from a melt, if the control of melt motion is performed exclusively via an externally applied variable temperature field, it will take quite a long time for the thermal front to propagate throughout the melt thus eventually causing local melt density variations and altering the thermal buoyancy forces. It has been well known that an externally applied steady magnetic or electric field can, practically instantaneously, influence the flowfield vorticity and change the flow pattern in an electrically conducting fluid [51-59,33]. Similarly, it is well-known that applying an electric potential difference to a flow-field of a homogeneous mixture will cause fractionation or separation of the homogeneous mixture into regions having high concentration of the constituents. This phenomena, known as flee-flow electrophoresis, has been extensively studied experimentally and, to a lesser extent, numerically [50] using classical EHD modeling. Nevertheless, there are no publications yet on actual algorithms for determining the proper variation of intensity and orientation of the externally applied magnetic and electric fields. This is not a
709
trivial problem because we are dealing with a moving electrically conducting fluid within which an electric current is induced as the fluid cuts through the externally applied magnetic field lines [11]. This induced electric current generates heat (Joule effect) as it passes through the fluid that has a finite electrical resistivity. In the case of solidification, the amount of heat generated through the Joule effect due to the externally applied magnetic field is often neglected compared to the latent heat of solidification and the amount of heat transferred in the melt by thermal conduction. The latent heat released or absorbed per unit mass of mushy region (where Tliquidus > T > Tsolidu s ) is proportional to the local volumetric liquid/(liquid + solid) ratio often modeled [59] as
f =
Ve
=
v,+v,
(
T - Tsolidus
/n
= 0n
(139)
Wliquidus -- Wsolidus
Here, 0 is the non-dimensional temperature, the exponent n is typically 0.2 < n < 5, subscripts g and s designate liquid and solid phases, respectively, while f 1 for T > Tliquidusand f 0 for T
p e - p ~ 1+
00
.
.
.
(140)
.
with a similar expression for the solid phase where the reference values are designated with the subscript "r". In this work, we assumed that electric conductivity and magnetic permeability do not vary with temperature. The EHD and the MHD systems of equations including solidification can be non-dimensionalized in a number of ways. The typical non-dimensional numbers are [33,60]: Reynolds hydrodynamic
Froude
Eckert
_ ~ O
R ~ - PrVrg r ~vr
FR2= Vr grgr
Ec=
V__~r 2 CrATr
(141)
710 Prandtl hydrodynamic
Stefan
Grashof
PR = ~tv~Cr Kr
STE = CraT~ Lr
a R = P~ff'rgrATrg3r (142) g2
Hartmann
Prandtl magnetic
Prandtl electric
/ )
vr
HT = grgrHr Or \ l'tvr
Pm - gvrO'r~'l'r Pr
~tvr PrbrA~)r
Coulomb
Electric field
Charge diffusivity
SE = qorA~)r PrV2
q org2r N~ - 8rAter
~vr D E = PrDor
(143)
(144)
where ~tv~,c~,A,l,r,~:~,~q,Lr,g ~ are the reference values of viscosity, specific heat, electric potential difference, heat conductivity, magnetic permeability, latent heat of liquid-solid phase change, and length, respectively. Also, mixture density and modified heat capacity can be defined as Pmix = f Pt + (1- f)Ps
(145)
~(Cg 0g) ~(Ceq 0s) Cmix = fPe ~ + (1 -- f)p~ 00 30
(146)
An enthalpy method [58,59] can be used to formulate the equivalent specific heat coefficient in the solid phase defined as c~q = c~
1 0L
STZ 00
(147)
so that latent heat is released in the mushy region according to equation (139). 9.1 EttD and solidification
EHD equations for phase-changing liquid-solid mixtures, where the solid phase is treated as the second liquid with extremely high viscosity, can be derived using Boussinesq approximation for thermal buoyancy [61]. We can also define mixture electric charge mobility
711 bmi x =
(148)
f b e + (1 - f)b s
and combined hydrodynamic and hydrostatic pressures in liquid and solid Pe-P+q~ o,~ F~
and
Ps=P+q9 Os F~
(149)
where q) is the non-dimensional gravity potential defined as g =-Vq~. Assuming equal velocities for both phases, the mass conservation is V-v=O
(150)
Linear momentum conservation for two-phase EHD flows with thermal buoyancy and Coulomb force
Pmix--~ + fpgV-(VV -I-ps
= f V I_~e
--
--
{
(1- f)ps v" (VV q- Ps!)
(151)
+-~-e2PeoteOg_
>]~ }
+ ( l - f ) V Fgvs(Vv+(Vv) * + [Re ~
PsO,.s0
g-
+SEqo.
Energy conservation for incompressible two-phase EHD flows including Joule heating can be written as [60]
Cmix _
O0
[
\
1
[_
- .Re----PT~If V" (KtVO)+ (1- f) V 9(KsVO)]
\ (152)
+Sc(qov+qo,mix_ Electric charge conservation equation including migration and diffusion is
712
E( bmix)] 1
~V. qo v+ Ot
-
_E -
ReP E
(153)
~ V 9( b m i x V q o ) ReD E
Since _E_E--V~, the electric potential equation resulting from equation (13) (154)
V-[(fe e + (1- f)~Zs)V~] = -NEq o must be solved simultaneously with the equations (150-153). 9.2 M H D and solidification
MHD two-phase solid-liquid flows can be modeled using a similar approach. The non-dimensional Navier-Stokes equations for phase-changing mixtures of two liquids (solid phase is treated as the second liquid with extremely high viscosity), can be formulated [33] so that the mixture mass conservation is V-_v=O
(155)
Linear momentum conservation for two-phase MHD flows with thermal buoyancy and magnetic force 63v Pmix-~+
f P e V " (---VX+
PeI)+ (1- f)P~V "(XX+ PsI)
=f V- -~(Vv+(Vv)* Ke
+(,-o
--
+
--
-~e2P
eae0_+ g
{-v [.vs(vv+(vv; tl+ ~ LR e
-
_
o ~+~.,,(~•215
Re
(156)
PmR~' e
PmRe
-
)
The non-dimensional hydrodynamic, hydrostatic, and magnetic pressures were combined to give pc-P +~-+ {.teH.H Pe FR PmR2e
and
~
P +~-y+ g~H-H FR PmRo 2
Ps
(157)
713
where tp is the non-dimensional gravity potential defined as g - - V q ~ .
Then,
the energy conservation for incompressible two-phase MHD flows including Joule heating can be written as [33] 00 Cmix
=f
[ \ + fpt?V" tcf0__v)+ ( 1 - f)ps v . [csq0v)
1 RePR V 9(Ks
+
1 HTE c (V x H)-(V x H
2 3 O"e PmRe
(158)
1 1 HTE c tV + ( l - f ) RePR V'(K:sV0)q--- 2 3 I-I).(VxI-I o s VmRe
The magnetic field transport equation for the two-phase MHD flow in its nondimensional form becomes [ 1, p. 150]
-pm~V x e
+~
VxH
(159)
O's~s
If electric conductivity and magnetic permeability are assumed constant, then _ v • (_v • H ) tgt
(1- f)/(~sgs)V2H PmRe
(160)
--
needs to be solved either intermittently [33] with the equations (155-158).
ACKNOWLEDGMENTS The author would like to thank Dr. Yimin Ruan and Dr. Owen Richmond of the ALCOA Technical Center for the ALCOA Foundation Grant, Dr. Martin Volz of Microgravity Program at NASA Marshall Space Flight Center for partially supporting a student assistant, Professor Akhlesh Lakhtakia of the Pennsylvania State University for stimulating technical discussions, and Mrs. Sheila Corl and Professor Hyung-Jong Ko from Kumoh National University of Technology, Korea for proofreading this chapter.
714 REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
19. 20.
W.F. Hughes and F.J. Young, The Electromagnetodynamics of Fluids, John Wiley and Sons, New York (1966). G.S. Dulikravich and S.R. Lynn, ASME FED-Vol. 235, MD-Vol. 71 (1995) p.49. G.S. Dulikravich and S.R. Lynn, ASME FED-Vol. 235, MD-Vol. 71 (1995) p.59. G.S. Dulikravich and S.R. Lylm, International Journal of Non-linear Mechanics, Vol. 32, No. 5 (September 1997) p.913. G.S. Dufikravich and S.R. Lynn, International Journal of Non-linear Mechanics, Vol. 32, No. 5 (September 1997) p.923. G.S. Dulikravich and Y.-H. Jing, ASME AMD- Vol. 217 (1996) p.309. S.-I. Pai, Magnetogasdynamics and Plasma Dynamics, Springer-Verlag, ViennaJPrentice Hall, Inc., Englewood Cliffs, N.J. (1963). L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, New York (1960). A.C. Eringen and G.A. Maugin, Electrodynamics of Continua I; Foundations and Solid Media, Springer-Verlag, New York (1990). A.C. Eringen and G.A. Maugin, Electrodynamics of Continua II; Fluids and Complex Media, Springer-Verlag, New York, (1990). O.M. Stuetzer, Physics of Fluids, Vol. 5, No. 5 (1962) p.534. J.R. Melcher, Continuum Electromechanics, MIT Press, Cambridge, MA (1981). G.W. Sutton and A. Sherman, Engineering Magnetohydrodynamics, McGraw Hill, New York (1965). A.S. Wineman and K.R. Rajagopal, Continuum Mech. Thermodyn., Vol. 7 (1995) p.1. P.G. Bergman, The Special Theory of Relativity, Handbuch der Physik, Bd. IV, Springer-Verlag, Berlin (1962). M.J. Marcinkowski, Acta Physica Polonica, A.81 (1992) p.543. A. Lakhtakia, J. of Advances in Chemical Physics, Vol. 85 (1993) p.311. H.A. Pohl, Dielectrophoresis; The Behavior of Neutral Matter in Nonuniform Electric Fields, Cambridge University Press, Cambridge, U.K. (1978). C.T.A. Johnk, Engineering Electromagnetic Fields and Waves, John Wiley and Sons, New York (1988). W.N. Cottingham and D.A. Greenwood, Electricity and Magnetism, Cambridge University Press, Cambridge (1991).
715
21. H.A. Haus and J.R. Melcher, Electromagnetic Fields and Energy, Prentice Hall, New Jersey (1989). 22. A.C. Eringen, Mechanics of Continua, 2nd Ed., Robert E. Krieger Publishing Co., Malabar, FL (1967). 23. A.C. Eringen, Ed., Continuum Physics-Volume II; Continuum Mechanics of Single-Substance Bodies, Academic Press Inc., New York (1975). 24. R.M. Bowen, Introduction to Continuum Mechanics for Engineers, Plenum Press, New York (1989). 25. S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North Holland Publishing Company, Amsterdam (1962). 26. G.S. Dulikravich and Y.-H. Jing, ASME IMECE, Dallas, TX (Nov. 1997). 27. A.J. Chorin, Journal of Computational Physics, Vol. 2 (1967) p. 12. 28. T.B.A. Senior and J.L. Volakis, Aproximate Boundary Conditions in Electromagnetics, lEE, London, UK (1995). 29. C. Hirsch, Numerical Computation of Internal and External Flows, Volume 1; Fundamentals of Numerical Discretization, Wiley-Interscience, New York (1988). 30. M.H. Carpenter, D. Gottlieb and S. Abarbanel, J. Comp. Phys., Vol. 108 (1993) p.272. 31. K.W. Thompson, Journal of Computational Physics, Vol. 68 (1987) p. 1. 32. K.W. Thompson, Journal of Computational Physics, Vol. 89 (1990) p.439. 33. G.S. Dulikravich, V. Ahuja and S. Lee, International Journal of Heat and Mass Transfer, Vol. 37, No. 5 (1994) p. 837. 34. V.J. Shankar, W.F. Hall and H.M. Alireza, Proc. IEEE, Vol. 77, No. 5 (May 1989) p.709. 35. J.S. Shang, AIAA paper 91-0606, Aerospace Sciences Meeting, Reno, NV (January 1991). 36. M.S. Tun, S.T.Wu and M. Dryer, J. Comp. Phys., Vol. 116 (1995) p.330. 37. T.J. Poinsot and S.K. Lele, J. of Comp. Physics, Vol. 101 (1992) p. 104. 38. R. Hixon and S.-H. Shih, AIAA paper, 95-0160, Reno, NV (1995). 39. T. Hagstrom and S.I. Hariharan, Math. Comput., Vol. 20, No. 10 (1994) p.155. 40. M.E. Hayder, F.Q. Hu and M.Y. Hussaini, ICASE Report No. 97-25 (also NASA CR 201689) (May 1997) 41. A. Jameson, AIAA Paper 91-1596, Reno, NV (January 1991). 42. C.L. Merkle and Y. Choi, Intl. J. Num. Meth. Eng., Vol. 25 (1988) p.293. 43. N.D. Melson, M.D. Sanetrik, and H.L. Atkins, 6th Copper Mountain Conf. on Multigrid Methods, Copper Mountain, CO (April 4-9, 1993).
716
44. P. Atten and R. Moreau, Journal de Mecanique, Vol. 11, No. 3, September (1972) p.471. 45. R. Poulter and P.H.G. Allen, 8th Intl. Heat Transfer Conf., San Francisco, CA, Vol. 6 (1986) p.2963. 46. M. Aoyama, T. Oda, M. Ogihara, Y. Ikegami and S. Mashuda, Journal of Electrostatics, Vol. 30 (1993) p.247. 47. W. Grassi and P. Di Marco, VII European Symp. on Material and Fluid Science in Microgravity, Belgium (1991). 48. J.S. Newman, Electrochemical systems, Prentice Hall, NJ (1991). 49. R.B. Schilling and H. Schaechter, Journal of Applied Physics, Vol. 38 (1967) p.841. 50. S. Lee, G.S. Dulikravich and B. Kosovic, AIAA paper 91-1469, AIAA Fluid, Plasma Dynamics and Lasers Conf., Honolulu, HI (June 1991). 51. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publication, Inc., New York (1961). 52. M. Salcudean and P. Sabhapathy, ASME MD-Vol. 20, ASME Book No. G00552 (1990) p.115. 53. H. Ozoe and K. Okada, International Journal of Heat and Mass Transfer, Vol. 32, No. 2 (1989) p.1939. 54. S. Lee and G.S. Dulikravich, International Journal for Numerical Methods in Fluids, Vol. 13, No. 8 (October 1991) p. 917. 55. G.S. Dulikravich, V. Ahuja and S. Lee, Journal of Enhanced Heat Transfer, Vol. 1, No. 1 (August 1993) p. 115. 56. G.S. Dulikravich, K.-Y. Choi and S. Lee, ASME FED-Vol. 205/AMDVol. 190 (1994) p. 125. 57. H. Hatta and S. Yamashita, Journal of Composite Materials, Vol. 22 (May 1988) p.484. 58. G.S. Dulikravich, B. Kosovic and S. Lee, ASME Journal of Heat Transfer, Vol. 115 (February 1993) p.255. 59. G.S. Dulikravich, V. Ahuja and S. Lee, Numerical Heat Transfer: Fundamentals, Part B, Vol. 25, No. 3 (1994) p.357. 60. V.R. Voller and C.R. Swaminathan, Numerical Heat Transfer, Part B, Vol.19 (1991) p.175. 61. D.D. Gray and A. Giorgini, International Journal of Heat and Mass Transfer, Vol. 19 (1976) p.545.
717
C O N D U C T I O N AND DIELECTRIC E F F E C T S IN E L E C T R O R H E O L O G Y C.W. Wu* and H a n s C o n r a d Department of Materials Science and Engineering North Carolina State University Raleigh, NC 27695
1. I N T R O D U C T I O N
Since Winslow's discovery [1,2] of electrorheology (ER) the subject has developed into a multidisciplinary field that has captured the imagination and attention of scientists and engineers worldwide. Early studies [3,4] on the interaction force between two adjacent particles in an ER suspension employed the point-dipole approximation, assuming no electric current flows through the suspension, i.e., both the particles and host liquid are non-conducting. This gives for the interaction force f =24na6aoKf(~dEo)2/R4 ,
(1)
where a is the radius of the particles, ~o the permittivity of vacuum space, Kf=~f&o the real component of the relative permittivity of the host fluid, R the distance between the particle centers, E o the applied field and ~d the polarizability induced by the particle/liquid dielectric mismatch, which is given by (2) * Visiting professor, on leave from Research Institute of Engineering Mechanics, Dalian University of Technology, Dalian 116024, People's Republic of China.
718 where Kp=E/% is the real component of the relative permittivity of the particles. 2~he point-dipole approximation however gives an order of magnitude lower estimate of the interaction force than actually observed [5,6]. Solution of the Laplace equation [7,8] gave a force more nearly that measured. Klingenberg and Zukoski [7] calculated the shear force of a single chain to be fs - 12So~a2I~d2Eo2(aIR)4[(2fH+fr) c~
" f~sin30 ]
(3)
where f//is the force component induced by the field component parallel to the axial line of the chain, fr the component induced by the torque attempting to align the line-of-centers with the field leading to the formation of the chain, f~ the repulsive component induced by the field component perpendicular to the axial line of the chain and 0 the angle between the applied field and the axial line of the chain. Chen, Sprecher and Conrad [8] gave for the axial attractive force between two adjacent particles (R/a~2.05) fa - 37"44Eo~a2t~d2Eo2exp{[ 14.84-6.16(Wa)]~2}.
(4)
Finite element calculations [9,10] gave the same magnitude for the interaction force, with the following empirical equation [10]" 61Ua - 4
fa = ~oa2K~d2Eo2 R/a -~d
~l-~d)/2(~)4
(5)
All of the above studies considered only the dielectric polarization in the ER suspension. They represent the so-called dielectric polarization model for ER response induced by the particle/liquid dielectric mismatch and predict that the electric field dependence of the attractive force and the resulting shear yield stress is quadratic. However, both the particles and the host liquids have in reality a fimte conductivity and during recent years it has become increasingly evident that the conductivity of the particles and host oil play an important role in the electrorheological response of suspensions, especially with dc or low frequency ac electric fields. Anderson [11], Davis [12], Chen and Conrad [13] and Conrad and Chen [14] suggested that the conductivities of the particles and host liquids could be included in the framework of the dielectric polarization model by using the complex polarizability
719
(6) where the relative complex permittivity K*=K-iK". K (usually designated K', but here for simplicity we omit the superscript ') is the real component, K"=o/c0% the loss component, o the conductivity which is assumed to be independent of the electric field, co the angular frequency of the applied ac field and i---~-l. The subscripts p and f refer to the particles and host liquid respectively. Within the complex polarization framework, Kim and Klingenberg [15] and Webber [16] predicted that the interaction force between particles should be proportional to 1~*12Eo2f(fl*), where f ( f ) is a complex function of 13".We will term this model for the ER effect the complex polarization model. When the electric field frequency co is very high, Equation (6) reduces to Equation (2), i.e., ~*=~d, and the dielectric effect dominates the ER response. In this case the complex polarization model reduces to the dielectric polarization model. When the field frequency co is very low (co-*0) The complex polarizability ~* given by Equation (6) reduces to
)/(%+2of),
(7)
where ~c is called conductivity polarizability or the conductivitymismatch parameter. When (~p>>Ot, Equation (7) gives ~c=l, in which case the particle/liquid conductivity mismatch dominates the ER response. The conductivities of the particles and the host liquid are assumed to be ohmic in the complex polarization models presented in [ 11-17]. These models still predict that the attractive force between particles and the resulting shear yield stress are proportional to the square of the applied field. Although the models include the conductivity in the ER response, they can only give an estimate of the shear yield stress for two limiting cases: (a) dc or low frequency ac field and (b) very high frequency ac. For example, Davis [17] provided a simplified analysis for a suspension of oxidized metal-particles in a weakly conducting oil. His prediction of the yield stress with ac electric field is in good agreement with the experimental results by Inoue [18], but one order of magnitude lower than those measured with dc electric field. Although one frequently obtains reasonable agreement between the measured rheology of ER suspensions and that predicted by the complex polarization model using the complex permittivities [11-15,17,19-20],
720
the model does not account for the fact that the electric field dependence of the flow stress is frequently less than quadratic, as observed in some experiments at dc field or low frequency ac field [ 14,18,21-24]. Also the model does not explicitly explain that the flow stress generally correlates with the associated current density [14,25]. These discrepancies pertaining to the polarization models were first addressed by Foulc et al [26,27] for the case when the conductivity of the particles is much greater than that of the host oil. A new concept in their study was that the host liquid exhibits non-ohmic (non-linear) behavior. They proposed that the effect of the field on the conductivity of such oils was given by a simplified expression of Onsager's theory [28] of(E)- j/E = of(0) [ (l-A) + Aexp~E/Ec],
(8)
where of(0) is the conductivity at low electric fields and A and Ec are constants which depend on the oil. In their model the conductivity mismatch F=Op/Of(0) between the particles and the host oil is the important electrical parameter. Further, to make the necessary calculations they defined two regions in the contact zone between spherical particles: (a) for a distance from the center line x>6 the sphere surface is equipotential and the electric current leaving the sphere is negligible and (b) for x<6 the electric field in the host oil is enhanced due to the increased conductivity of the thin oil film in this region and therefore most of the current passes through this region. The enhanced conductivity of the thin oil film results from the non-ohmic conductivity of the common non-polar oils used in ER suspensions. Their conduction model then predicts that the attractive force between the particles is proportional to Eon, where the exponent n ~ 1 at very high applied field and n ~ 2 at very low applied field. Qualitative experimental agreement with their conduction model was obtained for polyamide half-spheres (14 mm dia.) in mineral oil [27] and for a suspension of cellulose particles in mineral oil [22]. Tang, Wu and Conrad [29], and Wu and Conrad [30] derived the attractive force and current between conducting particles suspended in non-ohmic conduction oil using an approach (to be discussed below) which differs from that of Felici, Foulc and Atten [27]. They obtained good agreement between the predicted values of both the shear yield stress and current density and those measured experimentally. Furthermore in their conduction models, the force and current is derived as a function of the separation of the particles. In more recent work, Atten et al [31] have however extended their conductivity model to include separation of the particles. They proposed that the interaction
721
force decreases approximately linearly with small separation of the particles. However, it will be shown below that the models of Tang, Wu and Conrad [29] and Wu and Conrad [30] give a better agreement between the predicted current density and the rheology of ER suspensions and those measured. In their study of non-ohmic conduction, Davis and Ginder [32] give a simplified analysis for the shear modulus and shear yield stress based on the work of Felici, Foulc and Atten [27]. They predicted that the shear yield stress of an ER suspension with dc field is the 1.5th power of the applied field. Subsequently, Davis [33] extended the non-ohmic conduction model using an integral equation method, which gave good qualitative agreement with the field dependence of the measured attractive force between two particles [27]. Recently, Wu and Conrad [34] extended their work on the non-ohmic conduction model with dc field to a suspension of metal particles coated with a weakly conducting oxide film in a non-ohmic oil. Good agreement occurred between the predicted values of both the shear yield stress and current density and those measured. Subsequently, Wu and Conrad [3537] determined the dielectric and conduction effects on ER response at any frequency of the applied field. Their predictions of both the attractive force and the current density were again in good agreement with those measured by: (a) Boissy, Foulc and Atten [23] for two polymer particles in mineral oil, (b) Garino, Adolf and Hance [38] and Miller, Randall and Bhalla [39] for ceramics particles in silicone oil or dodecane, and (c) Inoue [18] for oxidized metal particles in silicone oil, respectively. In this chapter, section 1 gives a general review of the studies on the role of electrical properties in electrorheological (ER) response. Section 2 first introduces the general physical equations for the combined conduction and dielectric effects, and then gives the simplified conduction models proposed to-date for: (a) bulk conducting particles, (b) low conducting particles with a conducting film and (c) highly conducting particles with a low conductivity film. This is followed by some predictions by the conduction models under dc field and by comparisons between the measured ER response and that predicted. Section 3 introduces our recent work on the conduction and dielectric effects in ER response under ac field, and reviews the progress in studies of transient response in ER fluids. Finally, section 4 proposes some design guidelines for improved ER fluids.
722
2. CONDUCTION MODEL FOR ER RESPONSE UNDER DC FIELD 2.1 General differential equation for the potential Intheoretical considerations of electrorheology we first have to define the local field (or the potential) distribution to determine the interaction and current density between particles. If particles suspended in a liquid have neither free bulk charge nor free charge at the interface, the general differential equation for the potential in both the particles and the host liquid under ac field is [40,41]: V-(e*VV)=0
(9a)
or
V.(o*VV)=0,
(9b)
where ~*=E+o/io)is the complex permittivity, o*=i~o~*=o+ic0~ the complex conductivity, ~0 the field angular frequency, o and ~ the conductivity and real component of the permittivity, V the unknown complex potential, which is dependent on the frequency, time and location. Once V is obtained, the electric field can be calculated by E--VV. If o=0 and ~ is independent of electric field or the electric potential, Equations (9a-b) become the Laplace equation v2V=0.
(10)
The dielectric polarization models [7,8,10,11] are based equation. One of the boundary conditions of Equation (9a) interface of the particle and the host liquid is that the normal to the interface of the particle and the host liquid In a general form n
on the Laplace or (9b) at the current density should be equal. (11)
where Ep=OVp/Onand E~=OVf/Onare the electric fields normal to the interface in the particles and the host liquid respectively, and t is the time. For an ac applied field Eo(t)=Eoexp(imt) (the potential V can be assumed to be V=V*exp(icot), where V* is the unknown complex
723
amplitude depending only on the location coordinates), Equation (11) can be written as
Ep*(OVp/On)=ef*(OVf/On), or
(12a) (12b)
where n denotes the normal direction at the interface of two media, and again the subscripts p and f refer to the particles and host fluid respectively. Another boundary condition is that the electric potential should be equal at the interface of particle and the host liquid, i.e., Vp=Vr
(13)
If we assume that the particles and the host liquid are non-conducting or the frequency is very high, Equations (10-13) reduce to those describing the dielectric polarization model [7,8,10]. As mentioned above, the conductivity of the host oil in ER fluids generally depends strongly on the electric field. In dc or low ac field, the non-ohmic character of the host oil has a significant effect on the ER response. In this case Equation (9), together with its boundary conditions (11)or (12) and (13), gives a strong non-linear complex differential equation, which makes the exact solution extremely difficult, even if a numerical method is used where iteration is involved. However, from the view-point of engineering application it is generally unnecessary to get the exact solution requiring extensive computations. Of importance is that we obtain a reasonable estimate for the ER response with an acceptable accuracy. Once we have the local field distribution between the particles, we can easily calculate the attractive force between the particles, the shear yield stress and the current density flowing through a chain of particles or the ER suspension.
2.2 Conduction model for nearly-touching spheres In the following we will discuss only dc electric field except for a special note. Felici, Foulc and Atten [27] divided the region between two nearly-touching spheres into two distinct zones: (a) the "contact zone" and (b) the "non-contact zone". If the radius of the "contact zone" is denoted by 6, non-ohmic conduction of the host oil is considered in the "contact zone"(x_~6) and equipotential is assumed in the "outer zone" (x>6). The distance 6 is obtained by setting the conductance of the solid sphere equal to the conductance of the host oil. This gives
724
(a/5)ln(a/5)- F/~,
(14)
where a is the sphere radius and F=~p/C~f(0) the ratio of the conductivity of the sphere to that of the oil at low field. They then calculated the total axial attractive force between the two spheres for low applied electric field to be fa = 4ua2~fF2Eo2/[uln(a/5]-
(15)
Equation (15) gives a quadratic dependence of fa on F and E o. At high electric field when x_~5the enhancement of the conductivity of the host oil becomes significant because of the non-linear electric field dependence of the oil conductivity; see Equation (8). The axial attractive force in this case becomes
fa-- 2ua2efEcEo{ln[(10F/u)(~]2 Eo/Ec ]}2.
(16)
Equation (16) gives an approximately linear dependence of the force on the applied field and only a weak dependence on the conductivity ratio F. In a more simplified estimate they give the attractive force fa = 2ua2~f'EcEm,
(17)
where E m is the saturation field in the host oil due to the strong nonohmic behavior of the oil. They estimate that E m ranges from 30kV/mm to 40kV/mm for the mineral oil they used. Thus, they concluded that for two nearly-touching spheres, under low dc applied field, the attractive force is proportional to Eo2, while under very high dc field the force is proportional to the applied field Eo. Their analysis gave qualitative agreement with force measurements on two nearly-touching, large scale, semiconducting half-spheres (a=7mm). It should be pointed out that although their model and experiment show that when Eo>lkV/mm the dependence of the attractive force on the electric field is linear, the shear yield stress does not have a linear dependence on the field. This is because when a chain is sheared the particles in the chain will separate and thus are no longer nearly-touching (see section 2.3). The saturation field E m in the host liquid is an important parameter in the non-ohmic conduction model. Wu and Conrad [30] give the following empirical equation as an estimate of the saturation field
725 F o Ec E ~ / E o - 30(~) "1(~0-0)09
(18)
and Davis and Ginder [32] give for the radius of the saturation region of two nearly-touching spheres 5/a = ~]2Eo/Em
(19)
Felici, Foulc and Atten [27] gave the following expression for the relationship between the axial attractive force fa and the current I passing through the spherical particle:
fa = 4~;a2~p2Vo 4/I2,
(20)
where Vo is the potential difference between the two adjacent halfspheres. Qualitative agreement with Equation (20) was obtained for experiments on polyamide half-spheres (a=7mm) in mineral. Although the conduction model for two nearly-touching spheres presented a new concept for ER response, it could not predict the shear behavior (i.e., the shear yield stress and the shear modulus) of a general ER suspension, which is of interest in engineering applications. This question and the approach will be addressed below
2.3 Conduction model for separated spheres The strength of ER suspensions can be understood in terms of the static electric interaction between the particles. A single-row chain of the particles is the most basic and the simplest structure in ER fluids. Hence, most of the theoretical studies on ER response [6-12,29-37] are based on an analysis of the mechanical and electrical properties of a single chain. Two geometric arrangements for the single chain are used: (a) the chain is parallel to the applied field and (b) it makes an angle with respect to the applied field. Klingenberg and Zukoski [7] used the geometric arrangement (b) and gave an exact analysis for the restoring force of the two-sphere system. Most studies, especially in the nonohmic conduction models, consider the arrangement (a) to obtain first the distribution of the local field and then to estimate the shear modulus and the shear yield stress when the chain is sheared. Method (a) is simpler than (b) and has an acceptable accuracy [42]. All of the work on the conduction model covered in this section uses this method.
726
Assume that with application of a dc field E 0 the chains formed in an ER suspension consist of spherical particles with radius a and are distributed uniformly between the two electrodes. Due to the symmetry we need consider only two half-sphere neighboring particles as shown in Figure 1. The following two approximate equations then apply [29,30,34]: OpEp(x)= of(E)E(x)
(21a)
[R-h(x)]Ep(x) + h(x)E(x) - RE o
(21b)
where R=2a+s is the distance between the two half-spheres' centers, h(x) the gap between the two half-spheres at any location x, Ep(x) and E(x) the local field in the particles and in the host oil, respectively. Equation (21a) is the continuity condition of current density and Equation (21b) that of the potential. Wu and Conrad [36] have shown that Equations (21a-b) give a good estimate of the local field distribution.
of Ef
- -~'I h(x) _Io I
(a)
eo
(b)
Figure 1. Schematic of the conductivity model for two separated spheres [29,30]: (a) The geometry and symbols employed; (b) an area element in the horizontal plane, of is the conductivity of the host liquid, Op the conductivity of the particles, Ef the dielectric permittivity of the host liquid, epthe dielectric permittivity of the particles, a the radius of the particles, s the separation of the particles, and h(x) the gap between the two particles at location x.
727
It was found that most of the current passes through the contact zone, which dominates the main behavior of an ER fluid. Good agreement occurred between the predicted and experimental data [21] for the effect of applied electric field on the quasi-static yield stress XE of a zeolite/silicone oil suspension. Based on the work of Felici, Foulc and Atten [27], Davis and Ginder [32] predicted the shear modulus of an ER suspension with non-ohmic conductivity at dc field to be G= 3EoKf0EoE m
(22)
and the shear yield stress 4
1
XE = ~EoKfOE ~ . 5 ~
(23)
where ~ is the volume fraction of the particles in the suspension. Wu and Conrad [30] derived the saturation field Em in the "contact zone" to be that given in Equation (18) (note: they used the conductivity definition o=j/E, which will be used in the remainder of this chapter except where noted otherwise). Outside the "contact zone" they used the equipotential assumption. Figure 2 shows the maximum local field in the liquid layer Ef=E(x=0) versus the separation of the particles. It is clearly seen how 104 ~u~otential: 103 ~-
E/Eo:I+I/S
~
A=0.007
Ec:O.ZikV/mm
=. o
r.l.l 10 2
101
; Eo(kV/mm)= ~ :
10o ' 10-4
:
;4
-'
r=lo 7 " '
. . . . . . . . .
~
,
,A,,,,I
,
10-3
,,
,,,,,I
,
10-2
,,
,,,,,I
10-1
,
,,
,,,,
10 0
S=s/2a Figure 2. The ratio of the maximum local field in the fluid layer to the applied field v s the normalized separation for different applied fields [30].
728
the saturation field varies with the normalized separation S and the applied field E o. Figure 3 shows the normalized attractive force F (f~=na2~oK~o2F) between the two particles. If Em/Eo
(24)
If Em/Eo>I+I/S , a saturation field occurs in the liquid layer and the normalized attractive force is F m -- 66(F/A)~ l(Ec/Eo )n
(25)
in the range ofEo=l-10kV/mm , Ec=0.001-1kV/mm , A=0.0005-0.5 and F=103-109. We have n=0.92178Ec ~176 when Ec=0.01-1kV/mm and n = l when Ec=0.1-0.3kV/mm. Therefore, the general form of the normalized force can be written as F = min.{Fm, 0.955/S}.
(26)
The shear stress of an ER suspension is given by 3 (~oK~o2F(?) ~
lO~
(27)
~ " ' '
'''"'1
' '
]o3 ~102
:: -_
101
-
10 0
I/
10-4
'''"'!
'
'
....
"1
' '
'"'~"_~
F=]O 7 A=0.007 ~ Ec=0.21kV/mm ~ j F=fa/[:n;eoKfEo 2a21 1 . "~ =0.955/S
,
,
lO
, ~,~,,I
10 .3
..... ,
n ......
I
10 -2
,
,
, , ,,,,!
10-1
. . . . .
]00
S=s/2a
Figure 3. Normalized attractive force between two particles normalized separation for different applied fields [30].
US
729
The shear yield stress can then be obtained by maximizing Equation (27). Wu and Conrad [30] give the following estimate for the relationship between the saturation field and the saturation radius for any separation of the spheres: Em/E o - (l+S)/[S+1_41_(5/a) 2 ].
(28)
When 5/a<<1, we have (29a) or
(5 = ~
a~(I+S)Eo/E m -S
(29b)
For nearly-touching spheres (S--0) (5 = a%]2Eo/~,
(30)
which is same as that given by Davis and Ginder [32] for two nearlytouching spheres; see Equation (19). The average current density of a single chain is a
If a saturation field occurs in the liquid layer a
For two nearly-touching spheres a
J : ~of(~)Eo + ~ rxE(x)of(E)dx.
(33)
0,.6
If extreme saturation occurs in the liquid layer, the current density is approximated by J = 2(~f(Em)Eo
(34)
730
with an error of less than ~10% compared with Equation (32). Equation (34) gives a reasonable estimate for the current density of a common ER fluid chain. For an ER suspension with particle volume fraction ~, Equations (31-33) should be multiplied by 3~/2 for an ideal arrangement of the particles. Recently, Atten et al [31] presented an extension of their earlier conductivity model [27] to the case of small separation between two half-spheres (a=7mm) in the saturation regime of the liquid phase. However, their prediction does not show good agreement with experimental results; see Figure 4, which includes the prediction by the conduction model of Wu and Conrad [30]. The conduction model by the latter authors shows better agreement with the experiments than Atten et al's model. The non-ohmic conductivity parameters of the oil employed in Wu and Conrad's calculation were obtained by fitting the experimental data of Atten, Foulc and Banqassmi [43] to Equation (8), which gives of(0)=3xl013S/m, A=1.35 and E~=l.49kV/mm. The dielectric constant of the oil is Kf=2.2 and the conductivity of the particles is o p= 1.7xl0SS/m. Figure 5 further compares the dependence of the measured attractive force on the applied field of two half-spheres [27] and that predicted by the non-ohmic conduction model [30] and the polarization model [8]. Again the non-ohmic conduction model shows better agreement with experiment than the polarization model. At low dc field and large separation the attractive force is proportional to the square of the applied field, and at high dc field or small separation it is approximately proportional to the applied field. This behavior is in good accord with that predicted by the non-ohmic conduction model. Davis [33] uses an integral equation method to calculate the attractive force between two conducting spheres suspended in a non-ohmic oil and gives a prediction similar to that of Wu and Conrad [30]. However, his polarization model gave a quadratic dependence of the attractive force on the applied field. It should be pointed out that the non-ohmic conduction models [29.30,32,33] predict that the shear yield stress of ER suspension is proportional to approximately Eo 15 at dc applied field ranging from 0.5kV/mm to 5kV/mm, which differs from the prediction for the axial attractive force between particles. This is because the yield stress in shear occurs when the particles are separated by the shear, with the result that the saturation region of the local field in the liquid between the particles decreases compared with the case of nearlytouching particles. As mentioned above, one of the most important differences between the non-ohmic conduction model and the complex polarization model is that the former predicts that the shear yield stress dependence on the
731
1.2 Atten et al [31] Wu & Conrad [30] 0.8
Z
Experiment [31]"
0.6
9 it 9
0.4
@
1.357kV/mm 1.071kV/mm 0.714kV/mm 0.357kV/mm
0.2
0
100
200
s
300
400
500
(/~m)
Figure 4. Comparison of the measured attractive force [31] vs the particle separation for two polymer half-spheres (a=7mm) in mineral oil Elf-T50 with that predicted by the non-ohmic conduction models" ~ Wu and Conrad [30]; ...... Atten et al [31].
101
'
!
I
I
I
I
I
I rl
I
~
Wu & Conrad [30]
I
I
I
I
~
~~162
100 -
..'"
:_
-.
Z 10-1
--- :-s:O 1
lOCi 10-3 0.1
.
.
.
.
.
"ill
;
~
1
I
I
I
I
,
i
i/
10
E o (kV/mm)
Figure 5. Comparison of the measured attractive force for two nearly touching polymer spheres of radius a=7mm (s=0) and separated spheres (s=100ttm) in mineral oil Elf-T50 [31] with that predicted by the nonohmic conduction model of Wu and Conrad [30] and the polarization model (t$*=t$c=1) of Chen et al [8]: s=0; ...... s=100ltm.
732
applied field is ~Eo 15 while the latter predicts the dependence to be E02. Davis and Ginder [32] provided a qualitative comparison for several model ER suspensions. Most of the suspensions show t h a t the shear yield stress dependence on the dc applied field is ~Eo 15. Of great interest is the effect of temperature on the ER response of water-activated ER suspensions shown in Figure 6. The non-ohmic conduction model [30] gives a reasonable prediction of the ER response in the temperature range 20--160~ [14]. It is proposed [30] t h a t the peak in the ER response at about 100-110~ is due to the combined effect of the temperature p e r se plus its influence on the water content in the host oil. The conductivity of the oil decreases with decrease in water content, but increases with the temperature. As a result, the host oil exhibits a strong non-ohmic behavior at low temperature (high water content) and high temperature (>ll0~ In the contrast, the particles may loose some of their water content at high temperatures (i.e., >100~ giving a decrease in their conductivity opposite to the temperature effect on the oil conductivity. The combined effects of temperature and water content in both the particles and the host oil give the highest conductivity mismatch for the ER suspension at about 100-110~ It should be pointed out, the above non-ohmic conduction models predict that the interaction force between the particles increases continuously with the conductivity ratio of F-Op/Of(0). However, 800. . . . .
I ....
I ....
Zeolite/Silicone 0il ~=0.28
600 " W=6wt.%
400-/ / ~ 9 / ~
-
0-,
~1~ ~ ) l,
0
2kV
//"
" : 200 " -
t .... Open" Exp. Filled: Pred.
,
I , , , ,
50
./A
A
"'A A
A
\ .--'~
~, "~
~
Eo=lkV/mm u"O.. I , , ,
100 T (~
,
I
150
I
I
,
,
200
Figure 6. Comparison of the measured shear yield stress dependence on temperature for a zeolite/silicone oil suspension and that predicted [30].
733
experiments [44] show that the interaction force first increases, reaches a maximum and then decreases with F. Boissy, Atten and Foulc [45] and Wu and Conrad [46] found that when F is small the ER response is negligible and when F<
2.4.1 Low conductivity particles with a conducting film For low conductivity particles with a conducting film, the current passing through the particle mainly flows along the surface film. Felici, Foulc and Atten [27] have given an analysis for two nearly-touching dielectric particles with a conducting surface film. Tang, Wu and Conrad [49] developed a conduction model for dielectric particles with a conducting film. A schematic of their arrangement and conditions are shown in Figure 7. Taking glass spheres with a water surface film as an example, they give that the dependence of the axial attractive force versus the applied field is quadratic at low applied field or with large separation between the particles, but becomes linear at high applied field and small separation. This result is similar to the prediction for
734
y
I
I
h(x) E0
/xL
r
Figure 7. Geometry of two low conducting particles coated with a conducting surface film. o = conductivity, E - dielectric permittivity, the subscripts p, f and I denote the particle, host fluid and the surface film, respectively, a - radius of the particles, s - the separation the particles, R = the distance between the two sphere's centers, and h(x) - the gap between the two particles at location x. bulk conducting particles suspended in a weakly conducting oil presented in section 2.3. They further found that increasing E c gives a strong ER response, which is the same as indicated in Equation (25) for bulk conduction in the particles. Although the current density passing through the chain increases rapidly with increase in water film thickness, the attractive force does not increase as much. This indicates that a thick water film is unnecessary. The role of a conducting surface film on a dielectric particle is mainly to change the conductivity of the particle. Both experimental observation [50,51] and theoretical considerations [49] exclude the socalled water-bridge assumption [52], which proposes that the interfacial tension between the water and host fluid provides a source of the shear resistance in a water-active ER suspension. For example, in experiments when Pda=2.1 (the radius of the particles a=100-110ttm), the separation of the particles s=10ttm. This separation is 3 orders of magnitude larger than the water film thickness (~220/k) and thus no "water-bridge" could form between the particles. Nevertheless, the measured attractive force between the particles is still considerably large (see Refs.[49,51]).
735
In Figure 8 the experimental measurement of the current density for a single-row chain of humidified glass spheres (200-220/zm dia.) in silicone oil [50,53] is compared with that predicted by the conductivity models [49,30]. Rather good agreement occurs between the measured values and those predicted by Tang, Wu and Conrad [49] for dielectric particles with a conducting film (absorbed water on their surface). In Wu and Conrad's conduction model [30] a composite conductivity of the particles Op=8XlOsS/m was used, which also gives a good agreement with the measured result. Figure 9 compares the measured current density vs shear strain for a single chain of glass spheres in silicone oil with those predicted. Again good agreement occurs. A comparison of the measured shear yield stress of the single chain with that predicted by the non-ohmic conduction models [30,32,49] and the polarization models [7,54] is given in Figure 10. The non-ohmic conduction models predict the shear yield stress to be the ~l.5th power of the applied field, whereas the polarization models give the field dependence to be quadratic. In the polarization models, we assumed that Op>>Of, giving the conductivity polarizability at dc field f~= 1. Figure 10 shows that the non-ohmic conduction models predict the shear yield stress better than the polarization models. 100
_
,.
i i[lll i ! . . . . . . .
i
i
!
i
:
,
[
i
1
,
1
Experiments.: -
,
.
1
o ~ A
~_,1~ 0
_T
T
i
i.t
t
,
|
3 Spheres ~,~[ 4 Spheres ............................... ~ ~ ~ 5 Spheres ~ ~ ! -
! i ii I '!-...........J..........i-~-i-i~--i--i-'-'~-i -!....!---i......Theoreticati _ T h e o r e t i .......... .c...a tii-....... i i--i I 01
0.20.3
i ~. ~ i i
i
~
~
..... Tangetal[49] -----Wu & Conrad 13(
:
'
0.5 E o
2
'
'
3 45
(kV/mm)
Figure 8. Comparison of the measured current density of a static single chain under de field [50,53] with that predicted [30].
736 51_
. . . .
I
. . . . . . . .
~.~_. ~"
4 1: ~
~ ................i............J 0 \ I
1
. . . .
- .....
o
-I
3 Spheres ]-]
i
iA
i ......; ................................
:
~.,...................... !...................... i ....... o .........
o ?ii i,, i, :i i i i i 0
I ....
A 4Spheres/:t 0 5Spheres1
~~i
,~ 2 *'~
I
_Expefimen.tal_ : :._ _~
0.1
0.2
,,1
0.3
0.4
0.5
u Figure 9. Comparison of the measured current density of a sheared single chain of humidified glass beads v s shear strain with that predicted by non-ohmic conduction models under dc field" 9 Tang, Wu and Conrad [49]; Wu and Conrad [30]. I
L
:
1
1
!i
J
9 Experiment [51,55] J /, ~ ~ ......... Wu & Conrad [30] J ~Z'.,~";" E] " Tang etai [49] [IIIIZ]III]I]I::]I::I~]:I]L~~I!!!-_ [i:J - Davis & Ginger [32] J:::::::::::::::::::::..~ ~:::::::::[---J -- ---Klingenberget al [7] [........i - ~ .....!.............Z k if!___.~..- .- --Conrad et al [54] ..........!2~iiiiiiiiiiiiiii~iiiiiii--
.......i:i_..:i:!i:i::::ii:ii~ii!:i::::::i~iiii:iiliZ.ii~:i:2:1111!iiiiii:i~ii:ii:ii _:_:.i_i 0.1
0.6
~
,J'2" 0.8
i
1
2 E
o
3
4
( k W m m )
Figure 10. Comparison of the measured shear yield stress of a single chain of humidified glass beads [51,55] with that predicted by the nonohmic conduction models: Tang, Wu and Conrad [49], ..... Wu and Conrad [ 3 0 ] , - - - - D a v i s and Ginder [32]; and the polarization models: Klingenberg and Zukoski [7], Conrad, Chen and Sprecher [54]. In the predictions by the polarization models it is assumed that *= 13c=1, i.e., Op>>Of
737
2.4.2 Highly conducting particles with a low conductivity film For highly conducting particles with a weakly conducting surface film, the current first passes across the surface film and then continues in the highly conducting bulk (core). Wu and Conrad [34] developed a model for such particles. They define a combined conduction parameter A
a OI
= 6~f(0) ' where 6 is the thickness of the surface film, oI the
conductivity of the surface film and gf(0) the conductivity of the oil at low field. Taking silicone oil as the host liquid (with of(0)=2.4x1012S/m, Ec=0.21kV/mm, A=0.007), they found that increasing A (i.e., increasing o I or decreasing 5) increases the ER response under dc applied field. If the surface film is an ideal insulator and thick (i.e., ai=0), there is no ER effect under dc field. If 5 is too thin, electric breakdown will occur in the surface film when the applied field exceeds some value, which was observed in the experiments [18,34]. Therefore, a reasonable thickness of the surface film is desired to avoid breakdown. The shear yield stress, shear modulus and the current density of an ER suspension with a volume fraction ~ of particles suspended in silicone oil were determined to be given by the following empirical equations: T E (Pa)
= 19r
G ( P a ) - 55r J(pA/cm z) - 6xl 07r
(35) nG 1 22c~f(0)AnJ
(36) (37)
for the range 1010 12 and lkV/mm<Eo<4kV/mm, where nG=l.55Eo ~176n~=0.78Eo ~176 (E o is in kV/mm). Finally, we will compare theoretical predictions with the experimental results by Inoue [18] on a suspension in silicone oil of dried spherical aluminum particles with a so-called "non-conducting" thin oxide skin. The average diameter of the particles was 20#m with various thicknesses of surface film. With a dc electric field only the ER response for a surface film thickness 6=0.2ttm was reported. The reported composite conductivity of such particles was 101~ which gives A=83.3 for 5=0.2ttm. For the aluminum core we will take Fp-1018. The silicone oil used was dimethyl silicone oil with 100cst viscosity at 25~ Inoue [18] did not report the non-linear conductivity parameters for the
738
silicone oil. Here we will use the conductivity parameters of Dow Corning 200 silicone oil (50cst at 25~ determined by Wu et al [50]. Figure 11 compares the predicted shear yield stress with that measured [18]. Also shown are the analytical results by Davis [17] based on the complex polarization model. Good agreement occurs between the measured yield stress and that predicted by the non-ohmic conduction model of Wu and Conrad [34], whereas that predicted by complex polarization model of Davis [17] is about an order of magnitude lower. It should be pointed out that Davis [17] assumed that the effective conductivity of the host fluid is 108-107S/m, and thus gives ~*=~o=-1/2 (oi<
1;EOc EO 1"46
.
l02
11~ lO 1
_. . . . . . . .
#
. . ~ f f
10 o
10-1
I
--
1
Experiment [18] Wu & Conrad [34] .... Davis [17] (~c=0.5) . . . . . . . Davis [17] (~c=l) I
I
I
2
3
4
5
E o (kV/mm)
Figure 11. Comparison of the predicted shear yield stress [34] with that measured [18] for a suspension of oxidized aluminum particles in silicone oil (~=0.2) under dc electric field. Also shown are the analytical results by Davis [17] assuming ~*=~c=0.5 and ~*=~c=l (the upper-limit), respectively. The average particle diameter is 20;um and the surface film thickness 5=0.2/1m.
739
~c=1/2, which is still smaller than the measured value. Furthermore, the complex polarization model predicts that the shear yield stress is proportional to E02 not ~Eo 15, the latter being predicted by the nonohmic conduction model and supported by Inoue's experiment. Of interest is that Davis' analysis [17] provides a good prediction for the shear yield stress with ac field. The reason for this may be that with ac field (especially at high frequency) the conductivity effect will be weaker than the dielectric effect. The dielectric constant of the materials employed does not usually change considerably with the electric field and thus Davis' linear analysis seems more reasonable for ac field than for dc field. A detailed study of the frequency dependence of the ER response will be given in sections 3.2 and 3.3. Figure 12 compares the predicted yield stress with that measured [34] for ER suspensions of silicon particles (l~5~um diameter) with an oxide surface film in Dow Corning 200 silicone oil. The conductivity of the silicon is 4.35x104S/m [56], giving Fp=l.83x10 s. The conductivity of the oxide film was taken to be oi=4.8x10llS/m. The average diameter of the particles is 2.5~um and the surface film thickness 5=0.2;um. Good agreement between the predicted and measured values of the yield stress is indicated in Figure 12. Figure 13 compares the effect of the 100
,
,
,
,
,
,
rr
I
I'
",
,
,
,
,
,,_ _
- O x i d i z e d S i / S i l i c o n e oil _
6=0.2/~m
r
J
_
10 Exp. Pred. I
I
0.1
,,
I
I
I
i
I
i
i
]
I
1
_I
I
I
I
i
i
lO
E o (kV/mm)
Figure 12. Comparison of the predicted shear yield stress vs. applied dc field with that measured for a suspension (r of oxidized silicon particles in silicone oil. The particles are of average diameter ~2.51~m and coated with an oxide skin of thickness 5=0.2~um [34].
740
5o
i
i
i
i
i
i
i
i ]
Eo=lkV/mm
40 -
"~" 30
o Exp. --- Pred.
Oxidized Silicon/Silicone Oil,=0.23 :
o
20
10 ,
,
,
,
,
,
0.01
,,!
,
I
I
I
I
I
0.1
I
I
1
(t,m) Figure 13. Comparison of the predicted effect of the oxide film thickness 6 on the yield stress with t h a t measured for a suspension (r of oxidized silicon particles in silicone oil with applied field E o - l k V / m m [34]. 10-1
_-
1
i
!
i
i
i
i
I
]
- Oxidized Silicon/Silicone Oil - ~=0.23, 6=0.05ym ,~, 10-2 - T = 2 3 ~ : _ 1 0 -3
A A
9
10-4
A A
/x
e,i
o
F
10-5 0.1
I
I
0.3
,,
I
I
,
,
,
!
1
,
3
E 0 (kV/mm) Figure 14. Comparison of the predicted and measured current densities of a suspension (~=0.23)of oxidized silicon particles in silicone oil. The averaged particle diameter is ~2.5/tm and the surface oxidized film thickness 6=0.05/tm. o Experimental data without electrical breakdown in the suspension; h Experimental data after several breakdowns; Predicted by the non-ohmic conduction model [34].
741
thickness of the oxide film. Again, good agreement between prediction and experiment occurs. Figure 14 compares the measured and predicted current densities for the suspension of oxidized Si particles (5=0.05ttm) in silicone oil. When no electrical breakdown occurred in the suspension, the predicted and measured current densities are in reasonable agreement. But after several tests in which breakdown occurred, the measured current density became an order of magnitude higher than predicted. In contrast, a few breakdowns had almost no effect on the yield stress. The breakdown field of a suspension of oxidized conducting particles depends on the dielectric strength and the thickness of the oxide surface film. The thicker the film, the higher is the breakdown field. After several breakdowns, many of the particles no longer have a satisfactory film to provide the desired degree of "insulation". These particles are then responsible for the observed dramatic increase in the current.
3. C O N D U C T I O N AND DIELECTRIC R E S P O N S E U N D E R AC FIELD
EFFECTS
IN
ER
In section 2 we mainly introduced the non-ohmic conduction models which apply to ER response activated by a dc field. We also gave the framework of the polarization models which to-date have been applied to two limiting cases: (a) high frequency ac field (dielectric polarization) and (b) dc or low frequency ac field (conduction polarization). In this section, we will present investigations into the combined effects of conduction and dielectric properties on ER response at any frequency. Electric polarization consists of three components [57]" (a) orientational, (b) atomic and (c) electronic polarization. Each corresponds to the motion of different kinds of particles (molecules and ions, atoms, and electrons, respectively) with different characteristic times. Thus, as the electric field frequency increases, non-equilibrium effects will appear successively in the different components of polarization. The motion of molecules and ions in weakly conducting materials will lag behind changes in electric field in the frequency range of 102-106 Hz. However, atoms can follow the field at characteristic times of the order of 10-14s. At very high frequency the field may change so fast that even the electrons cannot follow. Since in ER suspensions the interested frequency range is usually less than 104 Hz and only few materials exhibit a Debye-type dielectric relaxation in this frequency range, we will not include the Debye-type effect in the following calculations. However, if we know the frequency dependence of the
742
dielectric constant or the conductivity of the materials, it can be easily considered in the calculations. 3.1
Complex
polarization
model
For the simplest analysis of the combined conduction and dielectric effects in ER fluids under steady ac field, the complex polarization model based on the point-dipole approximation gives a reasonable qualitative prediction for the ER response [15-17,58]. As mentioned above, these studies predict that the interaction between the two particles is proportional to the square of the complex polarizability [3*. Tang and Conrad [58] have calculated the I~*(r versus the frequency for conducting particles suspended in a host oil with ohmic conduction and showed that ER particles with high conductivity ratio Fo=Op/Of and low dielectric ratio FE=Kp/I~ will have a high polarizability at low frequency fields, which however decreases rapidly beyond a critical frequency. In contrast, ER particles with low Fo and high F~ have a low 113"(~o)12 at low frequency fields, but a higher 113"(co)12 at high frequency. lO 4
10 a 10 2
:
1 01
10 o lO o
10
--"
lO
far
lO ~
lO2
lO3
l~
Figure 15. Calculated diagram of active and inactive ER response regions for several normalized electric field frequencies ~2. For a given frequency the region to the left of the curve is the inactive ER region. Region I is the inactive region; Region II the conductivity domain, which is only active below a certain frequency; Region III the dielectric domain, which is only active above a certain frequency; Region IV the active ER effect region independent of the frequency [58].
743
According to the complex polarization analysis [58], the strength of ER fluids increases with increase in the I~*(co)12. If we empirically take I ~*(o))12 -0.5 as the boundary between an "active" and "inactive" ER effect, we obtain a diagram which defines the active and inactive ER response regions; see Figure 15. The critical frequency at which 113"(00)12 =0.5 is
where fc has a solution only in two cases: (a) when 08.243 or (b) 08.243) is conductivity dominant, where ER fluids only show a strong effect below a critical frequency fc" Region III (08.243) is dielectric dominant, which becomes active when the frequency exceeds f~. Region IV (Fa>8.243 and F~>8.243) is always active independent of the frequency. The above analysis is based on the assumption that the particles have the same dielectric and conduction properties in the bulk and on the surface. The polarization process becomes more complicated with surfactants around the particles [15] or composite particles, e.g., highly conducting particles coated with a thin dielectric surface film [17,18,34] or dielectric particles with a conducting film [49]. Tang and Conrad [58] also calculated 1~*(o))12 in the frequency domain for several conducting surface film thickness ratios. In a special case a new type of frequency response for the dielectric particles with surfactants was found, which can not be obtained by bulk polarization alone. Although the complex polarization analysis is supported qualitatively by experimental observations [15,20], it does not give a reasonable estimate of the magnitude of the shear yield stress of ER suspensions. In the following we will introduce a simplified analysis for the ER response considering both the dielectric and conduction effects under ac applied field, which gives better agreement with experiment.
744 3.2
Simplified
analysis
3.2.1 Analysis method Consider a single chain with an infinite number of spherical particles aligned in the direction of the applied electric field and surrounded by a nonpolar liquid. Two half-spheres in such a chain were shown above in Figure 1. Assume that the time variation of the applied voltage is V(t)=V 0cosc0t=Re(V0 ei~t)
(39)
where co is the angular frequency of the applied voltage and t time. Let E* denote the complex amplitude of electric field. Referring to Figure 1, along the y-axis (x=0), we have the following approximate equations with ac field [41] 2aE*p + sE*f= V0,
(40a)
O p E p * - O'rE*f,
(40b)
so that ~ _ (~176 E*f = 2ao,f + SO,p - 2a(of +ic0~f) + s(o +ic0e ) p
E* -
~176
P
2ao*f + SO*p
-
=
P
(~176163176 2a(of +ic0s + s(o p +ir163
(41a) (41b)
When c0=0 (dc voltage), O*p - o and o*f = of, the fields are induced by P the particle/liquid conductivity mismatch and Equations (41a-b) become
E*
~176
f-
2aof + SOp
E* -
~176
p
2aof + SOp
-
(42a)
(42b)
745
which can be also obtained from Equations (21a-b) when x=0. When o) is very high or o p =of =0, the field is determined by the particle/liquid dielectric mismatch and Equations (41a-b) become epVo E*f = 2aef + Sep
(43a)
E* = efV~ P 2aef + Sep
(43b)
The complex amplitude of the current density is j*= c~*pE p = o*fE*f =
~176 2ao*f + SO*p
(44)
Let a-2~eo/Of, f]=afKf (here f is the frequency of the applied electric field) .
Fe=Ep/~f,Fo=Op/(3f(0), (1 + S ) E o
E f
:
o,f/o,
+
s
=
S=s/2a, we then obtain
(I+S)E o ~+ if] +S F o + iQF~
(45)
where S=s/2a is the normalized separation of particles, Eo=Vo/(2a+s) the amplitude of the applied electric field, and ~ a non-ohmic parameter for the host oil. For ohmic conductivity of the host oil -
1,
(46a)
and for non-ohmic conductivity - 1 - A + Aexp( Exfi~-~c ),
(46b)
where E is the rms value of the local field in the oil. Equation (47) is a complex equation which can be solved by a computer. The electric field in the host liquid layer is Ef(x,t) - Re(Ef*e i~~ = ~E(x)cos(o~t+0 E (x))
(47)
746
IEf*l
where E(x)= ~r~ is the rms value of Ef(x,t) and 1 Im(Ef*) 0E = t a n [ ~ 1
(48)
is the phase angle shift from the applied field and is a function of the location x. Outside the contact center (x>0), an equation similar to Equations (42a-b) can be used to estimate the local electric field distribution between the two adjacent particles. However, a and s in Equations (4247) should be replaced by a'=a~/1-(x/a) 2 and s'=s+2a[1-~/i-(x/a)2]. Further, the normalized separation S should be replaced by S'=(S+I)A/1-(x/a) 2 -1. This simplified analysis gives an estimate of the local field with good accuracy [36]. Having obtained the distribution El* (x) of the complex amplitude of the local electric field, we can get the complex amplitude of the current density 9
$
$
$
(49)
J(x) =(:If (x)Ef (x).
The current density at location x at any time t is j(x,t) - ~/2 j(x)cos(~ot + 0j(x))
(50)
where j(x)= Ij(x)* I / ~ is the rms value of j(x,t) and
5-
-lr Illl(j (X)*).
(51)
is the phase angle of the current density shift from the applied electric field, which is a function of the location x. The rms value of the average current density along a chain is 2~ r
j = {~_
[ ~:a12
~x~j(x)eos(mt+~(x))dx]2dt}l/2
(52)
747
The attractive force between two particles is given by a
fa(S,t ) = ~soKf 1 f 2 ~ IEl(X)* 12cos2(eot+0E(x))xdx = e 0 g a 2 I ~ ~ 2 F(S,t),
(53)
where the normalized force F(S,t) is 1
F(S,t) = [ 2(E(~)/Erm~)2 COS2(cot+0E(~))~d~.
(54)
Here ~=x/a and E ~ is the rms value of the applied electric field. The rms value of the normalized attractive force is tl
O
Fr~s =
co ~ F(S,t)2dt
(55)
and the mean value of F(S,t) is 2n O)
Fmean
(9
2~
J F(S,t)dt
(56)
The shear yield stress is given by
Tnm=SfE,.m~ 2 max. (F,.m~(~,)q:+?2)
(57a)
or
TEmean--E:Ferms 2 m a x .
(Fmean(Y)q;~2)
(57b)
In most experiments, the measuring meters give the rms value of the ac signal; but some meters may give the mean value.
748
In a real ER fluid, many chains span the two electrodes, Equations (57a-b) should then be multiplied by the factor 3(]), where ~ is the volume fraction of the particles in the ER suspension. Conrad, Chen and Sprecher [54] suggested that a structure parameter As should also be included if the particles are not sphere-like and the structure consists of clusters of chains. If the chains of spherical particles in an ER suspension are considered to be ideal, single-row chains and to act independently, we can take A~=I. Calculation of the current density of an ER suspension differs slightly with ac field from that with dc field. With dc field the current passing directly through the liquid phase between two electrodes is negligible compared with that passing along the chains. However, at high frequency ac field this current can not be neglected. If the currents passing through both the chains and liquid phases are considered, the current density of the suspension is: 2n 0)
J-(~-
[ 0
I ~
nOx~j (x)cos(~ot+Oj(x))dx a0~3
3 ) ~ j (a)cos(mt+Oj (a))]2 dt}l/2 + (1-~r
(58)
or it can be roughly estimated by omitting the phase angle effects of the current passing through the chains and the pure liquid, giving 2yt 03
j _ ~{ ~ _ 3co
[ a21- u x ~ j (x)cos(o)t+Oj(x))dx]2dt}~/2+ (1_2~)j(a)
(59)
q where j(a) is the current density (rms value) of the pure host liquid under the applied electric field and is given by j (a) = ErmsqOf2+(2nf~:f)2' 9
(6O)
749
3.2.2 Ohmic conductivity of the host oil Figure 16 shows how the ER response changes with the normalized frequency g~ of the applied field for a single chain. If Fa=F, the frequency has no effect on ER strength. If F > F , the conductivity mismatch dominates the ER strength at low frequency of applied field, and if F~
Fmean='~ff2"~Frms 0.48(24 1[3" 12)exp([3'5 ) -
lO3 _=
I
I
I
I
I
I
I
l
l
l
I 1
10 3
l
l
r =ro=l~ l
o'10]
~
=]02 ~ :
7
r~=ro=lO . ~
L s=o.oo5 10 ~ 10-4
9
~ 10-2
i
r~=ro=s i 10 o
i
i
]o2
i
I
lO4
,
M
- r~=}, ro=]'O--.~
i
i
I
10 ]
0o
Q
Figure 16. The normalized attractive force between two particles versus the normalized frequency [35].
750
when [~'2- I~* COS0~[2 < 0.85, where 13 is the real component of 13",and (61b)
F m e a n ~ F r m s - 24 1[~* 12exp(2~ ,32 ) when 13'2 ~ 0.85. Here 1~*12=
(KpKf)2 +(~P'~f)2/(~ e~
(61c)
:)
The mean and rms shear yield stresses of a single chain are predicted by the following empirical equations when ~'2<0.85: TEmean~'I:Enns-
2.61EfErms21 ~* 12exp(~ '12 ),
(62a)
1.1~fEr~2exp(exp[3.1([~'2-0.04)2~
(62b)
and when [3'2zO.859 "~Emean---~3"~Erms=
Boissy et al [23,24] measured the attractive force between two polyamide half-spheres of radius 7mm immersed in the mineral oil ElfTF50. The reported conductivity of the solid spheres Op=l.7xl0SS/m.
]o8
I -
106
- ....
--
~
I
I
I
F~--10
I
I
I
!
I
3, F o - 1 0
- F~-5,
I ,-
I
...-~/
Fo-106
r'~-5, 1"o=10 . . ~ . . ~ s-o.oos
~_..2--:-.~ 10 ~ ~ I I 10-4 10-2
I
100
I
I
102
i
I
104
I
!
106
~
108
f~ Figure 17. The average current density of a single chain versus the normalized frequency [35 ].
751
The conductivity of the oil and, therefore, the conductivity ratio Fo= Op/Of was varied by adding an ionic surfactant (AOT) to the oil. The dielectric permittivity ~_ of the half-spheres decreased with the frequency of the applied efectric field. The dielectric constant of the oil is Kf=2.2. Figure 18 compares the measured attractive forces [23] with those predicted by our model [35]. In the experiments, the exact separation of the two half-spheres was difficult to observe and thus no data were reported on their separation. By taking the normalized separation S=0.005 (R/a=2.01), good agreement occurs between the predicted and measured attractive force of the two nearly-touching polymer spheres. It should be pointed out that in the high frequency regime (f>20Hz) the conductivity ratio has no effect on the ER response. This is indicated by both prediction and the experiments. Furthermore, the theoretical calculation shows that when the normalized separation S<0.01 there is almost no effect of the frequency on the attractive force at high frequency (compare curves a and a' in Figure 18). However, at low frequency the separation of particles has a significant influence on 10 o
10-1<
j ~
( "
dr '
Erms=0.202kV/mm ....~
a
"'-,
10-2 l :2!-! .... ,:, .............
-
S = 0 . 0 0 5 ( R / a = 2 . 0 1 ) -=-
i"',
~d
""'",
-
10-3 ...... 10-4 I _ . . . . . . . . I 0.01
O. 1
Fmean
:.
........ J
........ I
1
10
. . . . . . . ,! 100
....... 1000
f (Hz)
Figure 18. Comparison of the predicted attractive force between two polymer half-spheres immersed in mineral oil (Elf TF50) with that measured (radius of the particles a=7mm, S=0.005, i.e., tUa=2.01). Experimental: open symbols; theoretical: solid curve for Frm~, dashed curve for Fmean. Experimental [23,24]" O Fo=l.4xl04, EIF(j= 1.2x103, A ['0=300, ~ ['0=30. Theoretical [35]: (a) Fo=l.4xl04, (b) Fo=l.2xl03, (c) I"o=300, (d) Fa=30 , and (a') S=0.001, Fo=l.4xl04.
752
the attractive force, especially at a high conductivity ratio Fo. In Figure 18, we give two theoretical curves: (a) S=0.005 and (a') S=0.001 for Fo -1.4x104 and the two dc measured points given in Refs[23] and [24], respectively. The difference between the two measurements for the same condition is apparently from experimental error, which is also indicated but smaller in the high frequency regime. Figure 19 compares the predicted shear yield stresses vs. applied field for three ER suspensions with those measured (open symbols) [38] employing 400Hz ac applied field. Good agreement occurs between the experimental results and the predictions. For the BaTiO3/dodecane suspension, slightly different results in different figures in [38] were given. This may be due to measurement error. Figure 20 compares the predicted and measured shear yield stress dependence on the frequency of the applied field for the BaTiO3/dodecane suspension. Due to the small conductivity and high dielectric constant of the BaTiO 3 particles, there is no effect of frequency in the range from 10 to 4000Hz. This indicates that for the BaTiO3/dodecane suspension the transition frequency between the conductivity and dielectric domains is lower than 10Hz. |
lO00 r -
[
'
,:0.2 f=400Hz
!
i
!
2 BaTiO3 ~ z x SrTiO3
lO0
TiO2
10 Predicted [351 o/x r3 Experiment [38] 1
0.8
,
1
,
:
1
I
2 Eo
~-
,,J
3
I,,,
4
(kV/mm)
Figure 19. Comparison of the predicted (rms) [35] and measured [38] shear yield stresses v s the amplitude of the 400Hz ac applied field for three ceramic particles/dodecane suspensions" TiO2, Kp=70; SrTiO 3, Kp=270; BaTiO3, Kp=2000; I~=2. of=10llS/m were taken.
In all the calculations Fo=10 and
753
103
.......
I
i"
I
........
I
I
,,1r
Predicted [35] Experiment [38] ~" 102 =
O
O
0
0
O
BaTiO3/dodecane ,=0.2 Eo- 1kV/mm
101
100
I
10
I
I I I llJ
I
I
I
I I I lli
I
l
lO3
102
I
_ :" I I I II
lO4
f (nz) Figure 20. Comparison of the predicted (rms) [35] and measured [38] shear yield stresses vs the frequency of applied ac field for BaTiO3/dodecane suspension (Kp=2000, Kf=2, ~=0.2, F~-10 and of=10 llS/m).
3.2.3 Non-ohmic conductivity of the host oil In most of the following calculations we will take Dow Corning 200 silicone oil as an example of a non-ohmic host liquid. Experimental measurements [50] for this oil give the following non-ohmic conductivity parameters: of(0)=2.4x10"12S/m, A=0.007, Ec=0.21kV/mm. In the nonohmic conduction model we can not simply conclude in which case the polarization effect or conduction effect dominates the ER behavior. However, it is found that if F >>FE, the non-ohmic conduction effect will dominate
the ER behavior (please note F
is here defined as
ro=Op/Of(O)>>op/Of(E)). If F < F , the opposite occurs. Furthermore, a high local electric field occurs mainly in the "contact zone" of the two particles, which is same as predicted above. Another characteristic of the local field in a liquid with non-ohmic conductivity is that there exists a saturation regime near the contact zone. Figures 21a-b give the shear yield stress and current density versus the frequency for different values of Fo and F . If Fo>>F e, a higher shear yield stress occurs at low frequency than at high frequency. If Fo
754
about 1.2 times the mean value, which differs slightly from the attractive force. The reason for this may be that the shear strains at which yielding occurs varies with frequency. Of interest is that if the dielectric effect is larger than the non-ohmic conductivity effect the transition regime from conductivity domain to dielectric domain is larger 2.4 2 ~,
1.5 1.6
1 .
(a)
0.5
.
3
-
.
V/~~l
.~ . 3 .
1.2 0.8
k
0 ~
0"4
10-7
10-5
I
I05
~
I
10-3
i
10-1
101
103
f (Hz) I
i
I
i
10 50
I
/
1. F~=5, Fo=106; 2. F~=IO3, Fo=106 3. FE=103,Fo=I(P; 4. FE=103
103 -
(b)
1,2
,~ 101 10-1 _ 10-3 10-5
S=0.005 Erms=3kV/mm
. ~ I
I 10-3
I
I 10-1
I
I 101
I
I 103
I 105
f (Hz)
Figure 21. The ER response in the transition regime, conductivity domain and dielectric domain for a single chain (S=0.005, (a) the shear yield stress and (b) the average current density. (1) FE=5, Fo=106; (2) Fe=103, Fo=106; (3) FE=Fo=103; (4) rE-103, ro=10 [36].
Er~s=3kV/mm):
755
for non-ohmic conductivity of the fluid compared with ohmic conductivity. The reason for this is that if the frequency increases, the dielectric effect increases, i.e., the local field will increase. However, due to the non-ohmic behavior the conductivity of the host fluid also increases. It is the non-ohmic behavior that makes the transition regime from the conductivity domain to the dielectric domain larger than in the case of ohmic conductivity when the dielectric effect is larger than the conductivity effect. Obviously, the stronger the non-ohmic conductivity of the host fluid, the larger such a effect. However, if the dielectric effect is smaller than the conductivity effect (Figure 21a, case 1), the transition regime is similar to that for ohmic conductivity. 10 4
'
F~=5 ..
.-3.5 5E~:~rms
i
I
<102Hz
Fa=10~ df
,~ 103
i
~
1
103Hz =104Hz
(a) -i_--
2 /ECrErms
_lO 1 0.8
,
I
,
1
I
I
I
2
3
4
I
I
'
(kV/mm)
E rm s
lO 4
I
5 6
I
'
I
I
Fo=106 F~= 103
I
'__
_" 103 Hz 50Hz _
dc ~,
10 3
(b)
la
10 2
I
I
0.8 1 Erm s
I
I
I
i
2
3
4
5
(kV/mm)
Figure 22. The shear yield stress (rms value) dependence on the applied field for a single chain: (a) F~=5, F,j=10 a and (b) FE-10 a, I"o=106 [36].
756
Figure 22a shows the shear yield stress (rms value) dependence on the applied field when the dielectric effect is smaller than the conductivity effect (F <103Hz) the shear yield stress shows a Erms2 dependence. It is interesting that, although f=103Hz is in the transition regime (see Figure 22a), the shear yield stress still shows a quadratic field dependence. In contrast to Figure 22a, Figure 22b shows the case when the dielectric effect is larger than the conductivity effect. This gives a weaker ER response at low frequency compared with high frequency. The shear yield stress shows a field dependence of E ~ 151 , E r ~ 153 and Em~ T M for the frequency f=0, 50 and 103Hz, respectively. Figure 23 compares the predicted [36] attractive force (rms value, S=10 5) with that measured [23] on the two 7mm radius polyamide halfspheres in mineral oil with electrode spacing of 14mm. The conductivity parameters employed in the calculation are the same as given in section 2.5.1, i.e., A=1.35 and Ec=l.49kV/mm. The dielectric constant of the particles decreases with the frequency. When the frequency f=10, 50 and 1000Hz, the dielectric constant of the particles is Kp=34, 24, and 18, respectively. It is seen in Figure 23 that the predicted attractive force is in good agreement with that measured. The reason the measured force at low field and high frequency is slightly smaller than predicted is probably that the force was too small to measure accurately. Figure 24 compares the dependence of the attractive force (rms value) on the applied field. With ac fields of frequency f=50Hz and 200Hz respectively, Wu and Conrad [36] predict that the attractive force is proportional to E~m~2, which is in good agreement with experiment. With dc field they predict that the attractive force is proportional to E ~ 13 at S=10 5, compared with the experimental F n ~ E ~ m ~1.6 . If S=10 2 is taken, it is predicted that F r a t E r n a l 4 5 (dashed line in Figure 24), which is in slightly better agreement with experiment. However, with ac field there is no difference between the predicted and measured field exponent for this range of S. The reason the field dependence of the attractive force at de field depends slightly on S is that at high applied field the saturation of local field occurs over a large range of S. Thus over this range the attractive force has almost a constant value. At low field, however, only within small range of S does saturation of the local field
op=l.7xl0-8S/m,of(O)=3xlO-13S/m,
757
10o
" ''"'"1
' ''"'"1
] ''"'"1
- - Pred . [36] 10-1< l
~"
10-2
~-,
10-3 10
~
~
/
,
I , ,,,,,,I
-4
100
10-1
' ''"'"1
, ,,,,,u_
Exp . [23 ] 9 -:_ o O. 1kV/mm A 0.2kV/mm m m :_
,,,,,,,I
,
,
101
,
102
,,,...
103
104
f (Hz) Figure 23. Comparison of the predicted frequency dependence of the attractive force (rms value) between two polymer half-spheres immersed in mineral oil (Elf TF50) with those measured [23]. Radius of the particles a=7mm, normalized separation S=10 -5. Experimental[23]" open symbols; theoretical[36]" solid curves. 10o 10-1
_--
J'"
~
'
~
I
'
~
i
'
-
i
I _-
dr
50Hz
10-2
OHz
10-3 10-4 10-5
I
I
0.06
I
I
I
I
0.1
I
0.3
I
I
I
z
t
0.50.7
Erm s (kV/mm) Figure 24. Comparison of the predicted field dependence of the attractive force (rms value) between two polymer particles immersed in mineral oil (Elf TF50) with those measured (radius of the particles a=7mm. Experimental[23]: open symbols; theoretical[36]: solid curves for S=10 -5 and dashed curves for S=0.001. Note: when f=50 and 200Hz, the solid curves for S=10 5 and the dash curves for S=0.001 come together).
758
occur. Hence, in this case increasing S leads to a small decrease in the attractive force; see the dash line in Figure 24. A more detailed treatment of field saturation is given in Ref.[24]. Figure 25 compares the predicted gain-frequency characteristic of the polyamide half-sphere/mineral oil system and that measured, where I is the measured current and Vrm~ the applied field (rms value). At high frequency the predicted value is in good agreement with experiment, irrespective of whether the non-ohmic conduction of the host oil is considered. However, at low frequency, the prediction using the assumption of ohmic conductivity does not agree well with experiment. Although the predicted value considering the non-ohmic conduction is smaller than experiment, the tendency of the gain to increase with increase in the applied voltage is similar to what occurs experimentally. Possible reasons for this are: (a) experimental error and (b) the conductivity given in Ref.[23] of(0)=3xl013S/m of the oil at low field used in the calculation is a low estimate. Ref.[24] gave of(0)=l.21xl0q2S/m. If we take of(0)=1012S/m, the predicted gain is in good agreement with experiment over the entire frequency range. -120 k
~" - 160 " Pred.[36]: I-Non-ohmic ~/~ 5.6kV "- -200 [--|2.8kV 1~ -240 ~ I" 9~
~,~
_ ~ Exp.[23]: _ O 0.TkV : A 1.4kV 0 2.8kV _
~ ~:""
-28o
10 -4 10-3 10-2 10-1 10 0
101
10 2
10 3
LO4
f (Hz) Figure 25. Comparison of the predicted gain of two polymer particles immersed in mineral oil (Elf TF50 without AOT) versus the frequency with those measured. Radius of the particles a=7mm, the normalized separation S=0.10 "5. Experimental[23]: open symbols; theoretical[36]" solid curves for non-ohmic conductivity and dashed curve for ohmic conductivity.
759
lO3
-
9 BaTiO3/dodecane, ~=0.2, f=20Hz [59] _ 9 BaTiO3/silicone oil, ~=0.1, f=60Hz [39] -
lO2
Wu and Conrad [36]: / - - - BaTiO3/dodecane ./~L" B aTiO3/silicone o i l ~ . '"
101
_
i
..J
i
i
i
,il
0.1
_
l
i
9
i
i
i
J
I J_
1 Erm s
10
(kV/mm)
Figure 26. Comparison of the predicted shear yield stress (rms value) [36] with that measured by Miller, Randall and Bhalla [39] for BaTiOs/silicone oil suspension (f=60Hz and ~=0.1, Kf=2.8, Kp=2000, =10l~ of(0)=2.4x1012S/m, A=0.007, Ec=0.21kV/mm) and by der and Davis [59] for BaTiOJdodecane suspension (f=20Hz and r K~2.0, Kp=2000, ~p=10l~ of(0)=2.4xl012S/m, A=0.007, Ec=0.2 lkV/mm). Figure 26 compares the predicted [36] dependence of the shear yield stress on the ac applied field for BaTiO3/silicone oil and BaTiOs/dodecane suspensions and that measured [39,59]. Again, there is good agreement between the predicted and measured values for both suspensions. 3.3 E R r e s p o n s e s u r f a c e film
of c o n d u c t i n g
particles
with
a dielectric
In section 2.4, we discussed the ER response of particles with a surface film under dc electric field. In this section we will consider the ER response of highly conducting particles with a dielectric film under ac electric field. Inoue [ 18], Yu et al [47] presented experimental evidence that highly conducting particles with a dielectric surface film suspended in a weakly conducting oil show significant ER response. Davis [17] gave a theoretical analysis of this based on the point-dipole approximation. Subsequently, Davis and Ginder [37] proposed the following expression for the shear yield stress at high frequency ac field:
760
3
~/gla
- 2 *Q rm
(63)
2Ef
Wu and Conrad [34] developed a non-linear conduction model for highly conducting particles with a dielectric surface film under dc electric field; see section 2.4. Their recent work [37] shows t h a t this type of ER suspension gives an exciting ER response under ac field if the surface film has a high dielectric constant. For simplicity we will only consider ohmic conduction of the host oil in the following, because non-ohmic behavior is generally not very important at high frequency ac field. Figure 27 shows how the shear yield stress depends on the normalized frequency ~2=2~fe0t~o f. For the general case, the normalized shear yield stress ~E--~E/[E0t~rm~ 2] is given on the right side of the figure. To give some idea of the magnitudes involved, the dimensional shear yield stress is given on the left side for the specific parameters listed (i.e., coated metal particles suspended in silicone oil). It is seen that if the
f~ 20 -
i
,
-
I',
I._lO
-
FoI= 10 2
15-
i
i
l
i
i
'
~
-
Fo=IO~2 - 60
-
Fs=lOl o
_
~,~
_
0
_
40
I[,~
F I=IO 3
5 2 -
80
_
~) 10 -
"~
_ 100
3
FoI=2~
5/a=O.O1
I i I i 1 0 - 5 1 0 - 4 1 0 - 3 1 0 - 2 10-1
1 10 0
I 101
- _
i 10 2
20
10 3
fl=2n'fEf/af Figure 27. Shear yield stress (rms value) of a single chain of highly conducting particles coated with low conducting film vs the normalized frequency. The right ordinate is the normalized shear yield stress YE------TE /[e0KfErms 2] with Fa=(jp/(Jf=1012, F=~p/~f=lO 1~ FEI=ei/~f=103, 5/a=0.01. The left ordinate is the dimensional shear yield stress XE for the special case of coated metal particles suspended in silicone oil with (Jf=2.4xl012S/m, Kf=2.5, Erms=3kV/mm and f=50Hz [37].
761 conductivity ratio F~ of the surface film to the host oil is less than the permittivity ratio F J , the ER strength is higher at high frequency ac field than at dc or low frequency. For the special case of coated metal particles suspended in silicone oil with af=2.4xl012S/m, the transition frequency fc is about 10 -4-10-2 Hz. This transition frequency is proportional to the ratio of the conductivity af to the permittivity ef of the host oil, i.e., fc=ff2c(Jf/(2~tef). If we take of=10SS/m and s163 fc=l 100Hz, which is same as the estimate by Davis [17]. No effect of frequency on the shear strength occurs when the conductivity ratio F~ equals the permittivity ratio F I. However, it should be pointed out that when the ratio FI is high, the local field in the liquid layer between two particles increases. This increased local field will increase the conductivity of the host oil if it exhibits super-ohmic behavior, and the I~
20,
15
'
"', ..... i
........
J
- ~=2897.5 F,pl012 - F=IOI 0
@) 1 0 -
........
l
5/a=0 01 "
80
60
ro~=1.67 -_ 4O I~
5 O
I00
: --:
-
~
-_
-_ 20 ---
1
,
~ , ,,,,,I
'
10
I I tl
~1
100
I
I
~ i~,~l
0
1000
FEI=ei/e f
Figure 28. Shear yield stress (rms) of a single chain of highly conducting particles coated with low conducting film vs the permittivity ratio F~I=ei/ef of the surface coating to the oil. The right ordinate is the normalized shear yield stress YE---'~E][s 2] with ro=(~p/Of=1012, F~=ep/ef=lO 1~ I4o=0i/0f=1.67, g~=2897.5. The left ordinate is the dimensional shear yield stress ZE for the special case of coated metal particles suspended in silicone oil with of=2.4xl012S/m, Kf=2.5, Erms=3kV/mm and f=50Hz [37]
762
real conductivity ratio may be lower than suggested. In this case the non-ohmic conduction effect of the host oil should be considered especially at low frequency. Figure 28 gives the shear yield stress versus the dielectric constant ratio of the surface film to the host oil for a single chain of particles. The yield stress increases with the dielectric constant ratio, which indicates that conducting particles with a high dielectric constant surface film can give an exciting ER response. For example, if the volume fraction of the particles ~=0.4, the surface film thickness 8=0.01a and the surface dielectric constant ratio FEI=I~/~1000, and if we choose a structure parameter As=10 [54], we can get a shear yield stress of 100kPaI Even if we consider the structure parameter A~=I, one still obtains a shear yield stress of more than 10kPa. In their recent work [60], Wu and Conrad give the following empirical equation to estimate the shear yield stress of highly conducting particles coated with a dielectric film suspended in ohmic oil (As = 1):
8 grE
2
x E (Pa) - O.O06+(~FEI)O.6 ,
(64)
where A=5/a and the units of the applied ac field Eros are in kV/mm. Figure 29a compares the experimental shear yield stress [18] for aluminum particles with an alumina surface film with t h a t predicted [ 17,37]. The frequency of the applied field is 50Hz. Davis [ 17] assumed that the surface dielectric constant KI=5 for A1203 , but Randall et al [61] report I~=10, whereas Thurnauer [62] report Ki=8-9. In Figure 29a we give two cases, when KI=5 and KI= 10. Both values give predictions in good agreement with the measurements of the yield stress. Even though the surface conductivity ratio FoI may range from 1.67 to 100, no significant difference occurs in the predicted values of both the shear yield stress and the current density. This indicates that when the frequency f=50Hz the surface conductivity ratio has no effect on the ER response, which is seen more clearly in Figure 27. Figure 29b compares the measured [18] current density with that predicted [37]. Again good agreement occurs. Figure 30 compares the dependence of the measured shear yield stress on the oxide film thickness with that predicted [37,17]. It is clear that good agreement again occurs between the predicted and experimental values. The term "EG yield stress" is defined by Inoue [ 18]
763
to mean that when the surface film thickness is too small a field higher than the breakdown strength of the film material could not be applied. From the analyses by Davis [17] and Davis and Ginder [32] and the calculation by Wu and Conrad [37], the smaller the surface film thickness 6 the higher the shear yield stress. However, a decrease in the film thickness also raises the local field in the film, which increases the possibility of dielectric breakdown. In this case a high field can not be applied, the result being that the ER suspension exhibits a low shear yield stress. This can be seen in Figure 30. According to the above analysis, the surface film should have a high dielectric constant, a high dielectric breakdown strength and a reasonably small thickness. 1000
. . . . ~...... I............................................. ! .......................... ! ........ / ! .
...........
-!!!!!i!!!i~!i!!!!!!!!!~!!!!!!!!!!!~i~!!!!!!!!!!~!!!~!!!!!i!!!!!!!!!!!!!!!!!!!~!!!!!~i~;ii!!!~!~t!!!!!~!~!!!--....
...... i ...... i ...........................................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
" . . . . . . .
i!!!!:::!!!:!:-
! ....... -,~.-.~'-"- ....... i .................. i .............
...... i...... i................................ -/.---~----':~'.. ................................... i .............
:::::::-, i!::!!,'
-~ 100
...... , .............. <...:
t..--
.............. , ............
::::::::::::::::::::::::::::::::::::::::::::: i i:! (a)
::::::::::::::::::::::::
Wu
& Conrad
[371 (K,:5)
...... !...... !...........
Wu
& Conrad
[37]
...... i ...... i...........
i
10
Davis
[17] (K,=5)
i
i
i
I
"
(K,=10)
[-~-
i S=O.O01 ,' 10 ...........!.............................................~.........................................................
!i
(b) i
--
Pred.
1 -!!!{{!!{!!i!!!!!!!!!!!!:!!!!!!!!!!!!!!. -
0.8
1
[37]
(KI=5) [ _J
Pred.[371 (K,:10) [!!]!!!::
2
3
4
5
Erm s ( k V / m m )
Figure 29. Comparison of the measured ER response of oxidized aluminum particles suspended in silicone oil [18] with those predicted [17,37] (5=0.2/zm, a=10/zm, (I)=0.2, f=50Hz, of=2.4x1012S/m, Kf=2.5, ro=Op/~f=1012, r~-ep/ef=101~ and stress and (b) the current density.
FIo=oi/of=l.67): (a) the shear yield
764
.-. 800 E,,,~l,,,,i .... i , , , i , ~ , i , , , ' t , ' " l " ' ~ ' ~ i ---Wu & Conrad [37]~ .... Davis & Ginder[32]600 L! o o 9 Experiment [18] 400
: i'~'-"
200 ~0
Erms=3kV/mm 50Hz -
...
0
:".
0.2
.... /
""0
m
"
c
0.4
0.6
0.8
5 (Hm)
Figure 30. Comparison of the "EG" shear yield stress measured by Inoue [18] for oxidized aluminum particles suspended in silicone oil with those predicted by Wu and Conrad [37] (solid curve) and by Davis and Ginder [32] (dashed curve): a=10~um, r f=50Hz, ~f=2.4x1012S/m, Kf=2.5, KI=5 , Fo=Op/Of=1012, r = e p / e f = l O 15
I
I
-I
1~
and lqo=oi/of=l.67. i
Exp. [18] - Pred.[37] (K~=5) Pred.[371 ( K , - I _ ~ "
10
wooa~176176
5
f 9
0
I
0
S=0.001 E~b=400kV/mm I
0.1
I
I
0.2
I
_1
0.3
I
0.4
a qum) Figure 31. Comparison of the measured [18] a p p l i e d breakdown field for oxidized aluminum particles suspended in silicone oil with those predicted [37]: a=10/tm, ~=0.2, f=50Hz, of=2.4x10lZS/m, K~2.5, Fo=Op/Of=1012, Fe=13p/Ef=1010 and FIo---oi/of=l.67.
765
The dielectric strength of a surface coating depends on its density, temperature, thickness and the purity. Moulson and Herbert [63] reported that the dielectric strength of purified, high density alumina of lttm thickness is about 500kV/mm at room temperature and decreases with the specimen thickness and density of the alumina material. Whitehead [64] reported that the dielectric strength of hollow lead-glass spheres increases with specimen thickness, but that of clear ruby muscovite mica decreases with the specimen thickness. Figure 31 compares the breakdown applied field measured by Inoue [18] for oxidized aluminum particles suspended in silicone oil and that predicted by Wu and Conrad [37], by taking the breakdown strength of the alumina surface coating to be EIb=400kV/mm. Good agreement occurs between the measured breakdown applied field and that predicted. In our model, when the peak value of the maximum local field in the surface film reaches 400kV/mm, the corresponding applied field is considered to be the breakdown applied field for the ER suspension. 3.4 T r a n s i e n t r e s p o n s e in E R fluids
In Section 1 it was pointed out that for dc field the conductivity mismatch between particles and fluid is the dominant factor in determining the ER strength. Using pulsed dc excitation, Ginder et al [65-68] and Tang and Conrad [58] have demonstrated the separate roles played by the conduction and dielectric properties of the particles and host liquid in transient ER response. An interpretation of the experiments, based for simplicity upon a point-dipole analysis, sheds considerable light on the mechanism of ER activity. Recently, Davis [33] studied the ohmic (linear) ER response to a pulsed field and the nonohmic (non-linear) ER effects at high dc fields using an integral equation method. Based on the point-dipole approximation, Ginder, Davis and Ceccio [68] proposed a model to qualitatively predict the transient response. The most important prediction of Ginder, Davis and Ceccio [68] is that the shape of transient stress curves depend sensitively on the relative contribution to the particle polarization mechanism of the dielectric mismatch ~d=(Kp-Kf)/(Kp-2Kf) and conductivity mismatch ~c=(CJp-~f)/(~p-2(~f). If the ER response is mainly induced by a conductivity mismatch (~d/~c<>l), sudden increase in the field produces an instantaneous peak
766
in the stress, which then decays over a time t* because of the residual conductivity mismatch. These distinct qualitative behaviors enable a determination of the mechanisms of electrorheological transient response. The predictions of the ER transient response were fitted to the measured stress transients using nonlinear-least-squares curvefitting algorithm; the solid curves in Figures 32-34 demonstrate that qualitative fits were obtained. Good agreement also occurred between the values of ~d/~c and t* derived from independent fits of the unipolar and bipolar transients in amorphous aluminosilicate and barium titanate suspensions. For the amorphous aluminosilicate fluid the fit to the date gives ~d/~c=0.15, which suggests that ~d/~c<
(I) '
200
'
'
-'
'
300
,
r--'
, 5
7 = 5 s O~
1SO
S -1
#
200
100
0 20
.
.
.
.
40
'
,
'
,
'
rphous
lO
"'
e /
20
-lO I--~
,
Aluminosilicot
-20
-20
. . . .
'
"
'
- 40
2
I
2
' r--
!
"i
I)
I (TTT~
5.0 Hz
-
1.0
Hz
1
0 O' 0.0
0.1
I
I
I
1
0.2
0.3
0.4
0.5
time (s) (a)
--2
0.0
1
I
l
0.5
I .0
I .5
.
2.0
time (s) (b)
Figure 32. The time dependence of (I) the shear stress 1; (dots); (II) the current density and (III) the applied field E o in the amorphous aluminosilicate ER fluid. The sample was sheared in a Couette cell at a constant rate Y=5s"1 and was excited by a lkV/mm peak amplitude. The solid line in (I)is a fit to the stress by the dynamic ER response model [68]. (a) 5Hz unipolar square wave; (b) 1Hz bipolar square wave.
767
conductivity mismatch mechanism is associated with the finite time required for the growth of the shear stress in this system (see Figure 32a). That ~d/~c <<1 is expected in ER systems in which the transport of real charge carriers --ions, electrons, or holes -- to the interface between particle and host liquid promotes the ER effect[9,11,12]. Within the transient model of Ginder, Davis and Ceccio [68], such a situation arises from a substantial mismatch between the particle and host liquid conductivities. It may also result from surface conductivity effects due to absorbed polar impurities. When the field direction is reversed (Figure 32b), a finite time is required for the charge carriers to redistribute and thus for the ER response to recover. The parameters obtained by fitting 200 r
~5o
,
,
,
200 r
(I)
5 ~-'
I~ 100
~
4o
'
5 s
~
.
-1
'
1
(II)
i
1
,
'
1
80
i
Titanate
0
,.
Titanate
"
-40
-40
J
i
Barium
40
o
i
i
0
i
Barium
I
i 7
q
50
0
80
.... ,
~ (I)
100 ~
50
~
15o
7
8O 2
2
10
I)
1
I)
Hz
_ _ ~ ~
10
Hz
I
I 0
~
0
-I
o I
0.00
0.05
0.10
,
|
0.15
~ 2
0.20
.
0.00
i
|
i
I I
0.05
0.10
0.15
0.20
,
t i m e (s)
t i m e (s)
(a)
(b)
Figure 33. The time dependence of (I) the shear 1: (dots); (II) the current density j and (III) the applied field E o in the barium titanate ER suspension. The sample was sheared in a Couette cell at a constant rate Y=5s 1 and was excited by a 2kV/mm peak amplitude, 10Hz unipolar square wave. The solid line in (I) is a fit to the stress by the dynamic ER response model [68]. (a) 10Hz unipolar square wave; (b) 10Hz bipolar square wave.
768
the field reversal data again indicate f~d/f~c=0.16<>l. This is due to the high dielectric constant and low conductivity of the barium titanate particles. Since dielectric-mismatch polarization occurs almost instantaneously with the application of the field, the large dielectric mismatch gives rise to the initial particle polarization and to the shear stress maximum observed when the field is applied (see Figures 33a-b). Because a finite particle-liquid conductivity also exists, charge is carried to the particle surface on a time scale t*, finally screening the dielectric polarization and thus causing the shear stress to decay to its steady-state value. When the field suddenly returns to zero, 800
r"
600
L
,
,
,
(I)
'"
,
"y = 5 ,
-t
400 200 0
02
,
~ ~
o.1 o.0 -0.1
~-~
-0.2
,
,
'
10 1
.
.
o
.
.
.
!
0.00
0.04
Hz
0.08
..
|
0.12
I
0.16
0.20
t i m e (s) Figure 34. The time dependence of (I) the shear x (dots); (II) the current density j and (III) the applied field E 0 in the crystalline aluminosilicate, or zeolite, ER suspension. The sample was sheared in a Couette cell at a constant rate Y=5s1 and was excited by a lkV/mm peak amplitude, 10Hz unipolar square wave. The solid line in (I) is a fit to the stress by the dynamic ER response model [68]. The relatively high conductivity of this fluid is reflected in high current density in (II).
769
the dielectric-mismatch polarization disappears almost instantaneously. The residual polarization due to the screening charge caused by ~o gives rise to a smaller transient stress peak (see Figure 33a) that also decays due to charge transport on the time scale t*. The unipolar stress transient in the crystalline aluminosilicate system (Figure 34) indicates that both dielectric and conductivity mismatches contribute to ER response, since the fit to the ER dynamic response implies that ~d/~c~0.6. The dielectric mismatch gives rise to the initial fast component of the stress growth, while the slower growth toward the steady state stress can be assigned to the conductivity mismatch. However, if the ~d/~c=l, the ER response will not change after it reaches the initial value induced by the dielectric mismatch. The ER response of this suspension will not depend on the field frequency; see also sections 3.2.2 and 3.3. It should be pointed out that Ginder, Davis and Ceccio [68] did not consider the frequency dependence of the dielectric constants and the conductivities, which may affect the transient response in ER fluids. Also, they did not include the non-ohmic conductivity of the host oil, which may give rise to field-dependent transient phenomena. Further, Block et al [69] proposed that the rotations of the particles in shear flow may assist charge transport and thus affect the characteristic time t*. Also, the time-dependent growth of the chains or clusters of particles induced by the application of the field may influence the transient response, which was not considered. The observed transient stresses might be associated with the buildup of stress as the structures are sheared. The time dependence of the dipole moment was calculated based on the assumption that the particles were isolated in an external field which differs from the case of an ER fluid with a high volume fraction of particles. Finally, it should be mentioned that good accord was obtained by Tang and Conrad [58] between their calculated stress-time response using their simplified analysis based on the work of Ginder, Davis and Ceccio [68] and the measurements. Davis [33] also gave a good prediction of transient ER response using the integral equation method. 4. D E S I G N OF I M P R O V E D E R F L U I D S
Recent review articles on ER materials [70-75] give some guidelines for the design of desired ER fluids. To improve ER response, both the host oil and particles should be given attention. It is desired that the host oil has low and stable conductivity, which is not strongly super-ohmic,
770
and has high dielectric permittivity. For the particles, we desire that they have both a reasonably high conductivity and a high dielectric permittivity. To improve the electrical properties of the particles two methods can be used: (a) improve the bulk electrical properties, (b) improve the surface electrical properties. Our studies suggest the latter to be an exciting possibility. In Table i some design guidelines are given for the host oil. In Tables 2, 3 and 4, some recommended design parameters are given for ER suspensions activated by: (a) bulk conduction of the particles, (b) dielectric particles with a conducting surface film and (c) highly conducting particles with an "insulating surface film", respectively. The recommendations in Tables 1-4 are based on the theoretical and experimental considerations reviewed above. Table 1 Recommended design parameters for the host oil of ER suspensions " Conductivity of
9 Dielectric constant 9 Density 9 Viscosity
(a) As low as possible, usually <10l~ (b) Weak non-ohmic behavior. (c) Low water content. (d) Not sensitive to the ambient. High. Approximately that of the particles. Recommended 20-200Pa-s.
Table 2 Recommended design parameters for bulk conducting particles in ER suspensions 9 Conductivity Op 9 9 9 9 9
Dielectric constant Density Size Shape Possible Max. x E
(a) (~p=106-109S/m, making 107>Op/Of >104. (b) Not sensitive to the ambient. As high as possible. Match that of the host off. Diameter = l---20/zm Spherical, but this needs fin'ther study. l-SkPa (Eo=3-4kV/mm)
771
Table 3 Recommended design parameters for "insulating" particles with a conducting surface film 9 Particles:
(a) High dielectric constant (b) Density close to that of the host oil (~0.97-0.98) (c) Spherical shape, but needs further study.
9 Film:
(a) Conductivity o I =104--107S/m, or oi/of >104 (b) Thickness 5= 10 -3--l~um, or 5/a<0.1 (c) Uniform, smooth and continuous.
9 Possible max. shear yield stress ~E" DC Field: 1-5kPa (E0=3-4kV/mm) AC Field: Not clear, but appears to be smaller than with DC field.
Table 4 Recommended design parameters for conducting particles with "insulating" surface film 9 Particles:
(a) High conductivity, Op>104S/m. (b) High dielectric constant, Kp>1000. (c) Density approximately that of the host oil.
9 Film:
(a) Conductivity, o I =10s-10llS/m. (b) As high dielectric constant as possible. (c) High dielectric breakdown strength. (d) Small thickness: 5/a<0.05, (depending on its electrical breakdown strength)
9 Possible max. shear yield stress ~E" DC: AC:
~5-10kPa. ~10-100kPa.
772
REFERENCES 1. 2. 3. 4.
W. M. Winslow, U.S. Patent 25 (1947) No.2417850. W. M. Winslow, J. Appl. Phys., 20 (1949) 1137 C.A. Coulson, Electricity (Interscience Publishers, Inc., New York,1961) A. P. Gast and C. F. Zukoski, Advanced Colloid and Interface Sci. 30 (1989) 153 5. A. F. Sprecher, Y. Chen and H. Conrad, Proc. 2nd Int. Conf. ER Fluids, ed. J. D. Carlson, A. F. Sprecher and H. Conrad (Technomic, Lancaster, 1990), pp. 82-89 6. T.C. Halsey and W. Toor, Phys. Rev. Lett. 65 (1990) 2820 7. D.J. Klingenberg and C. F. Zukoski, Langmuir, 6 (1990) 15 8. Y. Chen, A. F. Sprecher and H. Conrad, J. Appl. Phys. 70 (1991) 6796 9. L.C. Davis, Appl. Phys. Lett., 60 (1992) 319 10. R. Tao, Q. Jiang, H. K. Sim, Physical Review E, 52 (1995) 2727. 11. R.A. Anderson, Proc. 3th Int. Conf. ER Fluids, ed. R. Tao and G. D. Roy (World Scientific, Singapore,1992), pp.81-92 12. L. C. Davis, J. Appl. Phys., 72 (1992) 1334 13. Y. Chen, and H. Conrad, Developments in Non-Newtonian Flows, ed. D. Siginer, W. Van Arsdale, M. Aitan and A. Alexandrou (ASME AMD, New York, 1993) 175, pp.199-208. 14. H. Conrad, and Y. Chen, Progress in Electrorheology, ed. K. O. Havelka and F. E. Filisko (Plenum Press, New York and London,1995), pp.55-86. 15. Y. D. Kim and D. J. Klingenberg, Progress in Electrorheology, ed. K. O. Havelka and F. E. Filisko (Plenum Press, New York, 1995), pp.l15-130 16. R. M. Webber, Progress in Electrorheology, ed. K. O. Havelka and F. E. Filisko (Plenum Press, New York, 1995), pp.171-184 17. L. C. Davis, J. Appl. Phys. 73 (1993) 680. 18. A. Inoue, Proc. 2nd Int. Conf. ER Fluids, ed. J. D. Carlson, A. F. Sprecher and H. Conrad (Technomic Publ. Co., Lancaster-Basel, 1990), pp.176-183 19. H. Conrad, and A. F. Sprecher, J. Statistical Phys. 64 (1991) 1073 20. W. J. Wen and K. Q. Lu, Appl. Phys. Lett., 67 (1995) 2147 21. H. Conrad, Y. Shih and Y. Chen, Development in Electrorheological Flows and Measurement Uncertainty, ed. D. Siginer, J. Kim, S. Sherif, and H. Coleman (ASME, New York, 1994), FED-Vol.2051, AMD Vol.190, pp.69-82 22. J. N. Foulc and P. Atten, Proc. 4th Int. Conf. Electrorheological Fluids, ed. R. Tao and G.D. Roy (World Scientific, Singapore, 1994) pp.358-371 23. C. Boissy, J. N. Foulc and P. Atten, ed. R. Tao and G. D. Roy (World Scientific, Singapore, 1994), pp.453-462 24. C. Boissy, P. Atten and J, N. Foulc, J. Intelligent Material System and Structures, 7 (1996) 599 25. H. Conrad, Y. Chen and A. F. Sprecher, Proc. 2nd Int. Conf. Electrorheological Fluids, eds. J. B. Carlson, A. F. Sprecher and H. Conrad (Technomic, Lancaster, 1990), pp.252-264
773
26. J. N. Foulc, N. Felici and P. Atten, C. R. Acad. Sci. Paris. 314 II (1992) 1279. 27. N. Felici, J. N. Foulc and P. Atten, Proc. 4th Int. Conf. Electrorheological Fluids, ed. R. Tao and G. D. Roy (World Scientific, Singapore, 1994), pp.139-152 28. L. Onsager, J. Chem. Phys., 2 (1934) 599 29. X. Tang, C. W. Wu and H. Conrad, J. Rheology, 39 (1995) 1059 30. C. W. Wu and H. Conrad, J. Phys. D: Appl. Phys., 29 (1996) 3147. 31. P. Atten, K. Q. Zhu, C. Boissy and J. N. Foulc, J. Intelligent Material System and Structures, 7 (1996) 573 32. L. C. Davis and J. M. Ginder, Progress in Electrorheology, ed. K. O. Havelka and F. E. Filisko (Plenum, New York, 1995), pp.107-114. 33. L. C. Davis, J. Appl. Phys., 81 (1997) 1985 34. C. W. Wu and H. Conrad, J. Appl. Phys., 81 (1997) 383. 35. C. W. Wu and H. Conrad, J. Phys. D: Appl. Phys., (1997) in print. 36. C. W. Wu and H. Conrad, "Dielectric and conduction effects in non-ohmic electrorheological fluids", Submitted to Phys. Rev. E, (1997). 37. C.W. Wu and H. Conrad, J. Appl. Phys., 81 (1997) 8057 38. T. J. Garino, D. Adolf and B. Hance, Proc. 3rd Int. Conf. ER Fluids, ed. R. Tao (World Scientific, Singapore, 1992), pp.167-174 39. D.V. Miller, C. A. Randall, A. S. Bhalla, et al, Ferroelectrics Letters, 15 (1993) 141 40. M. J. Chrzan, and J. P. Coulter, Proc. 3rd Int. Conf. ER Fluids, ed. R. Tao (World Scientific, Singapore, 1992), pp.175-191 41. M. Zahn, Electromagnetic Field Theory: A Problem Solving Approach (John Wiley and Sons Press, New York, 1979) 42. C.W. Wu and H. Conrad, Submitted to J. Materials Research, (1997). 43. P. Atten, J. N. Foulc and H. Banqassmi, Progress in Electrorheology, ed. K. O. Havelka and F. E. Filisko (Plenum Press, New York and London, 1995), pp.231-244. 44. H. Block, and J. P. Kelly, Proc. 1st Int. Symp. ER Fluids, ed. H. Conrad, A. F. Sprecher and J. D. Carlson (North Carolina State University Publications, Raleigh, 1989), pp.l-26 45. C. Boissy, P. Atten and J. N. Foulc, J. Electrostatics, 35 (1995) 13 46. C. W. Wu and H. Conrad, J. Rheol., 41 (1997) 267 47. W. C. Yu, M. T. Shaw, X. Y. Huang and F. W. Harris, J. Poly. Sci., 32 (1994) 481. 48. Y. H. Shih and H. Conrad, Proc. 4th Int. Conf. Electro-Rheological Fluids, ed. R. Tao and G. D. Roy (World Scientific, Singapore, 1994), pp.294-313 49. X. Tang, C. W. Wu and H. Conrad, J. Appl. Phys., 78 (1995) 4183 50. C. W. Wu, Y. Chen, X. Tang and H. Conrad, Int. J. Modern Phys. B, 10 (1996) 3315. 51. C. W. Wu, Y. Chen, X. Tang and H. Conrad, Int. J. Modern Phys., B, 10 (1996) 3327 52. Stangroom, J. E., Phys. Tech., 14 (1983) 290. 53. C. W. Wu and H. Conrad, "Electrical properties of electrorheological particle clusters", To be published (1997)o
774
54. H. Conrad, Y. Chen and A. F. Sprecher, Int. J. Modern Phys. B, 6 (1992) 2575 55. C. W. Wu and H. Conrad, "The shear strength of electrorheological particle clusters", To be published (1997). 56. G. E. McGuire, Semiconductor Materials and Process Technology Handbook (Noyes Publications, Park Ridge, NJ, 1988), pp.28-49 57. C . J . F . Bottcher and P. Bordewijk, Theory of Electric Polarization, II Dielectric in Time-Dependent Field (Elsevier, New York, 1978) 58. X. Tang and H. Conrad, J. Appl. Phys., 80 (1996) 5240 59. J. M. Ginder and L. C. Davis, Proc. 4th Int. Conf. ER Fluids, ed. R. Tao and G. D. Roy (World Scientific, Singapore, 1994), pp.267-282 60. C. W. Wu and H. Conrad, 6th Int. Conf. on ER and MR Fluids, Yonezawa, Japan, July, 1997 61. C.A. Randall, D. E. McCauley, C. P. Bowen, T. R. Shrout and G. L. Messing, Proc. 4th Int. Conf. Electrorheological Fluids, ed. R. Tao and G. D. Roy (World Scientific, Singapore, 1994), pp.60-66 62. H. Thurnauer, Dielectric Materials and Application, ed. by A. R. Von Hippel (Technology Press of MIT and John Wiley & Sons, New York, 1954), pp.179-189 63. A. J. Moulson and J. M. Herbert, Electroceramics: Materials, Properties, Application (Chapman and Hall Press, London, New York, 1990) 64. S. Whitehead, Dielectric Breakdown of Solids (Oxford, Clarendon, 1951) 65. J. M. Ginder and S. L. Ceccio, 65th Annual Meeting of the Society of Rheology, Boston, MA, Oct. 17-21, 1993. 66. J. M. Ginder and S. L. Ceccio, Polym. Preprints, 35 (1994) 315-320 67. J. M. Ginder and S. L. Ceccio, J. Rheol., 39 (1995) 211 68. J. M. Ginder, L. C. Davis and S. L. Ceccio, Progress in Electrorheology, ed. K. O. Havelka and F. E. Filisko (Plenum Press, New York, 1995), pp.281-294 69. H. Block, K. M. W. Goodwin, E. M. Gregson, and S. M. Walker, Nature, 275 (1978) 632 70. K. D. Weiss and J. D. Carlson, Advances in Electrorheological Fluids, ed. M. A. Kohudic (Technomic Publishing, Lancaster, 1994), pp.30-52 71. K. O. Havelka, Progress in Electrorheology, ed. K. O. Havelka and F. E. Filisko (Plenum Press, New York, 1995), pp.43-54 72. H. Block, and J. P. Kelly, A. Qin, and T. Watson, Langmuir, 6 (1990) 6 73. H. Block and P. Rattray, Progress in Electrorheology, ed. K. O. Havelka and F. E. Filisko (Plenum Press, New York, 1995), pp.19-42 74. F. E. Filisko, Proc. 3rd Int. Conf. on Electrorheological Fluids, ed. R. Tao (World Scientific, Singapore, 1992), pp.116-128 75. F. E. Filisko, Progress in Electrorheology, ed., K. O. Havelka and F. E. Filisko (Plenum Press, New York, 1995), pp.3-18.
775
R H E O M E T R Y OF E L E C T R O R H E O L O G I C A L F L U I D S Rex C. Kanu a and Montgomery T. Shaw b
aDepartment of Industry and Technology, Ball State University, Muncie, IN 47306 bDepartment of Chemical Engineering and Polymer Program, University of Connecticut, Storrs, CT 06269-3136 1. R H E O L O G I C A L M E A S U R E M E N T S
1.1 Introduction The purpose of this section on rheometry of ER fluids is to describe methods that have been used for rheological characterization of ER fluids in enough detail so that the experimentalist interested in studying ER fluids will have a good start on the designs that have been tried, along with an appreciation of their strong and weak poims. We will exclude engineering tests in this section unless they result in true theological information. Auxiliary characterization such as electrical and optical properties will be covered briefly. The inclination in rheometry is to adapt existing equipment to the task, and with ER fluids this has been followed successfully. Most investigators have taken commercial instruments and replaced existing fixtures with ones that are electrically isolated from each other and from the rest of the instrument. Both rotary and linear motions have been used successfully. Pressure-driven flows have generally required custom-built equipment. When working with ER fluids, two precautions are necessary: (1) potentially lethal voltage sources are needed for most ER experiments and safety interlocks thus become very important, and (2) high voltage connected to even well-isolated fixtures can create stray currents to parts of the instrument that should not be exposed to any currem. This may require auxiliary grounding of parts leading from the fixtures to the bulk of the instrument. For example, rather than counting on a ground through a motor shaft or a load cell, a separate ground lead should be used [1]. Janocha and
776
Rech [2] have described some additional problems with high-frequency, highvoltage fields. Aside from protecting the experimentalist and the instrument, one should be aware that the geometry of the fixtures, along with the dielectric properties of all materials in the electric field influence the direction and strength of the electric field. For example, if the classical parallel-plate geometry is used, then the field near the edges of the plates will be concentrated near the corners unless the sharp corners are rounded. The strength of the field will further be influenced by the nature of the dielectric surrounding the plates (e.g., air vs. fluid). At the electrical stresses typically required, air may break down [3]. According to Paschen's law, the breakdown voltage is a function of the product of pressure x distance. At 1 ram, a field of 1 kV will break down air, and fields of half this value will create corona discharge, adding to any observed current. Thus successful designs will reduce the field before encountering a fluid-air interface. Details concerning the design of cup-andbob viscometers have been described [4]. Suspensions are the most common ER fluids. Suspended particles in fluids subjected to fields polarize, and then migrate together by attraction of the electric dipoles. This process is indistinguishable to a more general process called dielectrophoresis. Dielectrophoresis forces result when there is a gradient in the electric field. The general relationship for dielectrophoretic force on a sphere is: F - 2~g o a3 ~2 - g l
VE 2
(1)
g2 + 2 9 1
where E is the intensity of the electric field, e2 is the permittivity of the sphere, el is the permittivity of the oil, e 0 is the permittivity of free space, and a is the radius of the sphere. Note that the direction the sphere moves depends on the sign of e2-~l and not on the direction of the field. Thus particles of high dielectric constant relative to oil will migrate to the regions of high field. Rheometers must therefore be designed to avoid large values of VE 2 "
A larger force by far is the electrophoretic force on a charged particle in the fields typically needed for ER experiments. By using ac fields, electrophoretic migration can be avoided. ER fluids also sediment and migrate due to the influence of gravity and hydrodynamic forces, respectively. Sedimentation necessitates reloading,
777
restirring or other precautions. Kawai et al. [5] used a system to circulate ER fluid from a reservoir through the rheometer, thus avoiding sedimentation. Banding in torsional flows has been observed [6], and other flows may feature similar hydrodynamic separation processes.
1.2 Simple Flow Observation The simplest ER experiment is often used as a demonstration: a field is placed across the fluid to solidify it, and then the voltage is reduced until the fluid slumps or flows. The geometry often comprises two parallel cylindrical wires. The electrical stress E between two parallel cylinders is spatially varying and is the highest near the electrodes, with a magnitude of approximately
E
AV
(2)
2a In da
where A V is the voltage between wires of radius a spaced at distance d. The mechanical shear stress o 21 is the highest also at the wires and will depend upon the volume of fluid that has solidified. A rough estimate is o" 21
_ d 2
Pg / 2 a
(3)
where p is the density of the fluid and g is the gravitational constant. By progressively lowering the applied voltage, a value of yield stress can be estimated. This simple idea has been modified for a parallel plate arrangement where the equations are now
AV
(4)
E
H and o 21 = H p g
(5)
where H| is the gap between the plates. Stipanovic and Schoonmaker [7]added a Teflon block above the fluid (Figure 1), which increases the shear stress to
778
cy21 = (Hwtpg + W) / wt
(6)
where W is the weight of the block, w is the width of the device and t is the depth of the fluid beneath the block. Assumed here is that the width of the block is the same as the width of the fluid, and there is no resistance to motion contributed by the fluid between the block and the electrode, a good| approximation because of the low electric field in that region. The Teflon block increases the stress range and gives one the ability to develop a ~ 2~ vs. E relationship over a| wide range of stresses without changing anything but the weight of the Teflon block. An alternative geometry for measurement of this kind is the annulus. Some advantages of the annulus would be the fairly well-defined electrical field combined with the ability to drive the fluid with gas pressure, which can be easily varied during the experiment. Such arrangements diverge, however, from the simplicity offered by the gravitational flow devices already described.
1.3 Sliding plate rheometers. This class of rheometers involves a linearly driven plate that also serves as the electrode. While the latter restriction is not imperative-the voltage of
PTFE Block j .
,,
z/ ,
,
2 m
/,
m
~
Figure 1. Gravity-operated apparatus for determining the yield stress of ER fluids (From Stipanovic and Schoonmaker [7])
779
a movable plate could be allowed to float between two other fixed electrodesmost investigators have found it convenient to arrange the movable plate as either the ground or the high-voltage electrode. Linear motions have been applied by a variety of methods, usually using a tensile testing machine.
+C~~
Stationary Pivot Point
J:D
jf-~,
~ ) ~
j
Microscope
To Microscope Stage
t E> Shear
Light
Field of View
"
Figure 2. Pendulum apparatus for applying small loads to ER fluids. Simultaneous observation is possible in this device. From Sprecher, Chen and Conrad, [8]. Another similar concept uses the weight of the plate itself by suspending the top electrode on a pendulum and moving the bottom plate underneath this swing (Figure 2) [8]. This arrangement has gives a nonlinear change between the movement of the bottom plate and the stress, which is advantage in the accurate determination of yield stress at low fields. Clearly if the bottom plate is confined to a horizontal plane, the gap will change slightly; compensation would be possible by mounting the bottom plate on another pendulum. Still another sliding plate design uses axial annular drag flow between a fixed, double cup and a light sleeve attached to the load cell mounted on the cross head of a tensile testing machine (Figure 3) [9]. This geometry lends itself to installation of optical paths in the field direction. Block et al. [10]
780
(+)
T
Teflon Insulator
Crosshead Motion
Crosshead
II 'QI ,an, !
!
ner Electrode
,1-rr -mm Gap (h)
ER Io - 5 0 m m
Fluid uter Electrode
Figure 3. Axial annular drag flow device attached to a testing machine. From Sprecher, Conrad and Carlson [9]. have described a horizontal sliding-plate rheometer that employs a very compliant, spring-leaf load cell to push very gently against the sliding plate. This apparatus was useful for determining the yield stress of the fluid, as opposed to steady flow properties. 1.4 Rotational Viscometric Flows 1.4.1 Torsional Flow between Parallel Plates The choice of geometry influences not only the homogeneity of the electric field, but the nature and uniformity of the deformation of the fluid. While
781
parallel plates can give a nearly uniform electrical field, the deformation field varies spatially. As important is the variation of the contribution of the material at various radii to the observed torque. The outcome is that the outermost edges of the fluid contribute the most to the torque reading, but may experience the most nonuniform electric field. This will lead to torque readings which will be difficult to interpret in terms of the stress in the fluid by the conventional methods [11]. In addition, particle migration by dielectrophoresis will be a problem. The advantages of the parallel plate geometry include ease of construction, small sample size, and the relative ease of adding optical paths. The normal force can also be measured. Under static conditions the attractive normal force F provides a measure of the dielectric properties of the fluid. The relevant equation is: F-
1 ~cR2~ 0~ f E 2 -~
(7)
where 'R is the radius of the plates, and g f is the effective permittivity of the ER fluid. Figure 4 shows the application of Equation 2 for an experimental ER fluid. Note that the normal force is quite linear in E 2 up to quite high fields. The value of g f derived from this plot is close to the value obtained with a time-domain dielectric instrument [12]. 1.4.2 Couette Flow The Couette geometry features reasonable homogeneity of both electric and deformation fields if the gap is small relative to the radius of the cup and bob. This advantage has attracted many advocates [4, 5, 13-27]. In addition, the Couette geometry offers large torque for a given diameter, making it particularly appropriate for measuring yield stress [ 18, 28]. The disadvantages are the usually large sample volume, and the inertia of the moving fixture. In many cases it is somewhat more cumbersome to introduce optical paths into a Couette cell than for parallel plates, although complex birefringence measurements on ER fluids have been performed in a Couette geometry with the light propagating in the neutral or vorticity direction, i.e., parallel to the axis of the cylinders (see section 3.2 for more details). This is virtually impossible with the parallel-plate geometry. While Couette flows generally have lower edge effects than parallel-plate flows, such effects apparently are not absent. They derive from a coupling of the normal hydrodynamic effects with fringing of the electrical field. A study by Janocha et al. [4] compared the response of Couette devices featuring
782
different bob geometries, including ones with conical bottoms and insulating flat bottoms. The use of an insulating bottom removed the influence of the contribution from the bottom area. While true with all ER geometries, these authors found that stirring the fluid with the field off between each measurement was essential to counteract the effects of dielectrophoretic migration of the particles. The Couette geometry is one of the best choices for the determination of yield stress. Because applications of ER fluids may call for holding of parts or fluids in position for a considerable time, the measurement of yield stress is of importance to the technology. The subject of yield stress itself, irrespective of the fluid, has generated lively debate; the reader is referred to, for example, Astarita [29], Bennington et al. [30], Bonnecaze and Brady [31], Harnett and Hu [32], Nguyen and Boger [28], Schurz [33], and Yoshimura et al. [18] to sample the arguments. For ER fluids, most investigators favor the application of a steadily increasing stress to the solidified ER fluid, finding the value that
1600 1200
55.35 Pa mm2/kV 2
g
8
0
13_
N
800
--
b
4OO
I 0.0
5.0
10.0
I
I
15.0
(E, kV/mm}
20.0
25.0
2
Figure 4. Attractive force between parallel plates is proportional to the static composite permittivity ~c of the ER fluid. Slope is equal to Sc So/2. From Jordan, McLeish and Shaw [ 1].
783
causes distinct flow. Torque-controlled drag-cup motors are most convenient for this type of measurement. Application of torque to a rotor using a Helmholtz coil working on a permanent dipolar magnet embedded in the bob enabled Woestman [19] to apply a torque that would drop slightly if the fluid yielded. This resulted in the fluid resolidifying. As this was repeated, the fluid's yield stress increased by over a factor of three, suggesting that the fluid acquires a more stable structure as the result of the disruption caused by the slippage. The ratio between the static yield stress (the point where motion started) and the dynamic yield stress (the point where the motion stopped) was about 1.2. 1.5 Pressure-Driven Flows Applications of ER fluids include valves wherein the ER fluid is required to resist pressure. There are also many designs of devices featuring pumping or squeezing flows that involve a combination of pressure-driven and drag flows. While one might hope that these flows could be fully predicted from rheometric drag flows, the nature of ER suspensions suggests that the coincidence of a solid surface at the maximum shear stress can lead to special characteristics. In addition, pressure-driven flows, especially in a thin channel, can provide opportunities for close examination of the fluid's structure by optical or other means. Because of the requirements for a uniform electrical field, experimentalists have focused on channel flows [34, 35]. Axial, pressure driven, annular flows find use in applications [36] but are less common in rheometry [37].
1.5.1 Channel (Slit) flows Two rheometers using slit flows have been described fully in the recent ER literature. One, due to Nakano et al. [35] uses a 10-mm-wide slit with a 0.6mm gap. The slit is 18 mm long, but only the last 10 mm are energized (this was increased to 20 mm in later work [34]). Fluid is feed from a cylinder, using a screw-driven piston, through a streamlined section into the slit. A pressure transducer gives a measure of the shear stress. By measuring pressure with and without the field, the cylinder and entrance pressure losses can be found because these areas are not energized. Transients upon applying the field can be related to the buildup of structure in the ER fluid. The apparatus describe by Weiss and Carlson [38] is similar, but has the added convenience that the fluid can flow back into the reservoir (a syringe) by gravity at the end of the experiment, requiring very little fluid.
784
V/A 77v////////A..i.GV/////~
_
.
~,.I
.
% 0
~
J
-=-V 0
Figure 5. Pressure-driven, annular axial flow device for use with ER fluids [37]. The annular gap is between parts A and B. This material has been reproduced from the Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science, 16 1974 by J. Arguelles, H. R. Martin and R. J. Pick by permission of the Council of the Institution of Mechanical Engineers. A somewhat different approach was taken by Thurston and Gaertner [39]. In their apparatus, an oscillatory pressure is applied along the slit. The slit is vertical with the driver piston at the stop and the bottom resting in an open cup
785
of ER fluid.
Because of inertia, this flow is complex; however, at low
frequencies in thin slits, the complex shear stress c~21 is given by the usual equation:
9 _p,
o-21
H/2L
(8)
where P* is the complex pressure, H is the slit height and L is the length of the slit.
1.5.2 Axial annular flows Arguelles et al. [37] have described an apparatus that exploits axial pressure-driven flow in an annular space between two cylinders, labeled A and B in Figure 5. The fluid, driven by air pressure of up to 500 psig (350 kPa), flows from the reservoir Co through holes C1 into volume C2 and then through the gap labeled h. It exits through opening O.
1.60therSound Propagation Korobko and Chernobai [40] depict a chamber in which a sending and receiving transducer are arranged to propagate sound parallel to the electric field. Both velocity and attenuation could be measured, although only velocity measurements were reported. Fields ranged up to 2 kV/mm. The sound velocity was linearly proportional to the field and did not show a threshold, even at particle volume fractions as low as 2 %. 2. E L E C T R I C A L MEASUREMENTS Of concern in this section will be measurements of the dielectric properties of ER fluids during flow, and preferably during the measurement of shear stress. The measurement of steady, transient or periodic current as a result of be considered a routine accessory of the rheometer. Of more value to the diagnosis of the structure of the fluid is the dielectric spectrum of the fluid during flow. In addition, and also of technical value, is the measurement of dielectric strength of the fluid, again under flow conditions. 2.1 Complex permittivity While several groups have measured dielectric properties of ER suspensions under quiescent conditions (e.g., [25, 41]), measurement of permittivity under flow conditions is far from trivial. A Couette cell designed to do this has been
786
described by Block et al. [17]. By using a spli.t-stator design, their apparatus could measure shear stress simultaneously and provide guard electrodes thereby. According to the descriptions, high voltage could be applied to the electrodes, but details were not provided. Separation of the high-voltage activating potential from the low-voltage measuring signal has been discussed by Placke et al. [42], but circuit diagrams were not provided. Their apparatus centered around a commercial rotational rheometer with a rotating cup and fixed bob.
2.2 Ultimate dielectric properties The ER effect obtainable in a fluid can be limited by the dielectric strength of the suspension. This may well be a function of shear rate. Inoue [20] apparently made measurements of the ultimate dielectric properties under flow. 3. O P T I C A L P R O P E R T I E S Of the methods for diagnosing structure of quiescent and flowing ER suspensions, optical methods have proven to be both adaptable and informative. Optical methods range from direct observation with a microscope to light scattering, to measurement of dichroism and birefringence. Many groups have been successful at combining one or more of these methods with the simultaneous measurement of rheological properties. Some of the methods will be explored below. 3.1 Direct observation under flow The observation of suspensions during continuous flow is not trivial; but, by using shallow depth of focus or sheets of light, individual particles, or sets of particles, can be observed. Another approach is to use dilute suspensions, with the appreciation that the structures . Exploiting one or a combination of these techniques, many researchers have arranged cells whereby the deforming suspensions can be observed [43]; a typical apparatus is illustrated in Figure 6. However, most of these do not simultaneously provide rheological information. Exceptions are the pendulum apparatus of Sprecher et al. [8] and Bossis et al. [6]. The observation of periodic oscillatory flow is somewhat easier than steady flow, because illumination can be synchronized with the motion of the particles. Equipment for such an experiment has been described by Bossis et al. [6], and this is reproduced in Figure 7. In this apparatus, motion is derived
787
I
I
0
Q
J
/ Delrinbase
I
I
~la/s slid.~ ' Stainless steel plates
l ,1\
Motor
Figure 6. Apparatus for observing with a microscope the deforming structure of ER fluids. Direction of observation is perpendicular to the field. From Jordan, Shaw and McLeish [ 1]. from the output of the accelerometer, whereas the current to the vibrator is proportional to the load. The light passes parallel to the field, which is applied to the suspension using tin-indium oxide coatings on the glass plates. The gap between the two plates is kept small, e.g., 100-200 ~tm.
3.2 Birefringence and Optical Dichroism. As light passes through a material, it is either transmitted, scattered or absorbed. The scattered and transmitted light can be measured directly; the absorbed light by difference. In anisotropic systems, such as aligned ER fluids, the absorption and scattering can be expected to depend on the direction of the particle columns with respect to the direction of the polarization of the light. Similarly, one expects the refractive index of the medium to be anisotropic, which will result in birefringence. Birefringence and dichroism can be interpreted as phenomena resulting from the real and imaginary components of the complex refractive index tensor, and thus are related. See Fuller [44] for more details, including typical optical arrangements.
788
Equipment for measuring birefringence under electrical fields has been described by Block et al. [22] and by Smith and Fuller [24]. Both arrange for the light to propagate in the vorticity direction in a Couette cell with the field applied in the direction of the velocity gradient. The apparatus of Smith and Fuller shown in Figure 8 is set up to give the dichroism, which shows dramatic changes on application of the electric field.
3.3 Light Scattering Systems can be chosen that absorb very little light, thus simplifying the interpretation of the measured transmitted and scattered light. From these measurements, inferences concerning the spatial variation in structure, and the changes with time (say, after applying the field or shearing the fluid) can be made. Unlike microscopy, the signal in a scattering experiment is independent of the bulk motion of the fluid, an important advantage in flowing systems. However, the information concerning structure is never as complete or certain as with direct observations.
I/ / JI Sine-wave generator ,
Phase detector
Camera
!\ i
'"
Microscope
Electromagnetic vibrator
1
l
I I
X o cos (~t - 6)
E ~_/
I , ; , ~
,
/
/
i~///
/i/
///Glass Plates ~ / 1
Z, , ~ / \ ) 1/
Micro mete r h i , " ","" ":"'~"/x,~ /
.
. . . . . . .
.
/
,/
//,
,"
,, ,~
Accelerometer
~ ~
Chopped light source
Figure 7. Design for observing the structure of ER fluids along the field direction while subjecting the fluid to oscillatory deformation. From Bossis et al. [6]
789
Xenon Arc Lamp
l
I
l
Monochrometer
f "~
Lens
i i 11
Polarizer
Wave Plate E-field ~ -I
CPU I
FlowCell I
I
PMT ~r4co
I ldc LPF
iI
Figure 8. Equipment for observing the dichroism developed by an ER fluid. From Smith and Fuller [24]. Suspensions such as ER fluids tend to be highly scattering. As a result of this, light that is scattered tends to be rescattered, which convolutes the signal even more. If the path length is large enough, but not large enough for the light to strongly absorbed, then the light that appears to be "transmitted" has actually been scattered many times; it diffuses through the suspension like a gas though a porous medium. At this limit, the observed transmittance no longer obeys Beers law [45, 46]. All information concerning structure is lost, save a single length scale corresponding to the mean-free path of the light. At larger path lengths, the transmittance begins to fall exponentially because of absorption, and obeys the rule" T ~ ~10l* e _L/labs
3labs
(9)
790
where T is the transmittance (ratio of transmitted light to incident light) labs is the absorbance length, 1" is the diffusion length and L is the path length. From the variation of T with L, both length scales (/*and labs) can be found. Path lengths up 7 mm for commercial ER fluids have been reported [45] using the equipment depicted in Figure 9. Deformation was accomplished by moving the first electrode back and forth. To derive more information about the structure of ER fluids, most researchers use ER fluids comprising particles and liquid of nearly equal refractive index. (To reduce multiple scattering, short path lengths are another option, but path lengths are limited by the particle size. A third methoddiluting the suspension-is not available because of the strong effect that particle concentration has on the behavior of ER fluids.) If the refractive indices of the particle and suspending fluid are matched, then the fluid is nearly transparent and the full two-dimensional light scattering patterns can be observed and interpreted in terms of well-established scattering theory [47]. An example of such equipment, with the light transmitted perpendicular to the field (vorticity direction in a Couette geometry) is shown in Figure 10. The path length through the oriented fluid was 2 mm. While deformation was available, torque could not measured simultaneously.
3.4 Other optical techniques Non-centrosymmetric media should generate a weak, but measurable signal at twice the frequency of an incident light beam. Such media could include ER
XY A1
Laser
"I
A2
-
f -
-
~
/
/
< \ \
L1 L2 Figure 9. Diagram of apparatus for measuring diffuse transmittance of ER fluids in the field direction. From Ginder [45].
791
Figure 10. Apparatus for measuring the time-resolved, two-dimensional light scattering for light propagating perpendicular to the field [26]. Reprinted with permission from J. E. Martin and J. Odinek, Journal of Rheology, 39(5) 9951009 (1995). Copyright 1995 Society of Rheology fluids subjected to DC fields [48]. The strength of the second harmonic generation (SHG) should reflect the nature of polarization in the ER fluid. Figure 11 shows schematically an apparatus for measuring SHG in an ER fluid. No provision was made for deforming the solidified fluid, but this could be added. Time and field are the principal variables in SHG-ER experiments. 4. SUMMARY
While ER fluid rheometry presents special problems, equipment has been developed that addresses these problems and opens up opportunities for investigating structural changes in other two-phase materials as well. The relationship between structure and stress in highly structured materials such as ER fluids is a constant challenge for rheologists, and techniques will continue to evolve. Imaging methods for studying the details of particle motion will undoubtedly be developed. One unfortunate aspect of ER fluids is that
792
presently NMR imaging techniques, used successfully for other suspensions, cannot be applied because of the magnetic fields produced by the current passing between the plates.
Sample Box
/
HIGH VOLTAGE
0 1--.
0
I
l IR Filter rY w <
IR Absorber
532nm IF PMT
._1
0 < i>-
BOXCAR COMPUTER
Figure 11. Equipment for studying second harmonic generation in an ER fluid. From Wu et al. [48]. REFERENCES 1. 2.
3.
T . C . Jordan, M. T. Shaw, T. C. B. McLeish, J. Rheol., 36 (1992) 441. H. Janocha and B. Rech in R. Tao and G. D. Roy (eds.) Electrorheological Fluids, Mechanism, Properties, Technology and Applications, World Scientific, Singapore, 1994, pp. 344-357. R. Bartinkas and E. J. McMahon (eds.) Engineering Dielectrics Vol. 1, Corona Measurement and Interpretation, Am. Soc. For Testing and Materials, Philadelphia, 1979.
793
4. 5. 6. 7. 8.
9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20.
21. 22. 23.
H. Janocha, R. Bolter, and B. Rech, Paper presented at the 5th International Conference on ER Fluids held in Sheffield, UK, 7/1995. A. Kawai, K. Uchida, K. Kamiya, A. Gotoh and F. Ikazaki, Advanced Powder Technol. 5, (1994) 129 G. Bossis, E. Lemaire, J. Persello, and L. Petit, Progr. Colloid Polym. Sci. 89 (1992) 135. A. Stipanovic and J. Schoonmaker, ACS Polym. Prepr., 35 (1994) 365. A. Sprecher, Y. Chen and H. Conrad, in J. D. Carlson, A. F Sprecher, and H., Conrad (eds.), Proc. 2nd Inter. Conf. ER Fluids, Technomic Publishing Co., Inc. Lancaster, P A , 1990, pp. 82-89. A . F . Sprecher, J. D. Carlson, and H. Conrad, Mater. Sci. Eng., 95 (1987) 187. H. Block, J. P. Kelly, A. Qin and T. Watson, Langmuir, 6, (1990) 6. G. V. Gordon and M. T. Shaw, Computer Programs for Rheologists, Hanser Publishers, New York, 1994. R. Kanu and M. T. Shaw, in Progress in Electrorheology, K. O. Havelka and F. E. Filisko (eds.) Plenum Press, New York, 1995, pp. 303-323. C. J. Gow and C. F. Zukoski IV, J. Colloid Interface Sci., 136 (1990) 175. H. Uejima, Japan. J. Appl. Phys., 11 (1972) 319. G. V. Vinogradov, Z. P. ShuI"man, Yu. G. Yanovskii, V. V. Barancheeva, E. V. Korobko, and I. V. Bukovich, J. Eng. Phys., 50 (1986) 429. W. M. Winslow, J. Appl. Phys., 20 (1949) 1137. H. Block, E. M. Gregson, A. Qin, G. Tsangaris, and S. M. Walker, J. Phys. E: Sci. Instrum., 16 (1983) 896. A. S. Yoshimura, R. K Prud'homme, H. M. Princen, A. D. Kiss, J. Rheol., 31 (1989) 699. J. T. Woestman, Phys. Rev. E, 47 (1993) 2942. A. Inoue, in J. D. Carlson, A. F Sprecher and H., Conrad (eds.), Proc. 2nd Inter. Conf. ER Fluids, Technomic Publishing Co., Inc. Lancaster, P A , 1990, pp. 176-183. T. Y. Chen and P. F Luckham, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 78 (1993) 167. H. Block, E. M. Gregson, W. D. Ions, G. Powell, R. P. Singh, and S. M. Walker, J. Phys. E.: Sci. Instrum. 11 (1978) 251. Y. Kojima, T. Matsuoka, and H. Takahashi, J. Appl. Polym. Sci., 53 (1994) 1393.
794
24. K. Smith and G. Fuller, J. Colloid Interface Sci., 155, (1993) 183. 25. M. Jordan, A. Schwendt, D. A. Hill, S. Burton, and N. Makris, J. Rheol. 41 (1997) 75. 26. J. E. Martin and J. Odinek, J. Rheol. 39 (1995) 995. 27. K. Tanaka, R. Akiyama, and K. Takada, Polym. J., 28 (1996) 419. 28. Q. D. Nguyen and D. V. Boger, Rheol. Acta, 26 (1987) 508. 29. G. Astarita, J. Rheol., 34 (1990) 275. 30. C. P. J. Bennington. R. J Kerekes, and J. R Grace, Can. J. Chem. Eng., 68 (1990) 748. 31. R. T. Bonnecaze and J. F. Brady, J. Rheol., 36 (1992) 73. 32. J. P. Harnett and R. Y. Z. Hu, J. Rheol., 33 (1989) 671. 33. J. Schurz, Rheol. Acta, 29 (1990) 170. 34. M. Nakano, R. Aizawa, and Y. Asako, paper presented at the 5th International Conference on ER Fluids, Sheffield, UK, July, 1995. 35. M. Nakano and T. Yonekawa, in R. Tao and G. D. Roy (eds.) Electrorheological Fluids, Mechanism, Properties, Technology and Applications, World Scientific, Singapore, 1994, pp. 477-489. 36. D. Brooks, Int. J. Modern Phys. B, 6 (1992) 2705. 37. J. Arguelles, H. R. Martin and R. J. Pick, J. Mech. Eng. Sci. 16 (1974) 232. 38. K. D. Weiss and J. D. Carlson, in R. Tao (ed.), Proc. Int. Conf. Electrorheol. Fluids, 1991, World Scientific, Singapore,1992, pp. 264279. 39. G. B. Thurston and E. B. Gaertner, J. Rheol. 35 (1991) 1327. 40. E. V. Korobko and I. A. Chernobai, J. Eng. Phys. 48 (1985) 153. 41. Y. Kim, and D. Klingenberg, in K. O. Havelka and F. E. Filisko (eds.), Progress in Electrorheology, Plenum Press, New York, 1995. pp. 115130. 42. P. Placke, R. Richert, and E. W. Fischer, Colloid Polym. Sci. 273 (1995) 848. 43. E. Lemaire, G. Bossis, Y. Grasselli, and A. Meunier, in C. Gallegos, A. Guerrero, J. Mufioz and M. Berjano (eds.), Progress and Trends in Rheology IV, Verlag, Darmstadt, 1994, pp. 140-142. 44. G. G. Fuller, Optical Rheometry of Complex Fluids, Oxford University Press, New York, 1995. 45. J. M. Ginder, Phys. Rev. E, 47 (1993) 3418.
795
46. J. M. Ginder and L. D. Elie, in R. Tao (ed.), Electrorheological Fluids: Mechanisms, Properties, Structure, Technology and Applications., World Scientific, Singapore, 1992, pp. 23-36. 47. T. C. Halsey and J. E. Martin, in R. Tao and G. D. Roy (eds.) Electrorheological Fluids, Mechanism, Properties, Technology and Applications, World Scientific, Singapore, 1994, pp. 115-128. 48. J. Y. Wu, L. K. Shen, and L. W. Zhou, in W. A. Bullough (ed.), Proc. 5th Int. Conf. Electro-Rheological Fluids, Magneto-Rheological Suspensions and Associated Technology, World Scientific, Singapore, 1996, pp. 698-703.
797
SOME APPLICATIONS OF NON-NEWTONIAN FLUID FLOW
J. w. Hoyt Mechanical Engineering Department San Diego State University
1. NON-NEWTONIAN FLUID FRICTION R E D U C T I O N OVERVIEW
1.1 Introduction Drag reduction in the turbulent flow of solutions of polymers or surfactants has been studied for almost 50 years, beginning with the discoveries of Toms [1] and Mysels [2]. The scientific and technical aspects of these non-Newtonian fluid flows are still under active study, since the underlying mechanisms involved in these flow effects, like those of turbulence itself, are only dimly understood. Nevertheless, a great deal of empirical information has been obtained, outlined in several recent books: Sellin and Moses [3]; Gyr and Bewersdorff [4]; and Choi, et al [5]. There are also numerous reviews: Hoyt [6]; Virk [7]; Shenoy [8]; and Kulicke, et al [9] among many others. A very comprehensive bibliography listing over 4,900 references has recently appeared: Nadolink and Haigh [10]. Conference proceedings are another excellent source of current information: the most recent being sponsored by the American Society of Mechanical Engineers (Hoyt, et al, eds., [11]). 1.2 Basic Drag-Reduction Concepts If the pressure loss per unit length in turbulent pipe flow is less than that found with a Newtonian fluid such as water, the fluid is said to be dragreducing. Solutions of high molecular weight linear polymers (above, say
798
50,000), surfactants forming aggregates of rod-like micelles, and fiber suspensions all reduce the pressure drop or fluid friction, and thus can be inferred to interfere somehow with the three-dimensional fluctuations u', v', and w' associated with turbulent flow. There are at least three amazing aspects of the drag-reduction effect: 1) Drag reduction only occurs in turbulent flow, defined (in pipe flow) as VD/v (the Reynolds number) larger than 2000 or so, where V is the bulk fluid velocity in the pipe, D is the inside diameter of the pipe, and v is the kinematic viscosity of the liquid. 2) Only a small amount of high polymers, surfactants, or fibers are required to dramatically reduce the friction. Solutions of less than I partper-million of the most effective polymers show large friction reductions, while surfactant and fiber concentrations of several hundred ppm give substantial reductions. 3) Smaller quantifies of additive are required for a given effect as the molecular weight is increased, or (in surfactants) the aggregate-forming ability is increased. The better additives have molecular weights in the millions, or form aggregates of similar molecular weight. If the polymers are degraded, or the aggregates broken up, by exposure to high shear stresses, the drag-reduction effect disappears. Savins [12] introduced the term "drag reduction", defined in pipe-flow as a percent: DR, % = ([AP~- APpI/APs} xl00 where AP~ is the pressure drop per unit length in a pipe flowing the solvent alone and APp is the pressure drop for the same flow rate, using additives. Another commonly found term is the "engineering friction factor", ~,, defined as: 1 = [D aP] / [Tp V2]
where D is the pipe diameter and p is the fluid density. Friction factor Reynolds number plots are extremely useful in demonstrating the presence (or absence) of drag reduction, since for a Newtonian fluid, the relationship was deduced by Karman and Prandtl. A convenient computing formula for the Karman-Prandtl (pure solvent) value is:
799
1/~/~ = 1.8 log [Re/6.9] Comparing values obtained with this relationship with data from additive-solution flow at the same Reynolds number leads again to the drag-reduction percentage, since DR, % = [{~s- Xp} / ~,~]x 100 The Karman-Prandtl relation is for a smooth pipe only; textbook expressions can be used for pipes known to be rough in nature. Care must be taken in examining literature values for friction factor, since chemists and occasionally other scientific workers may use the "Fanning" friction factor which has a value 1/4 that of the D'Arcy or engineering factor defined above. The maximum value of drag reduction which can be obtained by using additives seems to be about 80%; a more refined estimate has been suggested by Virk [13], which is often referred to in the literature as "Virk's asymptote". Some of the most effective water-soluble drag-reducing polymers such as poly(ethylene oxide) and polyacrylamide can be obtained in very high molecular weights (ca. 5 million or more) and their properties have been extensively studied in laboratory investigations. Their use in industrial or commercial applications has been extremely limited due to the breakdown or degradation of the polymer molecules caused by shearing or agitation in pumps or other mechanical devices. The fragile long chains (which are extended in dilute solution) are readily fractured by passing through pumps. Another factor is the fairly high cost of the polymers, which limits their application to the transport of high value products. As explained in the next section, these objections to the use of polymers to reduce pipe friction are overcome in petroleum pipelines, where there may be hundreds of miles between pumping stations, and an increased flow rate is highly advantageous. Drag-reducing aggregates of surfactant molecules are also broken up by passing through pumps, but unlike polymer molecules, the aggregates reform when the shear stresses are relaxed, returning to their original effectiveness. Economic considerations suggest that surfactant solutions are best fitted to recirculating flow systems. Thus surfactants may play an
800
important role in systems such as large-scale domestic heating or cooling, as described in a following section. The concepts outlined here in a very general form are given as an introduction to the more detailed applications of polymers and surfactants in fluid flows found in the following sections.
2. OIL-PIPELINE FRICTION R E D U C T I O N 2.1 Introduction
The use of high-polymer additives to reduce the pipeline friction of crude-oil and petroleum products has been a spectacular success, and currently forms an important part of industrial petroleum transport technology. Several factors account for this success. First, although the drag-reducing polymers are not inexpensive, their use in parts-per-million quantities to greatly increase the flow of a very high-value product is often much more economical than the capital and operating expense of installing additional pumping capacity. Secondly, the additive can be applied with relatively minor portable equipment, which can be easily removed in case flow augmentation is no longer needed. Another factor is that there is no need to remove or treat the additive-containing product in any special way - the additives disappear in the refining process, or are innocuous in other pipeline products. Finally, of importance in crude-oil transport in cold climates, the greatly reduced heat transfer of additivecontaining fluids helps keep the product viscosity at a lower level. These advantages have led to installations of drag-reducing additives at more than 80 locations worldwide (Motier, et al, [14]), since the initial trials in the Trans Alaskan Pipeline in 1979. Steady improvements in polymer properties have accelerated this use; Motier, et al, show that currently only about 1/14 the quantity of polymer (compared with the 1979 additive) is needed for the same amount of friction reduction. 2.2 Additive Characteristics
The commercial chemicals used as drag reducing additives are highly proprietary. Motier and Carrier [15] suggest that the additives are o~olefin polymers or copolymers; the molecular weights are extremely high 35 million or more. A surprising aspect of these high molecular weight
801
polymers is their resistance to shear degradation; in the 1.2 m dia Trans Alaskan Pipeline, high drag reduction is maintained over a 353 km section with initial polymer concentrations currently on the order of a few ppm. The additives are supplied commercially as 10% solutions in a carrier similar to kerosene. With this polymer concentration, the resulting fluid is extremely viscous; injection pumps require gas pressure boosting at the suction side to function properly. No particular injection nozzles or equipment seems to be required - the additive is simply pumped into the pipeline and disperses via turbulence. Higher polymer-concentration suspensions are under study to reduce the supply volume required. Although initially there was concern that additives might present a problem in crude-oil refining, the o~-olefin polymers appear perfectly acceptable to refinery processing, and at the current levels of pipeline application (1-2 ppm; Motier, et al, [141] would be practically undetectable anyhow.
2.3 Applications The first and probably the major success story in polymer drag reduction was (and is) the Trans Alaskan Pipeline System (TAPS). The highly competitive and proprietary nature of the petroleum industry has meant that very little detail regarding applications is available in the open literature, but Burger, et al [16] have given some actual test data on flowing Alaskan crude through 2.66, 5.25, 33.4, and the TAPS 119.4 cm dia pipes. From these d a t a , a rather convoluted scheme based on estimated molecular parameters was used to predict the TAPS performance from laboratory data on the smaller pipe sizes. Although the scheme worked well enough to warrant installation of drag reduction in the TAPS pipeline, the much simpler method described later in this Chapter, based on hydrodynamics, leads to even better predictions as shown in Figure 1. From Berretz, et al, [17]; Beaty, et al, [18]; Hom, et al, [19]; and Motier, et al [14], we learn that the initial success of drag reduction lead to the cancellation of plans to build two additional pumping stations on the TAPS pipeline. The 1.45 million barrels per day pumping capacity of the pipeline was raised to 2.1 mbpd by polymer injection. Above 50% drag reduction in the 1287 km long pipeline is achieved with 28 ppm or less polymer. The logistics of providing some 95,000 liters per day of 10% polymer solution in kerosene to remote locations in Alaska are impressive.
802
0.100
II
II
I
_ --
IIII
.
.
I
I
.<
d
o .w o
4-1
0.010
o -r4
4J O h~
0.002 I 000
.
,I
.
I O, 000 Reynolds
I
.
I 00,000
.
.
.
.
I , 000, O(
number
Figure 1. Pipe-flow data for Alaska crude oil with 10 ppm drag-reducing additive. Actual data for 3 pipe sizes, with scaling from the two smaller (2.664 and 5.25 cm dia) to the full-scale (119.4 cm dia) pipeline. Adapted from Burger, et al [16] tabulated data. I"1 2.664 cm pipe dia; O 5.250 cm dia; A 119.4 cm dia; E1'119.4 cm prediction from 2.664 cm dia data; (~ 119.4 cm prediction from 5.250 cm dia data; ---Karman-Prandtl friction line for Newtonian fluids. Similar formulations have now been applied to other crude oil pipelines (Beaty, et al, [18,20]; Lester, [21]; Motier and Prilutski, [22]; Motier, et al, [14], as a apparently viable commercial technique. The low installation cost, and impressive flow increase performance, together with the "use only when needed" feature has lead to increased acceptance in crude oil production, replacing expensive fixed assets which may have only a short usage requirement. Polymer additives have been used with great success in petroleum product pipelines. Carradine, et al, [23]; Motier, et al, [24]; Muth, et al, [25-27]; and Motier and Carrier [15] describe applications to diesel oil,
803
gasoline, and natural gas liquids pipelines. Drag reductions of over 40% appear to be routine, and this offers the possibility of eliminating marginal pumping stations. 2.4 S u m m a r y
The use of high-polymer additives to enhance flow in petroleum pipe lines has been a great commercial success. Additive effectiveness has been improved so that only a few parts-per-million are required to give friction reductions of 50% or more. The economics of polymer addition seem to be very favorable in the pipeline transport of high value products such as petroleum, reducing the need for pumpiv.g stations as well as increasing throughput.
3. SURFACTANT APPLICATIONS TO LARGE-SCALE HEATING AND
COOLING SYSTEMS 3.1 Introduction
Many regions are served with district-wide heating or cooling systems, which, in the case of heating for example, may involve transporting hot fluid over considerable distances. The heat source is usually lower-grade energy from electrical generating stations or industrial operations which would otherwise be rejected to the environment. Offsetting this low-cost source is the energy requirement to pump the hot fluid, typically several kilometers or more, to heat exchangers servhag domestic needs. The use of surfactants as drag reducers to reduce this substantial pumping requirement is under intensive study. Mound 7% of the heating requirement in Germany, 35% in Sweden, and as much as 40% in Denmark and Finland is centrally supplied, while district heating serves six hundred thousand households in Seoul, Korea. Hence the opportunities for energy saving are plentiful. Moreover, in new designs, smaller transmission pipes might be used, if drag reduction by surfactants achieves acceptance. 3.2 Surfactants - Basics
Surfactants are a broad class of surface-tension reducing chemicals, characterized by having both a hydrophilic (watersoluble) and a hydrophobic (water-repelling), often oil-soluble, component in the same molecule. In solution, depending on the detailed chemistry, surfactants
804
may be ionic or non-ionic. The anionic group forms the basis of household detergents, which are produced in enormous quantifies world-wide. Cationic and non-ionic surfactants are less well-known, but readily available industrial chemicals. Thirty years ago, Savins [28] showed that some anionic surfactants could greatly reduce the turbulent pipe friction as compared with the water in which they were dissolved. At higher shear stresses, the drag reduction disappeared and the friction was identical to water. Remarkably, when the shear stress was lowered, the solutions were again found to be drag-reducing. Many surfactants of various ionic classes have now been found to be friction-reducing. All studies show that pipe friction reductions of up to 80% can be observed, but at higher shear stresses, the drag reduction effect abruptly disappears, promptly reappearing when the shear stress is lowered. 3.3 Surfactant Micelles
Originally it was thought that drag reduction in surfactants was due to the formation of rod-like (or worm-like) "micelles", which caused the effect, but were then broken up and reformed as the shear stress exceeded some critical level and then was reduced. Ohlendorf, et al [29] have now shown that the rodlike micelles take appreciable time (up to hours) to form initially, and are not broken up in high shear-stress situations. Rather, the rods aggregate into much larger structures, which can align themselves in the flow direction, or form a network in the flow which results in a lowering of the turbulent friction. When the aggregates are exposed to shear stresses exceeding a certain level, they are dispersed (i.e. the aggregate binding forces are overcome) and the drag-reduction effect is lost. However, unlike long-chain polymers, the surfactant micelles will reform again (in a few seconds) into the drag-reducing aggregates when the shear stress is reduced below the critical level. Thus the drag-reducing ability of surfactant solutions remains essentially intact after passing through pumps, valves, etc., which would destroy the effectiveness of longchain polymer solutions. The detailed chemistry involved in micelle formation and the subsequent aggregation is quite complex. Apparently, micelles are initially roughly spherical, and begin to form at a surfactant concentration which depends on the surfactant itself, as well as temperature and presence of
805
counter-ions. This "critical micellular concentration" (CMC) is often on the order of a few hundred parts-per-million or less. At first, the hydrophobic regions of the individual molecules group together to minimize their area exposure. Electrostatic forces also play a role, but in the presence of counter-ions the initially spherical micelles (at least in drag-reducing surfactants) tend to rod-like form, at somewhat higher concentrations. Ohlendorf, et al [29] estimate the rod length for a typical drag-reducing surfactant micelle to be around 10 to 40 nm, depending on temperature. These rods form at a surfactant concentration from two to five times the "CMC", again depending on the temperature. Explanation of the existence of aggregations of these rod-like forms is less straight-forward. The two principal phenomena which suggest aggregation are shearthickening and drag-reduction itself. In Figure 2 we see how the viscosity of a typical drag-reducing surfactant solution suddenly jumps from approximately that of water to a value 10 times larger as the shear stress is increased. This astonishing behavior must be due to the micellular rods (which do not overlap in the solvent at rest) suddenly forming a network or a lengthy aggregate. Flow birefringence studies suggest the latter. That significant drag reduction occurs in turbulent flow is another indication of lengthy aggregates. In Figure 3 we note the pipe-flow behavior of a drag-reducing surfactant solution, with the engineering friction factor, ~, ,plotted as a function of the Reynolds number VD/v, where V is the average flow velocity in the pipe, D the diameter, and v the viscosity of the solvent (water). While the laminar pipe flow friction factor is somewhat higher than water (due to higher viscosity as noted before), in the turbulent region we notice a large reduction compared with the water solvent. This action is similar to that of high-molecular-weight polymer solutions, where it is thought that, in turbulent flow, the polymer coils unwind to form lengthy strands which act to dampen the u', v', W r velocity fluctuations in the flow. Similarly, turbulent surfactant flow also dampens these fluctuations, thus supporting the idea that aggregates occur, and that they are of extended length. Further, at a certain higher Reynolds number the friction factor abruptly returns to the pure water value, suggesting that the aggregates are dispersed by the more violent velocity fluctuations occurring as the Reynolds number is increased. Curves such as Figure 3 are typical of drag-
806
10 q ,mPa m,
Y. ! s e c
0.1
I
10
|
10
,i
J
10
|1
!
10 4
Figure 2. Viscosity versus wall shear rate for 0.1% solution of C~6 TASal a t 35~ adapted from Ohlendorf, et al [29]. reducing surfactant flow; individual curves are functions of temperature, surfactant type and concentration, counter-ion type and concentration, as well as pipe diameter.
3.4 Basic Laboratory Experiments Many surfactants of various ionic classes have been identified as friction-reducers, and Shenoy [30] has given an extensive review of earlier work. Work has continued in an effort to provide surfactants for districtheating with improved temperature range, lesser environmental hazard, lower cost, etc. Determination of the most useful surfactant for a proposed application is a complicated balance involving shear stress in the pipe, surfactant and counter-ion concentration, and range of operating temperatures. The drag reduction which is obtained is a non-linear function of all of these variables for a given surfactant. Table I lists many of the surfactant formulations under active study for large-scale district heating.
807
Table 1 Drag-Reducing Surfactants Cationic myristyltrimethylammonium salicylate cetyltrimethylammonium salicylate cetyltrimethylammonium chloride (Arquad 16-50) with NaSal counter-ion (Arquad 18-50) with NaSal counter-ion 12 with NaSal counter-ion (Oleyl-N(CH3)(C 2H40H)2C1 ) Ethoquad T 13-50 with NaSal counter-ion (tallow-tris hydroxyethyl ammonium chloride) Habon G hexadecyl dimethyl hydroxyethyl ammonium-3hydroxy-2-naptholate (n-alkyl; n = 16) Obon-G n - 18; also with added NaSal Dobon-G n = 22 C14TASal C16TASal C16TAC C16TAC C18TAC Ethoquad 0/
Zwitterionic N-alkylbetaines with N = 15 or 17, plus Na dodecylbenzenesulphonate
Selection of a surfactant compound for a given application is a balance involving cost, environmental considerations, and useful lifetime, as well as performance at various temperatures as a drag reducer, in pumps and heat exchangers, valves and meters, etc. Laboratory tests have focused on several of these factors. Figure 4 (Chow, et al, [31]) shows the effect of operating temperature on drag reduction performance of a typical alkyl trimethylammonium chloride (Arquad 18-50) with equal weight NaSal counter-ion. For this surfactant combination, the maximum drag reduction (70-80%) was obtained at higher Reynolds numbers as the temperature was increased from 30 to 90 ~ C. At 100 ~ C the drag reduction effect greatly decreased. A
808
0.080
0.040
~<
I
I
I
!
X
I
Water
o
0.020
0 o U. r 0 o
Symbol
U.
A
- ~
0.090 in.
•
0.008
o
? -
0.500 in.
0.004 10 3
10 4
10 5
Reynolds Number
Figure 3. Drag reduction with an equimolar solution of C~6 TABr and 1napthol to give a total concentration of 508 ppm. Adapted from White [51]. wide operating range is desirable in practical systems. Figure 5 (Schmitt, et al, [32])shows that, if protected from oxidation, temperature recycling from 30 to 120 ~ C gave reproducible drag-reduction results with a 1000 ppm solution of surfactant. Each temperature cycle extended over a period of 34 hours. Other work with C~4TASal and C16TASal solutions has been reported by Bewersdorff and Ohlendorf [33], Bewersdorff [34], and Harwigsson, [35]. Recent studies using Arquad and Ethoquad compounds include Lu, et al [36], Myska and Zaldn [37], and Usui, et al, [38]. Park, et al [39] found no effect of a centrifugal pump on the friction-reducing properties of several
809
[c,a I
I
I
iG
jlll
l
ji
CH 3
7",," N - -
.
.
.
.
it-
iii
i
_ i
i
_
Cl
I CH 3
C
o6 u
0
SYMBOL
0
-
T ( o C)
iL
i
i i
o +
.30
Z
50
o 9
70 80 90 1 O0
40
X o
6
D
--
0.243
iN.
f~ I
10"
10 5 Reynoi(:J$
Arcluod
18/50
-~-
NoSol
10 a
Number (2000
/
2000
IDIDm)
Figure 4. Effect of solution temperature on drag reduction for 2000 ppm surfactant. At 100~ much of the drag-reduction effectiveness is lost. Adapted from Chou, et al [31]. surfactant solutions, including C~6TASal and Habon-G, confirming earlier results of Steiff, et al [40] and Gaslievic and Matthys [41]. 3.5 Environmental Considerations
In spite of only moderate toxicity of most of the surfactants under study for district heating applications, the large quantities involved make environmental considerations very important. Harwigsson, et al [42] estimate that about 60% of the 7 x 105 m 3 of water circulating in Swedish domestic heating systems must be replaced annually due to leaks and maintenance. At 500 ppm of surfactant, this would mean about 200 tons of surfactant discharged into the environment each year. Therefore, more benign and easily biodegradable surfactants having drag reducing qualifies are being sought. Hellsten, et al, [43] have suggested Zwitterionic surfactants as being possible candidates. (zwitterionic surfactants contain
810
DR [%]
80-
L 6O
z.O i-
20 ,,...,
1
40
60
80
100
120 T[oC]
Figure 5. Results of a temperature recycling test with a +10~ rate of temperature change. An oxygen stabilizer, hydrazene, added to the 1000 ppm surfactant solution to avoid degradation. cycle lasted 34 hours, flowing the solution through a 12.7 mm Adapted from Schmitt, et al [32].
per hour has been Each test dia pipe.
both positive and negative charges.) Compounds investigated by Hellsten exhibited drag reduction, but only over a limited range of temperatures. A contrasting opinion is offered by Steiff, et al, [44], who present data suggesting that Habon-G and Dobon-G are environmentally acceptable in Germany, and solutions of these surfactants would require no further
811
treatment before discharge into a sewerage system. As a background in considering the environmental concerns of surfactants in district heating systems, it should be remembered that these systems usually contain substantial quantifies of other substances, such as the oxygen scavenger, hydrazene, so that evaluation of the effects of leakage, etc, becomes more complex. 3.6 Heat Transfer Considerations
As noted in the Section on oil-pipeline friction reduction, drag-reducing additives reduce heat transfer in pipes even more spectacularly than the friction. Thus in the long-distance transport of heated fluid with surfactants to reduce the pipe resistance, the overall heat loss also can be reduced, offsetting this advantage is the problem of extracting the heat a t the point of use. Some of the suggestions for overcoming this problem are listed below: Use a heat exchanger of the cross-flow type Increase the velocity in the heat exchanger Break up the micelles just before the heat exchanger Cross-flow or plate-fin heat exchangers may be advantageous in surfactant-solution heat exchangers, since the boundary-layer effect found in pipes is reduced. Hoyt and Sellin [45] found little difference in crossflow heat transfer in a drag-reducing polymer solution compared with water, when the results are correlated using a Reynolds number based on the increased viscosity of the polymer solution. In large-scale tests, however, as explained in Steiff, et al [44], it has been found that there is still a substantial reduction in effectiveness in these types of heat exchangers. Pollert, et al [46] concur in these results, but note that if the fluid velocity through the exchanger is increased, there seems to be no loss of heat transfer effectiveness. Kawaguchi, et al [47] suggests that bringing the surfactant solution to a temperature a few degrees higher before entering the heat exchanger could destroy the micelles and thus restore the heat exchange properties of the fluid to that of water. Others have suggested mechanically breaking
812
up the micelles by mixers or valves, before entering the heat exchange region. The practicality of these ideas remains to be tested. Contrary to expectations from laboratory results, field tests (Steiff, et al, [40], Pollert, et al, [46]) seem to show little or no influence of the surfactant additives on the overall system heat transfer. While this may suggest that the principal thermal resistance was not on the surfactant side of these heat exchangers, it cannot be assumed that this is the usual case. Matthys [48] has given a thoughtful review of current understanding of the heat-transfer problem in the flow of drag-reducing fluids, supplemented more recently by experimental work (Gasljevic and Matthys, [49]).
3.7 Large-Scale Demonstrations Brief reports on actual use of surfactants in district heating systems have become available. Steiff, et al [40] describe results from full-scale tests in Volkingen, Germany, which appear to confirm the energy savings expected due to decreased pumping-power requirements. Similarly, Pollert, et al [46] found in a demonstration in Kladno-Krocehlavy, Czech Republic, that pumping power was reduced by 40% when Habon-G was employed. Interestingly, the surfactant retained its properties in the system for the two winter-heating seasons studied. As mentioned above, no effect on heat exchanger operation was noted in these two investigations. Gasljevic and Matthys [50] give a very complete report on the use of surfactants in the cooling system of a large building. This application of surfactants is even more challenging than district heating since the many fittings, valves and heat exchangers add much more complexity to the flow patterns than the long straight runs expected in district systems. In this chilled water system, a reduction of about 30% was achieved in the required hydraulic pumping power. As expected, the heat transfer, both in the "chiller" (where the circulating water is cooled by refrigerant), and in individual room heat exchangers, was diminished. However, the reduction was less than might be expected, due to large heat transfer resistances found on the refrigerant and air sides of the heat exchangers. In other words, the principal barrier to heat exchange was not the chilled surfactant solution, but rather the other fluid components of the system.
813
A complex cooling system such as that studied by Gasljevic and Matthys involves numerous components which interact in complicated ways, making analysis difficult, and somewhat uncertain. Nevertheless, the field test was regarded as very successful, no doubt leading to further implementation of the use of surfactants in chilled-water cooling systems.
3.8 Summary The use of surfactants in district and building heating and cooling systems is an emerging technology with great promise for significant energy savings. While there seems no doubt that the pumping-power requirements of such systems can be greatly reduced by the use of surfactants, appropriate formulations to avoid environmental concerns and equipment designs to enhance heat transfer are important next steps in securing commercial acceptance. 4. SCALE-UP By scale-up, we mean predicting the pressure drop in large industrial or commercial pipelines based on results from laboratory-scale small pipes. Scale-up for the use of drag-reducing polymer solutions in long-distance off pipelines or central-station heating or cooling schemes is an important current problem. Astarita [52], in an elegant article tracing references back to Vitruvius (35 B.C.), declared that a non-trivial scaling theory cannot be constructed for non-Newtonian fluids. Many attempts to provide scale-up information for drag-reducing fluids have tended to confirm this gloomy conclusion, by introducing graphical or iterative procedures (Granville, [53,54]; Matthys and Sabersky, [55]; Sellin and Ollis, [56]; Taylor and Sabersky, [57]), or by admittedly empirical methods (Savins and Seyer, [58]; Gasljevic and Matthys, [59]). However, quite recently, accurate scale-up procedures for the same fluid (homologous scaling, according to Astarita) have become possible, based on better knowledge of the basic fluid dynamics involved in such flows. We start by recalling the velocity profile for drag-reducing fluids"
814
24
~
-
22
20 . - -
18
---
16
.
14
.
.
.
--
/o
AB 12
12 p p m I 10
---
'
J-
2
r
n
i1~
6 ppm
0
0.05
0.1
0.2
0.5
1.0
U ~ (fps)
Figure 6. Delta B as a function of u* for poly(ethylene oxide) in a one inch d i a m e t e r pipe. u+ =Alny §
815
where A and B are constants having the approximate values of 2.5 and 5.0, and AB is a function of the drag-reducing substance, its concentration, pipe roughness, etc. The problem has been that AB is a highly non-linear (and non-predictable) function which must be determined by experiment. Figure 6 shows a typical result. An assumption made in most scale-up techniques is that the value of AB is the same for equal values of shear stress (or friction velocity, u*) in both small and large pipes. Based on this assumption, Hoyt and Sellin [60] introduced a relatively simple scale-up calculation method which views the drag reduction as an "negative" analogy to the familiar pipe roughness employed in Newtonian fluid flow. An even more recent advance has obviated the need for such an analogy. Anderson, et al, [61] provide data from a large-scale experiment which demonstrates conclusively that, for equal values of shear stress, AB has the same value in both small and large pipes. Figure 7 is an example of measurements of AB plotted against the friction velocity (u*), for pipes differing in diameter by a factor of 6. The same polymer solution was delivered to the test pipes and the measurements made simultaneously, side-by-side, so that the only difference was pipe size. Similar results were obtained over a wide range of polymer concentrations. Scaling using the invariance of the AB-u* relationship proved to be very precise. Hence we can greatly simplify the scaling relations. Extending Prandtl's logarithmic velocity distribution law by including AB, we find that: 1/~/~,- 2 log [Req~,/2.51] + AB/~/8 where X is the engineering friction factor for flow in pipes and Re is the Reynolds number VD/v. Solving for AB, and letting subscript I indicate the small pipe, and 2 the larger, AB~ = q8[1/q~-2 log {Req~/2.51 } ] and AB2 = ~/811/~/X2/2 log {ReqX2/2.51}].
816
5O e 0
40
D,4.IS' 0.,1~6"
30 m <1
g
Oo 20
10 o
O
,
2
L
,
I
3
.
,'
9 !
,
4
I 9 1~
J . l , , , l ~ l
se7
!
]
,
t
,
1
2
10-1
.
~
,
!
s
9
I
~
I
4
9 J
. [ . l . l . t , l
s87
'
100
U,-ftJs
Figure 7. Delta B as a function of u* for p o l y a c r y l a m i d e p o l y m e r solution in two different size pipes. The m e a s u r e m e n t s were m a d e simultaneously, side-by-side, using 33 p p m solution from the same reservoir. Equating AB's and noting that Req~, - u*Dq8/v, where D is the pipe diameter: 1/~/s
- 1/~/)q +2 log [u*D2"48/2.51v] - 2 log [u*D1~/8/2.51v]
Hence 1/'4K 2 - 1 / ~/k~+ 2 log
[D2/D,].
817
4
0 U 0 c 0
o o
10-2
0
.--
u
%
+ _
@@ @ 0 0 ,
104
:
0=6.15" Meosured D=l.026"bteasured 0=6.15"Predicted ,
t,
2
t,l,t,l
!
3 45
1
t
t
@
t]
105
,
',
,
t
2
,,,I,~,1
!
3 + 5
1
!
I
II
106
t
,
t
2
.,,!,~,
3
ReynoldsNumber Figure 8. Friction factor as a function of Reynolds number for the data of Figure 7, together with scaling predictions for the 6.15 inch dia pipe flow, based on the data from the 1.026 pipe. Black dots are the actual test data for 6.15 inch dia. The friction factor k2 so obtained from a plot of ~a vs Re1 is to be replotted at the same shear stress, and so, equating u* from the Reynolds number expression given above"
Re2 = [ ~/X~/4K2] [D2/D~]Rel. This extremely simple scaling procedure yields excellent results as shown in Figure 8, where the data from a 1.026 inch diameter pipe test has
818
been scaled up to 6.15 inch diameter and compared with the four available test points. Similar agreement of scaling and actual test data was obtained for five other polymer concentrations, which suggests that the procedure is reliable, as well as being grounded in fundamental engineering relations. Astarita will be surprised! The above scaling procedures work very well if the smaller pipe is 1/2 inch (13 mm) or more in diameter. As shown in Hoyt and Sel!in [60], the velocity profile in pipes of lesser diameter is significantly different from that of larger pipes, thus voiding similarity scaling. If only very small diameter pipe-flow data are available, the empirical approach of Savins and Seyer [58] has given useful results. Savins and Seyer replace the actual shear stress of the polymer solution flow with that of the pure solvent. In effect, this modifies the velocity profile to approximate that of the larger-scale polymer pipe. If data from very small pipes are the only available source of scaling information, the Savins and Sayer scheme probably remains the only alternative. It has been used with considerable success in off pipelines (Lester, [62]).
5. FLOW TRACERS 5.1 Introduction
Many techniques with varying levels of sophistication have been proposed as methods of showing the flow path, vorficity, or velocities of turbulent flow in boundary layers or around various objects. The simplest idea is a dye streak, but dyes suffer from very rapid dissipation and are almost useless in turbulent flow. Hence very elaborate techniques, such as Particle Image Velocimetry, have been brought forward to provide turbulent flow information. However, the equipment cost of the lasers, shutters, cameras, etc., involved in these methods is beyond the reach of many investigators. By incorporating non-Newtonian properties into a dye-streak, a highly effective flow tracer has been developed. Such a tracer is of great interest since it allows one to follow turbulent and separated flows currently inaccessible with simple methods.
819
5.2 Tracer Development The non-Newtonian tracer has been developed at Bristol University (Hoyt and Sellin, [63-65]) using ideas based on drag-reduction technology. The basic concept is to incorporate both the shear-induced-state (SIS), sometimes referred-to as shear thickening" or "strain-hardening" flow aspects found in surfactant solutions, with the high "extensional viscosity" or "thread-drawing" properties of polymer solutions. These are then used to form (with a colorant) a tracer fluid which can be ejected into the turbulent flow as a dye-streak which resists dispersion and breakup while following the flow path. Key to the development of the tracer is the amazing viscosity- shear stress relationship of certain surfactant solutions. As shown earlier in this chapter, solutions of C~6TASal (as well as C~4TASal) exhibit a sudden order-of-magnitude jump in viscosity as the shear rate is increased from a fairly low value. This attribute is used in the tracer to stabilize the ejected strands of colored fluid, so that they remain coherent in rapidly changing stress fields. The tracer threads are held together axially by the high extensional viscosity provided by high-polymers such as poly(ethylene oxide) or polyacrylamide. This contribution is very important as otherwise the surfactant strands would tend to snap. A final component of the tracer is a small amount of white latex wall paint to act as a colorant. The paint also acts to add additional stability to the ejected tracer threads. Since the tracer is a mixture of surfactant, polymer, and paint suspension, the rheology is rather complex, and largely unknown. Most of the development work has been with the following formula, or slight variations thereof: 2% C16TASal 1/2% PEO Tap Water White Paint
500 250 1000 5
ml ml ml ml
The 2% C~6TASal solution is made up from 2% (by weight) cetyltrimethylammonium bromide, mixed with 2% by weight sodium salicylate. C~4TASal (made from myristyltrimethylammonium bromide and NaSal in the same manner) gives a more responsive tracer thread;
820
. /
mixtures of the two can be employed. As explained in a previous section of this chapter, more dilute solutions of these compounds have been studied for possible use in district heating systems. PEO is polyethylene oxide) (Polyox WSR-301, Union Carbide), a familiar drag-reducing additive. A~I/2% solution, made up by dispersing the dry powder in methanol, and gently mixing in deionized water, provides adequate elogational viscosity, evidenced by a high degree of "thread drawing" (noticed when pouring the solution from a container). Other high polymers such as polyacrylamide can be used. The tracer components are stored separately and mixed together shortly before use. The mixed tracer is dispensed from a rake of I m m ID hypodermic tubing as shown in Figure 9, or from individual tubes, fed by gravity or a small pump. The non-Newtonian character of the tracer is shown by the slight "die-swell" as the fluid leaves the hypodermic tubes.
Figure 9. Method of dispensing tracer from rake of hypodermic tubing. Figure 10 shows flow around a 51 mm dia cylinder at a Reynolds number of 11,500. A mirror above the flow channel allows the threedimensional characteristics of the flow to be observed simultaneously with the view from the side. The 3-D flow visualization obtained in this manner is at present unavailable from any other technique. A frame from a video recording using the tracer is shown in Figure 11. A much more diute solution of tracer has been used in order to show the smaller scales of the turbulent flow around a 51 m m dia cylinder at a
821
Figure 10. Side view of flow around 51 m m dia cylinder at Re = 11,500. Plan view is seen in mirror above. Reynolds number of 12,600. The turbulent character of the boundary-layer flow leaving the cylinder, as well as details of the vortices being formed, can be revealed by frame-by-frame examination of the video recording. This flow tracer method permits the use of ordinary film and cameras, as well as "home" video, and of course, visual observation. It is hoped that the technique will find a useful place in education, inasmuch as it gives more detail than presently available from computer simulations or other flow visualization methods. The only disadvantage is that, in a recirculating flow facility, drag-reducing polymer is returned to the test section.
822
Figure 11. Video frame of flow around 51 mm diameter cylinder at Re = 12,600.
DEDICATION
This Chapter is dedicated to the memory of Preston Lowery HI, Chen Liang, and Constantinos Lyrintzis, Engineering faculty colleagues and friends at San Diego State University, whose lives were snuffed out by a deranged graduate student, August 15, 1996.
ACKNOWLEDGMENTS
The support of the University of Bristol and the U.S. National Science Foundation in preparing this chapter is very greatly appreciated. The confidence in this work expressed by Dr. Michael Roco of the N SF, through Grant No. CTS-9508409, is gratefully acknowledged.
REFERENCES @
B.A. Toms, in Proc. 1st Int. Congress on Rheology, North-Holland Publ., 2, 1948, 135.
823
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
K.J. Mysels, Flow of Thickened Fluids. U.S. Patent 2 492 173 (1949). R.H.J. Sellin and R.T. Moses (eds.), Drag Reduction in Fluid Flows, Ellis Horwood, Chichester, 1989. A. Gyr and H.-W. Bewersdorff, Drag Reduction of Turbulent Flows by Additives, Klewer Academic, Netherlands, 1995. K.-S. Choi, K.K. Prasad, and T.V. Truong (eds.), Emerging Techniques in Drag Reduction, IMechE, London, 1996. J.W. Hoyt, Trans ASME, J. Basic Engrg., 94 (1972) 258. P.S. Virk, AICHE Journal, 21 (1975) 625. A.V. Shenoy, Colloid Polymer Sci., 262 (1984) 319. W.-M. Kulicke, M. Kotter, and H. Grager, in Advances in Polymer Science 89, Springer-Verlag, Berlin, (1989) 1. R.H. Nadolink and W.W. Haigh, Applied Mechanics Reviews, 48 (1995) 351. J.W. Hoyt, et al (eds.), Symposium on Turbulence Modification and Drag Reduction, ASME, FED-Vol.237, New York, 1996. J.G. Savins, Soc. Petrol. Eng. Journal, 4 (1964)203. P.S. Virk, H.S. Mickley, and K.A. Smith, Trans ASME, J. Applied Mechanics, 37 (1970) 488. J.F. Motier, L.-C. Chou, and N. Kommareddi, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 229. J.F. Motier and A.M. Carrier, in Drag Reduction in Fluid Flows, Ellis Horwood, Chichester (1989) 197. E.D. Burger, L.G. Chorn, and T.K. Perkins, Journal of Rheology, 25 (1980) 603. M. Berretz, J.G. Dopper, G.L. Horton, and G.J. Husen, Pipeline and Gas Journal (September 1982). W.R. Beaty, R.L. Johnston, R.L. Kramer, L.G. Wamock, and G.R. Wheeler, in Third International Conf. on Drag Reduction, Bristol University (1984) F.1. A.F. Horn, J.F. Motier, and W.R. Munk, in Drag Reduction in Fluid Flows, Ellis Horwood, Chichester (1989) 255. W.R. Beaty, R.L. Johnston, R.L. Kramer, L.G. Warnock, and G.R. Wheeler, Oil & Gas Journal, 82 (Aug. 13, 1984) 71. C.B. Lester, Oil & Gas Journal, 83, (1985) 51; 76; 107; 116.
824
22. J.F. Motier and D.J. Prilutski, in Third International Conference on Drag Reduction, Bristol University (1984) F.2,1. 23. W.R. Carradine, G.J. Hanna, G.F. Pace, and R.N. Grabois, Oil & Gas Journal, 81 (1983) 92. 24. J.F. Moiler, D.J. Prilutski, Z.-J. Shanti In, and R.J. Kostelnik, in Third International Conference on Drag Reduction, Bristol University (1984) F.3. 25. C.L. Muth, C.J. Stansberry, G.J. Hussen, M.H. Lewis, and M.S. Ziobro, Pipeline & Gas Journal, (June, 1985) 38. 26. C.L. Muth, T.U. Hannigan, R.S. Vruggink, and G.F. Pace, Pipeline & Gas Journal, (June, 1986) 26. 27. C.L. Muth, M.J. Monahan, and L.S. Pessetto, Pipeline Industry, (July, 1986) 43. 28. J.G. Savins, Rheologica Acta, 6 (1967) 323. 29. D. Ohlendorf, W. Interthal, and H. Hoffman, Rheologica Acta, 25 (1986) 468. 30. A.V. Shenoy, Colloid & Polymer Science, 262 (1984) 319. 31. L.-C. Chou, R.N. Christensen, and J.L. Zakin, in Drag Reduction in Fluid Flows, Ellis Horwood, Chichester,1989, 141. 32. K. Schmitt, F. Durst, and P.O. Brunn, in Drag Reduction in Fluid Flows, Ellis Horwood, Chichester, 1989, 205. 33. H.-W. Bewersdorff and D. Ohlendorf, in Turbulent Shear Flows 5, Cornell University (1985) 9.41. 34. H.-W. Bewersdorff, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237, (1996) 25. 35. I. Harwigsson, "Surfactant Aggregation and its Application to Drag Reduction". Ph.D. Thesis, Lund University, 1995. 36. B. Lu, Y. Talmon, and J.L. Zakin, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 169. 37. J. Myska and J.L. Zakin, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 165. 38. H. Usui, T. Itoh, and T. Saeki, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 159. 39. S.B. Park, H.S. Suh, S.H. Moon, and H.K. Yoon, hi Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 177.
825
40. A. Steiff, W. Althaus, M. Weber, and P.-M. Weinspach, in Drag Reduction in Fluid Flows, Ellis Horwood, Chichester, 1989, 247. 41. K. Gasljevic and E.F. Matthys, in Recent Advances in Non-Newtonian Flows, ASME, AMD-Vol.153 (1992) 42. I. Harwigsson, A. Khan, and M. Hellsten, Tenside Surf. Det., 30 (1993) 174. 43. M. Hellsten, I. Harwigsson, C. Blais, and J. Wollerstrand, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 37. 44. A. Steiff, K. Klopper, B. Zeidler, W. Althaus, and P.-M. Weinspach, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 235. 45. J.W. Hoyt and R.H.J. Sellin, Experimental Heat Transfer, 2 (1989) 113. 46. J. Pollert, P. Komrzy, K. Svejkovsky, J. Pollert, Jr., and J.L. Zakin, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 31. 47. Y. Kawaguchi, Y. Tawaraya, A. Yabe, K. Hishida, and M. Maeda, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 47. 48. E.F. Matthys, in Drag Reduction in Fluid Flows, Ellis Horwood, 1989, 129. 49. K. Gasljevic and E.F. Matthys, in Developments in Non-Newtonian Flows, ASME, AMD-Vol.75 (1993) 101. 50. K. Gasljevic and E.F. Matthys, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 249. 51. A. White, Nature, 214 (1967) 585. 52. G. Astarita, J. Non-Newtonian Fluid Mechanics, 4 (1979) 285. 53. P. Granville, in Proc. 2nd Int. Conf. on Drag Reduction (1977) BI.1. 54. P. Granville, in R.H.J. Sellin and R.T. Moses, eds. 3rd Int. Conf. on Drag Reduction, Bristol University (1984) C3,1. 55. E. Matthys and R. Sabersky, Int. J. Heat Mass Transfer, 25 (1982) 1343. 56. R.H.J. Sellin and M. Ollis, I & EC, Product R & D, 22 (1983) 445. 57. D.D. Taylor and R.H. Sabersky, Letters in Heat and Mass Transfer, 1 (1974) 103. 58. J. Savins and F. Seyer, Phys. Fluids, 20 (1977) $78.
826
59. K. Gasljevic and E.F. Matthys, in Proc. Symposium on Development and Applications of Non-Newtonian Flows III, ASME, San Francisco (1995). 60. J.W. Hoyt and R.H.J. Sellin, Experiments in Fluids, 15 (1993) 70. 61. G.W. Anderson, J.J. Rohr, and J.W. Hoyt, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 19. 62. C. Lester, Oil & Gas Journal, 83, No.5 (1985) 51. 63. J.W. Hoyt and R.H.J. Sellin, Experiments in Fluids, 20 (1995) 38. 64. J.W. Hoyt and R.H.J. Sellin, in Engineering Turbulence Modelling and Experiments 3, Elsevier, Amsterdam, 1996, 381. 65. J.W. Hoyt and R.H.J. Sellin, in Proc. Symposium on Turbulence Modification and Drag Reduction, ASME FED-Vol.237 (1996) 225.
827
PAPER COATING RHEOLOGY
D. W. Bousfield and A. Co Department of Chemical Engineering, University of Maine Orono, ME 04469-573 7 USA 1.
INTRODUCTION Paper and paperboard are often coated to improve the appearance of the sheet and the quality of the print of the final product. Products such as magazines, catalogs, labels, and consumer packaging are often coated. The coating is normally composed of a pigment such as kaolin or calcium carbonate, a latex binder, and a soluble binder such as starch. Coatings are formulated at high solids content in order to minimize drying requirements and to improve quality. The coating must be mixed, pumped, recirculated, and metered onto the moving paper web. The final metering or coating operation is the critical step in the process; the coated layer must be free from defects and near a target coat weight. The rheology of the coating is an important issue in the control, design, and operation of the coating process. A roll applicator followed by a blade on the same roll is a common method to apply coating onto the web. In the last 15 years, the short dwell applicator system became popular; coating is pumped to a pond in front of the blade and the blade meters on the final coating amount as depicted in Figure 1. The short dwell applicator has one advantage short contact time and, consequently, minimal water penetration into the web. Uses of fountain or jet applicators and "metered size presses" are recent trends to apply coatings. High coat weights are normally obtained by a rod metering step followed by an air knife component. However, the blade metering element is a common last step to control coat weight and is the focus of this chapter and much of recent research. There are a number of unique aspects of paper coating that make this operation complex. The web is rough, porous, and absorbent. In addition, it changes its physical properties upon contact with water. The coating step is often accomplished at speeds higher than 15 m/s. The final coating layer thickness ranges from 7 - 30 ktm. The web is compressible, along with the rubber backing
828
Coated web
~'\"\,\
T
Excess to pond
Figm'e 1. Schematic of paper coating operation with a short dwell coater. The region around the blade tip is enlarged to show the details of the blade tip. Coating is applied to the web in the pond. The blade controls the final amount of coating applied to the web. that supports the blade loading. The blade is slightly flexible, it is mounted and loaded in various ways, and it wears during operation. All of these factors, including the high speeds, cause paper coating to be a complex operation. Economic driving forces cause the coating to be applied at high solids. Application at high solids reduces energy costs for drying and is known to produce a better quality product in terms of light scattering and printing properties, as reported by Van Gilder et al. [1 ]. Coating of high solids gives the paper less chance to roughen upon contact with water and the coating layer does not sink into the paper web. However, high solids content naturally brings challenging rheological problems. There is a trade-off between solids and operational speeds. Figure 2 shows data from Taylor [2] that is typical in the industry; as the coating solids increase, the maximum speed for defect-flee operation decreases. In order to increase the speed of a specific coating line, the solids content of the coating often needs to be reduced. This reduction increases drying costs and lowers quality of the product. Some pigments have different "runnability" limits. Ground calcium carbonate pigments can often be coated at arotmd 70% solids, whereas delaminated kaolin clays need to be coated at around
829
16 -~
14-
E v
12-
9
,
,
I
9
9
.
I
9
.
9
I
9
I
I
I
9
9
9
I
.
,
9
.1
(J
o
10-
> .o
8.
E
6.
.E
4.
~
2-
1
x
o 0
0 0.38
9
"
"
I
0.40
"
"
'
I
"
"
'
I
"
"
"
I
"
0.42 0.44 0.46 Solids V o l u m e Fraction
"
"
I
0.48
"
'
"
0.50
Figure 2. The relationship between coating solids and the maximum web velocity. The open circles are data from Taylor [2]. The line is a model calculation from Bousfield [3]. 55% solids. In any case, there is a constant incentive to operate at high solids and not to have operational difficulties. The operational difficulties or "runnability" problems vary widely in the industry, but common problems are the scratches, streaks, or skips in the coating layer and the buildup of coating on the blade. The buildup on the blade in itself is not a problem, but when the buildup breaks off the blade and ends up on the web, problems in the calendering and printing operations develop. The blade buildups are ot~en called "whiskers", "stalagmites", "weeps", or "spits", depending on the operator and the nature of the building (dry or wet). Other operational problems are related to coat weight control, both in the machine and cross-machine directions. Another common problem is web breaks; these cause a loss of production and correlate to high blade forces. Many of these problems are related to rheology of the coating. 2.
R H E O L O G I C A L TESTS OF COATING COLORS The most common industrial test to monitor the viscosity of a coating formulation is an apparatus with a spindle rotating in a container of fluid. Although this type of apparatus cannot measure the "true" viscosity of non-
830
Newtonian fluids, it is used extensively in the coating industry because of its ease of use and its value as a tool for quality control. However, to understand the effects of the components of a coating formulation on its rheological characteristics and to relate these rheological characteristics to its performance in a coating operation, numerous researchers have made use of various standard rheological tests. These include steady-state measurements in shear flow, measurements in small-amplitude oscillatory shear flow, and transient measurements in shear flow. Of these measurements, viscosity measurement in steady-state shear flow is the most commonly used. Examples of recent works are Triantafillopoulos and Grankvist [4], Carreau and Lavoie [5], Purkayastha and Oja [6], Ghosh et al. [7], and Ghosh [8]. To obtain the viscosity function over a wide range of shear rates, one usually has to utilize several viscometers: the cone-and-plate system for low shear rates, the concentric cylinders configuration for medium to high shear rates, and the capillary type for high shear rates. An important experimental consideration is that there is sufficient time for the coating to reach its equilibrium configuration at a given shear rate. Also of interest is the yield stress of the coating colors. Its importance in the actual coating operation is minimal since the operation is performed at very high shear rates. However, it may play an important role in the recovery stage after the coating application. The behavior of the viscosity versus shear rate curve depends on the components of the coating colors. Figure 3 shows the viscosity curves of a starchcontaining formulation and a formulation with latex particles (Roper and Attal [9]). The curve of the starch-containing formulation exhibits a power-law shearthinning region and a region of almost constant viscosity at high shear rates. On the other hand, the formulation with latex particles shows a region with shearthickening or dilatant behavior. Beyond the power-law shear-thinning region, the viscosity rises sharply with increasing shear rate and then decreases slightly at higher shear rates. Roper and Attal [9] indicated that the dilatant behavior depended on the solids level and the latex particle size; the extent of dilatant behavior was increased with higher solid levels and larger latex particle size. Aside from the solid levels and the size and shape of solid particles, the polymer additives dissolved in the suspending medium also have considerable effects on the rheological behavior of the coating suspension. Examples are shown in Figure 4, which depicts the viscosity curves of Carreau and Lavoie [5] for kaolin suspensions with three different concentration levels of CMC (carboxy methyl cellulose). They attributed the large increase in viscosity to the interactions between kaolin and CMC, since the viscosity of the CMC solution itself did not change significantly at these concentration levels.
831
101
~,
I
I
,
I
I
I
I
I
1 0 o.
t~ n
(/) O o
.~> 10-1..
1 0 .2
10 -~
m
I
I
10 0
~)
101
I
1 2
10 3
Shear
I
10 4
I
I
10 s
1 7
10 s
Rate (l/s)
Figure 3. The viscosity functions of a strach-containg coating (open circle) and a coating with latex particles (open square). Data are from Roper and Attal [9]. 10
4
I
I
I
I
i
10-
I
[]
11'
2
A w,
10-
n
-~-
1
[]
10-
0
10-
10 "1.
-2
10 ;
1 -3
I
10-2
I 10-1
1
;0
1
01
1
;2
1
;3
7, to (1Is) Figure 4. Effect of CMC content on the viscosity and the dynamic viscosity for 44% vol. Kaolin suspensions. Data are from Carreau and Lavoie [5].
832
Also shown on Figure 4 are the dynamic viscosity versus frequency curves of the suspensions. These are obtained from small-amplitude oscillatory shear experiments. The dynamic viscosity rt'((o) is determined from the componem of the shear stress that is in-phase with the rate of deformation and rt"(oJ), from the component that is out-of-phase. Figure 4 shows that the viscosity curves and the dynamic viscosity curves coincide, that is, rl(8)--r/'((o~,:~. This is analogous to the Cox-Merz rule for polymer solutions or melts, which predicts that the magnitude of the complex viscosity (r rl * ((o)I-x/rt;' ((o)+ rt"((o)) is equal to the viscosity at corresponding values of frequency and shear rates. This empirical analogy can be useful in predicting viscosity data when only linear viscoelastic data are available. Alternately, linear viscoelastic data can be expressed in terms of the storage modulus G' (-rl'a, ) and the loss modulus G" (-rt'a,). Figure 5 shows the storage modulus of kaolin suspensions at various concentration levels of CMC, as reported by Carreau and Lavoie [5]. As in the viscosity curves, a small concentration level of CMC increases the storage modulus drastically, again due to the interaction between kaloin and CMC. Another observation is that the storage modulus approaches a constant value at low frequencies, a behavior that is more like a viscoelastic solid. Similar behavior was also observed for the coating formulations considered recently by Ghosh [8]. 10
3
a
,
...
,
I
,
1%CMC 2
10A
n 0.25 % CMC 1
10-
~00---0-0-(~
0 ~ ' 0 ~ ~ ~ CMc
I !
o
10
j 10"
10 "1
1() ~
101
10'
(l/s)
Figure 5. Effect of CMC content on the storage modulus for 44% vol. Kaolin suspensions. Data are from Carreau and Lavoie [5].
833
150"
L
'
....
J
' 50
G' 4O 100"
3O
(.9
~'
50 20
0-I1 0 -~
, " 1 0 "2
,' 1 0 "1 Frequency
, 100
I0 101
(Hz)
Figure 6. The dependence of the storage and loss moduli and phase angle on the frequency for a clay-based coating color containing 1.25 part of CMC per 100 parts of clay. Data are from Fadat et al. [10]. Figure 6 shows the linear viscoelastic data of Fadat et al. [10] for a clay-based coating color with a different grade of CMC. Here the behavior of the storage modulus is more like that of a viscoelastic liquid. The reduction of the phase angle 5 (-tan -~G"/G') with increasing frequencies indicates that the suspensions becomes more solid-like at higher frequencies. These behaviors are different from those exhibited by the coating suspensions studied by Carreau and Lavoie [5] and Ghosh [8]. Transient shear-flow experiments have been used to elucidate the characteristic times of structure formation or breakup in coating suspensions. A common industrial practice is to generate a hysterisis loop from shear stress measurements completed over a strain-rate sweep for a finite time interval. Although this type of measurement is valuable for quality control, it is of little use in characterizing the structure formation or breakup in the coating suspension. More useful transient experiments are the stress growth measurement upon the inception of steady shear flow, as described by Ghosh et al. [7] and Ghosh [8], and the stress relaxation measurement after a sudden sheafing displacement, as reported by Young and Fu [11] and Ghosh [8]. These experiments were able to yield estimated characteristic times of the coating suspensions studied.
834
Normal stress measurements at very high shear rates were conducted by Windle ad Beazley [12] using a jet-thrust technique for starch-based coatings. Calculation of the first normal stress coefficient from their normal stress data gives very low values, indicating that normal stress effects may not be significant at the high shear rates encountered in coating operation. This is corroborated by Triantafillopoulos and Grankvist [4], which did not observe any jet swell in their high-shear capillary flow experiments for several coating formulations. Due to the converging-diverging geometry in the coating operation, the flow is both shearing and extensional. Elongational viscosity may therefore play a significant role. Measurements of elongational viscosity at the high deformation rates experienced in a coating operation have not yet been reported. However, Carreau and Lavoie [5] claimed that they did not observe any significant elastic or extensional effect in the results from their lubrication equipment at high deformation rates. 3.
STRUCTURAL MODELS TO PREDICT RHEOLOGY The application of paper coating has a unique aspect in that the clearance between the blade and the web is only one order of magnitude larger than the pigments in the suspension. In some places, on the top of a "high" spot of the paper web, the coating layer is the same order of magnitude as the pigment. A number of coating defects and the "stalagmites" or "whiskers" described by the industry seem to be related to the particulate nature of the coating. Therefore, the process calls for a good understanding of the flow properties of the coating at the particulate level. Calculations using Stokesian dynamics were introduced by Brady and Bossis [13, 14] to relate the microscopic behavior of suspension to their macroscopic behavior such as rheology. Stokesian dynamics is a technique to calculate the trajectories of particles in a suspension undergoing flow. The technique has been shown to predict various phenomena. These include the increase of viscosity with increasing solids described by Durlofsky et al. [15] and Toivakka and Ekltmd [16], the reduction of the viscosity of a mixture of large and small spheres given by Chang and Powell [17] and Toivakka and Eklund [18], the increase of viscosity at high shear rates shown by Boersma et al. [19], the Brownian shearthinning nature of suspensions characterized by Phung et al. [20], and the influence of particle roughness reported by Bousfield [21 ]. The technique calculates the hydrodynamic force on every particle at a particular instant of time. Other forces such as electrostatic and colloidal forces can be included in the calculation. Once the net force on a particle is known, its acceleration is calculated. The particle velocities and positions are updated with
835
numerical techniques. In the limit of small particle Reynolds number, the particle velocities can be found through matrix techniques; the velocities must cause the net force on every particle to be zero. In a blade coating geometry, the clearance between the web and the blade can be in the order of 15 ~xn. Bousfield [22] showed that structures of particles will form in shear fields and that these structures can span the gap between the web and the blade. This mechanism is the same as those reported by Brady and Bossis [14] and by Boersma et al. [19]. When a cluster of particles forms in this geometry, large forces are transmitted between the web and the blade. This mechanism could be responsible for blade wear and blade deposits, if these structures exit the blade. Figure 7 depicts this situation: particles are forced closer together as the fast moving particles must past the slow moving particles near the blade. If the electrostatic or steric repulsive forces are not significant to keep the particles separated, particle clusters can span the gap between the blade and the web. A potential mechanism for the viscoelastic response of a highly dispersed suspension was modeled by Toivakka et al. [23] with a Stokesian dynamics calculation. When electrostatic or steric forces keep the suspended particles well separated, an energy minimum is obtained by an ordered packing of the particles. Any small flow field will disrupt the particle configuration and will push the particles closer together in some regions, as demonstrated in Figure 8. If the motion is slight, as in a small-amplitude oscillatory experiment, the suspension
Figure 7. Schematic of the results from a particle motion model showing the clustering of pigments between the blade surface and the web at high shear rates. This structure can transmit large forces between the web and the blade.
836
Static
i| I
-i =v
II
Figure 8. Slight motion caused by shear will bring some particles closer together. This compression of the electrostatic or steric repulsive forces can store energy and lead to a viscoelastic response of the suspension, even when dissolved polymers are not present.
Figure 9. A schematic showing the pushing away of a larger particle from a wall by smaller particles, as described by Toivakka and Eklund [18]. can store energy by the compression of the electrostatic or steric forces. This storage of energy leads to the possible elastic nature of a suspension, even when dissolved polymers are not present in the system. Example of this type of experimental system was reported by Fadat et al. [10] for coating suspensions. If the deformation is too large, particles move away from the static position and a non-linear viscoelastic result is obtained. The position of different sized pigments and particles in the coating layer has been predicted by Stokesian dynamics calculations. Chang and Powell [17] show how small particles can disrupt the formation of clusters and, thereby, reduce the
837
shear viscosity. Toivakka and Eklund [18] confirm this mechanism and show how small particles can end up closer to the blade surface during shear. Fine particles are able to be closer to the solid boundaries because of their size. In addition, when particles are exposed to a flow field, larger particles must move faster than the finer particles near the walls. This velocity difference causes the larger particles to push the smaller particles even closer to the walls. After a number of interactions, the larger particles migrate away from the walls. This would create a separation of particles according to size, with the finer particles near the solid boundaries. The coating layer could have gradients of particles from its surface to the bulk. This mechanism is illustrated in Figure 9. Bousfield et al. [24] show that the free surface after the blade may cause a similar separation of particles based on size, with the smaller particles being located at the top surface. These results may explain the high concentration of latex binder at the top layer of the coating observed industrially. 4.
M O D E L I N G OF BLADE COATING Initial attempts to model the blade coating operation are based on Newtonian fluids. Follette and Fowells [25], Bliesner [26], Turai [27], and others give simple analyses to relate shear rates, viscosity, geometry, and web velocity to the force generated on a blade during coating. Lubrication theory is used to reduce the complexity of the equations. The pressure distribution under the blade is found to be a strong function of the blade operating angle. Bliesner [26] and Saita and Scriven [28] used lubrication theory to relate the details of the blade geometry or blade deflection to the coat weight and blade loading. The force on the blade required to reject coating due to inertial forces was given by Eklund and Kahila [29]. This force can be obtained from a mechanical energy balance on the flow upstream of the blade. The resulting force F on the blade, for a roll applicator system, is estimated to be Y=
mV(1 + eosa)
sin a
(1)
where m is the incoming mass flow rate of the coating to the blade, Vis the web velocity, and cx is the operating angle of the blade. This force seems to be important for roll applicator systems, but it is not clear how to use this type of equation for a short-dwell coater. This expression indicates that the blade loading is not a function of the coat weight metered by the blade nor the viscosity level of the coating. Therefore, some other viscous forces must be important, in addition to this force, in the coating operation.
838
Pranckh and Scriven [30] used fimte element method to solve the two dimensional Navier-Stokes equation for the flow near the blade. The solution is in conjunction with a nonlinear beam equation that describes the deflection of the blade and a spring model that describes the deformation of the rubber backing roll and the web. Surface tension forces upstream from the blade and after the blade were included in the analysis. In the flow field, there is a stagnation line that separates the coating that is retm-ned and the coating that is metered onto the web. The pressure distribution near the heal of the blade and under the blade tip shows little pressure gradients in the vertical direction: this result indicates that lubrication theory is valid in this region. Again, the operating angle of the blade is found to be a critical issue in these flows. The pressure distribution shows a stagnation pressure upstream of the blade, which is close to 89 p V 2 , and a pressure increase near the blade heel due to viscous forces. This result links the pressure pulse due to the inertial terms with that of lubrication theory. The influence of the absorption of the coating into the porous web is first accounted for by Chen and Scriven [31]. The pressure distribution of a roll applicator and by a blade metering element from Pranckh and Scriven [30] is used to calculate the penetration depth of the coating. The influence of air being trapped into the web and the compression of the web are taken into account. Assuming a specific pressure distribution under the blade, Letzelter and Eklund [32] predict the filter cake thickness and the amount of dewatering. Bousfield [3] proposed a model to describe the formation of a filter cake on the web during blade coating. Lubrication theory was found to duplicate the pressure distribution of Pranckh and Scriven [30] and was used to describe the dewatering and subsequent filter cake buildup on the paper web. The model by Bousfield [3] is shown to predict the onset of operational difficulties of literature data, if dewatering parameters are known. These difficulties seem to be related to the growth of a filter cake which had a thickness near the blade-paper gap. In Figure 2, the data of Taylor [2] are compared with the predictions of the model. This show that high solids content increases the coating viscosity and reduces the amount of water needed to be removed to generate a significant filter cake. The influence of the shear thinning nature of coating was first modeled by Modrak [33]. A power-law model was used in conjunction with lubrication theory to describe the flow in a "rounded" blade. Viscosity data from a capillary viscometer was fitted for a Newtonian-like fluid, a shear-thinning coating, and a shear-thickening fluid. The power-law exponent varied from 0.675 to 1.3. The stagnation point moved downstream under the blade tip as the coating became shear thickening. The pressure distribution and the hydrodynamic litt on the blade were an order of magnitude higher for the shear thickening fluid, even though at a
839 shear rate of 3 • 104 S-1, the shear viscosities were similar. However, the shear rate under the blade tip must reach 106 s-1. For the three coatings, the shearthinning model did a reasonable job at predicting pilot scale data. Finite element methods were used by Triantafillopoulos et al. [34], Roper and Attal [9], and Isaksson and Rigdahl [35] to calculate the flow field in a short dwell coater pond and near the blade tip for a roll-applicator system. Free surfaces and inertial terms were included in the later analyses. The coating was described as a power-law model with various values of the power-law index. The circulation within the pond of the short dwell coater was calculated to change from a single vortex at high viscosity values to two vortices at low viscosity values. The shear thinning nature of the fluid was fotmd to reduce the pressure distribution under the blade and decrease the total blade loading. In addition, shear thinning fluids are found to produce a "plug flow" nature to the flow field, where most of the shear can occur in a thin layer near the moving web. The influence of blade angle and "slip" at the blade-coating interface are also discussed in Isaksson and Rigdahl [35]. The importance of viscoelastic properties of coating was first brought up by Windle and Beazley [12]. The discussion on viscoelastic properties was focussed on the extra normal forces generated during shear: these normal forces could contribute to the hydrodynamic lift forces during operation. An expression for the extra force on the blade due to normal forces was proposed. Two-dimensional solutions for a viscoelastic fluid are described in Sullivan et al. [36], Olsson [37], and Olsson and Isaksson [38]. The results by Olsson use a geometry that closely resembles the blade geometry of paper coating. An upper convected Maxwell fluid, an Oldroyd-B fluid, and a Giesekus fluid are used as constitutive equations. The pressure distributions and streamlines are significantly influenced by the presence of viscoelasticity. A recirculation region is predicted upstream from the blade tip and near the exit of the blade, as depicted in Figure 10. The pressure is predicted to decrease due to the extensional components of the flow field. This result would produce a lower blade force to obtain the same coat weight, as compared to a Newtonian fluid. The entrance region seems to be "clogged" with a vortex, resulting in less fluid that is able to pass under the blade. This result resembles the outcome of Sullivan et al. [36] where the force on the blade is reduced because of the influence of viscoelasticity. The K-BKZ constitutive equation was used by Mitsoulis and Triantafillopoulos [39] to describe the flow under a blade with the influence of a free surface. The K-BKZ constitutive equation has multiple relaxation times and predicts normal forces. The pressure distributions and the net flow rate under the blade are influenced by the viscoelastic nature of the constitutive equation.
840
Figure 10. The formation of recirculation regions for an Oldroyd-B fluid, as described by Olsson and Isaksson [38]. One vortex forms upstream from the heel of the blade. Another vortex forms near the tip of the blade. Both act to constrict the even flow of fluid under the blade. The importance and our current understanding of viscoelasticity in coating flows are reviewed in detail by Triantafillopolous [40]. The influence of various additives on the viscoelastic nature of the coatings are discussed. The importance of the viscoelastic nature of these coatings on the operation of blade coaters and the formation of defects are summarized. A clear direct link between the viscoelastic nature of the coatings and specific operational difficulties is not yet established. This direct link is difficult to establish because of the experimental difficulties that exist in terms of measuring blade forces and comparing one coating against another for identical operating conditions: changes in coating composition to change he viscoelastic nature of the coating also changes the water holding behavior. 5.
CONCLUSIONS With the recem advances in theological testing methods and computational tools, our understanding of the blade coating of paper has improved considerably in the last ten years. The importance of "water retention" on the ability of a coating to operate at specific conditions has been one significant step. However, paper coating still offers a number of challenges, as high speeds and high solids are desired. Still the small scale between the blade and the web requires a better understanding of the small-scale microstructures that can form under the blade. The role of viscoelasticity is still open to debate. A direct comparison of a complex fluid dynamics calculation and pilot scale results has yet to be
841
accomplished. These issues should become clear as effort is spent to understand the role of rheology in the blade coating process. REFERENCES
1.
R. Van Gilder, D.I. Lee, and R. Purfeerst, TAPPI J., 66 (1983) 49.
2.
J.A. Taylor, Proceedings of the 1987 TAPPI Coating Conference, TAPPI Press, Atlanta GA (1987) 1.
3.
D.W. Bousfield, TAPPI J., 77 (1994) 161.
4.
N. Triantafillopoulos and T. Grankvist, Proceedings of the 1992 TAPPI Coating Conference, TAPPI Press, Atlanta GA (1992) 23.
5.
P.J. Carreau and P.-A. Lavoie, Proceedings of the 1993 TAPPI Advanced Coating Fundamentals Symposium, TAPPI Press, Atlanta GA (1993) 1.
.
S. Purkayastha and M.E. Oja, Proceedings of the 1993 TAPPI Advanced Coating Fundamentals Symposium, TAPPI Press, Atlanta GA (1993) 31.
7.
T. Ghosh, P.J. Carreau, and P.-A. Lavoie, Proceedings of the 1996 TAPPI Coating Conference, TAPPI Press, Atlanta GA (1996) 303.
8.
T. Ghosh, Proceedings of the 1997 TAPPI Advanced Coating Fundamentals Symposium, TAPPI Press, Atlanta GA (1997) 43.
9.
J.A. Roper III and J.F. Attal, Proceedings of the1993 TAPPI Coating Conference, TAPPI Press, Atlanta GA (1993) 107.
10. G. Fadat, G. Engstrom, and M. Rigdahl, Rheol. Acta, 27 (1988) 289. 11. T.S. Young and E. Fu, Proceedings of the 1991 TAPPI Coating Conference, TAPPI Press, Atlanta GA (1991 ) 61. 12. W. Windle and K.M. Beazley, TAPPI J., 51 (1968) 340. 13. J.F. Brady and G. Bossis, J. Fluid Mech., 180 (1985) 105. 14. J.F. Brady and G. Bossis, Annual Rev. of Fluid Mech., 20 (1987) 111. 15. J.L. Durlofsky and J.F. Brady, J. of Fluid Mech., 200 (1989) 39. 16. M. Toivakka and D. Eklund, Nordic Pulp and Paper Res. J., 9 (1994) 143. 17. C. Chang and R.L. Powell, J. of Rheol., 38 (1994) 165. 18. M. Toivakka and D. Eklund, TAPPI J., 79 (1996) 211. 19. W.H. Boersma, J. Laven, and H.N. Stein, J. Rheol., 39 (1995) 841. 20. T.N. Phung, J.F. Brady, and G. Bossis, J. Fluid Mech. 313 (1996) 181.
842
21. D.W. Bousfield, Nordic Pulp and Paper Res. J., 8 (1993) 176. 22. D.W. Bousfield, Proceedings of the 1990 TAPPI Coating Conference, TAPPI Press, Atlanta, GA. (1990) 325. 23. M. Toivakka, D. Eklund, and D.W. Bousfield, J. Non-Newt. Fluid Mech., 56 (1995) 49. 24. D.W. Bousfield, P. Isaksson, and M. Rigdahl, J. Pulp and Paper Sci., 23 (1997) J293. 25. W.J. Follette and R.W. Fowells, TAPPI J., 43 (1960) 953. 26. W.C. Bliesner, TAPPI J., 54 (1971) 1673. 27. L.L. Turai, TAPPI J., 54 (1971) 1315. 28. F.A. Saita and L.E. Scriven, Proceedings of the 1988 TAPPI Coating Conference, TAPPI Press, Atlanta, GA (1988) 13. 29. D. Eklund and S.J. Kahila, Wochbl. Papierfabr., 106 (1978) 661. 30. F.R. Pranckh and L.E. Scriven, TAPPI J., 73 (1990) 163. 31. K.S.A. Chen and L.E. Scriven, TAPPI J., 73 (1990) 151. 32. P. Letzelter and D. Eklund, TAPPI J., 76 (1993) 63 33. J.P. Modrak, TAPPI J., 56 (1973) 70. 34. N. Triantafillopoulos, G.R. Rudemiller, and T. Fanfngton, Proceedings of the 1988 TAPPI Engineering Conference, TAPPI Press, Atlanta, GA (1988) 209. 35. P. Isaksson and M. Rigdahl, Rheol. Acta, 33 (1994) 454. 36. T. Sullivan, S. Middleman, R. Keunings, AIChE J., 33 (1987) 2047. 37. F. Olsson, J. Non-Newt. Fluid Mech., 51 (1994) 309. 38. F. Olsson and P. Isaksson, Nordic Pulp and Paper Res. J., 4 (1995) 234. 39. E. Mitsoulis and N. Triantafillopoulos, Proceedings of the 1997 TAPPI Advanced Coating Fundamentals Symposium, TAPPI Press, Atlanta GA (1997), 27. 40. N. Triantafillopoulos, Paper Coating Viscoelasticity, TAPPI Press, Atlanta, GA (1996).
843
RHEOLOGY OF LONG DISCONTINUOUS FIBER THERMOPLASTIC COMPOSITES S.G. Advani* and T.S. Creasy +
*University of Delaware, Department of Mechanical Engineering, Spencer Laboratory, Newark, DE 19716 +University of Southern California, Centerfor Composite Materials, VIlE 602 MC0241, Los Angeles, CA 90089-0241 1. INTRODUCTION The total cost of installed composite components is high enough to keep these materials out of many products that could benefit from them. This introduction discusses the motivation for using discontinuous fibers to reduce this cost, the status of research into the manufacturing behavior of these materials and the scope of this chapter in presenting these properties. 1.1 Motivation High performance composite materials are beneficial when compared with other candidate materials by specific strength. But applying advanced composite materials to mass market industrial products has a tremendous limiting factor: the traditional high cost of producing parts. The benefits of high strength composites emerge when controlled fiber placement optimizes the directional strength of the material. Traditional construction techniques involve significant labor and time costs. These costs must be reduced to increase the market for designed materials. Burdensome hand-layup is the most expensive fiber placement method m,.d must be avoided [ 1]. Tape laying machines, although automated, are too slow for rapid production cycles [2]. Many industries use injection molding, transfer molding or sheet molding of low to medium strength composites with particle or short fiber reinforcement. These techniques rapidly produce items such as electronic connectors or automotive panels. If high strength composites can be formed in a manner analogous to sheet metal stamping new markets may benefit from their use.
844
Thermoforming may allow rapid stamping of high strength composites [3]. In thermoforming a sheet of fibers impregnated with polymer heats up until the polymer melts. Two general methods form the sheet: stamping between dies [4] or applied differential pressure [5], both shown in Figure 1. Initially at rest, the sheet flows until it contacts the mold. A distribution of strain rates and total strain occurs over the area of the sheet. Sections of the sheet receive different displacement conditions as new features of die contact the sheet. Clamps hold the edges of the sheet; thus, the primary mode of deformation is transient biaxial elongation. Otherwise, the fonning may occur by other mechanisms common in continuous fiber systems, [6, 7]. The transients are important as the process time is short. Steady flow may only be achieved in limited regions if at all.
Figure 1. Sheet forming processes: a) matched die stamping employs precision tools on either side of the fluid sheet that ensure surface dimensions and quality, b) differential pressure uses pressurized gas on one side of the sheet to push the melt onto a tool that controls the quality of one surface.
But strong materials require high volume fractions of either continuous fibers, which cannot extend in the fiber direction [5], or well aligned arrays of longdiscontinuous fibers [4]. Continuous fiber materials have limitations in thermofonning [5]. If the fibers remain unbroken--necessary to avoid an area with a concentration of damaged fibers--the sheet conforms to the mold by interply slip and by drawing extra fiber length from the edge of the panel toward the center of the sheet. The sheet accommodates the entire distributed total strain in the direction of a given fiber by drawing in that fiber. For large deformations a
845
suitably large flange region provides the needed fiber length. This fiber drawing process produces buckling and wrinkling of the sheet [8]. If severe, buckling can intrude upon the region of the finished part and decrease the quality of the component. Discontinuous fiber systems allow the fibers to move past one another during forming. Since the fibers can move independently, different regions conform to their total strain. Creating a high strength discontinuous composite is challenging. Fibers must be long enough to provide strength close to that obtained with continuous fibers [9]. The discontinuous arrays must be well aligned to maintain the necessary high fiber volume fraction. Fibers conforming to these requirements interact in typical processes such as injection molding. Fibers will break and alignment will be limited from fiber entanglement and flow field conditions. Innovative methods are needed to create a high performance sheet product. An example long discontinuous fiber thermoplastic melt system (LDFMS) is LDF/PEKK. Here the matrix is poly-etherketoneketone (PEKK) [ 10, 11]. These 7 micron diameter, 5.6 cm long carbon fibers fall between typical short fiber fillers and continuous fibers in length and performance. LDFMS material is an attempt to approach high solid strength typical of highly aligned continuous fiber composites with melt formability similar to short-fiber filled polymers by allowing relative motion between adjacent discontinuous fibers. This material contains a high volume fraction (60 %) of aligned fibers with an average aspect ratio (a r) of 8000, [1 ]. The transient and steady state rheological properties of this fluid are expected to be different from the unfilled melt. 1.2 Status of the Field
A detailed review of relevant research in particle and fiber filled fluids follows in Section 2.1.2. In summary, the studies of filled fluids show that fillers increase the shear and elongation viscosity of the system when compared to the matrix polymer. Increasing particle aspect ratio magnifies this effect with viscosity increases of several orders of magnitude. In elongation the fillers--including particulates--induce a reaction that many investigators believe involves "local" shear of the fluid between the particles. However, the specific physics of a system and flow condition must be analyzed carefully. Similar transient response curve shapes may be caused by unrelated phenomena.
846
1.3 Scope of the Chapter This chapter covers the investigation of the elongational flow behavior of two LDFMS systems. It includes fabrication of a model system from nylon fibers and a low density polyethylene (PE) matrix with fiber aspect ratios of 25 and 100. Tensile extension of each LDFMS is compared to a micromechanics model of relative fiber motion with appropriate constitutive relations considered. The results are discussed with regard to the measured rheology of the polymers and implications for formability and future research. Section 2.1 reviews the literature of the shear and elongational properties of neat and filled polymer melts to show the generally understood properties of these systems. Section 2.2 discusses the material preparation steps for making long fiber reinforced melts. In section 3 the experimental procedures are listed for each device applied to the investigation. Section 4 contains the results of the experiments. Possible models of extension behavior of filled systems are presented in section 5 with a check of the data from section 4. Finally, the conclusions are summarized in section 6 with some suggested avenues for further study. 2. MATERIALS Polymer science has grown rapidly since the 1940s as new synthesis methods, experimental appliances and structural theories were introduced [12]. This section stmunarizes the outcome of research into the response of polymer melts in shear and elongation flows.
2.1 Traditional Fluid Systems It is good practice to study each constituent of a system independently if they are available as separate compounds. This is certainly the case for systems containing polymer melts. The polymer melt by itself has an array of properties that must first be understood. 2.1.1 Neat Polymers The fluid dynamics of polymeric melts can be described by a set of rheological properties called material functions [13]. These functions describe the manner in which the fluid response changes as a function of the flow conditions imposed upon the melt. The flow conditions can be the applied strain rate (~'yxfor shear,/~ for elongation) or the applied stress (xyx for shear, x E for elongation).
847
Shear Flows Material functions in shearing flow can be defined under both steady state conditions and as transient functions with time as an additional parameter. The transient functions will approach the steady functions for large times trader steady applied strain rate or stress. We begin with Newtonian material functions under steady flow conditions. Modified, these functions handle polymeric fluids under steady and also under transient conditions. Steady Flow The fluids of interest in sheet forming are usually non-Newtonian polymer melts. For these complex fluids, the entanglement of the polymer macromolecules produces a rate dependence of the steady viscosity [12, 14]. We define the rate dependent steady viscosity, rls (Tyx), such that Xyx - rlS(Tyx)Tyx
(1)
As shown by Figure 2, isotropic polymer melts behave as Newtonian fluids in creeping flows, ~'y~ << 1.0 sl; but, they then demonstrate "shear thinning" for large strain rate. Less common is "shear thickening" at high strain rate. At decreasing strain rate viscosity approaches a constant value called rl0, the "zero shear rate viscosity." The viscosity function of the fluid is defined in nondimensional form as rls (~'yx)/1"10. The Carreau relation provides a good description of the steady viscosity from Newtonian creeping flows to high strain rates (power-law region) [15]:
- rl0[1 + O#y )2 ]
(2)
The ~ (with units of time) adjusts the position of the "knee" in the TIs--Tyx curve. Here n controls the power law slope of the curve [13]. Polymer melts also exhibit elasticity which introduces a new stress in shear flow; this stress acts normal to the direction of shear flow. A cone and plate rheometer measures the normal force during a shear flow. This normal force is the total thrust parallel to the rotation axis of the cone and plate. Bird et al. [13] define the first and second normal stress coefficients in the form: -2 'lTyy-- '17XX -- l~/rl('il)~,X and "r, z z - X y y - gt2(~)i, 2 . For these coefficients Yyx is used since a reversal of the direction of the shear flow will not change the sign of the normal stress. The steady state first normal stress coefficient behavior is similar to the steady viscosity behavior.
848
Shear Thickening Newtonlan
_c Shear Thinning LCP
In~ Figure 2. Steady viscosity of Newtonian and non-Newtonian fluids as a function of shear strain rate. Newtonian fluids have a constant viscosity. Shear thickening fluids increase in viscosity as strain rate grows. Shear thinning fluids and liquid crystal polymers (LCP) have regions of decreasing viscosity for an increase in shear rate. Some LCPs have two regions of thinning connected by a Newtonian plateau between them as displayed here.
Transient Flow The above discussion assumed that steady-state behavior applied to the measured shear viscosity. Polymer fluids show time dependent properties when flow starts from rest. The way that transient stress (or equivalently transient + viscosity by rl~ =xy~/'~y~) grows to its steady-state value as the flow progresses delineates the transient material functions of the fluid. When a Newtonian fluid at rest goes into motion in shear, the shear stress moves from zero to "Cy~in a step function as in Figure 3. When the flow stops the stress immediately drops to zero. A viscoelastic fluid set into shear flow from rest takes finite time to reach steady-state. Stress and rls are now fimctions of Jfyx and time, x ~ (~(yx,t) and rl~ (J~yx,t) respectively. The plus sign represents a flow +
+
begun from rest at time "zero" with Xyx or rls increasing into positive time. As +x ~ Xyx and rls+ --~ rls. We call x + the flow continues Xy ~ the "stress growth" [131.
849
Figure 3 demonstrates stress growth for both linear and nonlinear viscoelastic fluids. For linear viscoelastic fluids the stress grows up to the steady stress level monotonically. A constant relevant to the flow, usually denoted by ~,l and called the "relaxation time," is a measure of the time required to reach the steady state response of the system, e.g., the Maxwell fluid model [16]. The relaxation time changes the arithmetic equation 1 into a differential equation on stress. Nonlinear models, such as the White-Metzner fluid, incorporate rate-dependent viscosity [17]: +
.+
"l:yx(~'yx , t ) +
G
"l:Yx(~/yx,t) - rl s (~'yx )~/yx
(3)
G is a constant stiffness term that models the elasticity of the fluid. The solution to equation 3 defines the transient viscosity as + rl~(~y~,t) = xy~(5/yx,t)/~y~. Monotonic stress growth function still confines this model to linear viscoelastic material functions.
--- Newtonlan Fluid --- Linear Viscoelastic Non-linear Viscoelastic .
.
.
.
.
"1
§
.
.
.
.
Stress Relaxation
|
~yx
~---- Flow Start-up ,
II
o
t1
Time
Figure 3. Transient stress in starting-up and stopping of simple shear for Newtonian, linear viscoelastic and nonlinear viscoelastic fluids. At time t - 0 a step value of Yyx is applied; at time t - tl, 5[yx - O .
More challenging to model is a material that exhibits a nonlinear viscoelastic response with stress overshoot. The material functions still carry the same § (~/y~ ,t); but, Xy +x approaches Xy~ with oscillations. At small notation, e.g., Xy~ strain rate this nonlinear viscoelastic fluid will act just like a linear one. When
850 +
~(y~ grows large enough it stimulates stress overshoot. As Figure 3 shows, Xyx grows larger than Xy~ and then falls to reach the steady value. As ~yx increases the ratio of the peak Xy+x to the steady Xyx increases. The peak value of Xy+x is the yield stress x y . A set of rl~/TIo fimctions for a nonlinear viscoelastic fluid is presented in Figure 4. The slowest ~/yx data form an "envelope" curve that is a maximum TI~ fimetion for the fluid. As ~yx grows, each rl~ curve falls under this envelope. The steady rls values reached at each strain rate are equivalent to the steady-state shear thinning property shown in Figure 2.
1.0 ~.0.1 ~
1000~0
+W
10,ooo~o
o.01 0.01 Time
Figure 4. Transient viscosity function for a nonlinear viscoelastic fluid in shear. This equation includes both shear thinning and stress overshoot with increasing strain rate. The first strain rate is within the Newtonian plateau since rl~ / rio - 1. The two highest rates fall within the power-law region of the Carreau curve so that rl~ / 110 < 1.
The appearance of nonlinear stress growth follows the change in the conformation of the polymer melt. At rest and without applied stress the long chain molecules reach a random entangled network, their preferred rest conformation, through diffusion processes that move the melt to a minimum energy state [14]. This rest conformation is depicted by the maximum valued characteristic relaxation parameter ~,l- When exposed to small ~/yx -- small
851
enough to produce linear viscoelastic response -- stress growth and relaxation are linear with relevant ~,1 9 That is, the strain rate is small enough that the melt can adjust to the flow by the same diffusion processes that allow attainment and maintenance of the rest conformation. Higher ~/yx stimulates a rearrangement of the conformation. The fluid cannot accommodate the greater strain rate with the diffusion processes. The stress grows until sufficiently large to change the mechanism by which the molecules adjust to the stress. The initial rest structure has a potential steady stress 9
+
*
comparable to xyx = rl0 ~/yx which is large since rl0>>rls(~/y~). As xyx ~ Xyx the stress level rises and the diffusion processes cannot adjust to the rapid flow. The high stress level supplies the energy necessary to break up the network and align the macromolecules to the flow. As the network tears apart the polymer becomes less resistant to the applied strain rate. A catastrophic realignment occurs. Stress falls to Xyx and becomes stable [14]. Elongational Flows Shearing flows are not the only deformation mode possible in a fluid. Shear free flows involve extension of the fluid without shearing. Before considering flows that may contain both extension and shearing components, it is worthwhile to study the sheerer flows separately when possible. Material functions definitions for elongational flow follow by analogy to the shear functions. Reports of steady elongational viscosity data are rare [18]. Many polymers seem never to attain steady elongation. With this and the transient nature of thermoforming in mind, we turn to the transient material functions. Elongation Flow Defined The volume of fluid remains constant during deformation under the assumption of incompressibility. The incompressibility effect renders elongation a three dimensional flow. Uniaxial elongation in the x direction makes the velocity components of V - v x ~ + Vy ] + v z k equal to: V x "-
~(,
~'y =
-~-ey
Vy--
1. --'~.Z
1.
2
(4)
852
This produces the strain rate tensor:
+ (w)r)
_
_
0 1. 0 --E 2
0 0
(5)
1. --E 2 _ With only terms along the diagonal nonzero, elongational flows are shear-free flows. We define the shear free elongational flow function [13] as: + -- 'l;yy + - rl~; (6) ~xx 0
0
where each term with a superscript "'+" is a function of strain rate and time. This equation defines rl~(/z,t) for measured stress difference under applied strain rate. A Newtonian fluid has the single-valued shear viscosity It. It can be shown that there is a single-valued Newtonian elongational viscosity given by the Trouton relation tie = 31.t (7) This equation also holds for unfilled polymer melts when t~ is small enough (creeping flow) to keep the response within the linear viscoelastic limit. In this + case TIE -- 3 rl~. Building upon the discussion of transient shear behavior above, one could presume that a shear thinning fluid in elongation shows increasing deviation from Newtonian behavior as ~ increases. These differences are discussed next. Measured Elongation Behavior Different shear thinning melts may show strain hardening or softening in elongation [19,20,21]. Figure 5 shows strain hardening for LDPE at 150 ~ [22land for a polystyrene melt at 170 ~ [23]. For the lowest strain rate in each chart rl~ follows the Newtonian curve up to 1000 and 100 s elapsed time, respectively. As k rises the non-Newtonian behavior occurs earlier in the flow. The slowest ~ test forms an envelope curve; but, it is a lower bound envelope for extensional viscosity. The shear and elongational functioning of unfilled polymer melts is complex enough on its own. When particles and fibers enter as well the results become convoluted and require extra care in interpreting the data.
853
2.1.2 Filled Polymers Analyzing the flow of filled materials requires giving careful attention to the relevant physics. With solid constituents come a plethora of plausible interactions. Impact, rotation, relative motion and non-hydrodynamic (friction) events may appear in varying degrees. Similar stress strain shapes may have different effects at their root. We select the particle aspect ratio as a starting point for classifying filled systems. For an ideal regular cylinder the cylinder length divided by the diameter, L / D , reasonably defines aspect ratio. For other shapes, e.g., grotmd minerals, the major particle dimension divided by the minor length determine L / D . Generally L / D varies from 1 to infinity (continuous fibers). Aspect ratio determines the maximum filler volume fraction based on the particle geometry and orientation. L / D and volume fraction together establish the types of interactions possible. Tightly packed spheres may occupy up to 74 percent of the volume; highly aligned cylinder arrays admit 78 and 91 percent for uniform square and hexagonal packing. Random three-dimensional arrays of low aspect ratio filler may have few particle/particle interactions at sizable volume fractions. But large L / D particles easily can reach each other and have significant interactions in random orientations at small volume fractions. The effect of increasing particle aspect ratio is discussed next for shear and elongation of filled melts. Measured Shear Behavior Parallel plate and capillary rheometers can evaluate filled systems with particle or short fiber fillers [24,25, 26, 27, 28, 29, 30, 31, 32, 33]. In Figure 6 glass fillers in a polyamide substantiate the effect of aspect ratio on shear flow. The stress obtained with the neat polymer melt in shear contrasts with the stress ensuing as filler aspect ratio increases at a fixed volume fraction of 30 percent [34]. Glass spheres raise the steady stress by 3.5 times, also, the time to reach steady state increases considerably. The fiber/polymer systems each introduce stress overshoot to the reaction. For aspect ratio 7.3 the stress falls to the same level obtained with spheres. At L / D 27 however, the stress overshoot is larger still and the steady stress is 11 times the neat polymer level. Laun showed that the stress overshoots were the result of the orientation of fibers initially perpendicular to the plate of the rheometer. When a sample of neat fluid passes through the overshoot and relaxes completely it can regain its original structure and again undergo the overshoot. When a sample of the filled material duplicated the test it did not return to an overshoot. Stress rose only to the level developed just prior to the relaxation. The physics effective here was fiber
854
rotation, not conformation realignment. Experiments with all fibers prealigned either parallel or perpendicular to the plate verified this [34, 35]. Barbosa et al. [36] produced an apparatus for studying fiber interactions and the resulting transients as short fibers orient to the flow. lOs
O.lO~/,~o.o3 1.0
/
/
'~.01
A !
gO
1S"1
I1. los_
104 10.2
,
,,J
10"1
:
,,J
100
i
''J
'
101
''d
102
'
''J
103
-
'
''
104
Time(s)
a) 10 10 10 9 i~" 10 8
g
, 10
+~10 6 10 5
10 "1
100
101
102
103
Time (s) b)
Figure 5. Elongational stress growth function at strain rates between 103 and 1.0 s1 for a) PE [22], b) PS [23]. The linear viscoelastic curve in extension forms a "floor" for the transient viscosity at increasing strain rate.
855 The shortest fibers elevated the steady stress to the same value as the spheres. These fibers were so short that they aligned to the imposed flow and had the same impact as the spheres. That is both spheres and fibers with L / D = 7.3 interrupt the streamlines of the flow by nearly the same amount [34]. The longer fibers could not align completely as they interfered with each other. Their imperfect alignment kept more of them interfering with the flow. The total stress level can modify this entanglement effect by providing more energy to align the fibers.
150
AspectRatio 9 27 He 7.31
~il ~, 100~ ~/
,
so
0
0
50
100
150
Figure 6. Effect of aspect ratio on transient shear response of a polyamide melt filled with 30 % vol random orientation solids [34]. ~yx =0.1 s1 for each sample. Particles raise the apparent viscosity from the neat level. Elongated particles introduce an overshoot in stress in addition to the higher steady stress.
Measured Elongation Behavior Figure 7 shows the impact of particles and volume fraction on elongation of PS presented in Figure 5. Both loadings of carbon black generate a large increase in the extensional viscosity. This increase scales with the particle loading. The trend of the transient viscosity growth also changed. The neat polymer showed strain hardening with increasing strain rate. The filled system behaves like a strain sottening (or shear thinning) material. That is, the elongational viscosity reaches lower values with increasing strain rate. Creating short fiber filled systems with controlled fiber orientation employs injection or transfer molding or even capillary extrusion [37, 38]. Variation in
856
fiber interactions and rate effects modify the quality of the composite [39]. The elongation of short-glass-fiber filled polypropylene demonstrated the effects of moderately aligned approximately 150a r fibers on rl~ [40]. The stresses fluctuated from + 5% to + 15% due to the fiber/fiber interactions from the distributed fiber alignment within the transfer molded specimens. 10 lo 10 9
~ '10 +KI
8
10 7
106
20% Carbon black
10 5
I
10-1
I IIIIIII
I I I llllll
100
9 0.(0063 II 0.02 9 0.O63
m o~
I I IIIIIII
101
I I IIIII1
102
103
Time (s)
a) 1010 10
"~10 8 10 7
0~0063 0.02 0.063
+t~
lOS ~ 10 5
25% Carbon black l.n_ 0.2 I I1111111 I IIIIIIII
10-1 b)
100
I I!111111
101
102
! III!111
103
Time (s)
Figure 7. Elongation of polystyrene with carbon black particles, aspect ratio = 1, at 170 ~ at particle volume fractions of a) 20 and b) 25 percent [23]. With a filler the linear viscoelastic limit curve again forms an envelope under which the curves at higher strain rate fall. The higher volume fraction shows this effect to a greater degree.
857
An interesting set of experiments were performed on glass mat/thermoplastic matrix (GMT) systems at 20 vol. percent fiber [41 ]. Squeezed in biaxial extension to 1/2 their initial thickness, GMT elongational viscosity showed no correlation to the zero shear rate viscosity of the various thermoplastic matrices (polycarbonate, polypropylene and polybutylene terephthalate). Where test temperature controlled the zero shear rate viscosity of two matrix polymers to match their zero shear rate viscosity the GMT samples produced differing elongational viscosity! When the authors considered a shear rate magnification effect on a local level, they obtained good correlation between the elongational data and the shear viscosity of each matrix fluid. They tried several magnifications by plotting elongational viscosity versus matrix shear viscosity. At a strain rate of 1000 times the biaxial extension rate the data from all GMT materials fell on a common line. All GMT samples contained the same fiber mat and should have the same shear rate increase at a fixed elongation rate. They conclude that local shear flow dominates the bulk flow condition. These results show that from particles to fibers, the fillers increase extensional viscosity and create a local flow that adds shearing to the deformation. In the present work we seek to use a basic micromechanics model [42] to obtain the relevant shear strain rate induced by the relative motion of the discontinuous fibers in elongation. From the observation of shear dominated flow in the elongation of the filled polymer a continuum approach will be used to model the system as nonlinear viscoelastic fluid subjected to shear. In order to test this approach we need to produce the LDFMS material. The next section describes two processes for making highly aligned long discontinuous fiber composites. The first is a model system that incorporates well behaved components; the second is a commercial product using a new high temperature polymer. 2.2 Novel Fluid Systems Making a highly aligned long fiber system at a substantial fiber volume fraction with control of fiber placement is a challenge. Early extension experiments with highly aligned fibers employed capillary rheometers and filler volume fraction of only 0.0078 % [43]. Production of highly aligned discontinuous fiber arrays is interesting for both academic study and industrial applications. In academic use they increase understanding of the basic principles behind the forming process. Improved strength plus easy processing with properties tailored for specific applications intrigues industry. Two materials were used. The commercial product, long-discontinuousfiber/poly-ether-ketone-ketone (LDF/PEKK), takes continuous fiber tows and
858
creates a highly aligned system with a distribution of fiber lengths and overlap. The matrix material is not that well characterized and the fiber interactions with the matrix are complex. However the material is available in usable form. The model material can be made with a simple matrix and a selected L / D ratio. However the price to pay for this control is the labor required to construct the fiber arrays. For the model system we required a simple polymer with a high degree of fiber control at 50% volume fraction. The next sections detail the idea of a novel technique for manufacturing these highly aligned arrays for academic research and several areas in which the technique may be applied to produce tailored composite sheets for rapid forming of panels.
2.2.1 Nylon/PE Model Material System Rheological models that describe the motion of filled systems usually are ideal cases easily considered but not often accomplished. In an attempt to control fiber volume fraction and L / D ratio, we added nylon fibers to PE to make a composite. The nylon monofilament, which was 0.51 mm diameter, came from 910 rn spools of 134 N test strength fishing line. PE film, 1 mil thick, acted as the matrix. The details of the process that merged the two components follow. Control of Fiber Aspect Ratio and Position A custom built loom allowed fabrication of discontinuous fiber preforms. The loom bed provided a workspace where preconsolidation with polymer stabilized the fiber array before the consolidation steps. Fiber mats were 108 by 311 mm. Consolidation Two consolidation steps formed the panel. The first pressing combined the fiber mat with 8 plys of the 0.025 mm thick PE film. A picture frame mold held the sandwich of bottom release film (Kapton), bottom four film plys, fiber mat, top four film plys and top release film. A Wabash press pressed the mold and its contents at 795 kPa, 125 ~ for 45 min. Pressure remained on the mold during the 45 min water cooling cycle that returned it to room temperature. Wherever the wett lifted some fibers above the others a razor blade cut the raised fibers. The bridging filaments below each cut held the array together. Pressing six of the preforms a second time under the same conditions produced six-ply test panels.
859
Sectioning and Bonding All L / D 100 tests and some L / D 25 tests at 0.01 S -1 used 22 by 311 mm specimens cut from the panels with the fibers parallel with the long dimension. These specimens employed a phase change from solid to melt and back to solid that allowed the grips to hold the specimen ends. An attempt at achieving a fully melted specimen used bonded specimens for the L / D 25 tests at 0.001 s1. These N/PE specimens were 75 mm long with 118 mm nylon extensions bonded to each end with high use temperature epoxy, which had a 100 ~ cure temperature. The nylon extensions provided a solid gripping surface outside the fiamace. These coupons would have no solid/liquid transition zones.
2.2.2 LDF/PEKK System The commercial material used in this study is a composite of a PEKK matrix with a high volume fraction loading of discontinuous carbon fibers. This polymer is a high temperature thermoplastic with a recommended processing temperature of 370 ~ [11]. The advantages of this system are high strength fibers at high volume fractions. The manufacturing process keeps the fiber alignment very high. Final structures made of this material will approach the performance of continuous fiber composite. In terms of this research the material has the disadvantage of high processing temperature and a sensitivity to degradation when exposed to air as a melt. These needs require extra attention during the experiments. Fiber Morphology A breaking strategy converts the continuous fiber tow to discontinuous fibers and creates a distribution of fiber lengths [44]. A lognormal distribution produced an average fiber length of 5.07 cm with the longest measured fiber at 16 cm. The fiber tow is Hercules AS-4 graphite with an average fibril diameter of 7 lam. Assembly of the Composite Fibers and matrix combine to produce a preimpregnated tape. Hand layup of the preimpregnated tape produced panels 42 cm square and 8 plies thick. The panels were consolidated under pressure at the processing temperature. This step formed the panels without shearing deformation and allowed some stress relaxation in the melt. Specimens cut from the panels measured 2.5 by 42 cm with all fibers aligned along the long dimension.
860
At this point we have two material systems to characterize. The model system has well known matrix properties and a simple fiber arrangement. The commercial system contains a new polymer and distributed discontinuous fibers. In the next section details of the test methods and equipment are discussed. Each system is characterized by rheometry of both the neat matrix fluid and the filled composite. 3. TECHNIQUES TO C H A R A C T E R I Z E LDFMS Rheological measurements of neat polymers by capillary and rotational devices are covered well in the literature [13, 45]. Many devices for measuring elongational viscosity exist [18, 21, 41, 46]. None work as well or are as versatile as shear rheometers. Most designs are optimized specifically to the fluid of immediate interest. The device described below was developed from the need to study elongation in long-discontinuous-fiber/poly-ether-ketone-ketone (LDF/PEKK) at 370 ~ The same device stretched the nylon/polyethylene (N/PE) so that data from both materials could be compared. Constant elongation rate tests provide transient tensile stress growth coefficient + data, rl~ ( t , ~) = XE ( t , ~)/k. These data describe the difficulty that will be encountered in forming a useful component at the high elongation rates desired for mass production. Controlled strain rate forming occurs in matched dies [3]. The fluid sheet becomes trapped at two or more points between the dies and stretches at a rate proportional to the closure speed of the dies.
3.1 Rheological Measurements Getting rheological functions from tensile experiments requires analysis of the specific specimen geometry and the displacement or load control scheme. Each method is discussed and analyzed in the next sections. 3.1.1 Constant Velocity Flow
Standard test practice for solids involves moving one end of the specimen at a constant velocity v. This is a reasonable approximation of constant strain rate for the small displacement of most solid specimens. The execution of the test is simple to do; electronic controllers always have a linear ramp function available. Screw drive test machines do this extension mode smoothly. Why is this style of control insufficient for rheology? We start with the strain rate relation [47]" ldL v = - - - - = -(8) Ldt L
861
where the specimen length L = L ( t ) = L o + vtsince Continuing:
v{l +
~(t) = -~0
L0J
-I -~:o {1 + ~:ot l_l
v is constant.
(9)
with g(0)=~o=V/L 0, which is the coveted constant strain rate. Equation 9 establishes strain rate as a decreasing function under a constant velocity scheme. An example from linear viscoelastic theory discloses the impact of this difference. For LDPE the tensile stress growth at 0.001 sl can be obtained from [13, 22]"
"on- i 3rl~
}e(t')dt'
(10)
With ~1 = 1000 s and t reaching a maximum of 7000 s for constant velocity extension, the strain rate decreases by 88%. The drop in tensile stress is slightly less, 85%, due to the memory effect of the relaxation modulus.
3.1.2 Constant Strain Rate Flow Obtaining extensional viscosity data from a polymer melt is a complex task with three major problems for the experimentalist. When the melt contains long fibers the difficulty increases and requires additional steps in producing values of elongational viscosity. The next two sections discuss the problems and their solutions that allowed calculation of apparent viscosity. Achieving Constant Strain Rate The first problem comes from the definition of rl~ and the displacement needed to measure it. We require the transient load data to solve the relation +(t,~) / ~ - F(t k) / A(t,k) / ~ (11) where the strain rate ~ is constant throughout the experiment. Equation 8 with v - L may be integrated to obtain ~t = ln(L / Lo) (12) Keeping ~ constant requires that the moving end of the specimen follows the displacement L ( t , ~ ) - Lo e~t (13) to account for the increase in gage length [47]. Thus, for constant k, both L and v are exponentially increasing functions of time. Since the specimen length increases exponentially a sophisticated control system is needed to accomplish the experiment. With the growing number of computerized function generators for driving test machines this is less of a problem than it was 10 or more years ago.
862
The second problem is the conversion of the transient load data F(t,/;) into tensile stress "c+ E . Engineering stress cannot be used for displacements of interest to thermoforming. Instead, assuming constant volume of a melt the cross sectional area decreases with extension as A ( t , g ) - Aoe-~t . Sectioning the sample after an experiment and weighing it as described below verifies this assumption. Thus equation 11 above determines the transient stress and viscosity. Third, the ends of the melt must be securely held during the test. Several methods do this with neat fluids and particle or short fiber filled systems [48]. For some L / D 25 tests detailed here we could use a short specimen with nylon extensions bonded to its ends with high temperature epoxy. This specimen type could only be used at the lowest strain rates. Larger L / D or faster testing rates need another technique. Adding long fibers to the melt makes each of these problems more challenging. Long fibers greatly increase the elongational viscosity and the force on the specimen is larger. In this study a phase change method held these specimens securely. The extreme ends of the specimen remained solid so that standard Instron grips could hold them. The specimen, when heated in the center, formed a melt/fiber system in a gage length of approximately 200 mm within a total specimen length of 305 mm. This solution to the third problem affected the first two as well. With a transition from solid to melt and back there is a distribution of strain rates from zero to a maximum and back to zero again along the major axis of the specimen. This distribution broadens with the presence of long fibers. Some fibers at each end of the melt zone will have some part of their length stuck in the solid or semisolid polymer. With a distributed strain rate, the change in cross sectional area is distributed also and must be determined before performing the conversion of load to stress and displacement to strain rate. Since no simple assumption of the area distribution is possible for these systems we determine the resulting strain rates and cross section from a post-experiment analysis of the specimens. Since there is a transition in strain rates from one end of the specimen to the other, this viscosity must be designated as an apparent value [49, 50].
3.1.3 Interrupted Flow Interrupted flow experiments provide a means of determining the source of nonlinear response (stress peaks) and the change in relaxation response as flow progresses from rest to steady-state. A variation of the constant /~ test, the interrupted flow experiments run at a constant strain rate for a time and then the
863
displacement abruptly is held fixed for a controlled amount of time. Upon resumption of the constant g flow, the transient response of the fluid depends upon the complete history of the specimen, that is, prior flow, hold period and resumption of flow. The objective of this experiment is to show that the stress growth is similar to the behavior of a nonlinear viscoelastic fluid. That is, the presence of the peak stress comes from the fluid dynamics of the melt alone and not from fiber-fiber interactions or changes in fiber orientation--as for random oriented short fiber + systems [34]. If the melt is solely responsible for the peak in XE, then, given sufficient stress relaxation, the LDFMS will again display stress growth to a transient peak [51]. The opposite also must apply, i.e., insufficient stress relaxation will inhibit the peak stress. It is possible to select a single interruption period that will provide sufficient stress relaxation at one strain rate and insufficient reduction at another rate.
3.2 Apparatus and Procedure Petrie [47] reviewed the factors that complicate construction of an apparatus to achieve constant rate extension of specimens. LDF/PEKK complicates the problem ft~her by requiting both high processing temperature and long specimen length. The melted zone must be at least three times the average fiber length ( i.e., totaling 16.8 cm) and ideally should be 1.0 to 1.2 times the largest known fiber length ( i.e., totaling 16.25 to 19.5 cm). This length avoids capturing a significant population of fibers that would otherwise bridge the melt zone. An electric tube furnace combined with a computer-controlled Instron model 1321 hydraulic test machine produced the rheological measurements collected here. This apparatus allowed elongation rates, ~, from 10-5 to 10-3 s"l for LDF/PEKK and 10-3 to 10"1 s"1 for N/PE . Processing requirements (exposure limit of the material at 370 ~ restricted the total test time during the lowest and the Instron hydraulic power supply's maximum flow rate limited the fastest attainable. Each test began with the furnace at room temperature. A consolidation device held the specimens under pressure during heating and testing. Both types of N/PE specimen used the same frame and insert. Four spring clips held the monolithic samples in place between the steel insert and frame. A single clip over the center of the 76 mm bonded coupon kept the system together. Bonded specimens used nylon shims in the comers of the frame to keep the insert from touching the nylon extensions. A high temperature film material (Upilex R) sealed and consolidated the LDF/PEKK specimens during the experiment. A vacuum drawn on the bag
864
protected the melt from degradation during the long heating times required with the fitmace used. With the heaters switched on, the specimen temperature rose to the test temperature. The ends remained well below the Tg of the matrix so that they provided a finn grip surface. After reaching the test temperature--following 20 min of heating--the sample rests for 5 min before the extension starts. With the techniques developed here, we evaluated the filled fluid systems proposed in section 2. In the next section the results of each experiment are shown. First the neat polymers are characterized, then the filled systems are extended and the data are discussed. 4. BEHAVIOR OF LONG DISCONTINUOUS FIBER SYSTEMS This section presents the experimental data for nylon/polyethylene (N/PE) and long-discontinuous-fiber/poly-ether-ketone-ketone (LDF/PEKK) in series. First, for each system, results for the neat polymer appear progressing to the filled system in elongation. 4.1 Nylon/PE Model System This system contains two well developed materials. The rheological testing of the LDPE confirms the specific properties of the batch of film used. Extension of the nylon fiber/PE melt demonstrates the impact of fixed length aligned fibers on viscosity of the composite.
4.1.1 Rheology of PE Film The general character of a LDPE melt is well known. The melt is isotropic, shear thinning and stable in air when close to its melting temperature. The first task of this research with PE was finding a reasonable melt temperature that does not degrade the melt during the extensional tests. Differential scanning calorimetry (DSC) [12] of the PE film shows gradual softening until 109 ~ Melting is complete at 121 ~ with no additional thermal effects found to 200 ~ Temperatures of 125, 140 and 200 ~ were considered for the role of standard temperature for all PE and N/PE experiments. The shear viscosity at 125 ~ was too high for easy use of the cone and plate rheometer. At 200 ~ PE degrades severely within an hour. PE viscosity is stable for 2600 s at 140 ~ with viscosity averaging 30970 + 650 Pa-s during the test. Steady Shear Viscosity The rotational and capillary viscometers discussed above produced the steady shear viscosity data presented in Figure 8. Also shown are the data of Meissner
865
[52]. The LDPE film corresponds to a Carreau curve with parameters 11o = 43,850 Pa-s, ~ = 5.40 s, n = 0.489. 2x10 5
osl_
104
I
l
:3o'c 150~
"~"~"0
PE 140~
n ffl
~'
103
102
10-4
10-3
10"2
10-1
100
101
102
103
t (s") Figure 8. Steady shear viscosity of PE at 140 ~ in air as a fimction of strain rate. Data are from rotational and capillary rheometers. Curve is a Carreau curve with parameters: rl0 = 43,850; ;~ = 5.40; n = 0.489. Data at 130 and 150 ~ from Meissner [52].
4.1.2 Rheology of N/PE Measured rheology of the N/PE system is limited to extensional deformation. With 60 % vol fibers that are 13 and 50 mm long, N/PE is incompatible with standard rheometers. Controlled g Flow Controlled g flow experiments on two types of N/PE produced the following set of data. The extensional viscosity shows the impact of aligned fiber filler on an isotropic melt in extension. Elongational Viscosity Nylon fibers with an aspect ratio of 25 raise the elongational viscosity of the system to 57 times 311o for PE. Figure 9 displays the measured transient elongational viscosity. For a total strain of 0.15 the average viscosity is 7.47 MPa-s. For strain rates covering one order of magnitude the viscosities are close. The lowest viscosity was reached by the panel with the curved fibers.
866
Increasing the aspect ratio to 1O0 raises the extensional viscosity at the lowest strain rate by another order of magnitude to 192 MPa-s. This is three orders of magnitude above the 31"10 value of 0.132 MPa-s. Figure 10 shows this and it also shows the drop in viscosity with increasing strain rate. Each rate increase by about 10 drops the viscosity by a factor of 5 to 6. These changes are similar to the function of a shear thinning fluid in the power-law region. 2x107 107 A
W !
n
n
m §
0 0.0457s-1 I I 0.0257 El 0.00261 106 0.15
0.30
~t Figure 9. Transient elongational viscosity of L / D 25 N/PE. The prior stress data are divided by the extension strain rate to obtain viscosity. The data show that the elongational viscosity is fairly consistent among the three tests. The wavy fiber plys in the 0.0257 s"1 strain rate test make the viscosity increase less and the peak occur later. The smooth curves are average curves for three repetitions; only one set of data points is shown for each strain rate to clarify the chart.
Figure 11 shows "steady" type viscosity data. The error bars at each point show the range of peak viscosity to the last viscosity attained at the end of the experiment. Points are connected with straight lines to group them by aspect ratio. The L / D 25 samples are rather fiat with strain rate--discounting the sample with wavy fibers. The 100 L / D specimens look like a power-law region response with decreasing viscosity.
4.2 LDF/PEKK System Experiments with LDF/PEKK follow the reporting sequence of the model material above. For this system, interrupted flow data follow the steady extension
867
results. These experiments explore the interesting nonlinear aspects of the response. We begin with the neat PEKK. 109 0.00437 s "1 108 -
.......
A
0.04-~7 s
-1
107
m
106 tll
+~
31"1o PE
105 104 [I 0
m
I
m
0.15
0.30
~t Figure 10. Transient elongational viscosity of L / D 100 N/PE at three strain rates. The slowest strain rate shows an increase of viscosity of approximately 3 orders of magnitude from 3r10 of PE.
109
~176
108
0 100 I-! 25
A
Io I
t~ IL
107
o. 106
II m
3rio
10 5
m
104 0.001
m
,
m
,
m
,,I
0.01
m
m
,
mmmm
,
mlmm
, ,I
0.1
mmm
PE m
. . . .
1
(s) Figure 11. Extensional viscosity of N/PE for two L / D ratios compared with neat PE; points are connected with straight lines to group the aspect ratios. The L / D 25 material, accounting for the wavy fibers in the middle sample, remains fiat in terms of the viscosity/strain rate ratio. L / D 100 material shows a powerlaw response to increasing strain rate.
868
4.2.1 Rheology of PEKK Polymer High temperature melts may degrade with long exposures to their processing temperatures. This would not be of concern in a rapid industrial process; but, slower processes and characterization experiments may be carried out over an extended time in order to improve the understanding of the material's behavior. First aging of PEKK is discussed, then the measured viscosity results are shown. DSC of PEKK illustrates the extra complexity of this matrix thermoplastic. There is a glass transition temperature at 148 ~ Soon after this the solid crystallization peak arrives at 201 ~ Further heating brings melting at 322 ~ which is complete at 360 ~ The recommended processing temperature is 370 ~ This temperature, with a few noted exceptions, was the thermal condition for rheometry and extension. In order to differentiate the effect of aging from other viscosity changes, PEKK melt was sheared at 1 s~ for 1 h in order to track the change in 11 with time for two environments: N2 gas and air. The PEKK aged in N2 had an increase in viscosity of 60 % during the hour with viscosity increasing from 470 to 749 Pa-s. The air exposure experiment was a severe test of the effect of air on the PEKK melt. The rotational rheometer was heated with air prior to insertion of the PEKK disk. After re-heating to 370 ~ the PEKK melt was allowed continued exposure to air for the 5 min stabilization time plus an additional 15 min. This exposed over one-half the surface of the 1.5 mm thick melt to air. After this conditioning the initial viscosity of the melt was 915.5 Pa-s, almost twice the viscosity of PEKK in an N2 environment. After 2000 s the experiment was halted because the viscosity rise could lead to damage to the rheometer. Viscosity at 2000 s was 2310 Pa-s. Steady Shear Viscosity The shear viscosity of PEKK measured from 2 x 10- 3 to 20 S-1 is in Figure 12. For strain rates from about 0.1 s-1 and higher, PEKK behaves as a shear thinning polymer melt. Below this rate PEKK demonstrates aspects of the behavior of a thermotropic liquid crystal polymer (LCP) [53, 54]. For strain rates less than 0.1 s-1 the viscosity rises from the one-decade-wide plateau and increases with decreasing shear rate. At the lowest rates, 2 x 10-3 and 2 x 10-2 s-~, the melt did not attain a steady shear rate within the processing time limit adopted for this study. The viscosity shown for the slowest rate is the average; the vertical bar indicates the range from 1/3 of the maximum to the maximum viscosity measured. The previously reported experiments with polyethylene at strain rates from 4 x 10-4 to 4 x 10-1 s1 verified the performance of the RMS-800 at low sheafing rate. As ~ decreased the noise level increased; but, the steady behavior is as
869
expected for PE. The slowest shear rate was selected as 1/5th the slowest rate used with PEKK. Here the PE did not reach steady-state within 3800 s; but, the viscosity grew towards the steady value without any unusual viscosity change. 10 s
Melt
10 s
- O - PEKK 3 7 0 ~ PE 140 ~ C
I
0,, 10 4 10 s
10 = 10 4
10 -a
10-=
10-1
10 0
101
10 =
(S "1) Figure 12. Steady shear viscosity of PEKK at 370 ~ PEKK has power-law behavior at low shear rates up to 0.2 s -~ and at high shear rates above 2.0 s -]. A plateau viscosity connects the two power-law regions at about 470 Pa-s. The PE data shown were collected with the same rheometer to test its performance at low shear rates.
The steady shear viscosity of PEKK measured from 2 x 10 . 3 in Figure 12.
to
20
S-1
is shown
Transient Shear Viscosity Figure 13 shows the change in transient viscosity growth as ~ is reduced from 20 to 0.02 s-]. At 0.02 s-] the viscosity grew from the expected plateau value of about 470 to 600 Pa-s to reach a peak of 5610 Pa-s. After the peak, viscosity fell to 2260 Pa-s, which is more than four times greater than the steady viscosity at 0.2 s-~. Note that at the highest shear rate some undershoot of ~1~ appears. Similar viscosity growth has been noted for thermotropic liquid crystalline polymers [55]. After a pre-shear at 1 s-~, a PEKK sample was allowed to relax for 30 rain. Then it was sheared at 2 • 10-3 s-~ for over 3500 s. The viscosity rose to a maximum of 136,000 Pa-s. Figure 14 shows the experiment from pre-shear
870
through two post-shear runs ('i, = 1.0 sl). The post-shear experiments returned the measured viscosity to the levels measured in the pre-shear plus an increase expected from the aging results above. Thus the significant rise in viscosity at 2 x 10 -3 sl cannot be attributed to aging because the increase is reversible and the exposure time in N2 was limited to the time in which viscosity would be expected to increase by 60 % at the most.
($-1)
?
lO 4
9
0.02
D A
1.00 20.0
v
+ ~. l O =
10 = 0
6
12
18
7 Figure 13. Transient viscosity of PEKK at 370 ~ N2 environment. The tests at 1 and 20 s~ strain rates respond as expected for shear thinning isotropic melts. The drop of shear rate to 0.02 sl produced a dramatic rise in viscosity.
4.2.2 Rheology of LDF/PEKK LDF/PEKK is the material system that started this research into long fiber systems. The results obtained with LDF/PEKK guided the sequence of experiments toward additional cases of deformation control. With the basic elongational tests finished, we looked interrupted flow as a means of separating fiber interaction and rheological components of the tensile response. Controlled ~ Flow Following the basic format used with N/PE, the next sections show the reactions of LDF/PEKK to similar experiments and additional tests.
871
,,
_
10 s A
,,,
~ (,-1) 1.0
,~
10 s
D
0.002
10 4
9
1.0 1st
II
1.0
m el
Pre-Shear Post-Shear
2nd Post-Shear
+t=-I 0 s 10 2 101 10 0
~
0.0
,
,
I
2,5
,
,
,
!
.
,
5.0
, , !
7.5
,
,
, ,
10.0
Figure 14. Transient viscosity of PEKK at 2 x 10 -3 s"l compared with pre- and post-shear transients of the same specimen, 370 ~ N2 environment. This shows that the viscosity increase at the low shear rate was a reversible function of the applied shear strain rate.
Elongational Viscosity of LDF/PEKK For each elongation rate, Figure 15 shows the normalized rl~-t behavior of the LDF/PEKK. At the slowest rate, T1E reached its maximum value. This value was used to normalize the rl~-t curves for all three rates. The -+ "rE - ~t data presented are remarkably like the response of a nonlinear viscoelastic fluid in shear. This becomes more apparent in the r l ~ - t curves shown here. The elongational viscosity is checked against neat PEKK data at the end of the controlled stress tests which immediately follow. Interrupted Flow In these experiments LDF/PEKK extended at a constant strain rate for a time and then the extension stopped with the total displacement held constant. Following a fixed rest period the flow resumed and the transient response recorded. Repeated appearance of the peak stress is a function of the rest time and applied strain rate. Also, the effect of slight fiber misalignment on the initial stress growth was noted. Panels are produced by hand layup of prepreg sheets. During the first extension step the fibers align to the drawing direction and
872
subsequent elongation started with an increased modulus of 41.3 GPa, up from the initial 14.7 GPa slope.
I
Q. 0.1
~o x oo 04
0.01
.3 -1
0 1.49x 10.4 s I I 1.66 x lOs 9 1.39x 10
Ip + UJ
0.001
9 0
I 100
,
I
.
200 Time
I
9
400
300 (s) +
Figure 15. Normalized elongational viscosity growth, rl~. , for three elongation rates of LDF/PEKK. The curve for the 10s rate reaches a value of 1 at 994 sec.
The data in Figure 16 show the result for three extensions of a single specimen at a target ~ of 0.001 s-1. For the first extension the specimen elongated 8% and the stress relaxed for 165 s. This time allows over 90% of x E to dissipate. After the 165 s relaxation period, x~ fell 92.8% to 12.4 MPa. Atter restarting the extension the specimen again passed over a yield peak, although, at 231 MPa, it is 5.0% lower than the peak in the first flow. Also, in the second flow the sample did not attain the desired near-fiat quasi-steady plateau after passing x y . Instead, 4-
a near-linear decreasing x~.-e results.
The lower x y is probably due to the
incomplete relaxation of x~. atter the first flow. The declining x~. atter the second peak is due to an error in the assumed gage length of the sample during the second flow. The three elongation test was repeated at the next slowest rate, 10-4 s1. The stress relaxation data above demonstrate that stress decay is much slower after drawing at this rate. Thus, for a fixed rest period, less stress will dissipate before the extension resumes and this should significantly influence the size and shape of the peak during the second and third extension. Figure 17 shows that the peak is less in the second and third flows.
873
250
i
I
Extension
i~ ~ First
200 13.
'~
150
Second
Third
v §
I.U
100 50 0 0.00
ii 0.05
0.10
0.15
.
. I, 0.20
0.25
Strain +
Figure 16. 1; E for three step elongation of LDF/PEKK with g of 10"3. Each stress relaxation period is 165 s. Peak stress returns after the second and third restart of extension.After the second extension, x E decays from 54.1 to 5.38 MPa, a 90 % +
decrease in 165 s. The third elongation step produces another peak in x E . By this time the error in the apparatus is significant throughout the elongation. The significance of the third extension is that a yield stress occurs again.
Since this is a new material with unique properties, two questions about the + test results come to mind: (i) What is the effect of the fiber alignment on the x E e behavior? (ii) To what extent might bridging fibers contribute to the peak stress or the magnitude of the stress growth slope? The effects of fiber alignment and fibers that bridge the gage length can be studied with the multiple extension test data by shifting the second pull curve so that it starts at zero relative e. After the first extension the stress growth slope grows from 14.7 GPa in the first extension to 41.3 GPa in the second. This shows that the minor misalignment that results from hand layup of the plies is significant in its effect upon the initial slope of the stress growth. No further increase in slope occurred in the third extension. On the prospect of bridging fibers adding significantly to the slope of the stress growth or the value of x y , the population of bridging fibers is much less than 2.6 % of the fibers in the cross section based upon the distribution data of Chang and Pratte [1 ]. The work of Bums [44] shows that at most 0.5 % of the fibers have a
874
statistical chance of avoiding fracture and bridging a 30.5 cm melt zone. So the bridging fibers alone could account for at the most 1/22 of the slope during the first pull based upon the modulus of AS-4 carbon fiber. Then, during the first pull, any bridging fibers break. The second and third pull stiffness and x~ are therefore entirely due to the properties of the fluid and the shear strain rate magnification within the material. 200 150
Extension - 0 - First , ~ Second
A !
Third 13.
IO0
v
,,
,
I
+~ J
50 f ' O' 0.00
'
.
.
.
0.05
.
"
6
.
9
0.10
i
0.15
i
0.20
0.25
Strain Figure 17. XE for three step elongation of LDF/PEKK with ~ of 10-4 The stress relaxation period is the same as for the higher strain rate test. Less stress relaxes and the peak stress is diminished. "~
~
The multiple flow test at 10-4 s-I ~, when combined with the prior discussion, supports the view that x y is a nonlinear viscoelastic effect under constant ~. Since x E had only decayed by 68.1% atter the 165 s pause the melt could not move to its preferred rest conformation before the next pull started. Thus, x y diminished. 4.3 Summary of LDFMS Results The data presented above show that LDFMS have the following properties in extension: Extensional viscosity is orders of magnitude greater than the zero shear rate viscosity of the matrix.
875
The increase in extensional viscosity generally increases as the aspect ratio of the aligned fibers increases. At L / D 25 N/PE had a "flat" extensional viscosity and at L / D 100 N/PE had a power law extensional viscosity. At an average L / D viscosity.
of 8000 LDF/PEKK had a power law extensional
At the highest two strain rates, LDF/PEKK in extension demonstrated a peak stress that could be recovered with sufficient relaxation of stress and that was diminished with insufficient stress relaxation. The next section proposes an analysis of the data that attempts to explain these properties by relating the shear and extensional flows through applied micromechanics.
5. MODELS OF R H E O L O G I C A L PERFORMANCE Fibers placed in a matrix can dramatically shift the deformation mode of the system. For the filled fluids discussed here the total data set of flow, relaxation with cessation of flow and behavior with restart of flow must be considered. Figure 18 shows micromechanics and constitutive domains for the aligned fiber systems of interest. The left figure shows a fiber/polymer cell in scale for a 60 vol ~ fiber loading. The cell is globally deformed in extension. A fluid element from the cell is displayed on the right to illustrate the applied global elongation and the generated local shear deformation defined by the development that follows. Micromechanics provide the local deformation. The constitutive relation must be appropriate to the superimposed deformations of the fluid element.
5.1 Micromechanics: Shear Cell Analysis The fiber/polymer cell in Figure 18 is the "shear cell" used in solid and fluid mechanics models. Next we develop the basic properties of the regular cell.
The first analysis of the effect of long fibers on extension of Newtonian fluids applied to dilute suspensions [42]. The shear strain rate is concentrated in the near field of the fiber. This effect increases the extensional viscosity of the system, which was verified by Mewis and Metzner [56]. A derivation for shear thinning fluids finds that the effect of the localized shear rate removes the fibers
876
as a factor in extensional viscosity [57]. That is, the viscosity drop in the fiber region compensates for the presence of the fibers. However, the systems employed here are not dilute. There is a substantial difference in shear rate distribution in a concentrated suspension. The shear rate is significant and of the same magnitude along the entire cell radius Ri < r < R 0.
Figure 18. The combination of micromechanics and constitutive relations provide the model for filled polymers. Micromechanics relate the strain rates and portions of the stress obtained from each mode. Constitutive relations relate the stress to the strain rate of each flow mode.
The relevant equations for the micromechanics start with the fiber pulling speed. Figure 19 illustrates the relative motion of the top fiber with respect to the lower neighbor fibers. Each fiber remains whole and therefore moves with the velocity of its centroid within the global elongation flow [58]. The relative velocity for ideally spaced fibers is VreI - ( x 2 - x l) - L / 2 . The Navier-Stokes equations yield the x component of the velocity vector: #-L ln(r / R~ (14) vx (r) - 2 In k where vx (R i ) is the inner boundary condition vx = ~ / 2 vx (R 0) is the outer boundary condition v x - 0
877
is the strain rate in s -~ L
is the fiber length is the radial coordinate such that R i < r < R o
&
is the cell radius, Ro > R i = fiber radius
k
is the ratio fiber to cell radii Ri / Ro , this is equivalent to where f is the fiber volume fraction
Then the shear rate is ~Vx_
r
kL
1
~rr - 21nk r
(15)
Figure 19. Relative fiber motion forms a shear cell in the Batchelor model.
The magnification scales directly with L / D. Continuing with the micromechanics, we compare the portions of the tensile stress each deformation contributes. This shows the effect of aligned fibers on the elongational viscosity for a Newtonian fluid. Pipes took a similar approach [59] but immediately eliminated the extensional load of the polymer. Here both effects are included. The total stress to extend the system is x E - ( F m + F f ) / 2 A; this is the sum of the force needed to extend the melt annulus (Fro) and the force required to pull one fiber from the annulus ( F f ) . The forces are divided by the
878
area of two shear cells (2A) as only half the fibers carry the tensile load with a good choice of the free body diagram [60]. For any volume fraction Fm = 2AfqE~. The fiber pulling force comes from the shear stress over the fiber surface within the cell [58] or F f = x , . x ( 2 ~ i ) L / 2 = rl'~rx/~.L. Using equation 15 the tensile stress is: XE
"-
t,}
3(1- f)rl~ + Ink tie -~
(16)
with tiE = 311. The first term shows the elongation component and the second term shows that the shear cell effect has a factor of (L / D)2. The portion of the total stress due to the sheafing within the cell quickly becomes orders of magnitude larger than the portion provided by elongation of the annulus. For L / D greater than 5 the annulus stress is less than one percent of the total measured stress. 5.2 Constitutive Relation: Giesekus Fluid Model Many constitutive relations exist for polymer melts. The objective here was finding a suitable relation for the combined flow of the filled system. If a relation does a reasonable job with both shear and elongational flows of neat systems it would be useful with a variable L / D as one or the other mode becomes dominant. The relation must incorporate the effects of conformation changes that accompany the restructuring of the polymer network under flow. The Giesekus model [61] provides realistic behavior for nonlinear viscoelastic fluids; it is based upon a network breakdown of interacting polymer molecules. The form of the model used is from Bird et al. [13]" + ~1~(1) - - 0 ~ { ~
9 ~} --O~ 2 {~(1) " 17 + 1; " T(1)} ----
110 +
(17) -
~
where ~ is the stress tensor, ~ is the strain tensor, 11o is the zero shear rate viscosity, ~l is the relaxation parameter, ~,2 ~ 1 / 1000 is the retardation parameter, and tx is the parameter that determines the degree of nonlinear behavior. This set of equations was solved with a 4th order Runge-Kutta program.
879 The strain rate tensor acts as the driving element of the solution to equation 17. These tensors arise from the micromechanics of the shear cell. The strain rate tensor for the elongational flow of the polymer annulus comes from the velocity vector: Y = Vr~.r + vo~o + Vx~x with v x = kx , v o = 0 and the boundary conditions on radial velocity of v~(r= R i ) = 0 and v~(r= Ro)< 0. These BCs allow the polymer to flow toward the center to correspond to the extension in X with the fiber acting as a solid inner boundary. To satisfy continuity V ' V - 1 r ---5__ r/9 (rVr)+ k = 0. And ultimately V - 2zk. ~(R2 - ,--7-
r ) ~', + kXex. The strain
rate tensor is: -
_R.2
0
- Vv + {Vv} r
0
0
R? e(-~-l)
0
(18)
/
0
0
2~
The sum of the diagonal terms is zero, which satisfies continuity. In shear flow the only velocity term is as defined above in equation 14. Thus:
= Vv + {Vv} r _
0
0
0
0
~L 2rlnk 0
~L 2rlnk
0
0
(19)
These tensors show that the magnitude of the strain rates are bounded by 2e in elongation and ( 1 / l n k ) ( L / D ) ~ in shear. For L~ D over 5 we rewrite equation 16 as XE
f
_fL] 2
5.3 Application to the Systems
Next the micromechanics and constitutive relations are applied to each material. The N/PE and LDF/PEKK use the micromechanics relations to verify the steady state data and the constitutive equation is applied to the LDF/PEKK system.
880
5.3.1 N/PE Analysis In the section of results we presented the apparent extensional viscosity of the N/PE in Figure 11. This chart placed the values of viscosity with respect to the average tensile strain rate. With the micromechanics discussed just above, we now determine both the shear strain rate and the equivalent matrix shear viscosity by equations 15 and 20. Figure 20 displays the resulting matrix viscosity from the micromechanics model compared with the neat shear viscosity of the PE film. The correlation is good overall with the larger L / D showing the best following of the PE curve with the power law behavior of the neat fluid followed by the N/PE result. The viscosity calculated is lower than the neat PE however. Two possible causes for this are insufficient consolidation force and less applicability of the model as L / D approaches 1. 5. 3.2 LDF/PEKK Analysis The LDF/PEKK system contains fibers of the same length as the L / D 100 N/PE but the aspect ratio is 8000. We find that the viscosity ratio TIE/TI is expected to range from 2.2 to 7.4 xl 0-". This is a reasonable range of values for the two high strain rate experiments; but, it is up to 2 orders of magnitude too low for the slower tests. An alternative viewpoint is provided by rearranging equation 20 and predicting the neat melt viscosity at the shear rate obtained with equation 15 for each experiment. For the two highest strain rates the estimated viscosity from LDF/PEKK covers the possible range of neat viscosity from PEKK data. As the strain rate decreases the over-estimate rises. At the slowest rate the viscosity estimate is about 68 times larger than the PEKK viscosity, TII,EXr, in the LCP region. At this point we can present two possible scenarios for this deviation at low shear rates. The sliding plate work of Ericsson et al. [62] suggests the first. Their work was with L / D 1000 and 2000 glass fibers in polypropylene (PP) thermoplastic at 14 and 28 % vol fraction. They found that at high shear rates the increased viscosity scaled with the square of the fiber concentration which indicates the response is dominated by hydrodynamic effects (shear of the fluid in narrow fiber regions at rates much greater than the bulk). At low shear rates they argue that non-hydrodynamic, i.e., fiber to fiber, interactions dominate the stress. They make no physical argtmaent for the change in behavior with strain rate; but, they note the increase in viscosity scaling at lower rates. The behavior of neat PEKK suggests the second possibility.
881
105
104
A
u)
103
0 -
Source PE
O N/PE L/D 100 I"! N/PE L/D 25
102 10 .3
10 .2
10 "1
r 1
10
102
103
(s "1)
Figure 20. Matrix viscosity of N/PE derived from the extensional viscosity data for the system. Solid line is the steady viscosity of the PE film.
This system could be responding to the thermotropic liquid crystal polymer (TLCP) character of the PEKK. We know the matrix material starts with TLCP structure and moves to isotropic as shear rate increases. There are at least two possible mechanisms for the transition. In the first, the TLCP structure "fails" first at the fiber surface where the shear strain rate is largest. In the second, the TLCP is "bound" to the fiber surface and becomes isotropic last. Bhama and Stupp [63] showed that carbon fibers in TLCP made a tremendous change in the formation of LCP structure. The fiber surface nucleated the crystal structure and increased the temperature of the nematic/isotropic transition. The LCP phase remained for several microns from the fibers surface after the rest of the fluid became isotropic above the transition temperature. LDF/PEKK has an average 1 lam thick PEKK layer around each fiber. Also, LCP capillary flow data have shown that a liquid crystal structure may preferentially form at a treated solid surface [64]. LCP layers up to 7 ~m thick formed a high viscosity layer with a lower viscosity orientation present in the flowing core. Carbon fibers are treated to control the interface with the polymer. The fiber surface morphology and surfactants may influence the rheology at shear rates where the LCP structure is stable. (One way to check this would be to add ground carbon fibers to PEKK and test it in a cone and plate rheometer. The small particles at low volume fraction could be run on the standard device.)
882
Under either mechanism this could create a "two fluid" system in the shear cell. A viscosity difference of 10 or higher quickly concentrates the deformation to the outer layer. This increases the shear rate for the outer fluid and increases the effective fiber volume fraction. Changes in this direction would reduce the difference in PEKK and system matrix viscosity as the strain rate decreases. Thus the interaction of the TLCP melt with the carbon fiber surface may make the shear cell model inapplicable at low shear rates. Some innovative techniques would be needed to assess this effect. More of this is discussed in the modeling that follows. Giesekus Fit to Steady Extension Data Since the elongation of the LDFMS engenders shear dominated response, a constitutive model that describes stress overshoot in simple shear might be applied to this material. Solution of the Giesekus equation by Runge-Kutta method allowed f'mding the parameters that reach x y at the required time with equal to the same ratio found by experiment. Figure 21 shows the best Giesekus model fit at all three strain rates. From the figures one notes that near quantitative fits of the data are possible at each k. x y/'r E
Giesekus Applied to Predict Interrupted Flow Previous experiments and analysis show that a single parameter Giesekus model suitably describes the behavior of LDF/PEKK in elongation at a fixed strain rate [65]. But, the relaxation response of the single parameter model was insufficient to describe the complex shitt of the fluid to larger characteristic times. That is, the model would always relax the stress faster than the actual fluid. For the model, the relaxation constant is less than ~,1 during the flow but can only return to ~'1 as its maximum value during the relaxation period. To predict the response to an interrupted flow experiment one must either model the relaxation more precisely--so that the model reaches the same residual stress as the fluid in the same time--or shorten the model's period of rest. Allowing the Giesekus model to relax for 165 s would relax away most of the stress at either strain rate. Instead, the relaxation period can be reduced to that required to get the same residual stress prior to restarting the flow. Figure 22 and Figure 23 show the predicted stress response to the resumption of flow when the hold period drops to 30 and 35 s for the 10.3 and 104 models respectively. Besides the error in the rheometer, in which steady strain rate flow was not
883
attained on the second extension, the model shows a good prediction of the degree of stress overshoot upon resumption of flow for both strain rates. 10 o
.
0 tO,
10 4
Ii
Source lu
~
-
10"=
Model
9
10-5
E!
lo-4
0
10-3
10 "=
0
100
200
300
400
Time(s)
Figure 21. Best Giesekus model fit with single parameter sets at all three strain rates.
250 200 13..
I,M
0
l~
000,~
150 100
Source Model LDFMS
50 0 i
0.00
...
i
0.05
J
0.10
o
0.15
Strain
Figure 22. Giesekus model applied to predict response to interrupted flow, 1.49 10-3 s-1. The single parameter fit covered the initial flow. The model relaxed to the same stress level as the experiment and the flow rate resumed using the same Giesekus parameters fit to the first extension.
884
200
150
C)O00 D O 0 0 0 v
lOO Source
u.I
50
Model O
LDFMS ,,
0 0.00 (~
=
9
0.04
i
0.08
I
0.12
,
0.16
Strain
Figure 23. Giesekus model applied to predict response to interrupted flow, 1.66 10-4 s"~. The single parameter fit covered the initial flow. The model relaxed to the same stress level as the experiment and the flow rate resumed using the same Giesekus parameters fit to the first extension.
6. OUTLOOK This section first summarizes the test and model results for both the nylon/polyethylene (N/PE) and long-discontinuous-fiber/polyetherketoneketone (LDF/PEKK) systems. Then the implications of applying these results to process models are discussed. The prospect of timber experiments with new fiber patterns is presented. Finally, a method of investigating microrheology and LCP/fiber interaction is presented. 6.1 Conclusions
The evidence discussed above leads to the following conclusions about the model micromechanics, the suitability of the constitutive equation and the LDFMS properties.
6.1.1 Micromechanics Scaling The N/PE model system shows that high volume fraction highly aligned fibers in a specific overlap pattern generate shear-dominant flow when extended. This effect makes the system's extensional viscosity greater than the neat polymer's by
885
changing the dominant strain rate from stretching to shearing. The induced shear strain rate, which rises directly as a factor of the fiber aspect ratio, can stimulate conformation changes (shear thinning, nematic/isotropic transition) in the polymer. For a fixed volume fraction of fibers, extensional viscosity increases as the square of the fiber aspect ratio. Finally, the extensional data follow from the shear properties of the matrix fluid by micromechanics. 6.1.2 Constitutive Model
A shear-thinning constitutive model applicable to sheared polymers was fit to the transient stress data from LDF/PEKK experiments. At the highest extension rates, which had the greatest chance of making the PEKK matrix predominantly isotropic, the nonlinear constitutive equation successfully predicted the size of the stress overshoot after stress relaxation that followed an initial extension at two different strain rates. The combination of micromechanics and constitutive relation can provide a method for predicting the forming loads-- especially for better understood polymers.
6.2 Implications for Process Application The results of these experiments bear upon the planning of finite element prediction of a forming process and upon the design of fiber filled systems for thermoforming. The models needed to predict forming of these systems are complex and computationally extensive. They must also address fiber rotation and deconsolidation. 6.2.1 Complex Models
The nonlinear behavior of systems with large L / D fillers in extension raises a computational challenge for f'mite element modeling of a forming process. Prior researchers have opted for a "solid equivalent" constitutive model technique that allows codes existing to predict stresses under applied deformation [66, 67, 68]. This strategy's shortfall is the over simplified stress to strain relationship, which is suited to Newtonian fluids under small deformations. These methods employ a basic relation of
[a] =[c][E]
(21)
where the CO9 compose the stiffness matrix with reduced terms by virtue of the symmetry of the material. For the solid equivalent method the right side of the equation becomes
[a(t)] = In]I?]
(22)
The stiffness matrix is now a viscosity matrix. Terms like Tlll and 1"122 represent the extensional and shear viscosities respectively. This expression
886
misses the polymer's nonlinear response and the effects of large deformations in contrast to the small deformations of solids. The Giesekus equation in a similar matrix form is f ([x],[~:],['i'],t)= g([~'],[~?],t) (23) which is more involved. Numerical techniques must include the transient effects since stress overshoots can be significant and we need the total stress history to deal with changes in deformation rates in simulated tool contact. 6.2.2 Fiber Rotation A short series of tests checked the effect of fibers oriented at an angle to the extension direction for LDF/PEKK. Fiber angles of five and 10 degrees and +45 laminates showed that rapid fiber rotation must be quantified if final fiber position in multiple ply laminates is to be predicted. The "hyperanisotropy" [69] of LDF/PEKK makes the material effectively unextendable in the fiber direction until the fibers rotate parallel to the applied deformation [70]. Then the loads rise rapidly as the shear cell effect occurs. Therefore, the off-axis fiber behavior is identical to continuous fiber systems. Numerical software must handle the transition from an interlaminar shear mode to shear cell approach as the fibers orient. 6. 2.3 Deconsolidation Two deconsolidation mechanisms occur in sheet forming. When melted in open air without any applied deformation, N/PE and LDF/PEKK deconsolidate. Residual stresses in the sheet separate the plys and lott the material. Tensile experiments show that forming enhances deconsolidation without an applied pressure. In the shear cell model, the relative motion of the fibers generates substantial normal forces. This normal force would push fibers apart. Raising the strain rate increases the effect. At the low strain rates used here the specimens remained consolidated within either the metal fixture or vacuum bag. Consolidation maintained the continumn so that the analysis could be completed. In some industrial forming processes no consolidation force keeps the sheet together. The fmal forming step reconsolidates the panels. Although this lowers the effective viscosity of the sheet with the addition of voids [71], the void content changes the fiber motions in ways that are very complex to model. 6. 2.4 Parameters that Affect the Process This is a summary of some parameters in applying long fiber reinforcement and goals in the control of the parameters.
887
Fiber Length The material should contain the shortest possible average fiber length for the application. This keeps the viscosity increase to the smallest level needed to make a good part. Fiber Length Distribution A random distribution of fiber lengths should improve the forming of the material. The strain rate achieved in the LDF/PEKK was much closer to the target rate than that in the N/PE model. Distributed fiber length raises the effective average fiber length [69] but some longer fibers can bridge gaps between the shorter fibers. Strain Rate Limitations Two effects restrict the applied strain rate. The first is deconsolidation. Normal forces must not overpower the consolidation force needed to maintain a continuum. The second is inertial effect in the polymer. Two experiments with LDF/PEKK at a target strain rate of 0.01 s~ and a target stress of 400 MPa provided a glimpse of this limitation. At 0.01 s"1 the material reached a peak stress of 380 MPa followed by a precipitous drop in the tensile load. When the moving grip stopped the specimen did not. It became larger than the grip to grip distance and went into compression. Similarly the 400 MPa test obtained a great strain rate that caused the computer to halt the test at the safe extension limit of the test frame. Again the specimen continued to flow although the test machine had stopped.
6. 2. 5 Microrheology Recent research into the microrheology of polymers shows that thin film and bulk properties may be very different [72, 73, 74, 75, 76, 77]. Since the flow condition of interest to this research is annulus flow, we suggest a microrheometer that simulates this condition. A capillary rheometer with a fiber inserted through the hole of the capillary. With the proper selection of the fiber diameter the device would span the range of bulk (no fiber inserted) to microrheology. Two methods of measuring the viscosity of the polymer are possible with this device. The first method would be to rtm the instrument as a capillary rheometer with the analysis of the data accotmting for annular flow. The fiber insert may be either stationary or moving. The difference in cross sectional area between the reservoir and the annulus assures that the polymer flows at peak velocities much greater than that of the fiber surface.
888 A second method for collecting the data is to use the filled capillary/polymer system in a "pull-out" test. The filled capillary/polymer system would be placed in a test machine and heated to the appropriate temperature. Then the fiber would be withdrawn from the capillary at various speeds as the data acquisition system records the load on the fiber. This device could also determine the effect of LCP structure on microrheology as discussed in the next section. 6.2. 6 LCP/Fiber Interaction
The effect of solid surface characteristics on LCP structure formation and stability is largely unknown although some experiments show dramatic effects [63]. The microrheometer proposed above would allow systematic study of these interactions. Fibers would be selected to change the surface from smooth and nonadhering (an untreated glass fiber) to smooth and adhering (a coated glass fiber), to rough (an etched glass or untreated carbon fiber), and finally to rough and adhering (a coated carbon fiber). Either method could measure any viscosity change that the fiber surface induces. REFERENCES
1. I.Y. Chang and J.F. Pratte, Journal of Thermoplastic Composite Materials, 4 (1991), 227. 2. R.M. Jones, Mechanics of Composite Materials, (1975), 1. 3. J.L. Throne, Thermoforming, (1987), 1. 4. R.K. Okine, D.H. Edison and N.K. Little, 32nd International SAMPE Symposium, 32 (1987), 1413. 5. P.J. Mallon, C.M. O'Bradaigh and R.B. Pipes, Composites, 20 (1989), 48. 6. F.N. Cogswell, International Polymer Processing, 1 (1987), 157. 7. A.S. Tam and T.G. Gutowski, Journal of Composite Materials, 23 (1989), 587. 8. D.W. Coffin, Flange Wrinkling and the Deep-Drawing of Thermoplastic Composite Sheets, (1993), 1. 9. D. Hull, An Introduction to Composite Materials, (1981), 1. 10. I.Y. Chang and B.S. Hsiao, 36th International SAMPE Symposium, 36 (1991), 1587. 11. I.Y. Chang, 37th International SAMPE Conference, 37 (1992), 1276. 12. F. Rodriguez, Principles of Polymer Systems, (1982), 1. 13. R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Volume 1 Fluid Mechanics, (1987), 1. 14. S. Matsuoka, Relaxation Phenomena in Polymers, (1992), 1.
889 15. P.J. Carreau, D. De Kee and M. Daroux, Canadian Journal of Chemical Engineering, 57 (1979), 135. 16. J.C. Maxwell, Phil. Trans. Roy. Sot., A157 (1867), 49. 17. J.L. White and A.B. Metzner, J. Appl. Polym. Sci., 7 (1963), 1867. 18. H.J. Mthastedt, Rheol., 24 (1980), 847. 19. F.P. La-Mantia, A. Valenza and D. Aciemo, Polym Eng Sci, 28 (1988), 90. 20. H.M. Laun and H. Schuch, J Rheol, 33 (1989), 119. 21. J. Meissner, Polym Eng Sci, 27 (1987), 537. 22. J. Meissner, Chem. Engr. Commun., 33 (1985), 159. 23. V.M. Lobe and J.L. White, Polym. Eng. Sci., 19 (1979), 617. 24. S. Barbosa, M.A. Bibbo and A. Miguel, Applied Polymer Symposium, 49 (1990), 127. 25. M.A. Bibbo and R.C. Armstrong, Proceedings of Manufacturing International '88, 105 (1988), 123. 26. A. Jamil, M. S. Hameed and A. Stephan, Polymer-Plastics Technology and Engineering, 33 (1994), 659. 27. O. Lepez, L. Choplin and P. Tanguy, Polym Eng Sci, 30 (1990), 821. 28. A.T. Mutel and M.R. Kamal, Two-Phase Polymer Systems, (1991), 305. 29. L.A. Utracki, Polym. Compos., 7 (1986), 274. 30. A. Vaxman et al., Polym. Compos., 10 (1989), 78. 31. S N. Maiti and P K. Mahapatro, Polym. Compos., 9 (1988), 291. 32. R. Hingmann and B.L. Marczinke, J. Rheol., 38 (1994), 573. 33. T.M. Malik et al., Polymer Composites, 9 (1988), 412. 34. H.M. Laun, Colloid & Polymer Science, 262 (1984), 257. 35. A.T. Mutel and M.R. Kamal, Polym Compos, 7 (1986), 283. 36. S.E. Barbosa et al., Composite Structures, 27 (1994), 83. 37. M.L. Becrafl, The Rheology of Concentrated Fiber Suspensions, (1989), 1. 38. S. Toll and P.O. Andersson, Polymer Composites, 14 (1993), 116. 39. N. Dontula et al., J. Reinforced Plastics and Composites, 13 (1994), 98. 40. M.R. Kamal, A.T. Mutel and L.A. Utracki, Polymer Composites, 5 (1984), 289. 41. S.M. Davis and K.P. McAlea, Polymer Composites, 11 (1990), 368. 42. G.K. Batchelor, Journal of Fluid Mechanics, 46 (1971), 813. 43. J. Mewis and A.B. Metzner, Journal of Fluid Mechanics, 62 (1974), 593. 44. J.S. Bums, The Influence of Aligned, Long-Fiber Array (ALFA) Reinforcements on Composite Manufacturing and Structural Performance, (1995), 1. 45. F.N. Cogswell, Polymer Melt Rheology, (1981), 1. 46. R.M. Patel and D.C.Bogue, J. Rheol., 33 (1989), 607.
890 47. C.J.S. Petrie, Elongational Flows: Aspects of the Behavior of Model Elasticoviscous Fluids, (1979), 1. 48. M.R. Kamal and A.T. Mutel, Polymer Composites, 5 (1984), 289. 49. R.K. Gupta, Flow and Rheology in Polymer Composites Manufacturing, 10 (1994), 89. 50. Secor, R.B. et al., J. Rheol., 33 (1989), 1329. 51. R.M. Christensen, Theory of Viscoelasticity, (1982), 1. 52. J. Meissner, Kunststoffe, 61 (1971), 576. 53. K.F. Wissbrun, Journal of Rheology, 25 (1981), 619. 54. A. Ciferri, Liquid Crystallinity in Polymers: Principles and Fundamentals, (1991), 1. 55. S.M. Guskey and H.H. Winter, J. Rheol, 35 (1991), 1191. 56. A.B. Metzner, Rheol. Acta, 10 (1971), 434. 57. J.D. Goddard, Journal of Fluid Mechanics, 78 (1976), 177. 58. A.J. Beaussart, J.W.S. Hearle and R.B. Pipes, Composites Science and Technology, 49 (1993), 335. 59. R.B.Pipes et al., Proceedings of the American Society for Composites, 159 (1992), 123. 60. R.B. Pipes et al., Journal of Composite Materials, 28 (1994), 343. 61. H. Giesekus, Journal of Non-Newtonian Fluid Mechanics, 11 (1982), 69. 62. K.A. Ericsson, S. Toll and J.E. M~tnson, Rheol. Acta preprint, (1996). 63. S. Bhama and S.I. Stupp, Polym Eng Sci, 30 (1990), 228. 64. J. Fisher and A.G. Frederickson, Mol. Cryst. Liq. Cryst., 8 (1969), 267. 65. T.S. Creasy and S.G. Advani, Developments and Applications of NonNewtonian Flows, 66 (1995), 123. 66. T.G. Rogers, Composites, 20 (1989), 21. 67. C.M. O'Bradaigh and R.B. Pipes, Composites Manufacturing, 2 (1991), 161. 68. D.W. Cot~n and R.B. Pipes, Composites Manufacturing, 2 (1991), 141. 69. R.B. Pipes et al., Journal of Composite Materials, 25 (1991), 1379. 70. N.J. Pagano and J.C. Halpin, J. Composite Materials, 2 (1968), 18. 71. S.F. Shuler et al., Polymer Composites, 15 (1994), 427. 72. B.A. Costello and P.F. Luckham, Materials Research Society, 289 (1993), 7. 73. I. Hersht and Y. Rabin, Journal of Non-Crystalline Solids, 172 (1994), 857. 74. E. Pelletier, J.P. Montfort and F. Lapique, Journal of Rheology, 38 (1994), 1151. 75. K.D. Danov et al., Chemical Engineering Science, 50 (1995), 263.
891
76. J.A. Tichy, Tribology Transactions, 38 (1995), 577. 77. M. Urbakh, J. Klafier and L. Daikhin, Materials Research Society, 366 (1995), 129. NOMENCLATURE
English ar H(~.) L/D
Aspect ratio of a particle, the major length divided by the minor length. The height of the relaxation spectnun as relaxation parameter increases from zero to infinity. Aspect ratio of a particle, the major length divided by the minor length.
Greek O~
13(x) rx(r) ~yx s
rio
Parameter controlling the non-linear response of the Giesekus model. Varies from zero to one. Axial density distribution with units of g/cm. Axial density distribution as a function of position along the specimen. Shear strain rate in sec ~. Shear strain rate in cylindrical coordinates for the shear cell model. Shear strain rate in cartesian coordinates. Extension strain. Extension strain rate in secf1. Zero shear rate viscosity; viscosity of a polymer melt in the linear viscoelastic strain rate range.
I"IE
Extension viscosity of a fluid in tension.
TlEapp
Apparent extension viscosity measured in a region with a strain
I"1S
rate gradient. Transient extension viscosity measured from the startup of flow from the rest state. Shear viscosity in steady shear flow. Transient shear viscosity measured during startup of flow from the rest state.
892
rls
Z'max ~p
Shear viscosity decay measured during the time following steady shear flow. Carreau equation parameter that controls the onset of shear thinning. Relaxation parameter in fluid constitutive equations. Relaxation parameter value at which 99.9 % of the area udner the relaxation distribution function H(E)is attained. Relaxation parameter value at which H(~) attains its peak value. Shear stress in units of Pa.
"~yx + 1;yx
Transient shear stress measured from the start up of flow until
1;yx
steady flow is obtained. Shear stress decay following flow. Deformation held constant during the decay. The peak value of the transient shear stress. A 'yield' value for
'[E + 17E
the conformation change in the polymer. Extension stress in tension. Transient extension stress measured from the start of flow from rest. Peak value of tensile stress. Stress relaxation measured atter extension.
893
THERMOMECHANICAL MODELING OF POLYMER PROCESSING J.F. Agassant, T. Coupez, Y. Demay, B. Vergnes, M. Vincent Centre de Mise en Forme des Mat~riaux, Ecole des Mines de Paris, URA CNRS 1374 BP 207, 06904, Sophia Antipolis , France
1.
INTRODUCTION
Polymer processing involves complex flow geometries, in the plasticating units (single or twin screw extrusion, injection) as well as in the shaping tools (die, mold). Thermoplastic polymer processes started only around sixty years ago and these processes have been firstly developed by trial and error. Polymer processing modeling is a recent story and appears as a useful tool, not only to limit the number of trials, but also to master all the thermomechanical parameters which will induce, for example, crystallization, macromolecule orientation and, as a consequence, end-use properties of the produced part. Modelling polymer processes appears as a challenge for several reasons : 9 molten polymers are highly viscous materials, which leads to important viscous dissipation ; 9 polymer processes may rarely be considered as isothermal; 9 molten polymers are non-Newtonian viscoelastic fluids, which means for example: - shear-thinning behavior for the viscosity ; - transient effects ; - normal stress differences in pure shearing flow. 9 the viscosity of molten polymers is temperature dependent, which implies that mechanical and heat transfer equations have to be solved simultaneously. 9 polymers present a low thermal diffusivity, which may lead to important temperature gradients, even when the polymer is flowing in very thin gaps; 9 most of the polymers are semi-crystalline materials and very specific structure developments (spherulites, for example) may be observed depending on the cooling rate and stress field encountered during the process. In this chapter, we will only consider the flow of an homogeneous molten polymer. Neither plastication mechanism nor crystallization kinetics will be taken into account. In a continuum mechanics approach, the different equations
894 to solve are the mass, momentum and thermal balances, linked by a constitutive equation and appropriate boundary conditions. These governing equations are now described in details.
1.1
Mass balance Molten polymers can generally be considered as incompressible materials, which leads to 9 V.u : 0
(1)
where u is the velocity field. When very high pressures are encountered (as for example in injection molding), the variation of the material density p in time and space has to be taken into account :
@ 0t + V.(pu) = 0
(2)
1.2
Stress balance The stress tensor o" is symmetrical and only force balances have to be considered: V.(r+ F - p?'= 0
(3)
where F represents the gravity forces and p?'the inertia forces. The Reynolds number Re compares inertia and viscous terms 9
Re= p Uh 7/
(4)
where U is the characteristic velocity of the flow, h the flow gap and 7/the viscosity. Generally, Re is negligible in polymer processing or, at a maximum, of the order of magnitude of several units (this is the case around gates and runners in injection molding or at high speed fiber spinning). To compare gravity and viscous forces, an equivalent Stokes number is defined:
St-
pgLh rl U
(5t
where g is the gravity and L the vertical dimension of the flow. In horizontal processes (extrusion, injection molding), gravity forces are negligible. This is no more the case when large vertical stretching distances are considered (fiber
895 spinning, film blowing). When both gravity and inertia forces are neglected, the momentum equation reduces to 9 V.o-= 0 1.3
(6)
Constitutive
equations
Molten polymers are generally viscoelastic, but several constitutive equations may be used, depending on the polymer, the flow geometry and the level of approximation one wants to use.
1.3.1. The Newtonian behavior a=-pl+27/
e
(7)
where p is the pressure, I the identity tensor and e the rate of strain tensor, defined as : = ~ (Vu + Vu t)
(8)
The Newtonian behavior is a crude approximation, but it may provide reasonable results when the rate of strain remains quite uniform within the flow geometry. It allows analytical calculations in simple flow geometries, which is very useful to test the validity of numerical methods.
1.3.2 The generalized Newtonian behavior : The viscosity 7/is a spatial function which depends on the temperatureT and on the second invariant of the rate of strain tensor" a = - p l + 2 rl(T, ~) [~
(9)
where the second invariant is expressed as 9
42
'""'
92
ld
(10)
Several functions may be proposed for the viscosity : 9 the so-called power-law :
11= K(T) ~ n-1
(11)
where n is the power-law index and K the consistency, which is only a function of temperature, following for example an Arrhenius law 9
896
E
1
1
K ( T ) = Ko exp ~ (~ - ~00)
(12)
E is the activation energy, which may vary significantly from one polymer to another, R is the ideal gas constant and Ko is the value of the consistency at the reference temperature To. The advantage of the power-law is to provide analytical solutions in a wide range of flow geometries, but its main drawbacks are an infinite viscosity at zero shear rate and the absence of Newtonian plateau at low shear rate. 9 the Carreau law [ 1] :
77 = r/0(T) [1 + (A
2]
may be considered as a relaxation time. n is, as previously, the power-law index and 770 is the temperature dependent viscosity of the Newtonian plateau. This expression is also used under the following form, called Carreau-Yasuda law [2], in which the parameter a describes the transition between the powerlaw region and the Newtonian plateau: (n-1)/a = 00(73 [1 + (~ ~) a]
(14)
These laws provide a precise description of the shear viscosity as a function of the shear rate. However, even the solution of a simple Poiseuille flow requires a numerical approach. 1.3.3 The viscoelastic behavior
Viscoelastic constitutive equations are numerous and, at that time, it is still a difficult task to select one which accounts for a large number of viscoelastic phenomena : existence of normal stress differences in pure shearing flows, transient phenomena in strain or in stress steps, strain hardening in elongational situations, extrudate swell... Two families of constitutive equations are encountered 9 the differential and the integral models. 9 The simplest differential model is the Maxwell model : (15.1)
cr = - p ' l + s
as
s+2~=2r/
e
(15.2)
where p ' is the isotropic part of the stress tensor, s the extra-stress tensor and 5 s / & the upper-convected derivative. It is to notice that this Maxwell model is
897 a generalization of the crude dashpot-spring linear model, but it may be also derived from the elastic dumbbell model (Rouse [3], Zimm [4]). The Oldroyd-B model [5] is derived from the previous one by adding a Newtonian contribution r/s" cy--p'l
+ 2 rls e +
s
(16)
It allows to account for the two normal stress differences in simple shear flow (a first normal stress difference, as for the Maxwell model, and a second one), but also for strain transition after a stress step. Jeffreys models [6] consist in introducing an additional function of the extra-stress tensor in equation (15.2) :
& f(s) s + A-~=
2 r/ e
(17)
For example, for the well known Phan Thien-Tanner model [7], we have : e2~ f ( s ) = (1 + - ~ tr s ) I
(18)
where e is a material function, and tr s is the trace of the extra-stress tensor s. Other functions may be introduced, for example in the Giesekus model [8] : f(s) = I +
a~ 77
s
(19)
These models may be generalized by using a spectrum of relaxation times (A,i ,r/i) instead of a single one. Multimode Maxwell or Phan Thien-Tanner models are expressed as [9] : S - ~. S i ,
~si f(si) Si + / ] , i - - ~ - 2 17i ~
(20)
1
They account simultaneously for shear viscosity, elongational viscosity and first normal stress difference in simple shear. 9 The simplest integral model is the Lodge model [ 10] 9 s-
f t m (t, t ,) C i 1 (t, t ,) dt' -
(21)
C>O
where m(t, t') is a memory function and C t I the Finger tensor. This model is equivalent to the Maxwell model by choosing :
898
m (t, t') =
exp( t
t)
(22)
Wagner [11 ] improved the Lodge model by introducing a damping function h of the two invariants I l and 12 of the Finger tensor 9 s = f t m (t, t ,) h (1 i, 12 ) C i I (t, t ,) dt'
(23)
-00
Different forms of the damping function may be found in the literature. For example, Papanastasiou et al. [ 12] proposed for h the following equation 9 h (I l, I2) =
1 1 + a [fl 11 + ( 1 - [ 3 ) I 2 -
3] b/2
(24)
where a,/3 and b are material parameters.
1.4
Energy balance equation
The most general form of the energy balance equation is 9 p
de
= - v.q +
(25)
e is the mass density of internal energy ; for an incompressible material, e is proportional to the temperaturede dT dt = Cp dt
(26)
where Cp is the heat capacity. More complex equations are proposed for compressible materials with phase transition [13]. q is the heat flux, which is proportional to the temperature gradient following the Fourier law 9 (27)
q =- k V T
where k is the heat conductivity. I~ = o'" ~" is the viscous dissipation, expressed as 9 .
O'" 8" = 7/ y 2
for N e w t o n i a n materials,
(28.1)
899
a " e" = K ~ n + 1
for power-law fluids.
(28.2)
It is to notice that for a viscoelastic constitutive equation, all the energy is not dissipated and equation (25) has to be modified [14].
1.5 Boundary conditions In order to solve mass, stress and thermal balance equations, we need to define relevant boundary conditions. 1.5.1 Mechanics and kinematics A zero velocity at the wall (sticking contact) is generally assumed in most polymer flows. This is reasonable for thermoplastics at low or intermediate flow rates. At high flow rates, flow instabilities are encountered, which may correspond to stick-slip transition [see Chapter 7, section 7.3]. When processing PVC, rubber compounds, or highly filled suspensions, slip at the wall may occur even at low flow rates. The determination of accurate slip velocity measurement methods and of relevant slip constitutive equations remain an open problem [see Chapter 4, section 4.2]. Depending on the problem to solve, a velocity profile, a pressure or a pressure gradient have to be prescribed in the inlet section. In the particular case of viscoelastic constitutive equation, the extra-stress components at the entry surface of the flow domain have also to be defined. At the die outlet, a zero pressure is generally assumed. In stationary free surface flows, a velocity vector parallel to the free surface, as well as a zero stress component perpendicular to the free surface, are considered. 1.5.2 Temperature and heat transfer A temperature profile has to be known at the entry surface of the flow domain. The more difficult problem is to determine accurate boundary conditions along the processing tools (extruder, die, mold) or along free surfaces. It is customary to fix the temperature at the wall (T = Tw) ; this is the case when precise and powerful thermal regulation systems are used. This could also be used as a first approximation to check a preliminary value of the temperature field. Generally, one imposes a heat flux q or a heat transfer coefficient hr" q - hr (Tw - Te)
(29)
where T e is the controlled or measured temperature of the tool. In a flat mold, for example :
900
km hr--- l
(30)
where km is the heat conductivity of the metal and l is the average distance between the cooling channels and the mold wall. When the geometry of the mold (or the die) is complex, l may be difficult to determine and it is preferable to develop a global computation in the polymer domain and in the tool (with, for example, an iterative loop between polymer flow and tool). For free surface problems (fiber spinning - cast film - film blowing ...), the determination of a realistic heat transfer coefficient remains a challenge because coupled and complex phenomena may be encountered : free convection, forced convection, radiation ... [13]. Very often, the heat transfer coefficient (or the heat transfer function, because it can vary for example between the die and the winding system in fiber spinning) will be considered as an adjustable parameter.
1.6
Scope of the chapter
Mass balance, stress balance, energy balance and constitutive equations with appropriate boundary conditions have now to be solved in the non trivial flow geometries encountered in processing equipments. Finite elements methods are generally used and the accuracy of the results will significantly depend on the precision of the mesh. In the next section, several complex 3D flow geometries will be considered. However, for some processing geometries, approximation methods may lead to simplified solutions with a reasonable accuracy. This will be presented in section 3 for confined flow situations and in section 4 for free surface flows.
0
DIRECT SOLUTION FOR POLYMER THE F I N I T E E L E M E N T M E T H O D
FLOWS USING
2.1 Viscous flow problem As shown in the first part of this chapter, the molten polymer can be considered as incompressible and the inertia and mass forces are neglected. A purely viscous isothermal polymer flow is then described by combining equations (1), (6) and (9), giving the following mixed velocity-pressure problem: Find (u, p) solution of" V.[2r/( ~ ) ~: (u)- Vp ]= 0
(31.1)
V. u = 0
(31.2)
901
+ boundary conditions The bounded domain f2 of boundary o~ is the region occupied by the fluid or more precisely the domain of calculation as shown in Figure 1. The above problem will be well posed if adequate boundary conditions are prescribed. For instance, in the extrusion die of Figure 1 (in fact, Figure 1 presents the internal volume of the die), the pressure is imposed at the inlet and outlet of the flow and a zero velocity is prescribed along the walls. The Newtonian law leads to a classical Stokes problem, considered here as a model problem to point out one of the difficulty in solving such a viscous flow problem : the treatment of the incompressibility condition. The Stokes problem has been early studied as a subset of the Navier-Stokes equations [15] and gave rise to the mixed theory, which general framework is now well established [ 16].
Figure 1"
2.2
Boundary conditions associated with a profile die extrusion flow problem
Variational formulation The finite element approximations are based on a weaker form of equations (31), also known as the virtual work principle. In order to simplify this presentation, we suppose that the velocity is prescribed everywhere along the boundary. Let Vand P (the Sobolev space V= (HI(~)) 3 and P = L2(~) for the Newtonian case) be respectively the velocity and pressure spaces. The problem to solve can be rewritten as 9
902
Find (u,p) ~ V x P , u = u 0 on bfl, u0 being given, such as :
~ 2r I
i: (u ) " e (u *) dr2 -
~pV.u*
df2 = O
rj p *V.u dr2 = 0
(32.1)
(32.2)
f2
V(u*, p*) ~ V 0 x P, where" V 0 = {u* ~ V, u * l ~ = 0} The velocity and the pressure must be computed simultaneously. Most of the numerical schemes are based on a particular choice of the pressure interpolation with different solution techniques. The penalty method associated with the reduced integration technique [17] corresponds to a discontinuous interpolation of the pressure, theoretically well understood in the robust augmented Lagrangian technique [18,19] used in 3D [20]. The mixed finite element formulation presented here is based on a simple and almost natural continuous interpolation of the pressure, entering in the mini-element family [21 ].
2.3 Finite element discretization The fundamental idea of the finite element method is to approximate spaces V and P by discrete spaces V h and Ph. The domain of calculation f2 is decomposed in a finite family of simple geometrical elements f2e. The numerical method will depend on the choice of the geometrical element. In 3D, hexahedral elements are often used, but restricted to relatively simple geometries. The general way to solve the mesh generation problems and to use automatic meshing methods is to use tetrahedral elements [22]. Moreover, it is possible to mesh complex three-dimensional geometries, such as extrusion dies for example, and to control the local mesh size, particularly at the very thin exit [23]. Figure 2 shows the surface of a mesh of the extrusion die example, composed of more than 50 000 tetrahedra. The accuracy of the numerical solution will depend on the mesh size parameter h which is defined as the maximum of the elements diameters. It will depend also on the polynomial interpolation order, defined as follows. Assuming a continuous velocity approximation, a pressure approximation which can be continuous or discontinuous, and tetrahedral elements (triangular elements in 2D), the discrete spaces are defined by 9 V h = {u h ~-. (C 0 (~))n, u h I ~ e E (pk (f~e))n}
(33)
903
Figure 2"
External view of the mesh of a die extrusion geometry and detail of the mesh in the exit section
ph = {ph E (C i (~), ph
I ~e ~ (pl (~e)}
(34)
where pk(f~e) and pl(f~e) are the set (or sub-set) of polynomials of degree k and l on element f~e. n =2, 3 is the space dimension. If i = 0, the functions are continuous, if i = -1, they are discontinuous. V h and p h cannot be chosen independently. The inf-sup Brezzi-Babuska condition must be checked (see [16] for a complete discussion of this compatibility condition). A well established technique to ensure this condition is to enrich the velocity interpolation with a bubble part [19, 24] (Figure 3). The discrete velocity space is augmented (in the hierarchical form) by the discrete bubble space denoted by :
B h = {bh ' bhlf~e ~ (H~(~e))3 }
(35)
The velocity takes the following form"
v h = u h + b h ~ (V h + B h)
(36)
904 The bubble has the advantage to be local and it can be condensed at element level without changing the bandwidth of the global system. This technique gave rise to the simplest mixed finite element : the mini-element [21 ]. In theory, the shape of the bubble function is free and it has only to vanish at the element boundary. We use a pyramidal bubble function (Figure 3) preserving the exact integration property of a first order tetrahedral element. The bubble can be chosen optimally [25] and it can be related to the stabilization technique [26]. The mini-element is used for the calculation of the three-dimensional examples presented in this paragraph. It has four unknowns per node. In two-dimensional calculations (viscoelastic examples presented later), we used the Crouzeix-Raviart element (also known as the P2+/P1). When coupled with an augmented Lagrangian technique [19], the pressure unknown can be locally eliminated, which leads to two unknowns per node. The main difference between mini and Crouzeix-Raviart elements is that the pressure is continuous for the first one and discontinuous for the second one. Despite the increase of degrees of freedom with the mini-element, the continuous interpolation of the pressure is, in 3D, largely less expensive in term of computational resources.
.;. . . . . . .Q"
Figure 3 9 3D mini-element with a pyramidal bubble function Under this stability condition, the numerical solution converges with respect to the mesh refinement to the real solution. The approximation in velocity and pressure are then dependent through the a priori following estimate : Ilu - uhll 1,f~ + lip - ph IIL2(f~) < C h i
(37)
For the mini-element, i - min(k, l + 1) = 1, k and 1 being respectively the interpolation order of the velocity and the pressure. It is then a first order element, since i = 2 for the second order Crouzeix-Raviart element.
905 Equation (37) can be rewritten by introducing the strain rate numerical error 9
[~~z (71 I ~(u ) - "C(uh) I 2 + I p - p h
I 2)dE').,]1/2 <__C'h
i
(38)
and finally this can also be related to the error in stress by using the constitutive relation"
l-j I
I 2dC ]1/2 _< C"hi
(39)
For the example of extrusion die, Figure 4 shows how the flow rate evolves with the mesh refinement for a fixed pressure drop. 700 ~. 600 O~
E
500
~
400 300 0
0
u. 200 100
9
0
!
5000
'
T
10000
'
I
15000
20000
Number of mesh nodes
Figure 4 9 Evolution of the calculated flow rate with the mesh refinement. When the mesh is composed of more than 10 000 nodes (for instance the mesh of Figure 2), it reaches an asymptotic value, independent of the mesh. In this example we used a linear interpolation of the velocity (plus a pyramidal bubble term for stability purpose) and a linear interpolation of the pressure, giving a first order element. Under these conditions, we obtain the results presented in Figure 5, which are almost free of numerical approximation. The pressure remains quite constant in the upstream die region and a high pressure gradient appears at the die exit. The goal of such a calculation is to compute the velocity profile at the die exit in order to predict the balance of the polymer flow and then to validate or modify the proposed geometry.
906
Figure 5 : Pressure field in the whole die (a) and velocity field at the die exit (b) 2.4 l t e r a t i v e
solver
An attractive feature of the mini-element interpolation is the possibility to solve the large linear discrete systems by an iterative method [27]. This is crucial for 3D applications for which the use of direct solvers is too much consuming, both in term of memory and in term of CPU. By using a static condensation technique [19], the internal velocity bubble terms vanish from the system. This elimination provides a new diagonal block matrix C. The remaining linear parts of the velocity and the pressure are then solution of the following system : B H _BC ) ( pV) = ( F )
(40)
where H and C are symmetric definite positive. The unknowns are thus the three components of the velocity and the pressure for each vertex. The size of
907
the system is then 4 N, N being the number of mesh nodes. The new block -C can be seen as the optimal stabilization matrix in the context of a stabilized P1/P1 method [28]. The form of the blocks remains almost the same in the non-linear case, with H = H(V) and F = F(V), V being fixed at each Newton iteration. The global matrix is symmetric indefinite and an iterative approach must be based on a residual method. The preconditioned conjugate residual method (PCR) for the Stokes problem has been introduced in [29]. The PCR method can be seen as a particular case of the generalized minimal residual method (GMRES) [30], which can be used with indefinite possibly nonsymmetric matrices. However, from the practical point of view, it can be seen as a preconditioned conjugate gradient extended to saddle point problems. The method used in [27] for 3D forming applications is based on the hybrid Orthomin/Orthodir form of the algorithm, as described in [31] for instance. 2.5 Thermal resolution 2.5.1 Convective diffusive equation Using equations (25), (26), (27), the heat transfer equation has the following form" pep
dT
= k AT+
(41)
The time derivative of the temperature can be decomposed in a heat variation at fixed point and a convective term 9 d T OT dt = oat + V T.u
(42)
From the mathematical point of view, equation (41) can be of different types, depending on the relative weight of each of its terms. When the flow velocity is zero or almost zero, the equation is dominated by the diffusive term. The problem is then elliptic for stationary problems or almost elliptic when the time variation is relatively small (cooling for instance). A standard Galerkin method will be appropriate for numerical solution. When time variation of the temperature is very fast, the equation tends to a parabolic type and its solution may exhibit shock. That is the case in injection molding at the contact between the cold wall and the hot polymer : the flow near the mold wall is slow due to the sticking contact since the temperature gradients become important. A relation must be respected between the time step and the mesh size, leading finally to adopt an adaptive anisotropic meshing technique [32]. Figure 6 shows an example of refinement of the mesh close to the wall in the case of the filling of a cavity. Convection is dominating in the main stream and the heat transport solution must be accurate. The standard finite element
908 methods often fail when the temperature gradients are important. Among the possible alternative method we can mention the streamline upwind technique [33], which consists to add diffusive terms in the standard Galerkin finite element method. These terms must be smaller of one order of magnitude to preserve the consistency of the equations. The standard discontinuous Galerkin method is well suited for hyperbolic equation, but restricted to purely convective term. The method of characteristics [34] is well suited both for convective and dissipative dominated problem.
Figure 6 : Anisotropic mesh of the filled area in injection molding
2.5.2 Finite element solution of the thermal equation The weak form of the equation is obtained by a standard Galerkin technique. The equation is multiplied by a test function, T*, chosen in an appropriate functional space T 0. T* is vanishing at the boundary and the equation is integrated by part. The problem to solve is thus 9 Find T ~ T(= HI(E~)) such as :
f~ pCp -~ dT T* dE~ =- ~ kVr.Vr* dE~ + f W T* dE~, VT* ~ T O
(43)
We assume that T is prescribed on the boundary. In the standard finite element method applied to these equations, we introduce one discrete space T h for the temperature field and the test functions.
pCp --~T*hd~ =-~ kVT h VT*hdE~ +
WhT,hdE2, VT,h ~ T0h
(44)
909
dTh
The evaluation of--d~ will differ from one scheme to another and will lead to symmetric or non-symmetric systems. For instance"
dTh
= lim dt at~o
Th(x(O)- Th(x(t-at)) At
(45)
will lead to the method of characteristics [34], which requires to compute at each integration point the upstream trajectory on a length determined by the time interval At. The advantage of such a method is to introduce only a mass matrix for the discretization of the convective term and then to maintain a global symmetric system. The drawback is to be non-local. It has been intensively exploited to perform the thermomechanical coupling in injection molding simulation [32]. On the other hand, a standard Galerkin method leads to a non-symmetric system:
f~pCp--~T*hdE~ 0Th + ~ VTh.uhT*hd~ + ~EkVTh.VT*hdE~- f
I~hT*h dE~ (46)
This method is quite natural but can show some instabilities and it is restricted to small temperature gradients. The other difficulty is to solve such a nonsymmetric system by an iterative solver (direct 3D solutions by means of Choleski factorization are too expensive to be really usable). The minimal residual method (which can be adapted to non-symmetric system) works when the gradient along the flow remains small enough. Afterwards, one can observe loss of convergence, even with the GMRES method [30]. The direct Galerkin method using a minimal residual method as iterative solution was used for the profile die extrusion application [35]. Very recently, we have proposed an explicit thermal solution algorithm. The heat flux is introduced in the general heat transfer equation as a new unknown variable, leading to a two fields problem" cgT
pCp Ot = - pCp V T.u - V . q + W
(47.1)
q =- k V T
(47.2)
These equations are solved explicitly by using a local Taylor-Galerkin scheme. The interpolation is performed by a piecewise constant per element for both the temperature field and the heat flux field. The advantage of this
910
discontinuous interpolation is to control the thermal shock between the cold wall and the hot polymer in injection molding application and to avoid any numerical instability.
2.6
Application to 3D mold filling
Let us introduce f2 as the cavity of the mold geometry. The sub-domains C2f = C2Xt) and ~2e = I2e(t ) are respectively the fluid domain and the empty space. We define the characteristic function of fluid domain ~lf2fby 9
~l~f(x't)=f
1
i f x ~ ~2f
0 if x ~ f2 e
(48)
As a consequence of the mass conservation applied to the fluid material, and whatever the density of the empty part, the fluid movement is described by the following transport equation 9 d (x, t) = 0,
Figure 7-
V x ~ ~2
3D finite element simulation of the filling of a grid cavity
(49)
911 The whole cavity is meshed. At each time step, the velocity and pressure fields are calculated everywhere using the 3D Stokes solver previously described. The empty part of the mold is represented by a zero velocity field and a zero pressure field. The interpolated fluid position (one constant per element) evolution is performed at element level by using a transport equation finite element solver, based on an explicit Taylor discontinuous Galerkin method. The basic idea of this scheme is that the successive time derivatives in the Taylor development of the characteristic function 1 can be calculated recursively by" at ll n f
= - V l o f .u
032 03/2 t[nf
O "-- V ~-~ a n f .U
69 3 0t 3 1 h i
02 =_ 7 - & _ f ~ a / . u ...
(50)
We have only to enforce the successive derivatives to be in the same approximate space, for instance the piecewise constant, the convective term being treated by a discontinuous interpolation technique.
Figure 8 9 3D double jets of viscous polymer
912
Figure 7 shows different steps of the filling of a thick 3D grid mold [36]. It has been run on a single middle size workstation. There are 9 827 nodes, which means 39 308 degrees of freedom and 50 739 elements. We can clearly see the evolution of the free surface and the formation of the weld lines. This method is not restricted to the flow front movement in confined mold filling problems. It can be used to compute accurately transient free surfaces as shown in Figure 8 where two jets of polymer collide each other, showing that large moving free boundaries can be well rendered by this approach.
2.7 Viscoelastic computation The direct calculation of viscoelastic flow remains one of the difficult topics in numerical analysis [see Chapter 3]. At that time, 3D computations are just emerging. For that reason, we restrict this presentation to the 2D numerical treatment of differential models, as the Oldroyd-B or the Phan Tien - Tanner model. The viscoelastic flow leads to a three-fields problem where the velocity, the pressure and the stress tensor are solutions of coupled equations. Let us consider the following problem derived from the Oldroyd-B model : Find u, p, s solutions of" V.[2r/s e (u) + s] - V p ' = 0
(51.1)
V.u = 0
(51.2)
s + ~ S~--20
e (u)
(51.3)
with boundary conditions 9u - u 0 on o ~ ; s = s o on c~'2i (inlet boundary). = 0 leads to a classical Stokes problem. In that case the three-fields formulation must be equivalent to the two-fields velocity/pressure formulation. For this reason, the unknown extra-stress tensor s belongs to the functional space S defined by: S = {s E ( L 2 ( ~ ) ) nxn}
(52)
The space 5 must contain the rate of strain space, denoted e (V). The finite element approximate space 5h must be chosen in order to be compatible with V h and p h previously introduced. An easy way to check these compatibility condition is to ensure that 5 h contains the discrete space e (Vh), for example by choosing S h = k( V h). The discontinuous Galerkin method is a consequence of the above construction. Indeed, if the velocity discrete space is based on C Oelements, the
913 gradient of the discrete velocity is then discontinuous. Consequently, the simplest discrete space containing the strain rate tensor is made of piecewise polynomial interpolation. In [13], the Crouzeix-Raviart element for the velocity/pressure discretization is associated with a second order interpolation of the extra-stress : W h = {u h E (C~
2, u h l ~ e E (P2+(f2e))2}
p h _ {ph E (C-1(~2)), ph
I~e ~ p l ( ~ e ) }
S h - {S h e (C-l(~)) 2x2, s h I f~e E (p2(~e))2x2}
(53.2) (53.3)
In this case, the standard notation p2+ means that the velocity is enriched with a bubble term. The bubble being made of a polynomial of degree three, the gradient of the velocity is of degree two. The variational problem can be written as : Find (u h, ph, s h ) E V h x p h x s h , u h = Uo on o3~, u 0 being given, such as: Sf~ 2rls ~(uh)" ~(uh*)df2 - S~P V'uh*d~ + fE~ sh ~(uh*)dE2 = 0
(54.1)
Sf~ ph V.u h d~ - 0
(54.2) 2uh.ne[sh]. .Ch do3~ _ f f2 2rls ~uh)"chd~2 e
V (u h,ph) 6 V h x Ph
Figure 9"
(54.3)
Comparison between calculated (top) and experimental (bottom) birefringence patterns in an abrupt contraction
914
where [s h] is the jump of s between two successive elements, n e is the normal to the element side and o~- is the reentrant side. Figure 9 compares, in an abrupt contraction geometry, the computed principal stress difference pattern to the experimental one, deduced from flow birefringence experiments [9]. The agreement is fair and allows to foresee the development of inverse methods for the characterization of molten polymer rheological behavior in non-trivial flow configuration, starting from flow birefringence patterns.
2.8
Conclusions Despite interesting and promising results, the use of direct 3D numerical simulation remains today generally limited to purely viscous fluids and requires high numerical resources. To solve industrial problems encountered in polymer processes, the use of approximation methods remains very often necessary. These methods, which will be presented in the next sections, require less computer resources and corresponding software packages are now well implanted in polymer processing industry. They may be roughly divided into two categories 9 - the confined flow approximation, which is adapted to flows into narrow channels or between nearly parallel plates, dominated by shear; - the thin film (or fiber) approximation, devoted to free surfaces, for which elongational flows are dominant.
3.
CONFINED FLOW APPROXIMATION
3.1 Kinematics approximation The confined flow approximation is also well known as the lubrication approximation [37-39], first applied to 2D flow simplification. It may be applied to flow situations were one dimension is small compared to the others. Let us consider for sake of clarity the schema of Figure 10, presenting the flow of a polymer between two nearly parallel plates, defined by the local distance h(x, y). The following assumptions are made : 9 the flow is steady state, laminar and isothermal; ~?h ~?h 9 the distance h(x, y) varies slowly in x and y directions 9 ~xx << 1, ~yy << 1 ; 9 the curvature of the surface is negligible compared to the local thickness. If these assumptions are verified, one can consider the flow as locally between two parallel plates, separated by the distance h(x, y). Thus, the full 3D velocity field can be simplified into a local 2D field :
u = (u(x, y, z), v(x, y, z), w(x, y, z))
u' = (u(z), v(z), O)
915
\ \
uT~ "%u
t.k.j
iX Xl ~
,..
. . . . . .
\\
\\ -
Figure 10 : Typical example of confined flow geometry It means that elongational components are neglected compared to the gapwise shear components and that hydrostatic pressure is constant throughout the thickness. The local continuity equation (equation (1)) is replaced by its integrated form : 3w
+ ~y + ~ z dz = o
or"
-~x
u dz
+ -~y
v dz + [w]
(55)
=0
(56)
Assuming no slip conditions on upper and lower walls, equation (56) may be written as"
~q~ Ox + ~Oy = 0
(57~
where qx and qy are the local flow rates per unit width in x and y directions, defined by : qx =
u dz,
qy =
v dz
(58, 59)
916 9 the use of a viscoelastic behavior in confined flow geometries does not alter greatly the velocity, nor the stress fields. Thus, we will just consider viscous behaviors in the following.
Remark
3.2
Restriction to isothermal Newtonian or power-law behavior 9 the Hele-Shaw equation If we consider the confined flow of a Newtonian fluid of viscosity 7/in the geometry of Figure 10, assuming the simplified velocity field, the Stokes equations reduce to"
_~ O~2U 0X "- ~ 0Z2 '
0p a21: Oy = 77 OZ2
(60,
61)
Integrating twice and using appropriate boundary conditions (no slip on upper and lower walls) lead to the expressions of the local flow rates per unit width:
qx = -
1 ae
1 aph3
(62, 63)
12 0 cgx h3, qY = - 12 0 0 y
Introducing equations (62) and (63) in equation (57) leads to the well known Hele-Shaw equation :
a ~xx [o3x h3] +
[0y
or
] = 0
7.(h 3 V p ) = 0
(64)
This equation can be generalized for non-Newtonian fluids under the form : V.(/~ V p ) = 0
(65)
where/3 may have different expressions depending on the selected viscous law. For example, for a power-law fluid, we obtain [40-41] : + 1)/n
/ 3 = 2 n + 1(
)
(1-n)/n Vp
(66)
3.3 Restriction to 2D flow : the Reynolds equation If we consider the same type of flow, but in a 2D situation (calender gap or journal bearing, for example), the integration of Stokes equation leads to 9
dp
12r/q
dx
= - h(x)3
(67)
917 where q is the flow rate per unit width in x-direction. For a power-law fluid, a similar expression is obtained"
dx =
2n
q
[h~)
(68)
If the change in gap h(x) is known, the above expression allows to calculate the pressure evolution along the flow direction and the pressure drop/flow rate relationship. For example, for a dihedron with initial gap ho, final gap hi, length L, and a linear variation of h between ho and hi, the pressure drop Ap is expressed as : ho+ hi Ap - 6 71 q L ho 2 h l 2
(69)
3.4 Heat transfer approximation The general temperature field T(x, y, z) corresponding to the flow presented in Figure l0 is defined by the heat transfer equation (see equations (41)). In a 2D flow where the temperature gradient across the flow thickness h is limited (low shear rate, wall temperature close to polymer temperature), it is possible to define a bulk temperature by :
T(x)
=
1
(70)
u(y) T(x, y ) d y
h(x) Y
where ~ is the mean velocity in x-direction. Using this expression, the energy equation can be written under the form [13] 9
dT u h(x) ~
_
1 + pep
~2k Nu = pCp
Tw-
h(x)
T
(71)
where Tw is the wall temperature and Nu the Nusselt number, which characterizes the heat transfer between the polymer and the wall 9 Nu
-
hy
h(x) k -
(72)
918 where hr is the heat transfer coefficient previously introduced in section 1.5. Equation (71) can be easily generalized to the 3D confined flow geometry presented in Figure 10 under the form : m
OT 1 Op c?T 10p 2k Tw - T -~h (-O-x-x + ~pcp Ox) + ~h (-~-y-y + ~pcp Oy) = pcp ~ Nu h
(73)
It is clear that these simplifications will give pertinent results only if the heat transfer is correctly estimated and the transverse temperature gradient is limited. They are generally useful for computing the flows in extrusion dies or in calender gaps, but inappropriate to the strong conditions encountered in injection.
3.5 First example : coat-hanger flat die geometry Coat-hanger flat dies are very often used for the production of films and sheets until a few meters width. In this geometry, the molten polymer is distributed through a coat-hanger, which design is supposed to ensure a perfect homogeneity of the flow rate at the die exit [42]. The flow in such a geometry may be described using the confined flow approximation. For a power-law fluid depending on temperature through an Arrhenius law, the set of equations to solve is : 2n+l Oh Op h OK Op h Ap+~ n + ~?y ~ - -~n ~x Ox+c)y
+ h
-
Ox2
1-n
1
+ 2 OP 0P 002x~y OxOy +
(1)l/n(2)(2n+l)/n[lo~x~
2n
( O~y~ l (1-n)/'2n tT~ox
u(x,y) = - h 2n+l v_ (x,y) - - h12.2n+ 1
1 l, l 12.+l I( y
= 0
+
Oy
(74)
(75)
(76)
m
u h (-ff~-x + ~pcp
)+
v h (
+ ~pcp
2k Nu ) = pcp
Tw-
h
T
(77)
919
K-
E 1 1 Ko exp ~ ( ~ - ~-~o)
(78)
This set of equations may be solved iteratively, using for example a finite volume method [43-45]. Figure 11 presents an example of computed result, corresponding to the extrusion of a polymer with a power-law index equal to 0.35. Flow lines and isobars are drawn on Figure 1 l a. The pressure drop is mainly consumed in the straining bar, located after the coat-hanger. The polymer flows preferably along the coat-hanger and is progressively distributed throughout the lips. The homogeneity of the exit flow rate may be evaluated by plotting the ratio of local flow rate qx to mean flow rate q as a function of the position (Figure lib). In the present case, the distribution is correct (+ 10 %), but it is important to point out the strong dependence on the power-law index. The temperature plays an important role in the flat die extrusion. Isovalues of bulk temperature for an initial temperature of 200 ~ and for a wall temperature of 200 ~ are shown in Figure 12a. Figure 12b indicates that the exit flow rate distribution is very sensitive to the wall temperature. It proves that it is possible to correct a flow rate heterogeneity just by changing the regulation temperature. 100 bars 90.1bars
.
70.3 bars
5o.s~
30.7"i
]
.zo.81"---T 10.9J--- 4 Po=l.o [ . . . . bar
t 1.2 C~
~" 1.1------, '
d
t.
i a) n-0.2 b) n:= 0.35 c) n=0.5
1,0
o ~- 0.9 ,~_. C
~' 0.8 A
B
Figure 11 9 Flow lines and isobars (a) and flow rate exit distribution (b) for the flow in the coat-hanger fiat die
920
200.0 *C
t i 1
"M~L
202.9
203.3
I
-
1.4 ,-~
1.3
l
I AF
IB
;, sS
1.2
O
O
c~
.
1.1
i
/
Wail femperafure: a) T : 180 ~ bl T = 200 ~ c) T : 220 ~
1.0 Q,P
"-
bx,......t , J/
09
~u
0.8
a%,," ..,,.,,"
/
s
O.7 A
Figure 12" Isovalues of bulk temperature (a) and exit flow rate distribution (b) for the flow in the coat-hanger flat die 3.7 Second example : mold filling of a box geometry For a large number of molded components, the confined flow approximation can be used. In injection molding, gapwise temperature gradients are very important, so that, contrary to extrusion, an average temperature cannot be used. Moreover, the rate of strain varies from zero to very large values (several thousand of s-l), and precise viscous model such as the Carreau equation must be used. Therefore, equations (60) and (61) are integrated twice without specifying a form for the viscosity 77, and /~ appearing in equation (65) is defined as : 13-
'z
(z - z*)
~
d z
(79)
z* is the value of z for which the velocity is maximum. It is equal to h/2 with a symmetric thermal regulation. In more general situations, it is defined by 9
921
~Z-Z* o
dz
(8o)
- o
The boundary conditions are 9 9 on the lateral walls, the velocity along n, normal to the wall, is set to zero, which means that 3p/On - O. The velocity component tangent to this lateral wall is not equal to zero. This is one of the limitation of Hele-Shaw models ; 9 the flow rate is given at the entrance, that is to say the pressure gradient is imposed; 9 at the flow front, the pressure is equal to zero, which means that the air is supposed to escape the cavity freely. When the cavity is not in a plane, the same kind of resolution is carried out in a tangent plane. As defined in paragraph 1.4, the energy equation is"
pcp
U
OT
+ V --~ + W
c)__~zI
c) 2T
= k t~Z2
9
+ W
(81)
Due to high transverse temperature gradient, the term w cgT/cgz cannot be neglected a priori, even if w is small in the confined flow approximation, w is not explicitly taken into account, but it can be evaluated by considering the local continuity equation. Boundary conditions are indicated in paragraph 1.5.2. The Hele-Shaw equation can be solved at each time step by finite difference [46-48] or finite element methods [49-52]. The position of the flow front can be evaluated with a moving mesh method [51], but a more simple technique consists in using control volumes built with the finite element description of the mid-surface of the cavity (Figure 13a).
Figure 13 9 Triangular mesh of the mid-surface and control volumes (a), and extension in the thickness within an element (b)
922
A fill factor a is associated to each control volume. If a = 0, the volume is empty. If a = 1, the control volume is full. If 0 < a < 1, the control volume is currently being filled, and the flow front is somewhere inside. The evolution of ct for each control volume ~ is obtained by'solving equation (82), which is deduced from the continuity equation for an incompressible material 9 ~9
& Jt~k h a d s
=-kf~a h u.n ot dl
(82)
The iterative resolution is usually the following 9 9 The pressure is linearly discretized on the finite elements. Equation (82) is written on each control volume, u being expressed as a function of the pressure gradient. A linear system with the nodal values of the pressure is thus obtained. 9 The temperature field is obtained by writing heat flux balance in control subvolumes (Figure 13b), and a temperature for each layer and each control volume is obtained. 9 Equation (82) permits to update the value of ct, and to obtain the new value of the flow front position.
Figure 14 9 Isovalues of the pressure at different time steps during the filling of a box shaped cavity (a-d) and fill factor at time 0.09 s
923 Figure 14 shows the isovalues of the pressure and the fill factor ~ for the filling of a box cavity. This kind of computation gives accurate results for the pressure, the temperature, and the flow front. The location of weld lines is usually correct. The equations can be generalized to compressible materials, for the packing phase, or for a better resolution of the filling of large cavities at high pressure.
3.7 Comparison between direct and approximated solutions 3. 7.1 Flow of a Newtonian fluid in a dihedron In order to check the validity of the lubrication approximations, we can consider for example the flow of a Newtonian fluid in a dihedron. An exact solution can be obtained using cylindrical coordinates [13]. The relationship between flow rate and pressure drop is 9 W Ap (si220 - 0 cos20) Ro 2 R12 Qexact =
17 (Ro2 _ R12)
(83)
where 0 is the dihedron angle and Ro and R1 the radius defining the entry and exit, respectively. The approximated solution using the lubrication approximation is given by equation (68). So, the ratio of flow rate between exact and approximate solutions is only a function of the angle 0" tg30 Qapprox 4 Qexact = 3 sin2 0 - 0 cos20 2
(84)
1,3
t,,.}
m X
1,2
o x 0
~
C,one
1,1
0
9
1,0
~
0
"
I
5
"
I
"
10
dron
I
15
'
'
20
Angle (degree)
Figure 15 9 Comparison between approximate and exact solution for the flow in a cone and in a dihedron
924
Figure 15 presents the evolution of the flow rate ratio as a function of the angle. It can be seen that the error made in using the lubrication approximation remains less than 5 % for an angle inferior to 10 ~ For an axysymmetrical conical geometry, this limit is roughly equivalent but further deviation is enhanced. These particular examples are representative of the global validity of the confined flow approximation. Whatever the real geometry and the polymer behavior, the upper limit of the relative angle between the walls must not exceed 10 ~ to ensure a correct description of the flow [53-54].
3.7.2 Particle tracking in injection molding We have seen in section 3.6 that the Hele-Shaw equation is now widely used to predict pressure, temperature fields, and flow front progression during cavity filling. Nevertheless, a direct resolution (see section 2) is the only technique which can be used when the geometry is complex. But even with simple cavities, more precise kinematics description can be necessary to predict accurately the flow, for example at the junction between the runner, the gate and the cavity, or at the flow front where the fountain flow takes place. Moreover, crystallinity ratio for semi-crystalline thermoplastics, crosslinking for thermosetting polymers or rubber compounds, reinforcing particles displacement and orientation need a precise knowledge of the thermal and kinematics history of each material point starting at the mold entrance. Particle tracking at the flow front has been specially studied. The pivot point method has been introduced by Manas-Zloczower et al. [55] and Dupret and Vanderschuren [56]. Heat diffusion and viscous heating in the frontal zone are neglected. The neutral line is defined as the gapwise position for which the actual velocity is equal to the mean velocity. A particle "below" the neutral line (that is between the mid-plane and the neutral line) will reach the flow front and jump instantaneously "above" the neutral line (between the neutral line and the mold wall). The position is determined by writing a local mass balance. A more precise description can be obtained in a reference frame linked to the flow front in isothermal conditions. Then, a steady direct two dimensional resolution is possible, either analytically [57] or with a finite element method [58]. Finally, the direct resolution in the laboratory reference frame described in section 2 can be used in isothermal and non-isothermal conditions. Figure 16 shows the evolution of a line of tracers obtained with the three methods [32, 59]. Direct 2D models permit to obtain the V-shape which has been experimentally observed by Schmidt [60]. After a certain time, when the tracer is stretched, the three approaches give similar results.
925
~ 0,8
F l o w front position 98
0,6 ~., ,-, 0,4
2,~r~ D #,, F i n i t e e. l e m e n t 2.D ~ A n a _ l y t i c a l Pivot
----------~ .........
0,2 0
,
,
,
,
i
0
,
,
,
,
i
l
,
~
- " ~ . .
"kN
,
,
i
2
,
,
,
3
,.i ,
,
,.
,
,
i
4 Flow length
,
,
,
~
5
'
'
'
'
I
6
~' 0,8
~
'
'
I
7
8
F l o w front position 9 2
0a
0,6 ~
0,4 ....... .........
l~ 0,2 0
,
,
~D # A n a l y t i c a l Pivot
~
i
0
0,4
N 0,2 ~
,
,
i
2
1
0,6
,
,
"'.~ I", ,
,
i
4
,_i
6
,
i
,
,
,
i
8 Flow length
,
,
~I
10
i
,.
,
,
i
12
,
,..
14
....
1
" 1
"-.,,.
~
2D
#
Finite
. . . . . .........
2D # A n a l y t i c a l Pivot
""-
F l o w front position 9 16 ....
0
'
'
'
'
I
0
] 0, 8
element
'
'
'
'
5
~
I
'
'
'
'
~ .......... _ ..-.......,
-
I
I
10 Flow length
I
I
15
\
,
0,6 0,4
2D Finite e l e m e n t 2D A n a l y t i c a l Pivot
. . . . . .........
0,2 0
'
0
' ....
'
I
8
'
'
'
F l o w front position 9 40 I
16
'
'
'
I
'
24 Flow length
'
'
I
'
32
'
'
I
40
'
'
'
48
Figure 16 9 Comparison of three methods for tracking a line of particles at the flow front
0
THIN FILM (OR FIBER) APPROXIMATION OF THE CAST FILM PROCESS
: EXAMPLE
Confined flows considered in the previous section are mainly shear flows. As the fluid is sticking at the wall, the flow is locally a plane Poiseuille flow
926
and this is the key point of the confined flow approximation. On the contrary, for stretching flows, the normal stress is zero at the free surface and hence the velocity is quite constant throughout the thickness. In the following, we describe one and two dimensional kinematics and mechanical approximations for the cast film process, considered as an example of free surface stretching flow. The same basic ideas can be developed for melt spinning and blown film processes. In the cast film process, the molten polymer is stretched between the flat die and the chill roll (Figure 17). The stretching process may be considered as isothermal, and cooling is achieved very rapidly on the chill roll. A shrinkage (neck-in) is observed in the transverse direction. The thickness is more important on the lateral edge of the film (dog bone defect), so that it is necessary to cut the edges after solidification. Challenge of modeling is to predict width and thickness of the solidified film. We first describe a twodimensional model for the isothermal Newtonian cast-film. Coupled equations for thickness and mean velocity in the thickness are derived from these approximations. Then, a one-dimensional model is presented by introducing additional approximations. This 1D model allows easy extension to viscoelastic constitutive equations, non-isothermal conditions or multilayer film processing. Finally, a stability analysis will be performed using different techniques both for the 1D and 2D models. Wo A v
exit
W(x)
Chill roll
Figure 17 9A schematic view of the cast film process 4.1 G e n e r a l m e c h a n i c a l a p p r o x i m a t i o n s for thin film 9 a 2D m o d e l
Figure 17 gives a schematic view of the cast film process. The molten polymer is outgoing from the flat die at x = 0 and is solidified on the chill roll
927
at x = L. The mean surface of the film is, for simplicity, assumed to be the plane z = 0 .
4.1.1 Kinematics Let us note h(x,y) and W(x) the film thickness and width. The polymer flows in the volume defined by :
1
1
{- -~ h(x,y) <_ z <- -~ h(x,y), 0 <_x <_L, -W(x) ~_y ~_ W(x) }
(85)
We assume, as for the confined flow approximations, that the thickness
h(x,y) is slowly varying in x and y-directions 9 Oh Oh << 1,--~-<< Ox oy
1
(86)
Let us note U ( x , y ) a n d V(x,y) the velocity components on the mean surface:
(87) (88)
u(x, y,z) = U(x, y) + o(z) v(x, y,z) = V(x, y) + o(z) where o(z) is a term of the order of magnitude of z. As w(x,y,O) = 0 on the mean surface, we have :
~w w(x,y,z) = z ~ (x,y,O) + o(z)
(89)
Using the incompressibility condition, we have"
c)w c)u Ov OU 8V c?z (x,y,O) =- -ffs (x,y,O)- -~y (x,y,O)=- -~x (x,y)- -~y (x,y)
(90)
4.1.2 Mass conservation and rate of strain tensor The continuity equation, naturally verified by the three dimensional velocity field, is for stationary 2D free surface flows replaced by an integrated form (this is to compare to equation (56) of section 3) :
a ~_~ u a~ h/2
dz
+
a ~_~ v ~ ha
ah
ah
dz + [ w - -~x u - -~yV
hi2 ]-h/2
=0
(91)
928 Using free surface conditions on upper and lower boundaries and estimating integrals by a mean value formula, the previous equation becomes 9
Ox (hU) +
(hV) - 0
(92)
As the flow is mainly extensional, the rate of strain tensor is (neglecting shearing in the thickness) : 9
~
Exx Exy 0 1~ -
g.xy Eyy 0 0 9
(93)
0 ezz 8U
9
9
I
8U
OV
8V
9
with" 8xx = o3x ; gXy = 8yx = -~ ( - ~ +-~-x ) ; ~ y = ay ; eZZ = -
8U
8V
a x - ay
4.1.3 Stress balance for a Newtonian constitutive equation Assuming a Newtonian behavior, we have"
8U
8V
(94)
f i z z - - 20 (-~x + ffyy ) - P
Using the zero stress condition normal to the free surface, and identifying the normal to the free surface to z-direction, we have :
8U 8V p = - 2r/(fiXx + -~-y)
and hence" ff =
with"
(95)
Gxx r 0 axy ayy 0 0 0 0
I I 8U
8U
8V
8U
(96)
8V
axx = 271 -~x + 271 (ff-ff + -~v ) ; i
1
8V
Gyy- 2~ Uy + 2~ (gx + 7y~ ; 8U 8V ~ y = 77 ( -~y + ~ ).
929
The stress tensor can be considered as a 2D tensor. Integrating the equilibrium equation (gravity forces are neglected) through the thickness and using the free surface condition, we obtain" V.(hff) = 0
(97)
4.1.3 Free surface and boundary conditions Equations (92), (96), (97) have to be completed with appropriate boundary conditions on the edge of the film of normal n 9 U nx + V ny = 0
(98)
(Txx nx + ffxy ny = Gxy nx + O'yy ny = 0
(99)
c)W
o3W 2 -1/2
with" nx = - Ox (1 + (-~x ) )
0 W 2 - 1/2
; ny = (1 + (--~x ) )
We add classical boundary conditions at the die exit and on the chill roll (at distance x = L): U(O, y ) = UO; U(L, y) = UL ; W(O) = WO ; h(O) = h 0
The draw ratio is defined as Dr =
( 00)
UL
u0
4.1.4 Results o f the two dimensional model For a given value of the thickness distribution h ( x , y ) , the velocity components U and V and the width W are solutions of a free surface elliptic problem (equation (97)), associated with boundary conditions (98), (99) and (100). For a given velocity field, thickness h is solution of a transport equation (92). This coupled system of equations can be solved iteratively by using finite element solvers for velocity and thickness, and a fixed point method to reach convergence [61, 62]. The preceding analysis may be easily extended to a viscoelastic Maxwell equation. The equations and the numerical resolution are detailed in [62] (another viscoelastic 2D model has been recently proposed by Debbaut et al. [63]). Figure 18 presents the final thickness distribution of the film. Figure 19 shows the shape of the film between the flat die and the chill roll [62]. When introducing a viscoelastic equation, the neck-in phenomenon is less pronounced and the dog bone defect remains located at the periphery of the film.
930
0,15 Newtonian gl (1) e-
0,10
c"
c-
O
r
0,05
rr'
Viscoelastic
0,00 0,0
9
I
'
I
0,1
0,2
i.
'
I
0,3
'
I
0,4
"
!
0,5
0,6
y/W o Figure 18 9 Final film section for a Newtonian and for a Maxwell fluid 1
~0
'
'
0,8 ..~
0,6
i
0,4 0,2 0,0 0,7
0,8 0,9 Reduced half-width (W(x)/W O)
1,0
Figure 19 9 Half-width of the film for a Newtonian fluid and for a Maxwell fluid 4.2 A d d i t i o n a l
mechanical approximations 9 a ID model A one-dimensional model may be obtained if the thickness and velocity are assumed to be varying with x only. In this case the model predicts neck-in and mean thickness of the solidified film.
931
4.2.1 Mass conservation Let us assume that h= h(x) and U = U(x). Equation (92) becomes 9
0 ~x (hWU) = 0
(101)
4.2.2 Equilibrium equation The stretching force F is defined by 9
F = h(x) W(x) tYxx
(102)
As we are neglecting mass and inertia forces, the balance equation (97) becomes 9
OF Ox = 0
(~03)
and hence F is a constant. The differential system of equations obtained by adding these kinematics hypothesis can be solved for Newtonian or viscoelastic fluids by using a shooting method on the value of the force F. The dimensionless parameter defining the aspect ratio of the film (A - W/W0) appears in this set of equations. Figure 20 compares the shape of the film with 2D and 1D Newtonian models. 1,0 0,8
_J
0,6
!
""
0,4 1D
model
0,2 0,0 0,4
9
I
9
I
9
0,6 0,8 Reduced half-width (W(x)/W 0 )
1,0
Figure 20 9 Shape of the film surface for Newtonian 1D and 2D models
932
For that particular free surface flow, the kinematics assumption (1D or 2D) and the constitutive equation (Newtonian or Maxwell) have a marked influence on the shape of the film between the die and the chill roll and on the final dimension of the film (thickness and width).
4.3 The stability of the cast film process 4.3.1 The draw resonance instability Draw resonance instability appears at large value of the draw ratio for isothermal cast film and melt spinning processes. If the draw ratio is greater than a critical one Drc, a periodic flow is observed. This critical draw ratio is close to 20 for isothermal melt spinning [64]. Onset of draw resonance instability for cast film induces periodical variations of both thickness and width [65]. 4.3.2 Computation of the critical draw ratio The stability of the one-dimensional solution is studied by using a linear stability method, identical to those developed for fiber spinning [66, 67]. It means that the time dependent set of equations is linearized and eigenvalues are computed. For a Newtonian fluid, the critical draw ratio is a function of the aspect ratio A (Figure 21). Increasing this aspect ratio leads to stabilize the process. Direct time dependent simulation is used to study the stability of the 2D solution. Figure 21 points out that the shape of the stability curve is not too different between the 1D and the 2D models. The 1D model is able to capture the influence of the film aspect ratio on the stability. 1,3 < ~
STABLE
1,1
o x__
0,9
to
0,7 o0
t~
E
.m ii
0,5
/
0,3 0,1
9
15
UNSTABLE,
I
I
I
I
I
20
25
30
35
40
45
Draw ratio
Figure 21 9 Cast film stability curve 9 influence of the film aspect ratio.
933
5. C O N C L U S I O N As shown in this chapter, it is nowadays possible to compute the flow of molten polymers in most of the complex geometries encountered in polymer forming processes, in both stationary (extrusion) and unstationary (injection molding - blow molding) conditions. This requires first precise volume meshing methods, starting for example from a CAD surface meshing of the tools. This necessitates also robust numerical finite element methods 9 mixed velocity/pressure Galerkin method for the mechanical problem, TaylorGalerkin method for the thermal problem, discontinuous Galerkin method for convection problems (viscoelasticity, for example). Iterative solvers and parallel computing reduce storage requirement and computation time, which allows to perform numerical simulations in complex industrial geometries, with refined meshing. These direct numerical methods do not necessitate any geometrical or kinematics assumptions, but they remain limited, at the present time, to purely viscous constitutive equations (in 3D). In addition, computation time is important and die or mold optimization, which necessitates a lot of successive numerical simulations, remains a difficult task. In that sense, approximation methods, which require however sophisticated mechanical and thermal treatments, based on kinematics, heat transfer and geometry assumptions, are still useful. In confined flow geometries, approximations consist generally in neglecting elongational components compared to the shear components throughout the thickness. As a consequence, viscoelasticity is a second order phenomenon in these shear dominant flows. The so-called Hele-Shaw approximations result in a single mechanical differential equation with the pressure as unknown. For example, most of the commercial injection molding software packages are based on this kind of approximation. In free surface stretching flows, on the contrary, shear components throughout the thickness are neglected compared to elongational terms. The so-called thin film approximation leads to differential equations with only the mean velocity components throughout the thickness as unknowns. Stationary, as well as time dependent solutions, may be obtained. In these elongational dominant flows, viscoelasticity has a first order influence. One cannot conclude at that time that polymer processing modeling is now fully completed. Several problems remain open : thermomechanical coupling with severe temperature gradients, 3D viscoelastic computations, adaptative meshing based on error estimation ... But the methods we presented can now be considered as practical tools to calculate flow rate and temperature distributions in polymer processing devices, to optimize processing geometries and, soon, to predict end-use properties of the produced polymer parts. This will require to incorporate new equations (crystallisation kinetics depending on temperature, temperature gradient and stress field, macromolecule or fiber
934
orientation...) and so to investigate in details the corresponding new scientific fields.
NOMENCLATURE cp
heat capacity -/
C t C Dr e
E F,F g h hT I 11,12 k, km K,K 0 l L m n n
Nu P p' P q q, qx, qy R Ro, R 1 Re S
T T* Te, Tw U, V, W U U*
Finger tensor matrix draw ratio mass density of internal energy activation energy of the viscosity force gravity flow gap or thickness, damping function, mesh size heat transfer coefficient identity tensor invariants of the Finger tensor heat conductivity consistency distance length memory function power law index normal vector Nusselt number pressure isotropic part of the stress tensor pressure space heat flux flow rates per unit width ideal gas constant radius of a dihedron Reynolds number extra stress tensor temperature test function controlled temperature, wall temperature velocity components velocity vector test function
935 m
U,V
U,V V W
f~
mean velocity velocities velocity space width
IIa
viscous dissipation characteristic function of the fluid domain
o~
fill factor
".L.
second invariant of the rate of strain tensor acceleration e 77, 7/0
F/e r/s A, P
o
0 f~
~-'~e ~e
V~
V
tr d dt 6 & 0 Ot
rate of strain tensor shear viscosity elongational viscosity Newtonian viscosity relaxation time density stress tensor angle of dihedron domain element empty part of the cavity fluid domain boundary of the domain
divergence gradient trace material derivative upper convective derivative partial derivative
Hl(~'2) = {u E L2(~')), ]Vu I EL2(~)} L2(~) - {u, I [u 12d~ < oo}
II. IIl,a = ( I lu 12da + I lV. 12da)l/2
936
REFERENCES ~
2. 3. 4. 5. 6. ,
8. 9.
10. ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
P.J. Carreau, Trans. Soc. Rheol., 16 (1972) 99 K.Y. Yasuda, R.C. Armstrong, R.E. Cohen, Rheol. Acta, 20 (1981) 163 P.E. Rouse, J. Chem. Phys., 21, 7 (1953) 1272 B. Zimm, J. Chem. Phys., 24, 2 (1956) 269 J.G. Oldroyd, Proc. Roy. Soc. London, A345 (1958) 278 R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths Eds., Stronham (1982) N. Phan-Thien, R.I. Tanner, J. Non Newt. Fluid Mech., 2(1977) 353 H. Giesekus, J. Non Newt. Fluid Mech., 11 (1982) 69 C. B6raudo, A. Fortin, T. Coupez, Y. Demay, B. Vergnes, J.F. Agassant, J. Non Newt. Fluid Mech., to appear A.S. Lodge, Elastic Liquids, Academic Press, New York (1964) M.H. Wagner, Rheol. Acta, 15 (1976) 136 A.C. Papanastasiou, L.E. Scriven, C. Macosko, J. Rheol., 27 (1983) 387 J.F. Agassant, P. Avenas, J.P. Sergent, B. Vergnes, M. Vincent, La Mise en Forme des Mati~res Plastiques, Lavoisier, Paris (1996) G.W.M. Peters, F.P.L. Baaijens, J. Non Newt. Fluid Mech., 68 (1997) 205 V. Girault, P.A. Raviart, Finite Element Approximation of the NavierStokes Equations, Theory and Algorithm, Springer-Verlag, Berlin (1986) F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag ( 1991) T.R.J. Hughes, W.K. Liu, A. Brooks, J. Comp. Phys., 30 (1979) 1 M. Fortin, R. Glowinski, Augmented Lagrangian Methods, Dunod (1983) M. Fortin, A. Fortin, Int. J. Num. Meth. Fluids, 5 (1985) 911 M.P. Robichaud, P.A. Tanguy, Comm. In Appl. Num. Meth., 3 (1987) 223 D.N Arnold, F. Brezzi, M. Fortin, Calcolo, 21 (1984) 337 T. Coupez, in" Numerical Grid Generation in Computational Fluid Dynamics and Related Fields, N.P. Weatherill et al. eds., Pineridge Press (1994) J.F. Gobeau, T. Coupez, B. Vergnes, J.F. Agassant, in : Numerical Methods in Industrials Forming Processes, Paul R. Dawson, Shan-Fu Shen eds., A.A. Balkema (1995) M. Fortin, Int. J. Num. Meth. Fluids, 1 (1981) 347 R. Pierre, SIAM J. Numer. Anal., 32 (1995) 1210 L.P. Franca, S.L. Frey, T.J.R. Hughes, Comput. Meth. Appl. Mech. Eng., 99 (1992) 209 T. Coupez, S. Marie, Int. J. S. A., to appear
937
28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58.
R. Pierre, Comput. Meth. Appl. Mech. Eng., 68 (1988) 205 A. Wathen, D. Silvester, SIAM J. Numer. Anal., 30 (1993) 630 Y. Saad, M. Schultz, SIAM J. Scient. Stat. Comput., 7 (1986) 856 S.F. Ashby, T.A. Manteufel, P.E. Saylor, SIAM J. Numer. Anal., 27 (1990) 1542 B. Magnin, Doctoral Dissertation, Ecole des Mines de Paris (1994) A.N. Brooks, T.J.R. Hughes, Comp.Meth. in Appl. Mech. Eng., 32 (1982) 199 O. Pironneau, M6thodes des E16ments Finis pour les Fluides, Masson, (1993) J.-F. Gobeau, Doctoral Dissertation, Ecole des Mines de Paris (1996) E. Pichelin, Doctoral Dissertation, Ecole des Mines de Paris (1998) J.R.A. Pearson, Mechanics of Polymer Processing, Elsevier, Londres (1985) V.L. Streeter, Handbook of Fluid Dynamics, Mc Graw Hill, New York (1961) Z. Tadmor et C.G. Gogos, Principles of Polymer Processing, Wiley, New York (1979) B. Vergnes, Doctoral Dissertation, Ecole des Mines de Paris (1979) R.T. Fenner, Principles of Polymer Processing, Mc Millan, London (1979) W. Michaeli, Extrusion Dies, Carl Hanser Verlag, Munich (1979) B. Vergnes, P. Saillard, B. Plantamura, Kunststoffe, 11 (1980), 752 B. Vergnes, P. Saillard, J.F. Agassant, Polym. Eng. Sci., 24 (1984), 980 B. Arpin, P.G. Lafleur, B. Vergnes, Polym. Eng. Sci., 32 (1992), 206 E. Broyer, C. Gutfinger, Z. Tadmor, Trans. Soc. Rheol., 19 (1975) 423 C.A. Hieber, S.F. Shen, Israel J. Technology, 16 (1978) 248 S. Subbiah, D.L. Trafford, S.I. Gticeri, Int. J. Heat Mass Transfert, 32 (1989) 415 G. Boshouwers, J. Van der Werf, PhD Thesis, Technical University Eindhoven, The Netherlands (1988) H.H. Chiang, C.A. Hieber, K.K. Wang, Polym. Eng. Sci., 31 (1991) 116 A. Couniot, L. Dheur, F. Dupret, in" Numerical Methods in Industrial Forming Processes, Balkema, Rotterdam (1989) 235 J.L. Willien, Doctoral Dissertation, Ecole des Mines de Paris (1992) A.M. Benis, Chem. Eng. Sci., 22 (1967), 805 W.E. Langlois, Slow Viscous Flows, Mc Millan, London (1964) I. Manas-Zloczower, J.W. Blake, C.W. Macosko, Polym. Eng. Sci., 27 (1987), 1229 F. Dupret, L. Vanderschuren, AIChE J., 34 (1988), 1959 J.M. Castro, J.W. Macosko, AIChE J., 28 (1982), 250 H. Mavridis, A.N. Hrymak, J. Vlachopoulos, Polym. Eng. Sci., 26 (1986), 449
938
59. 60. 61. 62. 63. 64. 65. 66. 67.
S. Karam, Doctoral Dissertation, Ecole des Mines de Paris (1995) L. Schmidt, Polym. Eng. Sci., 14 (1974), 797 S. D'Halewyn, J.-F. Agassant, Y. Demay, Polym. Eng. Sci., 30 (1990) 335 D. S ilagy, Doctoral Dissertation, Ecole des Mines de Paris (1996) B. Debbaut, J.M. Marchal, M.J. Crochet, Z. Angew, Math. Phys., 46 (1995) 679 J.R.A. Pearson, M.A. Matovich, I & EC Fundamentals, 8 (1969) 605 Ph. Barq, J.-M. Haudin, J.-F. Agassant, Intern. Polym. Proc. 8 (1992) 334 D. Silgay, Y Demay, J.-F. Agassant, C.R. Acad. Sci. Paris, t.322, S6rie lib (1996) 283 D. Silagy, Y Demay, J.-F. Agassant, Polym. Eng. Sci. 36 (1996) 2614
939
MODELLING
AND SIMULATION
OF INJECTION
MOLDING
F. Dupret, A. Couniot 1, O. Mal, L. Vanderschuren g, O. Verhoyen CESAME, Unitd de Mdcanique Appliqude, Universitd catholique de Louvain, avenue G. Lemaftre 4-6, B-1348 Louvain-la-Neuve, Belgium Tel :32 (0) 10 472350, E-mail : [email protected] 1
current address : Siemens-Nixdorf lnformation Systems S.A., LoB "Major Projects", chaussde de Charleroi 116, B-1060 Brussels, Belgium 2 current address : Shell Research S.A., avenue Jean Monnet 1, B-1348 Louvain-la-Neuve, Belgium
1.
INTRODUCTION
The last decades have been marked by the spectacular development of synthetic polymers, which are now commonly used in all of the major market sectors 9consumer appliances, automotive, industrial, electric and medical. This sensational success arises not only from the cost effectiveness of plastic parts, but also from the wide range and amazing balance of chemical, mechanical and electrical properties of polymers. The growth of the polymer industry has been conducive to the development of various processing techniques, among which injection molding remains one of the most widely employed. Injection molding is used for the mass production of identical parts obtained by introducing a polymer melt into a cooled cavity. A major advantage of this process is its suitability for the production, in a single operation, of ready-to-use articles of considerable geometrical complexity. As a result, its field of application is very broad, ranging from the production of small parts such as tiny watch gears or electronic components to large objects such as boats or automotive parts. Most polymers may be injection molded, including amorphous and semi-crystalline thermoplastics, thermosets, elastomers and fiber-reinforced plastics. Injection molding is considered to be one of the most complex polymer processing techniques. A sketch of a typical molding machine is shown in Figure 1. Primarily, two parts may be distinguished 9an injection unit comprising a hopper,
940
Figure 1. Schematic layout of a reciprocating screw injection molding machine, and description of the main processing stages. a rotating screw and a heated barrel, and a clamping unit containing the mold typically made of two plates. The injection unit melts the polymer and injects it into the cavity, whereas the clamping unit ensures the closure and opening of the mold together with the final part ejection. In the case of thermoplastics, which this chapter essentially addresses, the material is supplied under the form of granules or powders introduced into the hopper. During plasticization, the screw rotates and acts as a mixer and a pump : the material is taken from the hopper, melted, homogenized and conveyed towards the barrel head while, at the same
941
time, the screw moves backwards. During injection, the screw acts as a piston, moves forward and pushes the molten material into the mold. This operation includes the filling stage, during which the polymer is injected into the mold, and the packing stage, during which a high pressure is exerted by the screw to force in additional melt to compensate for subsequent shrinkage due to cooling and solidification. These steps are followed by the cooling stage during which the material solidifies, and the ejection stage where the mold plates are separated, thereby allowing the ejection of the part. We shall not describe this process any further ~ the reader will find a thorough description in [1,2]. Let us however mention three related developments which will be addressed in this chapter. The first one is compression molding, where a polymer charge, initially placed inside the cavity, is forced into the desired shape by squeezing it via the motion of one half of the mold relative to the other [3,4]. Compression molding offers several advantages, in particular the use of simple molds, the application of a low stress level during processing and a reduced material waste. The second process is Reaction Injection Molding (RIM), where two or more low viscosity liquids, which become reactive when brought together, are mixed just before filling, the polymerization taking place partly during but essentially after injection [5,6]. This process is typically applied for the production of large objects due to the lower pressures required. The third class of related processes is formed by Structural Reaction Injection Molding (SRIM) and Resin Transfer Molding (RTM), which are both used for producing continuous fiber composite parts by injecting a thermosetting resin into a mold filled with a fiber preform [7,8]. In all these cases, the design of a new mold is a delicate and time consuming task, often requiring dedicated trials and experiments. The aim of numerical modelling is to prevent this costly and lengthy approach by providing an easier way to apprehend the physics involved and to assess the influence of the key process parameters. Quite clearly, the ultimate objective is, given a polymer, an application and a desired type of geometry, to allow the numerical determination of an optimum mold design together with the most appropriate operating conditions. However, the numerical modelling of the filling and packing stages of the injection molding process (which represents the subject of this chapter) is plagued with various difficulties. For the sake of simplicity, let us restrict ourselves here to non-reinforced thermoplastics. During filling (Figure 2), the mold plates are maintained at a temperature lower than that at which the injected material solidifies. The problem is thus transient and 3D, in view of the front motion and the important heat transfer between the melt and the cavity walls. Another difficulty arises from the temperature dependence of the polymer viscosity, which couples
942
Te Jt e
jjjjjjjj..~W~alJ~jjjjjjjj Gate I ~ ~ ~ .................................
9
........
Front
i .......................
~
Figure 2. Sketch of the injection molding problem. the kinematics and the temperature distribution, especially close to the walls where the lower temperatures increase the viscosity and slow down the fluid or lead to the formation of a frozen layer. An additional difficulty relates to the theology of polymer melts, which exhibit a strong non-Newtonian behavior as a result of the very high shear rates prevailing during filling. Finally, the high temperature gradients and pressures involved induce significant compressibility effects, in particular during the packing stage. A comprehensive numerical simulation of injection molding therefore appears to be an outstandingly difficult task. Nonetheless, the geometry of molded parts is typically characterized by a small thickness compared to the other characteristic lengths or radii of curvature. As a consequence, Generalized Newtonian (GN) stresses may be assumed and the goveming equations can be considerably simplified over most of the flow domain. The rheology of molded polymers is analyzed in Section 2, while Section 3 details the flow and heat transfer model resulting from the thin cavity assumption. The numerical algorithm accordingly developed at the University of Louvain (UCL) [9-26] is presented in Section 4, and illustrated in Section 5 by means of simulation examples and experimental validations. In these sections, only the injection and compression molding of thermoplastic materials are addressed. Extensions of the model to the cases of reactive resins, semicrystalline polymers and the filling of fiber mats are considered in Section 6. The molding of fiber-reinforced polymers is investigated in the companion chapter "Suspensions : Modelling the Flow of Fiber Suspensions in Narrow Gaps" (which will be referred to as "Chapter FS" in the sequel).
943
It is worth noting that the numerical simulation of injection, compression and related molding processes has received considerable attention for more than 25 years. Pioneer studies were devoted to modelling [2,27-34] and numerical [3446] issues, in order to scale the problem, to understand basic effects (such as the formation of a solidified layer on the walls, the phenomena governing molecular orientation in the final part, the role of viscoelasticity, etc.) and to develop first simulation tools for 2D isothermal in-plane, or non-isothermal cross-sectional calculations. The mathematical model was further elaborated [9,12,17,47-65], with a view to correctly represent the fountain flow, to take into account the presence of abrupt changes of thickness, bifurcations and other geometrical features, to include polymer compressibility and to accurately predict the flow stresses. More and more efficient numerical techniques were concurrently developed to solve the general 2lAD flow and heat transfer problem (which basically consists of 2D pressure and 3D temperature calculations) [52,56a,64,66-81] or to perform full 3D simulations [82]. At the same time, the model was extended to reactive [5,6,25,32,49,58,72,83-86] and semi-crystalline [19,20,87-93] polymers, and to compression molding [3,4,34,76,94,95], SRIM-RTM [7,8,21,25,76,81,96102] and other [103,104] molding processes. 2.
RHEOLOGY OF INJECTION MOLDED POLYMERS
In continuum mechanics, constitutive equations relate the Cauchy stress
aij,
heat flux qi, specific internal energy U and entropy S to the past history of the material point thermodynamic variables [105,106]. For amorphous thermoplastics, the latter are the temperature T, the deformation gradient from initial to actual state and the temperature gradient. Additional variables are required in the case of reactive, semi-crystalline or fiber-filled polymers (see Section 6 and Chapter FS). It is convenient with fluids to consider the pressure p as an additional thermodynamic variable, to start from the specific enthalpy H instead of the internal energy, and to introduce a state equation for the specific mass p. The stress is decomposed as ffij - -P~ij + "cij, where "cij denotes extra-stress
and Sij is Kronecker's symbol. Amorphous thermoplastics obey viscoelastic rheology, which can be represented by a broad class of constitutive equations [ 106,107] among which the Leonov model [ 108-111 ] (which is of the differential type and was developed on the basis of irreversible thermodynamics) has been the object of particular attention in injection molding [30,53,54,57,60a,64,65,67,112]. Since discussing the selection of a viscoelastic model is outside the scope of this chapter, only the single mode
944
model will be presented. The elastic Finger strain tensor B (.e) and the symmetric plastic rate of strain d o are defined to represent the partially relaxed elastic deformation from initial to actual state and the irreversible relaxation rate of elastic deformation, respectively (without volume change effect). Letting t , x i and v i stand for time, Cartesian coordinates and velocity components, the constitutive equations for B (.e) and dip can be put in the general form [57,111] 9 v
#~(e),4 B(;e),j + de pB(e),~ ,~J 4" "ik '*kjp -- 0 and
(1)
,
(2)
d p - b / 4 0 ( n ( e ) d - n(.e)-d ) , v
where the convected derivative _Bij(e) is def'med to maintain d e t ( B ( e ) ) - I 9 V' o3B(.e ) rgn ('e ) _ (e) _ ~ -t- v k l~ iJ
--
oqt
(3)
oavi R ( e ) - B [ f c ) oavj 20qVkB.(.e ) "-'kj q- -~x k 3 & k u '
& ~
r;~Xk
while b is a positive function of T and the invariants I 1 - B (e) , 12 - B ( e ) - l , and
0-
O(T, p )
is a characteristic relaxation time for small amplitude motions.
The Einstein summation convention over repeated indices is used, and and Rv ij(e)-a
denote the deviatoric parts of R--ij(e) and B (.e)-I
R(e)d --ij
Extra-stresses are
defined as follows 9 "gij -- 2 ( B(e)d ""
V/012)
awlat,--itl(e)-d ~
ij
9
(4)
where the elastic potential W is a function of T, I 1 and 12 . Typically, W - 3/2 G ( 2 n + 1)-1 [(1- f l ) ( I 1 / 3 ) 2n+l + f l ( I 2 / 3 ) 2 n + l - 1] ,
where G - G ( T )
(5)
is an elastic modulus and fl and n are constants.
Injection and compression molding are dominated by shear, since most often thin parts are produced in practical applications. A quite accurate approximation of the flow is therefore obtained by assuming a developed shear flow theology, which provides the Criminale-Ericksen-Filbey ("CEF") model (see e.g. [106]) : V
r ij - 2 rl d ij + 4 Iii 2 d ik d kj - ilt 1 d ij ,
(6)
945 v represents the upper-convected derivative of v
Odo
d ij =
Ot
+ vk
~0
o5,~
Ox ~
& k d ~j
- dik
while the shear viscosity 0
dij,
Ov
J Ox k
(7)
,
and the 1st and 2 "d normal stress coefficients
~1
and I[/2 are functions of temperature, pressure and shear rate (~'), with t2
--
2 dqdq
(8)
.
In view of the numerical difficulty inherent to the viscoelastic problem, most of the effort in injection and compression molding modelling has consisted in neglecting the normal stress differences in (6), which gives rise to the g e n e r a l i z e d N e w t o n i a n model. In that case, a large mass of data indicate that the Cross and Carreau laws, which are particular cases of the Carreau-Yasuda law [113], .
-
+
,
(9)
with a - 1 - n for the Cross mode! and a - 2 for the Carreau model, are successful in describing the shear stress/shear rate relationship for a large variety of polymer melts. This was demonstrated by Hieber et al. [52,59,114,115], Chiang et al. [56,78], Chen and Liu [73], and Douven et al. [64], who obtained excellent pressure validations in PP, PS, ABS, PC, nylon and PMMA filling and post-filling experiments. In (9), rl0 - ~10(T,p) and the constant v 0 are the zero shear rate viscosity and shear stress, while n is the power-index (0< n < 1). The Cross model adequately fits the viscosity curve for polymers of broad molecular weight distribution [52,59,115] and should thus be generally preferred to the Carreau model. In the log-log diagram, both laws exhibit a Newtonian plateau at low shear rate and a power-law asymptote of slope ( n - 1) at high shear rate. It is interesting to compare the shear viscosity of the Leonov model to the Cross and Carreau laws. As an example, Simhambhatla and Leonov [111] propose to use b(Ii,I2)-(I2/I1)
m
in (2) for LDPE melts.
Equations (l-S) provide an equiv-
alent shear viscosity which exhibits the expected Newtonian plateau and powerlaw asymptote. Other values can be obtained depending on the selection of material fimctions. A last simplification is often introduced. In view of the short process cycles in injection and compression molding, the wall shear rate is generally quite high and
946
the power-law model, FI -- Tl~),.gl-n,~, n-1
,
provides a fairly good viscosity estimate to calculate the shear stress ( ~ - r/~') in the cavity. Let us point out that Verhoyen and Dupret [23] have shown that the kinematics of GN fluid flow in a thin cavity is relatively insensitive to the viscosity dependence upon shear rate, even at low flow rates, whereas stresses are highly viscosity dependent (see Section 3.3). The GN model has been extensively used in the simulation of molding processes, especially with power-law and Cross law viscosities [9,10,14,17,23,26, 31,37,40,41,52,56,64,66,69,72-74,76,78,79,104]. The present chapter focuses on these models. The calculations of flow stresses via viscoelastic rheology and GN kinematics is discussed in Section 3.4. In polymer melts, the heat flux can be modelled by the Fourier law, qi
-
- k o~/oqxi
where k - k (T,p)
(11)
,
is the thermal conductivity. An empirical role and data are
proposed in [56,64]. According to Douven [ 116], it is not necessary to consider anisotropic conductivity effects in injection and compression molding, even when molecular chains are highly oriented by shear, since heat conduction within the fluid is dominated by the transverse flux contribution (and thus only one component plays a non-negligible role). It should be noted that most of the available data do not take into account the pressure dependence of k, whereas comparison with experiments seems to indicate that this effect can be non-negligible (see Section 5). To complete the model, constitutive equations are needed for p , H , and S. During filling, most of the fluid is in the liquid state and, for amorphous thermoplastics, p is a function of p and T (with very fast relaxation). The SpencerGilmore state equation was often used in early studies [37,68]. Nonetheless, it seems that the empirical equation of Tait [56,60,64,73b,116,117], l i P - ~/Po
-
- C/Po
ln(1 + p / B )
(12)
,
where C is a dimensionless constant while l i P o and B are material functions, is more appropriate to model l i p in both the liquid and the glassy states. The glass transition temperature Tg is assumed to depend linearly upon p, while 1/p o and B are linear and exponential functions of T, Tg -
rg~ + y p ,
Po -1 - fll + f12 ( T -
Tg) ,
B -
fl3exp(-flnT),
(13)
947
with different coefficients
~i
used above and below Tg. This approach is not
completely satisfactory in the vicinity and below
Tg, where relaxation is much
slower. According to Kabanemi and Crochet [118], both stress and thermal expansion relaxations must be considered during the cooling stage. This is achieved by introducing the concept of fictitious temperature (i.e. the temperature of local thermodynamic equilibrium), together with additional relaxation functions (for pressure and for fictitious temperature). Finally, H is in principle a function of T, p and the shear elastic deformation. In the case of a GN fluid, shear elastic effects are neglected and
c)H/c?f where
-
T
Cp ,
OH/Op
,
-(1-~g/')/p
is the absolute temperature,
Cp
(14) is the specific heat at constant
pressure and o~ - - p - a c)p/OT is the thermal dilation coefficient. The second equation (14) is a direct consequence of the 2 nd law of Thermodynamics which, in its rational form, imposes from the Clausius-Duhem inequality the definition of a unique specific entropy from the differential dS - d H / T - d p / ( p T ) , a s well as constraining r/ and k to be positive (see e.g. [105]). An empirical rule governing the thermal dependence of Cp is proposed in [56a] and material parameters are given in [56b,64] for various polymers. The pressure dependence of Cp is governed by (14), since c?2H/Op31" - c?2H/31"Op. However, in practice, this effect is low. To close this section, it is important to recall that amorphous thermoplastics generally behave as thermorheologically simple materials [ 119]. The thermal dependence of viscosity, relaxation times, etc., is therefore characterized by a shift function a r, which expresses time-temperature equivalence and is defined by a WLF-type relationship [56,64,73b,119])" below
( T o - C 2 ) , the material is fro-
zen and a r vanishes, while above (To-C2), a r is given by the expression
ar -
exp(-C,(r-ro - Dp)(C 2 +r-ro)-'),
(15)
where To is a reference temperature, and C 1, C2 and D are material constants which depend on To (note that C 1 and C2 are more or less equal for all polymers when To - T g , with C 1 - 3 2 , C2 - 5 0 ~ Hence, in (2) and (9), 0 and 7/ can be assumed to obey the following rules :
O(T, p)/O(To, O) -
ar ,
rio(T, p)/rlo (T0,0) - ar 9
(16)
948
3.
MATHEMATICAL MODELLING
3.1 The lubrication approximation and its limits The aim of this section is to introduce the Hele Shaw model, which is basically valid in the flow of most thermoplastics and some thermosets in thin cavities. In particular, it applies during the filling and packing stages of injection and compression molding. GN rheology and creeping flow will be assumed (with negligible inertia and gravity effects [2]), while the influence of crystallization and/or curing will not be considered. 3.1.1 A s y m p t o t i c analysis
Scaling is very complex in molding processes in view of the many dimensionless numbers involved, and we refer to the book of Pearson [2] for more detail. Our aim is, following Van Wijngaarden et al. [31] and Dupret and Vanderschuren [9], to treat the lubrication approximation as an asymptotic model in order to provide a sound basis for the construction of the boundary conditions. Under the assumption of a low and slowly varying gap thickness 2h, asymptotic equations are obtained by letting the dimensionless number e = h o / L tend towards zero, where h0 and L are characteristic dimensions of the gap and the part, respectively. This approach clearly distinguishes two flow regions [12,33,120], namely the outer zone where the lubrication approximation is valid, and the inner zones which are formed by the flow front, side wall ... boundary layers, that will be investigated in Section 3.2. It is sufficient to consider a model problem, where the fluid is assumed to be incompressible, with a power-law viscosity and constant thermal properties, while the part is symmetric with respect to a planar midsurface (Figure 2). Inplane coordinates are denoted by x a , with Greek indices going from 1 to 2, while the gapwise coordinate is denoted by z or x 3. Latin indices go from 1 to 3. The injected or initial fluid temperature Ti and the wall temperature
Te are
given constants. The filling duration is denoted by it. The mass, momentum and energy equations read as :
Ov, = 0 , C~Xi
Ox i
Ox j
(17)
949
OXi)
pCp - - ~ +
with
71 -
/.if2 +
k
m o e bp-ar yn-1
In equations (17-20), p,
Cp,
(19)
,
(20)
. k,
n,
m o,
and a are material constants.
b
The gapwise velocity component v3 will also be denoted by w. Physical quantities are scaled as follows :
I
x a-Lx va-L'c
p
a, 1,
z-hoz va,
P
W-WoW
,
t=zt ,
,
P
P-PoP
,
(21)
h - hoh', ,
,
T=T i+
)
-T e T'.
The dynamic similitude assumption states that, when e--> 0, the dimensionless fields v a' , w' p' and T' (as fimctions of x~, z' and t ' ) tend towards given limits provided the function h'(x'#, t')
is fixed and the less d e g e n e r a c y
principle is applied to determine the scaling factors w0 and P0 and to remove any indeterminacy in the limiting process. Asymptotic equations are therefore obtained by assuming that dimensionless pathlines tend towards given limits, together with the fields p' and T ' , when the above conditions are satisfied. Writing the mass equation (17) in the form 3va' + w0"t" 0w'
= 0,
(22)
less degeneracy [120] provides w0 : w0 = eL/~" .
(23)
Defining y,2 _ (Ov,a/Oz,)2, it follows from (8)that y2 _(y,2 +o(eZ))/(e~:)2. The momentum equations become Po L oqx; Po F_L 03Z,
-
m~ E:2L,b'(E,g') n-1 moe C L~'(e~') n-1
~ c?
eb'p'-a'T'~ "n-1 Ova o~z'
+ 0 e2
'
(24)
e b > ' - a ' r ' y "n-1 &'P, + 0z'
+2Oz ~,
'T'y
&" +
,
950
where a ' and b' are defmed by the relations a"
-
a ( T i - W e) ,
b" -
bpo .
(25)
To remove any indeterminacy, a" and b" must be fixed asymptotically in such a way that the products b'p" and a'T" do not degenerate when e---)0. Further, considering equation (24.1), less degeneracy imposes to select P0 as P0 -
(26)
moe-ari/(El+n~n I . lk
l
Finally, the energy equation (19) takes the form - 7 - + w" + v a -Oxa
Gz
-
9
B r eb'p'-a~'y "n+l +
o32T" Oz
(27) ,
where the Graetz and Brinkman numbers (which represent the ratios between the advected or dissipated heat and the heat which is diffused in the gapwise direction), are defined as Gz
= pCp(eL)2/(k'c),
Br
-
mo e-aT~ L2El-n/('cn+lk(Ti - r e ) ) .
(28)
Both numbers must be asymptotically fixed when e ~ 0. It is also convenient to define the Nahme-Griffith number as Na = a ' B r . The lubrication model is obtained by letting e tend towards 0 in (22), (24) and (27), with w0 and P0 defined by (23) and (26) and the numbers n, a ' , b', Gz and Br fixed asymptotically. Retuming to dimensional quantities, the simplified system is av a
g4,
oaxa v cp
= 0
0 +
3p = 0 , 0z
,
v ~ -&--~ + w - - g
-
o
(29)
,nl
with 7/ given by (20) and ~' by the simplified expression
~2
_
(0 v~ / 0 z) 2 .
(30)
To consider the presence of a solidified layer along the walls, the viscosity law must be modified and additional dimensionless numbers are needed. In early
951
studies [2,28b,31,33], a no-flow temperature was often introduced but this concept is difficult to define rigorously. It is better to use a WLF-type viscosity law (equations (15) and (16.2)) throughout the entire cavity. Asymptotic equations are obtained in the same way as in the present section. 3.1.2 Hele Shaw formulation The Hele Shaw flow model results from integration over the gap of the simplified field equations. It is worth recalling the work of Berger and Gogos [37], Williams and Lord [40], Broyer et al. [41 ], Hieber and Shen [66], Van Wijngaarden et al. [31] and many other authors [35a,38], who developed the first numerical tools to simulate injection molding by this approach. Good reviews can be found in references [47,64]. In the sequel, this model is presented in detail in view of its connection with all subsequent developments. Non-synmaetric thermal conditions are considered on the cavity walls, according to the approach of Chiang et al. [56,78] (see also [16]). Compressibility effects are taken into account [ 14,56a,73b]. In order to separate the effects of pressure and temperature, the specific mass constitutive equation is approximated as follows : t9
--
Dr(1 -- ] ~ ( T ) - k - ) t ( T ) p )
where
(31)
,
is the specific mass at the temperature Tr and zero pressure, while
Pr
fl(T)
and 7(T)
when
T = Tr.
are material functions, the former being constrained to vanish Two kinds of gap average quantities will be used.
~p(xa,z,t) stand for any physical field, the averages ~ and r
Letting
are defined by
h
2h
-
f 0a ,
0co,
(32
-h
where
to(x~,z,t)
is a so-called profile function, which is proportional to the
velocity profile in the gap and will be defined later. Consider first the in-plane momentum equations (29.2) (which are not affected by compressibility). A first integration over the gap provides the relations O~ ot
Oz
_
Op
z - z0
3x~
71
(33)
where zo(x~ ,t) is the unknown level at which 3v~/Oz vanishes. From (33) and (20), the viscosity is thus provided by the relation
o"
mo
-
n-'
.
(34)
952
A second integration provides the in-plane velocity components from (33) and (34), taking the wall no-slip boundary conditions into account" _-
-
e-bp/n mo -1In to(z,T(.)) ,
II /0x ll ' ' ~
(35)
where the profile function to, which is defined by the integral to(z,T(.))
~hz (( - z o)1~ - Zo] l/n-1 e ar(()/n d (
-
,
(36)
must vanish on the walls. Hence, the additional relation
~o(-h,T(.))- o ,
(37)
can be used to determine the zero shear rate level z0. A third integration over the gap provides the average mass flow rate p v a 9
(38)
2h pv a = - S o Opioaxa , where the fluidity S o is given from (31) and (32) by the expression Sp
-
2hPr
II~p/~x=ll''n-' e -bp/n mo -1/n (1 - fiB + TBP) ~ ,
while ~
is obtained by carrying out an integration by parts 9
2h~-
~[Z-zoll/n+leaT(z)/ndz. -h
(39)
h
(40)
Consider now the mass equation
ap ~ ~ ( p v ~ )
oat
Ox~
+
(~)
-
0
(4~
From (38), integration provides the 2D pressure equation" c9 ~-(2hfi)where Sp
~
r (Sp O p ) ~,
(42)
,
is given by (39)and
fi - Pr ( 1 - ~ + ~ p ) .
It is also necessary to
integrate the mass equation (41) from z to h, in order to obtain w (which is present in the energy equation). Taking into account the no-slip wall condition w ( z - + h ) - +Oh / oat, one obtains the following relation"
953 pw
-
p-~
dr + -~a~hz"v~dr
Vo~
(43)
,
where the derivatives a*/o~ and O*/o~ are carried out while f'~ing z / h instead of z (as in the a/& and O / ~ derivations). This expression is convenient in numerical calculations because z / h (and not z) is used to position the nodes in the gap.
~_
//,,,./////////////,,1__/
"';~~~
Figure 3. Simplified model of the gapwise velocity component. In practice [9], the expression of w is often simplified by dropping the a*/& and O*/axa terms in (43), which consists in assuming a parallel, converging or diverging velocity profile in the gap (Figure 3). This approximation is useful because the exact calculation of w can be quite difficult since va varies very rapidly in the vicinity of the side-wall re-emrant comers, due to the presence of a singularity at these locations [20]. However, the presence of "solidified" layers [2,28b,31,33] along the cold walls reduces the accuracy of this approximation, which can be improved by taking the thicknesses c5+ and S- of the solidified layers into account. In the molten zone ( - h + ~-
~
with
z + gb ~
0
~- -
~
6-) +
(~++S-)/2
(h
,
~
~-
S0 -
5;
(60)+ v a
(~+-~-)/2
(60)
.
,
(44)
(45)
The thicknesses ~+ and c5- must be evaluated from the "no-flow" temperature Tm. Discretization nodes are optimally placed at the "solid"-liquid interface (where T - Tm), and the mesh or grid must be adapted to ~+ and S- [31]~ Finally, when compressibility effects are taken into account and the lubrication approximation is used, the energy equation takes the form (from (14))
954 0%
+
+ w--s
-
ccc
+
+
(46) -t- m 0 e b p - a T
~)n+l
The heat transfer in the steel mold can be simplified in the same way and the governing energy equation within the walls is
PsCs(V3T ~ +-oat O h ~o~) =
(z>h ,
or _
z <_ - h )
(47)
where Ps, Cs and k s denote the steel material properties. Equation (47) is coupled with the heat transfer in the gap. This model was improved by Chiang et al. [78], who superpose a steady 3D contribution obtained by averaging the successive molding cycles, on the transient contribution provided by (47).
3.1.3 Extensions Extending the Hele Shaw model to curved midsurfaces is very easy [9], since asymptotic analysis can show that curvature effects are negligible in Hele Shaw flows because the gapwise dimension is "infinitely thin" as compared with inplane dimensions. The goveming equations must be adapted, since the easiest approach is to use a unique Cartesian system of axes Ox i
to represent all the
vector and tensor components, together with curvilinear coordinates
(~1,~2,~3)
to represent the position. Any vector field (such as velocity) has 3 in-plane components v i and an additional out-of-plane component w. Letting ~a denote the in-plane coordinates and ~3 represent the distance to the midsurface, the inplane gradient of a quantity such as T is calculated by the matrix formula
]
Oxi
= A(ATA)-I
with A ( 3 •
DT
o~ =
Dt
~
&
-
O~1 c?~2
'
(48)
E "] ~
0xi
, while DT/Dt is calculated as follows"
~T (ip) + Vi-'---'X---t-
oaxi
W ~
c?~3
.
(49)
Another very important extension consists in considering other viscosity laws, such as the Cross law (see equation (9)). If a pressure-temperature decoupled
955
~e~o 103
. . . . . . . . . . . . . . . .
I
'
'
103
....
'r, "1
.....
"'''1
/'
' "
102
............................ Z................................ !............... .-,
lOl ............................. i............................... i/ ........ ::..
101
............................ + ............................... ~ '.t.~"......... .1
1~176 I
10 ~
............................ i....................
.............................................. i .........i............... ........ .~.....................
::j:i!..----':::i~
....
10 -1
1 0 -1
ii.....
10 -2 10 .2
10 4
,
10 -1
1//1/o
~/~o x
........ t
........ J
........ I
........ t
......
":
10~
10 ~
~:
........
~
........
i
........
~
' '"' ..... t
......
".-
10 ~
! 10 -1
,
10 ~ Z'/Z" 0
Z'/Z" 0
10
. . . . . . . . . . "1
........................... -:::.... -:............... ..:..~.................
10-2 10 -2
10 ~
..'--::..::b'""
.
.
.
.
.
.
.
.
.
.
.
.
10 -2 10 3
.
.
.
.
.
.
.
.
10 -2
.
.
1 0 -1
.
10 1
10 ~
!iii.i....iiiiiii
10-21 1( f3
102
10 ~
i ... f................!................i..............i 10-2
10-1
100
101
102
Figure 4. Approximation of the Cross viscosity law by an expansion with 2 terms (left) and 3 terms (fight: A3 = 1.78, B3 = 0.23) for a commercial grade PC (Makrolon 2805, n = 0.093). Top : shear stress; bottom : viscosity. numerical scheme is required, the approach of Verhoyen and Dupret [23] can be used. The quantities 7/o and Vo are approximated as follows 9 1 _ 7/o -
Z 1 k m,k(T) m2/C(p) '
-
m;(T) m~(p)
(50)
~'0
where the number of terms in the 1 st expansion must be limited to reduce the computational cost, while the 2 no decomposition is sufficient because ~:o is most often close to a constant. According to (29), (30) and (33), the shear stress can be written in the form
so that
oe
-
II~/ax~lllz-
Iz- zol/O
-
(5~
zol, ~/ll~
~:
9
(S;>
956 On the other hand, from (9) and (50), the reduced shear stress z / z o is a given function F n of the reduced shear rate (r/0 y/'c o )"
2"
(~0 ~'/T0))I
9o
1+
C/To
_Fn(T]O'~']
(53)
-"
"
This latter relation can easily be approximately inverted : 110 fl
_
Fn_ 1 Z
_
Ai
z
(54)
,
% where the first two terms produce the Newtonian viscosity plateau when z --->0, and the power-law asymptote when z --->oo (by selecting A1 = 1, B 1 = n, A2 = 1,
B 2 = 1), while a third term, which is calculated from a best fit (with
n < B 3 < 1 ), is useful to improve the approximation (Figure 4). Finally, from (51), (52) and (54), and using the lubrication approximation (equation (33)), the in-plane velocity gradient can be approximated by the sum
Ov,~ 3z
cgxa
mlk(T)rn2k(p)
ItAI 9
m;(T)~(p)
I
(55)
z _ z0lsi/n sign(z
z0)1.
In the case of symmetric temperature and velocity profiles (z o = 0), each term of (55) can be integrated as in the case of a power-law viscosity. The non-symmetric case requires to approximate [z - Zo[8i/n sign(z - z0) as a stun of powers of z and z 0. At each time step, the gap integrations can therefore be performed once and for all when the temperature profile is known. 3.2 B o u n d a r y conditions
3.2.1 Fountain flow The boundary conditions prevailing at the moving front during filling give rise to the fountain flow [121] (Figure 5). Due to the no-slip wall conditions, a strong deflection of the pathlines is observed near the front. Hence, the front region is continually fed with fluid coming from the core of the cavity, while the front feeds the regions located closer to the walls.
957
X3 I
|
_ m R . ~ L
r
Xt
Figure 5. Sketch of the fountain flow 9 relative motion of the fluid with respect to the front. Previous investigations of fountain flow first aimed at understanding the basic mechanisms governing its kinematics and its effect on the frozen molecular orientation [27,41]. Richardson [33] developed asymptotic expansions by matching the front and Hele Shaw flow regions. Pearson [2] identified two zones in the molded part, namely, the set of material points having experienced fountain flow and the remaining region (Figure 6). Numerical experiments were performed by several authors [45,48,51,54,55,122,123], in order to investigate the behavior of tracers submitted to fountain flow and the influence of shear thinning, capillarity and viscoelasticity on the front shape and the frozen orientation. The experimental validations performed by Castro and Macosko [32], Coyle et al. [50] and Behrens et al. [51] showed good agreement between measurements and predictions, without significant influence of non-Newtonian effects. The 2 no objective of investigating the fountain flow was to develop approximate models able to capture its basic kinematics without resolving all the flow details. We here present a singular perturbation analysis [12] of the model developed (independently) by Manas-Zloczower et al. [49] and Dupret and Vanderschuren [9]. Starting from the model problem of Section 3.1.1, a particular system of axes Px i is selected (Figure 5), such that P moves with the flow front, Px 1 is parallel to the front velocity and Px 3 is oriented in the gapwise direction. Asymptotic equations are established in each of the inner regions, which are all located in the front zone (in contrast with the single outer Hele Shaw region). Limit equations are obtained under the assumption of dynamic similitude and taking into account the matching conditions between inner and outer zones. The scaling used in the inner flow regions is based on considering a set of decoupled
958
(a)
~.o
~
08
~2.a
~...~..
\
-
.ok,
0.0
(b)
0.8
'~
max
~
'
0.6
t~ 0.4
o.5"~
0.2
0.8
0.0 L 50
o
150
100
250
200
300
x [mm] Figure 6. Cross-sectional space-time distributions obtained by (a) the isothermal and (b) the non-isothermal filling of a rectangular plate. The residence time of the material points is in seconds. The separation line (in bold) is not the line of maximum residence time when w is taken into account [20]. "infinitely zoomed snapshots" at any location and time on the from. More precisely, letting the origin of the axes move with the front and selecting any initial time during filling, physical quantities are scaled as follows : X i
{
--
hox
Vi = t
i
~-1'
,
t
v i9
,
p
= =
e "C t
,
' PO( p " + Ep*)
,
T
=
Ti + ( T i
_
Ze)T"
,
(56)
where the matching conditions impose (and this is a key point) to keep the same pressure and temperature scaling as in the outer zone. The mass equation is not affected by the scaling process. Its dimensional asymptotic form is thus exactly (17). In a similar way, all the terms defining ~' in (8) have the same order of magnitude and the dimensional asymptotic shear rate is thus exactly given by this equation. Also, the viscosity is not affected by the limiting process. The dimensionless momentum equations reduce to
959 o3p'/cgx[- O(e),
which means that their asymptotic dimensional form is
o ~ / ~ i - O. Therefore, the 0th-order term of the pressure expansion is constant
in every inner front region. In a further step, the 1St-order terms of the dimensionless momentum equations are considered : eb'p'-a'T" e
(onv[ +
+
which provides the asymptotic dimensional equations 9
O~+
t~ ((O~ i O~j))
0x~ =0xj 77 9xj+0x~
(58)
,
where p + denotes an additional pressure contribution in the front zone. The dimensionless energy equation is Gz - ~
+ v[
- e Br e b'p'-a'T" ~,n+, q. OqX;2 J
O(~)
,
(59)
where, for matching purposes, Gz, B r , a" and b' have the same value when e ~ 0 as in the outer flow. The dimensional asymptotic form of (59) is thus O T / O t - OT/Ot + v i anT/oqxi - 0 .
(60)
The temperature of each material point therefore remains constant when it crosses the front region. The matching conditions impose velocity, pressure and temperature "continuity" between the inner and outer regions [120,124] : lim ( v l , v 2 , v 3 , p , T ) xl-~-oo
HS , v 2 HS O, p HS T HS ) ' ' (xl =0) '
v1
(61)
where the superscript (t-/s) indicates the Hele Shaw outer solution. The order of magnitude of w in the Hele Shaw flow (equation (23)) imposes to match v 3 with 0. As different scales have been used for both x 2 and t in the inner and outer expansions, the relations (61) indicate that inner flows are two-dimensional and quasi-steady. This discussion indicates how to select appropriate front boundary conditions in the Hele Shaw model : 9 The asymptotic front region is infinitely thin and consists of a set of straight segments perpendicular to the midsurface. 9 The front pressure in the Hele Shaw flow is the atmospheric pressure when
960 surface tension is negligible [50]. The temperature of the material points is kept constant when they (infinitely fast) cross the front region. Hence, the temperature of a material point reentering the Hele Shaw region at the level z2 must be the same as when it entered the front region at the level z I (Figure 7) :
r(z 2) - r(z l) .
(62)
In fact, the characteristics of the simplified energy equation (46) are leaving or entering the flow domain depending on whether the material point is moving faster or slower than the front. Hence, one thermal condition must be imposed at the level z2 while no condition may be imposed at the level z~.
////
....../ / / / / / / w a " / f / / / S
~
z21
Zl~ w
Figure 7. Simplified fountain flow model. The front condition (62) is not affected by compressibility effects, since the front inner pressure is a constant. The pairs of levels associated through condition (62) are governed by mass conservation. Letting r.of denote the mass averaged profile function (see (36)),
p(T, Pa ) o)f
p(T, Pa)O) ,
-
(63)
the condition linking z 1 and z 2 reads as p
-
)
- 0
(64
Note that (_Of provides the front in-plane velocity vaf from (35). The above discussion shows that, according to the terminology of Garcia et al. [58], the "mass balanced front model" introduced in [9,49] relies on a sound mathematical basis and fits perfectly with the Hele Shaw approximation. Also, this approach can be used to model the effect of fountain flow when fiber reinforced polymers are molded (see Chapter FS). All the "simplified and averaged
961 simplified front models", which set the front temperature uniform and equal to the centerline [40b,69] or average temperature just upstream of the front, present mathematical inconsistencies which can induce heat losses at the front. However, these models generally behave quite well [58] since temperature variations are weak in the front region. Hence, no significant problem can be detected with the mesh or grid refinements typically used. Finally, the solution could be improved by solving the simplified inner flow problem and improving matching between the inner and outer solutions. This coupled technique is quite complex and, in this respect, the simplified model of Castro and Macosko [32], as adapted to 3D flows by Peters et al. [ 104], provides a good method to gain accuracy. 3.2.2
Wall boundary conditions
In Hele Shaw flows, the boundary conditions are treated in a different way along the cavity side walls (whose area is O ( h o L ) ) and on the larger upper and lower walls (of area
O(L2)).
T h e side walls conditions [124] can again be derived
from an asymptotic analysis. This theory will not be detailed. Using the scale L along the side wall and h in the other directions, it is easy to develop the inner flow governing equations and to conclude that the following conditions must be imposed on the Hele Shaw solution at the side walls : 9 free slip of the averaged flow, which provides the condition 0p / ~gn = 0 ; 9 no thermal condition, because the characteristics of the energy equation are tangent to the side wall. It is also possible to define along the side walls a classical displacement boundary layer thickness, which does not increase in the flow direction. The situation is totally different on the upper and lower cavity walls, since the objective is to approximate their thermal effect by means of boundary conditions. For that purpose, it is sufficient to consider the upper wall ( z - h). The heat transfer in the metal is governed by the 1D equation (47), with the following boundary conditions : 9 continuity of temperature T w at the solid-fluid interface (z = h); 9 balance of heat flux at the interface (the heat flux entering the wall is qw); 9 matching with the initial wall temperature where t f
Te 9 T - T
e ,
if
(t
< tf
denotes the time at which the front reaches location x~;
9 matching with the regulation temperature T e 9 T = T e , if (t > t f , z - h where
ec
, z >__h ) ,
+ ec),
is the distance between the interface and the thermal regulation
channels; e c can depend on position x a .
962
Zt
coolant
h+e c h + e(t)
o.(t)
h
wall
T
.
.
T ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ":::'::"":.'.".'5.'L':':.",':',':',':'.'L'.".':','L'.5"L . . . i ._
r
polymer z) Te
Temperature
Tw(O
Figure 8. Typical temperature profile and thermal penetration length. An accurate wall thermal boundary condition can be established from the theory developed by Dheur [16] (see also [9,26]), as long as a monotonic and smooth behavior of qw prevails ( T w is generally higher than T e and qw is positive and decreases with time with thermoplastics, while T w is lower than Te and qw is negative and increases with time in the case of thermosets). In that case, the temperature profile within the wall can be approximated by a linear function, { ~ -_ TeTW(i - (z - h)/e) + Te (z - h)/e ,
if
h >< hz + < he +, e
(65)
where e is the (unknown) thermal penetration length (Figure 8). The required boundary condition is established by means of the Galerkin method, using linear test functions and taking into account the flux conditions 9 - k s -a-T~ ( z = h ) = q w ,
-
- - ~ ( z - h + e) = 0
.
(66)
After some calculations, this yields the pair of equations e 0"7"w 3 k s ( T . PsCs-+2 e ~ w - Te)-qw 40t
-0
0
(67)
PsCs -~((Tw - Te )e 2) = 6ks (T w - Te ) ,
whichgovem T w, qw and e as long as e < e c. When e reaches single condition coupling qw and T w must be applied :
e c, a
963 -e~0Tw + ~ksl~Tw-Te)Z PsCs 3 0 t e
~ _ qw
=
0
.
(68)
The thermal behavior of the upper and lower walls is complicated by the fountain flow, which generates a thermal shock between the core polymer and the walls, with a resulting singularity on the front-wall intersection lines [28a]. This effect can be analyzed by means of the thermal shock theory between two semiinfinite rigid bodies of initial constant temperatures Te and TZ o [44], since the fluid motion and the non-uniform temperature field play no role close to the singularity during the first instants following the shock. This theory shows that Tw remains constant after the shock. In a similar way, when a thermal shock is considered between a semi-infinite body (the fluid) and a model body (the steel wall) where conditions (67) hold, Tw again remains constant after the shock [26] 9 T w - (or T e + [3 Tfo
)/(o~ +/~) ,
,
(69)
with
-(3S s sl8)
-(pc.kl
)
and where
Tfo
denotes the temperature of the fluid making contact with the
wall at time t f .
After the shock, the temperature profile inside the "semi-infi-
,
nite" fluid is easily calculated in terms of the complementary error function, while the thermal penetration length is
e -(6ks(t-
tf)/PsCs) 1/2 .
(71)
3.2.3 Bifurcations and abrupt changes of thickness The theory developed to build up a simplified model of fountain flow can be easily extended to the so-called singular regions, where thickness variations are abrupt or very steep and take place in a region of the same order of magnitude as the part thickness. Singular regions include edges, abrupt changes of thickness and bifurcations as in the case of T shapes (Figure 9). Inside the singular regions, the same scaling (56) is introduced and a fixed Eulerian reference frame is used in each "infinitely small and rapidly fading" inner zone, where asymptotic analysis shows that the scaled flow is essentially 2D and quasi-steady, while pressure is almost constant, and heat conduction and viscous heating are negligible. According to Couniot et al. [17] and Dheur [16], the following jump conditions must therefore be imposed" 9 Each singular region is infinitely thin, consists of straight segments normal to
964
(a)
11111/I////
(b)
I//I/111/I
/////
~/i//
I
I//11111/11111
(d)
(c)
/
f / / / / / /
///I/IS
//////,
IIII/
,,,..._ ,,....-
/ / / / / / / / 7 / / / / / / / /
7 ~ 7 / / / / / / I / / / / / /
Figure 9. Singular regions in Hele Shaw flow" (a)edge; (b)abrupt change of thickness; (c) bifurcation with separation; (d) bifurcation with junction. the midsurface(s), and is represented by a line on the midsurface. 9 Pressure is govemed only by the parabolic conservation equation (42) and is continuous across the singular line, without singular losses of head. 9 The temperature of a material point crossing the singular region remains constant (Figure 10). This condition has the same form (62) as in fountain flow modelling. The pairs of levels associated through condition (62) are determined by mass conservation. In the case of the separating flow in a bifurcation of upstream and downstream half-thicknesses h-, h( and h~, the levels z 1 and z 2 are linked by the condition ,~22 p (v, p)
or
if2h~ p (T ' p)09
P
(7"p) odz ,
,
dz - ff ~h-Q (T, p)o9 dz
(72)
(73)
depending on whether the upper or lower downstream branch is considered. This analysis shows that the edges play no role in the simplified model. In fact, the influence of the geometry reduces to the dependence of the thickness and the midsurface metrics upon position. Moreover, from (36) and (37), it turns out that the velocity and temperature profiles are not affected by an abrupt change of thickness as long as a power-law viscosity is used. The case of a bifurcation is more complex. Two situations are possible, viz. flow separation and junction (Figure 9), and the flow pattern is governed by the pressure gradient orientation
965
T(z2)=T(Zl )
2h2
2h-
~'/////////
'j~ 2h +
1
Figure 10. Thermal jump condition across a bifurcation. with respect to the singular line. Flow separation and junction can be simultaneously present along different segments of a bifurcation. In both cases, an importam consequence of (62) is that the temperature and velocity profiles are no longer symmetric with respect to the midsurface in the downstream branch(es), even when the upstream flow(s) is (are) symmetric [17,125]. This effect, which can influence the flow far downstream from the singular line, will be investigated in Section 5.1. It should also be noted that an internal thermal shock takes place in the case of flow junction. This theory has multiple extensions, in particular when dealing with semi-crystalline or reactive polymers, or with fiber suspensions, or with the filling of fiber mats (see Section 6 and Chapter FS).
3.2.4 Influence of the feeding system From the nozzle to the mold cavity, the delivery system in the injection molding process consists of a tronconical sprue, a set of trapezoidal runners, and of one or several gates which can take on various shapes. The role of the feeding system is to transfer mass and heat inside the mold and thus to supply boundary conditions downstream from the gate(s) for the calculation of pressure, temperature, species concentration, crystallinity degree, fiber orientation ... in the cavity. Hence, the importance of the delivery system is indirect. Nevertheless, its influence can prove crucial in obtaining satisfactory results in production devices [20]. Since the pioneer work of Williams and Lord [40a], Richardson [33] and Pearson [2], a number of more or less simplified flow models have been developed with the aim of balancing the delivery system when multiple cavities are simultaneously filled, or when multiple gates are used [11,56a,70,78,79,126,127]. However, oversimplification can lead to misleading results, since the temperature, species, crystal-
966
linity, fiber orientation ... non-uniformity, which is most often observed just downstream from the gate(s), is a consequence of the flow kinematics in both the runners and the gates. A typical example will be analyzed in Section 5.2. In general, the complexity of the role of the feeding system can be explained by the very low thermal conductivity and quasi-vanishing diffusion coefficients of common molten polymers, which result in a lack of mixing when the fluid crosses the runners and the gates. The flow in the gates is highly complex, since it is 3D and viscoelastic, with high Deborah number [2,22,79]. Model simplifications are hardly conceivable. Quasi-steady flow may be assumed and heat conduction may be neglected, while the effects of polymer compressibility and thermal dilation can be simplified. The main issue is to determine whether a model coupling extensional and shear viscosities is sufficient to represent the main transfer effects. Similar, but less critical, difficulties appear in the bifurcations of the runner system. To accurately calculate the flow in the runners, their trapezoidal shape must be taken into account since it induces transverse recirculations that can affect the heat and mass transfer. The simplified mass and 3D energy equations are established in the same way as in the mold cavity by means of the lubrication approximation, and all details will be omitted. Letting x a a n d z denote the transverse and axial coordinates, it is again convenient to introduce a profile function oJ, which depends upon temperature profile only [22]. The axial velocity component w is proportional to o~, which obeys the equation c)
e_aT
~x~
0(.0
[l ll
_
~
-1
,
(74)
with o~ = 0 along the boundary of the nmner section. Evaluation of the transverse velocity components va in the energy equation must be performed carefully, since mass conservation provides a single equation. The system is closed by taking the 2D c u r l of the non-simplified transverse momentum equations 9 3
[ ((); 0
0v/~
Ovr
rl~-~-
3.3 Geometrical aspects of the front motion From mass conservation, the front velocity vaf is given by the expression vaf
= pv a/p
.
(76)
967
-,,
-.j
x2
m~
'",
,
I
I
~
Front
p
~ ::-
.....
~:
Oblique side wall
t \
\
\
\
\
\
\
\
\
\
\
\
\
' \
\
~
\
~ \
\
\
~
m +1)
- ~t/(2 ......
1;
m
m, ...............
Horizontal side wall
0 *
p*
"m = -1/3
-1/4 ~
\
\
\
N
\
\
\
\
\
\
\
\
\
\
~
\
\
\
\
Figure 11. Hele Shaw flow past a comer" schematic diagram in the real and complex potential spaces (top); successive fronts calculated from the analytical solutions (bottom). Hence, from the front pressure condition and equation (33),
vcr is normal ev-
erywhere to the front line and the flow fronts are perpendicular to the side walls (in Hele Shaw flows, side walls are equivalent to reflection symmetry lines). In this section, the following problems will be addressed" 9 How does the front cross a side comer, or the vertex of a conical midsurface ? 9 How do weldlines form, and how does the front collide with the side walls ? 9 What is the effect of an abrupt change of thickness or a bifurcation ?
968
3.3.1 Effect of a side wall corner The flow front shape downstream of a side wall comer can be determined by means of conformal mapping in the Newtonian isothermal incompressible case. Consider in Figure 11 a uniform flow of speed v0 that nms past a side wall comer of angle ( 1 - m)zc (m is positive in the case of a re-entrant comer and negative with m > - 1 / 2
in the case of a salient comer). The thickness is uni-
form. Define the velocity potential r (with ~-~ - ~9r f = ~0+ i ~
and the stream function
- e~/~ o~/0X/~ ), in such a way that the complex potential
is an analytical function of the complex variable ~" = x 1 + ix z , with on F(t).
the front boundary condition r
by expressing ~" as a function ( ( f )
The solution can be found easily
since the functional domain of ~ is the
fixed region (r < 0 , V > 0). The complex velocity ~ = V1 + i V2 is given by the relation (77) Self-similitude of the solution at different times imposes that ~ be orthogonal to
(~/t - ~) along the front, or equivalently that
Re 0~-
-1
-0
0r
- 2t ,
if
t>_O, ~ - 0
.
(78)
The other boundary conditions are : Im(r
: o,
if
-
Im(~e-mn/) : 0,
if
lit-O,
if
Ifl
d~/df
~ eim~r/Vo,
-Kt<_r
o,
<-Kt, with
~<0,
(79)
~>0,
where 2hp K t / S o is the comer pressure and the constant K needs to be determined. After a few trials, the solution is found to be
( rot
~
1 2m + l
1
+
v0 t
~
2m + l
~
(8o)
v0 t
Hence, K - Vo2/(2m + 1) and the front velocity along the side wall Vfe is, from (77) and (80) with r - ~ - 0 ,
V fe
--
Vo/(Zm + 1).
969
Solutions are shown in Figure 11. When the front runs past a salient comer, or a cone vertex whose total midsurface aperture angle is lower than 2Ir, a stagnation point forms immediately at the comer or the vertex, and a material line joining the comer to the front extremity, or the cone vertex to the closest front point, starts to stretch out. This line is initially infinitely small and forms a weld line in the latter case. When the front runs past a re-entrant comer, or a cone vertex whose aperture angle is higher than 2zt, the velocity immediately becomes infinite at the comer or the vertex, which are continually crossed by material points. These conclusions are reinforced when the viscosity obeys a power-law. With a Cross or Carreau law, the flow in the vicinity of a stagnation point is governed by the Newtonian viscosity plateau. Analytical solutions of the flow in Hele Shaw cells were investigated in detail by Richardson [128] and Entov et al. [129], while Aronsson [130] develoved very simple solutions for the limiting case of a zero power-index fluid (n - 0).
3.3.2 Weld lines The flow past a cone vertex of aperture angle lower than 27r, that was analyzed in the previous section, exhibits a key aspect of the kinematics of weld lines in Hele Shaw flow. When two fronts collide, a single pair of material points meet and the weld line results from the fast stretch of a material line that initially reduces to this pair of points. The evolution of weld lines is complicated by the observation that these "lines" are in fact true deforming surfaces. Starting from a straight segment perpendicular to the midsurface at the location of the front meeting, the infinitesimal weld surface begins to stretch rapidly and becomes a material surface, that moves at the fluid velocity. Hence, the weld surface is normally strongly deformed by pressure gradients. Moreover, its extremities experience fountain flow, which typically induces a 3-fold weld surface in the front vicinity, while a 2-fold shape is generally prevalent elsewhere. More complex behaviors due to changes of flow pattern are possible. The traces of the weld surface on the cavity walls are governed by fountain flow, since they represent the successive positions of the material points that experience a thermal shock with the walls. These traces are always symmetric with respect to the midsurface, even when the velocity and temperature profiles are not. Beside the exceptional case of a flat weld surface, thermal diffusion will remove the temperature discontinuities across the surface. This will not be the case for any quantity (such as crystallinity degree, fiber orientation ...) that is not subject to diffusion. In a similar way, the derivatives of these quantities are normally discontinuous across the surface that separates the core fluid from the material points having experienced fountain flow.
970
v2~2+ vlf+2
(b) V+
+
d/~
+ v~)
~ffont
v;: , .
~__ x 1
Figure 12. Flow across an abrupt change of thickness" (a) sketch of the front velocities; (b) uniqueness of the intersection of the front and the singular line. When a front collides with a side wall, the kinematics are the same as when two symmetric fronts meet. Boundary layers are however different.
3.3.3 Abrupt changes of thickness and bifurcations The principles governing Hele Shaw flow in the presence of a singular line were developed in Section 3.2.3. Following Couniot et al. [17], precise angles are formed between the front segments and the singular line. Consider first an abrupt change of thickness, and let h-, vlf, V2-f , Sp and h +, V ~ f , V ~ f , + S o denote the upstream and downstream values of the half-thickness, the normal and tangent velocity components, and the fluidity, respectively (Figure 12). Isothermal flow and fluid incompressibility are assumed. Mass conservation yields h+v[:
-
h Vl:
(81)
,
while pressure continuity implies continuity of the tangent pressure gradient, which from (38) and (76) is written as (82)
2h +p v2+f / S; - 2h-p v]f /Sp .
Finally, uniqueness of the intersection between the upstream and downstream front segments and the singular line provides the relation
E V
+2]/v+ E 2
"k- V 2 f
2f
Vlf
q" V 2 f
21/v'
9
(83)
971 Equations (82) and (83) can be cast in a more convenient form. In the case of a power-law viscosity, the fluidity is given by (39), or equivalently, from (38), (40) and (76), by = while
m0 -1 ( n - l + 2)-nh
,
(84)
][vaf[I- Vlf 2 -FY2f 2 )1/2 can be eliminated by means of (83).
Hence,
after some calculations, the following relation is obtained :
(85)
v 2+f / h +2 -- V2T - / h -2 . Also, multiplying (83) and (85) side by side provides the relation
The 3 equations (81), (85) and (86) involve 4 tmknowns
--
+
+
(vlf ,v2f, vlf, v2f ),
which can thus be determined to within a multiplying factor. In particular, the angles
o~- and
t~ + formed by the upstream and downstream velocity direc-
tions with the singular line are functions of the ratio
tan2 o~- -
tan 2 o~§ =
(h+/h -)
only"
.
(87)
The same formulas apply whether h + is higher or lower than h-. Two patterns are thus possible, depending on which is the feeding and which is the fed branch. Also, it can be observed that a vanishing or infinite thickness ratio corresponds to the case of a lateral side of the mold. When
h + = h - , the angle
a + = a-
is
indeterminate and (87) is no longer valid. When h § is close to h-, a front curvature boundary layer is thus present near the singular line in the Hele Shaw flow. The m-plane dimension of this boundary layer is governed by the flow pattern and the thickness ratio, but not the actual thicknesses. Hence, although (87) is valid in theory, numerical solutions can exhibit angles o~+ and a - that differ from their theoretical values, except if a very strong mesh refinement is used. In reality, the flow in the singular region is governed by the theory developed in Section 3.2.3. The inner flow domain is partly or totally superposed on the front curvature boundary layer, which can be absent from experimental results. The analysis of a bifurcation can be handled in the same way. Two problems
972
must be set apart, viz. flow separation and junction. A different equation governs mass conservation in these cases, while pressure continuity and uniqueness of the intersection between the fronts and the singular line each provide a pair of equations. A total of 5 equations with 6 unknowns is thus found and the same conclusions can be drawn as in the case of an abrupt change of thickness. Verhoyen and Dupret [23] have shown that this theory can be extended to more general viscosities, including the Cross and Carreau laws. Indeed, the jump equations (81-83) remain valid, together with (33), (38) and (51) (from the lubrication approximation). Hence, the following expressions are successively found for the fluidity:
I1@/~11so -112~ 11,
(88
IlOplo~<,llso -Ilop/O~ III_".paz Iz~(C-zo )l. dr p dz [.z i' sign(~"- Zo)d~" h
= Ip~ Iz- ~01dz
(since 19 is a constant).
(89)
-h
On the other hand, from (51) and (54), the shear rate is -- TO 00 -1F,,-l(llop/Ox<,lll
z
-
zol/~o),
(90)
where the function F n is defined by (53). Hence, introducing (90) into (89) yields, in the isothermal case (with zo - 0 ) 9
So - 2p't'o 2h 170-1 where
II@/ex~l1-2Hn(h'ro -1 I1@/~<~11),
H n (~) - /~~ I~o~ Fn-' (~) d~
However, from (88), H0p/Oxal[ S o h - '
(~" > O) .
[l~o~l[-1 i~ continuous
(91) (92) across the singular
line, as is II0p/0xall2 So 2 h -2 v # (from (83)), and thus"
2 since
-I1 / 11 s;
h+v;f / S ; - h-v2f / S ;
,
(94)
in view of the pressure continuity gradient. Therefore, comparing (91) and (93)
973 shows that
H n (h [lOp/ oaxaIi/*0) is continuous across the singular line, since 7/0
and "ro are functions only of p. Hence, h IlOP/Oqxall/'r0 is itself continuous since H n is strictly increasing, and thus from (93) : Sp+ // h +3 - S p / h -3 , which also involves from (94) the continuity of
(95)
v2f/h 2
(equation (85)).
Further developments proceed as with a power-law viscosity. Similarly, it can be shown [23] that, when an unbalanced bifurcation is fed with incompressible isothermal fluid through the upstream branch, and the incident velocity is uniform and normal to the singular line, the motion of the fronts downstream of the bifurcation depends only upon the thickness ratio, but not on the viscosity law. These considerations have far reaching consequences and explain why the filling kinematics are, in general, only weakly dependent upon shear thinning effects and dominated by thermal effects. 3.4 Viscoelastic effects Since the aim of this chapter is not to analyze in detail the role of viscoelasticity in injection and compression molding, this section is devoted to a review of the existing literature. Viscoelastic effects are significant in critical regions, mainly the front zone, the edges, abrupt changes of thickness and bifurcations of the part, the gates, and the branchings of the delivery system. They also influence the flow stresses (via normal stress differences) and the cooling stage [118], and are directly connected to the generation of frozen-in orientation. Nevertheless, the GN model generally provides accurate results since the flow is dominated by shear, and also thanks to the rather low sensitivity of the gap-averaged kinematics upon shear thinning effects as shown in the previous section. Early studies (see the review of Grmela [47]) were limited by the tremendous difficulty of the viscoelastic problem in normal processing conditions [107]. Tadmor [27] and Dietz et al. [29] developed simplified models to evaluate fountain flow extensional effects and the resulting frozen orientation. Kamal and coworkers (see e.g. [70,122]) investigated 2D cross-sectional flows in the cavity with a White-Metzner rheology, but this approach is limited by the poor properties of this model [ 131 ]. Most of the effort bore upon using the Leonov constitutive equation (see Section 2), since its material constants can be determined from standard experiments. The 1D or 2D incompressible multi-mode model [108111] was used by Isayev and Hieber [30], followed by several authors [53,54,65, 67,112], to analyze the effect of the processing conditions on the flow and
974
N1 (MPa) 0.1
Direct -
Indirect
6 0.06
0.04
tO
0.02
0 -'
-0.5
0
0.5
1
Distance midplane (mm)
Figure 13. Predicted first normal stress difference after 4.7 s in the filling of a 80 • 50 • 2 mm PC rectangular plate (vf = 120 mm/s, T/ = 320~ Left" direct simulation; right" comparison between direct (solid line) and indirect (dotted lines) calculations 8 mm from the linear gate. (From [57]). residual stress distributions in junctures or in the front region. Experimental comparison by means of birefringence measurements showed reasonably good agreement, such as in the subsequent investigations performed by Haman [60], using the 2D compressible Leonov model [57], and by Chang and Chiou [63], using the K-BKZ model. In these latter studies, such as in the packing stage analysis of Nguyen and Kamal [61] and in [65], an attempt is made to extend the lubrication approximation to viscoelastic rheology by assuming a priori equal scales for the extra-stress components (without considering the less degeneracy principle), while simplifications are introduced more carefully in [60a]. The scaling problem still needs to be solved for viscoelastic flows in thin parts. An important step was made when Baaijens [57] demonstrated that simulating GN kinematics, and re-calculating the stresses from a viscoelastic rheology and this pre-calculated velocity field usually provides a fairly accurate estimate of the solution in shear dominated flows. This indirect method, which was further used by Douven et al. [64], Caspers [132] and Kabanemi et al. [62] to simulate the molding of Leonov and Wagner fluids, represents a key step forward since it is much easier to solve the indirect than the direct (coupled) problem. Figure 13 depicts a typical result from [57], where the agreement between indirect and direct calculations is obviously excellent.
975 4.
N U M E R I C A L SOLUTIONS
4.1 Review of available methods Beside pioneer studies performed by means of the finite difference (FD) method in order to solve the 189 problem [31,35a,40b], most of the effort in injection molding modelling has borne upon developing mixed algorithms, where the 2D pressure field is evaluated on the midsurface, while the temperature and other 3D fields are integrated by means of a hybrid scheme over the entire filled region. These techniques, which are detailed in the literature below and will not be discussed, will be classified according to the solution of the free boundary problem since the main difficulty to be overcome in the simulation of molding processes arises from the fact that the filled domain evolves considerably with time. The particular method developed at UCL is described in the next section. Two general approaches were followed, the 1 st consisting in using control volumes on a fixed mesh covering the entire midsurface, while the 2 nd class was based on tracking the flow front(s) and adapting the mesh (or grid) at each time step in order to cover only the filled domain. The l~t technique, which is very robust and by far the easiest to implement, does not involve a discrete front representation. Front phenomena are thus more difficult to model and strong mesh refinement is required in the front zone. This approach, which was 1~t used by Lord and Williams [40b] and Broyer et al. [41] to model 2D non-isothermal crosssectional, or 2D isothermal in-plane flows, has been the object of intense investigations from the Cornell group [56a,71,72,78]. It was further used to simulate compression molding [95] and extended to the K-BKZ model [63], see also [75, 76,79,80]. Following Chiang et al. [56a,78], this technique attaches to each node of the 2D mesh (surrounded by a given control volume) a variable which represents the volume fraction of the injected polymer within the control volume. Solving alternatively for this variable and the pressure field, and for the temperature field (using subvolumes in the gap and a streamline-upwinding technique) provides a reliable and efficient way to simulate the cavity filling, with an easy representation of all the geometrical details and the delivery system. Normally, at most one element should be filled per time step, but the VOF method developed by Voller and Peng [81 ] to simulate RTM shows great promise as to the removal of this limitation. Finally, the Taylor-Galerkin method developed recently by Pichelin and Coupez [82] to simulate the 3D free surface flow of a GN fluid represents a very important step forward obtained by means of an approach which can be associated with the control volume technique. The front tracking-mesh adaptation methods are based on mesh deformation or remeshing. In the former case, the 2D filled zone is mapped onto a fixed domain
976
over which the problem is discretized. There is no longer any free boundary, with the drawback that it becomes very difficult to address the complexity of typical industrial parts. Whereas conformal mapping was used in early studies [44,128a], the Thompson algorithm seems to behave better, see the results obtained by Gtigeri and co-workers [74,77,98,133]. Particular methods are developed in [77,98] to couple the 2lAD Hele Shaw model with full 3D calculations in the singular flow regions (front zone and junctures). On the other hand, the 1st remeshing algorithms were based on the generation of successive element layers according to the 2D front progression [52,66,73a], but this technique can hardly be extended to general shapes. The best approach, as used in the MOLDSYS software (see the next section) and in [80,104,132,134], consists in tracking the front(s), recovering as many elements as possible from a fixed mesh covering the entire midsurface, and generating new elements in the remaining portion of the filled area. To close this review, it is worth recalling the various techniques elaborated to simulate the packing stage [14,56a,60a,61,64,68,70,78,104,132] and to understand the fountain flow goveming mechanisms [48,50,51,54,55,70,77,122,123].
4.2 Description of the MOLDSYS approach The MOLDSYS software developed by Dupret and co-workers at UCL [9-23, 25,26] (see also Chapter FS) solves the free boundary problem on the basis of the following concepts : (i)front tracking and automatic remeshing, in order to capture the shape of the moving free boundary while providing a systematic method of adapting to geometrical events such as weldline formation or front split after collision with a side wall; (ii) Eulerian integration of the temperature or any other 3D field (such as crystallization degree, reaction degree, fiber orientation); (iii) extrapolation of these fields in the region located between the successive flow fronts, using an extrapolation mesh and taking the fountain flow conditions into account; (iv) 2D implicit FEM pressure calculation. The front region plays a critical role in molding processes in view of the key effect of fountain flow, which causes fast out-of-plane advection and material deformation in the front zone. As far as a general purpose algorithm is desired, in order to simulate injection and compression molding, RIM, RTM .... and predict crystallization, curing, fiber orientation and their effects on the flow, it is essential to accurately track the fronts. A remeshing strategy was selected instead of deforming meshes in order to adapt easily to the topological changes of the filled domain. Moreover, as in-plane diffusion is neglected in the lubrication approximation, the simplified energy equation (46) is such that, during any given time interval, the fountain flow thermal condition (62) has no influence on the tempera-
977
rare evolution upstream of the initial front of the time step, which can be treated as an outlet section (without thermal boundary condition). The characteristics of (46) are indeed leaving the domain along this "initial" flow front. On the contrary, the effect of fountain flow must be taken into account in the region located between the initial and final fronts, since characteristics are leaving or entering this region across the moving front depending on whether the material points move faster or slower than the front. The extrapolation mesh is therefore built on in the inter-front region and equation (62) is considered for the fluid layers that re-enter the domain after having experienced fountain flow. This method is general and can be applied to any field for which in-plane diffusion is neglected. To describe the time-dependent algorithm, it is sufficient to consider a single time step. Let M n, F n, ~ n denote the mesh, front(s) and discrete domain at time t,, and [Phi, [T~] the associated column vectors of nodal pressures and temperatures, while [Vn] and [Wn] stand for the discrete in-plane and out-ofplane velocity components along the front(s) and inside the domain at the integration nodes. Additional fields [ A n ] represent the nodal crystallization or curing degrees, or orientations ... if need be. While the flow fronts move from time t n to tn+ 1 , they scan the "inter-front region" An+1 . This area is discretized as a quadrilateral "extrapolation" mesh denoted by En+ 1. T h e sequence of operations needed to calculate these quantities at time tn+ 1 is organized as follows : 9 Front displacement so as to obtain the extrapolation mesh En+ 1 together with ^
provisional representations of the flow front(s) Fn+ 1 . 9 Tracking of front-front and front-side wall meetings, and front conditioning to generate the definite front(s) Fn+ 1 . Remeshing to generate Mn+ 1 on ~n+l" 9 Eulerian time integration of the temperature and other 3D fields on the previous domain soastoobtainprovisionalvalues
[7~n+1], ['4n+1] on M n. Extrapo-
lation of [7~n+l], [/~n+l] on the inter-front zone
An+1 , taking fountain flow
conditions into account. Correction of the temperature and other fields so as to obtain I T n +l ] , [an+l] on M n +l . 9 Implicit pressure time integration so as to obtain
[Pn+l]" Calculation of in-
plane and out-of-plane velocities [Vn+1] and [Wn+1]. 4.2.1
D e t e r m i n a t i o n o f the n e x t c o m p u t a t i o n a l d o m a i n
Front velocities are oriented along the boundary for vertices lying on the side walls, and along the bisectors at the front for other vertices. The fronts are
978
I
|
Figure 14. Extrapolation mesh in the presence of a bifurcation. moved using the Euler scheme [13]. A more sophisticated technique would be very difficult to implement since, during the time step, not only can the shape of the domain evolve, but so can its topology as flow fronts merge, divide, or meet the boundaries. Also, it was shown by Couniot [13] that the present remeshing algorithm is of a very low order and that there is no point in devising sophisticated techniques. The Euler scheme has the benefit that the area scanned by a front segment is represented by a quadrilateral, possibly degenerating into a triangle. The union of these elements defines the extrapolation mesh. In the case of thickness discontinuities or bifurcations, as velocities undergo a transformation when the material points cross the singularity (see Section 3.3.3), each quadrilateral originating from an upstream facet (relative to the singularity) yields one transformed quadrilateral per downstream facet (Figure 14). Managing the extrapolation mesh then becomes relatively easy [ 17]. The situation is more complex regarding front junctions. Computing the new position of the fronts features the difficulties of computational geometry [ 13], these stemming from the discrepancy between pure geometry and its numerical transposition. As a consequence, geometrical algorithms involve tolerances, which means that the transitivity of geometrical properties is lost (e.g., two straight lines which are parallel to a third line to a given tolerance are not necessarily parallel to each other to the same tolerance). The complexity of these algorithms is therefore often dramatic, particularly in the simulation of molding processes when a front tracking strategy is selected, since the size of the element sides must be controlled in order to obtain well-behaved meshes. To overcome these problems, the solution must be implemented as a mix of numerical and logical operations. In particular, the present front motion algorithm is
979
Completely filled
[
/t//tltl
I
Partially filled
Completely empty
Figure 15. Partition of the elements of the fixed mesh. based on the concepts of "virtual front nodes" and "events" [13], and uses as reference the fixed mesh. Basically, a virtual node corresponds to the intersection of one front segment with one side of a mesh element. As the front moves, the virtual node moves along the element side until it reaches a mesh vertex, which constitutes a first type of event. Other event types include the crossing of a mesh boundary or a re-entrant comer along the boundary. All the events are sure to be detected and simultaneous events can be processed asynchronously (which consists in removing or inserting virtual nodes in the front description). Hence, operations are mostly logical and not numerical.
4.2.2 Generation of the temporary mesh used for the next iteration The discrete front description provides a partition of the elements of the fixed mesh into three classes, viz. the elements which are (i) completely filled, (ii) completely empty, and (iii) partially filled (Figure 15). A simple approach would be to re-use as such the elements of class (i) and to generate additional elements in the remaining portion of the filled area ~ the so-called generation zone. This approach is workable with some enhancements. First, the discrete front description must be modified in order to handle front-side wall and front-front meetings, and filtered to ensure that the local front discretization scale corresponds to the local fixed mesh scale. Such conditioning can be achieved by redistributing the front vertices so as to preserve mass balance. As front collisions are treated by accepting a slight overlap between the colliding fronts, or a slight crossing of the side walls, it is easy to detect and remove the dead front segments. Front filtering must also remove the unfilled areas which are regarded as negligible and increase the filled areas which are still negligible (removing matter can lead to a halt or a non-natural motion of the fronts). The simulation time is finally adjusted on the
980
basis of a comparison between the total mass integrated at the nozzle and the mass currently present in the mold. After front filtering, the temporary mesh is generated. First, the fronts are located with respect to the fixed mesh and, if strictions are detected, some totally filled elements are considered as partially filled. Generation is then performed. As a 1 st step, triangles are built using a Delaunay triangulation that is constrained to use the boundary vertices of the generation zone. This method, which leads to optimal triangulations with respect to some criteria, can be implemented quite efficiently and extended to curved surfaces. Preserving imposed boundaries can be achieved in several ways [13]. As a 2 "d step, as many triangles as possible are merged two by two into quadrilaterals. The temporary mesh Mn+ 1 is built by merging the filled elements and the generated ones.
4.2.3 Integration of temperature and additional fields The temperature field is discretized by means of fimte elements along the midsurface and cubic splines in the gap [25,26] (full polynomials as in [9,11] are less stable):
T --- ~, T~i) (z,t) r
) ,
(96)
i
where the shape functions r
are linear or bilinear on the parent element,
while T~i) (z,t) is the temperature profile approximation at node x~ ) . Space discretization of the energy equation is performed by means of a hybrid scheme combining a S UPG technique along the midsurface [ 11,25] and collocation in the gap. Weighted residuals of equation (46) are l~t integrated over f~n, and are further considered for each node
z (k) in the gap. The original SUPG
method [135] consisted in defining the test functions ~i)(xa ) as" ~(i) -- ~)(i) +
where
A'i" v a o3~)(i) /oaxot ,
(97)
A~" = (l~'~lA~ +lVo[Ao)/(-~/~Va 2) is an element characteristic time (~
and 7"/ denote local coordinates, with - 1 < ~,7/< 1 for quadrilaterals and 0 < ~ , 0 , ~ + 7/<1 for triangles, while v~ and V,, A~ and A n are characteristic velocities and lengths at the element centroid). Vanderschuren [11 ] observed that using the same test functions at each level z
981
oJ and Op/o3ca . After some calculations, the following expression is found" Atva =
~i5
+
'
(98)
which replaces A~" va in (97). For triangular elements, the following ad hoc expression was found to take triangular symmetry into account :
- 2~]-511 IO~i
Atv a -
(1- q)
+ (1- r
~i
+ (r + 7/)i0~ - ~ l ] ~_-q2----_ ll_2oxo Op OX aOp "
(99)
After in-plane space discretization, the energy equation (46) reads as :
-1pCp OZ~ (k
j
OZ
m o ebp_aT~n+l -
I pc; ~-~m
)l ~~mfiOjdS o~
-
Oz
pCp -~ + va
~i)dS ,
(100)
where T(j) denotes the time derivative of T(j)(z,t). Following Mal et al. [25, 26], cubic spline collocation is performed in the gap by considering (100) for each node z (t) and expressing the derivatives o32T(j)/oaz2 in terms of the nodal values of T~j) and the end derivatives 31"~j)/cgz( z - 0 or h, or z - + h ) . The latter are then eliminated using the wall conditions (67) or (68) and the relation -kOT(j)/Oz ( z - +h)= + qw (a vanishing derivative is imposed along the midplane in the synmaetric case). When k is temperature dependent,
oqT(j)/0z
is evaluated at internal nodes by a 2nd-order FD approximation. Eventually, the semi-discrete energy equation takes the genetic form"
[r]- K([rl,[pl...)[rl
- Y(VI,[P]-.-)
,
(101)
where the matrix K (related to heat diffusion) is block-diagonal, each block being associated with a given midsurface node x~aj) and having dimension m (the number of nodes in the gap), while the right-hand side includes all the terms arising from heat advection, friction and compression. Moreover, K is of the form A-1B, where A and B are block-tridiagonal matrices, while in (101) f is of the form M-~q, where the mass matrix M consists of m equal blocks
982
whose elements are the integrals ~nm ~((i)~(j)dS. Eulerian time integration of (101) is carried out in MOLDSYS using the 3rdorder scheme developed by Dupret and Vanderschuren [9,11], as extended by Mal [25] to temperature and time dependent matrices K. The method is implicit for heat diffusion and explicit for other effects, with improved stability for heat advection. In the absence of diffusion, it reduces to the 4th-order Runge-Kutta scheme. During integration, the nodal values of p, 'i", va, w and p are frozen. The thermal penetration inside the walls can be imposed or calculated (equations (71) or (67)). In the latter case, according to Dheur [16], the best approach consists in introducing instead of e the unknown X - (Tw - T e)e 2, which is governed by the equation )~ - 6ks(T w - T e)/pscs. When abrupt changes of thickness or bifurcations are present, particular techniques must be applied. First, midsurface nodes are doubled, or tripled ... along the singular line depending on the number of adjacent branches, while the associated shape and test functions are split accordingly. Secondly, constraint (62) must be taken into account. As va and p are frozen during temperature integration, the relations (72-73) are themselves frozen and the correspondence between feeding and fed branches and layers can be established before integration. When a given node is fed with fluid from the adjacent branch(es), the temperature profile must be imposed, from the temperature of the feeding branch(es), by means of this correspondence (Figure 10). The different branches of the part can often be arranged in such an order that feeding branches are calculated before fed branches. In some circumstances, iterations are necessary. Time integration provides provisional values of [7~n+l] and [An+,] on the old mesh M , . Extrapolation in the inter-front region An+~ is performed in 2 steps using the extrapolation mesh En+ ~. First, quantities are evaluated at the nodes of
En+ 1 that belong to the provisional front
/~n+~ without considering fountain flow. This is achieved by least square approximation of the field gradients along F n . Secondly, the effect of fountain flow is taken into account for those fluid layers moving slower than the front. Since o9 and p are temperature-profile dependent (from (31), (36) and (37)), the condition (64) cannot be used directly to determine the pairs z 1 and z2 governed by condition (62). An iterative technique is thus implemented. At each node x~) of /~n+l, a l~t guess as to the profile function ~o(i) is obtained by extrapolation. When, at a given level z~2k) ,
983
the fluid is moving slower than the front \/r z
"\o9)')), the temperature is
z"
corrected by picking the temperature of the level z 1 determined by (64). The profile function is re-constructed and the sequence of operations is repeated up to convergence. Unacceptable wiggles were generated in the solution when applying this procedure, due to the thermal shock caused by fountain flow (see Section 3.2.2). After lengthy investigations [16], the algorithm was modified [25,26] by splitting the temperature field into a numerical and an analytical part (T* and Ta). The former is integrated and extrapolated using the above algorithm. The latter is provided by the thermal shock between two semi-infinite bodies and reads as
+
[
1
(lO2)
+ (T~,- T f o ) e f f c [ ( h + z ) ( 4 k o ( t - t f ) / P o C - ~ o ) - l / 2 ] , where Tw+ and Tw are the upper and lower shock temperatures (equations (69)
and (70)), while t f - t f (x~) is the time at which the front reaches x a , Tfo is the temperature of the core fluid meeting the walls at time
+
tf, and k~, p~, Cpo,
ko, Po and Cpo are the fluid properties at room pressure and temperatures Tw+ and Tw . A single contribution is considered in the symmetric case. The calculation of Ta requires to track Tfo and tf as functions of x a [25]. As the definite front(s) and mesh Fn+ 1 and Mn+ 1 differ from their provisional counterparts
/~n+l and
MnUEn+l, t h e n o d a l v a l u e s o f
must be re-evaluated on Mn+l, together with tf
ypi mg
te o,ate va,ues
[7~n+1] and
JAn+l]
and Tfo. This is performed
[an+l] on
pro
cedure is more stable than least square approximation [16]. When a given node of Mn+ 1 is located outside f~n u An+l, an appropriate extension of the ele-
ments of En+ 1 might be required [ 16]. In the presence of front-front collisions, the extrapolated values are averaged in the overlap area and the weights are obtained from the distances to the previous fronts F n provided by En+ 1 [ 10,16]. The elements of En+ 1 are generally quite flat, especially at the end of a short shot. Therefore, letting ~l denote the inter-front distance and SXn+1 represent a characteristic element size of Mn+ 1 in the direction normal to Fn+ 1, the ratio
984
(~l/~Xn+ 1 is often much lower than 1. A relaxation technique was developed by Dheur [16] to address the resulting discrepancy between theoretical and calculated temperatures, since in the numerical procedure the fountain flow condition (62) entirely affects the elements adjacent to Fn+1 (of size 6Xn+l), instead of the inter-front zone An+1 (of size 51). The "numerical" temperature profile T(j) (z,t,+ 1) is therefore corrected at any front node x(aj) by the formula T(j) - (1 - 0) T(j)o + 0 T(j)n ,
(103)
where T(j)o and T(j)n represent the extrapolated profiles obtained just before and after applying the condition (62), while the relaxation coefficient 0 is defined by O-min(51/Sx,,+l,1). The normal characteristic element length 5x,,+1 and inter-front distance 51 are estimated for any node of Fn+1 on the basis of the new mesh Mn+ 1 and the extrapolation mesh En+1 [16,25]. It should be noted [16] that the exact value of 0 is not important, since extrapolated values are already influenced by fountain flow before integration. Relaxation is a numerical stabilization procedure (likewise to upwinding), which is indispensable at low flow rate.
4.2.4 Calculation of the pressure and velocity fields The 2D mass equation (42), which governs the pressure field, is space discretized using Galerkin finite elements [9,14,66,136]. Details are unnecessary. The pressure equation couples the calculations in the part and the delivery system. Time integration is carried out by means of the implicit Euler scheme, since equations are stiff in view of the low compressibility of molten polymers. Temperatures are frozen duringintegration, together with ~ , /3, ~, /3B and ?'B (see (31), (32), (39) and (40)). The system is solved using the Newton scheme. The last substep is devoted to calculating the nodal velocities where needed. The front velocity (equations (63) and (35-37)) is discretized by means of 1D linear elements. A single unknown (the velocity norm) is calculated at each front node, using a least square technique applied along Fn+ 1. In-plane velocity components are given by (35-37). Iterations are necessary to determine the zeroshear rate level z0 in the case of a non-synmaetric flow profile. A specific drawback of the Hele Shaw approximation is that velocity tends to infinity in the vicinity of side wall re-entrant comers [20], while viscous heating causes temperature to tend to infinity at the same time. This is a model, not a numerical, problem which all the more perturbs the discrete solution as the mesh is refined.
985
Hence, calculating in-plane velocities is difficult and it is advisable to smooth the result [20]. Finally, w is calculated from (43) via the Galerkin technique. In practice, (44) is much more stable and generally produces satisfactory results.
5.
S I M U L A T I O N R E S U L T S AND E X P E R I M E N T A L VALIDATION
5.1 Results of numerical simulations Much can be learned by means of numerical simulation from in-depth analysis of results obtained on simple geometries. In this section, we consider the nonisothermal filling of a b i f u r c a t i o n composed of three 10 cm x 10 cm square facets (Figure 16); the cavity features a single point injection gate, located in the comer of one of the facets ~ which is identified as the upstream facet. To facilitate simulation, the fluid (a polycarbonate) was assumed to be Newtonian, with constant material properties (r/= 400 Pa.s, p - 1200 kg/m s , k = 0.215 W/m.K and Cp = 2200 J/kg.K). Hence, thermal calculations were decoupled from the kinematics. Details on the material data can be found in [17]. Several alternatives are examined, featuring an increasing discrepancy in the thicknesses of the downstream facets, designed so that the total volume remains constant. The influence of the geometrical asymmetry of the part is clear, since the thicker downstream branch gets filled first. Just after it gets filled, the front velocity in the thinner branch exhibits a sharp increase, as a consequence of flow redistribution.
(a)
(b)
(c)
Figure 16. Sketch of the test bifurcation and successive flow fronts, for a flow rate of 60 cc/s. Upstream, and horizontal and vertical downstream branch thicknesses : (a) 4/4/4; (b) 4/5/3; (c) 4/6/2 (in mm).
986
35000 p (kPa) 30(0
-
/6/2
25000 t = 1.3 s
t = 3.52 s
20000
15000 I
10000 0.0
,
,
0.5
1.0
I
,
,
,
,
,
1.5
2.0
2.5
3.0
3.5
t(s)
4.0
Figure 17. Evolution of the pressure at the spree : (a) 4/4/4 mm thickness distribution; (b) 4/5/3 mm; (c) 4/6/2 mm. The pressure evolution at the sprue is shown in Figure 17. As it is observed quite frequently on complex parts, the crossing of the bifurcation and the thickness discrepancy have a barely noticeable pressure effect before the end of the filling of the 1=t downstream facet. The latter event occurs at variable times as the asymmetry increases. A detailed examination of the velocity field would reveal that a slight recirculation is induced in the branch that gets filled first, since the pressure along the bifurcation is higher in front of the injection gate than at the other bifurcation extremity. Such a recirculation can have an important effect in the case of fiber-reinforced composite parts. The bifurcation induces a usually asymmetric downstream repartition of the flow and, along with the flow, the temperature profile across the gap is split. As a consequence, the downstream temperature and velocity profiles are asymmetric (see Section 3.2.3). The four phenomena governing temperature evolution in
Figure 18 (next pages). Layer by layer temperature field at the end of the filling, from T = 400 K (black) to T = 600 K (white) (approximate range). The bifurcation (in bold) is located at the top of the upstream branch and the bottom of the 2 downstream branches. Flow rates : (a) 60 cc/s and (b) 120 cc/s.
987
988
989
injection molding (fountain flow, heat advection, viscous heating and heat diffusion) do not occur at the same time. Fountain flow occurs first, when the polymer reaches a given point in the cavity. Its main effect is to make the temperature profile quite uniform. However, this situation is short-lived at a given Eulerian location, in view of the effect of heat transport (which generates asytmnetric profiles far downstream from the bifurcation) and the viscous dissipation taking place in the sheared fluid layers. Soon the temperature profile starts exhibiting a wellknown "seagull" shape. After a while, heat diffusion starts exerting an increasing cooling influence, especially when the local flow rate is low. Figure 18(a) shows the temperature field at the end of the filling, in different layers across the thickness, each identified by its relative distance to the midsurface. In the core (layers 3 to 5), temperature is relatively uniform and has a value close to injection temperature. Only at very few locations is this observation not valid, viz. in stagnation areas (visible at the comer in layers 2 and 6 of the upstream branch) and in the region filled last, where the temperature is higher due to viscous heating (see layer 3 of the vertical branch). On the other hand, near to the skin, where the cold mold effect is higher, temperature distribution appears more or less symmetric in the downstream branches, except in the bifurcation vicinity. This is easily understood in view of the flow and temperature profiles splitting at the bifurcation, where the inner skin layers are fed with a much hotter material than on the outer side. In fact, the asymmetry tends to increase when the front moves away, but it falls in competition with viscous heating or cooling. Finally, between the core and the skin, a noticeable effect is the temperature increase induced by viscous heating in layers 2, 3, 5, 6 of the thin vertical branch, as a consequence of the sharp increase in velocity occurring after the thicker branch has been filled. Also, the temperature profile appears to be highly nonsymmetric in these layers, due to the kinematic bifurcation. This effect is also visible in the horizontal downstream branch (layers 2, 3, 5, 6). All these observations are confirmed by results obtained for other flow rates (Figure 18(b)).
5.2 Experimental validations Experimental validation has been a constant concern since the beginning of injection molding numerical modelling. As the melt flow is by nature unsteady, non-isothermal and 3D, while the fluid is compressible and non-Newtonian, few analytical solutions exist and their use is highly restricted. Therefore, laboratory experiments are necessary to validate the models. In a 1~t step, experimental flow fronts and gate pressure measurements in 1D filling were compared to numerical predictions [35b,40a,42]. In the early 80's, the developmem of more sophisticated tools allowed to predict and validate the 2D front shape and pressure evolution
990
Sprue
(a)
Pressure
Heat flux
r
Optical fibers / Pressure - ~ ~ . . ~ e a t flux
Gate
--- ~ ~ _ _
pos. 5 pos. 4 pos. 3 pos. 2 pos. 1 after gate
gate (b)
/
Ill Ill -111 -la qi qH
57 9
I I
\ I
qll
17
51
365
Figure 19. (a) Sketch of the injected plate with a close-up of the gate and the locations of the sensors. (b) Finite element mesh representation (where use is made of the symmetry of the part), together with the thickness distribution. in the cavity [43,46,66]. Later, important steps consisted in introducing accurate characterization of the viscosity and other material properties, while extending the model to the process packing and cooling stages [52,56b]. Nowadays, experimental validation focuses on the prediction of the part's final properties (such as density, residual stresses, shrinkage and warpage), in some cases taking viscoelastic effects into account [60b,64,132,137]. It is clear that precise on-line measurements and material characterization, together with rigorous numerical tools, are required to obtain accurate validations. As an example, injection molding experiments were performed in the laboratories, and with the help and expertise of the staff of the "P61e Europ6en de Plasturgie" of Oyonnax (France), using a 420-ton hydraulic Billion press. The mold cavity was a 367x102x4 mm rectangular plate (Figure 19). In order to ensure linear filling, the gate was designed
991 311 309 ~ 307 305 ~
303 301 299 297 295 0
..................................... 0.2 0.4 0.6 0.8 1 1.2 1.4
1.6 1.8 2 time (s)
Figure 20. Filling of a rectangular PS plate. Comparison between predicted ( . ) and measured (---) mold surface temperatures for the first sensor in the cavity. with a large rib. The delivery system consisted of a 11 cm sprue and a 32.5 cm trapezoidal nmner. The steel mold temperature was regulated by water circulation in channels situated 10 mm inside the walls. The mold was equipped with transducers and optical fiber sensors. Pressure transducers were positioned at the inlet of the sprue, just upstream and downstream from the linear gate, and in the plane of symmetry of the cavity (40, 115, 189 and 264 mm downstream from the inlet). Heat flux transducers consisted of pairs of thermocouples separated by a steel cylinder (one positioned at the mold surface and the other 7 mm inside the mold), which could produce a very fast response time of 0.2 ms with a precision of 1 K. Heat flux transducers were placed at the inlet of the runner, just upstream and downstream from the gate, and in the cavity at distances of 40, 115, 189, 264 and 319 mm downstream from the inlet, and 20 mm from the plane of synmaetry. Although different polymers were injected, only the results obtained with PS (Dow Styron 678E, for which the material parameters are given in [20,25,116]), will be detailed. The injection and mold temperatures were 483 K and 294 K, respectively, while the injection flow rate was 250 cm3/s. The evolution of the mold surface temperature is presented in Figure 20. As filling progresses, the temperature at the sprue rises first, followed by all the sensors. The increase is sudden and accurately records the front motion. After the front has passed, the polymer-wall interface temperature remains approximately constant during the remainder of the filling. This effect, which is a direct consequence of the thermal shock theory presented in Section 3.2.2, was obtained in both the numerical predictions and the experiments, and for materials as different as PS, PC, PP and
992
Figure 21. Successive flow fronts obtained from simulation with a very simple runner and gate model (top), and from short shot experiments (bottom). PET, thereby validating the thermal model of Mal et al. [25,26]. Also, this shows that in-plane heat conduction within the polymer and the walls is negligible, whereas the effect of the thermal singularity caused by fountain flow is important. Short shot experiments were performed in order to visualize the flow front pattern (Figure 21). It is obvious that the gate, which was designed to ensure linear injection into the cavity, is not efficient. The polymer first enters the cavity center, before reaching the lateral extremities of the thick rib. Later on, a linear front forms, while the polymer flows faster close to the lateral side walls. This finally leads to a 'V'front shape, which grows when the flow rate is increased. Verhoyen [20] has proved that this effect, which sometimes generates a front weldline at the end of the filling and was observed with all other injected polymers, is a direct consequence of the non-uniform viscous dissipation taking place in the runner during filling, leading to a non-uniform and transient gate temperature profile. The resulting non-uniform fluidity in the cavity is responsible for the flow front 'V' shape (Figure 21). -.
993
6.
EXTENSIONS OF THE M O D E L
6.1 Chemical reactions In Thermoplastics Injection Molding (TIM), solidification results from the cooling, below the glass transition temperature Tg, of the injected polymer in
contact with the cold cavity walls. In reaction and thermoset injection molding (cf. Macosko [5], Kamal and Ryan [6], and the references therein), the irreversible curing reaction of a resin is used to build the solid stiff part. While reaction injection molding usually refers to mixing activated systems (such as polyurethanes or epoxies), thermoset injection molding refers to heat activated systems (such as polyesters). In this section however, the terminology of RIM is used in the general sense. The advantages of RIM compared to TIM are numerous. The low viscosity of thermosetting resins before curing allows one to have high fiber reinforcement contents, as needed for good mechanical properties, and also the injection of large parts (of several m 2) on classical machines. In addition, the solidification resulting from curing is easily controlled by changing the resin composition and using catalysts or inhibitors. The same modelling approach used in TIM may be used to analyze RIM provided some particularities are taken into account. First, additional thermodynamic variables must be introduced to represent the evolution of the resin composition. Each material property of the melt becomes a function of p, T, possibly ~', and of the different species mass fractions w i . Neglecting species diffusion, each
toi is governed by an equation of the form
Dooi/Dt-fi(p,T,y,to~),
where fi is the kinetics of species production. Secondly, as curing reactions are exothermic, reaction heat must be incorporated in the energy equation. Hence, the numerical procedure presented in Section 4.2 may easily be extended to simulate RIM processes. To this end, T and toi are solved simultaneously, thereby ensuring stability in this strongly coupled problem [25]. The modelling of RIM has been the object of constant attention in the literature. While simplified models were used in earlier studies [83], a major concern has been to understand the space-time distributions induced by fountain flow and their effect on the curing reaction. This problem was intensively investigated by Macosko and co-workers [4,32,49, 58], see also [84]. Other modelling issues relate to the effect of inertia when the viscosity of the injected polymer is very low [85], or to the molding of highly viscous reactive polymers [72,86]. A precise determination of the material properties of the reactive mixture is of primary importance for accurate simulation [3,5,6]. While the influence of the reaction on p and k can often be neglected, its effect on 7/ must generally be
994
(a)
z = h
0.5
z - 0
(b)
Figure 22. Reaction injection molding of an automobile front hood (of lm x lm x 3mm). Distribution of the degree of cure and temperature at end of filling (after 17 s) (a) on the wall; (b) along the midplane of the cavity. (From [25]). taken into account, especially in the vicinity of the so-called gel point where r/ rapidly tends to infinity. Even more critical is the accuracy of the model used for reaction kinetics. When the resin formula may be considered fixed, simple n thorder models can often be used, and all the information on the reaction is lumped into a single variable called the degree of cure. When the influence of a modification in the resin composition is to be investigated, mechanistic models are necessary. A most interesting and widely used example of this approach is the model developed by Stevenson [138] for unsaturated polyester, where the influences of initiator and inhibitor concentrations and types on the reaction are considered. This model is used in the example presented in Figure 22 (Mal [25]). In order to reduce computational time, use is made of the part symmetry in the simulations. The injection temperature is 356 K, the heating channels being kept at 423 K in order to activate the curing reaction. 6.2 P o l y m e r crystallization
For semi-crystalline polymers, the thermo-mechanical conditions prevailing during the different stages of the process interact with the development of micro-
995
structures that in turn affect the product end-use properties [139-147]. Polymer crystallization has been extensively studied and several kinetic models developed [24,148-151], most of them based on the Kolmogorov-Avrami-Evans approach [152,153]. However, little work has been performed to introduce these models into injection molding simulation. Dufoss6 [154] investigated the effect of nonisothermality on PE molding. The effect of shearing on PPS crystallization was analyzed by Hsiung et al. [87,91 ], who extended the Hele Shaw model to include crystallization by means of the Nakamura model [155]. More recently, Isayev et al. [88] simulated PP molding, taking shearing influence into account by combining the models of Liedauer et al [156] and Nakamura et al. [155]. The combined effects of temperature, shear and pressure were analyzed by Ito et al. [92] and Verhoyen et al. [20,93], in order to predict PP and PET molding. Crystallization models can be implemented in Hele Shaw simulation by using the scheme of Section 4.2. This requires (i) to introduce a kinetics model in order to integrate the crystallization degree a v as an additional 3D field, and (ii) to consider the effect of crystallization on r/, k, p and H. The goal of the present section is to summarize the investigations carried out by Verhoyen et al. [20,93], showing that combining crystallization modelling and simulation provides a very convenient methodology to better understand the crystallization mechanisms under high pressure and shear conditions. 6.2.1 The crystallization model
The model developed in [20,93] combines the consecutive Avrami model [19, 24] with the approach of Schneider et al. [157] and Andreucci et al. [158]. The former is an extension of the Avrami theory, which takes into accotmt induction time, primary and secondary crystallization, ultimate degree of crystallinity, and the effects of temperature and shear stress. The latter allows to separate the nucleation and crystal growth processes. In the sequel, subscripts 1 and 2 refer r to the primary and secondary mechanisms, while avi denotes the relative degree of crystallinity of mechanism (i). The kinetics model reads as DO1 Dt
-- 1 / t ~ ,
I DN1 = 1V Dt ' DS1 = 8 ~ G R 1 ,
LN
002 Dt
-lit
(104)
2 ,
DR1 = G N 1 Dt DV1 = G S 1 , Dt
if 0 1 > 1 ,
else
DR1 = O , Dt
(105)
996
) 9
Q 9 (a)
9
| " (
(b)
~':
Figure 23. Sketch of the crystallization model. Interactions affect the nucleation and growth of real spherulites, but have no effect on the fictive crystals.
Da~Cl c Dt = ( 1 - a V l ) DavC2 : Dt
DV~
Dt
(,
)( (106)
if Da v - Otv ( Wl DavlC + (1Dt Dt
W1
02>1 ,
else D C-aw = O, Dt
~ c) j Dav2 Dt
The parameters t oi represent the isothermal induction times, while nonisothermal induction times are calculated from the accumulated factors 0i [159], which reach 1 when induction is complete and the associated mechanism starts. Equations (105) refer to the model of [157,158] : the time derivative of the specific volume VI of the fictive nuclei is the product of their specific surface S 1 by the growth rate G, while DS1/Dt is 4~r times the product of the fictive specific diameter 2R 1 by G, and DR 1/Dt is the product of the specific number of fictive nuclei N 1 by G; the evolution of N 1 is governed by the nucleation rate /~ (Figure 23). As the fictive specific volume (V1) is also the fictive crystallinity degree, the Avrami postulate provides the relative crystallinity degree
C Ctvl
of the first mechanism. C
The relative degree for secondary
crystallization av2 is calculated by means of the differential form of the Avrami equation, with k 2 and n 2 as kinetic parameter and Avrami index. The real degree of crystallinity is calculated by combining the 1st and 2 nd mechanisms
997 (with the weight W 1 ), taking into account the asymptotic degree of crystallinity oo
tzv . The model of [157,158] was not used for the secondary mechanism, whose physical behavior is totally different. It is assumed that nuclei pre-exist in the melt, and that, in static conditions, the number of activated nuclei /V1 is independent of time (with instantaneous nucleation). Hence, /V1 is a given function of T, ~' and p, which can be determined experimentally. When the material undergoes a complex thermo-mechanical history, /V is obtained from the relation /V - max (D1Vm(t)/Dt, 0), in order to avoid any decrease of N 1 when the polymer is heated. This expression should be improved. This model can predict PET crystallization in typical laboratory apparatus (such as DSC or cone and plate rheometer) in both isothermal and non-isothermal conditions [20,93]. In principle, the dependence of PET material properties on txv should be specified. Detailed material functions are given in [20]. However, these effects may be neglected in PET injection molding since crystallization develops mainly during the packing and cooling stages.
6.2.2 Experimental and numerical results Injection molding experiments were performed using the equipment described in Section 5.2. The half-thickness of the plate was 3 ram. Processing conditions are detailed in [20]. The simulation was peformed by imposing a gate flow rate of 60 cm3/s during filling, while the experimental pressure evolution was prescribed during packing and cooling until the gate was frozen. A non-uniform wall temperature (from 293 K to 303 K) was imposed, corresponding to wall sensor measurements performed at the beginning of the filling stage. The results obtained by prescribing a uniform inlet temperature of 568 K are depicted in Figures 24(a) and 24(b), where the predicted and experimental crystallinity profiles are represented at two locations in the plate, together with the profiles of N 1 . The results obtained by taking only thermal effects into account, and only thermal and shear effects, are shown for comparison purposes. In all cases, a zero a v is predicted along the walls, indicating a quench of the polymer when it meets the walls, while a uniform crystalline core is obtained only if pressure-induced crystallization is taken into account (when only the thermal effect is considered, the core crystallinity reaches 3 %, which is far from the experimental 28%; when the shear stress effect is added, no significant difference is observed). However, the predicted and experimental crystallinity profiles do not fit well in Figure 24(a). As explained in Section 5.2, this results from the important temperature increase observed in an annular region located at the inlet of the gate and
998 0.6 i
(a)
II
0.4 "~
-~
~
i
"'
~ .
.
I
.
I
.
.
.
.
.
'"
.
I
.
~
cr
k1~6l ~O: ,
0
.............................. ..-.................. x .............. :.....-.........::...........................
rf/
0.3
.....
1020
i
-"-~176 -""--~
\~1014
0.2 r,.)
~.
0.1
I
0.0
__.~ _=_- : ...............k ........... : - ~----'~""i'~"-'" I I I -2 -1 0 1
-3
~176 ~
....- ,L. ! % ..
0.6
%e
.
.
~
"] 1012
.
.
.
1020
.
0.5 "~ o ,,,,,~
o.2
b ro
1018 101s
-/"
0.1 -
~,,~.
I" o~
.--
-.:.-:..:...-...~...(....-..-'=.....':...=...--"
?~1 \ v4
-
i
i
i
...............t..':..___.. L/
-1
0
1
2
3
1012 c_.._l
101~
0.6
102o
0.5
10 TM
(c)
0.4
, ,...r
r~
b
0.2
1016
t
0.1 L _ ~ ~ r 0.0 I -3
~
O
1014
0.0 t.7 .... _:..r ............ _1 -2
exl)
z cur
0.4 0.3
:tt:
~..o %LA .......~ - . . ~ . % - ~ l 10 lo 2 3
(b)
"9
~
i
% -
I
-2
-1
......
I.....
I
0
1
-
Z
0
1014 c~ i,,,~ 9
1012 L--..-I
101~
~
2
3
z [mm]
Figure 24. Gapwise crystallinity profiles at distances of 55 mm (a and c) and 310 mm (b) from the gate in the symmetry plane. Comparison between experiments (e) and numerical predictions with (a and b) uniform and (c) improved inlet temperature profiles. Dotted lines" only thermal effects considered; dashed lines" only thermal and shear effects; solid lines : all effects. (From [20]).
999 due to viscous heating in the nmner. Since nuclei are generated mainly upstream from the delivery system (i.e. where p is the highest), they are partially destroyed in the nmner under the action of the high peripheral temperature. Hence, their inlet distribution is not uniform, and this strongly affects crystallization. Also, the non-uniform inlet temperature profile again had a major impact on the front shape. A simple model was developed to evaluate the distribution of nuclei downstream from the gate [20]. The resulting profiles are shown in Figure 24(c), which exhibits a much better agreement between calculation and experiments.
6.3 Filling of fiber mats SRIM and RTM differ fundamentally from classical injection molding. Usually, in these processes, a thermosetting resin is injected into a cavity where a fiber reinforcement has been placed beforehand [7]. Using SRIM or RTM, it is possible to achieve, in one shot, complex composites with a high ratio of continuous fibers and excellent mechanical properties, while the low resin viscosity (< 0.1 Pa.s typically) allows the manufacturing of large parts (of several m 2) with low cost tools. In addition, good orientation control is obtained. Although the reinforcement considerably modifies the physics, the same simulation tool developed for TIM and RIM may be used to simulate SRIM and RTM provided some particularities are taken into account [21,25,80,96,97,100,133, 160-163]. The equations must be established via a multiphase analysis, in which the liquid flows through a solid porous medium (cf. Tucker and Dessenberger [8] and the references therein). For an incompressible Newtonian resin, stationary fiber reinforcement, negligible inertia and long range viscous forces, the resin flow is governed by the Darcy law, which relates the intrinsic fluid velocity v i to the pressure gradient O~/OqXi ,
v i = - / ~ ~ Kij Op/Oxj
,
(107)
where q~ is the porosity (i.e. the liquid volume fraction), 7"/ is the resin viscosity and Kij the permeability tensor. The Darcy law is useless in describing how the resin penetrates the bundles and wets each individual fiber. As in TIM and RIM, most SRIM or RTM parts are thin (see Section 3.1). Dimensional analysis thus leads to combining the mass and Darcy equations as
c~xa
1000
h
where
S ao -
I / ~ KaO dz .
(109)
-h
Greek indices mean that the quantity is limited to its in-plane components. Equation (109) has the same form as (42), the fluidity factor S being replaced by a fluidity tensor Sag. The procedure outlined in Section 4.2 is thus applicable, provided minor modifications of the front tracking algorithm in singular regions are introduced [21,25]. The in-plane permeability Ka# is generally determined by eigenvector-eigenvalue measurements in simple flows [7,97]. In addition, progress has been made in relating the reinforcement geometry and draping with Ka~ (taking into account in some cases the elasticity of the preform) [164-170]. To illustrate the effect of permeability, the injection of an automobile front hood has been simulated using two different kinds of reinforcement (Figure 25). The filling time is 141 s. Crosses indicate where vents should be placed. This result shows that permeability is a key factor in determining the geometrical and operating parameters for a given part. It should be noted that, as perfect draping along the borders or the junctions is difficult, high permeability zones are often found, resulting in easy flow channels or race tracking [162,171-173]. In SRIM and RTM, solidification generally results from a chemical reaction initiated by heat or mixing during filling. In order to reduce the cycle time, the mold walls are set to a high temperature so as to enhance this mechanism; hence, thermal effects must be taken into account in the model [ 100,102]. However, the heat transfer in the porous medium is much more complex than in Section 3.1, since hydrodynamic dispersion must be considered due to the tortuous nature of the flow paths. Assuming local thermal equilibrium between the fiber bundles and the resin, the energy equation [99,101] reads as :
(op, c,
p,c, v, O [Kij+Kii~o~I"
1 ap
@
(11o)
where P ph and Cph denote the specific mass and specific heat (with subscripts l and s referring to the liquid and the solid), Ki~ and Kia are the thermal conductivity and mechanical dispersion tensors, and ~ is the specific heat source due to chemical reactions. In (1 lO), K~ is a function of the conductivity
1001
(a)
(b)
(c) (d) Figure 25. Isothermal simulation of the filling of an automobile front hood" (a) finite element mesh; (b) example of temporary mesh; (c and d) evolutions of the flow front for a non-isotropic (Kxx/Kyy - 3 ) and an isotropic reinforcement. of the two phases and the reinforcement geometry [21,174], while
KiJ, which
formally accounts for the mixing convection heat transfer, is often considered to be proportional to Mal et al. [21] propose the equation"
Uv,II
Kid where
=" ~Pl Cl IIVml1-1 Zijkl Vk Vl ,
Lijkt
(111)
is a fourth order tensor. This relation can induce different mixing
effects parallel or perpendicular to the flow, in agreement with experimental observations [175]. Also, it is observed that hydrodynamic dispersion is invariant with respect to time scale changes, corresponding to a purely geometrical mixing model. As for Kij, the symmetries required for Lijkl are not clear. A complex dimensional analysis shows that, in SRIM and RTM, gapwise mixing is
1002
t = 70s (end)
31530~ 32.5~ 335~/'~ (a)
t = 70s (end~ " "
305 ,,,-~ 315 325/'~~ (b)
3
t = 141s
(end)
305
,,,,315~-,,7"~
33~.~@r" 3 4 ~
4
t = 141s
(c)
(end)
~
...315~
305
33~~, / 345~~~~,,~
5>
(d)
Figure 26. Simulation of the non-isothermal filling of an automobile front hood. Average temperatures (in K) at end of filling. Top : filling time = 70 s; bottom: filling time = 141 s; (a, c) no dispersion; (b, d) Lzz = 0.0017 mm. (From [21]). predominant when compared with in-plane contributions, and
tijkl
V k. v t
thus
has a single non-negligible component Lzzal3 v a v/3. ff it is further assumed that Lzzoc~
is transverse isotropic [21] ( L z z a o - L z z tSaO), the equations involve a
single mixing coefficient Lzz and are equivalent to the model of Tucker and Dessenberger [8]. Thanks to the material data provided in [99], realistic numerical experiments may be carded out. In particular, the non-isothermal filling of the automobile front hood of Figure 25 has been simulated by Mal et al. [21 ] in order to compare
1003
the results obtained with and without neglecting mechanical dispersion (Figure 26). Even though Lzz is small, its effect on the average temperature distribution leads to a difference of up to 10 K in the entry zone when filling is faster. It is essentially in the immediate vicinity of the gate that its effect is important, because mechanical mixing is very effective in the regions where thermal gradients and velocities are high. A complex balance between dispersive and conductive heat transfer takes place during filling and, the higher the flow rate, the less effective conductive heat transfer is (while mixing acts on higher thermal gradients). In conclusion, the global heat transfer is increased in the vicinity of the gate, while it is reduced in regions of lower velocities. Two closing remarks are in order. First, chemical reactions can be modelled using the methodology of Sections 4.2 and 6.1, as long as the kinetics and material data are known. The evolution equation of species concentration is similar to (110), with an additional term relating to curing kinetics, while molecular diffusion can generally be neglected and mechanical dispersion is governed by the tensor Lijkt. Secondly, it is easy to extend to SRIM and RTM the theory of abrupt changes of thickness and bifurcations developed in Sections 3.2.3 and 3.3.3. In the case of isothermal flow, tmiform material properties and a transverse isotropic permeability, mass conservation exactly provides (81), while pressure continuity across the singular line forces the continuity of the tangent velocity
(v2f +- v2f-),
instead of (82) (see Figure 12). At the front, equation
(83) remains valid and thus, after a few calculations, (81) and (83) show that, when h + ~ h - , the normal velocity component vanishes at the flow front +
(Vlf -Vlf
-0).
This simple effect, which holds true in the case of bifurca-
tions, can be observed in practice. purposes. 7.
It can also be exploited for computational
CONCLUDING REMARKS AND A C K N O W L E D G M E N T S
This chapter summarizes the models and numerical techniques that were elaborated to develop the MOLDSYS simulation software. Combining sound physical knowledge with rigorous mathematical approximations and accurate algorithms proves efficient in addressing the main issues of process modelling. Various polymer molding techniques, all based on the thin cavity assumption, have been implemented in the simulation program. The numerical method, which is based on front tracking, automatic remeshing and extrapolation (taking fountain flow into account), has been successfully extended to the filling of complex parts.
1004
Results prove the validity of this approach. Also, it is shown that the simulation tool can be used to better understand the tmderlying physics when complex materials are molded. Besides the authors of this chapter, several people participated in this research, including Marcel Crochet, Luc Dheur, Olivier Hansen, Kali Kabanemi, Natasha Van Rutten and Vincent Verleye, whom the authors wish to thank here for their contributions. The work was carded out within the framework of the European BRITE project RI1B-0087-F(CD), the "Multimat6riaux" project of the Walloon Region of Belgium, and the program of Interuniversity Attraction Poles of the Belgian state, and in collaboration with the Shell Research and Technology Center in Amsterdam (the Netherlands) and the "P61e Europ6en de Plasturgie" of Oyonnax (France). The experimental work was possible thanks to Rapha61 Favier's permission. Grants from the IRSIA (Belgium) and the FRIA (French Commtmity of Belgium) are acknowledged. The authors wish to thank JeanPierre Gazonnet, G6rard Dechavanne and Virginie Durand for their friendly help and advice in performing the molding experiments. The efficient page setting work of Victor Vermeulen was also appreciated. REFERENCES
1. Z. Tadmor and C.G. Gogos, Principles of Polymer Processing, John Wiley & Sons, 1979. 2. J.R.A. Pearson, Mechanics of Polymer Processing, Elsevier, London, 1985. 3. C.L. Tucker, in : A.I. Isayev (ed.), Injection and Compression Molding Fundamentals, Marcel Dekker, New York, 1987. 4. T.A. Osswald and S.C. Tseng, in : S.G. Advani (ed.), Flow and Rheology in Polymer Composites Manufacturing, Elsevier, Amsterdam, 1994. 5. C.W. Macosko, RIM, Fundamentals of Reaction Injection Molding, Hanser, Munich, 1989. 6. M.R. Kamal and M.R. Ryan, in : A.I. Isayev (ed.), Injection and Compression Molding Fundamentals, Marcel Dekker, New York, 1987. 7. C.D. Rudd, A.C. Long, K.N. Kendall and C.G.E. Mangin, Liquid Moulding Technologies, Woodhead publ., Cambridge, 1997. 8. C.L. Tucker and R.B. Dessenberger, in 9 S.G. Advani (ed.), Flow and Rheology in Polymer Composites Manufacturing, Elsevier, Amsterdam, 1994. 9. F. Dupret and L. Vanderschuren, AIChE J., 34 (1988) 1959. 10. A. Couniot, L. Dheur, O. Hansen and F. Dupret, Proc. Numiform'89, Balkema, Rotterdam (1989) 235. 11. L. Vanderschuren, Ph.D. Thesis, Universit6 catholique de Louvain, 1989.
1005 12. F. Dupret and L. Dheur, Proc. Oji Int. Seminar, Tomakomai, Japan (1990), Hemisphere (1992) 583. 13. A. Couniot, Ph.D. Thesis, Universit6 catholique de Louvain, 1991. 14. K.K. Kabanemi and F. Dupret, Proc. Numiform'92, Balkema, Rotterdam (1992) 357. 15. K.K. Kabanemi, Ph.D. Thesis, Universit6 catholique de Louvain, 1992. 16. L. Dheur, Ph.D. Thesis, Universit6 catholique de Louvain, 1992. 17. A. Couniot, L. Dheur and F. Dupret, IMA Conf. Series, New Series Nr. 42, Clarendon Press, Oxford (1993), 381. 18. V. Verleye, Ph.D. Thesis, Universit6 catholique de Louvain, 1995. 19. O. Verhoyen, F. Dupret and R. Legras, Proc. Numiform'95, Balkema, Rotterdam (1995) 1209. 20. O. Verhoyen, Ph.D. Thesis, Universit6 catholique de Louvain, 1997. 21. O. Mal, A. Couniot and F. Dupret, Composites Part A, 29A (1998) 189. 22. N. Van Rutten and F. Dupret, Proc. ASME'97, Dallas, Texas, MD-Vol. 79 (1997) 399. 23. O. Verhoyen and F. Dupret, J. Non-Newtonian Fluid Mech., in press (1998). 24. O. Verhoyen, F. Dupret and R. Legras, Polym. Eng. Sci., in press (1998). 25. O. Mal, Ph.D. Thesis, Universit6 catholique de Louvain, 1998. 26. O. Mal, L. Dheur, P. Pirotte, N. Van Rutten, A. Couniot and F. Dupret, Proc. Numiform'95, Balkema, Rotterdam (1995) 1165. 27. Z. Tadmor, J. Appl. Polym. Sci., 18 (1974) 1753. 28. H. Janeschitz-Kriegl, Rheol. Acta, 16 (1977) 327; 18 (1979) 693. 29. W. Dietz, J.L. White and E.S. Clark, Polym. Eng. Sci., 18 (1978) 273. 30. A.I. Isayev and C.A. Hieber, Rheol. Acta, 19 (1980) 168. 31. H. Van Wijngaarden, J.F. Dijksman and P. Wesseling, J. Non-Newtonian Fluid Mech., 11 (1982) 175. 32. J.M. Castro and C.W. Macosko, AIChE J., 28 (1982) 250. 33. S.M. Richardson, Rheol. Acta, 22 (1983) 223. 34. C.L. Tucker and F. Folgar, Polym. Eng. Sci., 23 (1983) 69. 35. M.R. Kamal and S. Kenig, Polym. Eng. Sci., 12 (1972) 294; 302. 36. J. Vlachopoulos and C.K.J. Keung, AIChE J., 18 (1972) 1272. 37. J.L. Berger and C.G. Gogos, Polym. Eng. Sci., 13 (1973) 102. 38. P.-C. Wu, C.F. Huang and C.G. Gogos, Polym. Eng. Sci., 14 (1974) 223. 39. Z. Tadmor, E. Broyer and C. Gutfinger, Polym. Eng. Sci., 15 (1975) 515. 40. G. Williams and H.A. Lord, Polym. Eng. Sci., 15 (1975) 553; 569. 41. E. Broyer, C. Gutfinger and Z. Tadmor, Trans. Soc. Rheol., 19 (1975) 423. 42. Y. Kuo and M.R. Kamal, AIChE J., 22 (1976) 661. 43. W.L. Krueger and Z. Tadmor, Polym. Eng. Sci., 20 (1980) 426.
1006
44. M.E. Ryan and T.-S. Chung, Polym. Eng. Sci., 20 (1980) 642. 45. C.G. Gogos, C.-F. Huang and L.R. Schmidt, Polym. Eng. Sci., 26 (1986) 1457. 46. A. Couniot and M.J. Crochet, Proc. Numiform'86, Balkema, Rotterdam (1986) 165. 47. M. Grmela, Polym. Eng. Sci., 24 (1984) 673. 48. H. Mavridis, A.N. Hrymak and J. Vlachopoulos, Polym. Eng. Sci., 26 (1986) 449. 49. I. Manas-Zloczower, J.W. Blake and C.W. Macosko, Polym. Eng. Sci., 27 (1987) 1229. 50. D.J. Coyle, J.W. Blake and C.W. Macosko, AIChE J., 33 (1987) 1168. 51. R.A. Behrens, M.J. Crochet, C.D. Denson and A.B. Metzner, AIChE J., 33 (1987) 1178. 52. C.A. Hieber, in : A.I. Isayev (ed.), Injection and Compression Molding Fundamentals, Marcel Dekker, New York, 1987. 53. N. Famili and A.I. Isayev, in : A.I. Isayev (ed.), Modeling of Polymer Processing, Recent Developments, Hanser, Munich, 1991. 54. H. Mavridis, A.N. Hrymak and J. Vlachopoulos, J. Rheology, 32 (1988) 639. 55. H. Mavridis, A.N. Hrymak and J. Vlachopoulos, AIChE J., 34 (1988) 403. 56. H.H. Chiang, C.A. Hieber and K.K. Wang, Polym. Eng. Sci., 31 (1991) 116; 125. 57. F.P.T. Baaijens, Rheol. Acta, 30 (1991) 284. 58. M.A. Garcia, C.W. Macosko, S. Subbiah and S.I. Gii~eri, Intem. Polym. Proc., 6 (1991) 73. 59. C.A. Hieber and H.H. Chiang, Polym. Eng. Sci., 32 (1992) 931. 60. A.A.M. Flaman, Polym. Eng. Sci., 33 (1993) 193; 202. 61. K.T. Nguyen and M.R. Kamal, Polym. Eng. Sci., 33 (1993) 665. 62. K.K. Kabanemi, A. Mt-Kadi and P.A. Tanguy, Rheol. Acta, 34 (1995) 97. 63. R.-Y. Chang and S.-Y. Chiou, Polym. Eng. Sci., 35 (1995) 1733. 64. L.F.A. Douven, F.P.T. Baaijens and H.E.H. Meijer, Prog. Polym. Sci., 20 (1995) 403. 65. S.C. Chen and Y.C. Chen, J. Appl. Polym. Sci., 55 (1995) 1757. 66. C.A. Hieber and S.F. Shen, J. Non-Newtonian Fluid Mech., 7 (1980) 1. 67. S.-F. Shen, Int. J. Num. Meth. in Fluids, 4 (1984) 171. 68. T.-S. Chung, Polym. Eng. Sci., 25 (1985) 772. 69. H. Alles, S. Philipon, J.-F. Agassant, M. Vincent, G. Dehay and P. Lerebours, Polym. Proc. Eng., 4 (1986) 71. 70. M.R. Kamal, in :L.T. Manzione (ed.), Applications of CAE in Injection Molding, Hanser, Munich, 1987.
10o7 71. K.K. Wang and V.W. Wang, in : A.I. Isayev (ed.), Injection and Compression Molding Fundamentals, Marcel Dekker, New York, 1987. 72. M. Sobhanie and A.I. Isayev, in : A.I. Isayev (ed.), Modeling of Polymer Processing, Recent Developments, Hanser, Munich, 1991. 73. B.S. Chen and W.H. Liu, Polym. Eng. Sci., 29 (1989) 1039; 34 (1994) 835. 74. S. Subbiah, D.L. Trafford and S.I. GiJ~eri, Int. J. Heat Mass Transfer, 32 (1989) 415. 75. P.A. Tanguy and R. Lacroix, Int. Polym. Process., 6 (1991) 19. 76. E.W. Liang, H.P. Wang and E.M. Perry, Adv. Polym. Tech., 12 (1993) 243. 77. B. Friedrichs and S.I. Gii~eri, J. Non-Newtonian Fluid Mech., 49 (1993) 141. 78. H.H. Chiang, K. Himasekhar, N. Santhanam and K.K. Wang, J. of Eng. Mater. and Tech., 115 (1993) 37. 79. P. Kennedy, Flow Analysis of Injection Molds, Hanser, Munich, 1995. 80. T.J. Wang, L.J. Lee and W.B. Young, Int. Polym. Process., 10 (1995) 82. 81. V.R. Voller and S. Peng, Polym. Eng. Sci., 35 (1995) 1758. 82. E. Pichelin and T. Coupez, to be published in Comput. Meth. Appl. Mech. Eng. (1998). 83. E. Broyer and C.W. Macosko, AIChE J., 22 (1976) 268. 84. J.M. Castro, Polym. Eng. Sci., 32 (1992) 715. 85. R.K. Mohammed, T.A. Osswald, T.J. Spiegelhoff and E.M. Sun, Intern. Polym. Proc., 9 (1994) 279. 86. G.A.A.V. Haagh, G.W.M. Peters and H.E.H. Meijer, Polym. Eng. Sci., 36 (1996) 2579. 87. C.M. Hsiung and M. Cakmak, Polym. Eng. Sci., 31 (1991) 1372. 88. A.I. Isayev, T.W. Chang, K. Shimoyo and M. Gmerek, J. Appl. Polym. Sci., 55 (1995) 807; 821. 89. G. Titomanlio, V. Speranza and V. Brucato, Intern. Polym. Process., l0 (1995) 1. 90. T.D. Papathanasiou, Chem. Eng. Sci., 50 (1995) 3433. 91. C.M. Hsiung, M. Cakmak and Y. Ulcer, Polymer, 37 (1996) 4555. 92. H. Ito, K. Minagawa, J. Takimoto, K. Tada and K. Koyama, Intern. Polym. Proc., 11 (1996) 363. 93. O. Verhoyen, N. Van Rutten, R. Legras and F. Dupret, in preparation. 94. C.C. Lee, F. Folgar and C.L. Tucker, J. of Eng. for Industry, Trans. ASME, 106 (1984) 114. 95. T.A. Osswald and C.L. Tucker, Int. Polym. Process., 5 (1990) 79. 96. F. Trochu, R. Gauvin and D.-M. Gao, Adv. Polym. Tech., 12 (1993) 329. 97. S.G. Advani, M.V. Bmschke & R.S. Pamas, in" S.G. Advani (ed.), Flow and Rheology in Polymer Composites Manufacturing, Elsevier, Amsterdam, 1994.
1008
98. B. Friedrichs and S.I. Gti~eri, Polym. Eng. Sci., 35 (1995) 1834. 99. R.B. Dessenberger and C.L. Tucker, Polymer Composites, 16 (1995) 495. 100. T.J. Wang, R.J. Lin and L.J. Lee, Intern. Polym. Proc., 10 (1995) 364. 101. C. L. Tucker, Polymer Composites, 17 (1996) 60. 102. G. Lebrun, R. Gauvin and K.N. Kendall, Composites Part A, 27 (1996) 347. 103. K. Himasekhar, L.S. Tumg, V.W. Wang, H.H. Chiang and K.K. Wang, Adv. Polym. Tech., 12 (1993) 233. 104. G.W.M. Peters, P.J.L. van der Velden, H.E.H. Meijer and P. Schoone, Intern. Polym. Proc., 9 (1994) 258. 105. C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer, Berlin, 1965. 106. R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1 : Fluid mechanics (2nd ed.), Wiley, New York, 1987. 107. R. Keunings, in : C.L. Tucker m (ed.), Fundamentals of Computer Modeling for Polymer Processing, Hanser, Munich, 1989. 108. A.I. Leonov, Rheol. Acta, 15 (1976) 85. 109. R.K. Upadhyay, A.I. Isayev and S.F. Shen, Rheol. Acta, 20 (1981) 443. 110. A.I. Leonov, J. Non-Newtonian Fluid Mech., 25 (1987) 1. 111. M. Simhambhatla and A.I. Leonov, Rheol. Acta, 34 (1995) 259. 112. A.I. Isayev and R.K. Upadhyay, in : A.I. Isayev (ed.), Injection and Compression Molding Fundamentals, Marcel Dekker, New York, 1987. 113. K. Yasuda, R.C. Armstrong and R.E. Cohen, Rheol. Acta, 20 (1981) 163. 114. C.A. Hieber and H.H. Chiang, Rheol. Acta, 28 (1989) 321. 115. C.A. Hieber and K.K. Wang, Proc. Antec'90 (1990) 943. 116. L. Douven, Ph.D. Thesis, T.U. Eindhoven, 1991. 117. J. Greener, Polym. Eng. Sci., 26 (1986) 886. 118. K.K. Kabanemi and M.J. Crochet, Intern. Polym. Proc., 7 (1992) 60. 119. J.D. Ferry, Viscoelastic Properties of Polymers (3ra ed.), John Wiley & Sons, New York, 1980. 120. M. Van Dyke, Applied Mathematics and Mechanics, Vol. 8, Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964. 121. W. Rose, Nature, 191 (1961) 242. 122. M.R. Kamal, S.K. Goyal and E. Chu, AIChE J., 34 (1988) 94. 123. T. Sato and S. Richardson, Polym. Eng. Sci., 35 (1995) 805. 124. H. Schlichting, Boundary-Layer Theory, McGraw-Hill, 1968. 125. W.F. Zoetelief, Ph.D. Thesis, T.U. Eindhoven, 1995. 126. B.H. Kim and M.C. Ramesh, J. Eng. Industry, 117 (1995) 508. 127. B.H. Lee and B.H. Kim, Polym.-Plast. Technol. Eng., 35 (1996) 147. 128. S. Richardson, J. Fluid Mech., 56 (1972) 609; 102 (1981) 263.
1009 129. V.M. Entov, P.I. Etingof and D.YA. Kleinbock, Euro. J. Appl. Math., 6 (1995) 399. 130. G. Aronsson, Euro. Jnl. Appl. Math., 7 (1996) 417. 131. F. Dupret and J.M. Marchal, J. Non-Newtonian Fluid Mech., 20 (1986) 143. 132. L.W. Caspers, Ph.D. Thesis, T.U. Eindhoven, 1995. 133. H. Aoyagi, M. Uenoyama and S.I. Gtigeri, Intern. Polym. Proc., 7 (1992) 71. 134. C.W.M. Sitters, Ph.D. Thesis, T.U. Eindhoven, 1988. 135. A.N. Brooks and T.J.R. Hughes, Comput. Meth. Appl. Mech. Eng., 32 (1982) 199. 136. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Vol. 1 (4t~ ed.), McGraw-Hill, London, 1989. 137. W.F. Zoetelief, L.F.A. Douven and A.J. Ingen Housz, Polym. Eng. Sci., 36 (1996) 1886. 138. J.F. Stevenson, Polym. Eng. Sci., 26 (1986) 746. 139. M.R. Kantz, H.D. Neuman and F.H. Stigale, J. Appl. Polym. Sci., 16 (1972) 1249. 140. M.R. Kamal and F.H. Moy, J. Appl. Polym. Sci., 28 (1983) 1787. 141. J.P. Trotignon and J. Verdu, J. Appl. Polym. Sci., 34 (1987) 1. 142. C.M. Hsiung, M. Cakmak and J.L. White, Polym. Eng. Sci., 30 (1990) 967. 143. C.M. Hsiung, M. Cakmak and J.L. White, Int. Polym. Proc., 5 (1990) 109. 144. M. Fujiyama, Intern. Polym. Proc., 7 (1992) 84. 145. M. Saiu, V. Brucato, S. Piccarolo and G. Titomanlio, Intern. Polym. Proc., 7 (1992) 267. 146. C.M. Hsiung and M. Cakmak, J. Appl. Polym. Sci., 47 (1993) 125; 149. 147. P. Jerschow and H. Janeschitz-Kriegl, Rheol. Acta, 35 (1996) 127. 148. H. Janeschitz-Kriegl and G. Eder, J. Macromol. Sci.-Chem., A 27 (1990) 1733. 149. I.H. Hillier, J. Polym. Sci., Part A - Polym. Chem., 3 (1965) 3067. 150. N. Billon and J.M. Haudin, Ann. Chim. Fr., 15 (1990) 249. 151. T. Ozawa, Polymer, 12 (1971) 150. 152. M. Avrami, J. Chem. Phys., 7 (1939) 1103; 8 (1940) 212; 9 (1941) 177. 153. U.R. Evans, Trans. Faraday Soc., 41 (1945) 365. 154. C. Dufoss6, Ph.D. Thesis, ENSMP, Sofia-Antipolis, France (1990). 155. K. Nakamura, K. Katayama and T. Amano, J. Appl. Pol. Sci., 17 (1973) 1031. 156. S. Liedauer, G. Eder, H. Janeschitz-Kriegl, P. Jerschow, W. Geymayer and E. Ingolic, Int. Polym. Proc., 8 (1993) 236. 157. W. Schneider, A. Ktippl and J. Berger, Intern. Polym. Proc., 2 (1988) 151.
1010
158. D. Andreucci, A. Fasano, M. Primiciero, M. Paolini and C. Verdi, Math. Mod. & Meth. in Appl. Sci., 4 (1994) 135. 159. W.L. Sifleet, N. Dinos, J.R. Collier, Polym. Eng. Sci., 13 (1973) 10. 160. R. Lin, L.J. Lee and M. Liou, Intern. Polym. Process., 6 (1991) 356. 161. A.W. Chan and S.-T. Hwang, Polym. Eng. Sci., 31 (1991) 1149; 32 (1992) 310. 162. B. Liu, S. Bickerton and S.G. Advani, Composites Part A, 27A (1996) 135. 163. Y.-E. Yoo and W.I. Lee, Polym. Composites, 17 (1996) 368. 164. N. Patel and L.J. Lee, Polymer Composites, 16 (1995) 386. 165. J. Van der Westhuizen and J. Prieur du Plessis, Composites Part A, 27 (1996) 263. 166. S. Ranganathan, F.R. Phelan JR. and S.G. Advani, Polymer Composites, 17 (1996) 222. 167. F.R. Phelan JR. and G. Wise, Composites Part A, 27 (1996) 25. 168. A.C. Long, C.D. Rudd, M. Blagdon and P. Smith, Composites Part A, 27 (1996) 247. 169. W. Chang and N. Kikuchi, Comp. Fluid Dyn., 7 (1996) 49. 170. D. Ambrosi and L. Preziosi, Composites Part A, 29A (1998) 5. 171. C.-J. Wu and L.-W. Houmg, Polym. Eng. Sci., 35 (1995) 1272. 172. D.A. Steenkamer, D.J. Wilkins and V.M. Karbhari, Composites Manufact., 6 (1995) 23. 173. D.R. Calhoun, S. Yalva~, D.G. Wetters, C.-H. Wu, T.J. Wang, J.S. Tsai and L.J. Lee, Polymer Composites, 17 (1996) 251. 174. M. Kaviany, Principles of Heat Transfer in Porous Media, Springer, New York, 1991. 175. R.A. Greenkom, Flow Phenomena in Porous Media, Marcel Dekker, New York, 1983.
1011
F l o w o f P o l y m e r i c Melts in Channels with M o v i n g B o u n d a r i e s A.I. Isayev a, C. Z o o k and Y . Z h a n g alnstitute of Polymer Engineering, The University of Akron Akron, OH 44325-0301, U.S.A.
1. I N T R O D U C T I O N
1.1 Significance of the Problem The development of numerical techniques to accurately approximate the flow of polymer melts in channels with moving boundaries is of paramount importance in polymer processing. In polymer processing, such a flow situation occurs in injection molding, extrusion and simultaneous injection/compression molding. In the case of injection molding, such a flow situation occurs in the non-return valve located on the front of the machine screw. During the injection stage of the molding process, the valve is required to close to stop the flow of polymer melt back into the screw region. Typically, a ring, ball or piston is utilized to close the flow passage into the screw to facilitate this shut-off. In extrusion, varying the geometry of the die by using choker bars or deformable lips allows the control of melt flow to obtain products according to desired specifications. In simultaneous injection/compression molding, polymer enters a mold and is compressed by a moving boundary that is perpendicular to the flow direction. In all of the examples, the boundary moves perpendicular to the dominant flow direction. Current research in our laboratory has focused on the simulation of the nonreturn valve during the molding process. Figure 1 shows a typical non-return valve used in an injection molding machine. This valves has a cylindrical ring which closes during the injection stage to close the passage into the screw. This closure stops polymer melt from flowing back into the screw. During the recovery stage, this ring opens to allow melt to accumulate in front of the valve/screw assembly. This melt will be injected into the mold during the next injection step.
1012
Valve
Ring o.~tflo~.~,( \
Outflow /
_Inflow
Inflow
Figure 1. Sketch of typical ring type non-return valve for the injection molding of thermoplastics. A plot of the pressure traces in front of the valve on the barrel wall (downstream) and behind the valve (upstream) in the screw section and screw displacement during the injection and recovery stages is shown in Figure 2. At the start of injection both pressures increase rapidly. The upstream pressure reaches a steady value, while the downstream pressure in the screw metering section decreases drastically. This decrease in pressure indicates the ring closing the flow passage into the screw. Thus, the valve is closed. The closing time of the valve can then be determined. For the recovery cycle, an oscillating pressure is observed in the upstream pressure transducer. This is caused by the movement of the flights of the rotating screw over the stationary pressure transducer. The downstream pressure measurement has a steady pressure due to the accumulation of polymer melt in the large reservoir (shot size) in front of the valve for the next injection stage. The pressure drop across the valve/screw determines the amount of resistance during the recovery stage, which affects the length of the recovery time. The ability to simulate this process of the valve opening during the recovery stage and closing during the injection stage will determine the time for the valve to open, close and the forces acting on the ring, which will help in the development of more efficient valves [1 ]. In order to understand the phenomena that takes place during the polymer processing with moving boundaries, an experimental slit die with a moving boundary has been designed (Figure 3). The polymer flows through the slit die as a wall closes the flow passage. The transient pressure drop across this wall is measured. To numerically simulate the transient channel flow with a moving boundary, computational code based on the f'mite element method has been developed using the Giesekus viscoelastic model. In addition, a commercial computational package is utilized to understand the transient polymer flow using a quasi-steady approach. The pressure drop from the die experiments and numerical simulations will be compared in an attempt to validate the numerical technique. The contribution of the elasticity of the polymer will be investigated
1013
30
14 Recovery
Injection Stage
25-
-12~
/
-10~
/
~, 2 0 -
-a 15--
6
~ r~ ~ 1==,4
10-
-4
5 1--3" A ~, i 0
,.
~
-2
i
i
i
I
2
4
6
8
I
I
I
I
I
10
12
14
16
18
20
t, S
Figure 2. Pressure measurements across valve and screw displacement versus time during the injection molding of HDPE with an injection speed of 2.54 cm/s and a melt temperature of 204.4~ by comparing viscoelastic simulations with generalized Newtonian-inelastic simulations using the Cross model. Changes in the velocity and stress components will be discussed to indicate the effect the moving boundary has on these variables. The usefulness of these methods in polymer processing will then be discussed. V,
I
I
L......... - = q
U I,._~ Inlet
zx]L.... ~ - - S J
m ovt n g !
L____v__~
XI~>
~Upstream mPressure
~
Transducer
DownstreamR Melt Pressure j~ Probe Transducer
Figure 3. Sketch of slit die with moving boundary. 1.2 B r i e f o v e r v i e w
of viscoelastic simulations
In the past two decades, polymer melt flow has been extensively researched. This research has led to an understanding of the basic flow of viscoelastic fluids in many different polymer processing applications. The advancement of this knowledge base has been expanded mainly through the implementation of computational simulations, which can only approximate realistic polymer
i
1014
processes. In particular, the simulations is carried out for two or three dimensional flow domains, complex geometry such as abrupt and tapered contraction flows, and flow with transient or unsteady boundary conditions. Nonisothermal boundary conditions have also had some limited development in extrusion, but not in the area of abrupt or complex geometry. Advanced numerical techniques available to simulate the complex flow of viscoelastic fluids include the fufite difference and the fimte element techniques. The books of Crochet [2] and Tucker [3] give a good review of these techniques. To properly model viscoelastic flow, one needs to understand numerical simulation techniques, the non-linear viscoelastic constitutive equations and the mathematical characterization of these systems of equations. Techniques used in past research include analytical solutions, decoupled fimte difference, decoupled f'mite element, and coupled/mixed f'mite element methods. In a decoupled approach, the goveming equations and comtitutive model are solved at different steps of the solution process. In a mixed/coupled approach, the goveming equations and constitutive equation are solved simultaneously. The choice of governing equations (energy, momentum and continuity) and the assumptiom which constrain the applicability of these equations, as well as the choice of constitutive equation, determine the type of viscoelastic flow a numerical technique can realistically simulate. Together, the goveming equations and constitutive equation form a set of nonlinear differential equations. This nonlinear set of differential equations is descritized into an algebraic system of equations, which can be solved with numerical techniques. The flow domain is divided up into elements or nodes and solved locally for each set of algebraic equations. The solutions from these local elements are then related globally to the entire flow domain to reach an accurate approximation. One technique to solve this system of equations is the finite difference method, although, much of the recent interest in numerically modeling viscoelastic flow has been concerned with the finite element method. In this type of approximation, the flow domain is discretized into a set of nodal points. The governing equations are then approximated at each node point using a Taylor series approximation. The constitutive equation is then solved separately using an iterative approach. Past work in this area has been conducted by many authors including Perera et al [4,5,6], Davies et al [7,8], Gatski and Lumley [9], Townsend [10,11], Tiefenbruck and Leal [12], Yoo et al [13,14], Phelan et al [ 15] and Yuan et al [ 16]. The method of choice for the simulation of viscoelastic flow is the finite element method. In this method, the flow domain is descretized into triangular or quadrilateral elements. The system of equations is approximated using the
1015 Galerkin method or method of virtual work. The equations are then solved by standard elimination techniques such as Gauss or Gauss-Siedel elimination. The solution of these equations gives approximations of the pressure, velocity components, extra stress components, etc. The two techniques to solve these equations are the decoupled and mixed/coupled methods. In the decoupled method, the continuity and momentum equations are solved for the Newtonian flow field. Then, the constitutive equation is solved separately using some type of streamwise integration. Usually, a Picard iteration, which is also called successive substitution, is applied to the constitutive equation to attain convergence. The extra stress contribution from the constitutive equation is then incorporated into the momentmn equations and then a new solution is found for the momentum and continuity equation. This iterative procedure is repeated until the convergence criteria is attained. Researchers to have used this method include Upadhyay and Isayev [17,18], Isayev and Huang [19], Bush et al [20], Lou and Tanner [21,22,23], Lou and Mitsoulis [24], and Hulsen and Van der Zander [25,26]. Since the streamlines start as flow boundary, the recirculation zones in contraction flows are solved by a numerical technique found in Upadhyay and Isayev [17,18]. The use of decoupled techniques has also been utilized for integral constitutive equations by Viriyayuthakom and Caswell [27], and Dupont et al [28] using particle tracking and Lou [29] using the control volume approach. Other methods include the Choleski decomposition method with a Picard iteration by Mitsoulis et al [30,31 ], streamline upwinding (SU) and Lesaint-Raviart methods by Fortin and Fortin [32], streamline-upwinding/PetrovGalerkin (SUPG) by Lou and Tanner [22], Barakos and Mitsoulis [33], and Lou [34], the adaptive viscoelastic stress splitting/streamline integration (AVSS/SI) scheme and adaptive viscoelastic stress splitting/streamline-upwind PetrovGalerkin (AVSS/SUPG) by Sun et al [35] and botmdary integral methods by Bush and Phan-Thien [36,37]. The use of mixed methods to simulate viscoelastic flow has received a large amount of attention due to the inability of these schemes to converge at high Deborah numbers. Early work on these techniques employed the method proposed by Kawahara and Takeuchi [38], where the Galerkin method and a Newton-Raphson solver converged only at low Deborah numbers. Researhers to have used this method include Crochet and Bezy [39], Crochet and Keunings [40,41,42], Davies et al [43], Dupret et al [44], Debbaut and Crochet [45], Keunings [46], Musarra and Keunings [47], Mendelson et al [48] and Kajiwara et al [49]. Other earlier research of Yeh et al [50] applied numerical methods designated as quadratic stress interpolation (QLL) and stress interpolation using bilinear polynomials (QQL) with Newtons method to solve the system of
1016
equations. The QQL method was found to converge at higher Deborah numbers and have a more stable convergence. This QQL method was also used by Brown et al [51]. Later, Marchal and Crochet [52] developed a more advanced mixed method where each element is divided into several bilinear sub-elements for stress to improve the algorithms convergence ability. In addition, some authors used the SU and SUPG methods introduced by Hughes and Brooks [53,54] to improve convergence. Other authors to have used this method include Debbaut et al [55], Pumode and Crochet [56], Hartt and Baird [57] and Yunm and Crochet [58]. Methods to improve the convergence of mixed numerical schemes have been researched by Brown, Armstrong and coworkers [59,60,61,62]. King et al [59] implemented a numerical method into the momenaun equations which makes the elliptic property of this equation explicit called the Explicitly Elliptic Momentum Equation (EEME). SUPG is also applied to this numerical scheme. This type of method has also been utilized by Burdette et al [60] and Coates et al [61]. One other formulation developed by Rajagopalan et al [62] is the elastic viscous split stress (EVSS) where the viscoelastic stress is split into the viscous and elastic stresses defined by x= _~ + ft. In addition, Sasmal [63] used this method. Other methods applied to the mixed solution technique include the spectral/fmite-element method by Beds et al [64], the timediscontinuous/Galerkin least-squares (TD/GLS) method by Baaijens [65,66], the Taylor-streamline upwind-Petrov-Galerkin/pressure correction formulation by Townsend and Webster [67,68], and the modified EVSS by Geunette and Fortin [69] and Azaiez [70]. Integral models have also been solved by mixed methods. An example of this is the paper by Papanastasiou et al [71 ]. All of these numerical techniques incur some difficulties in the approximation of viscoelastic flow. One problem studied by Holstein and Paddon [72] and Lipscomb et al [73] is the singularities which occur at the re-entrant comer of contraction flows. The extra stress terms can reach physically unrealistic values since the flow behavior of the fluid at these comers is not well understood. The gradients of the stresses at the boundaries can become excessively high. To overcome this problem, either the mesh size is increased in the region of the reentrant comer or a slip condition is applied at the boundary. Another source of instabilities is inherent in the constitutive equations themselves. Kwon and Leonov [74,75] have investigated the Hadamard stability and the positive def'miteness of the configuration tensor of different constitutive equations. Models proposed by Larson, Leonov and Giesekus, which are known to accurately predict viscoelastic flow, experience blow-up instabilities. These instabilities can be overcome by properly modeling the dissipative terms, adding a small
1017
Newtonian viscosity or using a multi-mode approach with respect to the relaxation spectrtma [17]. Van der Zanden and Hulsen [26] have also reported on these types of instabilities in the simulation viscoelastic flow. Joseph [76,77] investigated instabilities during the flow of viscoelastic fluids. He determined that hyperbolicity changes the system of equations from an elliptic to a hyperbolic type, thus making numerical solutions difficult. The simulation of the moving boundary requires the use of a transient numerical scheme. The basic understanding of the unsteady flow of constitutive equations with a time dependent history of the fluid deformation is well documented. The nonlinear response of viscoelastic fluid is different from the response typically seen for a Newtonian fluid. To more accurtely simulate this fluid, a viscoelastic model is applied in conjunction with a multi-mode approach to the deformation history. One type of flow situation, the sudden imposition of flow, occurs in many polymer process applications. The types of unsteady flow include a suddenly imposed velocity gradient, constant pressure gradient, a periodic pressure gradient, start-up flow with a wall moving parallel to flow, etc. In unsteady flows, the elasticity does not effect the f'mal velocity, but at the imposition of the velocity or pressure gradient a nonlinear response occurs in the velocities and stresses of the fluid. For a suddenly imposed velocity gradient, an overshoot in the pressure occurs. For the suddenly imposed pressure gradiem, a velocity overshoot is observed and is followed by a velocity minimum before a steady value is attained. Researchers to have investigated this phenomena include Fielder and Thomas [78], Waters and King [79], Chong and Franks [80], Chong and Vezzi [81], Townsend [82], Akay [83], Duffy [84,85], Balmer and Fiorina [86], Ryan and Dutta [87], Upadhyay et al [88], Isayev [89] and Kolkka and Ierley [90]. The solution of time dependent problems for the full set of governing equations has held much interest in the modeling of viscoelastic fluids. Along with the inclusion of a time variable, numerical stability and convergence difficulties are already confronted when trying to model the steady flow of a viscoelastic fluid. In addition, time dependent flows add an increased amount of computational time to solve problems. The numerical scheme typically used to approximate the time variable is a fmite difference approximation. Past publications applying transient simulations include the research of Townsend [10] using a De-Fort Frankel scheme, Keunings [91], Northey et al [92] using fully implicit integrator and semi-implicit integration, Keiller [93], Olsson [94] using a predictor-corrector method for time stepping, Rasmussen and Hassager [95] using a Lagrangian integral method and Baloch et al [68]. Results indicated that stable time dependent solutions were attainable, but flow occurring at high Deborah numbers
1018
were not presented. In order to understand the behavior of a viscoelasstic fluid, the experimental techniques that have been employed include streak photography, stress birefringence studies, multiple flash technique and laser doppled velocimetry. An excellent review of this research has been published by White et al [96] and lsayev and Upadhyay [97]. These methods work well when clear polymers are used in the experimental research. Unfortunately, fluids such as rubber compounds which cannot be seen through or polymers which can only be processed with industrial processing machinery cannot take advantage of these experimental methods. The only other experimental variable to measure is the pressure at various locations in the flow channel. Experimental work on pressure prediction to numerical simulations has been conducted by Isayev and Upadhyay [97] and Huang et al [98,19]. Results though indicate that the prediction of pressure is not as accurate as the prediction of the other field variables. In addition, this error can become significantly high for higher flow rates and in an abrupt contraction or expansion. In the present research, a decoupled finite element method previously used in two dimensional unsteady viscoelastic modeling is applied to the solution of polymer flow in channels with a moving boundary. The momentum and continuity equations are solved using the method of virtual work. The constitutive equation is solved by a Picard iteration technique. This formulation has been shown stability up to a Deborah number of 50 and a Weisenberg number of 4.2 by Upadhyay and Isayev [17], a Deborah number of 270 by Isayev and Upadhyay [98], and recently to a Deborah number of 845 by Isayev and Huang [19]. The only limitation in achieving a higher value of the Deborah number is the density of the mesh. A higher density mesh, hence, allows convergence at higher Deborah numbers. Added to this past f'mite element code is the simulation of transient flow with a moving boundary, which is closing the flow domain. This boundary moves in the direction perpendicular to the main direction of flow. In addition, the computational package FLUENT has been utilized to simulate the flow of a generalized Newtonian fluid based on the Cross model. The simulation using this package applies a quasi-steady approach with a velocity imposed at the moving boundary where the wall is closing the channel. Through these two paths of numerical simulations, the understanding of the effect the transient nature of flow with a moving boundary has on viscoelastic flow will be developed and understood.
1019
2. THEORETICAL The modeling of polymer processing can be grouped into three particular areas of research. One is based on the generalized Newtonian theory, which takes into account only the nonlinear shear rate dependent viscosity of the material. The other is based on the viscoelastic theory of polymer melts, which has many different forms, but is mainly concerned with incorporating the non-linear stress terms into some type of constitutive equation along with the nonlinear viscosity. To complete the theory of viscoelastic modeling, the viscoelastic-plastic fluid theory is also covered. 2.1 Generalized Newtonian Fluid
For most types of fluids, the viscosity of the material does not change as the shear rate on the material increases. For an incompressible Newtonian fluid the shear stress is related to the strain rate by the equation r~ = -ja~
(1)
where ~t is the Newtonian viscosity which is a constant for a given temperature, pressure and composition. Early rheologists used the empiricism that the viscosity could be a function of shear rate to model the shear rate dependence of the viscosity of suspensions, pastes, polymer melts and solutions. Thus, they incorporated the dependence into the previous equation as
(2) where r/(~;) is the apparent viscosity, which is a function of shear rate 2, pressure and temperature. Many generalized Newtonian equations have been postulated since the realization of this type of relation. The types of available models are the CrossArrhenius, Power Law, Klein, Carreau-Yasuda, Spriggs tnmcated power law, Eyring, Powell-Eyring, Sutterby, Ellis, and Bingham. Bird gives a good review of many of these models in chapter 4 of reference [99]. Two very useful generalized Newtonian relations, are the power law and the Cross-Arrhenius equations. The power law equation of Oswald and de Wade [ 100,101 ] is written as -n--1
r/=mg
(3)
1020
where m is a constant and n is the power-law index. This equation can only fit the high shear rate range of the viscosity curve called the power law region. This equation is useful since many analytical solutions are available, which can be easily understood. Many of these solutions can be found in the book of Bird [99]. In addition, this equation is capable of being implemented into numerical solutions easily. An example of this is a paper by Hieber and Shen [ 102] and also in Hieber [103] in which a finite element/finite difference is used to simulate the injection-molding filling process. Hieber [104] and Isayev and Upadyhay [97] also used the power law equation along with the finite element method to estimate the pressure drop and extra entrance length across planar and axisymmetric contractions. The main drawback of this equation is the inability to accurately model the Newtonian or low shear rate range of viscosity. The model by Cross [ 105, 106] is a three parameter equation of the form 7/ n(Y) =
o
i-n
(4)
where rio is the zero shear viscosity and ~* is the shear-stress level at which rl is in transition between the Newtonian limit rio and the power law region. The effect of temperature and pressure on the melt viscosity can be taken into account through an Arrhenius type dependence. Hieber gives a thorough description of these types of equations in chapter 1 of reference [ 103]. The modeling of the injection molding process by the Cross-Arrhenius equation has been well documented since the early 1980's. Hieber et al [107] used the Cross-Arrhenius type equation to model material viscosity to understand the effect of juncture losses resulting from the elasticity of the fluid. By comparing the numerical viscous simulation and experimental data, they were able to deduce the amount of juncture pressure from the elastic effects. Sobhanie, Deng and Isayev [ 108,109,110,111 ] have extensively used the Cross-Arrhenius equation for the injection molding of robber compotmds. An additional parameter was added to this simulation to account for the cure kinetics of the rubber compounds during injection molding. The numerical pressure predictions showed qualitative agreement with the experimental pressure data. Other researchers who have used generalized Newtonian models include Duda and Vrentras [112]. These authors used the Powell-Eyring equation to understand the effect nonlinear viscosity has on the amount of extra pressure loss
1021
in juncture regions. Kim-E, Brown and Armstrong [113] used a Carreau type viscosity equation to simulate the flow through an axisymmetric contraction. Results indicate steep velocity gradients occurring near the wall caused high shear rates and a shear thinning viscosity. These steep gradients effect the accuracy of the finite element approximations.
2.2 Viscoelastic Fluids The limiting factor in the ability of generalized Newtonian theory to predict polymer melt flow is the inadequacy of the equations to predict the elastic effects in a flowing polymer. These effects are extremely important in flows of polymers through sudden contractions and expansions. In addition, nonlinear stresses arising from polymer melts contribute to large pressure drops and steep stress gradients. Also, the elastic effects can dominate in start-up or transient flows of polymers. Thus, these elastic effects have a significant impact on the flow of polymer melts in many areas of processing. The modeling of viscoelastic phenomena employs two main types of constitutive equations to model polymer flow. The two types are the integral and differential type equations [99]. In the present paper, we will consider the differential models only. To characterize the flow of viscoelastic fluids, the dimensionless Deborah number defines the rate of straining on a fluid as De = .3U ~,
(5)
where 3U/b is the characteristic shear rate with U being an average velocity downstream of the contraction and ~, is the mean relaxation time defined as N
~rlkkk k = x_-,
(6) N
with TIk being defined as 770= 77~+~ = k, TI~ is a parameter similar to the lower Newtonian viscosity, and TIk, and Ek are the viscosity and the relaxation time in the k th mode. Most differential models are only an extension of the Maxwell [114,99,115] type equations from the 1860's. The Maxwell model assumes a linear relation for the viscous and elastic responses of the fluid. This relation can be written as
1022
Xxy+ G /)t - -tt~xy
(7)
where 17xy is the shear stress and G is the shear modulus. For steady-state motions, the equation simplifies to the Newtonian fluid. This equation shows time-dependent response upon an imposition of flow. In addition, other types of constitutve equations can be explained depending on the type of time derivative utilized for equation (7). For sudden changes with time, the equation simplifies to a Hookean solid. One can introduce the time derivative 5/8t in the convected coordinate system such as V
~-,__=a=$-Vv.a-~=.Vv
T
(8)
where ~ is the substantial time derivative, which translates with the material particle, Vv is the velocity gradient with Vv~ denoting the transpose. This system is called the upper convected coordinate system where the base vectors are parallel to material lines. Therefore, the vectors are stretched and rotated with the material lines, t, the time derivative, is written as D a ~ = b--Tx== ~-x=+ v__.Vx__
(9)
Limitations for this model exist which exclude it from properly predicting realistic polymer flow at high Deborah numbers. For instance, the model does not include any type of shear rate dependent viscosity. The first normal stress coefficient is not shear rate dependent either. The elongational viscosity becomes infinite at f'mite strains and moreover, the recoverable strain is over-predicted at high strain rates. In addition, the model cannot predict polymer behavior after the imposition displacements strains. In later developments, Oldroyd [116] proposed a quasi-linear differential model which was frame invariant called the Oldroyd-B model or the convected Jeffrey's model [117]. This model included a time dependent deformation tensor and is written as
~.,~-~+~ = 2no D+ X.,~D
(lo)
1023
For this equation, one needs to provide three parameters where rio is the zero shear rate viscosity, D is the deformation tensor, s is the relaxation time, and is the retardation time. This equation is capable of describing time-dependent flows, but is still unable to correctly predict rheological behavior. Equations of this type are limited to polymer flows with small deformation rates or low Deborah numbers. To account for these deficiencies more elaborate differential models have been developed such as the Maxwell corotational, White-Metzner, Gordon-Schowalter, Johnson-Segalman and Oldroyd-8 constant models which appear in Bird [99] and Larson [115]. These models were capable of describing more of the rheological properties of melts, but were still unable to describe all of the material properties correctly. In the mid-1960's, models were proposed that could describe polymer flows more accurately. All of these equations contain nonlinear stress terms. With the incorporation of this term, the constitutive equations are more suitable in predicting stresses in shear flow. One model which includes the quadratic stress term is the nonlinear viscoelastic equations developed by Giesekus [118-122]. This constitutive equation is based on the concept of a deformation-dependent tensorial mobility or drag. The equation is derived from the theory for concentrated solutions and melts using the dumbbell theory for dilute solutions. The assumption is made that the mobility is not dependent upon the individual configuration of each polymer segment. Instead, an average configuration of all the segments is used to relate the mobility tensor to the configuration tensors. This averaging of the polymer segment configuration bridges the gap between the molecular ideas from which the constitutive equation comes from and the treating of the polymer chains as a continuum of polymer melt or solution. This equation replaces the scalar mobility constants Bk with a non-isotropic mobility symmetric second order tensor 13to give an equation of the form ~.a+~-~-~= = 2rid
(11)
where $ is the elastic stress term, I"1 is the shear rate dependent viscosity, G is the shear modulus and D is the rate-of-deformation tensor. To achieve realistic predictions, two assumptions are made on the dependencies of $ and l~ on the elastic strain tensor C. First, the relation [G-5] of _ to ~ is of the neo-Hookean type dependence and is written as
1024
where ~ is the unit tensor. The shear modulus, G, is related to nkT, where n is the number of beads per unit volume, T is the temperature, and k, is Boltzmann's constant. The second assumption [120] is that a linear relation exists between 13 to C and is written as = ~ + oc(C- ~)= (1- ct)8 + txC
(13)
In this equation tx is an empirical constant of proportionality (mobility factor) and is related to the compressibility of the material, oc must satisfy the condition of 0 < cz < 1. The most general form of the constitutive equation can be written as
a
2
3
+ ~-~ + ~,~-~ = 2r/D
(14)
When oc equals O, equation (14) reduces to an isotropic mobility tensor and the UC Maxwell model is retained. When oc equals 1, the anisotropy is at its maximum and equation (14) produces results in shear and extension similar to those for the corotational Maxwell model. For the research presented in this paper, the assumption of incompressibility of the polymer is assumed and oc is equal to 1A. In addition, a multi-model approach is applied to equation (14) resulting in 1
l: + ~ ~ k
2
3
+ 2,k-~_~ = 2 r/kD
(15)
where k denotes the k th mode. The basic understanding of this model has been well documemed for many types of simple flows. Giesekus [120,121] published results on the predictuon for simple shear and simple extension which predict shear thinning and a nonvanishing first and second normal stress difference. In addition, this research has defined where the Giesekus model predicts real solutions by Yoo and Choi [123] and viscoelastic instabilities in shear by Oztekin et al [124]. Other f'mdings include the Giesekus and other models ability to predict Poiseuille flows by Schleiniger and Weinacht [ 125], sinusoidally undulating channel flow and TaylorCouette flow instabilities by Beds [126], shear flow and experimental comparisons by Vlassopoulos and Hatzikiriakos [127], exponential shear stress coefficient and elongational flow by Schieber and Weist [128], shear flow, and
1025
uniaxial and biaxial extension by Khan and Larson [129], steady and transient shear flow by Quinzani et al [130], and tmiaxial, biaxial and elongational flow by Isake et al [131]. Larson [132] has compared several different models including the Giesekus model for steady-state flows, start-up steady straining, stress relaxation following cessation of steady straining, single and double step strains, and elastic recovery, in sheafing, or uiaxial, biaxial and planar extension. Other work has discussed the compatibility of equations with equilibrium thermodynamics by Grmela and Carreau [133] and the restriction of the extra stress tensor and requirements that this tensor must be positive def'mite by Hulsen [134,135]. Other differential equations which are often used in the modeling of viscoelastic fluids are the Leonov [136,137] equations, the Phan-Thien Tanner (PTT) model [138,139],the f'mite extensible nonlinear elastic (FENE) model [140,141] and the Doi-Edwards [142] model. For practical purposes, constitutive equations quadratic in stress are the most popular for today researchers. More complex models can be derived, but the implementation of these models would probably be to cumbersome due to the complexity of the mathematical equations. In addition, to many parameters would need to be specified in order to implement the models. Excellent reviews of constitutive equations and the relations between the different models is available in the books by Bird et al [99] and Larson [115].
2.3 Viscoelastic-plastic fluids In the modeling of polymeric fluids, one other area of interest is the modeling of filled polymer systems. The inclusion of the interaction between polymer and filler adds additional assumptions to the constitutive model. These filled systems are characterized by the rheological behavior of the matrix, the particle characteristics, the dispersed state, the interaction between particles, and the interaction between the filler and the polymer matrix. In addition, the particle size effects the rheological properties of the filled polymer melt. For large particles, an increase in shear and elongational viscosity if observed. In systems with small particles, yield values in shear and elongational flow occur, as well as, other changes in the rheological properties. The approach to modeling these filled systems is either a continuum approach or a micro-mechanics approach. The first to propose a stress-deformation rate equation for a fluid with yield stress was Bingham [143]. This was subsequently followed by the equations of Prager and Hohenemser [144]. These authors proposed a constitutive model for the flow of an incompressible viscoplastic material as D = 0 for J2 -
(16)
1026
J2-Y - - - - - - x = 2riD J2 = -"
for J2 > Y
(17)
for rate-of-deformation tensor, where Y is the yield function, and gj is the deviatoric stress tensor def'med as x = ~o..-/3o= J
m
8..
(18)
~J
The second invariant of the deviatoric stress tensor is def'med by (19)
J2 = 4~-1;. q;
Subsequently, Slibar and Parsley [145,146] proposed equations based on thixotropic phenomena with time-dependent yield values. The rate of deformation tensor and deformation were both incorporated into the yield function Y. Later, a viscosity function was proposed by Harris [147] which assumed an initial viscosity could be recovered after a long enough time. The first to develop a viscoplastic rheological model with yield was Schwedoff [ 148]. Using a modified form of the Maxwell model, this equation is written as =0
when
(20)
x>__Xy
~,d x + (x - Xy)= rl~/ when dt
x>_Xy
(21)
In steady state shear flow, this model reduces to x = Xy+ rl~ when
x __.Xy
(22)
Later, HuRon [ 149] proposed a model which uses a yield criterion with a memory function. White [150,151] also developed a more specific viscoplastic model based on the combination of avon Mises yield criterion and a hereditary function. To more accurately model the thixotropy of fluids, Suetsugu and White [152], Montes and White [153] implemented a time dependent yield criterion. Isayev and Fan [ 154] used a von Mises yield criterion and the Leonov viscoelastic model to propose an equation for f'flled polymers. This model though does not predict thixotropy. Often, filled systems will have time dependent properties caused by the breadkown and build-up of particle structure. To account for this time dependence, Cheng and Evans [155] have developed an isotropic,
1027
incompressible, inelastic, thixotropic model where the particle breakdown and build-up process is included. Others to have developed this type of theory include Cheng [ 156] and Kemblowski and Perera [ 157]. More advanced models in filled systems include the work of Leonov [158] and Simhambhatla [159]. These authors developed a rheological model based on the Leonov viscoelastic model [136]. This model assumes that the total stress is the sum of viscoelastic stresses developed from micro-flow of the polymer matrix around flocs, and the stresses due to particle-particle interactions of the dispersed particle phase. In addition, Coussot et al [160] developed a similar model for low molecular weight matrices. Sobhanie et al [ 161] developed a viscoelastic-plastic model where the total stress is assumed to be the sum of the mean stresses due to micro-flow of polymer melt arotmd a particle and the mean stress due to particleparticle interactions. The Leonov model and a yield criterion are used to approximate the mean stresses in the polymer matrix and filler network. Yield is def'med by a function in terms of a differential equation with an internal parameter describing the evolution of scructure changes during floc rupture and restoration. In this model, the total stress ~ is represented by the equation m
x=x +x
p
(23)
m
where _x_ is the mean stress in the polymer matrix and ~P is the mean stress in the filler network. equation
The mean stress in the polymer matrix is represented by the
N
-1) "rm= P~ + 2 r/~D + 2~.~(Wk.lCk - Wk,2Ck
(24)
k=l
where ~ is the elastic strain tensor and Wk is the elastic potential. The mean stress in the filler network is represented by a scalar yield function p
Y
x =--x =
(25)
J2 =
The particle network is modeled by a Voight-Kelvin viscoelastic equation which assumes that there is an initial modulus go which can decay and recover based on the amount of damage to the filler network.
1028
3. N U M E R I C A L SIMULATIONS 3.1 Finite difference formulation for a generalized Newtonian fluid The first numerical approach used to simulating the flow of a polymer fluid in this paper is the finite difference technique. This numerical approach along with the finite element technique in the section 3.2 are applied since the equations to model two dimensional polymer flow are too complex for the derivation of analytical solutions. For viscous-inelastic simulations, the commercially available computational fluid dynamics code FLUENT is utilized. To more accurately model the nonlinear viscosity of the polymer melt, FLUENT was modified to incorporate a Cross model viscosity function. A quasi-steady analysis is applied to the viscous solutions to account for the time dependent nature of the moving boundary. The solution of this problem using finite difference code will provide an understanding of the dynamic response of non-return valves. The finite difference code FLUENT uses a pressure based segregated finitevolume method to solve the governing equations. Further aspects of this numerical formulation are found in the FLUENT manual [162] and the book of Patankar [163]. The incompressible form of the continuity equation is written as V.v=O
(26)
where v is the velocity vector and the isothermal, incompressible and steady form of the conservation of momentum equations with no body forces is written as V.~=O
(27)
where ~ is the total stress acting on the fluid. The model implemented is the Cross model defined by equation (4). To determine the shear rate dependent viscosity of this model the local strain rate is determined by
3ui ~3u~
= r'~ ~
+ Ox,
(28)
where i and j represent the direction of the velocity or displacement. The scalar invariant of this local strain rate is determined as the local shear rate. The two dimensional continuity (26) and momentum equations (27) are solved by integration. The differential form of the momentum equations are written as
1029
(OVx v OV ]
P v'-~"x + Y Oy )=--~-x + ~9-T-
(29)
v Ov,]
P vx-~x + '/9y ) = /gx + i)---~ for the x and y directions respectively. equation is written as
(30) The differential form of the continuity
~9v_._Lx ~)Vy Ox +--~y = 0
(31)
where Vx is the velocity component in the x-direction or axial direction, and Vy is the velocity component in the y-direction or gap wise direction. (Yijis the stress components in the indicated directions. The stress tensor ~ for the generalized Newtonian flow is written as
g=-p +n(V.v+V.v")
(32)
and the individual stress components are written as ~Vx Oxx = - p + rl Ox
(33)
~gvy o . = -p + rl ~)y
(34)
( Vx
~x, = -P + rlt--~-y + Ox )
(35)
where p is defined as the pressure. To approximate the differential equations the flow domain is divided using a finite-volume method as shown in Figure 4. The node points are located along the grid lines. The scalar and vector variables are stored at the center of each control volume. The governing differential equations are solved by integration across each control volume using the divergence theorem. In the solution of these field variables, the pressure at each control volume face is determined. In addition, variables such as the velocity components, which obey a momentum and mass balance across each face, must be determined. Across each face, there is an
1030
assumed average mass balance. The mass balance is applied to ensure that continuity exists between the variables in a given control volume and the variables of neighboring control volumes. node (i+l,j+l)
node,. (i,j+l)
face
node (i,j)
:
Cell Center Q location of variable storage
9
node (i+l,j)
Figure 4. Control volume scheme for use in FLUENT. In the solution of equations (29), (30) and (31) for each control volume, FLUENT uses several different interpolation schemes to give a more accurate approximation of the field variables across each control volume. The choice of interpolation schemes includes the power law scheme [163], blended Second Order Upwind/Central Difference [164] and QUICK [165]. The power law scheme uses a one-dimensional convection-diffusion equations for interpolation. The blended second order upwind/central difference and QUICK higher order interpolation schemes, which determine the faces values between control volumes based on the values stored at the centers of two control volumes and an adjacent control volume upstream of the first two cell centers. The higher order interpolations schemes result is more accurate approximations in the solution of the flow domain. 3.1.1 Numerical Scheme for Finite Difference Method For the solution of the flow domain, FLUENT uses an iterative procedure First the velocity components are solved for using an initial guess for the pressure. Mass balance is then applied to correct the pressure. The viscosity from the Cross model is then updated. The governing equations for each control volume is solved based on the values of neighboring control volumes. The procedure continues using a line-by-line solution of the equations until a satisfactory error in the residuals has been obtained. A quasi-steady approach is applied to solve for the transient moving boundary. This type of approach is applied to very viscous fluids from knowing that the Reynolds number is very low and applying the assumption that creeping flow is
1031
occurl'ing. In addition, the assumption is made that the flow is isothermal. In the simulations the position of the wall from fully open to eighty-five percent closed is divided into a given number of displacements. A control volume grid is built for each displacement using the FLUENT preprocessor. The boundary conditions are applied to each grid including the flow rate, material properties, and constant closing velocity of the wall. To approximate the velocity of the moving wall, the sides of the wall use a wall boundary velocity, which is parallel to the wall. The velocity of the bottom face uses a flow inlet condition with a uniform velocity across the entire face. The velocity boundary conditions are shown in Figure 5.
v, =vy =0.0 vy= wall v~ =vy--0.0 velocity WGII v~= walll<:::]v,=O.O ~7 v, =0.o[:~L[Ivy--velocity velocit t Inle~y i
vx =O. Op f
i
[[] Inlet condition'Exit J" ,with uniform x ' , , ~ _ . j ~/~vel~
v~ =vy =0.0 Figure 5. Boundary conditions for numerical simulations of moving wall and die. 3.2 Finite element formulation for a viscoelastic fluid
To fully understand the effect of the moving boundary, the viscoelastic rheological properties of a polymer fluid must be incorporated into the simulation. Thus, numerical methods such as the f'mite element method are used to properly describe the polymer melt flow. In order to simulate the closing of the slit die with the FEM, the continuity and momentum equations must be put into f'mite element formulations. The Giesekus constitutive equation will be solved by the streamline integration technique of Viriyayuthakom and Caswell [27] and Shen [166]. Finite element code using the FORTRAN programming language already exists for the Leonov model. This code was originally developed by Upadhyay and Isayev [97,18,17] and Huang and Isayev [19,98]. This code simulates the viscoelastic flow through tapered and abrupt planar contractions. Based on this code, quasi-three dimensional transient moving boundary f'mite element code for planar complex geometry has been developed by Zhang. As mentioned previously, the solution of this problem using the f'mite element technique will help to understand the dynamic response of a non-retum valve. For the f'mite element formulations, incompressible form of the continuity equation is written as
1032
V.v =0
(36)
and the isothermal and incompressible form of the conservation of momentum equation with no body forces is written as Dv
V.~= p Dt
(37)
where v is the velocity vector, p is the fluid density, and r is the total stress acting on the fluid. The operator D( )/Dt is the material time derivative b( )/bt + v. V(). The constitutive equation to be solved is the multimode Giesekus model with the incompressiblity condition a equal to 89 corresponding to the incompressibility condition. This equation is written as 1
"t" + - ~ k ~
2
0
+ ~,k~-_Tk = 2r/kD
(381
where ~.k is the relaxation time, Gk is the shear modulus and rlk is the viscosity of the k th mode. The extra stress tensor is incorporated into the Cauchy stress tensor and is written as r
+~
(39)
where p is the pressure, 5__is the unit tensor and r__L_is the extra-stress tensor. For the Giesekus model, the Cauchy stress tensor will be written as Npara ~---
""
--
=
n~k
where w is the extra-stress tensor of the k th mode with Npara denoting the total -'k
number of nodes and p is the pressure contribution to the stress. The continuity and conservation of momentum equations are solved using the virtual work f'mite element method with six-node triangular elements. The two dimensional momentum equations in the x and y directions are written as
(~v,~
aw
~yX1 aC~x ~r
(41)
1033
(Ovy
Ov,
Ovr~ o,=~ o,=.
(42)
respectively, and the continuity equation is written as
0vx 0vr 0x +-~-- = 0
(43)
where p is the fluid density, Vx is the velocity component in the x-direction or axial direction, and Vy is the velocity component in the y-direction or gapwise direction, oij is the stress components in the indicated directions. For the Cauchy stress tensor, the individual stress components are written as Ov,
(44) k
Ovy
o-,,,, = - p + r/,--~-- + ~ r,,,,,,,
(45)
O-y-- 7r/,C(Bv~ T +~)
(46) k
The extra stress terms x~j,k are the contribution of stress from the constitutive equation for different modes of relaxation. 3,v3,P3)
(Ul,Vl,P~.."........~
(u2'v2'P2)
Figure 6. Six-node triangular element for u-v-p formulation with triangular area coordinate system. To find an approximate solution, the flow domain is discretized into a series of sub domains or elements as shown in Figure 6. These subelements are applied to the flow domain for the solution of the continuity and conservation of momentum
1034
equations with triangular elements. The velocity components are approximated by second order polynomials with the node locations at the comers and midpoints. The pressure is approximated by f'~st order polynomials with the location of the nodes at the comers of the triangular elements. The velocity components and pressures are approximated by shape functions N and M, respectively. The shape functions can be written simply as, v~ = N__~u,v~ = N_yv,and p=M_.p. Further development shows that these shape functions are a set of coordinates, which determine the location of each triangular element and the subsequent node points in the flow domain. The relation def'lnes these coordinates by L = Ai / A = (ai + bi x +
Ci y)
/ 2A
(47)
where A is the total area of the element, Ai is the are the area coordinates as indicated in Figure 6 and ai, bi and ci are the coordinate values of the three comer nodes. The three area coordinates satisfy the following relation L, + L2 + L3 = 1
(48)
The area coordinates have the unique relation that L,= 1 when x = xj, and y = yj if i= j. Furthermore, the area coordinates have the relation that L,= 0 when x = xj, and y = yj if i ~ j. One other constraint is that 0 < Li < 1 w h e n x ~ x j and y ~ yj. The velocity components use the same quadratic shape functions, which are def'med by six different variables for each element such as N=[N,,N2,N3,N4,N5,N6]
(49)
with each Ni being def'med by the standard triangular quadratic basis relations of N~= L~(2L~-I), N2= L2(2L2-1), N3 = / _ o ( 2 L 3 - 1 ) ,
N4=4LIL2, Ns=4L2L3 ~ Nn=4L3LI
(50)
The pressures use the linear shape functions defined by three different variables for each element written as M_M_=[M~,M2,M3]
with each M being defined by the standard triangular linear basis relations of
(51)
1035
(52)
Ml = L1, M2=/_,2 & M3 = L3
The derivaties of the velocity components and pressure shape functions are defined as (9Vx
ON
0Vx
0N -
(53.a)
-
~gx - igx u, iOy ~gy u 0Vy 0N 0Vy 0N ~ ~ ~ ~ V 0x 0x v, by by -
(53.b)
Op
OM Op a M - - p__, Oy- ~- - p__ &- &
(53.c)
Equations (41), (42) and (43) are written into the finite element formulation using the method of virtual work in two dimensions as follows
~=6v=
~
+
=
i= v9 otw+v w+v '
dR
(54)
.l',,<~v, ~ +-~--)d=: .1",,<~Vy. p t . ~ + v.-~- + v,
(55)
and
(56)
J=~. 0x +
=0
respectively, where dR is def'med as the integration in a given region. Applying the Gaussian theorem written in the form of
~v~~) -gd (~v~.())---g-()~(avD The left-hand side of the momentum equation (54) can now be rewritten as
which can be written in the form
(57)
1036
f~(~v"r')a~-I,[ ~(~v')~x~ ~(~v') aR~y ~" ]
(59)
and equation (55) is
(~v,o.)+#
(60)
which can be written in the form
I~(~v,.~)~-I~
[~v o( ,)~. +.v,/}. ~y ~.
(61)
where F is the integration along the boundary of the domain and T1 and T2 are a simplified way to write the relations in the brackets. Substituting these relations into the conservation of momentum equations (54) and (55) results in the following set of equations
~v~ ~vx ~)~ = I, [~v o( ,)oo+ ~(~v~, Oy ~ ]dR+I,~v~.o~-~+v~-~+v, (62) and
~(~ v,. T2)dF
(63)
Introducing the shape functions for the velocity components from equations (53.a) and (53.b), the conservation of momentum equations (54) and (55) can now be written in the following form.
+
ay
Ox, dR+j'u_ u _ _ . t , - ~ +
u+
v
=
(64)
1037 and
L- ~
0y r
/
dR +~R__ Nr N ' p
/
"~
+
u + -~z
v _dR =
8Vr~r(Nr" T2)dI"
(65)
In addition, the continuity equation (56) can be written as
r_
R r(ON
(66)
o
by introducing the pressure shape function from equation (53.c). The stress components in the momentum equations are represented by the following equations with an artificial damping parameter co to control the stability of the equations. These equations are written as axx = - p " + 2
+co ~ - 2 ( o '
& + ~ ' r m-x' x,k
(67)
k
ayy=-
+2
a, =
+ co
+oJ)--~---2o90y +
+Z'r~yJ:k
- o~ 9 v7 -~
ay +
(68)
a Vy -~
oe
m-~
+
(69)
T.:
where m is the variable to be solved and m-1 is the variable from the previous solution. This set of equations can be simplified and written in a matrix representation to form a set of equations, which will be solved for each element to give the approximate solution of the flow domain. This matrix is written as
i ONr ON-~R
r m
r/'+m f N r ONdR
ax +
1038 fOv,, r - - I NrNdR + J-7--N NdR Rm
-
R
O3Vx
T
---~-N_ N_aR
T
2~~+~
r
-
0
o T
-
R-L~
m-1
~N~
R_L ~, ~+2--~---~~
o
o
1
r
~N
NdR
0
o
o
~ _NTNdR o
0
0
_r ,~ ~N L
0
o
o]
u
R
+p
m
o'x----
_
og~ N o
0
0
]
fi t" +
N r T2
(70)
0
In these matrix equations, the N r and N vectors indicate the shape functions for the velocity components with six constants and the M indicates the shape functions for the pressures with three constants. The terms which include the At terms indicate the f'mite difference approximation of the time stepping. The velocity components are the initial conditions for the time step. The fmite difference representations of the velocity components are given as
1039 m+l Ou
u
m
m+l
-u
- - Ot At
and
Ov
v
m -v
- - & At
(71)
in the x and y directions, respectively. Equation (70) can be written in a simpler form as
TM][(R,)q
u~
(72) {K31}Um {K32}V
TM
0-
J
where the K matrix is banded and symmetric. Therefore, the matrix set is solved using LU decomposition. The system of equations can be numerically integrated for each element in the flow domain using standard integration techniques. The integration across each element can be summed to give the contribution of the velocity components, stresses and pressure across the flow domain from the continuity and momentum equations. This summation can be written as NE
I F(x,y)dR = ~_, I F(x,y)dR R
(73)
i=1 R,
where NE is the total number of elements, i is the element number from 1 to NE, and Ri is the domain of the ith element.
3.2.1 Numerical Scheme for Finite Element Method The method applied to solve this system of equations is a decoupled scheme where the pressures and velocities are solved separately to the Giesekus constitutive equation in relation to the extra stress terms. This constitutive equation and the extra stresses are solved using a streamwise integration technique introduced by Viriyayuthakom and Caswell [27] and Shen [161 ]. In this technique, a Newtonian flow field is d e t e m ~ e d using the momentum and continuity equations. Then the the stress terms _~j,k are determined using the streamline integration of the constitutive equation and the known values of the previously found velocity field. Next, the pressures and velocities are solved again with the extra stress terms treated as a known variable. This iterative procedure is repeated until a converged solution is found and the error has been
1040
minimized to an assumed value. In the streamwise integration procedure, a streamwise coordinate s is introduced which is positive downstream along each streamline. The constitutive equations can be written in the form of dv
)r
o~
1
ak
2 2rhD
(
=k = Vv-~ + ~ 9( Vv - ~ - ~ - v. VL - ~-~'k~ - 2Gk AkL + Ak----= F Vv, [v_J--~--
(74)
by taking advantage of the convective term in the constitutive equation. The resulting equations are a system of first-order nonlinear differential equations, which are integrated using Euler integration. The integration is def'med as x
=x----'k + ~ - r l v v (n+l)
where ----k x
'~k )
(75)
,l(n)
and ------k denote the stress tensor at two consecutive points n+l and n,
respectively, and As is the arc length between the two points. The integration starts at the upstream boundary and continues along the streamlines of the extra stresses. This integration can be performed for any relaxation time ~,k as long as the streamwise coordinate is chosen such that ~ <<~,klV_J. However, if the relaxation time is small, the assumption is made that perturbation solution is determined by x = Gk[~+ ~,k(VV+ V : ) - 11
------'k
xlVv_J<
and a
(76)
which is related to the Giesekus constitutive equation. The accuracy of this extra stress tensor is directly related to the accuracy of the velocity and rate of strain tensors. This method does have difficulties when calculating the flow in the recirculation zones since the streamline for a vortex is closed and does not start or end at an inflow boundary. Thus, no initial conditions can be applied to solve this situation. The streamlines in these flow regions are solved separately. The assumption is made that the velocity components and strain rates are very small compared to the flow, which drives the recirculation. Thus, from the assumption that Z,k[VV_Jis small, equation (74)is solved for the stresses. If ~.k[VV_J is much larger than unity, the stresses are determined after the stresses of some of the
1041
neighboring nodes have been calculated. To apply the solution of the constitutive equation to the finite element method, the extra stress tensor is represented by a 7-point Gauss quadrature as shown in Figure 7, which has fourth order accuracy. For the triangular elements, three points are related to the comers e
d
Figure 7. Seven point Gaussian quadrature for a triangular element. of the triangle, three points are related to the midsides of the triangle and one point is related to the centroid of the triangle. The stress tensor from the Giesekus model is evaluated at the Gauss points using the streamline integration. Each streamline from an individual Gauss point is traced backwards until the botmdary of the elements. From this point, the values of the stress tensor are computed from the adjoining upstream element by interpolating quadratically at the side of this element. With these values determined, the integration is then performed along the trace path up to the original point. The elements in the recirculation zone are determined using equation (76) or the streamline integration as mentioned previously. At the walls the velocity is zero or defined by the velocity of the moving boundary and the values of the stress tensor are found from the constitutuve equation. The application of appropriate boundary conditions is required in order to numerically solve the system of equations. Boundary conditions required include the velocity and stress field at the upstream boundary. Knowing these values, the velocities and pressure field can be detennined and the streamwise integration can start. The flow field upstream is assumed to be one dimensional, steady, and fully developed. The equations ofmotion are written as 8P~ ~ -I- +col '~v3 -co ~r ~v~-~ 2 k+. ,~'~ ' 8Y2 9 8Y2 k=l tT~
(77)
1042
~9pm ~22~ - ~ ~gy k=l 33'
(78)
and
zl,.~ = G,~
/
1+ X, - 1
*'12., = G, 4 1 + X,
where Xk = 41 + 4"i'22L[ and ~ = 0 % .
*~" = G, ~ / l + X ,
/
- 1
(79)
Integration of (77) in conjunction with (74)
results in An' y =
+ to ~" - to ~"-' + ~'t',2,,
(80)
k=l
where A m is defined as the pressure gradient Am=-/)pm//)x. The velocity is found by assuming no slip at the boundary and integrating the equation v:(y) = Ji~'dy
(81)
In addition to the simple algorithm previously used to solve the viscoelastic simulations, the numerical scheme implemented in the current simulations incorporates the transient momentmn equations and the imposition of a moving boundary. This numerical scheme is shown in Figure 8. First, the time step is set at t=to. A mesh is then generated for the first flow domain for a fully open channel. The initial values for the velocities, pressure are assumed, after which a Newtonian solution is then found for the flow domain. Next, the constitutive equation is solved using the streamline integration. With the solution of the extra stress tensor found, the momenama and continuity equations are solved treating this stress tensor as a known. This iterative procedure is continued until a converged solution is found for the velocity, pressure and stress componems. Next, the time and displacement is incremented to the next step. This includes the movement of the boundary based on the imposed wall velocity for each time step. A new mesh is generated for this flow domain. Assuming that the displacement steps of the moving boundary are small between each time step, the velocity, pressure and stress components from the previous time step are reinterpolated into this new flow domain. The decoupled algorithm to solve for
1043
Initialization of v_~ pO, ~_oij~, for first time step
Generatemeshinitial
I
I
on ions I
solve Newtonian flow field
Solve for_~k from constitutuve [ equation using _v~'~
I
Apply boundary conditions I on v~l
I
Find f f and p" from momentum and continuity equations treating ff~ as known
Converged i f , pm, ~ij~ for ~iven time step
•
If tm = t~ , numerical solution finished
Figure 8. Flow chart of moving boundary.
Reinterpolate i f , pm, _.ffij~,for t~ mesh for t~+~ time step to solve transient process
"t
Generate new mesh for new wall heil~ht
't
Apply time step tm+l = tm + At and movement of wall boun m+l = x m + Ax
1044
the field variables is then applied with the appropriate boundary and initial conditions until a converged solution is found. This process is repeated until the time and displacement limits have been reached. Then, the velocity components, stress components and pressure are discretely known at each time step or position of the moving boundary.
103 -
IMM 102model Giesekus 101
'
10-1
'"
.... I
10o
' '"'"'1
' '"'"'1
10~
9 ' '"'"~
102
' '"'"'1
103
104
'
'"'"'
l0 s
7w, S 1 Figure 9. Viscosity versus shear rate at the wall for HDPE at 204.4~ Cross and Giesekus model fits.
along with
4. E X P E R I M E N T A L RESULTS 4.1 Material The polymer used in the experiments is a commercial grade high density polyethylene (HDPE) Marlex (Lot # HMN 4550-03) supplied by Phillips. To rheologically characterize this material for the Cross and Giesekus models, viscosity measurements were conducted on a Rheometrics mechanical spectrometer (RMS-800) in the plate-and-plate mode in the low shear rate range from 10 -1 to 10 2 S-1. For higher shear rates, experiments were conducted on an Instron capillary rheometer (Model 3211) for the shear rate range from 101 to 3x103 s -1 and on an injection molding machine (Van Dora Demag 170 ton HT series) fitted with a capillary type nozzle for the shear rate range from 2x103 to 4x104 s ~. Measurements have been made at three different temperatures including 182.2, 204.4, and 232.2~ A plot of the viscosity curve for 204.4~ against the shear rate at the wall is shown in Figure 9.
1045
4.2 Methods of investigation To test the numerical schemes implemented, a slit die with a moving boundary will be used. The experimental setup is shown in Figure 10. The slit die has a width over height ratio of ten and is attached to a Killion 1-inch extruder, which will provide the flow of the polymer melt during the closing of the die. A screen plate (not shown) is placed between the extruder and die to assure that the pressure increase during the closing of the wall does not effect the flow rate or screw characteristic curves. The pressure drop across the wall is relatively insignificant when compared to the pressure drop across the screen plate. ,_
8,89cm
2,54cm
7,62cm
Ptunger
E
_j_u
z Ft-ow- .N~e~io
~-.------4~
. . . . . . . .
~ /
CO ,,/3
Figure 10. Experimental Setup for the Slit Die with a Moving Boundary. To record the experimental data, two pressure transducers (Dynisco PT412-5M6/18 and PT435A-5C), as shown, measure the pressure drop across the die and moving wall as the wall closes. The upstream pressure transducer is located one inch in front of the entry of the wall region. The downstream pressure transducer is located one inch from the exit of the wall. The movement of the wall is controlled with an Instron T-500 tensile test machine and an LVDT (Schaevitz Eng. Type 300HR S/N 3307) with a maximum displacement of 1.27 cm over a volt range of 0 to 10 volts. The wall is 2.54 cm in length and 6.35 cm wide. This wall varies the height of the die across the whole width of the wall. The height of the flow channel is 0.635 cm with the wall fully open. To control the temperature of the die, a melt probe is placed next to downstream pressure transducer. The
1046
temperature control components (model E5CS-R1K JX-F) are from Omron Tatesi Electronics Co. Cartridge heaters (Watlow firerod 9651M G4J33) are placed symmetrically in the die. In addition, a band heater ( Watlow thinband STB5 1J2 T) is placed around the screen plate and adapter plate, which changes the flow geometry from the circular geometry of the extruder to the slit die geometry. Two inch Owens Coming lnsulwool insulates the slit die and the die to extruder adapter. Several processing parameters are varied in these experiments including the flow rate through die and the closing speed of the wall. A data acquisition system (1 KHz) is used to record data from the pressure transducers and LVDT for wall movement, which consists of an A/D converter module ADM 12-11 (Quatech) with in house software and an IBM compatible PC computer. In the experirnemal testing of the wall closing, the pressure drop across the moving wall is recorded as the wall moves from a fully open position to 85% of fully closed. The speed of the moving wall is held at a constant rate for each experiment. The flow rate is also constant for each experiment. The experimental test conditions are shown in Table I. Table I Extrustion and Die Experimental Conditions Melt temperature (~ 204.4 Barrel zone temperatures (~ 204.4, 204.4, 204.4 Die Temperature (~ 204.4 Flow Rates (g/min) 20,40,70 Wall Velocity (cm/s) -0.0287, -0.071, -0.154, -0.317, -0.635 IIII
I
4.3 Experimental observations Experimental results are presented for the transient flow of a polymer in a channel with a moving boundary. The experiments were conducted for the three different flow rates. For each flow rate, the velocity of the moving boundary was given five different values. The only exception is the wall velocity of-0.635 crn/s for the flow rate of 70 g/min. Results from these experiments show the time dependency of the pressure drop due to the moving boundary. For all of the experiments, the pressure increases as the gap height decreases, but some interesting results are obtained due to the viscoelastic rheology of the HDPE. The results for the experiments with the given flow rates and wall velocity are shown in Figure 11, 12 and 13. The experimental pressure measured at the upstream pressure transducer is plotted versus time on a log-log scale. The
1047
downstream pressure measurement is not included. The pressures measured at this location are very low and do not add any insight into the understanding of the effect the moving boundary has on the pressure. Figures 1 l a, 12a and 13a show the actual velocity of the wall versus time for each wall speed based on a specified flow rate. This velocity is not constant during the entire experiment. Typically, the velocity of the wall decreases at the end of each experiment before the wall reaches the stopping height of 85% closed. For the faster closing speeds, a variation in the velocity occurs at the start of the closing of the channel. The actual velocity of the wall is determined by the slope of the LVDT versus time curves for each experiment. For the flow rate of 20g/min with different wall speeds shown in Figure 1 l b, each pressure shows an overshoot above the steady pressure during the closing of the channel. The overshoot is greater for the higher wall speeds. In addition, each pressure trace shows an initial jump in the pressure at the start of the closing of the wall. In Figure 12b, the pressure traces for a flow rate of 40g/min show no overshoot for closing speeds of-0.0287 and -0.071 cm/s. In addition, the pressure traces without overshoot reach the steady state pressure after the wall has stopped moving. This indicates a delay in the development of pressure. The three fastest closing speeds do exhibit an overshoot in pressure during the closing of the wall, although, the overshoot is not as pronounced as the overshoot for the flow rate of 20g/min when comparing the magnitude of the overshoot to the steady pressure for a given flow rate. These results for 40g/min show an initial jump in the pressure at the start of the closing of the channel. For the flow rate of 70g/min, experimental pressure is presented for four different wall speeds. The upstream pressure indicates no overshoot for any of the wall closing speeds. This indicates a delay in the pressure development. These results for 70g/min also show an initial jump in the pressure at the start of the closing of the channel. The results indicate a relation between the closing speed of the wall and the flow rate of polymer through a slit channel. If the flow rate is low relative to the velocity of the moving boundary an overshoot in the pressure will occur. As this flow rate is increased for a given moving boundary velocity, the magnitude of this overshoot relative to the steady pressure will decrease. If the flow rate is increased to a larger value, no overshoot in the pressure, over the steady value, will occur and a delay in the development of the pressure will occur. Thus, the pressure continues to increase to the steady value after the moving boundary has stopped.
1048
0.0
- -
-~'-0.0287
-0.1 54
-0.2 -
J
-0.3 >
-0.317
-0.4 -0.5 -0.6 -
[ . . . J V w = -0.635 cm/s
-0.7 2
!
!
|
|
3
4
5
6
w
i
!
|
|
20
78910
30
Logt, s
le+7 -
-0.635 J t
!
-0 317 " -0.154 ~
".UJ ]~/]
le+6 -
A"0"071
-0.0287
ea~ o le+5
b !
!
!
!
3
4
5
6
!
!
!
[
7 8 910
!
20
30
L o g t, s
Figure 11. Actual moving wall velocity (a) and experimental (solid lines) and predicted (dashed lines) upstream pressure (b) as a function of time at the flow rate of Q=20 grams/min and various imposed moving wall velocities, Vw.
1049
0.0
;
--
-0.1 -0.2 -
~~.0~17
__.r'-
-0.0287
"0"154
-0.3 -
>
-0.4 -0.5 -0.6 -
~ , ~ Vw= -0.635 cm/s
-0.7 2
!
!
!
3
4
5
|
!
|
!
!
I
20
6 78910
30
Log t, s
le+7-
"0"635 t
~'0"317~ "0"154 .
.
.
.
.
.
4"0-071
.
~.
.
.
-0.0287 .
.
.
.
.
.
le+6 O
b
le+5 w
3
4
5
I
6 7 8 9 10
20
30
Log t, s
Figure 12. Actual moving wall velocity (a) and experimental (solid lines) and predicted (dashed lines) upstream pressure (b) as a function of time at the flow rate of Q=40 grams/min and various imposed moving wall velocities, Vw.
1050
0.0
-0.3 -
-0.5
-
-0.6
-
-0.0287
,.o.o71
-0.154
-0.2 -
-0.4
J
/ - 5
-0.1 -
V w = - 0 . 3 1 7 cm/s
-O.7
2
3
u
u
4
5
u
u
u
u
u
i
20
6 78910
30
Log t, s
le+7
-o.63s d It
/'
I
-0.317
-0.154
-o.o 1
-0.071
I !
t~
]
le+6
,J
/
!
J
o
le+5 i 2
" i
i
i
3
4
5
i
i
i
i
|
6 7 8 910
I
20
30
Logt, s
Figure 13. Actual moving wall velocity (a) and experimental (solid lines) and predicted (dashed lines) upstream pressure (b) as a function of time at the flow rate of Q=70 grams/min and various imposed moving wall velocities, Vw.
1051 5. S I M U L A T I O N RESULTS 5.1 Material Parameters From the rheological experiments for HDPE at 204.4~ the Cross and Giesekus model were fitted to the viscosity data. The fitted model parameters for the Cross model are 11o = 1500 Pa.s, x* = 8 x 1 0 4 Pa and n = 0.34. The density of p=960kg/m 3 is assumed, which is typical for HDPE. For the Giesekus model, the fitted model parameters are 111 = 775 Pa*s, r12 = 330 Pa*s, 113 = 39 Pa*s, 9~1 = 0.032 s, L2 = 0.0017 s, 9~3 = 0.000081 s, rls = 1.77 Pa*s and the characteristic relaxation time of the HDPE melt according to equation (6) is 0.0221s. Figure 9 indicates that these parameters fit the experimental viscosity data quite well for a wide range of shear rates. 5.2 Viscous Numerical Results The numerical results are presented for the transient flow of a polymer in a channel with a moving boundary. The computational package FLUENT is used in the simulations using the Cross model. A quasi-steady approach to accotmt for the moving boundary is used in this case. The movement of the boundary was broken up into fourteen displacements. An inlet velocity based on the flow rate, zero velocity of the stationary boundaries, the velocity of the moving boundary and a pressure equal to zero at the die exit are the boundary conditions applied for the simulations along with the appropriate material parameters. To account for the moving boundary, two types of analysis are employed. The first is an ideal velocity, where at a time less than zero the moving boundary velocity is zero. At a time greater than or equal to zero, the moving boundary has a velocity equal to the specified closing speed. When the moving boundary reaches 85% closed the boundary motion instantaneously stops. The ideal velocity is found by determining the first order slope of the LVDT when the wall is closing the slit channel. These transient boundary conditions are written as
Vw= 0.0
for t < 0 and t > t85~
Vw -- Vclosing for 0< t < t85~
Calculations are conducted for the quasi-steady steps of the fully open position and 85% closed position for a velocity of zero and with the specified closing speed velocity. Thus, an instantaneous starting and stopping velocity is assumed. The second type of analysis employed the actual velocity by determining the local slope at each quasi-steady position. This variation of wall velocity during the
1052
closing of the slit channel is shown in Figure 1 la, 12a and 13a. Results of the simulations of pressure traces with the ideal velocity are shown in Figures 1 lb, 12b and 13b for the flow rates of 20, 40 and 70 g/min, respectively. These figures indicate the pressure at each position during the closing of the moving boundary. In all of the graphs, as the flow rate increases for a given boundary velocity, the pressure increases, but not by a significant amount. As the boundary velocity increases for a given flow rate, the pressure increase and peak pressure increases drastically. This indicates that for a viscous quasi-steady simulation the velocity of the moving boundary has a dominant effect on the pressure as the moving boundary closes the slit channel. For the start-up and stopping of the moving boundary, an abrupt change is seen in the value of the pressure. Results for viscous simulations with the actual wall velocity are shown in Figures 14 and 15 for the various closing speeds and the flow rates of 20, 40 and 70 g/min, respectively. The simulated pressures show a sharp increase at the start of the closing of the slit channel. This increase in pressure is not as drastic in the case of the ideal velocity simulations since an instantaneous velocity is not assumed. As the wall closed fiulher, the predicted pressure increases until the end of the closing of the channel is reached. At this point, the pressure reaches a maximum value. The location of this peak depends on the value of the wall velocity at each quasi-steady position. Near the end of the closing simulations, the actual velocity of the wall decreases rapidly. Thus, lower predicted pressures are found when compared with the simulations which use the ideal velocity at the end of the closing of the channel. 5.3 Viscoelastic Numerical Results Viscoelastic numerical results are presented for the transient flow of a polymer in a channel with a moving boundary. The simulations use in-house code which is transient and accounts for the moving boundary. The Giesekus constitutive equation is applied with the fully transient momentum equations. The simulations include the elasticity of the polymer, the transient flow of the polymer and the movement of the wall boundary into the flow domain. The flow rate, zero velocity of the stationary boundaries, velocity of the moving boundary, steady viscoelastic entrance and exit conditions and flow exit into a zero pressure region are applied for the simulations along with the appropriate material parameters. To account for the moving boundary, the viscoelastic simulations employ the actual velocity of wall. Results of the viscoelastic simulations, which use the actual velocity are shown in Figures 14 and 15 for the flow rates of 40, 20 and 70 g/min, respectively. A time step of 0.005s was chosen in the simulations. This
1053
le+7 I -0.317 ~ r .
-0.154 .~
-0.071
t~
le+6 @
,
j
- 4 - - Viscous w/Vw= -Vw(t) Viscoelastic w/Vw= -Vw(t) Experimental Data
7 le+5
-
I
I
I
I
I
I
I
i
3
4
5
6
7
8
9
10
Log t, s Figure 14. Experimental and predicted pressure as a function of time for a flow rate of 40g/min and various wall velocities. time step is capable of capturing the time-dependent response of the polymer, which occurs for the multi-mode Giesekus model. The HDPE used in the experiments exhibits a low relaxation time compared to the external time of straining based on the flow rates and closing speeds. Thus, only the steady state response and not the time dependent response of the second and third modes will contribute to the predictions of the field variables. In addition, if the second and third relaxation modes were included in the simulations, the computational time would become excessively large. The viscoelastic simulations that use the actual wall velocity are shown in Figures 14 and 15. This velocity is similar to the velocity used in the viscous simulations, except that a smoother pressure development occurs since a greater number of time steps are included in the calculations. The simulations predict a sharp increase in the pressure at the start of the closing of the wall. The pressure gradually increases, until a peak pressure is reached near the end of the simulation. However, the peak pressure is lower than that for the viscous
1054
le+7 I --~'- Viscous w/Vw= -Vw(t) Viscoelastic w/Vw= -Vw(0 Experimental Data
,A
le+6 tm O a
le+5 !
I
I
I
I
3
4
5
6
7
Log t, s le+7 -
- Viscous w/Vw= -Vw(t) Viscoelastic w/Vw= -Vw(t)
_.l(~*,~ . . . .
9
r,
9
le+6
o -1 le+5
b
--
I
I
I
I
I
3
4
5
6
7
8
Log t, s Figure 15. Experimental and predicted pressure as a function of time for a flow rate of 20g/min (a) and 70/min (b) for the actual wall velocity with a set velocity of -0.317 cm/s. simulations that use the ideal and actual velocity. This pressure then reduces to a steady value. After the wall has stopped moving the simulated pressure has not reached the steady value as shown in Figures 14 and 15a. This would indicate that a small amount of relaxation occurs in the viscoelastic simulations. In Figure
1055
15b, a large peak in the pressure development does not occur. The pressure development for the viscoelastic simulation almost attain the steady pressure with no overshoot.
6. COMPARISON OF NUMERICAL AND EXPERIMENTAL RESULTS
Figures 11, 12 and 13 are the comparison of simulated and experimental pressure traces for the viscous simulations with the ideal velocity. Figures 11a, 12a and 13a are the experimental velocity of the wall during the experiments. Figures l lb, 12b and 13b are the comparison between the simulated and experimental pressure during the closing of the die channel. For the predicted pressure drops using the Cross model with a quasi-steady type simulation, the simulated pressures do not predict the experimental pressure very accurately at the beginning or end of the transient flow experiments. In most cases, for the flow rates in Figure 11, 12 and 13 the simulated pressures are higher than the experimental pressure at the beginning of the closing of slit die. At the start of the closing of the slit die, an initial jump is observed in the experimental pressure. This experimental pressure jump occurs for the three flow rates and the five closing speeds. The viscous simulation results predict a faster pressure jump than the experimental pressure. This is true for all simulations conducted. Then, the simulated pressure shows a much higher overshoot than the overshoot of the experimental pressure. The simulated pressure instantaneously goes to the steady pressure when the wall stops moving. However, the experimental data indicates that the pressure does not reach a steady value instantaneously, but after a short period of time reaches the steady value. The lower peak pressure and longer time to reach a steady value after the wall has stopped moving indicates the viscoelastic response of the HDPE. The lower peak pressure indicates a delay in the development of pressure due to the viscoelastic effect. The slow recovery to a steady value of pressure after the wall has stopped moving indicates the importance of the relaxation of the polymer. This is very interesting since the polymer has a relatively low relaxation time, but the time dependent effect of the moving boundary results in the viscoelastic effect becoming more pronounced. For the flow rate of 20g/min in Figure 11, the predicted and experimental pressure traces show overshoot at all closing speeds. For the flow rate of 40g/min in Figure 12, some more interesting fluid behavior is seen. For the wall closing speeds of-0.0287 and -0.0711 cm/s, no overshoot is seen in the experimental data, although, an overshoot is numerically predicted for all of the
1056 different closing speeds due to the quasi-steady approach taken and the elastic effects are ignored in these simulations. When no overshoot occurs, the pressure development is delayed and the steady value of experimental pressure is not reached until after the wall has stopped moving. This indicates the viscoelastic effects of the polymer are relevant in the time dependent experiments, which the viscous simulations cannot predict. For the other three closing speeds, an overshoot is seen in the experimental pressure, but the numerical predictions for the peak pressure are much higher. For the flow rate of 70g/min in Figure 13, the experimental pressure for the four closing speeds do not indicate any overshoot. In addition, when the pressure development is delayed, the steady experimental pressure is not attained until after the wall has stopped closing the slit channel. In addition, the time to reach the steady pressure value has a longer time span for the experiments with a slower wall velocity. Part of this delay observed in the experimental data could come from the slowing of the moving wall during the end of each experiment. In addition, the ideal velocity used in the simulations does not take into account the slowing of the moving wall. The simulated pressures predict an overshoot as in the previous numerical results. This reinforces the fact that the viscous simulations cannot predict the pressure accurately for the flow of a viscoelastic fluid, even though the relaxation modes of the polymer are relatively small. Thus, the viscoelastic effects appear to be extremely important in the flow of a polymer through a slit channel with a moving boundary. For the viscous simulations where the actual velocity of the moving boundary is applied, the predicted pressures indicate different results when compared with the experimental values as shown in Figures 14 and 15. The prediction of the pressure during start-up compares well with the experimental pressure since enough quasi-steady increments were chosen to approximate this region. The simulated pressures show good agreement with the experimental pressure after the start-up region except at the end of the simulations. As with the viscous simulations with an ideal velocity, the simulations predict a higher pressure even thought the actual velocity is employed in the simulations. Thus, the simulations based on the quasi-steady analysis cannot predict the experimental pressure at the end of the closing. For the viscoelastic simulations where the actual velocity of the moving boundary is applied, the predicted pressure shows different results. The predicted pressure compares well with the experimental pressure at the start of the closing of the channel. Then, the pressure gradually increases and reaches a maximum and after that reduces to the steady value of the pressure. The experimental peak pressure is not predicted accurately, although, the viscoelastic
1057
2 0.6
--
0.5
--
0.4
--
4
scale:
5
-
l cm/
~ 0.30.20.10.0
-''''
''
''
I ' ' ' ' ' '
8
'''
I'""''''''''
10
I'
12 x, cm
''
'''
''
' I ''
14
' ' ' ' '
''
I''
16
Figure 16. Predicted velocity profiles at five different cross sectional areas of the flow channel at Q=40g/min and Vw= -0.317 cm/s for viscous (solid lines) and viscoelastic (dashed lines) simulations. simulations that use the actual wall velocity predict a much lower pressure than the other simulations. The predicted peak pressure is always higher than the experimental pressure except in some instances after the wall has stopped closing the channel. The pressure jump at the start of the closing of the channel is predicted well since the actual velocity of the moving boundary is employed and a higher amount of time steps are used in this analysis when compared to the viscous simulations which used the actual velocity. The viscoelastic simulations offer the closest prediction to the experimental data. When an overshoot occurs in the experimental pressure as seen in Figures 14 and 15, an overshoot occurs in the viscoelastic predicted pressure. In addition, the relaxation of the pressure due to viscoelasticity of the polymer occurs in this simulation. When a suppression of the pressure development occurs in the experimental pressure, very little overshoot occurs in the predicted pressure. Unforttmately, the simulations were unable to predict this suppression of the pressure development. The predictions of pressure can be improved by using a smaller time step. This would allow the simulation to include all three relaxation modes of the Giesekus model. The velocity vector profiles of the simulations are shown in Figure 16. The v and u velocity components are shown in Figure 17a and 17b, respectively, for a flow rate of 40 g/min, and the second fastest closing speed of-0.317 cm/s. The moving wall is eighty percent closed for these figures. Figure 16 indicates that
lO58
0.6 --
0.5-
/
t~ 0 . 4 ff~ 0 . 3 0.2-
3
O.1-
a
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
1.50
V, cm/s 0.6
--
0.5
i
/5
0.4 if, 0.3 0.2
4 ........
/3
0.1-
b
0.0
"
0
'-''
,;'
1
'"":"'
"
-i 2
~
'
'
3
4
5
U, cm/s
Figure 17. Predicted profiles of v- (a) and u- (b) velocity component at different cross sectional areas of the channel at Q - 40g/min and Vw- -0.317cm/s for viscous (solid lines) and viscoelastic (dashed lines) simulations. the variation in the velocity magnitudes across the gap are close for positions 1 and 2 and slight differences occur at positiom 3, 4 and 5 between the viscous and viscoelastic simulations. It should be noted that the velocity magnitude at position three in Figure 16 has a value greater than zero at the wall due to the velocity contribution from the moving boundary. The velocity components in Figure 17 indicate a good corr~adson between the viscous and viscoelastic simulations for u-velocity components. However, some differences in the vvelocity component are shown. This good agreement would indicate that the
1059
elastic effects of the polymer have little effect on the velocity profile of the flowing polymer during the closing of the slit channel. Thus, the viscoelastic simulations that use the actual moving boundary velocity predict pressure better than the other simulations. The viscoelastic simulations predict a lower pressure which is much closer to the experimental results. When a suppression of the pressure occurs as seen in Figure 15b, the viscoelastic simulation predicts a minimal amount of overshoot. When an overshoot occurs, the viscoelastic simulations predict a lower pressure than the viscous simulations. For the viscous simulations which implement the ideal velocity of the moving boundary, the pressure development is poorly predicted. Thus, the variation in the wall velocity has an important impact on the proper prediction of pressure development. In addition, the predicted pressure overshoot is drastically higher than the experimental overshoot of pressure. For the simulations where actual velocity was employed, the viscoelastic simulations predict a lower pressure than the viscous simulations. In both simulations, predicted pressures are lower than in the simulations where the ideal wall velocity is used. The inclusion of the elasticity of the polymer allows for a much better prediction of the pressure development. The simulation predicts a pressure much closer to the experimental data.
7. CONCLUSIONS In the present study, experiments were conducted to measure the pressure drop during the flow of a polymeric fluid through a slit die with a boundary moving at a specified velocity, which is perpendicular to the direction of flow. The polymer flows through the die as the wall closes the flow passage. The polymer used in the experiments was HDPE. Experiments were conducted for three different flow rates and five different moving wall velocities. A data acquisition system was used to record the transient pressure drop as the wall closes the die channel. The flow rate was held constant for each experiment. The velocity of the moving wall was held constant, but was found to actually vary as the wall closes. Experimental results indicate that for the flow rate of 20g/min an overshoot occurs in the pressure as the wall closes the channel. For the flow rate of 40g/min, an overshoot does not occur for all of the closing speeds. This overshoot occurs only for the two fastest closing speeds. For the flow rate of 70g/min no overshoot occurred for any of the wall closing speeds. When no overshoot occurs, a delay is present in the development of the experimental pressure. This delay was found to be a combination of the change in the velocity
1060
of the wall and a delay in the development of the pressure due to viscoelastic effects. The prediction of the experimental pressure was simulated using three different types of numerical simulations. These simulations included viscous and viscoelastic simulations that employed an ideal and actual wall velocity. The viscous simulations used the generalized Newtonian fluid model according to the Cross model. The viscoelastic simulations used the Giesekus model with a timedependent stress evolution. The viscous simulations were conducted with the computational code FLUENT and used a quasi-steady approximation for the wall movement. The viscoelastic simulations were conducted with in-house FEM code, which is transient and accounts for the moving wall boundary. The simulations that assumed an ideal wall velocity predicted a sharper increase in the pressure development at the start-up of the wall velocity. In addition, these simulations predicted a peak pressure which was drastically higher than the experimental pressures. All simulations with the ideal wall velocity predicted an overshoot in the pressure at the end of the closing of the channel. The simulations which employed the actual wall velocity gave better predictions of the experimental pressure. The viscous and viscoelastic simulations predicted the start-up pressure very well. The prediction of the pressure overshoot for the viscous case was much higher that the viscoelastic predictions. For the simulations that used the actual wall velocity, a prediction of the pressure during the start-up of the closing of the channel by the moving wall was found to be accurate. The simulations, though, did not predict the delay in the development of pressure. A small amount of overshoot in the pressure was predicted even in the cases when no overshoot occurs in the experimental pressure traces. The use of the actual velocity improves the prediction of the peak pressure. The viscoelastic case was found to predict the transient pressure development better than the other numerical simulations. The viscoelastic simulations which used the actual velocity predict a lower pressure than the viscous simulations. This indicates that the elastic nature of the polymer as described by the Giesekus model helps in the prediction of pressure. These viscoelastic effects occur even though the HDPE melt would seem to not exhibit a significant amount of elasticity. The transient nature of the experiments make this elasticity become more pronounced. For a better prediction of the pressure smaller time steps must be implemented in the viscoelastic simulations. The comparison of the velocity magnitudes and components across the gap height for several different positions in the die at 80% closed indicate little difference. Thus, the elasticity does not effect the velocity profiles. For the simulations conducted, the viscoelastic FEM code can offer significant insight
1061
into the prediction of the experimental pressure. For polymers with higher amounts of relaxation, this code could possibly help in the prediction of the suppression of the pressure development. In these instances, a delay in pressure could possibly be predicted. Research is currently being conducted on the closing in the absence of flow through the slit or squeeze flow experiments. In addition, polymers which have much higher relaxation times will be incorporated into the current research on the channel flow of polymer with moving boundaries.
REFERENCES 1. A. I. Isayev, C. Zook, Y. Zhang and R.J. Scavuzzo, SPE Antec Tech. Papers, 41:1 (1995) 595. 2. M.J. Crochet, A.R. Davies and K. Waiters, Numerical Simulation of NonNewtonian Flow, Elsivier Science Publishers B.V., Amsterdam (1984). 3. C.L. Tucker, Computer Modeling for Polymer Processing, Hanser Publishers, New York (1989). 4. M.G.N. Perera and K. Walters, J. Non-Newtonian Fluid Mech., 2 (1977) 49. 5. M.G.N. Perera and K. Waiters, J. Non-Newtonian Fluid Mech., 2 (1977) 191. 6. M.G.N. Perera and K. Strauss, J. Non-Newtonian Fluid Mech., 5 (1979) 269. 7. H. Court, A.R. Davies and K. Waiters, J. Non-Newtonian Fluid Mech., 8 (1981)95. 8. A.R. Davies, K. Waiters and M.F. Webster, J. Non-Newtonian Fluid Mech., 4 (1979) 325. 9. T.B. Gatski and J.L. Lumley, J. Comp. Physics, 27 (1978) 42. 10. P. Townsend, J. Non-Newtonian Fluid Mech., 14 (1984) 265. 11. P. Townsend, J. Non-Newtonian Fluid Mech., 6 (1980) 219. 12. G. Tiffenbruck and L.G. Leal, J. Non-Newtonian Fluid Mech., 10 (1982) 110. 13. J.H. Song and J.Y. Yoo, J. Non-Newtonian Fluid Mech., 24 (1987) 221. 14. H.C. Choi, H.J. Song, and J.Y. Yoo, J. Non-Newtonian Fluid Mech., 29 (1988) 347. 15. F.R. Phelan, M.F. Malone and H.H. Winter, J. Non-Newtonian Fluid Mech., 32 (1989) 197. 16. X.F. Yuan, R.C. Ball and S.F. Edwards, J. Non-Newtonian Fluid Mech., 46 (1993) 331. 17. R.K. Upadhyay and A.I. Isayev, Rheol. Acta, 25 (1986) 80. 18. A.I. Isayev and R.K. Upadhyay, J. Non-Newtonian Fluid Mech., 19 (1985) 135. 19. A.I. Isayev and Y.H. Huang, Recent Adv. in Non-Newtonian Flows, 153 (1992) 113.
1062
20. M.B. Bush, J.F. Milthorpe and R.I. Tanner, J. Non-Newtonian Fluid Mech., 16(1984)37. 21. X.-L. Lou and R.I. Tanner, J. Non-Newtonian Fluid Mech., 21 (1986) 179. 22. X.-L. Lou and R.I. Tanner, J. Non-Newtonian Fluid Mech., 22 (1986) 61. 23. X.-L. Lou and R.I. Tanner, J. Non-Newtonian Fluid Mech., 31 (1989) 143. 24. X.-L. Lou and E. Mitsoulis, J. Rheol., 34:3 (1990) 309. 25. M.A. Hulsen and J. Van Der Zanden, J. Non-Newtonian Fluid Mech., 38 (1991) 183. 26. J. Van Der Zanden and M. Hulsen, J. Non-Newtonian Fluid Mech., 29 (1988) 93. 27. M. Viriyayuthakom and B. Caswell, J. Non-Newtonian Fluid Mech., 6 (1980) 245. 28. S. Dupont, J.M. Marchal and M.J. Crochet, J. Non-Newtonian Fluid Mech., 17 (1985) 157. 29. X.-L. Lou, J. Non-Newtonian Fluid Mech., 64 (1996) 173. 30. E. Mitsoulis, J. Vlachopoulos and F.A. Mirza, Polymer Eng. Sci., 24:9 (1984) 707. 31. E. Mitsoulis, J. Vlachopoulos. and F.A. Mirza, Polymer Eng. Sci., 25:11 (1985) 677. 32. M. Fortin and A. Fortin, J. Non-Newtonian Fluid Mech., 32 (1989) 295. 33. G. Barakos and E. Mitsoulis, J. Non-Newtonian Fluid Mech., 62 (1996) 55. 34. X.-L. Lou, J. Non-Newtonian Fluid Mech., 63 (1996) 121. 35. J. Sun, N. Phan-Thien and R.I. Tanner, J. Non-Newtonian Fluid Mech., 65 (1996) 75. 36. M.B. Bush and N. Phan-Thien, J. Non-Newtonian Fluid Mech., 16 (1984) 303. 37. M.B. Bush, J. Non-Newtonian Fluid Mech., 31 (1989) 179. 38. M. Kawahara and N. Takeuchi, Computers and Fluids, 5 (1977) 33. 39. M.J. Crochet and M. Bezy, J. Non-Newtonian Fluid Mech., 5 (1979) 201. 40. M.J. Crochet and R. Keunings, J. Non-Newtonian Fluid Mech., 7 (1980) 199. 41. M.J. Crochet and R. Keunings, J. Non-Newtonian Fluid Mech., 10 (1982) 339. 42. R. Keunings and M.J. Crochet, J. Non-Newtonian Fluid Mech., 14 (1984) 279. 43. A.R. Davies, S.J. Lee and M.F. Webster, J. Non-Newtonian Fluid Mech., 16 (1984) 117. 44. F. Dupret, J.M. Marchal and M.J. Crochet, J. Non-Newtonian Fluid Mech., 18 (1985) 173.
1063
45. B. Debbaut and M.J. Crochet, J. Non-Newtonian Fluid Mech., 20 (1986) 173. 46. R. Keunings, J. Non-Newtonian Fluid Mech., 20 (1986) 209. 47. S. Musarra and R. Ketmings, J. Non-Newtonian Fluid Mech., 32 (1989) 253. 48. M.A. Mendelson, P.-W. Yeh, R.A. Brown and R.C. Armstrong, J. Non.Newtonian Fluid Mech., 10 (1982) 31. 49. T. Kajiwara, S. Ninomiya, Y. Kuwano and K. Funatsu, J. Non-Newtonian Fluid Mech., 48 (1993) 111. 50. P.W. Yeh, M.E. Kim-E, R.C. Armstrong and R.A. Brown, J. Non-Newtonian Fluid Mech., 16 (1984) 173. 51. R.A. Brown, R.C. Armstrong, A.N. Beris and P.-W. Yeh, Comp. Meth. Appl. Mech. Eng., 58 (1986) 201. 52. J.M. Marchal and M.J. Crochet, J. Non-Newtonian Fluid Mech., 26 (1987) 77. 53. T.J.R. Hughes and A. Brooks, in Finite Elements in Fluids, eds. R.H. Gallagher, D.H. Norrie, T. Oden, O.C. Zienkiewicz, Wiley, Chicester, 1982. 54. A.N. Brooks and T.J.R. Hughes, Comp. Meth. Appl. Mech. Eng., 32 (1982) 199. 55. B. Debbaut, J.M. Marchal and M.J. Crochet, J. Non-Newtonian Fluid Mech., 29 (1988) 119. 56. B. Purnode and M.J. Crochet, J. Non-Newtonian Fluid Mech., 65 (1996) 269. 57. W.H. Hartt and D.G. Baird, J. Non-Newtonian Fluid Mech., 65 (1996) 247. 58. F. Yurun and M.J. Crochet, J. Non-Newtonian Fluid Mech., 57 (1995) 283. 59. R.C. King, M.R. Apelian, R.C. Armstrong and R.A. Brown, J. NonNewtonian Fluid Mech., 29, (1988) 147. 60. S.R. Burdette, P.J. Coates, R.C. Armstrong and R.A. Brown, J. NonNewtonian Fluid Mech., 33 (1989) 1. 61. P.J. Coates, R.C. Armstrong and R.A. Brown, J. Non-Newtonian Fluid Mech., 42 (1992) 141. 62. D. Rajagopalan, R.C. Armstrong and R.A. Brown, J. Non-Newtonian Fluid Mech., 36 (1990) 150. 63. G.P. Sasmal, J. Non-Newtonian Fluid Mech., 56 (1995) 15. 64. A.N. Beris, R.C. Amastrong and R.A. Brown, J. Non-Newtonian Fluid Mech., 22 (1987) 129. 65. F.P.T. Baaijens, Comp. Meth. Appl. Mech. Eng., 94 (1992) 285. 66. F.P.T. Baaijens, J. Non-Newtonian Fluid Mech., 48 (1993) 147. 67. E.O.A. Carew, P. Townsend and M.F. Webster, J. Non-Newtonian Fluid Mech., 50 (1993) 253.
1064
68. A. Baloch, P. Townsend and M.F. Webster, J. Non-Newtonian Fluid Mech., 50 (1993) 253. 69. R. Guenette and M. Fortin, J. Non-Newtonian Fluid Mech., 60 (1995) 27. 70. J. Azaiez,, R. Guenette and A. Ait-Kadi, J. Non-Newtonian Fluid Mech., 62 (1996) 253. 71. A.C. Papanastasiou, L.E. Striven and C.W. Macosko, J. Non-Newtonian Fluid Mech., 22 (1987) 271. 72. H. Holstein and D.J. Paddon, J. Non-Newtonian Fluid Mech., 8 (1981) 81. 73. G.G. Lipscomb, R. Keunings and M.M. Denn, J. Non-Newtonian Fluid Mech., 24 (1987) 85. 74. Y. Kwon and A.I. Leonov, J. Rheol., 36:8 (1992) 1515. 75. A.I. Leonov, J. Non-Newtonian Fluid Mech., 42 (1992) 323. 76. D.D. Joseph and J.C. Saut, J. Non-Newtonian Fluid Mech., 20 (1986) 117. 77. D.D. Joseph, Viscoelasticity and Rheology, Academic Press, Inc., N.Y. (1985). 78. R. Fielder and R.H. Thomas, Rheol. Acta, 6:4 (1967) 306. 79. N.D. Waters and M.J. King, Rheol. Acta, 9 (1970) 345. 80. J.S. Chong and R.G.E. Franks, J. Applied Poly. Sci., 14(1970) 1639. 81. J.S. Chong and D.M. Vezzi, J. Applied Poly. Sci., 14(1970) 17. 82. P. Townsend, Rheol. Acta, 12 (1973) 13. 83. G. Akay, Rheol. Acta, 16 (1977) 598. 84. B.R. Duffy, J. Non-Newtonian Fluid Mech., 4 (1978) 177. 85. B.R. Duffy, J. Non-Newtonian Fluid Mech., 7 (1980) 107. 86. R.T. Balmer and M.A. Fiorina, J. Non-Newtonian fluid Mech., 7 (1980) 189. 87. M.E. Ryan and A. Dutta, J. Rheol., 25:2 (1981) 193. 88. R.K. Upadhyay, A.I. Isayev and S.F. Shen, Rheol. Acta, 20 (1981) 443. 89. A.I. Isayev, J. Rheol., 28:4 (1984) 411. 90. R.W. Kolkka and G.R. Ierley, J. Non-Newtonian Fluid Mech., 33 (1989) 305. 91. R. Ketmings, J. Comp. Physics, 62 (1986) 199. 92. P.J. Northey, R.C. Armstrong, R.A. Brown, J. Non-Newtonian Fluid Mech., 36 (1990) 109. 93. R.A. Keiller, J. Non-Newtonian Fluid Mech., 43 (1992) 229. 94. F. Olsson, J. Non-Newtonian Fluid Mech., 51 (1994) 309. 95. H.K. Rasmussen, O. Hassager, J. Non-Newtonian Fluid Mech., 56 (1995) 65. 96. S.A. White, A.D. Gotsis and D.G. Baird, J. Non-Newtonian Fluid Mech., 24 (1987) 121. 97. A.I. Isayev and R.K. Upadhyay, in Injection and Compression Molding Fundamentals, A.I. Isayev Ed., Marcel Dekker, New York (1987).
1065
98. Y.H. Huang, PhD. Dissertation, University of Akron (1992). 99. R.B. Bird, Dynamics of Polymeric Liquids: vol. 1, Wiley, N.Y. (1987). 100. W. Oswald, Kolloid-Z., 36 (1925) 99. 101. A. de Waele, Oil Color Chem. Assoc. J., 6 (1923) 33. 102. C.A. Hieber and S.F. Shen, J. Non-Newtonian Fluid Mech., 7 (1980) 1. 103. C.A. Hieber, in Injection and Compression Molding Fundamentals, A.I. Isayev Ed., Marcel Dekker, New York (1987). 104. C.A. Hieber, Rheol. Acta, 26 (1987) 92. 105. M.M. Cross, J. Colloid Sci., 20 (1965) 417. 106. M.M. Cross, Rheol. Acta, 18 (1979) 609. 107. C.A Hieber, V.W. Wang, K.K.Wang and B. Chung, SPE ANTEC Tech. Papers, (1984) 769. 108. M. Sobhanie, J.S. Deng, and A.I. Isayev, J. Applied Polymer Science, Appl. Polymer Symp., 44 (1989) i 15. 109. S. Deng and A.I. Isayev, Rubber Chemistry and Technology, 64:2 (1991) 296. 110. M. Sobbanie and A.I. Isayev, in Modeling of Polymer Processing, A.I. Isayev Ed., Hanser, Munich (1991). 111. J.S. Deng, Ph.D. Dissertation, Univ. of Akron (1989). 112. J.L. Duda and J.S. Vrentras, Trans. of the Society of Rheology, 17:1 (1973) 89. 113. M.E. Kim-E, R.A. Brown and R.C. Armstrong, J. Non-Newtonian Fluid Mech., 13, (1983) 363. 114. J.C. Maxwell, Phil. Trans. Roy. Soc., A157 (1867) 49. 115. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterwirth Publishers, Stoneham, MA, (1988). 116. J.G. Oldroyd, Proc. Roy. Soc., A200 (1950) 523. 117. H. Jeffreys, The Earth, Cambridge University Press, (1929) 265. 118. H. Giesekus, Rheol. Acta, 5:1 (1966) 29. 119. H. Giesekus, Rheol.Acta, 21 (1982) 366. 120. H. Giesekus, J. Non-Newtonian Fluid Mech., 11 (1982) 69. 121. H. Giesekus, J. Non-Newtonian Fluid Mech., 12 (1983) 367. 122. H. Giesekus, J. Non. Equilib Thermo., 11 (1986) 157. 123. J.Y. Yoo and H. Ch. Choi, Rheol. Acta, 28 (1989) 13. 124. A. Oztekin, R.A. Brown and G.H. McKinley, J. Non-Newtonian Fluid Mech., 54 (1994) 351. 125. G. Schleiniger and R.J. Weinacht, J. Non-Newtonian Fluid Mech., 40 (1991)79.
1066
126. A.N. Beris, M. Avgousti and A. Souvaliotis, J. Non-Newtonian Fluid Mech., 44 (1992) 197. 127. D. Vlassopoulos and S.G. Hatzikiriakos, J. Non-Newtonian Fluid Mech., 57 (1995) 119. 128. J.D. Schieber and J.M. Wiest, J. Rheol., 33:6 (1989) 979. 129. S.A. Khan and R.G. Larson, J. Rheol., 31:3 (1994) 207. 130. L.M. Quinzani, G.H. McKinley, R.A. Bronw and R.C. Armstrong, J. Rheol., 34:5 (1990) 705. 131. T. Isake, M. Takahashi, T. Takigawa and T. Masuda, Rheol. Acta, 30 (1991) 530. 132. R.G. Larson, J. Non-Newtonian Fluid Mech., 23 (1987) 249. 133. M. Grmela and P.J. Carreau, J. Non-Newtonian Fluid Mech., 23 (1987) 249. 134. M.A. Hulsen, J. Non-Newtonian Fluid Mech., 30:1 (1988) 85. 135. M.A. Hulsen, J. Non-Newtonian Fluid Mech., 38:1 (1990) 93. 136. A.I. Leonov, Rheol. Acta, 15:2 (1976) 6. 137. A.I. Leonov, J. Non-Newtonian Fluid Mech., 25 (1987) 1. 138. N. Phan-Thien and R.J. Tanner, J. Non-Newtonian Fluid Mech., 2 (1977) 353. 139. N. Phan-Thien, J. Rheol., 22:3 (1978) 259. 140. R.B. Bird and R.J. DeAguiar, J. Non-Newtonian Fluid Mech., 12 (1983) 149. 141. J.R. DeAguiar, J. Non-Newtonian Fluid Mech., 13 (1983) 161. 142. M. Doi and S.F. Edwards, J. Chem. Soc. Faraday Trans. II, 74 (1978) 1789. 143. E.C. Bingham, Fluidity and Plasticity, McGraw Hill Book Company, New York, N.Y. (1922). 144. W. Prager and K. Hohenemser, Uber die Ansatze der Mechanik der Continua Zeitschrit~ fur Angewandte Mathematik und Mechanik 12 (1932) 216. 145. A. Slibar and P.R. Parsley, J. Appl Mech., 29 (1959) 107. 146. A. Slibar and P.R. Parsley, in Elasticity, Plasticity and Fluid Dynamics, Haifa ( 1962). 147. J. Harris, Rheol. Acta, 6 (1967) 6. 148. T. Schwedoff, Congres de Physique, 1 (1900) 478. 149. J.F. Hutton, Rheol. Acta, 14 (1975) 979. 150. J.L. White, J. Non-Newtonian Fluid Mech., 5 (1979) 177. 151. J.L. White, J. Non-Newtonian Fluid Mech., 8 (1981) 195. 152. Y. Suetsugu and J.L. White, J. Non-Newtonian Fluid Mech., 14 (1984) 121. 153. S. Montes and J.L. White, J. Non-Newtonian Fluid Mech., 49 (1993) 277. 154. A.I. Isayev and X. Fan, J. Rheol., 34 (1994) 35.
1067
155. D. C-H Cheng and F. Evans, Br J. Applied Physics, 16 (1965) 1599. 156. D. C-H Cheng, Br J. Applied Physics, 17 (1966) 253. 157. Z. Kemblowski and J. Perera, Rheol. Acta, 18 (1979) 702. 158. A.I. Leonov, J. Rheol., 34 (1990) 1039. 159. M. Simhambhatla, PhD Dissertation, The University of Akron (1994). 160. P. Coussot, A.I. Leonov and J.M. Piau, J. Non-Newtonian Fluid Mech., 46 (1993) 179. 161. M. Sobhanie, A.I. Isayev. and Y. Fan, Rheol. Acta, 36 (1997) 66. 162. FLUENT User's Guide, Fluent Inc., Lebanon, NH, V4.2 (1994) 854. 163. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, HcGraw-Hill, New York (1980). 164. J.P. Maruszewski, Fluent Inc. Technical Memo, TM-049, 1991. 165. B.P. Leonard, Draft for First National Fluid Dynamics Conference, Cincinnati, Ohio, July, 1988. 166. S.F. Shen, Int. J. Num. Meth. Fluids, 4 (1984) 171.
1069
FREE SURFACE VISCOELASTIC AND LIQUID CRYSTALLINE POLYMER FIBERS AND JETS S t e p h e n E. B e c h t e P , M. Gregory Forest b, Qi W a n g r and H o n g Zhou b
Department of Aerospace Engineering, Applied Mechanics, and Aviation, The Ohio State University, Columbus, Ohio ~3210 b Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3250 r Department of Mathematical Sciences, Indiana University-Purdue University at Indianapolis, Indianapolis, Indiana ~6202 1. I N T R O D U C T I O N
We provide a cursory history of thin filament modeling of fibers and fiber manufacturing processes (referred to as fiber spinning in the industry); extensive references can be found in our published articles. Our goal here is to identify those aspects of fibers and filaments which are accessible from slender I-D models, and which have been addressed in our own work. As such, this chapter will be a biased account of viscoelastic and liquid crystalline polymer fiber developments. An important issue is a unified derivation of I-D fiber models. Different industrial processes vary widely in the dominant competing physical effects. For example, [1] analyzed the competition between inertia and Newtonian viscosity; [2] considered viscosity and elastic relaxation. It is a fact of life that one does not know a priori what the dominant physics in the slender flow is, and moreover, it is highly likely that a slight change in the physical competition may be either beneficial or detrimental. For this reason, we have focused on a unified derivation together with a comparison of all the various leading order physical regimes; a similar point of view has been taken in [3] for Newtonian fibers where the number of possible dominant balances is manageable. For a wide range of inviscid [4], Newtonian viscous [5, 6, 7, 8, 9, 10],
1070
viscoelastic [11, 12, 13], and liquid crystalline polymer [14, 15] constitutive relations, together with surface tension, gravity, and inertia, we have developed a slender fiber perturbation algorithm that produces all onedimensional slender, axisymmetric, isothermal and thermal models in the literature, as well as several relevant new models. This approach posits a separable radial and axial structure at every order in perturbation, and therefore has one major advantage and disadvantage. The advantage is that the method is algorithmic, can be implemented with symbolic manipulations, and works whenever there exist separable slender flows. The disadvantage is that the method does not address whether the separable flows it describes are uniquely specified by the slender longwave approximation. For Newtonian flows this issue is resolved (private notes). But for more complex constitutive laws of the types considered here, it is an open problem to construct slender fiber flows which are not captured by our posited radial/asymptotic expansions. We have also addressed how to select the specific regimes and corresponding model which best fits a particular experiment or process. This modeling feature is essential both for direct simulations of a given experiment/process, and for inverse material characterization [16, 17, 18]. We refer to our original published work for details of these developments over the past decade. For this volume we survey a selection of leading order models for increasingly complex physical regimes. Similar to work by [19] and [20, 21], we have exploited the mathematical structure of transient models to deduce which quantities can, or must, be specified at upstream and downstream boundaries of the fiber spinline. In viscoelastic [13, 22] and liquid crystalline polymer [14] fiber models we have discovered a novel space-time change-of-type in which the governing equations for the fiber flow may experience a transition from a well-posed hyperbolic structure to a catastrophic elliptic structure. We have also identified this transient changeof-type with predictions for the onset of polymeric fiber failure [23]. The mathematical structure of the I-D models for our selection of physical regimes will further be shown below to reflect on the stability of two very different classes of steady states" cylindrical jets and noncylindrical fibers under tension.
1071 2. S L E N D E R F I B E R M O D E L S F O R I N C O M P R E S S I B L E I N V I S C I D , N E W T O N I A N , V I S C O E L A S T I C , A N D LIQUID C R Y S T A L L I N E P O L Y M E R F I B E R FLOWS
We refer the reader to our journal articles for the 3-D, incompressible axisymmetric free surface formulation and slender fiber perturbation theory. The fluid rheology is modeled in the 3-D equations by the constitutive (i.e. stress vs. rate-of-strain) relation one posits. The I-D models we survey below arise from the following constitutive assumptions for the stress t e n s o r T: Inviscid Fluids: T -- - p I ,
(1)
where p is the scalar pressure maintaining the incompressibility constraint, I is the identity tensor. Newtonian Viscous Fluids: T = - p I + 2~7D,
(2)
where r/is the zero strain rate viscosity, D is the rate-of-strain tensor given by
[Vv +
(3)
with V denoting the Eulerian gradient and v the fluid velocity. Viscoelastic Fluids: T - - p I + q?,
(4)
where
+A~DT=2y z)t
D+A2
.
(5)
z)t
Here '~1 is the relaxation time, A2 is the retardation-time constant, which is also called "polymeric viscosity", and - (~-~ + v . V ) ( . ) + Ft(.) - (.)f~ - a ( D ( - ) + ( . ) D ) ,
(6)
1072
where a is a slip parameter ( - 1 _< a <_ 1) (a ~ 1 is considered the most physical value, a = 1 is called the upper-convected Maxwell rate, a = 0 is the co-rotational rate), and f~ is the vorticity tensor given by [Vv - (Vv) ].
(7)
Liquid Crystalline Polymers: T = - p I + 2r/D + To~,
(8)
where To,. -
3ckO[(1 -
N I --~-)q - N ( Q . Q ) + N ( Q " q ) ( q + ~) I
+ 2A(Vv 9 Q)(Q + 5) ].
(9)
The orientational contribution to stress, To,., depends on flow and the macroscopic orientation tensor, Q; a discussion of Q is given later. Here r/is the solvent (Newtonian) viscosity; c is the number of polymer molecules per unit volume; k is the Boltzmann constant; 0 is absolute temperature: N is a dimensionless measure of the LCP density and characterizes the strength of the intermolecular potential; and the parameter )~ is the relaxation time of the LCP molecules associated with rotation of the dumbbell molecules. The LCP constitutive law (8), (9) requires a dynamical equation for Q as it couples to flow : ~ q - ( V v . Q + Q . Vv T) - F ( Q ) + G(Q. Vv), I
F ( Q ) - - ~ { ( 1 - -~)q - N ( Q . Q ) + N ( Q " Q)(Q + 5)}, G(Q, Vv)
_
(lo)
I ~2 D - 2(Vv" Q)(Q + ~)
where a a is the drag or friction parameter taking on values between zero and one; Crd = 1 is the isotropic case where the friction tensor is proportional to the identity. As a d decreases, this corresponds to increasing the ratio of resistance encountered perpendicular to the dumbbell axis versus resistance along the axis; the limit a a = 0 is the most anisotropic limit in which the friction is zero along the molecular axis. In equation (10) note that F characterizes the orientation dynamics independent of flow,
1073
whereas G describes the flow-orientation interaction. These macroscopic constitutive and dynamical equations are derived in [24] in the Doi closure approximation [25] which allows one to decouple the molecular-scale probability distribution function. Later we recall an efficient representation for
Q. We return to the general development: As we vary all possible combinations of physical effects which enter at leading order, for all regimes which are self-consistent a typical structure emerges in, the leading order I - D models in the axial coordinate z and time t:
Ndiag qt
+ Mf~u qz - f(q, qzz, b),
(11)
where q is a d-dimensional vector of unknown functions of (z, t) which decouple at leading order, e.g. the leading order axial velocity v, free surface radius r axial normal stresses J~zz, T~, and orientation order parameter s; Ndiag is a d x d diagonal matrix; Mf~u is a d x d matrix; b denotes body forces. In all I-D models which follow from this approach, the physical unknowns which decouple are those which are uniform in the fiber cross-section to leading order. In all the models presented below, the following dimensionless parameters are used throughout: R-
pzg r#o
(Reynolds number)
F - gt20z~ (Froude number) W-
proz~to 2 (Weber number)
A1 - ~1 to
(Weissenberg number)
A2 - & to
(dimensionless retardation time)
Ca
Pe-
-
(12)
~,to = rFo pCz~
Kto
Bi - hro K
R/W
(capillary number)
(Peclet number) (Biot number),
where a is the surface tension of the material interface, p is the density of the fluid, r0 and z0 are respectively the characteristic scMes in radiM
1074 and axial directions, to is a characteristic time, C is the specific heat per unit mass, I( is the thermal conductivity, and h is the heat loss coefficient. The ratio e = r o / z o is the s l e n d e r n e s s r a t i o , 0 < e < < 1, which is the perturbation parameter in the asymptotic expansion. In this section we present the governing equations and, where possible, compute characteristic speeds. Solutions to these equations are presented in later sections. 2.1 Model 1" Inviscid fibers dominated by surface tension and inertia In the inviscid model [4, 6] the axial velocity v and the free surface radius r are decoupled at leading order, based on the inviscid constitutive assumption (1)-
1 o
v,
v
~
r
0
1-[o]
(13)
The characteristics are given by dz d t = 81,2 -
i v Jr v / 2 W ( ~ , i -
(14)
x/~.
Hence the model equations are elliptic, and ill-posed as an initial value problem. This model has classical (Hadamard) catastrophic instability; it results fi'om the longwave approximation of the free surface Euler equations and, of course, is not a property of the full 3-D equations. Refer to [5, 13, 22, 23] for physical interpretations and cautionary remarks on ill-posedness in slender models. Refer to [4] for similarity solutions which yield predictions for the conical shape of an inviscid drop tip. 2.2 Model 2: Viscous fibers dominated by viscosity, surface tension and inertia In this model, we promote viscosity and gravity to leading order. The model equations are derived with the constitutive law (2):
0 r
~
__ 1
w
r
v
][v l I z
~
r
0
~zz+~r162
1 02
]
.(15)
This model, together with physically-motivated ad hoc regularizations, has been applied extensively to the study of viscous drop formation [7, 8, 9, 10].
1075 2.3 Model 3: Viscoelastic fibers dominated by surface tension, inertia, viscosity and relaxation (a Maxwell fluid)
In this model we promote elastic relaxation to leading order, so that now surface tension, inertia, viscosity and relaxation are dominant effects. The model equations are deduced from the constitutive law (4) and (5) with A2 - 0. The axial (~'zz) and radial (Try) extra stress components now couple to the free surface radius (r and axial velocity (v) at leading order. Letting u - (r v, Tzz, T~), we find (16)
N ut + M Uz - f ( u ) ,
where N -- diag[0, 0 , - T z z , - J ] , . ] ,
(18)
f - (0, O, -Tzz, - T ~ ) , v -~ 2 1 (/)2 w v 0 AI(1 - 2a)Tzz - 2Z r
M-
AI(1 -[- a)~'rr
0
0 -B A1 v
0 B 0
0
A1 v
-]- Z r
'
(19)
where
B-
fot
(20)
Z = qr~ tofo'
(21)
f0 is a characteristic force, and B-k2is the Reynolds number. The characteristics of the quasi-linear first order system in Model 3 can be calculated explicitly:
dz dt = s j -
[ v / v-4-~
j-1,2 j - 3,4.
(22)
Here A is given by 1
/ k _ A I R 3 r q_ B (J'zz + 2 rrr) - - ~ in which the first term (AI-~ r
r
(23)
is the "gain" from high viscosity, the second
term (B (Tzz + 2 2~r)) is the "penalty" for compressive stresses (for which
1076
it is negative) , and the third term ( T~ 1 oF) is the "penalty" for high surface tension. The system (16) is genuinely hyperbolic if and only if A > 0, indicating well-posed evolution. In particular, the system (16) is: 9 Hyperbolic type (3, 1) (meaning 3 positive and 1 negative characteristics) if and only if A > (Ov) 2, where s'l, s2, s3 > 0; 5"4 ~ 0. In this case, there are exactly 3 upstream boundary conditions and 1 downstream boundary condition. Therefore, the system is well-posed for modeling fiber spinning processes as long as this inequality on the solution is maintained. Moreover, the Riemann variables prescribe what process quantities may be imposed upstream and downstream. 9 Hyperbolic type (4, 0) if and only if 0 < A < (~v)2 where all four characteristic speeds sj are positive. In this case all information moves downstream and one can only impose upstream boundary conditions. This same system therefore describes extrusion flows as long as the above constraints are satisfied. 9 Mixed elliptic/hyperbolic type if and only if A < 0, indicating illposedness. The equations are not applicable for dynamical evolution in this regime. The boundary between well-posed hyperbolic evolution and ill-posed mixed-type behavior is A = 0, which is not an invariant condition (i.e. the sign of A may change along exact solutions). [22] studies families of exact model solutions which yield all varieties of hyperbolic to elliptic/hyperbolic transitions. In Figure 1 we show one possible space-time boundary of this well-posed/ill-posed transition, corresponding to the exact model solution: (~ - - r
V --
VO~
Tzz(t, z ) - e-t/AiAo(z- vot),
(24)
T~(t, z) -- e-t/a~[Ao(z -- vot) + A~ where A ~ qSo, vo are constants and the function Ao(s) is arbitrary except that Ao(s) =/=O , - A ~
1077
10-
.
61
n
2
4
6
8
10
Figure 1" The regions in the (z,t) plane of hyperbolic and mixed elliptic/hyperbolic type which evolve from the Cauchy d a t a with the discriminant A - A ( e c ) + e-88 v0t), where v0 - A1 - 1, A(z) - 2 + sin(z) _> 0, ~ >- 0, z x ( ~ ) - 2wA~ ~o ( ~- A o 1) _ - 0 . 5 , so t h a t the system is hyperbolic at time t - 0 for all z _> 0. A(ec) - - 0 . 5 implies t h a t the system becomes mixed elliptic/hyperbolic as t --+ oc for all z > t. From [22].
1078
2.4 Model 4" Viscoelastic fibers dominated by surface tension, inertia, viscosity, relaxation and retardation (a Johnson-Segalman fluid) When an elastic retardation effect is included, the extra stress can be decomposed into two parts, A2 rr _ Tp + 2r/~ll D
(25)
where Tp is the polymer part of the stress, A2 is the retardation time and 2 q ~ D is the Newtonian part of the stress. Substituting the decomposition into the generalized Oldroyd Fluid-B model, the constitutive equation (5) becomes a Maxwell-type equation for Tp, rrp 4- )kl - ~
= 27/(1 - ~11 D.
(26)
This is the Johnson-Segalman formulation for viscoelastic flows. From this formulation, the leading order model is a 4 x 4 degenerate parabolic system, involving the same unknowns as in Model 3: ut + C(u)u~ = f(u) + Duzz,
(27)
where u -- (4), v, 7"zz, ~,.,.)t, v
~2 1
+
0
0
-B
B
Tzz)
C(u) -
(28)
0
-2a~z
0
Z(1-A1) a Z , r + " A1
0 0
0 0
O0
0 0
0 0
0
0
V
'
A,, T1 + 6 BZ-~11dz Vz f(u)
O0
v
0
0 3 B Z ~A1 0 0 D
- 2Z(1-~)_A~
-
.
_T_~ A1
T~ A1
(29)
1079
All the physical parameters are the same as before. Due to the decomposition of the extra stress, the range of A2 is restricted to 0 _< A2/A1 _< 1. 2.5 M o d e l 5: L C P inviscid fibers d o m i n a t e d by i n e r t i a , s u r f a c e t e n s i o n , gravity, a n d m o l e c u l a r - s c a l e o r i e n t a t i o n effects" p o l y m e r k i n e t i c e n e r g y , an e x c l u d e d volume potential, and anisotropic drag
From [14, 15] the uniform leading order behavior in the LCP fiber crosssection is given by the special uniaxial representation for the orientation tensor Q: 1 3'
Q-sdiag[
1 2 3' 3 ].
(30)
where s is the scalar order parameter, which is restricted to the range 1 < s < 1 From this representation, the slender fiber equations for this regime take the form [14]: 2
~
~
"
ut + C ( u ) U z - f,
where u -
(31)
v, s) t,, f - (0, 1/F, -ad/AU(s)) t, and
(0, v
6"~.(U ~ __ '~.w't
1
we ~
]
2aU(s)
o
r
r2 V
0
(32)
-~U'(~)
-h(~)
v
The eigenvalues of the coefficient matrix, which determine the characteristic speeds, are ~1
m
V~
~,3 - v + qA(~, r ~, w, m), A(,. r ~. w. -
- ~[h(.)U'(.) ~[h(~)U'(~)- u ( ~ ) - c_~], ~<~
where with
N)
(33) -
u(.)]
1
2Wr
f U(s)ds is an effective intermolecular excluded-volume potential,
N(1-s)(2s+l)],
U ( s ) - s[1 - --~
(34)
h(s)-(1-s)(2s+l),
(35)
1080
3ck0
(36)
which parametrizes the molecular kinetic energy per unit volume relative to inertial energy per unit volume; and Catcp is the LCP capillary number, (7
= 3ck07-----
(37)
The discriminant A(s, ~b; a, W, N) may be positive, zero, or negative depending on the solution variables s and ~, and depending on the parameter values a, W, N. In particular, this regime has the possibility to evolve from a well-posed hyperbolic initial-boundary value problem into an illposed elliptic-hyperbolic region at a critical space-time location, signaling a rapid departure from slender fiber behavior. Note the strong similarity between this LCP model system (31)-(37) and the Maxwell model system (16)-(23). When A > 0 the quasilinear system is strictly hyperbolic. W h e n / k - 0 the system is degenerate, with three identical characteristics and only two independent eigenvectors. When A < 0, then the system is of mixed elliptic-hyperbolic type, and is therefore ill-posed as an evolutionary system. In the context of general solutions, the sign of A varies through a competition of the solution variables s, ~; at particular values of s, ~ the hyperbolic versus elliptic behavior is governed by the parameters N and Cal~p. 2.6 M o d e l 6: T h e highly a n i s o t r o p i c d r a g limit of M o d e l 5
This model is formally obtained fi'om Model 5 by letting ad = 0 in the leading order equations; refer to [14]. The nonlinear classification in Model 5 is unchanged, since the terms proportional to ad only contribute to the nonlinear lower order terms in the equation for s. Thus, the criteria for hyperbolic versus mixed hyperbolic-elliptic type are as given in Model 5. In this model one loses the selection of discrete branches of equilibria determined from the zeroes of U(s); the intermolecular potential is effectively constant in this limit so all values of s are allowable equilibria. The parameter values for which A(s0) < 0 correspond once again to ill-posed equilibria, as discussed in Model 5.
1081 2.7 Model 7" Viscous generalization of Model 6
This model couples viscous terms, both Newtonian and orientationdependent, to Model 6, retaining its highly anisotropic drag limit (ad = 0) [14] The leading order system is given by r + V~z + ~Vz/2 = o,
(02~), + (02~)z
_
1(~2
~
1
+ ~r
(38)
+ ((,7~sf<~ + ~u(~))r
st + VSz -- Vz(1 - s)(2s + 1),
where fluff(s) is a I-D effective viscosity consisting of a Newtonian elongational viscosity (gr/) and an orientation-dependent contribution (2a)~s2),
~ss(~) - a~ + 2~,~ ~.
(ag)
2.8 Model 8" The most general L C P fiber regime, with a r b i t r a r y drag, L C P relaxation, inertia, viscosity, surface tension, gravity, and L C P kinetic energy
We now couple arbitrary anisotropic drag effects to Model 7. Let u - (r v, s) t. The leading order equations can be written as N d u t + Muz - G(u) + (Uz, Cuz)(0, 1,0) t + 'q~ffr
1,0) t,
(40)
where Nd -- diag[1, O2, 1],
(41)
G(u) - (0, s
(42)
o C
r
Cr/~SS(,) 0
o 0
2a)~sr 2
v W
0
2ogU(8) ~
(43)
0 r2
1
M
2(~Asr 2 ,
V(~2
- ( 1 - s ) ( 2 s + 1)
0 - o z U ' (8)(~ 2
v
The form (40) amplifies various properties of the equations:
(44)
1082
i) The Nd, M and G terms correspond to quasilinear 1-D behavior along characteristics -3-i dz -- aj, where aj are the eigenvalues of N d l M . ii) An indefinite nonlinearity in derivatives of u appears in the second (axial momentum) equation, in the way of a quadratic form with symmetric off-diagonal C. If this quadratic form is diagonalized, one finds the sign of these nonlinearities in derivatives is indefinite. iii) A degenerate viscous regularization term, rl~llr appears only in the axial momentum equation. This yields a viscous Burgers equation for the axial velocity v with additional nonlinear derivative terms, coupled to quasilinear equations for the free surface r and order parameter s. 2.9 Model 9: Thermotropic LCP fibers
In this model we couple temperature (0) through an energy equation. We refer to [26] for the derivation, and a discussion of the application of this physical regime. The leading order equations in the melt-phase including all effects in Model 8, are: (r
+ (v62)z _ 0,
(r
+ (O2 2)z -
+ 1
+ ( ~2
0)v: + (45)
st + VSz - Vz(1 - s)(2s + 1) - Ac0)U(s), 2Bi ( 0 - 1 + A0) Pc(Or + vOz) __ -1~ (~2 O z ) z - --~
where A(0) is the scaled LCP relaxation time, A ( 0 ) - Ae~(a/~
(46)
R~ff(s, O) is an effective 1-D flow-orientation Reynolds number, R e ; f ( 8 , O)
--
3
egCl/~
-~- 2 ol 0 A(0) 82,
(47)
consisting of a Newtonian contribution and an orientation contribution (2o~OA(O) s2); and U ( s ) - s ( 1 - N / 3 ( 1 - s)(2s + 1))defines the uniaxial bulk free energy, f U(s)ds. The dimensionless parameter a characterizes the molecular stress relative to inertial stress; ad/A parametrizes anisotropic drag on polymer
1083
molecular motion relative to the solvent; N is a polymer density parameter that enters prominently in the intermolecular excluded-volume potential; Pe is the Peclet number which is a measure of specific heat relative to thermal conductivity; Bi is the Biot number characterizing the heat loss relative to thermal conductivity; g is the Griffith number quantifying the degree of viscosity variation with temperature. The nonisothermal parameter A0 is chosen as AO
--
1 -- Oa/Omelt
(48)
,
which is a measure of the degree of nonisothermality. When Oa -- Omelt, then A0 = 0 and our model reduces to the isothermal case. In [26] we have varied this parameter to study the energy effects on our previous isothermal LCP spin model predictions [15]. One can also formulate (45) into the general vector form (11) (see [26]).
3. L I N E A R
STABILITY
OF CYLINDRICAL
JETS
Insight into the structure of the 1-D slender fiber models of the previous section can be deduced by linearizing the models about constant cylindrical jet solutions when they exist (i.e. when gravity is negligible). This calculation can then be compared against Lord Rayleigh's analysis of the capillary instability of circular cylindrical jets [27]. The comparison clarifies the nature of the slender, longwave model for each selection of dominant physics. First we recall exact 3-D results. Another discussion along these lines may be found in [28]. Rayleigh's [27] linear stability analysis of a cylindrical inviscid jet with radius r0, constant axial velocity v0, surface tension a, and constant mass density p shows that an infinitesimal perturbation of the jet free surface of wavelength 27r/k grows exponentially with rate
7(k)-
pr~
Io(kro)
'
(49)
where I0 and I1 are the modified Bessel functions of the first kind of order 0 and 1, respectively, indicating a longwave instability for all wavenumbers k such that 0 < k < 1/ro, with a finite cutoff at the wavelength of the jet circumference 27~r0.
1084
To compare this dimensional result with slender asymptotic growth rates, we nondimensionalize the wavenumber k and growth rate 7(k) as follows"
k--,
~(k) - ~ , to which yields
(50)
zo
I1 ~n,(k)-
k (1 - ~'2d)I1(~)
W e
Io(k'e)
'
(51)
where -
(52)
ro/zo
represents the radial to axial aspect ratio and W is the Weber number. In nondimensional form, the exact linearized growth rate a/vi~c(k) for a viscous fluid jet is described by the transcendental equation [29, 30]
(7~visc)2 ~(]r Jr- ")/visc~~2 [2 r --- 2T4/(1 -(kc)
- 1]
2) - ~2~'4 7(~;~)- f(~,~) ,
(53)
where 7 ( : ) - : i0 (~-) /l(Z) '
(54)
(~5)2 __ (~6)2 -t- ZYvisc /~ ~2.
(55)
Looking forward to a comparison of slender 1-D models and these exact 3-D formulas, consider the slender longwave limit Ik,el < < 1. The exact inviscid rate (49) has the expansion %,~,(~) - v/g ~ + o(~2/~2),
(56)
while the viscous rate has the expansion [30]
#visc(k)
--
3~:2 / [3~2, 2 ~2 2R + ~ [ ) - ~ + ~ + ~
(57)
1085
|
.
.
.
.
|
.
Inviscid,..E'~
3.5
xact e2.5 c-
2
o
c,1.5
er
.
.
.
.
.
.
.
.
.
.
.
.
0.5
Oo
I
I
I
I
2
4
6
8
10
12
k
Figure 2: Nondimensionalized Rayleigh growth rate versus wavenumber k for inviscid or viscous jets. The exact inviscid Rayleigh growth rate, together with the exact viscous Rayleigh growth rate, the leading order approximate inviscid growth rate (56) and the leading order approximate viscous growth rate (57) where W - 1.0, e - 0.1, R - 10, is provided in Figure 2. As shown in Figure 2, the inviscid and viscous growth rate share the same cutoff. Now we investigate linear stability of constant cylinder equilibrium solutions to all leading order slender fiber models presented in the previous section (with ~1 __ O) and display how these models approximate the Rayleigh instability. The constant solutions always have radius r - 1, axial velocity v0 - 1, with the remaining constant unknowns specific to the model. For simplicity, we drop the superscripts. We posit that the perturbation 5u of the constant solution is of the form (~U ~
r
c,
(5s)
where c is a constant vector. The linearized growth rate is given by the real part of u, 7 - Re(u). The I - D linearized growth rate of each model is given as follows (recall all the tilde overbars have been dropped)"
1086
Model 1" Inviscid fibers dominated by surface tension and inertia:
,7 nv(k) - k ( 2 w )
(59)
Model 2" Viscous fibers dominated by viscosity, surface tension and inertia:
~/visc (~) --
3k2
2R +
I k?-R] (3k2~2
k2
(60)
Note that these slender model growth rates agree, as they must, with the leading order slender longwave approximation of the exact 3-D inviscid and viscous growth rates (56) and (57) (Figure 2). Model 3: Viscoelastic fibers with Maxwell fluids: The growth rate of Model 3 is given by a fourth-order algebraic equation [22] for the four values 7j(k) (j - 1 , . . - , 4). There is one trivial factor of this characteristic polynomial, 71(k) - -1/A1, corresponding to linearized exponential decay at the relaxation rate (A1)-1. The three remaining rates, 7j(k), k - 2, 3, 4, are the roots of the following cubic polynomial: 1
k2
2
7 3 + ~117 + 2WAI
k2 -
2WA1
=0,
(61)
where
A=
6
Ca
(62)
A1.
Note that the sign of the linear term in 7 fixes the characteristic type of the constant solution. Application of Descartes's rule of signs shows there is always one positive real root, which we denote 72(k), of (61), independent of the sign of the linear term, while the two remaining roots V3,4(k) are complex with negative real part. A simple asymptotic balance for k ~ ec determines the behavior of 72 (k) : A -1 72(k) ~
/A
A>O 6w
k~/2w~
as k---~ oo,
A < 0 as k ~ oo, k2/3( i2w n 2) 1/3 A - 0 as k ~ o o .
(63)
1087 Model 4" Viscoelastic fibers with Johnson-Segalman fluids: As in Model 3, there is one linear factor:
,7,~odd4(k ) _ _ 1 .
(64)
A1 The three remaining growth rates satisfy the cubic polynomial equation: 1 3k2A2 k2 6 k2 ~3 _[_ (All -+- AIR )3/2 4- 2WA1 (~aa - A1)~/- 2WA1 -- 0.
(65)
The growth rates in the range of short waves are given by
( i
(WA1) -1 m -~- A 2 -~- RWA2
~/Js "
a~ ( - A 6RA2
+ j A2 + RWA~
-t-O(]r
+ O(k- 1)
A ~_ 0 for ]~-1 e,o 0 (66) A < 0 for k -1
whereA-- 3 1 andR1 RA1 2W BZ" In Figure 3 we plot the growth rates for the leading order inviscid, viscous Rayleigh results and the growth rates for Maxwell, Johnson-Segalman models with small (Ca = 1.0) and large (Ca = 10) capillary numbers Ca. The remaining parameters are A1 = 2.0, A2 = 1.0, R = 10.0, e = 0.1, and the Weber number is determined by the Reynolds number and the capillary number through W = R/Ca.
Model 5: LCP inviscid fibers with anisotropic drag" With a nonzero anisotropic drag parameter present at leading order, there are three branches of constant solutions parametrized as in Figure 4. The linearized growth rates are given by the roots of the cubic equation to follow, which we present in a suggestive form with the drag terms isolated on the right hand side: ,),3 + / k ( s j ) k 2 ~ _
ad U' (sj)(k2(2W)-i _ .y2), A
(67)
where A(sj) for equilibrium solutions is given by (where for consistency
r
1)
A(sj) -- c~h(sj)U'(sj)- (2W) - 1 - ct h(sj)U'(sj)
Calcp 2
'
(68)
1088
u
m
.
i.
m
m
m
9
Inviscid Leading Order,' 3.5 ." / 9
9
9
..'" ,'Maxwell (large Ca) /
9
~2.5
-I~
x:
/
-'/"/ "
." 9
/
/
/
2
o
o~1.5
0.5 O0
m
m
m
i
i
m
2
4
6
8
10
12
k
Figure 3: Nondimensionalized growth rate versus wavenumber k for leading order inviscid or viscous, Maxwell or Johnson-Segalman jets. The lower part of the figure corresponds to Maxwell model with small capillary number (dash-dotted curve), Johnson-Segalman model with small capillary number (solid curve), and the leading order viscous Rayleigh result (dotted curve), respectively. and
sj (j -
s~-0,
1, 2, 3) are zeros of
s2,3-~
1+3
U(s)given by
1-~-~
.
(69)
The longwave and shortwave asymptotic behavior of (67) for Model 5 are" 3'21 ~ -i-k(2W) _ ~1 - ~ A k2 for k ,,~ 0 '
~
"~3 ~ - ~ u ' ( s j )
and
I
")'3,1 e,o
'
'
(70)
aAh(sj)k2 for k ~ O,
+ ~
-t-~q--/k(Sj)-
~,~ ,.., "2WA --~-U'(~;)(A(~;))
~.~)h(sj)U,2(sj) for k-1 ~
2AA(sj -1 - F
'
(9(k-2), for k -1
,-., o.
0,
(71)
1089
0.5
U/>0 i i
~ U ~ <0 Ok
$1
o
8/3 \ 3
U'>O
U'< 0
-0.5
U'>O
,
-10
,
,
,
I
2
,
,
J
,
I
,
,
4
,
,
I
6
,
,
,
'
I
8
,
,'
,
,
10
N Figure 4: Real order parameter equilibria s versus LCP density N. The solid curve indicate where U(s) = 0 and U'(s) is positive while dashed curves are where U(s) = 0 and U'(s) is negative. There is a turning point at N = 8/3 where two real equilibrium branches s2 and sa are born. At N = 3, s2 collides with Sl and U'(sl) and U'(s2) change sign, which shall be responsible for transitions in the orientation-dominated linearized modes as we analyze stability of these equilibria.
1090 Model 6: LCP inviscid fibers with highly anisotropic drag: In this model with Od = 0, any equilibrium order parameter so is allowed in - ~1 <_ so <_ 1. However, for comparison with all other models where the critical points of f U(s)ds are the selected equilibria, we only analyze these discrete equilibria. The growth rates are given by (67) where ad = 0:
(72)
"/2,1-Ar-kRe(~/-A(8j))
(73)
9
Clearly, when U'(sj) < 0 (see Figure 4), then A(sj) < 0 and the equilibrium is catastrophically unstable. This is, of course, consistent with the classification earlier that shows the full nonlinear equations are elliptic when A < 0. When U'(sj) > 0, then A(sj) > 0 only if Catcp is sufficiently small, Catcp < 2h(sj)g'(sj). In such cases (in particular for the prolate phase, s2 (1 + 3x/1- 8/(3N))/4 ), when LCP surface energy dominates surface tension, then the growth rates are identically zero, indicating neutral stability. -
Model 7" LCP viscous fiber with highly anisotropic drag" Now this model couples Newtonian and effective LCP viscosity to the previous model. Intuitively, one expects a "viscous regularization", i.e., the previous model growth rates should be lowered. The linearized growth rates 71,2,3 are easily calculated in closed form: ~1 ,2 - - -~(--r]eff(80) k2 -j"
"73
-
O, 74
-
O, "/5
-
O.
2 k4 - 4A(so)k2), ~ff(so)
(74)
(75)
Again we evaluate so at the critical points of the 1-D Maier-Saupe potential, sj. The vanishing of 73 reveals the orientation instability, realized through growth in the uniaxial order parameter s, is also suppressed when ad -- 0 at leading order, i.e., the highly anisotropic drag limit.
1091
We turn now to the Rayleigh-dominated rates. Note that ~/1 + ~2 ~1~/2 - - k2m(sj). With regard to stability, the sign of A(sj) is critical:
-k2rl~ff(sj) < 0, and
If A(sj) > 0 then Re(")/1,2) a r e both negative for all k, with the remarkable result that orientation contributions to the free surface energy, in the limit of hyper-anisotropic drag ( a d - 0), completely stabilize the surfacetension-driven Rayleigh instability. If A(sj) < 0 then the growth rate Re(~/2), which reflects the Rayleigh instability, is positive and bounded for all k, with qualitative behavior similar to the viscous Newtonian 1-D Model 2 with surface tension and inertia. Therefore, for such equilibria with sufficiently large Calcp, the Rayleigh instability survives independent of the drag anisotropy. This regime suggests that the degree of drag anisotropy plays a critical role in the orientation-flow coupling, in particular with regard to the possibility that orientation contributions to free surface energy may suppress the Rayleigh capillary instability. This remarkable suggestion is made in the mathematical limit, aa = 0, corresponding to zero drag along the dumbbell axis relative to the drag orthogonal to the molecular axis. Model 8" LCP fibers with all effects coupled at leading order: This is the most general regime of LCP fibers, with all effects coupled at leading order. The linearized dispersion relation for Model 8 is
dU'(sj)k 2 =0. 2W~
(76)
In Figure 5 we compare the growth rates for all LCP slender models.
4.
FIBER
SPINNING
STEADY
STATE SOLUTIONS
AND THEIR
STA-
BILITY
In Figure 6 we show the numerical solutions of the viscous Model 2 with
1092
3.5
L
'
'
'
....
'
'
/"
'
'
'
'
-i
2.5
2
1.5
1
0.5
0
~
Model
N
6 with s m a l l C a
\
-0.5
-
\. \
I
\
'
"-
" "-"
" --
Model-7
" ~
with- s m a l l C - a
-
!
'/ --1
-
-1.5 0
, 1
, 2
I .... 3
, 4
I 5 k
, 6
, 7
I 8
J 9
10
Figure 5" Growth rate as a function of wavenumber k for different models. 1 W' 1 and T" 1 The upstream boundary different values of the parameters ~, condition is chosen as v ( O ) - 1,
(77)
while the downstream boundary condition is v ( 1 ) - 10.
(78)
Figure 7 shows a typical steady state profile for a Maxwell liquid filament (Model 3), while Figure 8 gives a typical steady state profile for a JohnsonSegalman liquid filament (Model 4). In Figure 9 we present a family of steady state solutions of the liquid crystalline Model 8 reflecting variations due to changes in the initial degree of orientation, s(0) = So. Note that there are uniform responses in 0, v, s to
1093
changes in so. Physically, we predict that lower so yields steady processes in which the filaments are thinner and faster at each interior spinline location. Figure 10 shows the qualitative dependence of the steady spinning solution on the draw ratio, which in our nondimensional model coincides with the take-up velocity. We have considered a range from slow spinning (Dr = 2) to fast spinning (Dr = 40). From 10 one observes that a higher takeup speed leads to higher axial velocity, thinner fiber radius, and higher degree of orientation for all z > 0. In [15] we have also studied how the steady spinning solution responds to variations in Reynolds number (R), Weber number (W), and Froude number (F). Our numerical results indicate negligible quantitative changes in the free surface r axial velocity v, and uniaxial order parameter s due to variations in W and F, and still very small quantitative changes due to variations in R. Sensitivity to these hydrodynamic parameters becomes evident, however, when we analyze stability of these steady states [15]. For further studies on the quantitative influence of LCP parameters (e.g., anisotropic drag coefficient, LCP kinetic energy relative to inertial energy, and dimensionless relaxation time), refer to [15]. In Figure 11 we present linearized stability results of the steady isothermal LCP spinlines of Figures 9 and 10. Figure 11a displays the maximum growth rate curve for the s0-parametrized family of Figure 9, from which we conclude that the entire Figure 9 family of steady states is linearly stable. Figure l l a further indicates that a weaker upstream degree of orientation leads to more stable steady states. The maximum linearized growth rates as a function of Dr, for the steady state family of Figure 10 is plotted in Figure llb, from which we deduce that faster processes are less stable in this region of parameter space. This might seem intuitively obvious, but we refer to a study [5] of slender Maxwell fluid models where the maximum growth rate curve oscillates about zero with varying Dr, creating alternating windows of stable and unstable draw speeds. Figure 11c provides the full two-parameter neutral stability curve: the critical draw ratio, Dr*(so) for each so, above which the steady state is linearly unstable. We deduce that the critical draw ratio for all steady states of Figure 9 is between Dr - 27.5 and Dr - 30.5. Most notable, however, is the shape of this neutral stability curve, indicating there is preferred initial degree of orientation (around so = 0.25) associated with
1094
the maximum stable drawing speed (around Dr - 30.4) for this twoparameter family of steady states. Now we present steady fiber spinning solutions to Model 9. Consistent with our nondimensionalization, the upstream conditions on fiber radius, velocity, and temperature are fixed:
r
1, v ( 0 ) - 1,0(0)- 1.
(79)
The downstream boundary condition is selected by the assumption that axial thermal conduction is negligible downstream, i.e.,
0(r Oz
(1) - 0 .
(80)
The remaining boundary conditions are free processing parameters to be specified/varied in the simulations below" s(0), v(1).
(81)
The upstream degree of orientation (s(0)) is a function of spinneret design, whereas the take-up speed (v(1) - draw ratio - Dr) is a measure of process speed and throughput. In Figure 12 we present typical steady state solutions of Model 9 due to changes of A0. Here a two-phase model has been posited [26], where the governing equations for the temperature above the glass transition temperature (0 > 0g) are given in Model 9, and in the solid phase (0 _ 0g) we employ a rigid cooling LCP fiber model so the velocity is constant (fixed by the take-up speed) whereas the orientation and energy equations are maintained. (The parameters used here are for academic illustrations, and are not taken from experiments.) Figure 13 depicts the critical draw ratio Dr* as a function of the nonisothermal parameter A0. Three different forms of N have been used, the constant value N - 4 and then two temperature-dependent forms [26]. These forms are chosen so that the intermolecular potential f U(s)ds has only the isotropic critical point for sufficiently high temperature, then passes to a double-well potential for lower temperature. As shown in Figure 13, the critical draw ratio grows with A0. As A0 increases from 0 to 0.4, there is 9.3% gain in Dr* for N c~ 9.7% gain for N linear, and 9.5% gain for N quadratic. These predictions clearly indicate that the cooling process increases stable spinning speeds. Note that for A0 - 0, N c~ -" 4, N linear : 2.1, and N quadratic - - 2.52, the qualitative differences in these isothermal steady states are negligible,
1095
"'111111/I/1[[
1,...
.
12 .
.
.
~
0.8
0.6
10
>
6
0.4
0.,~ .....................
1 ..,
0.8
0.{
0.2 ~
0.4
z
9
0.6
0.8
o
,
1 (3.,'}
%
II iii|I |1|
Ill
0.2
0.4
0.6
0.8
o:6
o'.8
12
. ."""/. ////////,//////////,,,,:i~
O.z
1
0.~
......
11111111111111|I
..............
CO
1
i
0.2
0.4
!
i
I
0.6
0.8
1
z
(b) ~
o'.2
o:4
z
12
/
10
0.!
/-.:y
0.6
if:: > 6
.../.! i /
0.4
........;:;r i ....
0.2 ........... .,.,.,.,~.,~,-r~ . . . . .
Oo
0.2
0.4
0.6 z
0.8
1
o -0
0.2
0.4
0.6
0.8
z
Figure 6: The variation of fiber radius 4) and axial velocity v due to the changes of: (a) l / F , where 1IF goes from 0 to 5 in increments of 0.5, with 1 / W - 1 / R - 1.0; (b) l / W , where 1/W goes from 0 to 5 in increments of 0.5, with 1 / F - 1 / R - 1.0; (c) l / R , where 1 / R goes from 1.0 to 5 in increments of 0.5, with 1/F - 1/W - 1.0.
1096
20 0.8
15 >,, ,,,,,,,,,,
0
~50.6
o10
L
>
0.4
0.2 0
!
0 0
,,,
0.5 axial coordinate
I
0.5 axial coordinate
1
0.5 axial coordinate
1
30 25 03
-0.5
20 -I
9-~ X 15 L
-I .5
10 !
0
0.5 axial coordinate
1
-2
0
F i g u r e 7: A t y p i c a l s t e a d y s t a t e profile for a M a x w e l l l i q u i d f i l a m e n t in a fiber spinning process.
The parameter
1, A1 - 0.05, 0 ( 0 ) - v(0) -
values are B -
1, T,.,.(0) - 0, D r - 20.
W -
F -
Z - a -
1097
20 0.8
|
15 >.,
to,)
,,,..,.,
rj
~0.6
o10 (1) >
0.4 0.2 0
0 0
I
0.5 axial coordinate
I
0.5 axial coordinate
300
r,/)
200
(l)
L
~-4
,,,,,... ~
-2
..,,,
•
"0
100
L
-6 0
~
0
~
0.5 axial coordinate
--
--8
1
"'
0
~ '"
0.5 axial coordinate
Figure 8" A typical steady state profile for a Johnson-Segalman liquid filament in a fiber spinning process.
1
1098
Figure 9: Steady solutions of LCP fiber spinlines reflecting variations due to changes in the initial degree of orientation, s(0). We vary s(0) = so from .05 to .95 in increments of .05. Arrows indicate the direction of increasing so. Figures 9a,b,c display the family of solutions, fiber radius r so), axial velocity v ( z ; so), orientation s ( z ; so), respectively. All parameter values are fixed at order one values" 1 / R - c~ - 5, 1 / W - 1, 1 I F - 1 , N - 4, A 1, a = 0.5. Boundary conditions are: 4)(0) = v(0) = 1, s(0) = .5, D r = 10 = v(1).
1099
Figure 10: Variations in the steady spinning state due to changes in draw ratio, Dr. All other parameter values and initial data are the same as Figure 9. D r - v(1) is varied from 2 to 40, in unit increments.
1100
//i /
-1.6 //
-1.7 /
-1.8
/
/
/
t
/
/
/
J
1. / i
/
/
l/max
//
-1.9
/J
/
/
-2.1
/
(b) 0.4
i
I
l 9
0.6
I
i i l I l
......................
0.2
/
/
/
//
/
/
/
// j/
/
/
j/
l/max
I
//
/
,
,
,
~
i0
0.8
.
.
.
.
:
D?
80
.
2O
.
.
.
:
3O
~
.
40
.
/ - ....... \
3o /
/'
\,
/
\ \
:
\
/
\ ',
29.5
'\
Dr* 29
\
28.5
(c) 0
0.2
0.4
0.6
0.8
80
Figure 11: Linearized stability information. Figure l l a concerns the s0parametrized family of Figure 9; Figure 11b concerns the Dr-parametrized family of Figure 10; Figure 11c summarizes the neutral stability boundary for a full two-parameter variation of the spinning steady state. Each data point in Figures l la,b depicts the maximum real part, Uma~ (vs. so in Figure 11a, vs. D r in Figure l lb), of all linearized growth rates for that respective steady state; a negative value of U,ia~ indicates linearized stability.
1101
1
10
: ""--:----:; ....-.....:...... -.... ;
0.9
B
0.8
J/#I/Y
....... ................
0.7 0.6 :i!~i
i
0.5
, ~ :i!;i:i:!:~:~:~''
~) ....
0.4
0-30
0:2
....0:4
0:6
~
0:8
0'2
o14
Z
z
0'6
0'8
1
(b) 0.8 0.6
..,~iii~!~
o:i~~~,,,~ ~ii!i!:~il!i,i:i~:~ .,
~ii~
0.4 0.2 0.65 O0
012
014
z
016
018
1
f
06;
0:2
0:4
z
0:6
. . . 11
0'8
(d) Figure 12" Steady state variations with respect to the nonisothermal parameter A0, 0 <_ A0 <_ 0.9, in increments of .045. All parameter values are fixed at order one values" c~ - 5, R 0.2, 1 / W - 1 / F A - 1, N4, a d - - 0 . 5 , P e - ~ c - c v 1, 0 g - 0 . 8 . Boundary conditions are: r v(0)- 0(0)- 1, s(0)- 0.5, v(1)- 10.
1102
43
9
9
n
i
42
41
~
I,,--
D
40
.1~i" 5~ J
...
.. : 1~:'~
~ ~...
39
""
... I
... ...
.~ .....
8
......
,
0
i
l
0.1
I
0.2 Ae
I
0.3
0.4
Figure 13" Critical draw ratio as a function of the nonisothermal parameter A0: effects of a temperature-dependent excluded-volume potential. The solid curve corresponds to N c~ = 4.0, the dotted curve corresponds to N linear = - 3 -t- 5 . 1 / 0 , and the dashed curve corresponds to N quadratic = 25 - 43.9/0 + 21.42/02. Here the material parameters are the same as in Figure 12. The A on the left margin of the solid curve corresponds to the critical draw ratio, Dr*, for the isothermal steady state of Figure 12.
1103 whereas the effect on process stability is on the order of 2%. Therefore, the dimensionless LCP density parameter N has a bearing even in isothermal stability of spinning states. 5. A P P L I C A T I O N S
OF S L E N D E R F I L A M E N T M O D E L S
We now discuss several applications of slender filament models, beyond the modeling of steady fiber manufacturing processes [1, 2, 3] and drop formation [7, 8, 9]. 5.1 O n s e t of p o l y m e r i c filament failure
In this section we use slender, one-dimensional asymptotic balance equations to model stretching and the onset of failure in a polymeric liquid filament. In one experiment designed by J. Matta to measure elongational flow properties of dilute polymer solutions, the fluid is slowly extruded from a vertical nozzle and forms a drop at the nozzle tip. As the drop reaches a critical mass it begins to fall, pulling a filament behind it. This experiment is described in reference [31]. Markowich and Renardy [32] modeled the Matta filament/drop experiment with simulations of a slender 1-D quasilinear parabolic integro-differential equation, derived from the 3-D Jeffreys (or Johnson-Segalman) constitutive law, which combines elastic relaxation and retardation. Their numerical simulations predict a qualitatively similar filament behavior for all parameter values reported. That is, an initial uniform relaxed filament always fails at the ceiling, as indicated in their model by an excessive localized growth in axial strain at the ceiling. Further, they report that enhanced elasticity delays the time of failure, while increased surface tension advances the failure time. The model we analyze here derives from a Maxwell viscoelastic constitutive law, which isolates the effect of elastic relaxation from elastic retardation. The Eulerian forms of both the Maxwell and Jeffreys 1-D slender axisymmetric approximations have been given in previous sections. The key distinction between the 1-D Jeffreys model and the 1-D Maxwell model is that the Maxwell equations may be either genuinely hyperbolic or mixed elliptic-hyperbolic, whereas the retardation terms from the Jeffreys constitutive assumption yield a quasilinear parabolic system. As a result the Maxwell model predicts a wider variety of filament behavior than the Jeffreys model, both in the location and nature of approach to failure, for
1104
the same class of initial/boundary data. The onset of filament failure is indicated by the 1-D Maxwell filament model through a hyperbolic to mixed elliptic-hyperbolic change-of-type. Our rationale for this interpretation is that a filament break must be accompanied by arbitrarily small axial length scales to resolve the rupture; coincidentally, the change-of-type in our model corresponds to a localized rapid generation of small lengthscales. Since the radius of the filament does not simultaneously vanish at the change-of-type location, the aspect ratio of radial to axial lengthscales is no longer small, in violation of a fundamental assumption of the I-D model. Thus, although our model breaks down at the change-of-type location, the way in which the model becomes invalid coincides with behavior associated with filament failure. Depending on the particular process conditions and material properties, the changeof-type location may occur anywhere from the ceiling to the drop and does not necessarily occur at the location of minimum radius. The diagnostic for change-of-type in the mathematical model is the discriminant defined in equation (23) which is positive for well-posed evolution and which vanishes at the change-of-type space-time location (z*, t*). This discriminant function consists of a sum of three terms, each with clear physical importance. There is an inertia/relaxation term which is strictly positive, a stress term which is positive in tension and negative in compression (the sum of these first two terms must be nonnegative [33)], and a surface tension term which is strictly negative. The sign of the discriminant may change as the initial/boundary data evolve, and the evolution toward change-of-type involves a delicate balance among these competing physical expressions. But there is no simple, consistent pattern in the transition away from a slender flow. Rather, a variety of mechanisms is possible and the model predicts various approaches to failure. This is in contrast to the results of Markowich and Renardy [32] for the Jeffreys model whereby the filament always fails at the ceiling due to an unbounded axial strain accompanied by a vanishing radius. In this study, we convert the Eulerian I-D Maxwell filament model to Lagrangian form and adapt it to model the filament break experiment of Matta. We always pose uniform initial data such that our I-D Maxwell model is a well-posed genuinely hyperbolic system for finite nonzero time, and satisfies a slenderness condition for finite nonzero time. Thus, our model allows the study of the competition of the included physical effects
1105
0
0
486.57
243.29
O
o O~-'J .,..4 M
~,
o ,=; .0
243.29
'
486.57
,v--.q 0
'It'
--.4
9,,.4 o .,=4 r .,,.~ "0
o
.....
o
~.29 axial c o o r d i n a t e
~88.~7
Figure 14: Onset of failure preceded by uniform thinning and uniform change-of-type: radius ~b and axial force r - T,~) - ~ / W of the filament, and discriminant A at the time of failure t* = 9.8470. The parameter values are B A - a - 1, W 1 - 1 , ~1 _ 10, Z 1, vol100 . The number of mesh points n - 800 and the time step A t - 0.625 • 10 -4. Hyperbolic to elliptic-hyperbolic change-of-type occurs at the drop interface z* - L ( t * ) - 486 957, with -~-z( d~ z*-, t* ) -- --2.5199 • 10 -6 very small, so that the change-of-type is uniform. At the moment of change-of-type the centroid of the filament is located at Zero(t*) = 358.68.
1106
o 0
0.5002
1.0004
0
0.5002
1.0004
o
r
0 axial
0.5002 coordinate
1.0004
Figure 15: A very brittle onset of failure: radius, axial force, and discrimi1 nant at the time of model failure t* - 0.951 x 10 -2 9 B - A - a - 1, W = 114, T1 _ 20, Z - 20, v o l - 10/~; the number of mesh points n - 3200 and the time step At -- 0.5 x 10 -5. Change-of-type occurs at z* - 0.78181 x 10 -3 d A( z *-~ t* ) near the ceiling; note T - 9 3 . 2 9 8 and ~ ( z *+ t*) - 1070.1 are large and of opposite sign, indicating a cusp-like behavior in A near (z*, t*). At the m o m e n t of change-of-type the centroid of the filament is located at Zcm(t*) = 0.50100. The suggested failure is brittle, with only 0.04 percent strain at the m o m e n t of change-of-type.
1107
during the evolution from a slender uniform filament until the slenderness assumption breaks down. In particular, the model fails whenever there is a hyperbolic to elliptic transition, since at that time the model experiences exponential growth in all lengthscales. The filament continues to stretch up to and even after the change-of-type, and we conjecture this transition equates to the onset of filament failure. Thus our model captures the evolution into non-slender flow. As long as the slenderness approximation is maintained and our assumption as to the competing physical effects remains valid, we can confidently predict details of the onset of failure: whether failure is approached through uniform flow and profile or through a sharp, localized gradient; where the onset of small scales occurs; whether the onset of failure is ductile (i.e. the filament at onset of failure has stretched significantly from the original length) or brittle (i.e. there is essentially no elongation before onset of failure); and whether there is little mass redistribution or concentration of mass. These are the aspects of polymeric filament behavior observed in the model. Figure 14 depicts a ductile failure with significant mass redistribution; whereas Figure 15 shows a brittle failure with little mass redistribution. Of importance to the application of this study, after the critical time of ill-posed change-@ type our model is invalid. That is, if we apply the model at all lengthscales, the full pdes are ill-posed. Effectively, the characteristic axial scale goes to zero immediately after the elliptic change-of-type with no argument for the radial scale converging to zero, so the slenderness constraint is immediately violated. As a result, we make no claims to capture the detailed structure of breaks. 5.2 N e w t o n i a n fluid characterization for dynamic surface tension
To address overspray of agricultural pesticides (due to spray drift and drop rebound from leaf surfaces) leading to groundwater contamination, the U.S. Department of Agriculture (USDA) needs to measure the physical properties of the spray solution which control droplet formation and drop rebound, namely surface tension and elongational viscosity; regulations are based on these properties. Due to the presence of surfactant additives, significant decay in surface tension of the pesticide solution, from an initial value near that of pure water (72 dyne cm -1) to as low as 30 dyne cm -1, occurs in the first 2-5 ms from the creation of the free sur-
1108
face, which coincides with the timescale of drop formation, atomization, and drop rebound. Since surface tension dominates the hydrodynamic instability leading to droplets and drop rebound, accurate methods for the determination of surface tension on the millisecond timescale are necessary; slow or static techniques such as the Du-Nouy ring effectively yield the equilibrium surface tension, which may be as low as 20 dyne cm -1. Similarly, the presence of the surfactant also introduces a flow dependence in the elongational viscosity which must be understood. With USDA researchers in Wooster, Ohio, we, in work described in [16], combine an inverse formulation of their model for an oscillating free surface jet with elliptical cross-section [12, 39] with USDA experiments of oscillating jets to measure the dynamic surface tension and elongational viscosity of jets of surfactant solutions with ages on the order of 1 ms. The oscillating jet technique is the only technique which measures the surface tension of a very new surface on the time scale of milliseconds. The oscillating jet is a liquid/gas system in which new free surface is created constantly, as in spray formation and drop rebound. The surface in the oscillating jet over the axial domain where the measurements are made has existed much less than the time necessary for the equilibrium surface tension value to be reached. To briefly describe the oscillating jet phenomenon, when a fluid exits an elliptical orifice the resulting free jet has a chain-like appearance, with the jet cross section oscillating in an elongational flow between perpendicular ellipses as it goes down the jet axis (see Figure 16). Essentially, the jet is an effective mass/spring oscillator; the tension of the free surface (the "spring") pulls the elliptical cross section to a circle to minimize area, but the inertia of fluid motion in the cross section (the "mass") causes an overshoot in the perpendicular direction, and so on. The wavelength of the oscillation is related thereby to the average surface tension over that length, and the amplitude decay to the average fluid viscosity. The particular contour of the free surface between the maximum amplitudes is indicative of the evolution of surface tension and viscosity within the wavelength, and can be exploited in the inverse problem. In the oscillating jet, the elongational motion is transverse to the filament axis, with high strain rates (on the order of 103 [16)], but small strains. This is in contrast to the flow under tension of [18] exploited in the next section, in which the elongational flow is in the direction of the filament axis, with much larger strains at lower strain rates.
1109
The USDA apparatus utilized in [16] records on video tape simultaneous perpendicular views of the oscillating jet free-surface profile, and hence affords the seven measurements shown in Figure 17. These measurements, in combination with an analytical model for the oscillating jet which assumes the surface tension and viscosity are constant over the wavelength over which the measurements were taken, produce in [16] the average values of surface tension and viscosity over the corresponding surface age interval. In [36], free-surface measurements made with the USDA apparatus and the model of [16] are utilized to characterize the surface tension of a battery of fluids.
Figure 16" The oscillating jet phenomenon; from [40]. In [17] equations are derived for oscillating jets which allow for surface tension and elongational viscosity to vary in space and time. Three specific relations for the decay of surface tension with surface age are investigated" exponential decay, the thin-film diffusion model derived by [35], and the algebraic form proposed by [41]. For each form the direct problem is solved, i.e. the governing integro-differential equations are numerically solved with the particular specified function for the decay of surface tension with age to obtain the free surface profile. In all forms initial and equilibrium values of surface tension and decay rates were selected consistent with experimental values reported in [35] and [41]. See Figure 18. Important qualitative differences in the oscillating jet behavior predicted under the three forms are discovered, with a striking similarity: in all forms the nearfield decay of surface tension, occurring in the first few wavelengths of oscillation or over a fraction of the first wavelength determines the amplitude and wave-
lll0
length of the farfield oscillation. This has powerful consequences in the inverse formulation of the problem, since rapid surface tension decay can be deduced from measurements of the jet profile far downstream of where this decay takes place. B R rain
-iV-
B R max
M R mtt(
~,
Ru rain
E R min _~t_
E R mcu
Figure 17: USDA measurements taken of the oscillating jet" from [16]. The inverse formulation of the analysis of [17] provides the means to resolve dynamic surface tension and elongational viscosity within a wavelength of oscillation and on millisecond timescales. 5.3 N o n - N e w t o n i a n fluid c h a r a c t e r i z a t i o n for elongational viscosity and relaxation
In [18] we combine recent advances in spinline measurements with mathematical analysis to produce a new capability in material characterization in elongational processes. Three experimental ~dvances documented for the first time in [18] are (i) the ability to measure the kinematics (filament radius and slope, velocity, and velocity gradient) pointwise along the filament accurately enough that one can treat them as known, differentiable functions in the inverse problem for material properties, (ii) quantitative
1111
75 \
\~
,,, 60
\~
E ~ 55 \x \X
gso g 9~
9
x '\\
45
.
\
\
. .
\
\
\\
4o
\
.~ 35 ..... Z.~_~.~
25
0
i
i
0,005
0.01
i
0,015 surface
age
(s)
....
i
i
i
i
0,02
0.025
0,03
0,035
0,04
1,8 1,6
A /I,,
1,4
t
_1,2 i -e.•
1
\ \
~u}0 , 8
I.
/?',i,,
/,, /~,
/,' I,
/,,/,',
/,~
,, t
I I I,' I', /'," t',,,Ii,'x,, I /,'t ,,,/
I
t
I;
~',l', \
\,,/.',' '\ ~
\,',/,,' \
\,I,
~. '
.~_ 0,6 9~" 0 , 4 0.2 01
i
0
2
i
i
4 dimensionless
6 axial
distance
z
i
/
8
10
12
Figure 18: (a). Surface tension vs. surface age. For all three decaying forms - 72.8 dyne cm -1 at 0 ms and cr - 31.4 dyne cm -1 at 0.031ms. The constant case is for a - 72.8 dyne cm -1 ( ); exponential decay ( - - - ) ; Brazee model ( . . . . . . ); Hua model ( . . . ) . (b) Free surface profiles ~1 with the variable surface tension given (a) and ~ ( 0 ) 1.5, 01,z(0) - 0 .
1112
measurement of the axial force in the filament at the die exit, and (iii) an accurate steady flow-rate control. In the analysis we admit many possible physical effects. In the forward problem the known material properties and boundary conditions dictate the dominant physical effects. In the inverse problem considered in [18], the determination of which physical effects are dominant is part of the solution. The apparatus employed in [18], shown in Figure 19, was constructed in collaboration with Prof. K.W. Koelling of the Department of Chemical Engineering at the Ohio State. An infilsion/withdrawal syringe pump is used to pump the fluid through a stainless steel delivery tube, controlled to a specified value which ranges from 1.5 x 10 -s to 2 x 10 -1 ml/s. This system avoids the variability in the volumetric flow rate of other types of pumps, due to for instance the difficulty in controlling the pressure level in a reservoir with falling fluid level, or the periodicity of a reciprocating pump. The fluid is ejected from the delivery tube in the form of a vertical filament which is taken up tangentially by a rotating drum of 5 cm diameter. The take-up drum is mounted on a movable assembly which allows the fiber length to be varied fi'om 0-19.6 cm. The drum is also connected to a flexible torque transmission cable which allows the driving motor to be placed away from the apparatus in order to minimize vibrations. A scraper is attached to the bottom of the drum to remove excess fluid.
Figure 19" The experimental apparatus. The diameter of the filament as a function of axial position is recorded
1113
using a charged coupled device video camera positioned on a constant speed vertically traveling stage. A limitation to resolution in edge detection, especially for transparent fluids, is the variability in apparent edge location due to non-uniform illumination; the advance here in the resolution of the free surface is due in large part to the use of a diffuse background illumination system (Lumitex Inc.). Images are recorded using a Super VHS video cassette recorder. Two other techniques to measure the fiber diameter were also evaluated: still photography, and using a laser based diameter measuring device; the video setup was found to provide comparable resolution coupled with greater ease in recording measurements. The video camera captures an image every 1/60 of a second. Each digitized frame captures 448 pixels in the axial direction; to avoid possible aberration in the camera lens only the central 400 pixels of the frame are utilized. The digitized image files are processed using an Optimal Zero Crossing edge detection algorithm. The velocity of the camera translation is computed by tracking an identifiable fixed point in successive images. Enough frames are stored so that each axial location on the free surface appears in the core 400 pixels in three different frames. Using the position of each successive image and the velocity of travel of the camera, the images are concatenated to give the entire profile of the jet. The raw data in terms of pixels is converted to length units with the calibration factor 0.001254 cm/pixel. The error in image capture and edge detection procedure is no more than one pixel, or 12.54 #m. This resolution enables the apparatus to measure on-line the kinematics of the deforming filament as a function of axial position accurately enough so that they and their derivatives can be considered as known functions in the inverse problem for material properties. The delivery tube deflects downward due to the tensile force in the filament. This deflection, measured by a displacement transducer, is used, together with analysis, to deduce spinline force at the tube exit (a filament under take-up conditions is referred to as a spinline in the terminology of the textile and reinforcing fiber manufacturing industry). There are several possibilities for the zero force state in the calibration of the transducer, such as the delivery tube full of stationary fluid, and the state with a container placed just below the issuing jet. These two choices have drawbacks" the first does not take into account dynamic and pressure effects in the tube during filament formation, and the second fails due to the elasticity of the fluid studied in [18], which leads to an upward thrust exerted on the tube.
lll4
To avoid these errors, two sets of surface profiles and tube deflections are recorded for each experiment, one for the filament with the take-up force, and one, for the calibration, without windup falling only under the effect of gravity. Analysis contained in [18] allows one to deduce the axial force in the filament with windup at the tube exit from comparison of the two sets of measurements. Reference [18] demonstrates how the measurements provided by the apparatus can be exploited to give quantitative information about rheological material properties in isothermal elongational flows under applied tension. The profile and force measurements fl'om a particular experiment are coupled with a mathematical model formulated as an inverse problem, to deduce material properties. Referring to [18] for details, with the fiber profile and upstream axial force experimentally known, the momentum and constitutive equations decouple: the momentum balances reveal the evolution of stresses that must be present in the jet to balance the effects of inertia, gravity, and surface tension and produce the experimentally observed measurements; and the constitutive equations deduce what stresses would be produced by the measured kinematics and upstream boundary conditions for a proposed rheological model of the fluid. The inverse problem consists of searching through parameter space in the proposed rheological model until the stresses generated by that model match those that must be there to satisfy momentum considerations. Specifically, in [18] the elongational rheology of a Boger test fluid is characterized within the assumption of a single-relaxation-mode Giesekus constitutive model. Significantly different values of the relaxation time and mobility parameter were obtained in the elongational characterization than were obtained via shear rheometry: the shear rheometer produced a relaxation time of 0.9148 s and a mobility parameter of 0.00035, whereas in the elongational flow the relaxation time was found to be 1.323 s with one inversion technique and 1.303 s with another, and the mobility factor was found to be zero, so that in the elongational flow (but not shear flow) the proposed Giesekus model collapsed to an Oldroyd-B fluid. The solvent and polymer viscosities deduced from the elongational and shear experiments were the same. ACKNOWLEDGEMENT
This work was funded in part by the Air Force Office of Scientific Re-
1115
search, Air Force Materials Command, USAF, under Grants F49620-97-10001 and F49620-97-1-0003, and the National Science Foundation, under Grants CTS-9319128 and CTS-9711109. The US Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the US Government. REFERENCES
o
.
3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15. 16.
17. 18.
M. Matovich and J. R. Pearson, Ind. Eng. Chem. Fundam. 8 (1968) 512. M.M. Denn, C.J.S. Petrie, and P. Avenas, AIChE. J. 21 (1975) 791. W.W. Schultz and S.H. Davis, J. Rheology 26 (1982) 331. L. Ting and J.B. Keller, SIAM J. Appl. Math. 50 (1990) 1533. M.G. Forest and Q. Wang, Siam J. Appl. Math. 54 (4) (1994) 996. S.E. Bechtel, M.G. Forest, and K.J. Lin, SAACM 2 (1992) 59. J. Eggers and T. Dupont, J. Fluid Mech. 262 (1994) 205. J. Eggers, Phys. Rev. Lett. 71 (1993)3458. X.D. Shi, M.P. Brenner, and S.R. Nagel, Science 265 (1994) 219. D.T. Papageorgiou, ICASE Report 93-45 (1993). S.E. Bechtel, J.Z. Cao: and M. G. Forest, J. of Non-Newtonian Fluid Mech. 41 (1992) 201. S.E. Bechtel, M.G. Forest, D.D. Holm, and K.J. Lin, J. Fluid Mech. 196 ( 988) 241. S.E. Bechtel, K.D. Bolinger, J.Z. Cao, and M.G. Forest, Siam J. Appl. Math. 55 (1) (1995) 58. M.6. Forest, Q. Wang, and S.E. Bechtel, Physica D 99 (4) (1997) 527. M.G. Forest, Q. Wang, and S.E. Bechtel, J. Rheology 41 (4) (1997) 821. S.E. Bechtel, J.A. Cooper, M.G. Forest, N.A. Petersson, D.L. Reichard, A. Saleh, and V. Venkataramanan, J. Fluid Mech. 293 (1995) 379. S.E. Bechtel, M.G. Forest, N.T. Youssef, and H. Zhou, accepted for publication in J. Appl. Mech. V.V. Ramanan, S.E. Bechtel, V. Gauri, K.W. Koelling, and M.G. Forest, J. Rheolo~v 41 (1997~ 283.
1116
19. D.D Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York (1990). 20. A.N. Beris and B. Liu, J. Non-Newtonian Fluid Mech. 26 (1988) 341. 21. B. Liu and A.N. Beris, J. Non-Newtonian Fluid Mech. 26 (1988) 363. 22. M.G. Forest and Q. Wang, J. Theor. Comp. Fluid Dyn. 2 (1990) 1. 23. Q. Wang, M.G. Forest, and S.E. Bechtel, J. Non-Newtonian Fluid Mech. 58 (1994) 97. 24. A.V. Bhave, R.K. Menon, R.C. Armstrong, and R.A. Brown, J. Rheology 37 (1993) 413. 25. M. Doi, J. Polym. Sci. Polym. Phys. Ed. 19 (1981) 229. 26. M.G. Forest, H. Zhou, and Q. Wang, Univ. of North Carolina at Chapel Hill, Dept. of Math., Preprint Series # 97-11. 27. L. Rayleigh, Proc. Lond. Math. Soc. 10 (1879)4. 28. F.J. Garcia and A. Castellanos, Phys. Fluids 6 (1994) 2676. 29. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, (Clarendon Press, Oxford, 1961). 30. S.E. Bechtel, C.D. Carlson, and M.G. Forest, Phys. Fluids 7 (1995) 2956. 31. J. Matta and R.P. Tytus, J. Appl. Polym. Sci. 27 (1982) 397. 32. P. Markowich and M. Renardy, J. Non-Newtonian Fluid Mech. 17 (1985) 13. 33. D.D. Joseph and J.C. Saut, J. Theor. Comput. Fluid Dynamics 1 (1990) 191. 34. W.D.E. Thomas and L. Potter, J. Fluid Mech. 76 (1975) 625. 35. R.D. Brazee, M.J. Bukovac~ J.A. Cooper, H. Zhu, D.L. Reichard, and R.D. Fox, Transactions of the ASAE 37 (1994) 51. 36. D.L. Reichard, J.A. Cooper, S.E. Bechtel~ and R.D. Fox, Atomization and Sprays 7 (1997) 219.. 37. R.S. Hansen, M.E. Wallace, R.C. Woody, J. Phys. Chem. 62 (1958) 210. 38. R. Defay and J.R. Hommelen, J. Coll. Sci. 13 (1958) 553. 39. S.E. Bechtel, J. Applied Mech. 56 (1989) 968. 40. L. Rayleigh, Proc. R. Soc. Lond. 41 (1890) 281. 41. X.Y. Hua and M.J. Rosen, J. Colloid and Interface Science 141 (1991) 180.
1117
NUMERICAL
SIMULATION
OF
MELT
SPINNING
OF
POLYETHYLENE TEREPHTHALATE FIBERS Kyoung Woo Kim a, Sang Yong Kim, and Youngdon Kwon b Department of Fiber and Polymer Science, Seoul National University, Seoul 151-742, Korea aFiber Research Center, Sunkyong Industries, Su Won, 440-745, Korea bDepartment of Textile Engineering, Sung Kyun Kwan University, Su Won, 440-746, Korea
1. INTRODUCTION Fiber spinning is one of the major polymer processing operations, by which fine and uniform filaments with required modulus and strength can be produced. There are three types of spinning processes such as melt, wet and dry spinning. Of all these, the melt spinning is the most widely applied for the production of synthetic fibers, where severe elongational deformation is usually accompanied. Thermally stable polymers such as polyesters, polyamides and polyolefins are employed for manufacturing fibers via melt spinning. Especially, the melt spinning of polyethylene terephthalate(PET) has been intensively investigated due to its commercial importance. In the typical melt spinning operations, undrawn filament yarns (UDY) are made at the spinning speed of 1---2 km/min. Then, the process of drawing in the solid state by 3---5 times of the original length follows in order to enhance the mechanical properties, and thus, results in fully drawn yarns (FDY). Herein, the melt spinning process is performed in
1118
two steps; one is spinning and the other is drawing. Later, for the improvement of productivity, spin-draw process was developed, which yields so called partially-oriented yarns (POY) at the spinning speed of 3 --- 3.5 km/min. They constitutes draw-textured yarns (DTY). If one further increases the spinning speed, the drawing process hardly affects mechanical properties of the fibers, and then the melt spinning can be carried out in one step to get the final products. This high speed spinning process was developed in the 1970's by modifying the functionality of winders. In the 1980's, high speed spinning of PET, PP and Nylon has been thoroughly investigated, and commercial products manufactured at the speed of 6---8 km/min[1] become available. Under such high speed spinning, one may occasionally observe some instability like melt fracture and draw resonance, which deteriorates the filament quality and reduces its productivity[2-3]. One can also notice neck-like deformation as the spinning speed increases, and vast amount of research has been carried out to clarify the origin of such phenomenon[4-5]. One of other peculiarities in high speed spinning is the crystallization in the spinline, where according to the order of their appearance three regions such as flow deformation, orientation crystallization and plastic deformation can be defined[6]. Flow deformation occurs immediately after the exit of the spinneret, and molecular orientation formed in the spinneret channel becomes relaxed. Below this is the orientation crystallization region, where abrupt increase in the orientation of amorphous region induces crystallization and the filaments become thinner and necking may occur. Finally, in the plastic deformation region, filaments are frozen and cold drawing under the additional effect of air drag due to the increase in the filament speed is also possible.
1.1. Structure and Properties of High Speed Spun Filaments 1.1.1. Morphology anal Physical Properties of Filaments As was explained in the previous section, the main objective of the high speed spinning is to produce fibers with the required high molecular orientation and crystallinity in one step process, that is, the spinning without the drawing.
1119
In the case of PET fibers spun at the speed of 3---4 kin/rain, even though molecular orientation is high, the fibers are in low crystalline or amorphous state[7,8]. Tensile strength and elasticity of the fibers are also very low, but the breaking elongation is high, thus the fibers cannot be applied for practical use without some additional processing like drawing followed by heat treatment. However, if the spinning speed reaches 5---6 km/min, such produced filaments possess as high as 30---40% of crystallinity and reasonable mechanical characteristics. In general, with the increase of spinning speed, higher the tensile strength of the filaments becomes and lower the breaking strain. When the speed exceeds 5 km/min, necking in the cold-drawing process after spinning disappears and the first yield point in the stress-strain curve becomes obscure. Hence, the stress-strain curve becomes similar to that of FDY. The tensile strength achieves maximum at the spinning speed of 6 ---7 km/min and then decreases if it is further raised[9]. The high speed spun fibers have higher crystallinity and crystal orientation than the low speed spun fibers with the treatment of cold or hot drawing. They possess low total orientation due to lower molecular orientation in the non-crystalline region and thus exhibit poorer mechanical properties than the low speed spun and drawn fibers. Thermal stability of high speed spun fibers is very high, and the shrinkage in the boiling water is usually very low. Dye uptake of high speed spun fibers is better than that of low speed spun and drawn fibers[ 10]. 1.1.2. Structure Distribution in the Fiber Cross-section
The cross-section of fibers spun at low speed and drawn are relatively uniform in its morphological structure, but sometimes the birefringence in the core region is somewhat higher than that in the sheath region[l 1]. This phenomenon may result from the thermal effect on the surface during drawing. On the other hand, the birefringence in the skin region of fibers spun at high speed is higher than that in the core region, and thus, the high speed spun fibers have a distinct skin-core structure. When observing the cross-section of high speed spun fibers under the interference microscope[12], one may examine that the distribution of birefringence along the fiber cross-section is uniform for the fibers spun
1120
at the speed below 6 km/min, however, the birefringence is higher in the skin than in the core for the fibers spun at the speed above 7 km/min. From the dyeing test, it has been also found that the skin region shows a darker image than the core region[13]. From these investigations, one can conclude that the cross-section of high speed spun fibers are not homogeneous, but exhibits the distinct skin-core structure.
1.2. Structure Formation in High Speed Spinning 1.2.1. Attenuation during Spinning The shape of molten polymers extruded from the spinneret nozzle attenuates under the influence of take-up tension. The attenuation is very slow in the low speed spinning, but in the case of high speed, the diameter of fiber decreases rapidly until it reaches the point of 50---100 cm below the spinneret[4]. This behavior is similar to plastic deformation of metal or neck formation during the cold drawing of solid polymers. Therefore, it is often called neck-like deformation or necking. The appearance of necking in the spinline is reported in the high speed spinning of PET, PA and polyolefin. Necking seems to result from the high tensile stress concentrated in the narrow region of spinline. This is one of the unique characteristics of high speed spinning which may play an important role in determining the regime of internal structure formation. Study on the condition of the neck formation in the solid material was initiated via the application of the Consid~re construction[14,15], and a number of numerical studies[16,17] followed. The theory states that the position of neck formation is usually located at the point where the following equation based on the Consid~re construction from the true stress-strain curve holds" dGt d/l -
6t /1 or
dGt de -
r l+e where at is the true stress, /1 the draw ratio, and e the strain. If the condition
(1)
dat at dA)A is valid in the entire region, smooth attenuation should be expected;
1121
however, if equation (1) is satisfied at one point, fracture may occur after necking. On the other hand, if it holds at two points, the spinline stabilizes and cold drawing will follow[18]. Necking phenomena at high speed spinning have been investigated since Perez's study[19]. It has been reported that crystallization starts near the neck point, which was confirmed through examining the data of X-ray scattering for the PET filaments. Necking has been explained in terms of the spinning instability and draw resonance[20]. On the other hand, it has been also studied in view of the structure formation, and it is claimed that necking occurs at the point where the viscosity drops abruptly. Thus, necking point moves upward along the spinline as the take-up speed increases[21]. Haberkorn et a1.[22] showed that crystallization rate is a function of take-up speed for Nylon 6 and 66 in the high speed spinning. According to their explanation, neck-like deformation occurs in the region above the solidification point of the fiber, and then the polymer crystallizes after neck formation. Thus, it is asserted that crystallization does not induce neck-like deformation and the decrease of local viscosity due to the heat of crystallization cannot be the cause of the necking. A theoretical study of the neck-like deformation has been conducted after the hypothetical division of the neck deformation into qualitative and quantitative deformations. Ziabicki[23] explained that an inflection point on the curve of velocity variation along the spinline becomes a criterion for neck formation and the cause of necking is the viscosity increase due to cooling of the melt. However, the viscosity increase is only a necessary condition for the stable spinning but not a sufficient condition for necking. Even the existence of inflection point in the velocity distribution curve may not be sufficient for the occurrence of necking, thus it can be said that necking occurs if the radius variation exhibits two inflection points[24]. The condition of neck formation is, thus, the appearance of two inflection points, where
R"--
d2R
dz2 -- 0
Here R is the filament radius and z the axial coordinate.
(2).
1122
1.2.2. Orientation Crystallization Behavior Since the fibers crystallize within 5---10 ms under high tensile stress in high speed spinning, they show structure and properties quite different from those of low speed spun and drawn fibers. Orientation of polymer molecules starts to develop in the amorphous region under high take-up forces, and then this oriented region crystallizes. Thus, crystalline region in the fibers possesses high degree of orientation, and amorphous region remains in the state of low orientation. The crystals are in the form of bundle-like fibrils rather than chain-folded lamellae[25-27], and the fibrils grow with the increase in take-up speed. Heuvel and Huisman[28] demonstrated the dependence of molecular orientation on the take-up speed. The results are as follows: below the speed of 2 kin/rain, the low oriented molecules in the fibers do not crystallize, at 3.5 km/min the molecules are oriented but do not crystallize, and above 5 km/min the distance between molecular chains becomes close enough to form crystalline state. Necking also appears in the spinline under the condition whereby the molecules are able to crystallize. In other words, necking is not probable for the noncrystallizable polymers such as atactic polystyrene or even for crystallizable polymers when the crystallization is suppressed by rapid cooling or low tensile stress. Effect of crystallization on necking has been discussed in view of heat transfer and deformation mechanism for some crystallizable Newtonian materials[29]. Crystallization is responsible for the direct increase of viscosity, and hence, increases necking intensity. On the other hand, crystallization heat lowers the local viscosity indirectly, and decreases necking intensity[23]. 1.3. Numerical Simulation
An attempt to numerical simulation of the melt spinning process was first made by Ziabicki [3 0] and Kase et a1.[31,32], and they were followed by Prastaro[33], Yasuda[34,35], and Denn[36]. In their approaches, the analysis neglected the effect of non-isothermal crystallization and molecular orientation. Therefore, even though the results describe quite satisfactorily the experimental data in the case of low speed spinning process without crystallization, the numerical scheme in high speed
1123
spinning with high degree of crystallization necessitates some modification. Shimizu et al.[4] and Katayama and Yoon[37] constructed a model with the energy equation including the effect of crystallization and applied it to numerical simulation of the high speed spinning. Spruiell et a1.[38-40] extended this procedure in the melt spinning process of Nylon. They included in their consideration the effects of temperature and orientation to the crystallization kinetics. In the high speed spinning, there exists viscosity distribution in the fiber cross-section and it results in the distinct skin-core structure. Since such viscosity variation may become a cause of necking, it should be also taken into account in the numerical simulation. Shimizu[4] calculated the temperature distribution in the radial direction using 1-dimensional numerical modeling. Hutchenson et al.[41] calculated the radial temperature distribution by solving energy and continuity equations in the cylindrical coordinate. Bell and Edie[42] performed the same work with the application of a finite element method. 1.4. Objective of This Work To investigate the high speed spinning process in terms of the numerical analysis is the main objective of this work. 1-dimensional numerical analysis enables one to predict the diameter and temperature variation along the spinline, but with this simplified method, it is not possible to describe the occurrence of discontinuous peak in the temperature distribution caused by the crystal formation and the formation of skin-core structure. In this work, the spinning process is assumed to be in cylindrical symmetry and the 2-dimensional numerical analysis has been carried out with a finite element method, neglecting the variation in the angular direction. Solving the energy and momentum balance equations simultaneously, we could estimate the temperature distribution in the fiber and describe the skin-core formation when calculating the crystallinity distribution. With the method of finding free surface employed in the typical finite element method, it is hardly expected to describe the neck formation in the high speed spinning and it is not easy to include in the formulation the effect of cooling, convection and air drag on the free surface.
1124
Therefore, in this study, the temperature and fiber diameter distributions are estimated by the relatively simple 1-dimensional numerical analysis. Then, with the boundary conditions specified by this simple calculation, solutions of the 2-dimensional problem are obtained, which give information on the fiber morphology and variation of some related properties. Effect of take-up speed, cooling air velocity and flow rate on the fiber structure and on the skin-core formation is also estimated.
2. NUMERICAL SIMULATION
2.1. One-Dimensional Numerical Analysis 2.1.1. Governing Equations Generally, there exists transfer of heat as well as momentum with possible variation of the material density in the polymer processing operations. Thus for detailed analysis, it is necessary to take all of continuity, momentum, energy and rheological equations into consideration. In the mathematical formulation, the followings are assumed: (1) The cross-section of a fiber extruded from the circular orifice remains circular. (2) The fiber spinning is performed in a vertical direction. (3) Flow is in a steady state. (4) Property variation in the fiber cross-section is neglected. (5) Surface tension of the fiber surface is neglected. (6) Heat generation due to viscous flow is neglected. (7) Heat conduction in the spinning direction and radiation is neglected. With these assumptions, a set of governing equations in the fiber axis direction(z-axis) is as follows[4,25,26,43]:
Continuity equation:
W-
xD2 4 Ov
(3)
where W is the mass flow rate, D the fiber diameter, density and v the fiber velocity. Momentum equation:
dF de - ~
+ _ ~ Oai,,C/v 2 __ Wg v
0
the fiber (4)
where F is the tensile force acting along the fiber, par is the air density,
1125
of which the value is 0.815x10 3 g/cm3[20], Cf is the surface friction coefficient and g is the gravity acceleration. Energy equation is reduced to the following including the effect of crystallization heat during the high speed spinning, which is shown as the second term in the right hand side of equation: Energy equation:
dT de-
zcDh( T - T a i r)
+
AHI dX C, de
(5)
where T is the temperature, h the heat transfer coefficient, lair the air temperature, Cp the specific heat of the polymer, A Hf the heat of fusion of polymer and X the crystallinity. For PET, A HT = 29 cal/g[4]. PET shows the behavior of constant viscosity at low shear rate(<100 s l) in the temperature range of 250---290~ but at higher shear rate(>100 s-~) it shows the shear thinning behavior within the same temperature range. To accommodate this non-Newtonian behavior, the power law model shown in equation (6) is adopted for this analysis. Rheological equation of state: a = r/~ ~
(6)
where a is the tensile stress, ~e the elongational viscosity, n the exponent of the power law equation and the value of 0.9 is chosen in this work.
2.1.2. Crystallization Kinetics For the analysis of crystallization in the high speed spinning process, the following Avrami equation with the modification introduced by Nakamura et a1.[44] is employed:
xo~X - 1 -
e x p _[ - ( f_ ~tK( T)dt),o]
(7)
where Xoo is the maximum crystallinity, K(T) the crystallization rate constant, t the time, na the Avrami exponent. The Avrami equation is thought to describe the initial crystallization stage only, but in this research we assume that it holds in the emire range of crystallization process with na = 1 specified[37]. The transformation of the time derivative of the cry stal l inity to the space derivative yields:
1126
d X - X~176n vflf {
xooX }
(8).
Half time of crystallization(tv2) defined by Ziabicki [3 0] becomes: 1. = 1 e x p [ - ( 4 1 n 2 ) ( T - Zmax )2] tl/2 t~/2 Do
(9)
where Tmax is the temperature at the maximum crystallization rate, Do is the half-width obtained from the crystallization rate-temperature curve, and t~/2 is the minimum value of tl/2 which coincides with the half time at Tmax.
The crystallization of polymers may depend not only on the thermal condition but also on the molecular orientation in the amorphous region, and when we include this effect, tl/2 becomes[30]:
1 _ tl/2(fa) --
1 {Af2 a tin(0) exp + Bfa + ""}
(10)
where fo is the orientation factor in the amorphous region and A, B, .-. are parameters which are in general functions of temperature. When the orientation is very low, i.e. focal, the second and higher order terms in the parenthesis of the right hand side of equation (10) may be neglected. Combining equation (10) with (9) with the following relation taken into account: K( T, f~) -
ln2
(11)
tl/2
results in the final form for the crystallization rate constant:
Do
+ Aft2
(12).
The values of parameters in the above equations have been specified by Ziabicki[44] and Alfonso et a1.[46] as:
A =
C1 -
AT:
TOm- T
C2zIT
(13)
(~T) 3
where C1=3.09•
(14) l~ C 2 = - 1 . 5 5 •
t~/z=42s, T~176
D0=64~
2.1.3. Boundary Conditions and Supplementary Relations Boundary conditions for extrudate velocity, temperature and crystallinity at the spinneret exit( z = 0) and the take-up point( z = L) are given as:
1127
v=v0, T= v-- VL
To, X = 0
at z = 0 at z = L
where v0 and To are the output velocity and
(15) the temperature at the die
exit, respectively, and vL is the take-up velocity. The heat transfer coefficient was modeled by Kase and Matsuo[32] for the forced convection by cooling air as:
0
where
D
[1 + ( 8 a;r) ]0 u
(16)
k~g~ is the thermal conductivity of air and its value is 5.50 • 10 -4
cal/cms~
v~
the velocity of cooling air, ReD is the Reynolds
number for the fiber diameter defined as: vD ReD = 0.29
(17).
The elongational viscosity ~e is in general a function of temperature and crystallinity as assumed by equation (18) and the surface friction coefficient is expressed in terms of Reynolds number[7] : 7e = 0.73exp
T + 273 exp a ~
Cs= 0.5R@ ~
(18) (19)
where the constants, a and b are assigned to be 4 and 2, respectively[38]. Such empirical formulae as the following equations are also employed in this study[33-37]: o = 1.356 - 5x 10-4T
(20)
C~ = 0.3 + 6x 10-4T
(21).
Birefringence zJna manifested by the molecular orientation amorphous region may be calculated by the formula[47,48]: Aria = Coba
in the (22)
where Cop is the stress-optical coefficient, the value of which is 7.8 • 10 -9 mZ/N. The orientation factor fa in the amorphous region and total birefringence A n can be estimated using the equations:
f_
Aria
An = 1
(23)
xooX )Aria + x~o f c A o
(24).
1128 Here fc is the orientation factor in the crystalline region and z/~ and z/~ are the intrinsic birefringences of the amorphous and crystalline regions, respectively. Their values for PET obtained by Dumbleton et a1.[49,50] are ,d o = 0.275 and ,d o = 0.22. Equation (24) fits rather well with the experimental data for the low speed spun fibers, but some deviation from the data can be observed for the high speed spun fibers. Therefore, for its remedy Katayama and Yoon[37] proposed the following relation, derived on the basis of the rubber elasticity theory: z/n =0.211 - e x p ( - 1.65x 10 -6 T+a273 )]
(25).
Numerical solutions for each process condition are then obtained with the application of above relations and material constants to the 1-dimensional formulation. 2.2. Two-Dimensional Finite Element Analysis
2.2.1. Governing Equations Equations for the 2-dimensional finite element calculation of the high speed spinning are formulated in the cylindrical coordinate system (r, 8, z) under such assumption that the flow in this spinning process is axisymmetric. Figure 1 shows the melt spinning procedure and coordinate system employed in this analysis.
RO
Die
i
z:0 R
Godet Roller
'~_
z=L
Figure 1. Schematic representation of the fiber spinning process.
1129 Following approximations are assumed to be valid in constructing the goveming equations: (1) The cross-section of spun fiber is always circular. (2) The fiber spinline is in a vertical direction. (3) The flow is in a steady state. (4) Heat generation due to viscous flow is neglected. Let u, v and w denote the components of a velocity vector v in r, 0 and z-directions, respectively, assuming that the flow is axisymmetric. Then v vanishes, and u and w become functions only of r and z, that is,
v = (u, v, w) = [u(r, z),O, w(r,z)]
(26)
which yields the following expression for the strain rate tensor D:
D = l
bl,, r
0
89 ( U, z -[- W, r)
0
_u
0
0
w,~
(u,~+w,~)
where u , is defined as - Ou ~,
(27)
etc.
The second invariant of D is defined as:
lid = u, 2~ + ( uT) 2 + w , 2~+ 1 ( u , ~ + w , ~ ) 2
(28)
and thus, the total set of equations in the 2-dimensional formulation can be expressed as the following. Momentum equation: -- D, r "[- rrr , r+ Z'rz,Z _+_ ~'rr--~r r O0 Af_ f r :
P( UZr ?"-+- WU, z) in r-direction
__ 1), _~_ rrz r_~_ ~'rz _~_ rzz z_~_f z = P( 1,1W, -~- WW, z) in z-direction
(29a) (29b)
where p is the pressure, fr and fz are r and z-components of gravity, respectively, r0 the components of the stress tensor in the cylindrical coordinate, and the second assumption assigns f~ = 0 and fz = p g. Energy equation:
pC~(uT, r+ wT, z ) - k(T, r~ + + T,r-'~" T , zz) -[- (/V (30)
where k is the thermal crystallization heat.
conductivity
of
polymer
and
I;V
is
the
1130
Continuity equation: u,~ +
__u + w,~ = 0 Y
(31) n-1
Rheological equation of state: 7(IID) = rj~(4IID) 2
(32)
where n is the exponent of the power law equation and 77 is the viscosity function. Other relations for the crystallization kinetics and the temperature dependence of the elongational viscosity are given by those used in the 1-dimensional numerical analysis. 2.2.2. Finite Element Method In solving the above equations by the Galerkin method, the following weak form is made with the biquadratic polynomial for the interpolating functions ~. [51,52]:
fx2r~J'i [ -- P, r + rrr, r + r,~ + r,, -r too _ O( uu, ~ + wu, ~) d~ = 0
(33a)
fs~rgri [ - P'
(33b).
+ r'~ '~ + r'~ r + r= '~ + O g - p(uw' ~ + ww' ~)] d~ =
Here ~ is the domain in the (r,z) plane in which this problem is defined. Momentum equations are then modified with the application of the product rule and the divergence theorem: fs~ [rgi. ~ ( - P + rrr) + rgi, zrrz + ~[l'i(--P + Z'00)
(34a)
+ or!if'i( uu, r -F WU z) ]df2 -- jo Q r~i~J r & fQ [ 7"~J'i, rrrz q- 7"~i,z( -- p Jr rzz ) Ji- pT"~l"i( ZlW, r -t- WW, z ) ] d ~
= f r~ipgds
(34b)
foar~i~Szds
where s denotes the boundary 0~2 of the domain ~ . ~r and ~z are the components of air drag acting on the fiber surface in r- and z -directions, respectively, and the related boundary conditions are expressed as:
r'~r = (__ p_Jf_ rrr)nrq- rrznz "-- tYrrnr'+" crr~nz
(34c).
Here a0 is the total stress tensor component and n~ the component of a unit vector normal to the fiber surface. Substitution of the rheological relation into the above equations yields:
1131
f,
I"
(35a)
+
r+WU,
Lo
f Y[ 7]~i, r( U, z "~- W, r) -~- ~i, z ( -- P "~ 2 7]W,z) -lt- i0 ~J'i( UW, r ~- WW, z)]d,~ (35b). f~
= J r~, og d ~ + Jo,.car~i
~ z
ds
Applying the product rule and the divergence theorem results in the following form for the energy equation:
f~ [oCprgi( uT ~+ wT,~) + kr( ~ri,rT, r+ ~i, zT, z)]d~ f'l
(36)
l'~
= J,.c2 [ r g i f V d f 2 + k [J
O.(2
Y~i T , n ds
where T,~ = T,~ n r =
hk ( Z - T air)
(37)
and the quantity T,n refers to the convective heat transfer. Employing the bilinear interpolating function r we can express the continuity equation as"
~ r~i( U, r+ Zt
w,~)dO = 0
(38).
Here each approximating solutions are in the form of:
~( r, z) = z ~ bli~i( ~, ~) , 1,#"
~( r, z) -- z~= Wi~i( ~, ~) I
(39)
B
where ~e and ~" are coordinates in so called the master element and M is the number of nodal points. Substitution of the above relations for the governing equations gives us a system of nonlinear algebraic equations. In order to solve this set of nonlinear equations, we employed the simple Newton iteration method. In this numerical computation, the iteration does not stop until the obtained values lie within a desired tolerance limit.
2.2.3. Boundary Conditions Boundaries of a fiber in the spinline assigned in this finite element
1132 method are illustrated in Figure 2. The line D-A in the figure denotes the center line of the fiber, i.e. the line of symmetry axis.
A
Die Exit
Fiber Center , , ~ Line
B
Free Surface
C Godet Roller Figure 2. Definition of boundaries in the flow domain. The boundary conditions can be stated as follows: At the nozzle exit: z - - 0 (A-B) (40) where w0 is the volumetric flow rate divided by the nozzle cross-section and To is the temperature at the exit. Free surface : r = R(z) (B-C)
u(r,z)=O, w ( r , z ) = wo, T ( r , z ) = To On the fiber surface, normal components of both velocity and stress vanish and heat transfer results only from convection, thus, the boundary conditions become" (41) ni r O (42)
n i Vi ~- 0 Yl i T , i =
h - -~ ( T - Tai,,)
where n~ is the unit normal vector on the free boundary and
(43) v, is the
component of velocity vector. Equation (42) is the necessary condition for the boundary surface, and in this case the function for the fiber diameter determined from the 1-dimensional numerical analysis is used instead. Temperature variation obtained from the 1-dimensional computation is
1133
also assumed as the boundary condition instead of equation (43) for 2-dimensional calculation. Take-up p o i m : z = L (C-D) w( r, L ) = wr~ '
OT Oz
-
0
(44)
where wL is the take-up velocity. Fiber center line : r = 0 (D-A) Since there is neither flow nor heat transfer in the radial direction on this center line, the following conditions are valid: u=0,
aT Or - 0
(45t.
3. RESULTS AND DISCUSSION 3.1 Behavior of Power Law Fluid Since the power law fluid assumption is used in this study of the high speed spinning, the dependence of fiber diameter, velocity, temperature and crystallinity on the power law index is first estimated. Figures 3-6 show the results of calculation when the tensile stress acts only on the amorphous region, which is the only deformable region in the fiber. As shown in Figure 3, when the index n is small, the crystallinity becomes very low and thus, the crystallization process is slow and the onset point is located rather far from the spinneret. Figures 4 and 5 demonstrate that diameter variation becomes slow and the neck-like deformation and small plateau caused by exothermic crystallization disappear as the index n decreases. Velocity distribution depicted in Figure 6 shows some singular point (a point of unsmoothness) when n is small, as if it was under a vigorous solidification condition. Hereinafter, n - 0.9 is assigned to the whole numerical analysis, since this value gives the best fit for the behavior of PET.
1134
50 1.0 (Newtonian)
6 km/min
4O
~
~
~
r~
30 20 n=0.7
10
0
I
I
I
50
100
150
200
Axial distance (cm) Figure 3. Crystallinity along the spinline at various power-law indices.
0.1
6 km/min
n
0.7 0.8 0.9
0.01
?5
(Newtonian)
0.001 0
I
I
I
50
100
150
200
Axial distance (cm) Figure 4. Fiber diameter along the spinline at various power-law indices.
1135
300
6 km/min
200
n shift(cm) 0.7 30 0.8 20 0.9 10
//~
~D .+.a
100
._
0
I
I
[
50
100
150
200
Axial distance (cm) Figure 5. Temperature distribution along the spinline at various power-law indices.
6 km/min 6 n
0.7 0.8 0.9 1.0 (Newtonian)
4 o CD
>
2
0
I
I
[
50
100
150
200
Axial distance (cm) Figure6. Velocity distribution along the spinline at various power-law indices.
1136
3.2. Attenuation along the Spinline Of all factors affecting fiber dimension and properties, such spinning conditions as take-up velocity, quench air velocity and flow rate seem to be the most important. Figure 7 shows that fiber diameter attenuates slowly at such low take-up speed as 4 km/min. However, as one increases the spinning speed, it decreases rapidly to its final value, as if a neck-like deformation occurs in the spinline. This is because the crystallization proceeds rapidly under the high speed and the extrudate solidifies quickly. Hence, the deformation is concentrated on the narrow region in the amorphous state. Figure 8 demonstrates the effect of quench air speed on the fiber diameter at the spinning speed of 6 km/min. From this result, we can draw a conclusion that the quench air speed does not have a significant influence on the fiber diameter. Effect of mass flow rate on the fiber diameter is illustrated in Figure 9. It is evident that the fiber diameter, and hence, the distance of the onset point of crystallization from the spinneret increases with the increase in the flow rate. 0.1 V ai r --'- 2 0
m / m i n
W = 1.5 g/min, hole
0.01
~
4 km/min
9 v,,,,q
"Q~
6
\ ~ ~ _ 0.001
I
0
50
:__,__:,
.
.
.
.
.
.
.
.
I. . . . .
100
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
,
I
150
200
Axial distance (cm) Figure 7. Diameter distribution along the spinline at various take-up speeds (symbols are for experimental results). [] ~ 4 km/min A ... 5 km/min o--- 6 km/min
1137
0.1
v = 6 km/min W = 1.5 g/min, hole
0
~--
0 m/min 20 3O
o 0.01 cO
?5
0.001 0
50
100
150
200
Axial distance (cm) Figure 8. Diameter distribution along the spinline at various quench air speeds (symbols are for experimental results). [] 0 m/min 9 20m/min o--- 30 m/min 0.1 v = 6 km/min Uair = 20 m/min r
(D r
\~"~'~ ',,\ "\
2.0 g/rain, hole ~_1.5 1.0
\
0.01
?5
.o _ -----
0.001 0
.
.
.
.
.
.
.
.
.
.
.
.
.
_ . . . . . . . . . . . . . . . . . .
I
I
I
50
100
150
200
Axial distance (cm) Figure9. Diameter distribution along the spinline at various mass flow rates(symbols are for experimental results). n 1.0 g/rain 9 hole A ... 1.5 g/min, hole o--- 2.0 g/min 9 hole
1138
In Figures 10-12, we can observe the temperature profile along the spinline as a function of take-up speed, quench air velocity and mass flow rate, respectively. The temperature decreases slowly (slow in a sense not of time but of distance), as the spinning speed increases. Similarly to the previous result on the fiber diameter, the change of quench air speed rarely alters the temperature profile as shown in Figure 11. We can examine the significant effect of flow rate in Figure 12. As the flow rate increases, cooling becomes very slow due to the increase of fiber diameter, and therefore quick temperature variation is suppressed. Figures 13-15 are the results for crystallinity variation. Figure 13 clearly shows that the rate of crystallization and the final value of crystallinity increase as the spinning speed increases. Again, the effect of quench air speed on the crystallinity may be regarded as negligible. Concerning the effect of flow rate, even though the crystallization starts later, the equilibrium value of crystallinity increases with its increase.
300
-••
v air = 20 m/min ', \
~
W = 1.5 g/min, hole
200
"- x\ "",, \ \ / / - 100
shift(cm) 4km/min 0
5
10
" ~,
20
% . %~ ".o 9
0
I,
I
I
50
100
150
200
Axial distance (cm) Figure 10. Temperature distribution along the spinline at various take-up speeds. 4 km/min ... 5m/min --- 6 km/min
1139
300
-• _
~
v = 6 km/min W = 1.5 g/min, hole
~
200 0 m/min
100
0
50
100
150
200
Axial distance (cm) Figure 11. Temperature distribution along the spinline at various quench air speeds. 0 m/min .-. 20 m/min --- 30 m/min 300
-•',•
v = 6 km/min v~i~ = 20 m/min
~
\,,,
200
\, \ \
100
-
~
"'",,%"",,2.0 g/min, hole ~
I
0 Figure 12. Temperature flow rates. 1.0 g/min 9 hole
50
I
I
100 150 Axial distance (cm)
distribution along the ..- 1.5 g/min- hole
200
spinline at various --- 2.0 g/min 9 hole
mass
1140
50 6 km/min tl'" #
40 ! o~..i
I I I I I I / I /I
30
o~..~
t~
20
,;~ ..~
I
'
I
r,.)
:
,i"U //
10
Vair
~" 2 0 m / m i n
W = 1.5 g/min, hole 0
50
100
150
200
Axial distance (cm) Figure 13. Crystallinity along the spinline at various take-up (symbols are for experimental results). [ ] ~ 4 km/min 9 5km/min o--- 6 km/min
speeds
50 40
o,.-i
! f
30
.~.,i
t~ ra~
20
0 m/min 30
r,.)
v = 6 km/min W = 1.5 g/min, hole
10
0
50
1
I
100
150
200
Axial distance (cm) Figure 14. Crystallinity along the spinline at various quench air speeds (symbols are for experimental results). [] 0 m/min 9 ..- 20 m/min o--- 30 m/min
1141
50
2.0 g/min, hole ,'7 .....
40
~
1.0
/// '///
30 4~
]'13- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
'~
20
'J;,/ !,
10
v = 6 km/min
;. ,
U ai r =
...
0
50
20
i
i
100
150
m/min
200
Axial distance (cm) Figure 15. Crystallinity along the spinline at various mass flow rates (symbols are for experimental results). [] 1.0 g / m i n - h o l e 9 1.5 g/min, hole o--- 2.0 g/min- hole
3.3. S k i n - C o r e F o r m a t i o n in the Fiber Cross-Section
One of the most important characteristics of high speed spinning is the distinct structural difference formed along the fiber cross-section due to different thermal and stress history applied during the spinning process. This is so-called the skin-core structure, and its formation mechanism can be explained with the help of results obtained in the 2-dimensional simulation. Figure 16 shows the temperature change at the skin and core regions of the fiber cross-section. Fibers extruded from the spinneret are cooled by the ambient quench air. Fiber skin has direct contact with the cooling air, and thus, it is cooled down more rapidly than the fiber core. This temperature difference between the skin and core induces the difference in crystallization process and as a result the skin-core structure forms. Figure 17 demonstrates the difference between the crystallinities in skin and core.
1142
Calculated results on radial distribution of the temperature in the fiber cross-section at various axial positions, are also presented in Figure 18, which simply illustrates monotonic decrease of temperature with radius. However, the temperature difference between the skin and the core becomes insignificant at the point far from the spinneret, where the temperature of the fiber becomes close to that of the surrounding air. Figure 19 shows the radial distribution of crystallinity at various points in the axial direction. The crystallization starts at the skin near the spinneret exit, then it proceeds slowly along the spinline, and crystallinity finally reaches the equilibrium distribution. It may be interpreted as that the crystallization process is initiated by the cooling and also the following stress development in the skin, and thus the crystallization propagates from the skin to the core.
300 v = 6 km/min
200 O
skin
100
0
1
I
50
100
_
l
150
200
Axial distance (cm) Figure 16. Temperature variation along the spinline in the skin and core of the PET fiber.
1143
skin core .l..,a
~" 0.5 (D ~ 1,,,,i
v = 6 km/min 0
.
I
0
50
I
100
150
200
Axial distance (cm) Figure 17. Relative crystallinity along the spinline in the skin and core of the PET fiber.
300 v = 6 km/min z = 20 cm 30
200
45 ~' 100
75 115 0
I
I
I
[
0.2
0.4
0.6
0.8
1
Normalized radius (r/R) Figure 18. Radial distribution of temperature at various positions spinline.
in the
1144
~
z =60 cm
o
v
v = 6 km/min
0.5
35 i
0
0.2
i
33
-----------'-~
-~
-----------
0.4 0.6 0.8 Normalized radius (r/R)
1
Figure 19. Radial distribution of relative crystallinity at various positions in the spinline.
3.4. Orientation Distribution in the Fiber Cross-section At the spinning speed of 6 km/min, it is thought that the molecular orientation develops mainly in the skin region before the point of necking, then after the completion of necking the orientation in the fiber core increases to the amount comparable to that in the skin. However, at the speed of 8 km/min or above, the stress concentration in the skin region prevents the formation of high degree of orientation in the amorphous region of fiber core, hence the distinct skin-core structure is formed. In Figure 20, one can see the calculated results of radial distribution of refractive indices in both parallel (n/l) and perpendicular (n• directions. While n• does not vary much along the radius, n// exhibits strong dependence on the radial position.
1145
The birefringence obtained from such data in Figure 20 as is plotted on Figures 21-23. At low take-up speed it does not show much variation along the radial direction, but at 6 km/min its dependence becomes noticeable (Figure 21). From Figure 22, we can observe somewhat significant effect of quench air speed on birefringence. As can be seen in Figure 23, the increase of the mass flow rate, the difference between the birefringence in the skin and the core becomes larger, since the increase of fiber diameter magnifies the structural difference between those regions.
1.68
1.54
1.67
g/•
x~ 9 ,,,,-i
~
1.535
1.66
9 ,-,,i
~
v = 6 km/min
1.65
v~i~ = 30 m/min W = 2.0 g/min, hole 1.64
0
0.2
I
q
q
0.4
0.6
0.8
1.53 1
Axial distance (cm) Figure 20. Refractive index profile across the fiber cross-section for PET fibers. parallel (n/l) --- perpendicular ( n l )
1146
140 Pair
= 30 m/min
W = 1.5 g/min, hole .
x
120
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
_
5
80 0
I
I
1
I
0.2
0.4
0.6
0.8
1
Axial distance (cm) Figure21. Birefringence profile across the fibers at various take-up speeds. 4 km/min ... 5 km/min
fiber cross-section
for PET
--- 6 km/min
140 0 rn/min
x
30
120
em cD o~.~
100 v = 6 km/min W = 1.5 g/min, hole 80 0
0.2
1
I
I
0.4
0.6
0.8
1
Axial distance (cm) Figure22. Birefringence
profile
across
the
fibers at various quench air speeds. 0 m/min ... 20 m/min
fiber
cross-section
--- 30 m/min
for
PET
1147
140 1.5 "" X
120
10
................... 2.0 g/min, hole
N
loo
.~,,i
v = 6 km/min Uair-- 3 0 m/min 80 0
0.2
I
I
I
0.4
0.6
0.8
1
Axial distance (cm) Figure23. Birefringence profile across the fiber cross-section for PET fibers at various mass flow rates. 1.0 g / m i n - h o l e ... 1.5 g/min, hole --- 2.0 g/min- hole
4. CONCLUSIONS In this study, we have performed the analysis of high speed spinning process, employing some hybrid numerical methods combining both 1and 2-dimensional formulations. The radius of the extrudate and the temperature on its surface are calculated under the 1-dimensional flow assumption, and then with these results specified as boundary conditions, 2-dimensional simulation is conducted to estimate the temperature profile and the morphological structure development along the fiber cross-section. Including the effect of crystallinity in the elongational viscosity and assuming the tensile stress acts only on the amorphous region enabled us to observe the neck-like deformation in the spinline. From the result of numerical simulation, it can be concluded that the take-up speed plays the most determining role in the fiber structure formation. Even though the flow rate shows strong effect on diameter and
1148
temperature variation along the spinline, its role diminishes in the case of crystallinity and birefringence of the final products. The formation of the fiber skin-core structure is simulated in this numerical procedure. Effect of spinning conditions on the formation of skin-core structure is estimated by observing the birefringence distribution in the cross-section. The distinct skin-core structure builds up due to the combined effect of high speed take-up and quench air. As the quench air velocity increases at high speed spinning and the fiber diameter becomes thicker, this structural inhomogeneity becomes severer.
REFERENCES
1. T. Kawaguchi, High-Speed Fiber Spinning, Chapter 1, John Wiley and Sons, 1985. 2. R. J. Fisher and M. M. Denn, AIChE J., 22 (1976), 236. 3. J. C. Hyun, AIChE J., 26 (1980), 294. 4. J. Shimizu, N. Okui and T. Kikutani, High-Speed Fiber Spinning, Chapter 7, John Wiley and Sons, 1985. 5. H. Yamada, T. Kikutani, A. Takaku and J. Shimizu, Sen-I Gakkaishi, 44 (1988), 177. 6. M. Matsui, Sen-I Gakkaishi, 38 (1982), P-508. 7. A. Ziabicki and K. Kedzierska, J. Appl. Polymer Sci., 6 (1962), 111. 8. K. Nakamura, T. Watanabe, K. Katayama and T. Amano, J. Appl. Polymer Sci., 16 (1972), 1077. 9. T. Kikutani, Ph.D. Thesis, Tokyo Institute of Technology, 1982. 10. K. Kamide and T. Kuriki, Sen-I Kikai Gakkaishi, 38 (1985), P-268. 11. G. Vassilatos, B. H. Knox and H. R. E. Frankfort, High-Speed Fiber Spinning, Chapter 14, John Wiley and Sons, 1985. 12. J. Shimizu, N. Okui and T. Kikutani, Sen-I Gakkaishi, 37 (1981), T-135. 13. J. Shimizu, Sen-I Kikai Gakkaishi, 38 (1985), P-243. 14. S. A. Silling, J. Appl. Mechanics, 55 (1988), 530. 15. B. D. Coleman and D. C. Newman, Polymer Eng. Sci., 30 (1990), 1299. 16. Y. Tomita and K. Hayashi, Int. J. Solids Structure, 30 (1993), 225.
1149
17. Y. Tomita, Appl. Mech. Review, 47 (1994), 171. 18. S. I. Krishnamachari, Applied Stress Analysis of Plastics, Chapter 3, Van Nostrand Reinhold, 1993. 19. G. Perez, High-Speed Fiber Spinning, Chapter 12, John Wiley and Sons, 1985. 20. S. Kase, High-Speed Fiber Spinning, Chapter 3, John Wiley and Sons, 1985. 21. K. Fujimoto, K. Iohara, L. Owaki and Y. Murase, Sen-I Gakkaishi, 44 (1988), 53. 22. H. Haberkorn, K. Hahn, H. D. Dorrer and P. Matthies, J. Appl. Polymer Sci., 47 (1993), 1551. 23. A. Ziabicki, J. Non-Newtonian Fluid Mech., 30 (1988), 141. 24. A. Ziabicki and J. Tian, J. Non-Newtonian Fluid Mech., 47 (1993), 57. 25. A. Ziabicki and L. Jarecki, High-Speed Fiber Spinning, Chapter 9, John Wiley and Sons, 1985. 26. J. M. Schultz, Polymer Eng. Sci., 31 (1991), 661. 27. J. M. Schultz, Polymer, 32 (1991), 3268. 28. H. M. Heuvel and R. Huisman, High-Speed Fiber Spinning, Chapter 11, John Wiley and Sons, 1985. 29. A. Ziabicki, J. Non-Newtonian Fluid Mech., 30 (1988), 157. 30. A. Ziabicki, Fundamentals of Fibre Formation, Chapter 2---3, John Wiley and Sons, 1976. 31. S. Kase and T. Matsuo, J. Polym. Sci., A3 (1965), 2541. 32. S. Kase and T. Matsuo, J. Appl. Polymer Sci., 11 (1967), 251. 33. A. Prastaro and P. Parrini, Textile Research J., 45 (1975), 118. 34. Y. Yasuda, H. Sugiyama and H. Yanagawa, Sen-I Gakkaishi, 40 (1984), T-370. 35. Y. Yasuda, H. Sugiyama and H. Hayashi, Sen-I Gakkaishi, 40 (1984), T-227. 36. M. M. Denn and D. K. Gagon, Polymer Eng. Sci., 21 (1981), 844. 37. K. Katayama and M. G. Yoon, High-Speed Fiber Spinning, Chapter 8, John Wiley and Sons, 1985. 38. R. M. Patel, J. H. Bheda and J. E. Spruiell, J. Appl. Polymer Sci., 42 (1991), 1671. 39. K. F. Zieminski and J. E. Spruiell, J. Appl. Polymer Sci., 35 (1988),
1150
40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52.
2223. J. H. Bheda and J. E. Spruiell, J. Appl. Polymer Sci., 39 (1990), 447. K. W. Hutchenson, D. D. Edie and D. M. Riggs, J. Appl. Polymer Sci., 29 (1984), 3621. W. P. Bell and D. D. Edie, J. Appl. Polymer Sci., 33 (1987), 1073. A. Dutta, Text. Res. J., 57 (1987), 13. K. Nakamura, K. Katayama and T. Amano, J. Appl. Polymer Sci., 17 (1993), 1031. A. Ziabicki, Kolloid Z., 175 (1961), 14. G. C. Alfonso, M. P. Verdona and A, Wasiak, Polymer, 19 (1978), 711. I. Hamana, M. Matsui and S. Kato, Melliand Textilberichte, 50 (1969), 382. I. Hamana, M. Matsui and S. Kato, Melliand Textilberichte, 50 (1969), 499. J. H. Dumbleton, J. Polymer Sci. A-2, 6 (1968), 795. T. Murayama, J. H. Dumbleton and M. L. Williams, J. Polymer Sci., A-2, 6 (1968), 787. E. B. Becker, G. F. Carey and J. T. Oden, Finite Elements : An Instruction Vol. 1, Prentice-Hall, 1981. M. J. Crochet, A. R. Davies and K. Walters, Numerical Simulation of Non-Newtonian Flow, Elsevier, 1984.
1151
PHYSICAL MECHANISMS MEDIA
O F F O A M F L O W IN P O R O U S
K o n s t a n t i n G . K o r n e v ~ , A l e x a n d e r V . N e i m a r k b , a n d A l e k s e y N. R o z h k o v ~
Institute for Problems in Mechanics, Russian Academy of Sciences, 101(1) Prospect Vernadskogo, Moscow 117526, Russia kornev@ipmnet, ru, rozhkov@ipmnet, ru b TRI/Princeton, 601 Prospect Ave., Princeton, NJ 08542-0625, USA aneimark@triprinceton, org 1. INTRODUCTION The cell structure of foams can be observed in many daily situations. We encounter many examples of foam irregularity, instability, chaos and the like in everyday practice. The specific properties of foams have been attracted the attention of researchers over the past few centuries. At the present time, there exists a substantial collection of books and reviews which cover a wide range of fimdamental questions. Books and reviews written by Boys [1 ], Mysels, Shinoda and Frankel [2], Bikerman [3], Ivanov [4], Kraynik [5], Wilson [6], Kruglyakov and Exerowa [7], Stavans [8] and Prud'homme and Khan [9] are only some of them. In confining systems such as porous materials, foam has been recognized as a fluid with a tmique hierarchical structure and rheological properties. In recent decades, due to novel applications, foams in porous media have become an intriguing subject of fimdamental, systematic studies by physical scientists. In particular, because of their extremely efficient blocking action, foams open wide horizons for applications in problems of groundwater/soil remediation, selective blockage and other processes of sweeping out a given porous material of either oil or another liquid. Foams found various applications in industrial technologies dealing with fiber structures, such as dyeing and finishing of textile fabrics, paper coating, resin-impregnation of fibrous mats and fabrics. However, while technologically effective, foams distributed in porous media or residing in contact with highly dispersed materials remain enigmatic. In this chapter, we discuss
1152
some physical problems associated with foams in porous systems. We do not attempt a complete review. Previous reviews of Raza [10], Marsden [11 ], Heller and Ktmtamukkula [12], Schramm [13] and Rossen [14] give a useful guideline in this branch of studies. Many important problems remain beyond the scope of this survey. Among them are mechanisms of foam generation in pores, interactions between foam lamellae and the wetting films clinging to the pore walls (Kheifetz and Neimark [15]), a role of disjoining pressure acting in thin films (Derjaguin, et al [16]; Neimark and Vignes-Adler [17]), transformations of lens into lamella (Ivanov [4]; Kruglyakov and Exerowa [7], bubble transport through smooth capillaries and physics of dynamic wetting (de Gennes [18]), mechanistic models of foam evolution in the bulk and in porous media, which are based on the population balance method (Falls, et al [19]), and also percolation models of foam flow in pore networks (see the latest review of Rossen [14] and references therein). We focus on the physical mechanisms which govern foam flow in confining systems. 2. FOAMS IN POROUS MEDIA
Although bulk foam, as a matter for scientific inquiry, has a great history, the problem of foam patterning in confining systems, such as porous media, drew the attention of scientists only recently. Apparently Fried [20] was the first to show that foam, because of its unique structuxe, reduces gas flow in porous media. In recent decades, it has been recognized that foam within porous media indeed possesses a unique structure and rheological properties. In particular, in both preand in situ-generated foams, pore sizes impose some constraints on the foam texture. If the characteristic bubble size is much smaller than the characteristic pore size, the foam confined in pores does not differ from bulk foam. In the opposite case, the foam in a porous medium is a system of thin liquid films (lamellae) spanning the pore channels. The transition from the bulk phase to the new "pore-confined" regime of foam patterning is very complex, and is discussed, as a rule, for ideal homogeneous capillaries or bead packs (Hirasaky and Lawson [21]; Nutt and Burley [22]). However, a more reasonable conceptual model of natural porous media is a network of interconnected capillaries of different sizes, which may contain constrictions and enlargements. Recent experiments (Bazilevsky, et al [23]) show that the motion of pregenerated foam through such a system demonstrates specific characteristic patterns.
1153
Figure 1. The sequence of frames depicts the displacements of the lamellae in the enlargement of the channel. The flow direction is from the bottom upwards. The white and black arrows indicate the positions of the pertinent lamellae at different moments of time. The channel consisted of 10 cubic cell elements, an 8 x 8 m m 2 cross-section, and 10 narrow elements 8 r a m long, and a 2.5 x 2 . 5 r a m 2 cross section. The main peculiarities of the phenomenon are as follows (Figure 1). Just after the beginning of their injection, the lamellae move freely along the narrow part of the channel. If the broad part is free of foam, the lamella, reaching this point, stops for the present, not being able to overcome the enlargement of the channel. Under the action of pressure supplied by the gas, the lamella swells, forming a bubble. The bubble either disappears, and the bubble swelling process restarts when the next lamella reaches this point; or, after touching the wall of the enlargement, the lamella jumps to the free zone and stops, clinging to the walls of the wide pore somewhere. The following lamella behaves similarly. The advancing bubbles jump sequentially from the narrow to the wide part of the channel and come to a standstill in the engagement with the walls or other lamellae. This process continues until the wide part of the channel is filled entirely with foam, whereupon the lamellae are free to pass through it. In the next enlargement, the process of foam filling repeats itself. And all the wide parts of the channel are gradually filled with bulk foam. In doing so, the specific film channels of approximately the size of the narrow part of the original channel arise. With all the broad parts filled, the lamella motion proceeds as follows. The drift of lamellae along the narrow zone alternates with periods of motion through
1154
the effective channels in the enlargements. In the last case, lamellae are subjected to transformation, and their motion is jumpy. (It is related to the tendency of the lamella system to minimize its surface energy.) No processes of rupture or birth of the moving lamellae are observed. Due to the manner of the foam generation, the distance between the drifting lamellae is of the same order as the channel period. When lamellae are passing each other, the bulk foam in the enlargements also undergoes deformations. However, its structure is not changed. It seems to be the same before and after each lamella passes. Thus, the experiments show that, during the motion of foam through the channel of an abruptly varying cross-section, a portion of it is used to form the effective channel. Such a channel plays the role of a transport path for another portion of foam. In the process, the foam lamellae demonstrate an astonishing stability. This and other visual observations (Owete and Brigham [24]; Chambers and Radke [25]) speak in favor of a crucial role of the solid skeleton in foam patterning in porous media. Even excluding for the moment the hydrodynamic features, foams in porous media or in contact with highly dispersed materials, have a unique structure by virtue of the thermodynamic reasons. Indeed, in the absence of external restraints, an interface between the gas and liquid phases tends to minimize its surface free energy by adopting a particular ordered structure. In the presence of solid surfaces, however, the interface between the two fluids takes up configurations whose equilibrium and stability depend upon the areas of contact and the interfaeial free energies of the surfaces separating the various phases. The solid skeleton imposes to the bubble system a multilevel hierarchical structure, in which the surface and capillary forces are self-consistently interconnected. The wetting films covering the pore walls interact with the lamellae, while the general conditions of foam equilibrium include primarily the conditions of equilibrium between these two type of thin films. In many instances, the thermodynamic properties of lamellae and the wetting films may dominate the foam behavior in porous media. This type of interactions especially concerns the strong foams, in which all bubbles are bounded by the lamellae (bubble-bubble interfaces) and the wetting films (bubble-solid interfaces) (Ettinger and Radke [26]; Kovscek and Radke [27]; Rossen [14]) (see Figure 2). Namely the strong foams play a major role in applications due to their strong blocking action.
1155
9
9
iii
Figure 2. The sketch of the gas and liquid distribution within a strong foam in a granular medium.
3. STICK-SLIP MOTION OF FOAM L A M E L L A Pore channels in real materials are not uniform. Alternating constrictions (pore throats) and enlargements cause variations of the Laplacian pressure across a lamella moving through the channel. Under these conditions, the resulting motion of the lamella will be irregular. Indeed, just after the lamella has left the pore throat, it bulges forward and resists the movement. But as it reaches a convergent part of the pore, the lamella bulges backward and accelerates. Visual experiments (Chambers and Radke [25]; Bazilevsky, et al [23]) do reveal that, when pushed at a constant pressure drop, very often the lamellae do not move steadily, but by an alternation of slides and periods of rest. This type of motion is called the stickslip motion (Rabinowicz [28]; Bowden and Tabor [29]). Specific features of foam patterning within a porous medium can be revealed on the simplest yet expository model of a one-dimensional foam, referred to as a bubble chain or train, immersed into a wavy channel (Komev [30]; Komev, et al [31 ]; Dautov, et al [32]). Imagine such a channel as a rigid capillary with a radius
r=ro+~C~
(2~x
2 ~'
(1)
1156
where the x - axis coincides with the axis of the symmetry of a capillary, to, ,~, and J are some characteristic scales of the porous medium. Assume, for simplicity, that the pore aspect ratio is a low ,~/ro << 1. This assumption, however, can be altered without change in most of the physical conclusions.
I ~I
a)
90 ~
Plateau border
b) ----~+i ~. 2
ai
X
Figure 3. a) Scheme of lamella distribution in a bamboo-like channel. Dashed lines are attributed to the initial positions of lamellae, and bold faced lines represent the lamellae under a load. b) Specification of the input parameters needed for calculation of the bubble volume variation and capillary force.
1157
Consider the train consisting of N lamellae, whose centers of chords are prescribed as points x = a ~ , a 2 , . . . a ~ , and assume that in the initial undeformed state, the foam is perfectly ordered. In such an ordered foam, the lamellae are connected by links (bubbles, or train carriages) of length K2 with an integer number K (K2 is the "carriage length"). Thus, the capillary forces tend to fix the lamellae at the pore throats. However, the elastic forces, caused by gas compressibility, compel the lamellae to shift into new equilibrium positions. Competition between these forces does result in the equilibrium states, which are observed in the experiments with foams in porous systems. The situation resembles 'commensurate - incommensurate' phase transitions in solid state physics (Pokrovsky and Yalapov [33]; Bak [34]) and related problems, in which a struggle between the binding and pinning forces occurs (Frenkel and Kontorova [35]; Josephson [36]; Kulik [37]; Seeger and Schiller [38]; Barone and Paterno [39]; Lichtenberg and Lieberman [40]). Under a load, the i -th bubble is deformed. And its length, with an accuracy of O ( 6 / r o ) , becomes a , - a,_, (Kornev [30]; Kornev, et al [31]; Dautov, et al [32]). The resulting elastic force acting upon the i - th lamella can be written with the same accuracy as
f e = xr21Pi+l - Pi )
(2)
Here P, is the gas pressure in the i-th bubble. Making use of the equation of the state of the ideal gas, we have with the same accuracy the following equation
Pi +1 (ai +1
-
ai ) = Pi (ai - ai-1
) ="=
PgK2 ,
(3)
where P~ is the initial gas pressure in an individual bubble. It is more convenient to rewrite equation (3) by introducing a new unknown fimction, p, - displacement of the i - th lamella from its initial position at the throat (Figure 3). Then a~ = i K ~ + p, + 3./2 and equation (3) takes the form
PgK2 Pi+l-Pi
= p
2K
,
(4)
i+1
Note that the capillary force has the form, similar to equation (2), but, instead of the pressure differential, the Laplacian pressure drop,
1158
R-1
ii+l =
(P'+I - P i ) 4y
(5)
should be inserted. Here 2y is the lamella tension, /~.,+,is the radius of curvature of a lamella (Figure 3). Then the capillary pressure drop can be written with the same accuracy of O(6/r o) as
Pi+l-Pi =
8Jr~ sinl2~r;i)/ 2r 0
(6)
The stick-slip motion of the bubble tram can be qualitatively described by using the following model
d2p i m~ = Of 2
~rrgAP
(7)
where m is an effective mass of the lamella, i is the dimensional time, and AP is the pressure drop across the lamella. The latter is originated from i ) the 'elastic' pressure drop, equations (2)-(4), ii) the Laplacian pressure drop, equation (6), iii ) the dynamic pressure drop, iiii ) and the extemal pressure drop. For clarity, consider only a quasi-static motion so that all the deviations from the Newtonian friction are assumed to be small. The main peculiarities of the stick-slip phenomenon can be elucidated by analyzing the motion of an individual lamella. In dimensionless variables, equation (7) written for a single lamella takes the form
e 2 d2pi dt---2- +
dPi = Ap-2n-psin -~Pi '
2 where e2= 2mh~Pg/9r/;ppr o2 2
(8)
and a dimensionless time is introduced as follows
t-2~Pgi/32rL~. Here rl.,~ is an apparent viscosity of a lamella and h~ is the wetting film thickness. The parameter
4y6 p = Pgro;~
(9)
1159
serves as a measure of the intensity of the capillary forces with respect to the driving forces (e.g., in the case of a bubble chain, these forces will be the elastic forces). The parameter zXp denotes an external pressure drop sealed by P~. Equation (9) has a very broad range of interpretations. It is widely used and describes various physical systems, depending on the choice of the parameters (Landau and Lifshitz [41]; Barone and Paterno [39]; Lichtenberg and Lieberman [40]; Zaslavsky and Sagdeev [42]). In application to the lamella motion, the parameter G is usually small. Therefore, the first term in the left hand side of equation (9) dominates the second only for initial instants of time. For an analysis of the lamella dynamics during a long time interval, the inertial term may be dropped and, as a result, equation (9) takes the form
dt
= A p - 2~r/.t cjn P i ' A p = C o n s t
(10)
.
This equation can be completely analyzed, because it has the explicit solution (Aslamasov and Larkin [43]; see also Barone and Patemo [39])
,
X~-~
.
tan
+
2~/~ Ap ]['
(11)
where the period T is expressed as r
=
27/"
.
(12)
IJAp 2 - 4~r2p 2 Notably, the characteristic time of translation over a period of the channel depends upon the applied pressure drop. Thus, the periodic motion represented by equation (11) looks like a stick-slip motion. In the vicinity of a threshold, i.e., when the applied pressure drop is of the same order as the Laplacian barrier, 6p ~ 2z/~, the lamella spends the main portion of the time in a slow creep. After overcoming the maximum pinning force (i.e., after reaching the point p - rc/2 ), the lamella suddenly jumps an enlargement and then the process of lamella motion repeats itself. As the pressure drop increases, the jump time becomes negligibly small, and the lamella moves almost
1160
steadily. The lamella velocity, averaged over the period T, becomes the following nonlinear function of the external pressure drop
0.12
2~./,t~p=0.99 ...... -2~Z~ =0.5 - - - 2~/gp =0.1 0.4 --.
. . . . . .
,
O.O
_ . . . . . . . . . . .
I
- . . . . . .
,
---
i
~
9. . . . . . .
,
0.4
i
,
I
,
,
0.8
tff Figure 4. Transition from a stick-slip motion to a sliding motion ( ~ 1 T d ~r 2n = ~Ap 2 - 4n. 2/z 2 <,b> =~ 0 r= T
= 10 -3 ).
(13)
with the expected threshold zXp~ p . Formula (13) displays the increase of the apparent viscosity of the lamella along with the increase of the rate of shear, as determined by the lamella speed (Figure 5). This is the so-called 'shearthickening' effect (Barnes, et al [44]). The averaging procedure which leads to equation (13) somewhat disguises the cause of the 'shear-thickening' effect. The physics of such a rheological behavior is as follows. In the vicinity of the critical pressure drop, the capillary forces drive the lamella almost entirely. Figuratively speaking, in this critical regime, a 'bare' external pressure drop is needed only to compel the lamella to shift from its equilibrium position. Once shifted, the lamella drifts almost autonomously. Only in a high-speed regime, does the apparent viscosity of the lamella tend to its bare value. Therefore, this 'shear thickening' behavior of a lamella is caused solely by irregularities of the channel. If we increase the pressure drop ft~her and further, we will inevitably arrive at the range of the validity of a Non-Newtonian friction law. The Bretherton friction law
1161
[45] is one of the possible candidates for description of the lamella friction in a Non-Newtonian range of flow. We recall that Bretherton [45] analyzed the motion of long bubbles in a cylindrical capillary of the radius Rand computed the thickness of the film between the bubble and the capillary wall h~ and the dynamic pressure drop across the length of the bubble. Both depend on the capillary number Ca (3Ur//cr), where U is the bubble velocity, 1/ is the viscosity of the wetting film, and cr is the surface tension. Bretherton pointed out that the dynamic pressure drop across such a bubble, additional to the Laplacian pressure drop, is given by =
(3r/U)2/3 cr
B =3.58.
0.05 <~>
f
J
J 0.02
/J
0.01
,
GO
I
I
0.01
,
I
j
,
i
,,,
0.02
9
I
0.03 Ap
Figure 5. 'Shear-thickening' behavior of an individual lamella. The increase of viscosity with the increasing rate of shear in an averaged steady flow (p = 10-~). This result helps to explain the lamella mobility. In particular, Hirasaki and Lawson [21] automatically spread Bretherton's analysis to a chain of bubbles separated by lamellae. It is assumed that the bubble train moves through a
1162
uniform capillary in a piston-like manner so that the distance between the lamellae remains the same. The Plateau border, in their analysis, is modelled as a region of constant curvature, i.e., the Plateau border is similar to the liquid ahead of and behind an isolated bubble, except that the radius of the curvature of a meniscus can be less than the tube radius. Hereby, the characteristic length of the hydrodynamic perturbations becomes of an order of the size of the Plateau border, i.e., it is much smaller than the length of the bubble. Within the regions of uniformity, the wetting film does not move yet resides at the corresponding gas pressure P, . Because of the difference in the gas pressures within the adjacent bubbles, there exists a transition region, the Plateau border, in which the liquid undergoes an action of the respective pressure drop. So, the liquid in the vicinity of the Plateau borders flows from one bubble to another, 'beating' the lamella. As a result, the total dynamic pressure drop per lamella becomes proportional to (Ca)2'3cr/r~ , where r~ is the radius of curvature of the interface at the Plateau border. It should be stressed that the theory which deals with a uniform capillary and which diminishes the effect of gas compressibility underestimates the foam viscosity under all experimental conditions (Figure 6). In addition, even the tendency in dependence of the gas-phase velocity is predicted incorrectly. To resolve the contradictions, Falls, et al [46] modified the Hirasaki-Lawson theory to account for the contribution of pore constrictions to the apparent viscosity. The Laplacian contribution to the dynamic pressure gradient was approximated as v P ~ n14o-/r,, where n~ is the number of lamellae per unit length and r, is the radius of the pore throat. As a result, the Hirasaki-Lawson apparent viscosity is modified by augmenting the Laplacian contribution. The latter is defined through the Hagen-Poiseuille law as, rico,= 4onI/Vgrt, where vg is the gas-phase velocity. Thus, the expected result is as follows: the apparent viscosity must vary with the -1 power of the gas velocity at low speeds. Falls and coworkers studied two regimes of foam generation: the first is the so-called controlled-bubble-size regime, where the bubble size was unchanged in situ, and the liquid was transported within the lamellae; the second is the so-called packgenerated-bubble-size regime, where one bead pack was used to generate the foam for a second. It has been demonstrated for both flow regimes that the apparent viscosity measured at low speeds varies as -1 power of the gas-phase velocity. Moreover, involving the Laplacian contribution of the apparent viscosity, the authors managed to fit the Hirasaki-Lawson theory to the experimental data even at higher speeds. They showed that, at higher rates, the viscosity varies as - 1 ! 3 power of the velocity in the first flow regime, and
1163
changes to - 2 / 3 power of the velocity in the second. The latter result means that the effect of the surface tension gradient governs the foam resistance in the second flow regime (Hirasaki and Lawson [21 ]).
10
o
/
ZX C17:~Pt=0
8
/
/
/
/
/
_
6
4
0
2
4
6
8
10
12
Figure 6. Typical rheological curves. Foams from 0.5% Hostapur SAS carbon number fractions. The length of the pack- 100 cm, the permeability-8 darcy, the pressure at the exiting end -7 bar. AP, denotes the threshold pressure drop (after Hanssen and Dalland [47]). Thus, the key assumption in treating the experimental data concerns the form of the contribution of the pore constrictions to lamella resistance. This term dictates the tendency in the total changes of the apparent viscosity as a function of the gas velocity. In the Bretherton flow regime, the 'shear thickening' behavior can be changed by the 'shear-thinning' (i.e., the reduction of viscosity with increasing rate of shear in an averaged steady flow). In particular, this fact can be checked by performing an analysis similar to that expressed by equations (10) and (13), but modified in accordance with the Bretherton theory (Komev [30]).
1164
I
I
I
I
I
I
I
I
I
i
i
I
a) ,90 >l-v/ 0 o 50 m >' 10" Ld V) "I" 12. I (r
**,/- OATA THEORY,
rl# + Tleon
kZ w
THEORY,J ' ' " ' " ' ' " -
""-.
rl# ONLY
<
I 0.1
5
vn.ocrry (cm/,) 104
I
i
*
m
i
i
i m I
I
b) DATA,
F
Xf
i--
THEORY,
TI. + Tlcon
0
10 3
q.. ,..
"" "
I--7' i.d
"-....
THEORY, 1"Is ONLY
10 z
0.01
9
DATA, Xf:l
/
"-. "-.
,
,
,
,
,
,
.
.
.
.
.
.
.
.
0.1
GAS-PHASE VELOCITY (cm/,)
Figure 7 Apparent foam viscosity in glass bead packs as a function of gas-phase velocity. Physical conditions: the capillary pressure Pc ~ 1900 dynes cm -~ , the
1165
ordinary gas-phase relative permeability k~
= 0.056.
a) The controlled-bubble-size
regime, r b/rca p ~ 3.8, where r b is the effective radius of a foam bubble and Gapis an equivalent capillary radius, b) The pack-generated bubble-size regime, r~/r~,, ~ 0.92. The parameter x e = S~/S~ is the ratio of the moving-gas saturation Se, estimated from the measured residence time of flowing gas, and the gas saturation, Se. For the pack-generated bubble-size regime, xe = 1; r/, is the Hirasaki-Lawson viscosity. (After Falls, et al [46]). Therefore, one of the possible explanations of the experimentally observed 'shear thinning' behavior (Figure 6) lies in a nonlinearity of the friction law as it might be expected from the Bretherton theory (Falls, et al [46]). The above mentioned results are applicable only for the following range of capillary numbers 10-4 < Ca < < 1 (Schwartz, et al [48]; Olbricht [49]) and for 'short' bubble trains. The bubble train is named 'short' when the length of the sample is on the same order of magnitude as the correlation length 1 N~o, ~ 2~rK ~ ' K# < <1 of a train (Dautov, et al [32]). (The Falls-MustersRatulowski experiments fall into this range). In the opposite case, the effect of chain elasticity cannot be ignored and the theory must be modified. 4. MOTION OF BUBBLE TRAINS
In order to examine the foam flow mechanisms, consider a less-studied alternative wave regime caused by oscillations of the gas pressure within the bubbles (Kornev and Kurdyumov [50]; Kornev [30]). There are at least two areas of applicability of the wave mechanism of foam transport: acoustic flows, for which the high frequency modes are important, and transport of the bubble trains itself. Illustrating these flow regimes, we shall first consider the specific features of the propagation of the acoustic waves through a perfectly ordered onedimensional foam, i.e., through a bubble chain. The standard starting point in any analysis of a wave picture, lies in a specification of the corresponding dispersion relations (Kittel [51]). It is instructive, therefore, to consider such a formula for our case. For the model expressed by the linearized version of equations (8) and (4), the respective relation has the following form (we neglect for a time the lamella friction)
1166
=
2
+ 2~r2K/~ ' ~
=
Pliquid
hK2,'
(14)
where co is the frequency, k is the wave number, and h is the lameUa thickness. This equation implies that there are two critical values for the frequency, below and above which the acoustic wave will not propagate through the chain. The lower critical boundary corresponds to the natural frequency of the individual lamella. This frequency is reached when the wave length tends to infinity (or k ~ 0), i.e., when the bubble train oscillates as a whole, and when all the lamellae vibrate at the throats of pore channel m unison. Another critical frequency selects the range of wave numbers admissible for the theory (when the wave length becomes comparable with the distance between adjacent lameUae, the theory fails to describe the triple substrate-gas-lamella interactions). Thus, the region, transparent to propagation of the acoustic waves, is bounded from above and below by well-defined values of the frequency. It should be stressed that the above discussed theory does not confine itself to the pore channels with small pore aspect ratios. Such a small-amplitude analysis is valid for all the channels within which foam resides as perfectly ordered. At a glance, because the screening effect is originated from the Laplacian blocking action of the lamellae, we should conclude that any perturbation will be suppressed if its frequency lies outside the admissible gap. It should be repeated that the above presented analysis relates only to the waves of a small amplitude, and it cannot be extended directly to a nonlinear case. Considering the stick-slip motion of lamellae, we should focus on the family of wave-propagating solutions of the respective nonlinear model, equations (8) and (4). This kind of dynamics occurs when we initially excite just one lamella, while all the others are initially at rest. In its general features, the stick-slip motion of the lamellae resembles the Frenkel-Kontorova mechanism for the flow of dislocations in crystals (Frenkel' and Kontorova [35]). The excited lameUa initiates a wave propagating through the bubble chain in a 'falling dominos' type of motion towards the other end of the chain, which is then reflected, moving in the opposite direction towards the initially excited lamella (Figure 8). In such a motion, practically no energy is lost during the flow, because the surface energy of the lamellae is compensated almost entirely by the elastic energy of the bubbles. Again, focusing on the dynamics within a short time interval, we find a soliton like that of the Frenkel-Kontorova kink (Frenkel' and Kontorova [35]; Lonngren [52]; Dodd, et al [53]). A scenario of the generation of a soliton is as follows.
1167
Just after the external pressure drop has overcome the threshold G - 4Vr--~ (Komev [30]; Dautov, et al [32]), the domain wall, which separates the stagnation zones ahead of and behind the front of the wave, depins and runs forward. Therefore, in soliton dynamics, the competition between the inertial, elastic, and pinning forces is important, and the viscous forces do not crucially change the character of the soliton propagation during a short period of time. If a bubble train moves for a long period of time, the inertia forces are negligibly small with respect to the three major forces - the elastic, viscous, and capillary forces. These forces struggle among themselves and drive the displacement waves. Notably, if the external pressure drop is maintained at a long time interval, the number of the domain walls changes. In fact, the number of the domain walls is proportional to the number of the channel periods over which the front domain wall has skipped. Therefore, the longer is the bubble train, the greater is the number of domain walls, and, consequently, the greater is the bubble train resistance. ta il lam e Ila
124 ;.122 ~-
N=80; I<=1 #=10 .3
120 ~1.18 1.18 L
middle lamella
1.14. ~
1.12 1.10 1D8 1.00 1.04
head lamella
102 1J00
=
3.g5
9
~
3.g8
I
~,
3.97
I
3.g8
=,
I
3.98
= ~ ~ r ~ ' ,
I
4.00
time
Figure 8. Wave mechanism of foam motion. The head and tail lamellae move m unison, while the middle lamella drifts in antiphase. Numerical experiment based on equations (8) and (4), e -- 0 (after Musin [56]).
1168
Illustrating a priority of the collective effects in the bubble train friction, we shall use a somewhat simplified model of lamella interactions. For clarity, a linear elastic spring will represent the elastic interactions of the lamellae. Moreover, because each individual domain wall plays a key role in the chain friction, we shall focus on the motion of a solitary wave. Then, in a continuous limit, equation (10) is written as (Kornev [30]) c~ p = 0 2 p
/~ sin p + q~.
3t
K
ds 2
(15)
Here q' is a dimensionless external force which represents the total action of the external pressure gradient onto each lamella in the chain. In the vicinity of the threshold, the profile of a domain wall can be adequately approximated by the Frenkel-Kontorova kink as p = 4 t a n - I exp[-(x- Xo)] +OL~]---~) , X =
.
(16)
Then, rewriting equation (15) in a wave fixed reference frame, the velocity of the domain wall is found as the solvability condition to equation (15) (Langer [54]; Kornev [30])
-
ooL-d-x)dx
= q~ ? dd~xX , --o0
(17)
where x - ~ - ~ ( s - vt) and the function p in equation (17) is expressed by equation (16). Substituting equation (16) into equation (17), we have qJ
Thus, equation (18) may be treated as an equation of the motion of an individual effective lamella along an active channel. The friction coefficient of such a macro-lamella is affected by the binding and pinning energies of the chain. A dependency like equation (18) has been recently reported by Braiman, et al [55] in a different comext. Their and Musin's [56] numerical experimems indicate that
1169
the friction coefficient grows with the number of lalnellae in the chain (the number of moving domain walls) and scales similarly to equation (18). 5. WEAK FOAMS. FLOW OF 'SOLUTIONS' OF BUBBLE CHAINS As shown in the previous section, during its motion, a bubble train is subdivided into a system of domain walls. The domain walls separate the regions within which the lamellae are displaced only slightly so that the principal part of the dimensionless displacements, p,, remains at an almost the same constant level. The size of each wall is much greater than the pore size. Therefore, on a macroscale level of a sample, the train may again be imagined as a onedimensional coarse-grained foam in which the domain walls serve as macrolamellae. Such lamellae interact between themselves via an renormalized elastic potential. The potential incorporates all the microlevel elastic and pinning interactions and displays the collective properties of the native chain (Lonngren and Scott [52]; Dodd, et al [53]) . It is important that the distance between neighboring effective lamellae crucially depends upon the history of loading. For instance, if a piston is applied to the tail end of the chain, and if such a load is maintained during a long period of time which is sufficient to shift the head lamella of the chain, then the number of domain walls will be a well-def'med constant. This constant is a universal characteristic of the given bubble train, the pore channel and the applied pressure drop. The resulting structure of the chain on the macroscale level of the sample will be periodic, i.e., the domain walls will form a new perfect lattice. But if it is granted that the load is continuously redistributed over the train (the situation which is most likely to occur in a natural porous medium), then the chain acquires, generally speaking, a nonregular structure: in the coarse-grained foam, the cell size will alter from one cell to another. A viewpoint which designates the bubble chain as a macroobject enables one to construct a hydrodynamic theory for a so-called weak foam. For the latter, there is a very small fraction of trapped bubbles, so that the gas and a system of bubble trains flow together through a porous medium. The gas mobility is reduced due to an enhanced friction of the bubble trains. We encounter a similar situation in the hydrodynamics of polymer solutions (Doi and Edwards [57]). The action of bubble trains on a free gas, which is flowing through a porous medium, resembles the action of polymer molecules on a flowing solvent. Because the bubble train resistance is accumulated almost entirely within the domain walls, and, at the same time, each domain wall is much greater than a pore size, the hydrodynamics of a weak foam may be considered as the hydrodynamics of an ensemble of the
1170
bubble chains 'dissolved' into a Darcian fluid, provided that the free gas flow obeys the Darcy law ( i.e., the relation between the pressure gradient and the gas velocity is linear). In a non-uniform flow, where each bubble train undergoes the action of an extemal pressure gradient which is imposed by the carrying free gas, any train will be stretched or contracted (because each macrolamella in the chain acquires a velocity imposed by the kinematics conditions). But, due to an inherent elasticity, the chain will tend to restore its equilibrium length. By restoring its equilibrium state, the chain draws into motion the surrounding gas and, consequently, other chains. Therefore, the chains which are stretched only slightly will flow to the regions in which the chains are more extended. As a result, a net restoring force appears. A schematic picture of the phenomenon is presented in Figure 9, where a dumbbell plays a role of a foam filled macrobubble or a bubble train.
/crv-rrr
O
2-rcrZ-
"-
Figure 9. The sketch elucidating the mechanism of creation of a restoring force in a gradient flow. The above discussed scenario of foam flow can be theoretically described by making use of equation (18) as a basic formula of a microscopic theory. Then, phenomenologically expressing qJ in the formula (18) through the pressure gradient, the flux of a foam within the sample can be written as (Kornev and Kurdyumov [50]; Kornev [30])
t
J = -cfl jds < u(s,t)u(s,t) > - V O 0
,
(19)
1171
here c is the concentration of the bubble trains, 1 is the length of the train, and p is a phenomenological constant. In equation (19), the vector u(s) denotes a vector which joins two adjacent macrolamellae, and arclength s is measured along the train. The angle brackets denote an average over the orientations of the macrobubbles which link the macrolamellae. It is convenient to rewrite equation (19) in the following form
/,
(20)
J = J o - cp jd,sS. V P , 0
Jo = - c - ~ - V P , S=
<
1 u(s,t)u(s,t)-;l
>,
I ij =6ij , S ' u = S i j u j , S : u = S i j u i u j ,
where tensor S is the tensor of the order parameter. One can see that the first term on the right hand side of equation (20) may be included into the ordinary Darcy law associated with the flow of flee gas. Thus, equation (20) is written in a form in which it is convenient to treat the foam motion in a porous medium as a flow of a "solution' of bubble chains (Doi and Edwards [57]). For such a hypothetical system, the detailed derivation of the constitutive equations has been presented by Kornev and Kurdyumov [50]. The system of these equations contains a generalized Darcy law for gas flow in the presence of a foam
v = - k f VP
-L,
L =J-
Jo '
(21)
and a kinetic equation which describes an evolution of the "blocking force' L due to the action of the velocity gradients. In a one-dimensional case, e.g., for a foam motion in a porous tube, the model has the following form Ov
(22)
Ox DL
DL
~ - l - V
.
Dt
v =-k
.
Dx 3t:'
L .
~-L f 3x
(23)
.
r .
(24)
Here v is the gas velocity, v is a relaxation time, and k~ is an effective seepage coefficient. The system of equations (22)-(24) may also be easily elucidated in
1172
the terms of the so-called lamella "break-and-reform' mechanism of foam motion (Holm [58]). Namely, equation (24) expresses the balance of forces for the moving foam. The viscous force on the left hand side of the equation is balanced by the pressure gradient and a blocking force. The latter arises by vim~e of a blockage of the gas channels by 'valve' lamellae. In the first approximation, the blocking force is directly proportional to the number of 'valve' lamellae. We thus can express the lamella density through the function L and, ignoring the generation rate of lamellae, we may treat equation (23) as a kinetic equation for the lamella population balance in a moving foam. The parameter r should thus be treated as a lifetime of a 'valve' lamella, and it may be represented as 1
1
1
r
rt
rh
where r, is a 'thermodynamic' lifetime of a lamella, and r, is its hydrodynamic traveling time. So, each 'valve' lamella blocks a gas path until it rips due to inherent thermodynamic or hydrodynamic instability. Within the framework of the above formulated bubble-train-flow-mechanism, the relaxation time can be obtained by analyzing a random walk of a bubble train as a whole. To describe this random walk, the following h y p o t h e s i s has been used: once formed, the system of active channels cannot be destroyed. Then the picture of a random walk resembles a reptation of a polymer molecule through the obstacles (de Gennes [59]; Doi and Edwards [57]; Kornev and Kurdyumov [50]). A bubble train can change its active channel only by moving through a network of active channels in a worm-like manner. In fact, only its ends participate in movement, but the rest of the macrolamellae are effectively trapped within the existing (at the given moment) active channel. The model in equations (22)-(24) reproduces the main features of the flow of a foam through a porous medium (Figures 10-11). The one-dimensional flow is governed by the parameters r and b = Lok I r ! H , where Lo is a boundary value of the blocking force L, and H is the length of the sample. These parameters can be extracted from the experimental data by using a piecewise-linear approximation (Figure 10) as follows AP kf H = ( l + b ) v , v --->0, AP H kf H =v+ b, v ~o
(25) (26)
1173
where AP is the applied pressure drop. The point at which the straight lines intersect has the coordinates
vr/H
=1, APrkf I H 2 = l + b ( 1 - e - 1 ) = l + b
(27)
.
W 5~
11 2/ 31
4
2
0
2
4
6
8
10
12
14
16
AP Figure 10. The velocity as a function of the pressure drop. 1-b--1, 3-b=5,
2 - b = 3,
4-b=10.
The coordinates of the imersection poim can be used as the fitting parameters. An additional parameter needed to specify the model is the breakthrough time T* = t*r . At this instant, a from of the foam first reaches the exiting end of the sample. The dimensionless parameter t* can be found from the following transcendental equation
1174
1=
f
t
H 2
b
1 - exp(-(1
1 + b + 1 +-~"
+
1+ b
b)t*) "
(28)
The system of equations (25)-(28) is closed so that the parameters b , r and ke can be found in each individual experiment. Then, these parameters can be plotted as functions of the degree of water saturation, properties of the foaming agent, etc. Despite the fact that the model contains a small number of the physical constants, the theory cannot be directly spread onto the two- or three-dimensional flows (because the parameter b depends upon the boundary value of the order parameter S). The similar problem occurs in the theory of polymer viscoelasticity (Bird, et al [60]) and also remains unresolved.
10-
t* u
I
I
m
0
1
2
"
II
4
1
6
l
b
8
1
AP Figure 11. The breakthrough time as a function of the pressure drop. 1 - b = 1, 2-b=3,
3-b-5,
4-b=10.
1175
6. STRONG FOAMS As follows from the previous sections, the main feature of foam rheology concems its 'pseudoplasticity'. The foam pseudoplasticity is very sensitive to the foam texture inside a porous medium and to a fraction of the pores blocked by lamellae. The physical reasons of the blocking action of weak and strong foams are different. For a weak foam, even in the absence of a start-up pressure gradient, an enhanced friction of bubble chains will result in an apparent pseudoplasticity of the gas-foam system. But for a strong foam, when the bubble trains flow through a small fraction of the pore channels, the blocking effect is originated from at least two mechanisms. The first is caused by trapped lamellae which do not participate in the movement. Such a blockage leads to a permeability that is orders of magnitude lower than that for the single phase flow. The second contribution to a reduction of the gas flux in the presence of a strong foam concerns an enhanced friction of the bubble trains. Thus, the total gas mobility, i.e., the coefficient in the generalized Darcy law, is the product of permeability divided by apparent viscosity. (If the liquid also participates in the movement so that its saturation changes during the process and alters the fraction of the trapped pores, then, instead of the absolute permeability, the relative permeability to gas has to be considered). In addition to a nonlinearity of the gas mobility, a start-up pressure gradient plays the key role in the treatment of the hydrodynamic effects in strong foams. This is the pressure gradient which is required to depin some bubble trains and to create a network of active channels (Cottrell [61]; Read [62]; Hirth and Lothe [63]; Suzuki, et al [64], Komev [30], Dautov, et al [32]). Thus, the term plasticity, in its own physical meaning, must be attributed to strong foams. Both, the bubble trains and the corresponding network of active channels play the same role as the one prescribed to the dislocations and the network of dislocations in solid state physics. In fact, studies have shown that the trapped gas saturation of strong foams in porous media can be as high as 80% (Rossen [14]). Therefore, the trapped foam forms an elastic field around the network of active channels. In a short period of time, the trapped foam resides as effectively motionless lamellae. However, the diffusion processes need to be considered to characterize the long term behavior of a foam (Falls, et al [65]; Cohen, et al [66]). In particular, a mechanism like the Nabarro-Herring-Lifshitz mechanism [67-69] of a diffusion-induced plasticity might play an important role in the motion of trapped foam. Indeed, under a pressure gradient, the gas will diffuse through the lamellae so that the gas pressure within the bubbles will change with time. Because of a difference in the Laplacian and gas pressures, the
1176
trapped lamellae will creep as well. A sophisticated analysis of this effect is highly desirable. There is an important evidence of self-organized criticality in foam flows through porous media. Namely, the typical manifold reduction of the gas mobility speaks in favor of the fact that, during the flow, the network of the active channels remains nearby in the same state, as it would be at the percolation threshold (Rossen and Gauglitz [70]). Scaling estimates of the gas mobility reduction by foams have been obtained by operating the ordinary percolation theory (de Gennes [71]; Rossen and Gauglitz [70]; Entov and Musin [72]). However, the critical behavior of the network of active channels has been assumed ad-hoc. Though most researches believe that the transport phenomena in strong foams obey the percolation laws, the physics of the hydrodynamic processes remains unclear and puzzling. 7. FOAMS IN FIBER SYSTEMS
Foams have been found various applications in industrial technologies dealing with fiber structures and fibrous materials, such as dyeing, printing, mercerizing, and finishing of textile fabrics, paper coating, resin-impregnation of fibrous mats and fabrics [73-78]. In these processes, the usage of foams instead of bulk liquids, as a vehicle for delivering small amounts of liquid solutions to fiber surfaces, leads to substantial energy savings because of small amounts of residual solvent to be removed at the drying step. Another class of processes, involving bubble generation in fiber structures, is fabricating fiber reinforced composites. Herewith, bubble formation during the stage of liquid resin impregnation causes a negative effect of non-uniform polymer distribution in the products (Judd and Wright [79]), and the mechanisms of air entrapment and bubble interactions within a fiber network have been studied aiming to diminish this phenomenon at technological conditions (Mahale, et al [80, 81 ]). Interactions of bubbles with fibers and fiber networks have certain specifics compared with capillaries and pore networks in solid-wall materials. The major difference is that in contract with granular materials, where the sizes of pores and grains are commensurable, the typical diameter of fibers, which constitute a skeleton of a pore structure, is commonly smaller than the typical diameter of voids/pores between the fibers. In fibrous materials, it is not easy to identify single pores, their shapes and dimensions. The definition of pore sizes and their distribution in fibrous materials is a matter of convention. The most rational way to introduce the pore dimensions in a real fiber system is to consider a model system of "effective" pores, in which some characteristic processes would occur
1177
obeying the same peculiarities as in the fiber system under consideration. For example, the effective pore sizes are estimated from the experiments with capillary equilibrium of immissible fluids (commonly, wetting liquid and gas) and with steady or quasi-steady forced flow of a non-wetting fluid (commonly, gas) within and/or through a fibrous sample. An advanced technique of liquid porosimetry has been developed by Miller and Tyomkin [82] for determining pore volume distributions in fiber systems and other materials. The method is based on the consecutive, quasi-equilibrium wetting fluid - gas displacement under precisely controlled pressure at isothermal environment. Therewith, the effective radius of a pore, where the liquid and gas phases coexist at given pressure, is operationally related to the mean radius of curvature of the equilibrium meniscus between the phases through the Laplace equation. During the process of gas-liquid displacement in fibrous materials, bubble formation occurs mostly due to a hydrodynamic instability of the wetting film clinging to the fibers. This is the common mechanism of lamella formation in any porous media [27]. However, the fiber structure causes some peculiarities. It is likely that in case of most of fibrous materials, we deal with strong foams with the bubbles commensurate to the pores. This conclusion follows from the experiments of Gido, et al [83], who examined the flow of foams through fibrous mats by characterizing the bubble sizes before and after injection. The authors observed that the size of the output bubbles exiting the fiber system was independent of the bubble size of the input bulk foam. For fibrous mats with different pore structure, the output bubble sizes were found correlated with the pore size distributions measured by the liquid porosimetry [82]. This result can be interpreted assuming that the foam bubbles, residing within the fiber network during the foam flow through this network, are grouped basically into two configurations: a system of immobilized bubbles strongly pinned to fibers and system of unpinned bubbles, which are formed in the bubble trains sliding along the active channels confined by the immobilized bubbles. The bubbles are immobilized when they are transpired by several fibers which intersect or are not collinear. These bubbles are crucified at fiber crossing and/or stretched by differently oriented fibers. The mobile bubbles are commensurate with the pore constrictions in order to pass through them without essential deformation. The movement of lamellae in the active channels within a fibrous structure should be like the lamella stick-slip motion within porous solids described in detail above. However, foam flow through a fiber mat is more difficult to formalize than the flow in a solid-wall porous body. At present, no quantitative approach exists to describe foam flow in fibrous materials. This problem is still awaiting its solution.
1178
While considering equilibrium distribution of bubbles and foam flow in fiber networks, we have to account for a specific behavior of wetting films on fiber surfaces. In particular, the lamellae (bubble films) should coexist with the wetting films and the Plateau borders at the intersections of lamellae and/or lamellae and wetting films. The equilibrium configurations of such a complex system are determined by capillary and surface forces (the latter is expressed via an additional so-called Derjaguin's disjoining pressure [84], [15], [16]). The fiber surfaces are commonly convex. This means that the capillary pressure acting on the liquid-gas interface of the wetting film covering the fiber is positive and tends to squeeze liquid out of film into the regions of fiber crossings. This tendency leads to a reduced mobility of wetting films in fiber systems compared with ordinary capillary systems. These effects have been considered in literature as related to the liquid spreading and drop residence on fibers first by Carroll [85, 86], who accounted for the capillary forces only, and then in great detail by French researchers from the de Gennes group [87-91 ]. Brochard was the first to emphasize a central role of long-range intennolectdar forces in residence, stability, and spreading of films and drops on fiber surfaces [87, 88]. Di Meglio [89] experimentally observed wetting films stabilized by the Van der Wa,~s forces, and proved that mass transfer between the drops residing on a fiber occurs through these films. Similar effects should be important in phenomena involving the bubbles on fibers and in fiber networks, however their description is lacking in literature. A theory of foams in fiber systems cannot be advanced without a solution of a chain of particular problems: equilibrium shape of a bubble residing on a fiber, transition zone between the bubble lamella and wetting film coating the fiber, slippage of a bubble along a fiber, bubble crucifixion at a fiber crossing, motion of a system of contacting bubbles transpired by a fiber ("bubbles on a spit"), etc. 8. CONCLUSIONS The motion of foams through porous media is a challenging problem in physicochemical hydrodynamics. The basic mechanisms of foam transport reviewed in this article contain some, but certainly not all, of the relevant physics of foam flow in porous media. Foam flow in porous media is a multifaceted process in which, on one hand, foam texture strongly governs foam rheology, and on the other hand, foam texture is in turn regulated by the porous medium through the capillary pressure. We have analyzed the main features of this process on examples of foam motion in model pore channels. The modem theories of the foam lamella transport in pore channels of varying cross-section and the models
1179
of the "weak foam" flow are discussed in detail. Careful analyses of the flow on the scale of individual pores or channels are useful in exposing effects of various physical parameters on foam motion and in identifying flow-induced patterns. In addition, the basic physical mechanisms of foam microhydrodynamics tmderlie a variety of technological processes in oil recovery, groundwater/soil remediation, textile manufacturing, etc. REFERENCES
1. C.V. Boys, Soap Bubbles and the Forces Which Mould them. Soc.for Promoting Christian Knowledge, E. and J.B.Yotmg, London,1890.Reprinted in Doubleday Anchor Books, New York, 1959. 2. K.J. Mysels, K. Shinoda and S. Frankel, Soap Films, Studies of their Thinning and a Bibliography, Pergamon Press, New York, 1959. 3. J.J. Bikerman, Foams, Springer-Verlag, New York, 1973. 4. I.B. Ivanov (ed.), Thin Liquid Films: Fundamentals and Applications, Marcel Dekker, New York, 1988. 5. A.M. Kraynik, Ann.Rev.Fluid Mech., 20 (1988) 325. 6. A. Wilson (ed.), Foams: Physics, Chemistry and Structure, Springer-Vedag, New York, 1989. 7. P.M. Kruglyakov and D.R. Exerowa, Foam and Foam Films, Khimia, Moscow, 1990. 8. J. Stavans, Rep.Prog.Phys., 56 (1993) 733. 9. R.K. Prud'homme and S.A. Khan (eds.), Foams:Fundamentals and Applications, Marcel Dekker, New York, 1995. 10. S.H. Raza, Soc.Petr.Eng.J., 10 (1970) 328. 11. S.S. Marsden, Foams in Porous Media - SUPRI TR-49, US DOE, 1986. 12. J.P. Heller and M.S. K u n t a m ~ l a , Ind.Eng.Chem.Res., 26 (1987) 318. 13. L.L. Schramm (ed.), Foams: Fundamentals and Applications in the Petroleum Industry, Advances in Chemistry Series 242, 1994. 14. W.R. Rossen, in [9], p.413. 15. L.I. Kheifetz and A.V.Neimark, Multiphase processes in porous media. Khimia, Moscow, 1982. 16. B.V. Derjaguin and N.V. Churaev and V.M. Muller, Surface Forces, Nauka, Moscow, Nauka, 1985; Surface Forces, Consultants Bureau, New York, 1987. 17. A.V. Neimark and M. Vignes-Adler, Phys. Rev. E, 51 (1995) 788. 18. P.G. de Gennes, Rev.Mod.Phys., 57 (1985) 827.
1180
19. A.H. Falls, G.J. Hirasaki, T.W. Patzek, P.A. Gauglitz, D.D. Miller and Y.Ratulowski, SPE Res. Eng., 3 (1988) 884. 20. A.N. Fried, The Foam-Drive Process for Increasing the Recovery of Oil, Report US Bureau of Mines R.I.5866 (1961). 21. G.J. Hirasaki and J. Lawson, Soc.Petr.Eng.J., 25 (1985) 176. 22. C.W. Nutt and R.W. Burley, in [6], p. 105. 23. A.V. Bazilevsky, K.Komev and A. Rozl~ov, in Proceedings of the ASME Symposium on Rheology & Fluid Mechanics of Nonlinear Materials, Atlanta, November 17-22, 1996. 24. O.S. Owete and V.E. Brigham, SPE Res.Eng., 2 (1987) 315. 25. K.T.Chambers and C.J. Radke, in N.Morrow (ed.), Interfacial Phenomena in Petroleum Recovery, Marcel Dekker, New York, 1990, p. 191. 26. R.A. Ettinger and C.J. Radke, SPE Res.Eng., 7 (1992) 83. 27. A.R. Kovscek and C.J. Radke, in [ 13], p. 113. 28. E. Rabinowicz, Friction and Wear of Materials, Wiley, New York., 1965. 29. F.P. Bowden and D.T. Tabor, Friction and Lubrication of Solids, Claredon Press, Oxford, 1986. 30. K.G. Kornev, JETP, 80 (1995) 1049. 31. K.G. Kornev, V. Mourzenko, and R. Dautov, in E.P. Zhidkov (ed.), Proceedings of the Conference on Computational Modelling and Computing in Physics, JINR, Dubna, September 16-21, 1996. 32. R. Dautov, K. Kornev, and V. Mourzenko, To appear in Phys.Rev. E. 33. V.L. Pokrovsky and A.L. Talapov, Zh.Eksp.Teor.Fiz., 78 (1980) 269. 34. P. Bak, Rep.Prog.Phys., 45 (1982) 587. 35. I. Frenkel' and T.A.Kontorova, Zh.Eksp.Teor.Fiz., 8 (1938) 1340. 36. B.D. Josephson, Adv. in Phys., 14 (1965) 419. 37. I.O. Kulik, Zh.Eksp. Teor.Fiz., 51 (1966) 1952. 38. A. Seeger and P. Schiller, in W.P.Mason and R.N.Thurston (eds.), Physical Acoustics, Academic, New York, 1966, Vol. IliA, p.361. 39. A. Barone and G. Paterno, Physics and Applications of the Josephson Effect, Wiley, 1982. 40. A.J. Lichtenberg and M.A. Lieberman, Regular and Chaotic Dynamics, 2end ed., Springer-Verlag, New York, 1992. 41. L.D. Landau and E.M. Lifshitz, Mechanics, Nauka, Moscow, 1965. 42. G.M. Zaslavsky and R.Z. Sagdeev, An Introduction to Nonlinear Physics: From Pendulum to Turbulence and Chaos, Nauka, Moscow, 1988. 43. L.G. Aslamasov and A.I. Larkin, Zh.Eksp.Teor.Fiz.Pis'ma, 9 (1969) 150. 44. H.A. Barnes, J.F. HuRon, and K. Waiters, An Introduction to Rheology. Second ed., Elsevier, 1993.
1181
45. F.P. Bretherton, J.Fluid Mech., 10 (1961) 166. 46. A.H. Falls, J.J. Musters, and J. Ratulowski, SPE Res.Eng., 4 (1989) 55. 47. J.E. Hanssen and M. Dalland, in [ 13], 319. 48. L.W. Schwartz, H.M. Princen, and A.D. Kiss, J.Fluid Mech., 172 (1986) 259 49. W.L. Olbricht, Ann.Rev.Fluid Mech., 28 (1996) 187. 50. K.G. Komev and V.N. Kurdyumov, JETP, 79 (1994) 252. 51. C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1956. 52. K. Lonngren and A. Scott, Solitons in Action, Academic Press. N.Y., 1978. 53. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, 1984. 54. J.S. Langer, in G. Grinstein and G. Mazenko (eds.), Directions in Condensed Matter Physics, World Scientific, Singapore, 1986, p. 165. 55. Y. Braiman, F. Family, and H.G.E. Hentschel, Phys.Rev.E., 53 (1996) R3005. 56. R. Musin, Ph.D. Thesis, Moscow, 1997. 57. M. Doi and S.F. Edwards, The Theory of Polymer Dynamics. Claredon Press, Oxford, 1986. 58. L.W. Holm, Soc.Petr.Eng.J.,.8 (1968) 359. 59. P.G. de Gennes, J.Chem.Phys., 55 (1971) 572. 60. R.B. Bird, R.C. Armstrong, O. Hassager, and C.F. Curtis, Dynamics of Polymeric Liquids, Wiley, New York, 1977, Vols. 1, 2. 61. A.H. Cottrell, Dislocations and Plastic Flow in Crystals. Oxford University Press, London, 1953 62. W.T. Read, Dislocations in Crystals, McGraw-Hill, New York, 1953. 63. J.P. Hirth. and J. Lothe, Theory of Dislocations, Wiley, New York, 1968. 64. T. Suzuki, H. Yoshinaga. and S. Takeuchi, Dynamics of Dislocations and Plasticity. Mir, Moscow, 1989. 65. A.H. Falls, J.B. Lawson, and G.J. Hirasaki, JPT, Jan (1988) 95. 66. D. Cohen, T.W. Patzek, and C.J. Radke, J.Colloid Interface Sci., 179 (1996) 357. 67. R.N. Nabarro. Report of the Conference of the Strength of Solids, Phys.Soc.London, London, 1948, p. 75. 68. C.J. Heri~g, J.Appl.Phys., 21 (1950) 5. 69. I.M. Lifshitz, Zh.Eksp.Teor.Fiz., 44 (1963) 1349; The Collected Works of II'ya Lifshitz, Nauka, Moscow,1987. Vol.1. 70. W.R. Rossen and P.A. Gauglitz, AIChE J., 36 (1990) 1176. 71. P. G. de Gennes, Revue De L'lnstitut Francais Du Petrole, 47 (1992) 249. 72. V.M. Entov and R.M. Musin, Preprint IPM RAN, Moscow, No 560 (1996).
1182
73. T.F. Cooke and D.E. Hirt, in Foams, ed. R.K. Prud'homme and S.A. Khan, Marcel Dekker, NY, 1995, p.339. 74. C.C. Namboodfi and M.W. Duke, Textile Res. J., 49 (1979) 156. 75. Gregorian, R.C., Text. Chem. Color., 19 (1987) 13. 76. D.E. Hirt, R.K. Prud'homme, L. Rebenfeld, AIChE J., 34 (1988) 326. 77. D.E.Hirt, R.K. Prud'homme, L. Rebenfeld, Textile Research J., 61 (1991) 47. 78. E.L. Wright, Textile Research J., 51(1981) 251. 79. N.C. Judd and W.W. Wright, SAMPE, 14 (1978) 10. 80. A.D. Mahale, R.K. Prud'homme, and L. Rebenfeld, Polymer Eng. & Sci., 32 (1992)319. 81. A.D. Mahale, R.K. Prud'homme, and L. Rebenfeld, 4 (1993) 199. 82. B. Miller and I. Tyomkin, J. Colloid & Interface Sci., 162 (1994) 163. 83. S.P. Gido, D.E. Hirt, S.M. Montgomery, R.K. Prud'homme, and L. Rebenfeld, J. Dispersion Sci. & Tech., 10 (1989) 785. 84. B.V. Derjagum, Kolloid Zeits, 69 (1934) 155. 85. B.J. Carroll, J. Colloid & Interface Sci., 57 (1976) 488. 86. B.J. Carroll, Langmuir, 2 (1986) 248. 87. F. Brochard, J. Chem. Phys., 84 (1986)4664. 88. F. Brochard-Wyart, C.R. Acad. Sc. Paris, Serie II, 303 (1986) 1077. 89. J.-M. di Meglio, C.R. Acad. Sc. Paris, Serie II, 303 (1986)437. 90. F. Brochard-Wyart, J.-M. di Meglio, and D. Quere, C.R. Acad. Sc. Paris, Serie II, 304 (1987) 553. 91. D. Quere, J.-M. di Meglio, and F. Brochard-Wyart, Revue Phys. Appl., 23 (1988) 1023.
1183
FLOW OF NON-NEWTONIAN FLUIDS IN POROUS MEDIA Shapour Vossoughi University of Kansas Department of Chemical and Petroleum Engineering Lawrence, KS., USA 1. INTRODUCTION Non-Newtonian fluid flow through porous media has become increasingly important in a wide range of disciplines and industrial segments. This is the result of availability of a wide variety of polymers that have interesting fluid flow properties, and there is a growing demand for their industrial use. Catalytic polymerization process, the injection of polymer and surfactant solutions into petroleum reservoirs to enhance oil recovery, food processing, and fluid flow through riving tissues are examples of the vitality of understanding the nonNewtonian fluid flow through porous structures. This section deals with the different aspects of non-Newtonian fluid flow through porous media and will bring together the different treatments of the subject matter commonly practiced in different disciplines. It will cover the complexity of both non-Newtonian fluid flow behavior and flow through porous media. Anomalous behavior of non-Newtonian fluid flow through porous media could be due to the fluid, the nature of porous media, or the interaction of fluid and porous media. Therefore, flow study should be carefully designed to distinguish between these effects or at least to acquire knowledge of which aspect of the non-Newtonian fluid flow is being studied. In this chapter, the nature of non-Newtonian fluids that are commonly employed will be studied, followed by a look at the nature of the idealized porous media and the geometrical complexity of true porous media. Fluid and porous media interaction, such as adsorption, mechanical entrapment, and inaccessible pore volumes, will have direct effect on the flow and will be analyzed and quantified. Microscopic and macroscopic view of the flow will be studied next; and, finally, the predictive models presently available for the study of the non-Newtonian fluid flow through porous media will be investigated. This will cover models based on hydraulic radius concept, friction factor/Reynolds number relationship, and empirical methods.
1184
2. NATURE OF FLUIDS Non-Newtonian fluid flow through porous media is not limited to the polymer solutions. There are a wide variety of non-Newtonian fluids that could be of interest. Following are samples that have already established their association with porous media.
2.1 Polymer Melts and Polymer Solutions Polymers come in a variety of forms with respect to their average molecular size and the geometrical shape of their molecules. This leads to a wide variety of rheological behavior termed as "non-Newtonian". To illustrate the variety of viscoelastic responses, Ferry [1 ] sampled polymer from seven different groups; four of them were uncross-linked polymers and the other three were cross-linked. Among the uncross-linked ones, he picked up amorphous polymers of low and high molecular weight, amorphous polymers of high molecular weight with long side groups, and amorphous polymers of high molecular weight below its glass transition temperature. Among the cross-linked polymers, he selected a lightly cross-linked amorphous polymer, a dilute cross-linked gel, and a highly crystalline polymer. He showed that each of these seven structural types has a characteristic viscoelastic behavior. He then concluded that there is a strong correlation between the viscoelastic behavior and the molecular structure of the polymer. A macromolecule chain in a solution is capable of assuming a variety of configurations by rotating around its chemical bonds. This makes the polymer solutions behave differently from their parent polymer gels. The polymer solutions might behave significantly different depending on their level of polymer concentrations, and interaction between polymer molecules increases as polymer concentration increases. The effect of the polymer molecules entanglement on the solution viscosity may become highly significant in the case of a concentrated polymer solution. The polymer solution viscosity becomes an increasingly nonlinear function of the polymer concentration for the concentrated polymer solutions. Injection of polymer solution into oil reservoirs to enhance oil recovery has become a common practice. Oil recovery during the primary stage is due to the depletion of the initial energy stored in the reservoir. Over seventy percent of the oil is still trapped in the rock pores after the primary recovery is depleted. During the secondary recovery stage, a fluid, such as water or gas, is injected into the reservoir to sweep the oil out and push the oil bank toward the production well. Waterflooding is a common secondary oil recovery technique [2]. Water, because of its low viscosity, tends to finger through the oil zone and creates early water breakthrough, but when a small amount of polymer is added to the water to increase its viscosity it creates a more stable front. Synthetic polymer, such as partially hydrolyzed polyacryalmide, and biopolymer, such as Xanthan, are the two types of
1185
polymers most commonly used for this purpose. Understanding the flow of aqueous polymer solution through porous media is essential for producing a reliable model for reservoir simulation.
2.2 Boger Fluids Constant viscosity fluids that are highly viscous and highly elastic were first introduced by Boger [3], hence, they are called Boger fluids. The first report of the fluids which exhibited elastic properties but remained Newtonian in viscous behavior was made by Giesekus [4]. The Boger fluid is prepared by dissolving a small amount of polymer in a highly viscous solvent. These fluids can be divided into two general groups; aqueous solutions consisting of a small amount of polyacrylamide dissolved in corn syrup [5,6]; and an organic-based Boger fluid consisting of a small amount of polyisobutylene dissolved in a mixture of kerosene oil and polybutene [7-9], or high molecular weight polystyrene dissolved in a solvent composed of low molecular weight polystyrene in dioctylphthalate [9]. The two classes of Boger fluids seem to have some differences in their rheological behavior. For example, Kemielewski, et al. [10], observed significant differences in the drag ratio measured for the two classes of Boger fluids over the same Weissenberg number range. The general reported characteristics of these fluids are; high viscosity, approximately non-shear thinning; high relaxation time; and optically clear. The Boger fluids are ideal model fluids for studying viscoelastic behavior in the absence of shear thinning. 2.3 Micro and Macro Emulsions Micro and macro emulsions are mixtures of two immiscible liquids in the form of small droplets of one phase into the other. The size of the droplets in the microemulsion solution is much smaller than those in the macroemulsion. Macroemulsions are turbid and thermodynamically unstable. The two phases will eventually separate into the original two immiscible liquids.On the other hand, microemulsions are translucent and thermodynamically stable. These fluids have been known for many years, and a wealth of literature is available on their properties and on their production techniques [ 11-13]. Micro and macro emulsions and their flow through porous media are frequently encountered in the oil industry. The produced crude oil is often in the form of emulsion with water, and the emulsion is broken to separate the oil from the water before it is shipped. Therefore, the flow of macroemulsion through porous media is an important aspect of the fluid flow near the production well. On the other hand, injection of microemulsion into the oil reservoir has been practiced to enhance oil recovery [ 14]. Microemulsions, sometimes called micellar solutions, are formulated for a specific crude-oil/reservoir-brine system to achieve ultra low interfacial tension of less than 10.3 dynes/cm. These microemulsions consist of hydrocarbons,
1186
water, surfactant, cosurfactant (such as alcohol or another surfactant), and electrolytes [ 15]. In many cases, polymer is also added to increase the viscosity of the micellar solution, since higher viscosity is desired to achieve a stable front and minimize viscous fingering effect. Micellar solutions are generally injected in various slug sizes to economize the chemical flood process. The micellar slug is then chased by a polymer solution to maintain its mobility control. Loss of surfactant due to the adsorption to the rock surface and dispersion at the front and at the back of the slug are the main reasons for the eventual breakdown of the slug effectiveness to mobilize the residual oil. Microemulsion can also be generated in situ as observed in alkaline flooding of oil reservoirs. It has been established [16-18] that some components of crude oil, such as organic acids, asphaltenes, and resins, react with alkaline solutions and form micellar-type structures. Rheological properties of an emulsion depend strongly on its composition. Its viscosity could change an order of magnitude in a narrow range of its concentration change. This is believed to be caused by the structural change of the microemulsion. The presence of electrolytes, such as salt, enhances the non-Newtonian behavior of the emulsions, and the effect of polymer addition on the viscosity of the microemulsions can be quite significant [19]. This enhanced viscosity is shear sensitive, and the viscosity recovery after the removal of the high shear is extremely slow. In general, suspensions and emulsions do not exhibit the same level of viscoelastic behavior as polymer solutions and polymer melts. However, emulsions of gel-like structure may exhibit marked viscoelastic behavior [13].
2.4 Suspensions Suspensions, sometimes called slurries, differ from emulsions in that one of the two phases is solid. The dispersed phase, which is a solid, is finely ground and mixed into a liquid. If the dispersion lasts, the mixture is called a suspension. Surfactants are usually added to make the suspensions more stable by preventing agglomeration of the solid particles. Suspensions at high volume concentrations are affected by many factors, such as hydrodynamic interaction between particles, doublet and high-order agglomerate formations, ultimately mechanical interference, and surface chemical effects between the particles as packed bed concentrations are approached. There is a critical concentration, after which, the viscosity increases sharply [20,21]. The abrupt increase in the viscosity levels may be caused by the strong inter-particle forces between the solid particles. Vossoughi and AI-Husaini [22] studied rheological behavior of the coal, oil, water slurries, and the effect of polymer as an additive. All the systems they studied were pseudoplastic and showed shear thinning behavior. Slurries made of finer coal particles were more viscous than those made of coarser grains. They studied the dynamic properties of coal slurries with and without polymer, using a small
1187
amplitude oscillatory test. In general, for all slurries tested, the dynamic viscosity was observed to be a decreasing function with frequency, and the modulus of rigidity was found to be an increasing function with frequency. Their study was limited to the frequency range of 0.24 to 6.0 cps. Flow of suspensions and slurries through porous media involves many challenges unique to this class of non-Newtonian fluids, and they need to be met. For example, in the filtration of suspensions or drilling mud infiltration near the wellbore in the petroleum drilling industry. 2.5 Gels Placement of gel in petroleum reservoirs to improve oil recovery has become an accepted practice. The technology is known as permeability modification or profile modification. Water, or any other fluids that are injected into the reservoir to displace oil, tends to pass through the more permeable zones leaving behind a significant amount of oil in the reservoir. Gelling solutions are injected following the waterflood to plug the already swept zones. Resumption of waterflood, forces the water to find a new path which leads to additional oil recovery. Gelling solutions are typically an aqueous polymer solution and some kind of heavy metal ions, such as chromium or aluminum, as cross linkers. Gels are highly nonNewtonian and their rheological properties are unique for each gel system [23]. A somewhat different process for profile modification, known as combination process, consists of the sequential injection of polymer solution and aluminum citrate solution [24]. The permeability reduction is believed to be due to the formation of layers of polymer/aluminum ion structure onto the wall surface of the pores. The adsorbed polymer molecules from the first polymer treatment acts as a base for the buildup of the structure. To enhance creation of the base and to increase adsorption of the polymer molecules to the rock, the rock was first treated by cationic polymer before the injection of the first anionic polymer cycle [25]. In-depth Permeability modification for the oil reservoirs with high permeability variation has also been achieved by injecting colloidal dispersion gels into the reservoir [26-28]. This consists of polymer with aluminum citrate crosslinker injected as a homogeneous solution. Above examples clearly reveal the complexity of the non-Newtonian fluid flow through porous media associated with the gel treatment. The fluid flow behavior is not simply a function of the rheological properties of the fluids involved, but the interaction between the porous media and the fluids plays an important role. 2.6 Foam Foams are a dispersion of a gas phase into a liquid phase, are unstable, and break easily. However, addition of surfactants can increase their stability and prolong the life of the foams almost indefinitely. Foams are injected into the petroleum
1188
reservoirs for the same reasons as gels. The profile modification for the case of foam injection is aimed toward the displacement processes where gas is being used as the displacing phase such as carbon dioxide flood, steam flood, or any other gas injection schemes [29,30]. Flow of foams through porous media has been studied in the literature [31,32]. The apparent viscosity of the foams flowing through sandpacks was measured [33] and was found to be significantly higher than the viscosities of the constituent fluids. This is one major factor for its effectiveness in profile modification. In some studies the permeability reduction for foam flow was found to be an order of magnitude larger than the permeability reduction predicted just based on the gas/brine mixture viscosity [34]. Foam rheology is crucial in designing an effective permeability modification for a given reservoir. Foams are non-Newtonian pseudoplastic fluid. Viscosity measurement in a capillary viscometer revealed that the data fit the Ostwald and de Waele (power law) relationship [35]. It was noticed that the bubbles do not behave as rigid particles but flow and slip at the same time. Therefore, a fixed-slip velocity could not be determined from the data. 3. N A T U R E OF POROUS MEDIA
Any solid body containing space to hold a fluid can be considered a porous medium. This can be as simple as a pipe or as complex as riving tissue. Only the interconnected pores, which could contribute to the flow, is of interest to the study of the fluid flow through porous media, but not all the dead pores should be ignored. Those dead pores, which have connection to the main flow path, could act as sink or source for the flow. Experimental investigation of the fluid flow through porous media is frequently carried out on idealized porous medium to avoid the complexity of the true nature of porous medium. 3.1 Idealized Porous Media Idealization, or simplification, of porous media is aimed toward studying a particular aspect of fluid flow through porous media. For example, capillary tube models are ideal to study viscous behavior of the fluid flow through porous media, and the Hele-Shaw model is ideal for studying interface instability based on perturbation theory. One should be careful in selection of idealized physical model and its capability to reflect the physical phenomenon of interest. A bundle of capillary tubes will not be able to reflect the viscoelastic nature of a fluid and, similarly, in studying two-phase flow through porous media in the absence of capillary effect, one should pick up a physical model with large pores to allow neglecting capillary forces. Following are a few examples of idealized porous media studied in the literature.
1189
3.1.1 Hele-Shaw Model One hundred years ago, Hele-Shaw [36] observed that streamlines in an inviscid flow can be visualized by making the gap of the two parallel plates small enough so that a sheet of water as thin as the boundary layer could only flow through. He added localized color to visualize streamlines. His remarkable observation put him, as an experimentalist, ahead of the mathematicians to experimentally generate streamlines for the geometries that fluid flow equations had not been solved yet. To show his remarkable observation, two of his photographs generated for the flow of water through a sudden enlargement are scanned and presented here. Figure 1 is when the gap between the two plates is large, i.e. a thick sheet of water is flowing
Figure 2. Sudden enlargement (thin sheet) [36] through the gap, and Figure 2 is when the gap is very small. This clearly shows that by reducing the gap, a truly two-dimensional inviscid flow can be produced. A Hele-Shaw model can be simply constructed by placing two parallel plates very close together. Incompressible fluid flow through porous media and Hele-Shaw model becomes analogous [37] and the average velocity components will be identical if permeability in the Hele-Shaw model is defined as,
1190
h2
(i)
12
where, k is permeability and h is the spacing between the two plates of the HeleShaw model. Hele-Shaw cell has been used to study interface instability related to the viscous fingering in porous media [38-45]. For example, Wiggert and Maxworthy [45] injected air into a Hele-Shaw cell saturated with silicon oil. The unstable immiscible interface resulted in a series of viscous fingering patterns that were photographed and digitized. Their Hele-Shaw cell consisted of two glass plates separated by a gap of 0.21 cm. Viscous fingering is an interface instability which occurs when the displacing phase is less viscous than the displaced phase. The creation of viscous fingers and their growth have been visualized in Hele-Shaw models as well as sand packs. Figure 3 is an example of the existence of the fingers even in laboratory systems where porous media are much more uniform and much more homogeneous than the actual reservoirs. Figure 3 is reproduction of one of the photographs taken by van Meurs [46] in his transparent three-dimensional glass beads model where oil was displaced by water.
Figure 3. Viscous fingering [46] Addition of a small amount of polymer to the displacing water has been shown to have significant effect on the stability of the front [47] and, consequently, produces much higher recovery efficiency. 3.1.2 Porous Media Made out of Glass Rods A two-dimensional flow that more closely mimics the actual reservoirs can be physically modeled by assembling glass rods parallel to each other and having a flow perpendicular to the axis of the rods. Vossoughi [48] used such a model to
1191
visualize the flow path in porous media and to study the non-Newtonian flow distribution in a porous bed. He used a regularly spaced matrix of 6 mm diameter glass rods in a triangular arrangement. Figure 4 is a schematic top view of the bed geometry used in this study.
@
d
~
X
0 T0 C) Flow
Figure 4. Schematic of glass-rods bed geometry [49] The cylindrical rods were positioned in a rectangular box of 5.2 by 6.5 cm inside cross section. The box contained 40 rows of cylinders, each row containing 9 or 10 cylinders in width. The dimension of the smallest opening, d, varied randomly between about 0.045 and 0.079 cm. Therefore, the bed is considered homogeneous on a macroscopic scale, but on small or microscopic scale there are considerable variations in pore diameter. A similar model with cylinders of shorter length and larger diameter was used by Kyle and Perrine [50]. 3.1.3 Porous Media Made out of Glass Beads Visualization of three-dimensional flow in porous media can be achieved by physical models made out of glass beads. Vossoughi [47] used a packed glass beads model in his study on viscous fingering in immiscible displacement in porous media. The model was made up of two transparent plates of 18 by 24 inches. The plates were spaced V2 inch apart, and two sizes of beads were studied. The larger bead size was 0.47 cm and the smaller was 0.15 cm in diameter. Use of glass-bead packs to study fluid flow through porous media is a common practice among investigators. Arman [51] used glass-bead pack in his study of relative permeability curves. Naar, et al. [52] used a laboratory five-spot model, made up of glass beads, to study the areal sweep efficiency of waterflooding. Rapoport, et al. [53] performed waterflooding in oil-wet glass-bead packs to
1192
establish that a laboratory-developed relationship between linear and five-spot flooding behavior holds regardless of the wettability or of the porous medium. In general, flow of Newtonian and non-Newtonian fluids through granular and non-granular packed beds is of considerable interest in many disciplines such as filtration, chemical processes involving flow through catalyst bed, and oil recovery operations. Packed beds are mainly used to study flow parameters in terms of pressure drop and flow rate. Visualization of flow distribution in displacement processes using non-glass beads pack can be achieved by X-ray shadowgraph technique developed in early 1950's [54]. In this technique, an X-ray absorber is added to either displaced or displacing phase. The phase containing the X-ray absorbent casts a shadow on a piece of film upon exposure to an X-ray beam. The density of the shadow is proportional to the saturation of that phase. 3.2 Complexity of True Porous Media The porous media presented above are over-simplified compared to the pore geometry of the actual rocks. Even the non-granular packed beds do not reflect the many complex features associated with naturally occurring rocks. It is realized that the naturally occurring rocks are composed of a variety of particle sizes with various angularity, particle size distribution, and particle arrangement. The pore space configuration is obviously different from that obtained by packing uniform spheres in a bed. Furthermore, part of the pore space is naturally filled by clay and cementing material. Cementing materials may block part of the pores and reduces, or, in some cases, eliminates interconnectivity. Figure 5 is composed of photographs of impregnated rocks revealing the complex pore geometry for the two cases of fine and coarse intergranular sandstone [55]. The complex pore geometry is arising from many factors in the geological environment of the deposit. The complexity in pore configuration becomes even more pronounced for the case of carbonate rocks. Fractures and vugs are frequently encountered in carbonate rocks as shown in Figure 6. In this figure, four typical carbonate reservoir rocks are depicted with a) vugular porosity, b) vugular with pinpoint porosity, c) fractures, and d) with conglomerate [56]. 4. FLUID/POROUS MEDIA INTERACTION
In addition to the complexity of the pore geometry, rock/fluid interaction becomes more pronounced in the case of naturally occurring rocks. Surface wettability could make a significant difference in flow behavior of wetting phase versus non-wetting phase. Rock/fluid interaction plays a more important role in injecting of polymer solution into a naturally occumng rock. In the case of naturally occurring rocks, the role of adsorption and entanglement of polymer molecules, pore blockage by polymer molecules, and inaccessible pore volumes to the polymer molecules
1193
become significantly more important and will contribute much larger share to the non-Newtonian flow behavior through porous media.
4.1 Adsorption and Mechanical Entrapment Adsorption of polymer molecules onto the rock surface plays a significant role in the fluid flow behavior of polymer solution through porous media. Adsorption is not always desirable and the loss of polymer molecules causes dilution and the apparent viscosity of the polymer solution decreases with time. This could eventually lead to the breakdown of its effectiveness as a mobility control additive. Adsorption of the macromolecules will also reduce the pore size available for the fluid to flow. The level of adsorption differs from one type of polymer to the other. Polyacrylamide-type polymers, commonly used in oil displacement processes, adsorbs strongly on mineral surfaces. The level of adsorption can be reduced by partially hydrolyzing the polyacrylamide with a base such as sodium hydroxide, potassium hydroxide, or sodium carbonate. The amount of polymer adsorbed per unit surface area increases with polymer concentration and with its molecular weight [57]. Rowland and Eirich [58] found that the thickness of the adsorbed polymer (for uncharged polymer) is in the order of the average diameter of the free polymer molecule coils in solution, i. e. approximately proportional to M ~/2, where M is the average polymer molecular weight. Michael and Morelos [59] studied adsorption of partially hydrolyzed polyacrylamide and sodium polymethyl methacrylate on kaolinite and observed higher adsorption with lower pH of the polymer solution. They concluded that the mechanism of adsorption was hydrogen bonding. Similar observation is also reported by Schmidt and Eirich [60]. There is a great deal of evidence in the literature that shows adsorption of the polymer molecules is affected by the presence of electrolytes. Polymer adsorption increases with increasing concentration of electrolytes [61-64]. Smith [62] reported an adsorption of 200 gg of a partially hydrolyzed polyacrylamide per square meter of surface area of silica powder when the polymer solvent was a 2% NaC1 solution. Mungan [63] measured an adsorption of 400 gg/rn ~of silica powder for Pusher 700 in 2% NaC1 solution. Mechanical entrapment of the polymer molecules is equally important in fluid flow behavior of macromolecule solutions through porous media. Entrapment mainly occurs at the pore throat, where the diameter becomes same or smaller than the diameter of the approaching molecule. Adsorption of the polymer molecules will definitely enhance entrapment by further reducing the pore diameter, especially near the pore throat. Dominguez [65] performed an experimental investigation of polymer retention in the absence of polymer adsorption by performing his experiments in porous media made of TEFLON | powder. Cores were prepared by compressing the powder. The
1194
Figure 5. Cast of pore space of typical reservoir rock. (a) Fine intergranular sandstone; (b) coarse intergranular sandstone. [55] Teflon powder cores have also been used by others. For example, Mungan [66] and Lefebre du Prey [67] used Teflon cores for their study of wettability effects and Sarem [68] used them to study polymer retention. Domingues [65] measured polymer retention for his Teflon core within the range of 10 to 21 gg/g of Teflon powder. Other reported values in the literature [64, 69-72] for polymer retention in
1195
Figure 6. Typical carbonate reservoir rocks (From Core Laboratories, Inc.) [56] porous media are from 6 gg/g to as high as 160 gg/g of sand grain. Pye [73] and Sandiford [74], in their experimental investigation of the flow of polyacrylamide solution through porous media, observed a reduction in water mobility of 5 to 20 times more than would be expected from the solution viscosity alone. The additional resistance to fluid flow is mainly attributed to the adsorption
1196
and mechanical entrapment of the macromolecules. This effect is characterized by introducing a resistance factor, F, defined as,
Fr=z
<2) p
where, ~, is the mobility and the indices w and p stand for water and polymer, respectively. Mobility is defined as, k (3)
where, k is permeability and kt is viscosity. The permeability reduction due to the adsorption and/or mechanical entrapment is partially irreversible. This is clearly demonstrated by Burcik [75] in his flow experiment of partially hydrolyzed polyacrylamide through slices of Berea sandstone. He observed a reduction in brine mobility after the injection of the polymer and the reduction persisted even after injecting 100 pore volumes of brine. This effect, subsequently called residual resistance factor, F is defined as the ratio of the initial brine mobility to the mobility of brine after the porous medium has been exposed to the polymer solution. ~w
(4)
, wp where, )~wis the mobility of brine before injection of polymer and Lw~is the mobility of brine after the bed was exposed to polymer injection and all the mobile polymer molecules have been displaced by subsequent brine injection. Residual resistance factor in the range of 1.08 to 15 has been reported in literature [76-79]. A value of residual resistance factor of 15 means that the porous medium permeability has been permanently reduced by a factor of 15. 4.2 Inaccessible Pore Volume
Inaccessible pore volume is a phenomenon associated with the flow of polymeric fluids through porous media. This occurs when the size of the polymer molecules in the solution becomes the same order of magnitude as the pore sizes. This causes some of the pores which are smaller than the polymer molecules to be excluded from the polymer flow. This phenomenon is well established and well recognized
1197
in the oil displacement processes where polymer solution is used as the displacing phase. Unfortunately, it is not well documented in other disciplines when polymeric fluid flow through porous media is discussed. Dawson and Lantz [80] studied inaccessible pore volume in a consolidated sandstone core by injecting polymer solution into the core. Two types of polymer were studied. One was polyacrylamide-type polymer (Pusher 700, The Dow Chemical Co.) and the other was a polysaccharide (XC biopolymer, Xanco, Div. Of Kelco Co.). The cores were initially saturated with brine, then polymer solution was injected continuously to satisfy the polymer retention. At complete equilibrium, i.e. injected polymer concentration being the same as the effluent polymer concentration, a slug of different polymer solution was injected. Injection of the initial solution was resumed and continued till the end of the experiment. They monitored the effluent polymer and the effluent salt concentrations and produced concentration profiles as given in Figure 7. It is evident from Figure 7 that the salt peak is arriving at 1 PV fluid injected as expected. However, the polymer peak is arriving at 0.76 PV of fluid injected. The early arrival of polymer clearly indicates that not all the pores are available for polymer to flow. This corresponds to an inaccessible pore volume of 24%. Inaccessible pore volumes have been reported in literature to range from 1 to 30% depending on the nature of the polymer and the porous media involved [81,82]. Polymer Retention on Rock Satisfied
---~I
I
0.5 _ Before Injection of Pulse
Pulse Size
o
~
t~0.4 ii
,o-o'o,~,
-
d #
dO. 3 . ,o ~,
.6" Polyacrylamide \ ,
t- 0.2 -
#
:
j
Jo p'o #
~0.1
~ L 0.0 0.6
$
~.g,.., o, 0.7
0.8
0.9
1.0
1.1
1.2
Fluid Injection, P V
Figure 7. Early arrival of polymer peak due to inaccessible pore volume [80] Adsorption and entrapment of polymer molecules in porous media, on the other hand, cause the effluent polymer concentration to lag the solvent flow. The combined effect, therefore, could totally or partially mask one effect by the other.
1198
That is, even if the polymer concentration profile appears at the same time as salt concentration, this does not necessarily indicate that there is no inaccessible pore volume and/or no adsorption and entrapment. It is possible that the dilution due to adsorption and entrapment has been compensated and, therefore, concealed by the inaccessible pore volume effect. Inaccessible pore volume is an important, unavoidable phenomenon associated with polymeric fluid flow through porous media and should not be ignored. Any realistic fluid flow model for such a system should take into account all the important fluid/rock interaction phenomena such as adsorption, entrapment, and inaccessible pore volume.
5. MICROSCOPIC VIEW Study of fluid flow through porous media at the microscopic level provides information at the pore level. This information can be used to generate a predictive model for the macroscopic behavior of the flow system. This obviously requires significant simplification of the flow geometry. The idealized porous media presented earlier can be taken advantage of for this purpose.
5.1 Photomicrography Technique Photomicrography is a powerful technique to visualize flow path at the pore level. Chatenever and Calhoun [83] are the pioneer in microscopic studies of the dynamic fluid behavior in porous media. They used an idealized porous media consisted of a layer of spheres sandwiched between two transparent parallel plates. Simultaneous flow of oil and water was observed and photographed to reveal the flow channels created by each phase. Later, Amoco Production Company [84] (formerly Stanolind Oil and Gas Co.) studied fluid distribution in cylindrical sand packs. The wetting phase was simulated by Wood's metal and the non-wetting phase by colored plastic. At various saturations the Wood's metal and the colored plastic were solidified in place in the core. The face of the core was magnified and photographed at each time that it was shaved off. This produced a graphic threedimensional view of the wetting and non-wetting phase distribution when the photographs were projected at motion picture speed. Vossoughi [48,49] applied streak photography technique to visualize the flow path and to measure velocity profile at the pore throat. In this technique, the velocity profile can be established by measuring the length of the streaks of known duration. He used an idealized porous media consisted of glass rods as described in Section 3.1.2. Flow visualization was achieved by adding approximately 2 cm 3 of aluminum particles to 106 crn~ of solution. The aluminum particles had an average diameter of 20 microns. Light was allowed to enter from a narrow slit at the side while pictures were taken from the top of the bed through a 12X microscope focused in the plane
1199 of fight. The fight was interrupted at known time intervals with a high-speed spoked wheel. Therefore, velocity of fluid particles can be calculated by simply measuring the length of an individual streak and dividing this measurement by the known time interval. The schematic of the photomicrography assembly and the plan view of the optical system used in this study are presented in Figures 8 and 9, respectively. The photographs were projected onto a screen to achieve a magnification of approximately 100X. The actual magnification was determined by projecting a photograph of a precision steel rule.
@m a
era
I~ Camera Stand
-
9j
Phototubre ------.-....~~}
9
U J
Microscope Stand
LI
Microscope
"
! '"
Light Slit
S!
-]
,d
Flow Direction
r Glass Rods Box
\ Figure 8. Schematic of photomicrography assembly [48] Fluid flow distribution with pore opening was studied for Newtonian as well as Non-Newtonian fluid. Pure glycerol of 13.6 poise viscosity was used as a Newtonian fluid. The non-Newtonian fluid was a polymer solution of 2% Separan AP 273 (a partially hydrolyzed polyacrylamide supplied by Dow Chemical Company). Relaxation time of the polymer solution, as estimated from normal stress measurements, was approximately 0.1 sec. Viscosities of the two fluids, measured with a Weissenberg Rheogoniometer, are presented in Figure 10. The polymer solution used in this study was highly pseudoplastic with power-law indices as given on the figure. Figures 11 and 12 are typical photographs taken at three different flow rates for the Newtonian and the non-Newtonian fluid, respectively. 5.2 Flow Distribution with Pore Size
As mentioned earlier, Vossoughi [48] generated velocity profiles at the pore openings from the photographs produced by streak photography technique. He
1200
Fluid Outle d
q
-
0 0 0 0
0 0 0
0 0 0 0
0 0 0
0 0 0 0
0 0 0
L
0 0 0 0
Fluid Inlet
,..__. ~.~
Glass Rods Box Chopper Disk Variable Speed Transmission and Motor
Light Beam
Adjustable Slit
Converging Lens Source Figure 9. Schematic plan view of optical system [48]
100
IIIIII
I
I
I III
r
Newtonian
.,..B
O 13_
:i. ~O
Z:l
~
0 o
Polymer
> (-.
]
Power Law {
<
0.1
1
m=7.5 n = 0.43
10 100 Shear Rate, 7, secl
1000
Figure 10. Viscometric behavior of the experimental fluids [49]
1201
Figure 12. Typical photographs for non-Newtonian fluid flow [48]
1202
plotted the projected streak length, as measured at local magnification of approximately 100, versus the position between the two cylinders. However, the scale was chosen so that the profiles present the dimensionless interstitial velocity versus position. He studied local velocity profiles for four different flow rates of 0.685, 1.717, 3.42, and 6.84 cm/s, 3 and he observed the relative distribution of the flow among the pores being independent of the flow rates. Figures 13 and 14 present velocity profiles at one pore opening generated for Newtonian and polymeric fluid flow, respectively.
,o I
~
.
Ai=0.360cm
i
:o
~
~
d = 0.0526 cm
.~~E to..- o~176176176 iS0
i =0.354cm o
NCL = 0.0889 cm
Ai = 0.382 cm
,w.......~...~....t
NCL = 0.17 cm
zx Q = 0.685 cm3/s n Q=1.717 cm3/s
o Q = 3.42 cm3/s 9 Q = 6.82 cm3/s
Figure 13. Experimental velocity profiles for Newtonian fluid [48] In these plots, data points of all the four flow rates (whenever streaks are detected at that particular pore opening) are included. The three profiles in each figure are for three locations of the same pore. The profile designated by "d" is for the pore throat (i.e. minimum opening between the two cylinders), and the one designated by "NCL" (stands for Not Center Line) is for locations away from the pore throat. The small variation in "d" values in the two figures is probably caused from the small movement of the cylinders from one run to another. The fines in Figures 13 and 14 are eye-fitted curves through the data points. The graphical integration of the
1203
o
Q = 3.42 cm3/s
"O5
9
Q = 6.82 cm3/s
r-
zx Q = 0.685 cm3/s
E
i5
[]
~'Ai = 0.236 cm 1
Q=1.717cm3/s
0 d = 0.0496cm
f ~_ ~,1: " n
0
A~ = 0.257 cm NCL = 0.089 cm
t f
%1,,
~
n
~,,~
0
1
A~ = 0.226 c NCL = 0.1016 cm
Figure 14. Experimental velocity profiles for non-Newtonian fluid [48] areas under the curves are shown by A~. It should be noted that the area under the curve, which is proportional to the flow going through that pore, should be the same for all the profiles generated for a given pore regardless of the position of the profile with respect to the pore throat. The sum of the individual flow rates in every pores of a single row should equal the total flow rate. This provides an independent check on the consistency of the data. In general, this was within +5 % of the total flow rate determined by the pump calibration. The total number of velocity profiles at various positions and flow rates studied in this work [48] was in excess of 400 and included approximately 5000 individual data points. The range of variables covered in this study is given in Table 1. The range of shear rate, given in Table 1, is within the power-law region of the viscometric behavior of the polymer solution presented in Figure 10. Therefore, the power-law parameters estimated from Figure 10 provide a reasonable approximation to the viscous properties of the polymer solution during the photographic experiments. Table 1. Range of variables for velocity measurements Flow Rate
(c~/s) .,,
Minimum Maximum
0.68 6.8
Reynold' s Number Polymer Newtonian 3.2 x 1 0 -4 7.5 x 105 7.5 x 104 1.4 x 10.2
Wall Shear Rate, (s -~) 3.3 33.4
To predict flow behavior based on the viscometric properties of the fluids
1204
involved, Vossoughi [48] made further assumption that the flow geometry at the pore level could be adequately simulated by flow through parallel plates. This assumption was justified based on two observations: 1) Most of the pressure loss will occur at the narrow openings between cylinders in a given row. 2) The narrow openings between cylinders are small compared to the diameter of the cylinders (0.05 cm compared to 0.6 cm). Thus, as shown by Figure 15, fluid flow at pore level was approximated by flow between parallel flat plates with spacing d (the narrow openings between the cylinders), but undetermined length in the flow direction.
Figure 15. Schematic for parallel-plate analysis [49] The equation of motion is then easily solved for creeping flow in the absence of any elastic effects. Flow of power-law fluid in the z-direction between the parallel plates produces [48,49]" 1
u-(
l+n
n ;~L d n + l )(m )n(2 ) n
n+l 2x [1-(--~-)
n
]
(5)
or,
u -
[1 _(9_) 2x n+l o]
(6)
Umax where, C is inserted as a tortuousity factor. Equation (6) reduces to Newtonian fluid for n = 1. Equation (5) can be integrated to produce the volumetric flow rate per unit depth of the pore.
1205
d12
q-2
Iudx
(7)
o
For power-law model, u is substituted from Equation 5 and then integrated, =
q P
2n 1 + 2n
(
kp m CAL
d l+2n ) ,1- (_~_.) ,
(8)
Equation (8) reduces to Newtonian fluid by setting n = 1 and m - It, 2 Ap d)3 qN -- ~ (#CA L)(-2
(9 )
Equations (8) and (9) predict the volumetric flow rate being a function of pore opening to the power (l+2n)/n and 3 for power-law and Newtonian fluid respectively. For the power-law exponent of n = 0.43, as given in Figure 10, Equation (8) predicts a dependency of the volumetric flow rate on pore opening to the power greater than 4. Figures 16 and 17 compare the predicted and the measured flow distribution with pore opening for the Newtonian and polymeric fluid, respectively. Experimental data in Figure 17 indicate that the flow is distributed among the pores more uniformly than what is predicted from power-law model. It is even less than the dependence on pore size for the Newtonian fluid. 6. MACROSCOPIC VIEW Non-Newtonian fluid flow through porous media can also be studied from macroscopic point of view. In this approach attention is shifted from the pore level to the global behavior of the fluid flow. Flow parameters, such as pressure drop and recovery efficiency, are measured to characterize the flow process. Single phase and two-phase flow of non-Newtonian fluid through porous media will be considered here. Single-Phase Flow Single-phase flow of non-Newtonian fluids through porous media became a topic of interest in research with the work published by Christopher and Middleman [85], Sadowski [86], and Sadowski and Bird [87] in 1965. Interest has grown and a large number of publications has appeared in the literature since then. The research work has mainly been focused to devise a model to predict the resistance to flow of nonNewtonian fluid through porous media. A majority of the papers deal with 6.1
1206
0.4
i
i
I
i
I
I 3
I
0.3 Theory
0 q
5
o2 0
0
8 0.1
OD" 1
O I 2 d/d min
Figure 16. Flow distribution with pore opening for Newtonian fluid [49]
0.4
0.3
q Q
I
I
I
I
I
I
i
_ Theory d4
0.2
0.1 1
2 did min
I 3
Figure 17. Flow distribution with pore opening for polymeric fluid [49]
1207
viscometric behavior of non-Newtonian fluids and partially or totally ignoring viscoelastic properties of the fluid. Some authors, such as Jennings, et al. [88], Smith [62], and Harvey [89], believed that the effect of elasticity of the low polymer concentrations of polymer solutions in porous media is insignificant. Sadowski and Bird [87] reasoned that no significant elastic effect would be observed provided the fluid relaxation time is small compared to the transit time through a pore length. It is well established that the effect of elasticity could be significant if the geometry of the media is such that the transit time of fluid flow through a contraction or expansion in a tortuous channel is comparable with the relaxation time of the fluid. This was initially shown experimentally by Marshall and Metzner [90] by flowing polymeric solutions through a sintered bronze disk. They observed upward deviation from Newtonian line in friction factor plotted versus Reynolds number. They correlated their data with a dimensionless number called Deborah number, NDou, and provided a limiting value for the Deborah number above which the additional resistance due to elasticity is expected to be felt. Deborah number is defined as the ratio of two time scales. One time scale is the characteristic time of the fluid, such as relaxation time, and the other is the characteristic time of the flow, such as the time required for the fluid to pass through one pore length.
Of
(io)
NDe b -- Op r
in which Of is a relaxation time of the fluid and 0pr is a measure of the rate of elongational deformation in the flow. Intuitively, elastic effects are expected to be felt when these two time scales become same order of magnitude. Vossoughi and Seyer [49] defined Deborah number as following:
NDeb =
relaxation time of fluid Inverse of velocity gradient in flow direction
(ii)
caused by sudden constriction They estimated the Deborah number by considering the elongation rate of a fluid element moving from point A in Figure 4 to a point at the minimum opening [91 ]. Thus 0u z
v/~0~-0 (12)
1208
where, v is superficial velocity defined as volumetric flow rate divided by cross sectional area of bed, ~ is the area porosity at minimum spacing of a row, and Az is the length defined by Figure 4. Therefore, Op r
~sA
z
"--
(13)
%;
Substituting into Equation (10)
NDeb =
Ofv
CsaZ
(14 )
Equation (14) allows estimation of the Deborah number from a knowledge of the bed geometry and the fluid relaxation time. The choice of Deborah number and even the choice of characteristic times are rather arbitrary. Some investigators have used Ellis number instead, to correlate their pressure data [87]. There is no rigorous treatment in the literature for additional pressure loss in porous media caused by fluid elasticity. Kemblowski, et al. [92] provided a good review of the subject and produced a table for the limiting values of Deborah number published in the literature. The values, above which additional pressure drop due to elasticity has been observed, range all the way from 0.05 to 3 with various definitions of Deborah number. Vossoughi and Seyer [49,91] presented data of friction factor plotted versus Reynolds number for the Newtonian and polymeric fluid through their idealized porous media consisted of glass rods. The upward deviation of the polymeric data from the Newtonian data is clearly demonstrated in Figure 18. The data were also presented in the form of friction factor multiplied by Reynolds number plotted as a function of Deborah number. This is presented in Figure 19. In this figure data from Marshal and Metzner [90] and the theoretical curve derived by Wissler [93] are also included. In a different study, Vossoughi [47] observed similar behavior for the flow of polymer solution through glass beads pack.. The friction factor- Reynolds number data of this study are reproduced in Figure 20. Two beds, one packed with large glass beads and the other with small glass beads, were employed in this study. The characteristics of the beds were presented earlier in Section 3.1.3. Data presented in Figure 20 follow the linear dependency of friction factor with Reynolds number for the Newtonian fluid. However, upward deviation from the straight line for higher Reynolds number is evident for the polymeric data.
1209
10 5
i
i
1 10-1
I 10 0
10 4 10 3 f 10 2 101 10 0 -
[] Polymer 0 Newtonian f=l/N Re
10-1 10-5
1 10-4
I 10-3
t 10-2
101
NRe
Figure 18. Friction factor - Reynolds number plot for glass rods bed [49] As before, deviation from linearity can be also demonstrated by plotting the product of friction factor by Reynolds number versus Deborah number. This is presented in Figure 21. As observed from Figure 21, different polymer concentrations tend to show independent trends of upward deviation. The polymer solutions of lower polymer concentrations reveal earlier upward trend. This upward deviation is not caused by adsorption, or plugging effects. Because, if this was the case, the more concentrated solutions would have appeared to show higher upward deviation at an even smaller Deborah number. 100
i i i liitil
9 - El - zx 10 _ O
i
i i i iiiii1
i i itiiil
i i i~
Separan 273 (48) Carbopol (90) Polyisobutylene (90) ET 597 (90)
1
-I
- f ' N a e = 1 + 90N~)EB
f-NRe 1.0
zx
- [:n:IED n
1 ~
-
m
m
f- NRe = 1 + 1 0 N b E B0.1
i i 1-3
IIIIIII
1-2
I
I IIIIIII
t
1-1
t t titttl
i
i it
10
NDeb
Figure 19. Effect of Deborah number for glass rods bed [91 ]
1210
r 107
I
00
I
!
I
i
o ~
t10 6 0 9
10 5
~
10 4
E a ~
0
o 10 a t--
.o 102 o
=m. t,...
Large Small Beads Beads 9 O Tap Water 9 g~, 100 cs Dow Coming 9 i-I 12500cs Dow Coming 9 V 0.05% Separan AP 273 (~ 0.1% Separan AP 273
u_ 1011 10-7
I 10-6
I 10-5
I 10-4
I = 10-3 10-2 Reynolds Number, NRe, Dimensionless
Figure 20. Friction factor- Reynolds number plot for glass-bead packs [47] 6.2 T w o - P h a s e F l o w
Two-phase flow through porous media is of significant interest in oil industry for displacement of oil through the reservoir. Injection of non-Newtonian fluids, such as polymer and micellar solutions, to displace oil is presently in practice. Displacement process is an unsteady-state process because of the continuous changes of the phase saturation with time. Displacement performance can be predicted either by numerical solution of the partial differential equations involved, or by the Buckley-Leverett model [94]. The Buckley-Leverett model, also called frontal advance equation, is an old and systematic approach which can be solved easily with graphical technique. The derivation of Buckley-Leverett model can be achieved either by material balance, as discussed in detail by Willhite [95], or by the method of characteristics applied by Sheldon, et al. [96], Scheidegger [97], and Vossoughi [47]. The advantage of latter approach is its built-in sharp displacing phase saturation gradient which must be entered arbitrarily in the other approach. The method of characteristics is a powerful tool to deal with a system of linear partial differential equations commonly encountered in the propagation of the plane waves. Roughly speaking, a characteristic path is a propagation path along which a physical disturbance, or entity, is propagated. The continuity equation for a onedimensional two-phase flow through porous media is given by, (15)
1211
50
I
I
or)
I
I
i
i
.9 c'10 11) E
I
CI $
0
x
1
i
u
O
_
m
t
Small Beads 9 0.05% Sep. AP-273 am 0.1% Sep. AP-273 9 0.2% Sep. AP-273
O 0.05% Sep. AP-273 o 0.1% Sep. AP-273 . . . . . Newtonian Fluid Behavior
t.-
.
I
Large Beads
l
mm
l
,
_o .,....o... 9 l_c~..li~. 4.~.,. oo 9 or'&"
0.5 ,, i 5x10 -4 10 -3
l
i
I
10-2 Deborah
i
i
Number,
i
n
.~...~
tr--~'rI
l
i
I
i
10-1 1 NDe b, Dimensionless
I 10 15
Figure 21. Effect of Deborah number for glass-bead packs [47]
Oqo=-oAOSo Ox
(16)
&
where, q is volumetric flow rate, S is volume fraction of the respective phases, t is time, x is position, and subscripts w and o stand generally for the displacing and displaced phases or simply water and oil respectively. Fractional flow rates are defined as,
L
qw
(17)
qo ----I
(18)
where, f is fractional flow rate and q is total volumetric flow rate. By substituting Equation (17) into (15), or Equation (18) into (16), the following partial differential equation is obtained. ~?S q ~?S ~+~~(s)~-0 where,
(19)
1212
df
f ' (S) : - dS
(20)
The subscripts are omitted, because similar equations are produced for displacing and displaced phases. Since phase saturation is a function of position and time, total derivative of S becomes,
as
as
. s - --d
+x
dt
(21)
Equation (22) is the matrix representation of Equations (19) and (21). 1
#A
(22)
&
dt
~
as
Along the characteristic curves following determinants are identical to zero.
q f'(S) ~A =0 dt
1
dx
0
=0 dt
(23)
dS
(24)
1213
q f ' (S)
oa
=0 dS
(25)
dx
From Equation (23),
d x : q f , (S) dt ~A
(26)
and, from Equation 24, dS = 0
(27)
That is, S = Const.
(2 8 )
which also satisfies Equation (25). Therefore, S is constant along the characteristic path, the equation of which is given by Equation (26). Equation (26) is identical to the equation derived by Buckley and Leverett [94] using the material balance concept. In derivation of Equation 26, no assumption was made concerning rheological behavior of the solutions involved. Therefore, by proper selection of fractional flow function, Buckley and Leverett model is applicable for Newtonian as well as non-Newtonian fluids. Equation 26 implies that at constant volumetric flow rate, q, a given saturation plane, S, moves at constant speed, dx/dt. Therefore, a plot of x, the position of each saturation plane, versus qt/~A must yield a set of straight lines with slopes equal to the first derivative of the fractional flow function evaluated at that specific saturation. Vossoughi [47,98] performed linear displacement tests in a transparent rectangular bed packed with either 0.15-cm or 0.47-cm diameter glass beads. Clear silicone liquids, trade named Dow Coming 200 Fluid with kinematic viscosities of 100, 1000, and 12500 cs, were used as oil phases. Water, various concentrations of glycerol solutions, and several concentrations of aqueous polyacrylamide-type polymer solutions were used as displacing phases. He measured the position of the zero-water-saturation plane, which corresponds to the tip of the longest finger, from the moving pictures generated for each displacement test. The position of this plane, Xo, was plotted against qt/#A for Newtonian and non-Newtonian displacing phase. Figure 22 is an example of the linear advance of the zero-saturation plane of a
1214
polymeric flood. Similar linear behavior was obtained for Newtonian as well as non-Newtonian displacing fluids. To explore the possible effect of polymer solution elasticity on the displacement process, Vossoughi [47] undertook the following approach. Two polymers, one more elastic than the other, were chosen. These polymers were Separan AP-273 and Pusher 500 from Dow Chemical. The former one had a molecular weight in the range of 10 million and the latter in the range of 2-3 million. The concentrations of the two polymer solutions were adjusted to obtain a similar viscosity level at shear rates encountered in the displacement tests. The relaxation time defined by Bueche theory was chosen as a criterion to compare the elasticity effect of the two polymer solutions. Assuming a molecular weight ratio of Separan to Pusher of approximately 3, this produced a relaxation time ratio of 7.9 [47]. That is the relaxation time estimated for 0.1% Separan AP-273 solution is approximately eight times that of the 0.2% Pusher 500 solution. Displacement behavior, such as advancing rate of the zero saturation plane, saturation profile, and breakthrough recovery efficiency, are compared in Figures 23 and 24 and Table 2, respectively. Data indicate similar behavior with small differences attributed to the small variations in viscosity levels. Therefore, the effect of polymer solution elasticity on the displacement process was concluded to be insignificant for the two polymers studied in this work. Table 2. Comparison of the Separan and Pusher displacement recoveries Flow rate, Cm~/s 0.16 0.64
Recove_ry efficiency, % of recoverable oil 0.1 ~ Separan 0.2% Pusher 28 31 18 17
7. METHODS OF PREDICTION
In spite of various attempts recorded in the literature, presently, there is no model available to predict the behavior of the non-Newtonian fluid flow through porous media. Most of the attempts have been focused on pressure drop prediction with tittle progress. It is because of the complex nature of the porous media coupled with complex rheological behavior of the non-Newtonian fluids. To this, the fluid-rock interaction that was discussed earlier in this chapter, should be also added and kept in mind. In general, theoretical treatments of the non-Newtonian fluid flow through porous media that are commonly practiced fall into three categories" 1) Those that are
1215
I
I
I
I
I
I
45 40 ,[3 35 30 IE o 25 x
o,,
2O
q,, Y
15
O Small Beads
10
O,~
D Large Beads
,e
,e I
0
5
I 10
I 15
I 20
I 25
I 30
35
qt/~A, cm Figure 22. Linear frontal advance of polymer displacement tests basically an extension of Newtonian fluid flow through porous media; such as hydraulic radius concept, Darcy's law adaptation, and friction factor- Reynolds number relationship; 2) simple converging - diverging geometry coupled with some kind of nonlinear constitutive equation; and, 3) empirical methods.
7.1 Hydraulic Radius Model This is the most common treatment of the non-Newtonian fluid flow through porous media, realizing that it only considers the viscous behavior of the fluid and is insensitive to the fluid elastic component. The method has been exhaustively covered in the literature and Kemblowski, et al. [92] have provided a good review of the topic. Here, the technique will be only applied to the idealized porous media composed of glass rods as presented in Section 3.1.3. The treatment here is parallel to those of earlier works [85,100] but with parallel plates approximating the walls of the capillary pores. As depicted in Figure 15, fluid flow through parallel glass rods was approximated
1216
~
m
m
I
jf/t
// /, //
40
/
// #
O
35 // //
o
30 Z
s
l
E 25 O
#
X
20 --O
0.2% Pusher 500 q = 0.159 cm3/sec
- - .A - -
0.2% Pusher 500 q = 0.635 cm3/sec
--0--
0.1% Sep. AP-273 q = 0.159 cm3/sec
15
10
- - - ~ - - - 0.1% Sep. AP-273 q = 0.635 cm3/sec 0 m
0
1
2
3
4
5
6
7
8
9
10 11
12 13 14
Q t / ~A, cm
Figure 23. Frontal advance rates comparison [98] by Vossoughi [48,49] as a rectilinear flow through parallel plates separated by a distance equal to the pore throat. His approximation was justified based on the observations that most of the pressure drop occurs at the pore throat and the diameter of the glass rods were an order of magnitude larger than the pore throat (0.6 cm compared to 0.05 cm). The choice of coordinate system and the direction of flow for this derivation are defined in Figure 25.
1217
100
I
I
I
I
I
I
~90
~8o m 70 ff 60 o 50 4-, 40 m 30 ~
I 9
9
I
I
I
o. 1% Separan AP-273 0.2% Pusher- 500
9 o
0
9
9 9
9
.-. 20 9 I
NlO i 10
9
9
9
o9
ooo
I i vt i 9 15 20 25 30 35 Position in the Bed, x, cm
40
Figure 24. Saturation Profiles comparison [98]
Y |
Figure 25. Rectilinear flow through parallel plates
t v 45 5"0
1218
The volumetric flow is obtained by integration of the linear velocity in the zdirection as given by Equation 29.
lR
qi - 2 f f uzdydx
(29)
00
where, q~ is volumetric flow rate; u z is linear velocity in z-direction, 1 is the length of the channel in transverse direction, and R is half of the gap between the plates. Since flow is assumed rectilinear, u, is a function of x only, qi (30) o
Integrating Equation (30) by parts, R
R
Iu -u xl -IX .z o
(31)
0
Assuming no slip at the wall,
UzXI~--0
(32)
From Equations (31) and (32), Equation (30) can be written as, R
q__Li= - I xduz 2l o
(33)
or,
R
2-ii _ - X ( ~ x ) d x o
The equation of motion for the laminar flow of Figure (25) produces,
(34)
1219
P~-PL) x
AP --
v-(
M~
'
X
(35)
M~
where, z represents xz-component of the stress tensor, z=. From Equation (35) the wall shear stress, %, is obtained. AP
Vw =~R
AL
(36)
Dividing Equation (35) by Equation (36) and solving for x yields, R X----T
(37)
Tw
or, R dx - ~ d'c Tw
(38)
By substitution of Equations (37) and (38) into Equation (34), Equation (39) is produced, qi _ ~--
21
R2 ~ (dUz )d. c 2 T
"t:w o
(39)
dx
or, qi --Um -21R
(40) ,./72 "~" w 0
T
where, u mis average velocity and dUz/dXis shear rate and will be designated by y; therefore,
um =
T2 Ty w o
d~"
For the assumed steady flow, z is a function of 7 only. For Newtonian fluids,
(41)
1220
v --~ty
(42)
and for a fluid approximated by the power-law model, T - mlTl"
(43)
where, ITI is the absolute value of 7, m and n are power-law indices. Let us define F as an average nominal shear rate, (44)
um
1-' --
R Equation (41) in terms of F becomes,
F
'r 2 w
r'}' dr
(45)
0
To modify Equation (45) for the multi-cylinder bed, it is sufficient to introduce v/~)~ in place of u m,
Um ~ ~ ~8
(46)
Where, v is superficial velocity, defined as volumetric flow rate divided by the cross sectional area of the bed, and ~ is the surface porosity. Surface porosity is calculated as, ~d
where, D is the diameter of the cylinder, and d is the minimum opening between two adjacent cylinders of the same row. Therefore, Equations (44) and (45) become,
G
-
V
1
=-~lv3
% r
T, o
7dv
(48)
1221
where, subscript q~indicates that the equation has been modified for the porous bed. Similarly, equation (36) can be modified for the porous bed as follows, AP
-~R T0- E
(49)
where, L is replaced by L' to include tormousity of the flow path, i. e., L'=CL
(50)
C is geometric constant which accounts for tormousity of the path of the fluid particles. The magnitude of C could be approximately estimated as, C = (zR)N
L
(51)
where, ~R is half the circumference of the cylinder and N the number of rows. The more accurate value of C must be determined experimentally by fitting the pressure data to the model. Replacing L' in Equation (49) from Equation (50) gives, AP
zo
--R CL
(52)
Equation (48) can be integrated using Equation (42) for Newtonian fluid, _TO
3p
(53)
Solving for g, T0
P - ~ 3F 0
(54)
The expression found for g is defined as Darcy's viscosity. Substituting for z, and F, from Equations (48) and (52),
1222
q~R 2 Ap
r
3C
(55)
Lv
From Darcy's law, l.l-k
Ap Lv
(56)
An expression for the permeability of the bed is obtained by comparing Equations (55) and (56). k - ~0~R2 (57)
3C As stated earlier, C must be evaluated experimentally. For power-law model, Equation (43) is solved for 7, y
--
_(L) 1/n (58)
m
where, the minus sign is chosen because of the choice of coordinates. Equation (58) is substituted into Equation (48) and then is integrated,
vo
l+2n
(59)
m
or, ,.co - m[l + 2n Fo] " n
(60)
Further, % and F, can be written in terms of k and C using Equations (48, 52, 57). An ,r ~, - . , / ~
/ COs -~
Fo - v / a/3Ck~s
L
~61~
(62)
1223 Equations (61) and (62) are substituted into Equation (60) and the resulting equation is solved for v n,
Vn --
(3k / CCs )1,2 (3Ckr
mp (63)
m(l+2n)n
L
n
From modified Darcy's law [85] for the flow of power-law model fluids through porous media, Equation (64) is obtained,
(64)
where, H is the viscosity level parameter. An expression for H can be obtained by comparing Equation (64) with Equation (63). H - m 1 + 2n).
7(
,,
)(1-.)/2
(3CkO
(65)
It is evident from Equation (65) that H = m = g when n = 1 for a Newtonian fluid. Using Ergun's definition of friction factor [1 O1], one obtains,
f=
2dO ~2Ap pv2L
(66)
where, f is friction factor and p is the density of the fluid. The Reynold's number, NRo, is arbitrarily defined so that, f - 1 / NRe
(67)
Therefore, from Equations (57, 64, 66) and knowing that R = d/2, one could obtain an expression for Reynold's number, NRo.
NR = Pdv2-~
24C0 H
(68)
1224
For a Newtonian fluid, n = 1, and therefore, NR ~ _
pflv
(69)
24C0,r
The experimental data of friction factors versus Reynold's numbers are compared with the above derivation, i.e. equations (66,68,69) in Figure 18. A value of 0.682 was used for C in generating the data from pressure drop measurements. This was determined experimentally using measured values of surface porosity and permeability in equation (57). The inertia effect is felt around Reynold's number equal to unity. The upward deviation due to elastic properties of the polymer solution is observed long before the inertia effect is felt. It should be realized that the opening between parallel plates are taken equal to the minimum opening between the two neighboring cylinders of the same row. This gives the highest possible value for a characteristic shear rate; therefore, smallest value for viscosity. In fact, in the bed, there is a range of values of viscosity which occurs and, therefore, the bed would have a somewhat higher value of average viscosity. In terms of f - NRo, the low value of viscosity causes higher NRo which makes the data points of polymeric solutions systematically shift to the fight of Newtonian in Figure 18.
7.2 Converging-Diverging Channel The analysis presented above reflects only the viscous behavior of the fluid and does not take into account the elastic effect of the solution. Wissler [93] considered flow of a viscoelastic fluid through a converging-diverging channel and performed a third-order perturbation analysis of the flow. The system considered was a converging-diverging flow between plane walls as shown in Figure (26).
111111. .ILLL I
Figure 26. Converging-diverging plane walls [93]
1225
Analysis was performed in a cylindrical coordinate system and conditions along the z-axis were considered to be uniform. The r and 0 components of the equation of motion are as following,
p ( ~12r
O~ r + Ur
]d00qVr
]202
r 30
r
__ _ _ ]_ G~T rr -Jr 1 ~ q~rO cgr 3r r 30
p(~VO
St
~0
120 ~ 0 r 30
+ Ur --Sr
1 c?p t- 1 o~ (r2T
- - ; 3-0 - - ~ -'~F
) --
( T rr -- T O0) +Pgr
(70)
VrVO r )1 3 ~ oo rO) -] r cOO t- Pgo
(71)
The equation of continuity becomes, 1 cO ( F V r ) ' k1- -Ov - ~o : O r3r r 30
(72)
The volumetric flow rate is obtained by integrating the velocity component in r direction. (73)
q - 2Io~rVr(r,O)dO
A constitutive equation is needed to describe the rheological behavior of the fluid involved. Wissler [93] utilized a nonlinear Maxwell model as following, c)z ~j ~J + ~ [ at
3 r ~j + vk c) x k
3 v~_ 3 x~ rkj
3U
- ~~'ki 3 x~
2 +-~(~'~l%)6ij] - 2rleij
(74)
where, )~ is the fluid characteristic time, and 1] is a constant. He then applied perturbation analysis by constructing solutions of the following forms,
1226 ot~
Vi -- E
~n vi(n)
n=0 oo
p - E ~-p(.)
(75)
n=0 oo
~ ij -- E ~ n z ij (n) n=0
where, n is the order of approximation. The lowest n corresponds to the Newtonian fluid which is well characterized. He arrived at solutions for extra stress, extra force, vorticity, and stream functions for each order of approximation. He further evaluated the net force acting on the fluid at the surface of the channel and intuitively extrapolated his solution to the pressure drop measurements in porous media as following, APviscoelasti c -- Apviscous [1 +
A( -
tJp
)2 ]
(76)
where, q/q~is the mean interstitial velocity, Dp is the average particle diameter, and A is a constant. Equation (76) can be equally presented in terms of friction factor - Reynolds number as given in equation (77).
fNR~-[I+A(r
Zq): p ]
(77)
From the definition of Deborah number given earlier by equation (10), fNRe
--
1 + A N 2 Deb
(7 ~ )
Experimental data presented earlier in Figure 19 seem to follow the trend of the above prediction. Other investigators [102-104] have also observed significant increase in flow resistance for polymer solutions flowing through porous media. It is believed that elongational flow regime is dominant in porous media. This is because of the rapid changes of cross-sectional area of the pore space in the direction of the flow. James and McLaren [103] attributes the increased flow resistance to the increase in
1227
elongational viscosity of the solution. This is directly related to the coil-stretch transition of the polymer molecules. Coil-stretch transition occurs when the strain rates to which the macromolecules are subjected exceed a critical value [105]. This concept was successfully applied by Vorwerk and Brunn [106] to predict the increased flow resistance for a random bed of spheres using three adjustable parameters. Others [107,108] believe that the formation of transient networks of polymer molecules is responsible for the increased flow resistance.
7.3 Empirical Methods Empirical methods are basically application of Darcy's law with an apparent viscosity, or mobility, to be correlated with the properties of fluid and porous media. An example of this approach is the work done by Hejri [76]. He studied flow of a biopolymer (Xanthan gum) through porous media made out of sandpacks and tried to correlate the observed polymer mobility with the predicted values. Darcy's law can be extended to the non-Newtonian fluid by introducing apparent viscosity for the viscosity term,
V
--
kho qL
w
m
(79)
where, k is permeability, v is superficial velocity, and 1] is the apparent viscosity to be determined experimentally. In the shear-thinning region, power-law model can be applied to describe the viscous behavior of the polymer solutions, - m y (n-l)
( 80 )
Shear rate, 3' appearing in equation (80) is the average shear rate within the porous media. There are different models to predict the average shear rate within a porous media. In general they are a function of the superficial velocity and of an unknown function of fluid and porous media properties, i. e.,
y = f(k,r
(81)
Substituting equations (80,81) into equation (79) yields,
v"-
~, Ap P L
where,
(82)
1228
k
(83)
A*p - m [ f (k'(~'n)ln- 1
is the polymer mobility constant to be determined experimentally. It is evident from equation (82) that a plot of pressure gradient, Ap/L, versus superficial velocity, v, in log-log scale should produce a straight line with slope of n and y-intercept equal to the inverse of L'p. Figure (27) is a typical plot generated by Hejri [76] for the flow of 3000 ppm Xantan gum through a sand pack of 2896 md permeability. The flow rate studied covered a wide range of Darcy velocity from 1.59 x 10.4 cm/s to 1.41 x 10.2 (0.45 to 40 ft/d). Data presented in this plot were generated by changing the flow rate from low to high and vice versa; therefore, there was no hysteresis in the pressure drop data. Similar plots were generated for polymer concentrations of 1000, 1500, and 2000 ppm and sandpacks of different mesh sizes with permeability to brine ranged from 894 md to 17,394 md. The permeabilities to brine after the sandpacks were exposed to the polymer solution (residual permeability, kwp) ranged from 525 md to 15260 md. Polymer solutions were prepared with 30,000 ppm KC1 and 1500 ppm formaldehyde as biocide. The polymer in this study was supplied by Pfizer Incorporated as a broth and is commercially referred to as Flocon 4800. The values of the flow behavior index, n, calculated from the core experiments were somewhat different from the power-law exponent generated from viscometric data for the same polymer solution. Polymer solutions were less shear thinning in the sandpack as compared to the rheometer. A linear relationship was obtained between the two flow behavior indices. The experimental values of polymer mobility constants generated above were correlated against predicted values based on four different models from literature. These models are based on the capillary bundle model with variation in the definition of the average shear rate in porous media. They produce different functions of f(k, ~), n) appearing in equation (81). These functions can be extracted from the models as following: 1) Teeuw and Hasselink [109] model for a power-law fluid, f (k,d?,n) =
3n+l
1
n
( 8 k ~ ) 1/2
(84)
1229
10 2 _
I
m
I I llllll
I
I I llllll
I
I I lll___
m
m
~_.
_
"ID
ml01 (.9 L_
kwp = 2,596 md Conc. = 3,000 ppm 9 High - to- Low Rate 13 Low- to- High Rate
t_
n
lO0 lO-1
I
I i llllll
i 10 0
I i llllll
I 101
_
-
I I IIIII 10 2
D a r c y V e l o c i t y , ft/d
Figure 27. Pressure gradient versus Darcy velocity in a sandpack [76] 2) Willhite and Uhl [77] model for a power-law fluid, n
3n + 1)n_1
f ( k , O , n ) - ( 4n
(85)
(0.5kq}) 1/2
3) Jennings, et al. [88] model for a Newtonian fluid,
f(k,r
1
(86)
(0.5kr a/2
4) Modified Blake-Kozeny model [85], [12(
f(k,r
9n+3
n
1 )-n]l-n (87)
( 1 5 0 k q } ) 1/2
In general, the predicted values of the polymer mobility based on the above models were poor compared to the experimental values. Hejri [76] correlated his experimentally measured polymer mobility constants with those predicted by the above models. His empirical correlations are as following,
1230
~p - 1.076~;~ 92
(88)
/~p -- 1.964A~
(89)
Z p --
2.685Z~ 884
(90)
/~p --
1.638X~934z
(91)
where, the numeral subscripts on ~ refers to the corresponding capillary bundle model presented above. Hejri [76] further derived an expression to predict the average shear rate in a porous medium from the porous medium properties (1% and ~), flow behavior index in porous media, n, and power-law parameters (n and m) for the polymer solutions derived from steady shear measurements. In this derivation, it is assumed that the apparent viscosity (viscosity in porous media) is equal to the bulk viscosity (viscosity in rheometer) at a given value of shear rate. Equations (79,82) are rewritten for the porous media by replacing k with kp and n by no respectively.
kwe Ap
V -- ~
m
(92 )
71 L Vn c
_A;Ap
(93)
s
where, kw~is the residual permeability, and no is the flow behavior index in porous media. From equations (92,93), apparent viscosity, 1] can be derived as, kwp
-- ,~p*vl-nc
(9 4)
Assuming the apparent viscosity in a porous medium, rl, is the same as the bulk viscosity, g, represented by power-law model given by equation (95), an expression for the average shear rate in porous media can be derived as presented by equation (96).
1231
] . / - m y "~-1
m~p 9' -
(9 5 )
1
1-n~ v X-.v
(96)
where, n~ is the flow behavior index, and m, consistency index determined from viscometric measurements, ~p* and n are determined from porous media experiments as presented in Figure (27). DEDICATION
This chapter is dedicated to Dr. F.A. Seyer, the author's former M.Sc. and Ph.D. advisor. REFERENCES
1. J.D. Ferry, Viscoelastic Properties of Polymers, John Wiley & Sons Inc.,New York and London, 1961. 2. G.P. Willhite, Waterflooding, SPE Textbook Series Vol. 3, Soc. Pet. Eng., 1986. 3. D.V. Boger, J. Non-Newtonian Fluid Mech., 3 (1977/1978) 87. 4. V.H. Giesekus, Rheol. Acta, 6 (1967) 339. 5. D.V. Boger and R. Binnington, Trans. Soc. Rheol. 21 (1977) 515. 6. D.V. Boger and H. Nguyen, Polym. Eng. Sci., 18 (1978) 1037. 7. G. Prilutski, R.K. Gupta, T. Sridhar, and M.E. Ryan, J. Non-Newtonian Fluid Mech., 12 (1983) 233. 8. R.J. Binnington and D.V. Boger, Polym. Eng. Sci. 26 (1986) 133. 9. J.J. Magda and R.J. Larson, J. Non-Newtonian Fluid Mechanics, 30 (1988) 1. 10. C. Kemielewski, K.L. Nichols, and K. Jayaraman, J. Non-Newtonian Fluid Mech., 35 (1990) 37. 11. D.O. Shah, editor: "Macro and Microemulsions," ACS symposium series 272, American Chemical Society, Washington, D.C., 1985. 12. H. Bennett, J.L. Bishop, Jr., and M.F. Wulfinghoff, Practical Emulsions, Vol. 1" Materials and Equipment, Chemical Publishing Company, Inc., New York, 1968. 13. K.J. Lissant, editor: "Emulsions and Emulsion Technology, Part I, II, and III," Surfactant Science Series, Vol. 6, Marcel Dekker, Inc., New York, 1974. 14. M. Baviere, editor" "Basic Concepts in Enhanced Oil Recoverey Processes," Elsevier Applied Science, London and New York, 1991. 15. F.H. Poetmann, Microemulsion Flooding, Secondary and Tertiary Oil
1232
Recoverey Processes, Interstate Oil Compact Commission, Oklahoma City (1974) 67. 16. H.Y. Jennings, Jr., Soc. Pet. Eng. J., 15 (3) (1975) 197. 17. W.K. Seifert, and W.G. Howells, Analyt. Chem., 41 (1969) 554. 18. P.A. Farmanian, N. Davis, J.T. Kwan, R.M. Weinbrandt, and T.F. Yen, ACS Symp. Series, 91, American Chemical Society, Washington D.C. (1979) 103. 19. G.A. Pope, et al., Soc. Pet. Eng. J. (Dec. 1982) 816. 20. C. Castillo, and M.C. Williams, Chem. Eng. Commun., 3 (1979) 529. 21. D. Bouchez, A. Faure, G. Scherer, L.A. Tranie, and G. Antonini, COM: The French Program, preparation, stabilization, and handling of COM, IV International symp. on Coal Slurry Combustion, Orlando, Fla., 1982. 22. S. Vossoughi and O.S. A1-Husaini, Rheological Characterization of the Coal/Oil/Water Slurries and the Effect of Polymer, 19u' International Technical Conference on Coal Utilization & Fuel Systems, March 21-24, 1994, Clearwater, Florida, USA. 23. R. Karrat and S. Vossoughi, Rheological Behavior of the Gel Systems used in Enhanced Oil Recovery, Theoretical and Applied Rheology, Volume 1, edited by: P. Moldenaers and R. Keunings, Elsevier Science Publishers (1992) 478. 24. R.B. Needham, C.B. Threlkeld, and J.W. Gall, SPE 4747 presented at the Third Annual Symposium on Improved Oil Recovery, Tulsa, OK, April 22-24, 1974. 25. J.C. Mack, and M.L. Durall, SPE 12929 presented at the 1984 Rocky Mountain Regional Meeting of SPE- AIME, Casper, Wyoming, May 21-23, 1984. 26. J.C. Mack, and J.E. Srrfith, SPE/DOE 27780, presented at the SPE/DOE Ninth Symposium on Improved Oil Recovery, Tulsa, OK, April 17-20, 1994. 27. J.E. Smith, SPE 28989 presented at the SPE International Symposium on Oilfield Chemistry, San Antonio, TX, February 14-17, 1995. 28. J.E. Smith, J.C. Mack, and A.B. Nicol, SPE/DOE 35352 presented at the 1996 SPE/DOE Tenth Symposium on Improved Oil Recovery, Tulsa, OK, 21-24 April, 1996. 29. G.J. Hirasaki, J. Pet. Tech., 41(5) (1989) 449. 30. T.W. Patzek and M.T. Koinis, J. Pet. Tech. 42 (1990) 496. 31. O.S. Owete, L.M. Castanier, and W.E. Brigham, Proc. AIChE Annual Meeting, Las Vegas, Nevada, September 1982. 32. G.J. Hirasaki and J.B. Lawson, SPE 12129 presented at 58~ Annual Tech. Conf. And Exhibition of Soc. Pet. Eng., San Francisco, 5-8 October 1983. 33. R.J. Treinen, W.E. Brigham, and L.M. Castanier, SUPRI TR 48, US Department of Energy, Report DOE/SF/11564-13 (DE 86000260), 1985 34. J.T. Patton, et al., "Enhanced Oil Recovery by CO2 Foam Flooding," Second Annual Report for U.S. DOE under contract NO. DE- AC21-78MC03259 (Nov. 1980). 35. J.T. Patton, S.T. Holbrook, and W. Hsu, Soc. Pet. Eng. J., 23(3) (June 1983)
1233
456. 36. H.S.S. Hele-Shaw, Nature 58 (1898) 34. 37. R.A. Greenkorn, Flow Phenomena in Porous Media, Marcel Dekker, Inc., New York, 1983. 38. M.A. Nobles and H.B. Janzen, Trans. AIME, 213 (1958) 356. 39. R.L. Chuake, P. van Muers, and C. van der Poul, Soc. Pet. Eng. J., 188 (1959); and Trans. AIME, 216. 40. J.L. Mahaffey, W.M. Rutherford, and C.W. Matthews, Soc. Pet. Eng. J. (March 1996) 73; and Trans AIME, 237. 41. R.A. Greenkorn, J.E. Matar, and R.C. Smith, AIChE J., 13 (1967) 273. 42. M.R. Todd, and W.J. Longstaff, J. Pet. Tech. (July 1972) 874. 43. S.D. Gupta, J.E. Varnon, and R.A. Greenkorn, Water Resour. Res. 9(4) (1973) 1039. 44. B. Habermann, Trans. AIME, 219 (1960) 264. 45. J. Wiggert, and T. Maxworthy, Physical Review E, 47(3) (March 1993) 1931. 46. P. van Meurs, Trans. AIME, 210 (1957) 295. 47. S. Vossoughi, Viscous Fingering in Immiscible Displacement, PhD Dissertation, U. of Alberta, Edmonton, Alta., Canada (1976). 48. S. Vossoughi, Non-Newtonian Flow in a Porous Bed, M.Sc. Thesis, U. of Alberta, Edmonton, Alta., Canada (1973). 49. S. Vossoughi and F.A. Seyer, J. of Can. Pet. Technol. 16(3) (1977) 110. 50. C.R. Kyle and R.L. Perrine, Can. J. Ch. E., 49 (Feb. 1971) 19. 51. I.H. Arman, Relative Permeability Studies, MS Thesis, U. of Oklahoma,, Norman, OK (1952). 52. J. Naar, R.J. Wygal, and J.H. Henderson, Soc. Pet. Eng. J. (March, 1962) 13. 53. L.A. Rapoport, C.W. Carpenter, and W.J. Leas, Trans. AIME, 213 (1958) 113. 54. R.L. Slabod, and B.H. Caudle, Trans. AIME, 195 (1952) 265. 55. W.F. Nuss, and R.L. Whiting, Bull. Am. Assoc. Petrol. Geologists (November 1947) 2044. 56. J.W. Amyx, D.M. Bass, JR., and R.L., Whiting, Petroleum Reservoir Engineering Physical Properties, McGraw-Hill Book Company, New York, 1960. 57. A. Silberberg, J. Physical Chemistry, 66 (1962) 1884. 58. F.W. Rowland, and F.R. Eirich, J. Polymer Science, Part A-1, 4 (1966) 2401. 59. A.S. Michael, and O. Morelos, Ind. Eng. Chem., 47(9). 60. W. Schmidt, and F.R. Eirich, J. Phys. Chem., 66 (1962). 61. W.P. Shyluk, J. Polymer Sci., part A-2, 6 (1968) 2009. 62. F.W. Smith, J. Pet. Tech. (Feb. 1970) 148. 63. N. Mungan, Revue de L'Institut Francais du Petrole, 24(2) (1969) 232. 64. M.J. Szabo, SPE 4668 presented at SPE 48 ~ Fall Meeting, Las Vegas, Nevada (Sept. 1973).
1234
65. J.G. Dominguez D., Polymer Retention and Flow Characteristics of Polymer Solutions in Porous Media," M.Sc. Thesis, U. of Kansas, Lawrence, KS (1974). 66. N. Mungan, Soc. Pet. Eng. J. (Sept. 1966) 247. 67. E.J. Lefebre du Prey, Soc. Pet. Eng. J., 13(1) (Feb. 1973). 68. A.M. Sarem, SPE 3002 presented at the 45" Annual Fall Meeting of the Society of Petroleum Engineers of AIME, Houston, TX (Oct. 1970). 69. A.C. Uzoigwe, F.C. Scanlon, and R.L. Jewett, SPE 4024 presented at SPE 47 ~ Annual Fall Meeting, San Antonio, TX (Oct. ! 972). 70. R. Dawson and R.B. Lantz, Soc. Pet. Eng. J. (Oct. 1972) 448. 71. G.H. Hirasaki, and G.A. Pope, SPE 4026 presented at SPE Fall Meeting, San Antonio, TX (Oct. 1972). 72. N. Mungan, J. Can. Pet. Tech., 8(2) (1969) 45. 73. D.J. Pye, Trans. AIME (1965) 911. 74. R.B. Sandiford, J. Pet. Tech. (August 1964) 917. 75. E.J. Burcik, Producers Monthly, 29(6) (June 1965). 76. S. Hejri, An Experimental Investigation into the Flow and Rheological Behavior of Xanthan Solutions and Xanthan/Cr(III) Gel System in Porous Media, PhD Dissertation, U. of Kansas, Lawrence, KS (1989). 77. G.P. Willhite and J.T. Uhl, Correlation of the Flow of Flocon 4800 Biopolymer with Polymer Concentration and Rock Properties in Berea Sandstone, WaterSoluble Polymers for Petroleum Recovery, G.A. Stahl and D.N. Schultz (editors), Plenum Press, New York (1988). 78. F.D. Martin, et al., SPE 11786 presented at the 1983 Soc. Pet. Eng. International Symposium on Oilfield and Geothermal Chemistry, Denver (June 1983). 79. W.B. Gogarty, Soc. Pet. Eng. J. (June 1967) 161; and Trans. AIME, 240. 80. R. Dawson and R.B. Lantz, Soc. Pet. Eng. J. (Oct. 1972) 448; and Trans. AIME 253. 81. B. Shah, An Experimental Study of Inaccessible Pore Volume as a Function of Polymer Concentration during Flow through Porous Media, M.Sc. Thesis, U. of Kansas, Lawrence, KS (1978). 82. D.S. Hughes, et al., Soc. Pet. Eng. Res. Eng. J. (Feb. 1990) 33. 83. A. Chatenever and J.C. Calhoun, Jr., Trans. AIME, 195 (1952) 149. 84. Fluid Distribution in Porous Systems - A Preview of the Motion Picture, Stanolind Oil and Gas Co. (1952); subsequently reprinted by Pan American Petroleum Corp. and Amoco Production Co.. 85. R.H. Christopher and S. Middleman, Ind. Eng. Chem. Fund., 4 (1965) 422. 86. T.J. Sadowski, Trans. Soc. Rheol., 9 (1965) 251. 87. T.J. Sadowski and B. Bird, Trans. Soc. Rheol., 9 (1965) 243. 88. R.R. Jennings, J.H. Rogers, and T.J. West, J. Pet. Technol. (March 1971) 391. 89. A.H. Harvey, An Investigation of the Flow of Polymer Solutions through Porous Media, PhD Dissertation, U. of Oklahoma, Norman, OK (1967).
1235
90. R.J. Marshal and A.B. Metzner, Ind. Eng. Chem. Fund., 6 (1967) 393. 91. S. Vossoughi and F.A. Seyer, Can. J. Chem. Eng., 52 (Oct. 1974) 666. 92. Z. Kemblowski, M. Dziubinski, and J. Sek, Flow of Non-Newtonian Fluids through Granular Media, from: "Transport Phenomena in Polymeric Systems," R. A. Mashelkar, A.S. Mujumdar, and R. Kamal (editors), Ellis Horwood Series in Physical Chemistry, John Wiley & Sons, New York (1989) 117. 93. E.H. Wissler, Ind. Eng. Chem. Fund., 10(3) (1971) 411. 94. S.E. Buckley and M.C. Leverett, Trans. AIME, 146 (1942) 107. 95. G.P. Willhite, Waterflooding, SPE Textbook Series Volume 3, Society of Petroleum Engineers, Richardson, TX, 1986. 96. J.W. Sheldon, B. Zondek, W.T. Cardwell, Jr., Trans. AIME, 216 (1959) 290. 97. A.E. Scheidegger, The Physics of Flow through Porous Media, University of Toronto Press, 3~dedition, Chapter 9 (1974). 98. S. Vossoughi and F.A. Seyer, Ind. Eng. Chem. Fundam., 23(1) (1984) 64. 99. S. Middleman, The Flow of High Polymers, Continuum and Molecular Rheology, Interscience Publishers, John Wiley & Sons, Inc., New York (1968) 148. 100. J.G. Savins, Ind. Eng. Chem., 16(10) (October 1969) 18. 101. S. Ergun, Chem. Eng. Prog., 48(2) (1952) 89. 102. D.L. Dauben and D.E. Menzie, J. Pet. Tech. (August 1967) 1065. 103. D.F. James and D.R. Mc Laren, J. Fluid Mech., 70 (1975) 733. 104. W.M. Kulicke and R. Haas, Ind. Eng. Chem. Fundam., 23 (1984) 308. 105. P.G. de Gennes, J. Chem. Phys., 60 (1974) 5030. 106. J. Vorwerk and P.O. Brunn, J. Non-Newtonian Fluid Mech., 41 (1991) 119. 107. J.A. Odell, A.J. Muller, and A. Keller, Polymer, 29 (1991) 119. 108. S. Rodriguez, C. Romero, M.L. Sargenti, A.J. Muller, and A.E. Saez, J. NonNewtonian Fluid Mech., 49 (1993) 63. 109. D. Teeuw and F.T. Hesselink, SPE 8982 presented at the 5th International Symposium on Oilfield and Geothermal Chemistry, Stanford, California (May 1980).
1237
FLUID DYNAMICS OF FINE SUSPENSION FLOW
Y. A. Buyevich
Center for Risk Studies and Safety, University of California Santa Barbara, 6740 Cortona Dr., Santa Barbara, 6.4 93117
1. INTRODUCTION Flows of colloids and suspensions of fme particles are important for a wide variety of multifarious applications encountered in nature and in various fields of modem industry. First of all, engineers and researchers are commonly interested in global hydraulic characteristics of such flows. However, more subtle peculiarities featured by such flows may be of great practical concern as well. For instance, it is imperative to obtain a certain information on the kinetics of flow stratification which would allow to evaluate the rate of particle deposition on channel walls, the intensity of phase separation in vertical flows, etc. The information of such a kind should also be of crucial significance in modifying many widespread industrial processes with an ultimate purpose in view to improve quality and performance characteristics of those processes. However, theoretical study of suspension flow usually meets with formidable difficulties of principal nature, even in cases where only comparatively sketchy hydraulic characteristics pertaining to a smooth laminar flow as a whole are under question. These difficulties, which are briefly indicated and discussed below, are in the first place due to the obvious lack of a sufficiently reliable hydrodynamic model even for laminar suspension flow. Such a model is at present not available, despite considerable efforts undertaken during the last decades in the field of both suspension hydrodynamics and rheology. Correspondingly, the main intended objective of this Chapter consists in putting forward some relatively simple considerations that hopefully will prove themselves both beneficial and advantageous in the matter of developing the needed fluid dynamic model for laminar flow of concentrated suspensions. A
1238
tangible tentative approximate version of such a model is presented and discussed in this Chapter as well. When dealing with colloid and fine suspension flows, there is a strong temptation to consider the dispersed medium under question as a fictitious homogeneous fluid characterized by some effective properties of its own, such as density and viscosity, in which case any flow is presumed to be governed by a single set of mass and momentum conservation equations. The effective properties are then implicitly regarded as quantities dependent on the fictitious fluid composition, and they may be identified on either theoretical or empirical grounds, or both. However, such a simple picture fails to adequately portray the flow in numerous cases where the fictitious fluid composition happens to be unsteady or non-uniform, so that its effective properties cannot be looked upon as invariable, but are instead certain, and not always known, functions of time and coordinates. First of all, this occurs in different flow processes that by their very nature imply separation of the suspension phases, or of different particulate species of the dispersed phase, such as sedimentation, crossflow microfiltration, and field-flow fi'actionation. However, this often occurs also in colloid and suspension flow in cases where separation is out of the question, and flow stratification arises as a consequence of uncontrollable natural factors. In such cases, the fictitious fluid properties can be again regarded as functions of the local mean suspension concentration. To find the concentration field, and thus to determine the said properties as fimctions of coordinates and time, an analysis based on the concept of the only fictitious fluid is clearly insufficient. An alternative continuum approach must instead be employed for this purpose, within the framework of which the suspension phases are viewed as separate interacting co-existing media with their own properties. This means that at least two sets of averaged field conservation equations have to be used to describe the suspension flow, one for each of the phases. Unknown variables of these equations are the mean particle and ambient fluid velocities, the mean interstitial pressure, and the mean suspension concentration. A serious stumbling block commonly confronted when attempting to find out the suspension concentration distribution on the basis of this continutun approach is caused by the fact that stratification of suspension flows is governed not only by regular forces experienced by suspended particles, but also by particle diffusion due to different physical reasons. Hence it follows that the conservation equations that do not include the impact of diffusional effects can hardly be anticipated to be helpful in the matter of determining the wanted concentration distribution. This expectation comes to be true, if random particle and fluid fluctuations that give rise to diffusional phenomena and that contribute to
1239
stresses acting in flow of the suspension phases are completely left out of account. As has been discussed in [1 ], the intrinsic inconsistency of conventional fluid dynamic schemes of suspension flow which do not allow for stress contributions due to the fluctuations consists in the fact that their governing equation do not permit the concentration field to be found alongside the fields of mean phase velocities and fluid pressure. The latter fields can be obtained merely on condition that the former one is given beforehand. However, the mere fact of the concentration field having been prescribed in advance decreases by unity the number of unknown variables in the governing equations, so that one of these equations cannot actually be satisfied and, in fact, has to be discarded. The physical reason of this deficiency is usually due to the occurrence in a flow of a transverse force experienced by suspended particles in the direction normal to the flow streamlines. This force causes the particles to migrate in the same direction and, inasmuch as the corresponding particle flux is not compensated for by any diffusion flux, it must eventually give rise to the flow stratification. The simplest example is provided by flows of not neutrally buoyant suspensions in inclined channels. In such flows, the lateral component of gravity as corrected for buoyancy brings about the initiation of both a region which is filled with close-packed particles and a region which is entirely devoid of particles. Under steady conditions, the whole flow domain must consist of only such limiting regions, without any region containing suspended particles in between [ 1]. In vertical flows, the similar particle migration is stipulated by the inertial lateral lift force acting on particles that rotate in a sheared flow [2]. When being not compensated by diffusion, this migration virtually results in the accumulation of particles in either a central or a peripheral flow region, depending on whether the particles are heavier or lighter than the suspending fluid. Our primary purpose consists in demonstrating in the present Chapter what has to be actually done in order to develop a consistent workable scheme aimed at tackling stratified flows of colloids and freely dispersed suspensions. The term "finely dispersed" insinuates that collisions of suspended particles are totally irrelevant in the interparticle exchange by momentum and energy which is therefore assumed to be carried out via the velocity and pressure fields of the intervening fluid. The preference is given to simplicity and clarity of presentation, rather than to strictness of our development which seems unnecessary at its present stage . For this reason, we shall address only the simplest possible type of dispersions, that is, a suspension of identical solid spheres in an incompressible Newtonian fluid which exhibit steric and hydrodynamic, but not any molecular mterparticle interactions that are usually
1240
specific to colloidal systems. The spheres are assumed to be free of embedded dipoles of any physical origin. Besides, when needed, we shall prefer sometimes to use certain semi-empirical, albeit intuitively perceptible, relationships and considerations stemming from other cognate fields of science, rather than to stick to rigorous derivation of the wanted relationships. The Chapter is organized as follows. As a beginning, we shall briefly review averaged conservation equations and relevant constitutive rheological equations for the phases of a suspension at neglect of random particle and fluid fluctuations. After that, we shall explain the influence of thermal particle fluctuations on the rheology of Brownian suspensions with the help of a specific "thermodynamic" constituent of the interphase interaction force. We shall also consider an alternative way to describe this influence by means of specifying appropriate contributions to the effective stress tensors that affect flow of the suspension phases. In the next place, we shall address hydrodynamically induced particle and fluid fluctuations that may obtain as a result of two possible physical mechanisms. The first mechanism bears upon random displacements of particles caused by relative motion of neighboring particle layers in shear flow. The second mechanism is due to the relative fluid flow working at random fluctuations of the suspension concentration, and thus originating peculiar "pseudoturbulent" fluctuations. The impact of both shear-induced and pseudoturbulent fluctuations on suspension flow is again described by introducing pertinent contributions to the effective stress tensors. Because a comprehensive presentation of these topics takes a considerable space, it is impossible to address application of the theory developed to various particular flow problems, which have thus to be considered in the future. 2. FIELD CONSERVATION EQUATIONS We start with formulating averaged field equations of mass and momentum conservation for the interpenetrating co-existing continua that model the continuous and dispersed phases of a suspension with no fluctuations. Such equations are to be obtained by using some smoothing procedure, such as those implying time [3] or volume [4] averaging of local conservation equations that are presumed to be valid within the phase materials. Unfortunately, however, these averaging procedures result in seemingly incongnlent forms of the field equations, as exemplified in [5]. For this reason, we prefer to use the technique of averaging over the ensemble of physically possible configurations of the assemblage of suspended spheres as developed in [6] and as recently discussed in [7]. This technique leads to a set of field equations for mass conservation
1241
3t
'
3t
+v.(ow)-0
(1)
and for momentum conservation
cpf ~t9+ v . V / v = V . c r - f - G p f V O C9 ~ + w . V I w = f - qkppV ff~ CPp(~
(2)
(3)
where v and w are the mean fluid and particle velocities, ~b and e = 1 - ~b are the particle concentration by volume and the void fraction, respectively, and (I) is the potential of an external body force field. Vector f describes the average mterphase interaction force per unit volume, and ~ is the effective stress tensor. What happens to be of principally significance is that effective stresses appear only in the equation (2) of continuous phase momentum conservation, whereas averaged stresses that might be expected to affect mean flow of the dispersed phase identically tum to zero [6,7]. Note, however, that this general conclusion is valid only for disperse systems in flow of which the impact of fluctuations on the phase stresses is negligible, so that it may be ignored. 3. CLOSURE OF FIELD EQUATIONS According to the ensemble averaging technique, both interphase interaction force and effective stresses that influence the continuous phase flow are expressible in terms of integrals over the surface of a chosen "test" suspended sphere. The corresponding mtegrands involve stresses wlfich are averaged over particle configurations that are conditioned by the test sphere center having been positioned at a prescribed point. These conditioned averages can in principle be determined by solving a problem for mean suspension flow around the test sphere. Averaged equations that govern this flow contain integrals of stresses obtained by averaging over configurations that have centers of two spheres fixed in space. As explained in [6,7], when continuing such a process, we arrive an a practically infinite chain of interconnected equations, and of hydrodynamic problems, for flow around different nmnbers of fixed spheres, and a familiar task arises of cutting off this chain. An approximate method of resolving this task implies viewing the test sphere as one immersed in a fictitious medium whose properties vary with the distance from the test sphere center, but coincide with those of the suspension as a whole
1242
as this distance goes to infinity. Within the framework of a certain approximation, the exact type of this variation is shown in [6, 7] to be dictated by the pair distribution function for suspended spheres. As a result, depending on a model used to describe this function, we obtain different rheological models for the suspension. In practice, the pair distribution function is influenced by suspension flow, and so cannot be regarded as an equilibrium property. This fact was well recognized by Batchelor who succeeded in calculating relative viscosity of a dilute suspension for flow of pure elongation [8], and also for simple shear flow in the limit of strong Brownian motion of the spheres, when the suspension approximately behaves like a Newtonian medium [9]. In the generalized case of an arbitrary flow, averaged stresses are not Newtonian, and they depend on the type of flow, due to hydrodynamic interactions that affect space distribution of suspended particles. If the suspension is not dilute in the sense that not only binary hydrodynamic interactions are essential, one might presume simultaneous interactions of many particles to work similarly to strong Brownian motion in rendering the structure of a flowing suspension statistically isotropic, and in thereby making the suspension behave as a Newtonian fluid. It is such a point of view that is actually adopted when describing the pair distribution function with no allowance made for effects that flow is likely to exert on the suspension statistics. In what follows, we are going to neglect the effects on the short-range order in a flowing suspension as produced both by the flow itself and by particle random fluctuations. Correspondingly, we shall use some representations for the pair distribution function resulting from the equilibrium statistical mechanics of dense systems of hard spheres. Having represented the conditional averaged stresses that act at the test particle surface in terms of the unconditional ones, we must fitaher proceed to expressing the latter stresses, as well as the interphase interaction force that appear in equations (2) and (3), through unknown variables of the field equations and their derivatives. This has been suggested in [6,7] to achieve with the help of the self-consistent field theory. According to such an idea, the averaged stress tensor divergence and the interphase interaction force vector are first presented as linear combinations of relevant vector quantities that are likely to completely determine the situation at any point within the flow domain. This enables us to close the governing equations, and ultimately, to solve the flow problem around the test sphere and to calculate the stresses at the test sphere surface. After that, the original integrals representing the stress divergence and the interphase interaction force can be calculated to yield expressions for the said quantities as functions of tmknown coefficients involved in the aforementioned linear
1243
combinations. The last coefficients have to be found afterwards from selfconsistency conditions which require these expressions to be identical to the original linear combinations. This leaves us with a set of algebraic equations in which the wanted coefficients serve as unknown variables. Solving these algebraic equations determine all constitutive rheological equations for fine suspensions, and so makes for final closure of the suspension fluid dynamic theory. 3.1 Effective Medium Model The simplest possible rheological model comes about as a result of the supposition that the averages conditioned by positioning the test sphere center at a certain point are indistinguishable from the unconditional averages. The fictitious medium the test sphere is assumed to be immersed into is then uniform, and its properties are precisely the same as those of the suspension as a whole. This model corresponds to entirely ignoring the non-overlapping property of hard suspended spheres, which are thus permitted to overlap. It is sensible to expect this model to be approximately valid for moderately concentrated suspensions in which the overlapping of the spheres is unlikely to produce a significant effect. Similar models according to which discrete particles are supposed to be inserted into an effective homogeneous medium were repeatedly formulated during the last decades on purely empirical grounds. As a matter of fact, such models were developed not only in connection with suspension rheology, but also while treating effective properties of various dispersions and composite materials, such as thermal conductivity, electric and magnetic permeability, moduli of elasticity, etc. Unsteady suspension flows have been recently considered in detail within the framework of a simple model of this kind in [10], and here we are going to exhibit main conclusions of that paper. According to [10], the effective stress tensor displays relaxation phenomena the type of which depends on a sign of a newly introduced quantity joe that has the density dimensionality
'I
4
5
Pe = -~ P - -~ Pp + -~ M (3p - qkpp
1 M =~ 1-2.50
, p = ~pf + r
(4)
The stress tensor is expressible as follows:
cr=-pi+
, o'~=21ufM
pe a2 1+ T~
+w"
ev ,
- F,S M
(5)
1244
where p is the mean interstitial pressure, I is the unit tensor, a is the sphere radius, ktf is the ambient fluid viscosity, and e~ is the strain rate tensor corresponding to the continuous phase mean velocity field. Quantity M plays the role of relative suspension viscosity attained in steady flow, whereas Tu is a specific relaxation time. If p~ is positive, as is the case for colloids and for the majority of suspensions encountered in practice, an adjunct relaxation relationship can be formulated as [10]
--
cgt
+ w-
ev
-
ev,,
-
ev
'
ev's
21ufM
= ~
(6)
Hence it follows that the strain rate tensor relaxes to its value which would establish itself in steady flow characterized by a given actual tensor of mean viscous stresses. In the opposite case of negative p~, which can be realized for gas-solid mixtures, we have another relaxation relationship, instead of that in equation (6),
According to this relationship, the mean viscous stress tensor now relaxes to its steady value corresponding to a given strain rate tensor. These relaxation phenomena come about as a natural consequence of the frequency dispersion effect for the suspension viscosity. This effect means that the effective suspension viscosity, as it manifests itself in unsteady monochromatic suspension flow, happens to be dependent on flow frequency. The frequency dispersion may be quite insignificant as far as the averaged stresses are concerned. However, as is demonstrated below, it proves to be rather important in the matter of influencing an inertial part of the total interphase interaction force that is inherent in dense suspension flow. The interphase interaction force attributed to a unit volume of the mixture is a sum of a few contributions having different physical meaning f = f d +f/~ + fF + fi +fb
(8)
Expressions for the viscous drag force, the hereditary Basset force, and the Faxen force due to flow non-uniformity are of exactly the same form as the corresponding force constituents for a single sphere in an unbounded fluid [ 11 ], save for the fluid viscosity and density being substituted by the effective suspension viscosity and density. They are expressible as
1245
f_9r uzM (v_w) a 2 fB -
(9)
9~2( .tllfMll/2i(~ VII p Jra2 )
+ w-
dr'
(v - w)t - t' 4 t - t'
(10)
--00
f F -- 3~4 luf M Av
(l])
The inertial force equals to
fi -
/(
p - - ~ pp + 3~bPe ~ - + w
v-w,
Supposedly, this force describes not only the effect of fluid virtual mass acceleration and the effect of accelerated motion of the ambient fluid itself, but also the influence on the interphase interaction force of the frequency dispersion of effective suspension viscosity. An equation for the effective buoyancy force contains two terms
fb-qkp
l + r b ~-~-+w.
V.+
~-~-+w.
w
(13) 2
o~pp
pfa 2
rb = -~ e p f + r pp u f M
The first term within the curly brackets describes a contribution to buoyancy caused by the suspension being under action of an external body force field that is characterized by potential O. This contribution is sensitive to changes in this potential with time and (or) in space. This may be important for a centrifugal force field that influences flow of suspensions rotating with either varying or constant angular velocity. However, this contribution does not evince relaxation if the specific acceleration of the external field is invariable, as is the case with gravity. The second term in equation (13) represents a buoyancy force contribution stipulated by an additional effective body force field that makes its appearance owing to particle inertia. It is worth noting that an additional term describing a relaxation effect associated with this buoyancy contribution makes its appearance [10]. This term can be shown, however, to be of a higher order of
1246
magnitude that other terms retained in equations (9) - (13), and for this reason, it must be omitted from equation (13). At v = 0 and in the dilute limit ~ ~ 0, force (8) can be easily demonstrated to reduce to the force that is experienced by a single Stokesian sphere moving in an unbounded fluid and that was previously evaluated in [11 ]. To get a deeper insight into the nature of the inertial phenomena that affect interphase interaction, we also reproduce an equation of motion for a single suspended sphere that can be straightforwardly deduced from equations (3), and (8) - (12). This equation can be written down in the following form dw
mp ~ = - m dt
a
d ( w - v) dt
+m-
dv
+R
(14)
dt
Here, operator d/dt is understood as that of full time differentiation along the averaged particle trajectory (or alternatively, along the dispersed flow streamlines). Vector R stands for a sum of all forces experienced by the sphere except for those proportional to the full time derivatives of the continuous and dispersed phase velocities (that is, except for the inertial force (12) and the second contribution to the buoyancy force (13)). The following effective masses are introduced:
m = c r n f + q~mp ,
ma = k m f ,
1/
(15)
k =-~ e +
pf where mp and my are the sphere mass and the mass of the fluid replaced by the sphere, respectively. In the dilute limit, m~ tends to m/2, and m tends to mf, so that equation (14) with coefficients (15) reduces to the well-known equation for unsteady motion of a body in an unsteadily flowing unbounded fluid [11, 12]. The first term on the right-hand side of equation (14) describes a combined effect of the virtual suspension mass acceleration and of the frequency dispersion phenomenon. Coefficient k can be conventionally interpreted as an effective added mass coefficient. It depends not only on suspension concentration, but also on the ratio of particle and fluid densities. This is due to the fact that the suspension cannot apparently be regarded as a truly homogeneous effective medium that surrounds an accelerating suspended particle when evaluating the effective suspension mass that is carried along with this particle. The second term in the right of equation (14) is due to the inertial force that makes its appearance in the suspension momentum conservation equation when this equation is formulated in a coordinate system that moves with velocity v, as
1247 is explained in [12]. This force must not be dependent on the suspension being a two-phase system. Accordingly, coefficient m that appears in this term represents the mass of the suspension that is replaced by a sphere, as if the suspension were a homogeneous one-phase medium. The determination of the added mass coefficient was a predominant subject of many papers, beginning with that by Zuber [13], representative examples of which have to be found in [14-18]. Nonetheless, conclusions inferred in these and other papers on the subject with respect to the added mass coefficient can hardly be unambiguously compared to our formula for k as listed in equation (15). This is due to the fact that practically all such conclusions commonly refer to a system of particles (most usually bubbles) immersed into a potential or almost potential flow, where the frequency dispersion of viscosity and the ensuing stress relaxation is of no consequence. At the same time, the contribution to k appears to be quite significant for fine particles in viscous flow, and it comes about as a result of the stress relaxation and which is not taken into accotmt in the majority of available papers. (Admittedly, this contribution correspond to the term in equation (15) that is proportional to effective density p~ involved in the definition of relaxation time T, in accordance with equation (5) .) At any rate, the added mass coefficient as detennined by equation (15) increases with ~ considerably faster than similar coefficients calculated in [1318]. This point deserves attention in the future work. It is significant that it is the mean suspension density, but not that of the pure ambient fluid, that is involved in equation (13) expressing the effective buoyancy force. As has been discussed in [10], this inference brings to an end a recent rather hot debate on which density must be used in the mentioned context while treating fluidized beds and other suspension flows (examples of different controversial arguments used in this debate can be found in [19, 20]). For future reference, we write down an expression for the sum of inertial and buoyancy forces for a suspension in the gravity field. We have
fJ +fb = r
+W.
- + w-V v+kq~pf -69t
/ (v - w) - Cpg
(16)
g being the gravity acceleration. In conclusion to this subsection, we enumerate main assumptions made when deriving the above constitutive equations. First of all, as has already been indicated, we 1) have utterly ignored the impact of possible random fluctuations of both suspended spheres and ambient fluid, and 2) have addressed only suspensions of low or moderate concentration in flow of which the nonoverlapping property of hard spheres may be justifiably overlooked.
1248
Besides, certain other simplifications have been made in the calculation, which can be summarized as follows. 1. The effective stress tensor was proven in [6] to contain a contribution that depends on the angular velocity of sphere rotation in shear flow. Such a contribution has been ignored, which is permissible to do if the strong inequality holds true [7]
(Pp/Pf )(a2y/vf ) <<1
(17)
y being a characteristic mean shear rate. 2. Only moderately unsteady flows have been considered [10], meaning that another strong inequality is valid that is akin to that in equation (17),
(18) co being a characteristic flow frequency. In practice, inequality (18) is commonly not at all restrictive, except for flows generated by sound and shock waves. A generalization of the above constitutive relations to such wavy highfrequency flows has been recently contemplated in [21 ]. 3. At last, the length scale over which averaged quantities, and in particular the suspension concentration, vary significantly has been assumed to be much larger than radius a of suspended spheres. Moreover, a possible contribution to the mterphase interaction force proportional to the suspension concentration gradient has been completely left out of account, despite the known fact that, in the generalized case, such a contribution must undoubtedly be included in the set of the relevant vectors describing the local situation near the test sphere [22]. 3.2 Other Models
Within the framework of the effective medium model presented, relative suspension viscosity M is defined in equation (4). This expression for M was more than once derived by different theoretical means, and in particular in [23]. It diverges as the volume particle concentration approaches 0.4. Correspondingly, this causes other suspension properties to diverge as well. This divergence is due to the fact that we have allowed the suspended hard spheres to overlap, so that the local volume concentration of particles near the test sphere is not influenced at all by preassignmg this sphere position (curve 5 in Figure 1). This can hardly be looked upon as a reasonable approximation for highly concentrated suspensions. Thus, to expand our model into a region of higher concentrations, we have to consider more sophisticated representations for the
1249
pair distribution function of suspended spheres which prohibit sphere overlapping. Some such representations have been discussed in [6], and also used in [24] to numerically calculate the relative viscosity of suspensions and emulsions as it manifests itself in steady shear flows. First of all, the mere fact that suspended spherical particles do not overlap results in the appearance of a "forbidden" sphere of radius 2a concentric with any given particle into which centers of other particles cannot penetrate. Then, it seems quite natural to approximate the pair distribution function as a step function. This function identically equals the distribution function for one particle (which coincides with the particle number concentration) everywhere outside the forbidden sphere, and it turns to zero inside this sphere. (Such an approximation can be proven to correspond to neglect of steric interactions in groups of particles containing more than two particles.) This pair distribution function corresponds to a local particle volume concentration in the vicinity of any suspended sphere that monotonously increases with the distance r from the sphere center (curve 3 in Figure 1). The local volume concentration equals zero at the sphere surface, and it reaches r as the said distance becomes three times as large as particle radius a. Correspondingly, the effective viscosity and density of the fictitious medium the test sphere is assumed to be immersed into smoothly change from their values specific to the pure ambient fluid to the effective viscosity and mean density of the suspension as a whole [6]. Now, let us approximate the local volume concentration near the test particle as a step fimction which is equal to zero within a spherical layer a < r < z a and r at r > za, 1 < Z < 3. Then we automatically arrive at a model according to which the test particle is separated from a homogeneous fictitious medium by a layer having a thickness (Z-1) a that is filled with pure fluid. The fictitious homogeneous medium is again characterized by the same properties as the suspension, whereas properties inside the concentric layer are those of the ambient fluid. A corresponding step-wise local concentration function is shown at Z = 2 by curve 4 in Figure 1. Models of such a kind were also repeatedly used in numerous works on effective properties, such as viscosity, elastic moduli, effective thermal conductivity and diffusivity, of suspensions, emulsions and composite materials. Let us consider the next logical step in approximating the local concentration variation in the vicinity of the test sphere of a dense suspension. This variation detenmnes also the properties of the fictitious medium this sphere is assumed to be immersed into [6]. This step may involve using different expressions for the pair distribution function that have been developed in the statistical physics from various approximate models of an assemblage of identical hard spheres, such as that by Kirkwood, different versions of Percus and Yevick model, hyperchained
1250
model, etc. The local particle volume concentration distributions that stem from two of such models at different values of mean suspension concentration are plotted in Figure 1 as well.
1
5
0.5
0
4
!
1
2
3
4
r/a
Figure 1. Local particle volume concentration near test sphere surface related to mean concentration as a function of relative distance from this sphere center; 1, 1" - model by Percus and Yevick at # = 0.4712 and 0.2612, respectively; 2, 2" model by Kirkwood at # = 0.484 and 0.2314; 3 - smoothed distribution corresponding to neglect of interactions of more than two spheres; 4 approximation of the last distribution by a stepwise function; 5 - spheres are permitted to overlap. Figure 2 illustrates the concentration dependence of the relative suspension viscosity in conformity with the models used to approximate the pair distribution function in Figure 1. The curves presented in Figure 2 have been calculated by Yendler [24] who also compared them with experimental data on steady-flow suspension viscosity as available by the late seventies. According to his conclusions, function M as defined in equation (4) gives satisfactory results up to ~ 0.2, the smoothed distribution is good enough up to # z 0.3, and the curves that correspond to either Kirkwood or Percus - Yevick model may be used through # z 0.4 - 0.5. Similar conclusions have been driven in [24] with respect to the relative viscosity of emulsions, the viscosity ratio of the phase materials being arbitrary. Supposedly, the model of suspension viscosity in steady flows must become somewhat dubious at very high concentrations. This is largely due to two reasons. First, occasional direct contacts of suspended spheres, or at least the formation of very thin lubrication films of intervening fluid separating pairs of
1251
spheres that come closer together, are hardly avoidable in highly concentrated suspensions, even if there are no interparticle collisions in the generally accepted sense. Secondly, the inner structure of a flowing highly concentrated suspension must be especially sensitive to various structure-ordering processes, up to those of the formation of crystalline patterns. This means that the actual pair distribution function as it establishes itself in dense suspension flow must be essentially different fi'om its approximations resulting from equilibrium statistical models, and it must depend on flow parameters. In particular, one can expect the occurrence of shear-thinning even in flow of a suspension composed of identical hard spheres without molecular colloid interaction, if suspension concentration is large enough.
M 6
.....,
,
,/; 5
4
2
I
0
0.2
__
!
0.4
Figure 2. Relative suspension viscosity as a function of particle concentration by volume according to different approximations for the pair distribution fimction that are illustrated in Figure 1; notation is the same as in Figure 1. Both the aforementioned points cannot in principle be addressed within the framework of the model being developed because we have from the very beginning completely neglected 1) forces that may arise as neighboring particles come to a contact, and also 2) the flow influence on the pair distribution function. This necessitates making use of some other independent, either theoretical or empirical, representation for the relative viscosity of highly concentrated suspension that does not emerge from our theory.
3.3 Provisional Approach Unfortunately, constitutive equations for unsteady suspension flow have been considered so far only for moderately concentrated suspensions where the non-
1252
overlapping property of suspended particles may be ignored [10]. Since the relative suspension viscosity is determined by the formula given in equation (4) which is approximately valid merely at ~b< 0.2 and which diverges at ~ ~ 0.4, this leaves us with a severe problem how to generalize these equations to suspensions of higher concentrations. To do this, we notice that all the constitutive equations as listed above appear to be fully determinate, granted that the only function, that of relative suspension steady-flow viscosity, is known. Correspondingly, we may hope to get a supposedly reasonable evaluation of these equations in unsteady flow of dense suspensions on purely semi-empirical grounds. Namely, we may retain the concept of an effective fictitious medium surrounding any suspended particle, but on condition that another suitable formula for the relative suspension viscosity is used. A simple example is provided by the function suggested in [10] M = (1- ~b)-5/2
(19)
which can be proven to be sufficiently adequate up to # = 0.3. At any rate, this function satisfies the Enstemian limit M oc 1 + 2.5 # at # << 1, and it behaves in a correct manner as # increases. By way of example, we shall illustrate the dependence of the added mass coefficient on both suspension concentration and density ratio of the phase materials for a suspension whose relative viscosity is determined by equation (19). In this case, we obtain from equations (4), (15) and (19)
l+
2
2
(20)
,
,of
For suspensions composed of particles that are lighter than the suspending fluid, and in particular, for suspensions of gaseous bubbles in a liquid, the added mass coefficient is illustrated by curves in Figure 3, a. This coefficient proves to be a rapidly increasing function of suspension concentration, and this function's recline slightly increases as the density ratio grows. For reasons that have been pointed out in Subsection 3.1, this recline is considerably larger than those in other existing models that treat particles in a potential flow. To show this, the added mass coefficient as follows from the Zuber's model [13] k=l
1+2~ 21-~b
(21)
1253
as well as the expansions of this coefficient valid for dilute suspensions k = _1(1 + 2.76~b), 2
1 (1 + 3.32~)
(22)
that were obtained in [15, 16], respectively, are also shown in Figure 3, a. 12
a.
.
.
.
.
.
6
b
4
-
0
,
,
,
,
,
0.2
,
,
0.4
,
q5
0
0.2
0.4
~b
Figure 3. Dependence of added mass coefficient k = ma/mf on ~b for suspensions whose particles are lighter than the ambient fluid (curves 1 on a correspond to tc = pp/pf = 0 , 0.5 and 1, in ascending order), and similar dependence of k ' = mdmp for suspensions with heavier particles (curves on b correspond to K= 1, 2, 5, 10 and 0% in descending order); curve 3 presents the formula by Zuber [13], and curves 4 and 5 give the added mass coefficient for dilute suspensions according to [15] and [ 16], respectively. It is worth noting in this connection that conclusions following from known works on the virtual fluid mass of spheres suspended in a potential flow are sensitive to the shape of the final sphere velocity distribution attained as a result of the action of the impulsive forces that are assumed to generate the sphere acceleration [17,18]. In contrast to this, equation (20) does not altogether involve any assumption of this kind. It holds true in the general case of an arbitrary moderately tmsteady flow that is supposed to satisfy inequalities (17) and (18). If the density of suspended particles is larger than that of the ambient fluid, it seems more judicious to relate the suspension additional mass to that of a sphere, but not to the mass of fluid replaced by the sphere. Dependence of k ' = mdmp on ~b at values of 1r that are larger than unity is illustrated in Figure 3, b.
1254
A couple of important effects that seem to be specific to gas-particle mixtures, as well as to suspensions of very heavy particles in light liquids, are worth mentioning. First, coefficient k" tends to decrease in a dilute or moderately dense suspension when the suspension concentration increases. Secondly, for suspensions of particles in a gas, this coefficient may even become negative in a range of small and moderate concentrations. This means that the acceleration of a single particle in such a suspension is not at all hampered by any virtual mass of the suspension that is carried along by the particle, but rather the surrounding suspension motion facilitates the particle acceleration. Dependence of k" on # at large ~c that illustrates these effects is demonstrated by the curves plotted in Figure 4.
0.1
0
b
r
0.2
Figure 4. Variation of the virtual suspension mass for moderately concentrated suspensions of heavy particles; the curves correspond, in descending order, to ~r = 10, 30, 100 and 1000. This conclusion is by no means connected with our having used approximate equation (19) for the relative suspension viscosity, within the scope of the semiempirical provisional approach as proposed in this subsection. It also holds true if the expression for M is used that results from the theory in [10], without having invoked any additional empirical assumption and that is listed in equation (4). It seems rather tempting to specially address this issue in more detail in future work. No matter how the relative suspension viscosity has been defined, the interphase interaction force as expressed by equation (8) proves in the generalized case to be insuiticient to ensure the dispersed phase momentmn conservation equation (3) to be satisfied for any monotonous suspension concentration distribution. For instance, let us consider the lateral component of this equation in an arbitrary uni-directional steady suspension flow. It is not
1255
difficult to see from equations (9) - (13) that merely the buoyancy contribution to the lateral component of f differs from zero in such a flow, so that the wanted equation component reduces to a very simple form
where gt stands for the lateral component of the acceleration associated with an external field of body forces that act on the suspension. It is easy to see that equation (23) can never be fulfilled except for ~ - 0, if the suspension is not neutrally buoyant, and if gt differs from zero. Hence a conclusion seemingly follows that the steady uni-directional suspension flow under consideration is impossible in the sense that the flow domain must always separate into two regions. As has been already mentioned in Introduction, one of these regions should be filled with pure fluid, where ~ = 0, and equation (23) transforms into an identity. The other region should be occupied with close-packed particles, where equation (23) is irrelevant because certain direct interparticle forces arise which must be explicitly included in the dispersed momentum conservation equation. Obviously, this conclusion is wrong since it contradicts numerous experimental observations, if for no other reason. In practice, the regular force that appears on the left-hand side of equation (23) causes a particle convective flux in the direction of this force. Such a flux modifies any initial spatial distribution of suspended particles. However, it not necessarily leads to complete flow stratification with the formation of a region totally devoid of particles, and of another region containing close-packed particles. Indeed, the particles experience random fluctuations of different physical origins. Particle displacements due to the fluctuations must result in the occurrence of a specific particle migration, and a net particle flux stipulated by migration must oppose the convective flux. Supposedly, the combined action of these two fluxes prevents, or at least hampers, the flow domain separation into the regions mentioned, and so it can result in establishing a steady suspension concentration distribution. It is such a steady distribution that has actually to be dealt with when considering steady flows. If the particles undergo successive displacements that can be regarded as mutually independent, the migration flux may be appropriately described as a diffusion flux down a concentration gradient. It is rather tempting in such a case to replace the dispersed phase momentum conservation equation by a convective diffusion equation for suspended particles, as has been done more than once in various papers on suspension flow. However, as will follow from the subsequent analysis, this can be formally done only for one-dimensional suspension flows, in which the suspension concentration depends on a single coordinate, and where it
1256
is only the corresponding lateral momentum conservation equation component that has to be substituted by the diffusion equation. It is quite clear that a scalar diffusion equation can never be substituted for the vector momentum conservation equation in an arbitrary flow. This means that the impact of the fluctuations on suspension flow must be treated in terms of additional forces or stresses that should be incorporated in the momentum conservation equations, but not in terms of fluxes. This point will be discussed in more detail later on. In what follows, we shall consider 1) particle fluctuations caused by different physical mechanisms, and 2) the origination of additional normal stresses in suspension flow due to the fluctuations. For evident reasons, we start with studying the influence on suspension rheology of thermal Brownian fluctuations, with a mind to afterwards generalize some results obtained from such a study in order to describe effects caused by hydrodynamically induced fluctuations. 4. ASSESSMENT OF THERMAL FLUCTUATIONS There are great many papers that deal with the problem of Brownian motion in a system of many interacting suspended particles, as well as with the elucidation of the influence that this motion is likely to cause on the suspension rheological properties. Recent examples are provided by the beautiful papers by Brady [25, 26] who has successfully addressed the statistics of a crowded assemblage of Brownian spheres when placed into a thermal bath characterized by temperature kT in energy units, and also the Brownian motion effect on the rheological behavior of concentrated suspensions. However, within the framework of the present treatment, we are surely unable to reproduce conclusions of the type of those drawn in [25, 26]. This is due to the fact that we have given up analyzing the influence of both suspension flow and particle fluctuations on the pair distribution function, if for no other reason. Thus, here we confine ourselves to a more modest aim of obtaining simpler conclusions pertaining only to normal stresses caused by the thermal fluctuations. Within the scope of our treatment, the effective suspension viscosity is regarded as a given fimction of suspension concentration, no matter whether it is influenced or not by particle Brownian motion, or either by other physical factors that happen to originate a short order in the suspension flow. As an alternative to ingenious methods used in [25, 26] and in other papers of the same type, we are going to resort to the method that is based on introducing a "thermodynamic" constituent of the interphase interaction force and that was thoroughly substantiated by Batchelor [27]. When applied to suspensions of interacting particles, this method represents a certain generalization of the classical thermodynamic force method devised by Einstein for random migration
1257
of a single particle in an unbounded fluid due to the particle interaction with fluid molecules. As has been stated in [28], the influence of Brownian motion on suspension statistics and rheology consists in smoothing out probability density function for suspended particles, in contrast to a tendency of suspension flow to make these functions non-uniform in their respective phase spaces. This influence is partially revealed through mutually interdependent thermodynamic forces experienced by different particles. They affect mean stresses in suspension flow in two ways. First, they cause an immediate contribution to these stresses. Secondly, they change the probability density functions, and in particular, the pair distribution function, thus implicitly affecting the stresses (and the effective suspension viscosity). Before proceeding further, we wish once again to outline some limitations inherent in the present paper. For reasons given above, below we concentrate only upon the stress contribution due to the first mechanism, and we make use of the analysis earlier forwarded in [27] for dilute suspensions and generalized to concentrated suspensions in [1]. However, it is worth noting that the failure to allow for the changes in probability density functions results in a slight difference of the resultant relative suspension viscosity from its representations found on the basis of more sophisticated schemes, even for suspensions of low and moderate concentrations. Thus, the coefficient at q~2 in the Taylor expansion of function M as defined in equation (4) equals 25/4 = 6.25, whereas the same coefficient was found to be equal to 6.2 for shear suspension flow dominated by intensive Brownian motion [9], and to 7.6 for flow of pure elongation without Brownian motion [8].
4.1 Thermodynamic Force Random migration of a particle in a suspension may be considered as a diffusion-like process if two conditions are satisfied. First, the migration time scale must be much larger than the duration of independent displacement steps made by the particle in succession, and this scale must, in its turn, be large as compared to a viscous time az/vf that characterizes propagation of local disturbances caused by the particle over distances of the order of particle radius a [28]. Secondly, the configuration of neighboring particles surrounding the particle under question must remain approximately invariable during time that it takes for this last particle to accomplish a large number of successive displacement steps, and for the particle diffusivity to attain its asymptotic value [27]. The former condition is needed because the particle motion is nonBrownian over times smaller than the viscous time, due to the transient development of the flow field near the particle. The latter condition is needed to
1258 approximately regard the fluctuating force, that is exerted on the particle by the surrounding fluid and that depends on positions of all the neighboring particles, as a stationary random process. To begin with, we shall briefly reproduce the reasoning employed in [27]. If a suspended particle is at equilibrium when an external force field is applied, the Boltzman distribution must be valid, and the particle Gibbs free energy must be spatially uniform. This means that an external force acting on the particle must be statistically compensated by the thermodynamic force Ft which is equal to the particle chemical potential gradient taken with a negative sign. The thermodynamic forces produce a particle convective flux that is precisely equal to the actual diffusion flux. Furthermore, the motion of the particle due to the external forces and to its diffusion caused by thermal fluctuations can be justifiably presumed to result in additive particle displacements. For this reason, the particle diffusion flux must be determined by exactly the same thermodynamic forces even under non-equilibrium conditions where there are no external force [27]. By its very definition, the thermodynamic force in an isothermal flow can be expressed in a general form as
(oPPl
F, = -k---8-;, J p,r
Vrt -10PPl
:P J n,T
Vp
(24)
where p P is the particle chemical potential. In accordance with equation (24), the thermodynamic force consists of two parts, the second one representing the buoyancy force. Since the buoyancy force has been already accounted of in equations (13) and (16), there is no need to retain it here. Hence, we are left with (25)
On p,T
p,T
As shown in [27], a similar force
F/=-
On JP, f
~, Oqk ) p,T
V~ = _n_n_Ft no
(26)
acts also upon each molecule of the ambient fluid, where flf and no are the chemical potential and the number concentration of the molecules. It follows
1259
from equation (26) that the thermodynamic forces act simultaneously on both phases of the suspension, and moreover, that their sum is zero for any suspension volume, that is, nF t + noF [ - 0
(27)
Equation (27) proves that the thermodynamic force per unit volume, ft = nFt, has to be regarded as an additional constituent of the interphase interaction force. In fact, equation (27) is a manifestation of the Gibbs - Duhem relation written out for colloid particles dissolved in a fluid. The chemical potential of hard spheres undergoing the Brownian motion can be calculated with the help of the standard methods of statistical mechanics. The calculation is trivial for very dilute suspensions of non-interacting spheres where ~tp coincides with the chemical potential of an ideal gas, in which case ~tp = bt,p + kT(ln 0 -~b)
(28)
the first term on the left standing for the inner chemical potential of a sphere. If a suspension is dilute, but the mterparticle interaction is taken into account, the application of the well-known group expansion technique seems to be natural. The latter leads to an expression for the chemical potential in the form of a series in powers of q~ that resembles the standard virial expansion~ Each term of this series is associated with the collective interaction between the particles being members of various possible groups composed of the fixed numbers of particles. Allowance for merely pair interactions conforms to retaining also the second virial coefficient of the expansion. This gives rise to the term of 8q~kT in the series for ktp, which has been previously calculated in [27]. The group expansion technique is unlikely to be of much use, however, when applied to crowded assemblages of suspended spheres, since this technique does not in fact give a tangible opporttmity to advance in a region of large concentrations. In such a case, the calculation is handicapped by the fact that no reliable rigorous statistical physical theory of dense gases and liquids has been elaborated to date, so that an approximate model has to be used. It seems sensible to make use of one of versions of the Pereus- Yevich theory of liquids because it makes it possible to obtain analytical results for assemblages composed of hard spheres. Following [1 ], we employ the version of this theory worked out by Carnahan and Starling [29], which gives the following equation of state for a dense gas of identical rigid spheres
1260
PV
=
NkTG(qk),
G(qk) = 1 + + ~2 _ ~3
(29)
3
where P is the osmotic pressure, and V is the volume occupied by the system of N rigid spheres. Roughly speaking, equation (29) brings to light corrections to the classical equation of state for an ideal gas that are due to sterie interactions between spheres having fufite volume of their own. It should be pointed out once again, however, that this equation fails to account for the formation of a crystalline phase in dense sphere assemblages. This failure calls in question applicability of this equation of state to suspensions of high concentrations that are close enough to a hypothetical concentration associated with the close-packed state. Equation (29) serves as a sound basis while evaluating the particle chemical potential and calculating the thermodynamic force for spheres suspended in a fluid that plays the role of a solvent. The difference of the wanted chemical potential from that of the same spheres forming a dilute system can be quite straightforwardly written down as follows:
(30)
A,u p = - k T dln QN
ON where QN is the configurational integral of the spheres divided by V N. Using equation (29) and an alternative representation of the equation of state in terms of the derivative of the configurational integral over volume yields O I n ( V N QN)__~___P _ N G(~) cTV
kT
(31)
V
Solving this equation results in [1] ar
lnQN = - N d 0
(32)
r =- fG(~)- 1 ON
p,T,No
,I
o
-~
G(O)-I ( c7 _/ d ~k- N
~
p , T ,N o
and this determines the chemical potential difference as defined by equation (30).
1261
Note that the differentiation is performed in equation (32) under the condition that not only p and T, but also the total number No of solvent molecules are maintained constant. By allowing for the expression of the osmotic pressure function as shown in equation (29) and for evident relations aN r
cr N + cro N o '
(O-N-/
= ~(1-~) p,T, No
(33)
N
where o- and o0 are the specific volumes of one particle and one molecule, respectively, we derive from equation (32) cTlnQN) ON
_ 8 - 50 - - (---i~ 0 p,r, No _~)2
(34)
Hence and from equations (28) and (30) follows the resultant formula /u p - 1.1,p + k T F ( q k ) ,
8 - 5______L~
F(~b) - l n 0 - r + (1 - ~)e ~b
(35)
which finally determines the thermodynamic force as defined by equation (25). The third term on the right-hand side of equation (35) is due to the steric interaction of hard spheres having a finite volume. Determination of the thermodynamic constituent of the mterphase interaction force requires thermodynamic force Ft be multiplied by the particle number concentration that is expressible, in its turn, as ~b divided by o-. Thus, this constituent can now be formulated in the following form: r, = , 1~, _- --0 l~, = - kZ 0 a ( ( r v 0 cr cr d~b
(36)
Equation (36) brings to completion the calculation of the thermodynamic mterphase interaction force constituent acting in a non-uniform Brownian suspension. The convective flux caused by this force constituent must identically equal the diffusive flux down a suspension concentration gradient, and so it can be substituted for this diffusive flux. To do this, the thermodynamic force constituent has to be incorporated into the total interphase interaction force alongside the other parts of this force listed in equation (8).
1262 4.2 Normal Stresses Due to Thermal Fluctuations
Next, we shall consider an alternative representation of the thermodynamic force that act on all particles in a unit suspension volume. This representation proves to very useful in our subsequent analysis. To start with, we reduce equation (36) to another form by means of introducing a new function of suspension concentration, L(~, that satisfies the following differential equation:
d
=
dF(~b) a---g-
(37)
where F(r is the function defined in equation (35). Integrating equation (37) at an evident initial condition yields
-
2 In(l-O) +3 1-20 r 1-0
+
1 15-8~-~ 2 2 (1-~) 2
-
(38)
Using equation (38) enables us to reformulate equation (36) as -f t =
kT
d
cr d e [r L(r
V~b- - V H ,
H - kTo.r L(~)
(39)
It is clear that the scalar quantity H may be identified with the effective isotropic normal stress that is caused by the particle thermal fluctuations and that manifest itself in mean flow of the dispersed phase. Physically, it describes a momentmn flux density produced by the particle thermal fluctuations, and by its very definition, such a density must be interpreted as a stress. Instead of adding the thermodynamic constituent to interphase interaction force (8) to describe the impact of Brownian motion on suspension flow, we can alternatively include the isotropic particulate stress H into momentum conservation equation (3) of the dispersed phase. Simultaneously, we must include a component of - FI to mean interstitial pressure p involved in accordance with equation (5) in the defirfition of the effective stresses that appear in momentum conservation equation (2) for the continuous phase. This amounts to introducing the following stress constituents that are caused by Brownian motion and that must be added to the stresses acting in a suspension flow without fluctuations: c~f=HI,
c~=-HI
(40)
1263
/2 G
L
0
0.2
0.4
O
Figure 5. Functions of suspension concentration that determine the osmotic pressure of a pseudo-gas of rigid spheres and the normal stress that affects dispersed phase flow of a suspension of the same spheres without collisions and that is due to thermal fluctuations. It is significant that function L(r is different from the osmotic pressure function, G(~, introduced in equation (29), as follows from the curves plotted in Figure 5. Correspondingly, normal particulate stress H that arises in flow of a suspension of collisionless spheres differs from the osmotic pressure for a pseudo-gas of the same hard spheres which are presumed to exchange momenttma and energy by direct collisions. Apparently, this difference has to be attributed to the distinction between the spheres that form a pseudo-gas and the spheres that are suspended in and interact through a fluid. As evidenced by our calculation, this distinction proves to be significant. The presence of the fluid that plays the role of a solvent considerably affects statistical properties of the spheres that resembles molecules of a solute, rather than molecules of a dense gas. It is hardly surprising therefore that the thermodynamic forces as calculated for a pseudo-gas and those for a solution of hard spheres happen to be different as well. Since kT identically equals the mean kinetic energy associated with one degree of freedom of a single sphere, the expression of H in equation (39) can be rewritten in terms of the variance of any i-th component of the particle velocity. (41) Here w' is understood as a particle fluctuation velocity that comes about as a result of isotropie thermal fluctuations and that actually represents the difference
1264
of the actual random velocity of a particle from the averaged velocity of the dispersed phase. 4.3 Some Generalizations
Equations (40) and (41) can in principle be used on semi-empirical grounds to describe normal stresses originated by particle fluctuations of another physical origin. With this purpose in view, they are be reformulated in the form
8I = fi, 8p =-fi,
fi- ppr
(42)
where an asterisk denotes the operation of diadic multiplication. In the case of isotropic Brownian motion, (w',w') -= (w~2) I, and the formulae in equation (42) reduce to those in equations (40) and (41). Equation (42) results actually from a quite plausible hypothesis that the stress tensor originated by particle fluctuations must be proportional to the only tensor available that characterizes the fluctuation intensity and that is quadratic in fluctuation velocity components. Moreover, there is altogether no reasons to expect that the scalar coefficient as defined in equation (42) is sensitive to the cause that drives particles to fluctuate, provided that the particles 1) behave themselves as statistically independent entities, and 2) do not come in direct contacts with one another in consequence of their fluctuating motion. Thus, we can supposedly retain the same value of this coefficient as that fotmd when investigating thermal fluctuations. However speculative, the hypothesis under discussion seems to be credible enough to offer a real opportunity of evaluating normal stresses that are originated by hydrodynamically induced fluctuations of statistically independent collisionless particles. In conclusion to this section, we are going to propose, also on semi-empirical grounds, a generalization of the above results to flows of very dense suspensions whose concentration approaches that of close packing. Formally, equation (40) can be rewritten in terms of a so-called Enskog factor Z that is extensively used in the statistical physics
I-l=nmp[1+ 4~Z(~)] (w/2),
Z(~)- L(~)- 1 4~
(43)
Roughly speaking, the physical meaning of this equation is as follows. Frequency of interactions increases in a crowded particulate assemblage, and momentum transport inside hard spheres is carried with a practically infinite rate. The Enskog factor describes the influence that these effects are anticipated to produce on averaged momentum transfer in a concentrated suspension.
1265
In flow of highly concentrated suspensions, particles are likely to come in frequent contacts one with another, and the implicit mechanism of interparticle exchange through intervening fluid may be expected to give way to a direct exchange mechanism carried out by particle contacts. It is natural to expect in such a case that function L(r must gradually tend to osmotic pressure function G(~) defmed in equation (29) as the suspension concentration increases. An equation for the corresponding Enskog factor then reads Z(~b) = G(~b)- 1 _ 1-0.5_______~ 4r (1-~) 3
(44)
Making a precise choice between these extremes requires a careful examination of both particle contacts and long-range hydrodynamic interactions in groups of many particles, and so presents a formidable task. As far as the authors know, this problem has been rigorously addresses only in connection with the dynamics of dilute gas-solid mixtures by Koch [30]. Fortunately enough, the difference between functions G(~) and L(~) as evidenced by Figure 5 is not too large, especially so for dilute and moderately concentrated suspensions. For this reason, there is no urgent practical need to discriminate between these functions. As has already been mentioned, the model by Carnahan and Starling [29] can hardly be applied to the description of the momentum transport augmentation in flow of a highly concentrated suspensions whose concentration approaches that of close-packed state. Therefore, in order to get a rough idea about flow of such a suspension, we can make use of the expression for 1-I listed in equation (43), in which the Enskog factor is empirically approximated from a suitable model of very dense gases. In particular, we can employ the approximate "geometrical" model of dense gases as developed by Enskog [31]. The Enskog factor is then expressed as
4~
, GE(~)= 1 -
(45)
where ~ is understood as a maximal particle concentration by vohune that is attributed to a hypothetical close-packed state. The exact value of this concentration is unknown, and it has to be regarded as an empirical constant. When formulated in terms of the Enskog factor, the expression for the normal stress tensor as advanced in equation (42) reads
1266
fI -
nmp [1 + 4r
w')= pp r [1 + 4r
(w',w')
(46)
Depending on concentration of a given suspension flow, different relationships may be used for the Enskog factor, and in particular, those listed in equations (43) - (45). In what follows, we shall check the equations mentioned by applying them to the determination of particle distribution in a simple steady Couette flow of a suspension of identical neutrally buoyant spherical particles between concentric rotating cylinders. If suspension concentration is sufficiently high for contacts between suspended particles be essential, equations (44) or (45) has to be used for the Enskog factor, instead of equation (43). In such a case, as before, equation (46) describes the normal stresses that act in flow of the dispersed phase. However, there is altogether no proof in this case that fluctuations cause the contributions to the total stresses affecting the continuous phase flow that must precisely equal the corresponding stresses as occur in the dispersed phase flow, when being taken with a negative sign. Therefore, we are obliged to confine ourselves to calculating only the dispersed phase normal stresses, whereas the said contributions to the continuous phase stresses remain undetermined. It should be emphasized in conclusion that both methods, which have been developed above to describe the normal stresses produced by fluctuations of statistically independent collisionless particles, are completely equivalent. This means that either the thermodynamic constituent of interphase interaction or the normal stresses as defined in equation (42) can be introduced into the momentum conservation equations for flow of the suspension phases with equal success. However, it seems wrong to incorporate both the said stresses and the thermodynamic contribution to mterphase interaction simultaneously. Apart from these stresses, Reynolds-like stresses may appear as a result of averaging of the left-hand side of equations (2) and (3), as it has been shown in [32] when studying stability of fluidized beds. 5. MODELING SHEAR-INDUCED FLUCTUATIONS Particles of sheared suspensions are well known to exhibit random displacements both across and in the direction of flow streamlines, due to relative motion of particulate layers that move in parallel to one another with different velocities. These displacements are illustrated in Figure 6. They produce particle self-diffusion in shear flow of macroscopically uniform suspensions, and also they cause the particles to migrate down a suspension concentration gradient, if any. Shear-induced particle self-diffusion investigated in [33], apparently for the first time. After that, it was studied in more detail in
1267
[34] and in some other papers. Shear-induced mutual diffusion of particles was proven to induce the flow stratification, meaning that a non-tmiform particle distribution develops in a shear flow as a result of the combined action of external forces and particle mutual diffusion [35]. There are known many attempts to work out a comprehensive and reliable model to treat suspension shear flows in general, and to determine particle distributions specific to various concrete types of shear flow in particular. Representative examples illustrating different trends of thought can be found in [36, 37]. The authors of [36] developed a fluid dynamic model of shear suspension flow, in which it was suggested to describe the shear-induced migration in terms of a special particle diffusion equation introduced on purely empirical grounds. In contrast to this, the authors of [37] deduced the steady particle distribution in a pressure-driven one-dimensional flow in a channel from the requirement that the total normal stress be constant in directions normal to the mean flow. The model stemming from this requirement was well supported by their numerical simulations performed on the basis of Stokesian dynamics technique, and these simulations were, in their rum, in qualitative agreement with experiment. While setting aside a more detailed discussion of these issues until later, we focus our attention on evaluating the normal stresses that are due to shearinduced fluctuations within the framework of the approach being developed. According to equation (46), this means that we have to calculate the tensor of averaged particle velocity products, or particle velocity variance tensor (w'*w'), in shear suspension flow as a function of suspension concentration, shear rates, and particle radius. This has been done in [38], and we reproduce the main results obtained in that paper. 5.1 Particle Velocity Variance Tensor As follows from a simple consideration based on the theory of dimensionality, and is also confimaed by conclusions drawn in [33, 34], random particle velocity components must scale with ya, y standing for a characteristic shear rate value. This proves that the particle velocity variance tensor must be quadratic in the spatial derivatives of the dispersed phase mean velocity. Moreover, this tensor must be independent of rotation of the suspension as a whole. This means that its components may depend on those combinations of such derivatives which have a bearing on the deformation rates in the dispersed phase, but not on combinations that describe rigid-body rotation of this phase. As such, the desired tensor is solely dependent on tensor ew that describes strain rates for the dispersed phase flow. In other words, the wanted tensor depends exclusively on the symmetric part of the spatial derivatives tensor for the true deformation rates mentioned
1268
above. To obtain a concise expression for this particle velocity variance tensor, we make use of some tensor covariance methods as characteristic of the rational mechanics. The most general rational mechanical representation that satisfies both the aforementioned requirements is of the form ( w ' * w ' ) = A e w .e w + BI2(ew)l , I:z(ew)=(1/2)ew'ew
(47)
/2 standing for the second tensor mvariant, and A and B being scalar coefficients independent of the dispersed phase mean velocity field. To find a relation between A and B, and thereby to reduce the uncertainty inherent in equation (47) to a single unknown scalar coefficient, an analysis of a simple shear flow sketched in Figure 6 appears to be helpful. y
1
ly
wx--]~
Figure 6. A sketch of particle interaction in shear flow; explanation in the text. Consider the situation in the vicinity of a spherical particle 1 of a dilute suspension in a reference frame having its origin at the particle center. Another sphere 2 that comes from infinity and that passes near the first sphere undergoes a displacement in a plane containing both the second sphere trajectory and the line connecting the sphere centers, as shown in Figure 6. The maximal displacement occurs in the symmetry plane that is normal to the above indicated plane, and at the same time, contains the first sphere center. The theory of similarity suggests this displacement to be fully determined by an original aiming distance r (the far upstream coordinate of the second sphere center) scaled with particle radius a. Let us define angle rp as the angle formed by the direction of shear (y-axis on Figure 6) and a line that is drawn fi'om the reference frame origin (the point in which the first sphere center is situated) to the point of maximal displacement in
1269
the plane of symmetry. It appears pretty obvious to assume from a simple syrmnetry reasoning that the maximal displacement is independent of the said angle, provided the relative velocity of the spheres being the same. Then this displacement length is expressible as l = being the relative aiming distance. Displacement components along the y- and z-axes in the symmetry plane equal cos (p and = 1 sin qg, respectively, and the initial relative velocity of the particle centers is u = 7'a cos (p. Next, let us focus attention on flow of a macroscopically homogeneous suspension (~ = const) under a condition of uniform shear (7'-- const). In order to evaluate averaged moduli of the first sphere's displacement components as caused by interactions of this sphere with all the other spheres whose centers lie within a cylinder r < R = a R, and which happen to intersect the symmetry plane for a unit time, we need the help of the following integrals:
1,(r/a),r/a
ly=l
n'/2
R
lz
R,
{lyoclz } ~dq9 fr dr(ycos~)Lsinqgjoc Ic~176 {re~4} fl * ( ~ 2 d na4ocr 1/2 0
(48)
0
where the integral that stands in the parentheses is independent of any dimensional parameters. Since the averaged displacements along the y- and z-axes are attributed to a unit time, they have to be regarded as convenient measures of corresponding velocity components for shear-induced fluctuations. Note that a result similar to that in equation (48) could readily be obtained from the theory of dirnensionality because there is the only quantity of velocity dirnensionality, y a, that can be constructed on the basis of available dimensional parameters. Evaluation of the averaged displacement components as based on equation (48) bears upon dilute suspension flow in which the chosen sphere's interaction frequency with all spheres that are passing by is in fact presumed to be linearly proportional to both suspension concentration and shear rate. This means that particle fluctuation velocity components in dilute suspensions is actually calculated by taking corresponding mean displacements of the order of a, which occur on the average as a result of a single pair interaction, and then by dividing them by the mean time that elapses between successive interactions of the chosen sphere with its neighbors. By its very definition, the last time is equal to the inverted pair interaction frequency. The indicated physical meaning of equation (48) allows us to generalize this equation to concentrated suspensions. In doing so, we avail ourselves of a semiempirical approach that has already been mentioned in brief in the preceding
1270
section. In a dense gas, as the particle volume fraction grows while the particle number concentration is held constant, the collision frequency is known to increase faster than is required by the last proportionality in equation (48). This increase in collision frequency occurs due to the forcing the particles into a confined space, and it is usually described with the help of the Enskog factor. Although relative particle displacements in a sheared concentrated suspension are affected by long-range hydrodynamic interactions involving many particles, main contributions to the displacements may again be viewed approximately as the result of particle pair interactions. As a first approximation, we hereupon conceive elementary particle displacements as taking place in consequence of two particles occasionally drawing close to each other, the influence of the other particles being assumed to cause a secondary effect on pair interactions. This amounts to viewing the complicated multiparticle interactions involving the chosen sphere in a sheared concentrated suspension as a sequence of certain effective pair interactions of this sphere with its neighboring spheres. Within the scope of such an approximation, the pair interaction frequency in a suspensions may be replaced by the collision frequency for the corresponding pseudo-gase composed of hard spheres. Then the same factor Z(~) has to be used to account for the increase in pair interaction frequency in concentrated suspensions due to the particles being crowded in a confined space. Thus, the last quantity in equation (48) has to be multiplied by the Enskog factor. Obviously, components of the particle velocity variance tensor (47) represent, by their very definition, averaged products of displacement velocity components. The last velocity components equal the displacements per unit time, equation (48), multiplied by the Enskog factor. Hence, we arrive at a general estimate (Wx2) oc (w~,2 ) oC(Wz2 ) oc~ 2 Z2(~) 2' 2a 2
(49)
If the suspension is macroscopically inhomogeneous and (or) its shear flow is not uniform, but length scale L characteristic of either the inhomogeneity or the non-uniformity considerably exceeds particle radius a, the above estimate must be still valid. Corresponding corrections can be shown to be of the order of a/L by expanding n and ?, in equation (48) into their Taylor series. Further, when having applied equation (47) to the plane shear flow illustrated by Figure 6, we obtain ,
1
2
(50)
1271
whence, taking in account also equation (48),
A+B-(zc/ 2 B 4) 1/2
~2 = 4
(51)
Using equations (49) and (51) enables us to reduce equation (47) to the following form:
(w,,w,)=C~2x2(~)a2[(rc2/4-1)ew .ew +I2(ew)]
(52)
where C stands for an unknown coefficient having the order of unity. This coefficient may slightly depend on the suspension concentration, due to the semiempirical nature of the model used to evaluate the shear-induced particle fluctuation velocity, as well as owing to the fact that multiparticle interactions have not been considered in an explicit form. A more sophisticated theory is clearly needed to reveal such a dependence, in which multiparticle interactions would not be replaced by a hypothetical sequence of pair interactions as far as their impact on the said fluctuation velocity is concerned. It is worthwhile to note that, by using simple rational mechanical considerations, we have arrived at a rather informative inference pertaining to the longitudinal fluctuations in the flow direction (that is, along the x-axis), in spite of the fact that these fluctuations did not enter our reasoning in a direct explicit way. Namely, as follows from equations (50) and (52), the velocity variance for these fluctuations must be precisely equal to that for fluctuations in the direction of shear (that is, along the y-axis). As a reasonable first approximation, fluctuations of different particles can be regarded as statistically independent. Then, on accord with the arguments forwarded in the end of the preceding section, the normal stresses generated by shear-induced fluctuations can be evaluated from equation (46). 5.2 Particle Distribution in Rotational Couette Flow
To provide an application example, and also with a view in mind to compare theoretical predictions with available experimental data, we now consider steady flow of a neutrally buoyant suspension between concentric rotating cylinders. The interphase interaction force altogether disappears in the flow trader consideration. On the other hand, equation (46) completely determines the normal stresses produced by particle fluctuations, if the particle velocity variance tensor is known. In the generalized case where both shear-induced and thermal fluctuations are relevant, this tensor can be presented as a sum of corresponding
1272
variance tensors for the shear-induced fluctuations and for the thermal fluctuations when taken alone. Such a superposition approximation corresponds to an assumption that fluctuations of the types mentioned are mutually independent, which appears to be quite credible in view of fluctuations of those types being originated by different physical causes. The radial components of momentum conservation equation (2) and (3), to which stresses due to thermal and shear-reduced fluctuations are added, then reduce to
r 2 dr r2r = 0 ,
r=pfM(~)y,
Y=drr
(53) d---~ r162
+CO
(~)(ya) 2
=0
where r is understood as the averaged shear stress in the flow under consideration, w is the mean suspension velocity which coincides with the mean velocities of both suspension phases, and C is a new constant that differs from C in equation (52). Solving the first equation (53) yields F
~ Y - M(r
'
R1 < r < R2
(54)
where R I and /?2 are radii of the internal and the external cylinders, respectively, and F is an integration constant that can obviously be related to the torque applied to the rotating cylinder. Next, we have from the second equation (53) and equation (54)
~b[1 + 4~bZ(~b)] ~pp+ C M(~b) (Fa)2 ( - ~ )
= const
(55)
the constant to be determined from, say, a condition stipulating that the suspension concentration averaged over the gap between the cylinders is given. It is not difficult to see from equation (55) that Brownian motion makes for the flattening of particle distribution in the gap, a distribution which otherwise may be rather steep. If we confine ourselves to non-Brownian suspensions, the term with kT/mp must be dropped out, so that constant C in fact disappears from equation (55) as well. As a result, we arrive at the following equation that determines the particle distribution within the gap:
1273
R 2 - - t F{~3 [I + 4~Z(~)]} 1/4
(56)
M(i~ Z(#~)) 2
Here q~ is a dimensionless constant that is a single-valued fimction of the total contents of particles within the gap. Because constant C has vanished from the analysis, equation (56) does not involve adjustable parameters at all, provided ftmctions M ( # and Z(~ are given. In our calculation, fimction M ( ~ as defined in equation (19) and the Enkog factor as determined by equations (44) and (45) have been employed. Solutions to equation (56) at different values of constant ~F are plotted in Figure 7. Experimental dots obtained in [38] are shown in Figure 7 as well.
1
'
1///oI/~'I'/__ ''/'
1
,
,
~_
a
0.8
.'
O
0.6
"
0.6
9
aa
9
0.4 0.2
0.2
9
0
t 0.4
I ~0
04 f ~ 0.4
I q}
0.6
0.2I 4/ 0.2
I
0
0.6
Figure 7. Concentration profiles in rotational Couette flow, equation (56), at ~ = 0.65 and with Z(~ defined from equations (44) and (45) (a and b, respectively), and experimental data by Phillips et al. [36]; a, curves 1 - 7 correspond to q~ = 1, 0.875, 0.75, 0.625, 0.5, 0.375 and 0.25, respectively; b, curves 1 - 4 correspond to qJ = 0.7, 0.6, 0.5 and 0.4, respectively, and a thinner line presents the same curve as curve 5 shown in a. The agreement between theoretical curves and experimental dots is not bad at all. Nevertheless, the dots apparently corresponds to a dependence of # on relative coordinate r/R2 which is more gently sloping than the dependence given by the theoretical curves. The dots are all the more so gently sloping in a range of high concentrations. These curves conform to the experimental data somewhat better, if allowance is made for the singularity of osmotic pressure at # - ~ , in conformity with equation (45). Comparison of the curves plotted in Figure 7, a with those presented in Figure 7, b proves, however, this singularity to be
1274 important only at concentrations which are sufficiently close to that of close packing. It is worth emphasizing once again that no free adjustable parameters have been used when drawing any curve in Figure 7, but for close-packed state concentration ~,. The agreement of theoretical predictions with experimental evidence as witnessed by Figure 7 can therefore be regarded as a weighty agreement in favor of the model developed. This notwithstanding, this agreemem is not altogether as conclusive as it might initially seem because of two approximations used. First, equation (19) has been assumed adequate to represent the relative suspension viscosity, and secondly, equations (44) or (45) have been chosen to represent the Enskog factor. One might argue that viscosity must increase infinitely as the particle volume fraction tends to its close-packed state value, so that model predictions in Figure 7 may be somewhat fallacious. At any rate, Brady [26] has obtained a theoretical limit for the relative suspension viscosity of Brownian according to which M(r --} (1 - #~)-2 at ---} ~ suspensions that proves to excellently agree with experimental data by many researchers. For instance, the well-known Krieger's formula [41] gives
1-
~m/_1.82
(57)
and the exponent is this formula differs quite negligibly from the exponent of 2 appearing in the Brady's theoretical result. Moreover, the osmotic pressure function and the Enskog factor must also diverge in the indicated limit, so that singularities of M ( ~ and Z(~ are likely to cancel each other out. Moreover, comparison of the curves plotted in parts a and b of Figure 7 evidences that the effect of divergencies at ~ ~ ~ is not as high as it might have been anticipated.
5.3 Diffusional Representation The second equation (53) that stems from the radial component of the vector momentmn conservation equation for the dispersed phase can be reduced to a diffusion-like scalar equation. In is not difficult to show that, to arrive at such a diffusion-like equation, we have to multiply the second equation (53) by the effective particle mobility for Stokesian sphere in a concentrated suspension, and then to multiply it also by particle mass rap. The particle mobility is expressible in terms the hindered settling function, K(~, that represents the ratio of particle sedimentation velocity in a concentrated suspension to the terminal velocity of a single particle in an unbounded fluid [27, 40]. Namely, it equals the mobility of a
1275
single particle, (6z,ufa) ~ multiplied by K(O). The hindered settling function is related to the relative suspension viscosity by an equation [40, 41 ]
K(r - (1 -
(58)
~)2
M(#) The resultant diffusion-like equation that replaces the second equation (53) looks as follows (it differs by a factor of (1- ~2 from the equation cited in [38]):
(1-#)2 d f M(qk) dr 0[1+4r
I
kT 6zc-pfa
+ C~b22,2 (~b) (7' a) 2 mp ]1 =0 6n'pfa
1!
(59)
This equation can be rewritten in a more familiar form, that is,
(DR + D,h)dq~ dr =0
(60)
where effective particle diffusivities caused by the thermal and the shear-induced fluctuations are introduced by means of the following relationships:
DB
= #)_____~2 (1d {q~[1+ 4#Z(q~)]} D~
(61)
M(~) dq~
Ds~ _ ( 1 - ~ ) 2
d - M(~--------~d~ {#3 Z2 (~)[1 + 4r162 } Ds~
(62)
where the following dimensional quantities are introduced
kT (ya)2mp 2C 7'2a 4 D~ - 6 rc,uf a ' Ds~ = C 6 rc/.tf a = --9 K
vf
(63)
Equation (60) involves two terms that must be interpreted as net diffusion fluxes produced by the thermal and the shear-induced fluctuations. In equation (63), DR~ is the familiar coefficient of Brownian diffusion for a single fine sphere immersed into an unbounded fluid, whereas D,h ~ plays the role of a characteristic scale for the coefficient of particle mutual diffusion caused by shear-induced
1276
fluctuations. Concentration dependence for the coefficients of mutual Brownian diffusion and of mutual shear-reduced diffusion is described by equations (61) and (62), respectively. The dependence of coefficient of mutual Brownian diffusion on concentration is illustrated in Figure 8, the curves plotted in this figure corresponding to M ( ~ taken from either equation (19) or equation (57) and to different formulae for Z(r In the dilute limit ~b ~ 0, this coefficient tends to DR~ and there is altogether no difference between coefficients of Brownian mutual diffusion and self-diffusion. However, such a difference arises and becomes more and more pronounced as concentration increases. Suffice it to say that the long-time coefficient of self-diffusion noticeably decreases with concentration [40, 42], whereas the mutual diffusion coefficient is proven by the curves in Figure 8 to substantially increase with concentration, in dilute and moderately concentrated suspensions. ,
,
,,
,
,
2 o
2*
.
.
.
.
0.2
.
_
r
.
0.4
. . ,
,
0.6
Figure 8 Relative coefficient of mutual Brownian diffusion as a function of particle volume fraction; curves 1, 2 - M ( ~ is taken from equation (19), and Z(~) is defined in accordance with equations (43) and (44), respectively; curves 1", 2" - M ( ~ is determined by the Krieger's formula (57), and 2'(~ is again taken from either equation (43) or equation (44); curve 3 presents the Einstein's formula (64) for dilute suspensions. The Batchelor's formula [27] for the Brownian mutual diffusivity
DB/D~
= 1 + 1.45~
(64)
is illustrated by curve 3 in Figure 8. This formula is well corroborated by experimental evidence. Because curve 1 goes sufficiently close to curve 3, it may be said that curve 1 is also supported by observations.
1277
Curves 1 and 1" conform to using formula (38) when expressing the nonnal stress in Brownian suspension flow. They essentially deviate from curves 2 and 2" that correspond to the usage in this context of the osmotic pressure function, equation (29). Roughly speaking, curves 2 and 2* stem from the model developed in [25], according to which the assemblage of suspended particles is regarded as a pseudo-gas. In contrast to this, curves 1 and 1" arise out of our modeling this assemblage like a system of molecules of a solute. The mere fact that it is curve 1, but by no means curve 2, that satisfactorily agrees with experimental data obtained for dilute suspensions serves as an implicit argument in favor of our model. As the suspension concentration increases beyond the range of moderately concentrated suspensions, the mutual diffusion coefficient begins to decrease, and it tends to zero as ~b~ ~ . This is undoubtedly due to an infinite increase in viscosity in compliance with formula (57), and to the corresponding decrease of particle mobility as suspension concentration grows. Since a relative role played by direct particle contacts in momentum transfer must grow with suspension concentration, one can justifiably expect that the actual concentration dependence of the mutual Brownian diffusivity will gradually undergo a smooth transition from curve 1 to curve 2* as concentration increases. It is worthwhile to mention in conclusion that a similar analysis can be applied to calculation of the Brownian mutual diffusivity in colloids whose particles exhibit molecular interactions of different origin [43], and also electrostatic interaction [44]. The theoretical results derived in the papers cited seem to be in a satisfactory agreement with experimental data. In principle, this encouraging fact makes it possible to evaluate the particle diffusion flux, and thereby to approach a difficult task of developing a rheological model for colloids of interacting particles. Figure 9 presents similar results for the dimensionless coefficient of mutual diffusion of particles caused by shear-reduced fluctuations. This coefficient appears to be rather negligible in dilute suspensions. However, it rapidly increases with concentration, so that the particle migration generated by these fluctuations becomes essential for highly concentrated suspensions. In the case of shear-reduced diffusion, the distinction between the mutual diffusivity and the self-diffusivity is even more drastic than that for Brownian diffusion, since it touches upon the basic scaling of these coefficients. Indeed, in conformity with equation (63), the coefficient of mutual shear-induced diffusion scales with x 7' 2~vf , x being the density ratio. In contrast to this, the selfdiffusion coefficient is well known to scale with ?'a2 [33, 34].
1278
It has been noted already that there exist a number of attempts to model particle distribution in shear-driven flows with the help of an equivalent diffusion equation. Most commonly such attempts are based on an unwarrantable assumption that the diffusion flux of suspended particles down a concentration gradient is governed by the self-diffusion. Such an assumption is evidently wrong, and without doubt, it is the mutual diffusion coefficient that must be used in the indicated context. Moreover, in view of a so substantial difference between these coefficients, conclusion drawn from models employing the selfdiffusion coefficient can be anticipated to be inadequate, both quantitatively and qualitatively. A more detailed discussion of some shortcomings specific to such models can be found in [38]. 102
.
.
.
.
.
'. . . .
,
2 ~t
o
ol
10-2
10-4
|
0
I
012
I
0.4
i
,
0)6
Figure 9. Concentration dependence of mutual coefficient of shear-induced particle diffusion; notation is the same as in Figure 8. On the whole, the method of attacking shear-induced particle migration as developed in this section provides a succinct springboard needed to study both particle distribution in shear-driven suspension flows of different types and influence of this distribution on the flow hydraulic characteristics. This method is based on the necessary and quite obvious requirement that the dispersed phase normal stress must satisfy the components of the dispersed momentum conservation equation in directions perpendicular to the mean motion, without invoking an independently stated diffusion equation. In simpler cases, like the one related to the rotational Couette flow of a neutrally buoyant suspension that has been considered above, this requirement reduces to the condition of the particulate stress being constant in the said directions. As such, our model has phenomenological similarities with the model proposed in [37], although the normal stress calculation is different.
1279
6. MODELING PSEUDOTURBULENT FLUCTUATIONS Apart from shear-induced fluctuations, suspended particles are involved in a hydrodynamically induced chaotic fluctuating motion of a quite different physical origin. This so-called "pseudoturbulent" motion is specific to fluidized beds and to vertical suspension flows at large, as well as to pressure-driven suspension flows in which the mean velocities of the suspension phases happen to be different, so that a relative fluid slip flow occurs. The underlying physical mechanism of the pseudoturbulent fluctuations has been extensively discussed in a number of papers, and in particular, in [45]. In this mechanism, mean relative fluid flow interacts with random fluctuations of suspension concentration, that is, with fluctuations of the total number of particles within any given mixture volume. Such concentrational fluctuations are due to various chance causes, and they are inevitably present in any suspension flow. Since the viscous drag experienced by particles is a strongly nonlinear function of local concentration, concentrational fluctuations are bound to cause the suspending fluid to exert a fluctuating force on any given particle. This fluctuating force accelerates the particle either in the direction of relative slip flow or in the opposite direction. An additional fluctuating force that acts on the fluid in an external body force field also appears due to density fluctuations which are closely connected with the concentrational fluctuations. These forces make the particles and the fluid fluctuate in the direction of slip flow, and also in the direction of external body force. In vertical suspension flow, such as flows accompanying fluidization and suspension sedimentation, the two indicated directions coincide, and only vertical fluctuations initially originate. Due to fluctuations of fluid pressure and relative velocity, as well as to both hydrodynamic and collisional interparticle interactions, the kinetic energy of original fluctuations is redistributed in such a way as to excite fluctuations in the other directions. As a result, an anisotropic fluctuating motion establishes itself in both suspension phases. If suspended particles are large enough for the interparticle exchange by momentum and energy to be mainly performed by means of direct collisions of the particles, the pseudoturbulent fluctuating motion is approximately isotropic. In this case, the particles fluctuate almost as statistically independent entities, like molecules of a gas undergoing thermal motion. Properties of the pseudoturbulent motion can then be investigated by analogy with the kinetic theory of dense gases, as was suggested, seemingly for the first time, by Jackson [46]. Although study of pseudoturbulent fluctuations in coarse dispersions with interparticle collisions certainly goes beyond the intended scope of this Chapter,
1280
we point out that various phenomenological approaches exist which are based on such an analogy and which are primarily aimed at incorporating the effects of the pseudoturbulent fluctuations on mean suspension and fluidized bed flow. Such approaches are exemplified in a number of works, and in particular, in [47]. A more sophisticated analytical kinetic theory approach is discussed in [48]. Another limiting case pertains to quiescent suspensions of fine particles which exchange momentum and energy exclusively through random fields of ambient fluid velocity and pressure, and collisions do not play a noticeable role in this exchange. It is only this case that we are going to address below. A significant feature of the pseudoturbulent motion generated in finely dispersed suspensions consists in the fact that suspended particles are by no means statistically independent. Long-ranged hydrodynamic interactions of the particles result in the formation of group containing many particles that fluctuate in a correlated manner. Such groups keep to incessantly originating and then disintegrating, so that the fluctuating motion can be phenomenologically viewed as an ever altering collection of intermittent flow patterns which are reminiscent of fluid moles in one-phase turbulent flow. In order to find out relevant statistic properties of the particulate groups, it is necessary to develop an efficient model of multiparticle hydrodynamic interactions. A generally accepted analytical approach to modeling flow of quiescent suspensions consists in summing up all such interactions involving different numbers of particles. In practice, such a summation has been effected only for pairwise interactions, with the help of a nice renormalization procedure as first invented by Batchelor to circumvent singularities and to get convergent results for macroscopically homogeneous dilute suspensions (see, for example, [8]). Unfortunately, some theoretical conclusions that have been obtained to date while treating fluctuations by means of different versions of this renormalization procedure seem to be unacceptable from the physical point of view. Thus, analytical studies and numerical simulations of a sedimenting suspension have shown that the variances for particle fluctuation velocity and coefficients of particle pseudoturbulent self-diffusion in different directions increase unboundedly with the size of the vessel that contains the suspension, if the spatial distribution of particles is supposed to be random [49-51]. The last inference is not supported by experimental data which prove the quantities mentioned to be insensitive to the container size if it exceeds a certain critical size [52,53]. These experiments found no systematic variation in the particle velocity fluctuations with container size at all. A possible explanation of this controversy might be founded on a stmmse that a short-range order establishes itself in a sedimenting suspension, so that a nonrandom suspension microstructure sets in. In particular, Koch and Shaqfeh [54]
1281
have suggested that changes in pair correlations as induced by the sedimentation process might lead to a screening of the hydrodynamic interactions. In such a case, velocity fluctuations could be finite and independent of container size for sufficiently large vessels. Indeed, these authors have concluded that there is a net deficit of particles in the neighborhood of any particle, and that this deficit provides for such a Debye-like screening. However, recent numerical simulations performed by Ladd [55] have demonstrated that there is no such particle deficit at all, and correspondingly, that the hydrodynamic interactions of particle pairs are not screened by the changes in the pair distribution function at long distances. Moreover, Ladd has drawn a conclusion that the particle velocity fluctuations essentially depend not only on the size, but also on the shape of the container. An alternative mechanism, that admittedly results in making the particle velocity variances and self-diffusivities insensitive to container size and shape, has been proposed in [56]. According to this mechanism, the pseudoturbulent fluctuating motion of particulate groups brings about additional viscous dissipation of kinetic energy. This dissipation gives rise to a corresponding effective average force that efficiently retards fluctuations of any particle, and thereby make the fluctuation intensity finite, irrespective of how large is the container size. The concept of effective force that slows down the fluctuations is certainly semi-empirical. However, it can be readily understood on the basis of the familiar ideas specific to various versions of the well-known mean-field approximation. Below, we shall briefly describe the model as developed in [56] for vertical suspension flow. 6.1 Friction Force Due to Energy Dissipation Let us evaluate the viscous energy dissipation that is supposedly caused by the pseudoturbulent fluctuations of particulate groups. These groups are separated by changeable boundary interlayers, and they incessantly exchange particles between themselves. Within the scope of a semi-empirical model that we are about to disclose, however, such groups may be regarded without loss of generality as statistically independent, whereas the behavior of particles that are contained in any group at any given moment of time may be looked upon as fully correlated. This means, in particular, that the fluctuation velocity of a group is the same as the velocity of any of its particles. If we make use of a characteristic particle correlation length as evaluated in [54], we shall arrive at the following estimates for group's linear dimension, L g , the group's volume, Vg, and the average number of particles within one group, Ng: a
Lg = c L -~ ,
a 3
Vg = cV ~-~ ,
1
Ng = c N 02
(65)
1282
where numerical coefficients of the order of unity are introduced. Energy is supplied to the pseudoturbulence from the relative fluid flow as it operates at eoneentrational fluctuations, in compliance with the mechanism described in [45]. In the steady state, this energy input is exactly compensated by the viscous energy dissipation, so that the pseudoturbulence intensity is maintained at a certain invariable level. There are two contributions to this dissipation: the one that is associated with the fluctuating drag forces acting on individual particles, and the one due to the random field of variable shear rates as generated by the fluctuating motion of the groups. The immediate objective of this subsection is to approximately evaluate the second contribution, for which purpose we shall use a simple consideration based on the theory of similarity. It is evident that the fluctuating shear rates are of the order of W/Lg, W standing for a characteristic value of the RMS particle fluctuation velocity. When again making use of the effective medium model as formulated in Section 3, we are able to evaluate the energy dissipated by the random shear motion of the particulate groups in a unit volume per unit time. By the order of magnitude, this dissipated energy equals ktfM (W/Lg)2. The total excessive viscous dissipation of energy, AE, attributed to one group is equal to this last quantity multiplied by group volume, meaning that its is of the order of/afM W2Lg ~ ~ M W 2 a/qk. Now, we want to describe this viscous dissipation by means of introducing an effective retarding friction force, F~ = - a mpw', that acts on one particle on the average. It is fairly clear that AE must be equal to the work accomplished by such friction forces experienced by all Ng particles within a group per unit time, when taken with an opposite sign. Then, using equation (65) yields an order-ofmagnitude equation to solve for unknown coefficient a
AE oc /uf M W 2 -a ocOtmpNg(w'2)ocotppW 2 a3
(66)
whence we obtain, as a final result,
Fr = - o t m p w ' ,
a = c a ppa2 luf CM(qk)
(67)
where c~ is a new coefficient of the order of unity. Such an introduction of retardation friction forces exerted on the particles is in accord with standard methods of the mean-field theory. These forces are of key significance since they can be proven (and will actually be shown below) to prevent the particle velocity variances and self-diffusivities being divergent in the
1283
dilute limit. It is evident that, as far as the divergence of the said quantities is concerned, the dilute limit plays the same role at a given container size as an infinite increase of the container size does at suspension concentration kept constant.
6.2 Stochastic Equations The Langevin equation for a group containing Ng particles can be formulated as
mp Ng dw' = Ng (F' + F r) dt
(68)
where F~ is defined by equation (67) and F' is the fluctuating force experienced by one particle within the group. The differentiation is performed in equation (68) along the group trajectory. Summing up the Langevin equations (68) for all groups within a unit volume of the suspension and using equation (67) yield the following new equation:
CPP 0---~= nF' - aqkppw'
(69)
Quantity w' represents a sum of random velocities of particulate groups in a unit volume which are assumed to be not correlated. It is quite evident that the group number concentration can be made as small as is desired, by appropriately defining the coordinate length scale. Then, if also a coordinate system moving with the dispersed phase mean local velocity is used, the partial time derivative can be substituted for the total derivative that appears in equation (68). This has been actually done when formulating equation (69). Two supplementary stochastic equations can be derived from the continuum mass and momenttma conservation equations for the continuous phase as formulated in equations (1) and (2). These stochastic equations read [56]
+u. v)r = (1- r v. v' (70) (1 -
qk)pf(~t +u. V) v' =-Vp'+jufMAv'-nF' -Fn'-pfg~b'
where the convective coordinate system connected with the dispersed phase motion is used again. To simplify the matter, we have ignored time and space
1284
dependence of the mean flow variables when formulating these equations. Thus, strictly speaking, equation (70) has formally a bearing upon fluctuations occurring in a macroscopically uniform steady suspension flow. The fluctuations of mean flow variables that appear in equation (70) are also sums of corresponding independent random contributions associated with different particulate groups. In accordance with the central limit theorem of the theory of probability, their variances equal the corresponding variances for one group divided by the group number concentration that may be presumed large as compared to unity. Thus, equations (69) and (70) can be thought of as the ones related to the fluctuations of a single group. Since stochastic equations (69) and (70) are linear in fluctuations, it is natural to employ the correlation theory of stationary random processes, an informative review of which is given in [57]. According to this theory, all fluctuations are represented as Fourier-Stieltjes integrals over frequency co and throughout the entire wave-number space k. Using these representation reduces differential equations (69) and (70) to a set of linear algebraic equations in which the role of tmknown quantities is played by random measures that are involved in the Fourier-Stieltjes integrals. These last equations enable us to express all the random measures as quantities proportional to the random measure for random concentrational fluctuations, d Z~. Correspondingly, all spectral densities of interest turn out to be proportional to the spectral density of the concentrational fluctuations, q-'r (co,k), with proportionality coefficients being functions of co and k. Using tile correlation theory of stationary random processes allows various two-time two-point correlation functions to be found by standard means characteristic of this theory [57]. An expression for the fluctuation force is to be determined from equations (8) - (13). For the sake of simplicity, we take into account only the contributions to the interphase interaction force that are due to viscous drag and to gravity corrected for buoyancy. After a simple manipulation, we then arrive at the following formula: 1 dM
nF'=qkPp[~r(v'-w')+~~ tcr dqk u~'
_ ~x - 1 g~, ]
x
r = - ~2 a 2
'
9 vf
(71)
It is also not difficult to get from equation (3)
Fn' : -ppg~k'
(72)
Using equations (71) and (72) in the algebraic equations for random measures that obtain when the aforementioned Fourier-Stieltjes integrals are inserted into
1285
equations (69) and (70), and then solving these algebraic equation, result in final formulae for the random measures of the pseudoturbulent fluctuations. In particular, it is not difficult to derive general expressions for the random measures of random fluid and particle velocity fluctuations v' and w'
dZv =
o+u.kk ck k
H
S-
dZ~ (73)
_ dZw
E { c, o + u . k _k ico+a+E ,~k k
r
[s_(S'k)k]
}
where scalar and vector functions involved are introduced in accordance with the following equations:
(io+a)E e - terM , H = i (1- qkl Pf ( ~ + u "k ) + ttf M k Z + ~kPP i o + a + E (74) S=
ico+a
ico + a + E
B+C,
B=~u+c.dlnM dO
C=(tc-1)r ~bM
Equation (73) gives an opporttmity to derive equations for the spectral density tensors of fluid and particle velocity fluctuations which express them as quantities proportional to the scalar spectral density of concentrational fluctuations. In order to be able to calculate the variances of particle velocity components by integrating corresponding components of the spectral density tensor over co and k, a trustworthy representation for the spectral density of concentrational fluctuations is necessary. The wanted representation was discussed at length in [56] where both particle velocity variances and selfdiffusivities were studied. In what follows, we are interested in the determination of particle velocity variances, but not of coefficients of particle pseudoturbulent self-diffusion. In this ease, a detailed discussion of the eoneentrational fluctuation spectral density proves to be excessive. As will become clear later on, only the variance of coneentrational fluctuations is actually needed to get reasonable estimates for the particle velocity variance under certain simplified conditions. The wanted expression for this variance follows from [58] where the classical Smoluehowski's combinatorial theory of fluctuations in dilute particulate systems has been generalized to concentrated systems. This expression reads
1286
(75) equation (65) having been taken into account when obtaining the second equality. At last, let us indicate sufficient conditions which must be fulfilled to allow us to make use of a simple equation (75), instead of a much more complicated representation for the concentrational fluctuation spectral density. As has been discussed in [56], for this purpose in view we have to 1) replace fluid and particle velocity spectral densities by their values at small frequencies, 2) assume the integral of the concentrational fluctuation spectral density over frequency to be dependent on modulus k of wave-number vector k, but not on this vector direction (that is, not on angular coordinates of a spherical coordinate system in the wave-number space), and 3) ignore viscosity effects. The terms that involve frequency may be dropped out of equations (73) and (74) if pseudoturbulent motion is quasi-stationary in a certain sense. This condition of psedoturbulent fluctuations being quasi-stationary amounts in fact to an assumption requiring a characteristic fluctuation frequency scale to be much smaller than friction force coefficient a determined in equation (67). The characteristic frequency scale for fluctuations must be of the order of vrMLg-2 ~ vf M~ 2/a2, whereas coefficient a ~ vf M ~ 2/1r 2. Thus, the condition trader discussion must hold good for suspensions of sufficiently low concentrations, all the more so, the larger the density ratio. In the generalized case, however, both frequency scale and friction force coefficient are of the same order of magnitude, and the said condition presents an approximation the adequacy of which has to be specially checked. The second condition requiting the integral of the concentrational fluctuation spectral density over frequency to be independent of the wave-number vector direction states that the concentrational fluctuations are nearly isotropic. This represents a reasonable approximation that has its direct analogies in the theory of fluctuations in molecular systems. At last, ignoring viscosity effects amounts to the neglect in the expression for H in equation (74) of the term containing fluid viscosity gf. This term can readily be shown to effect only quite a negligible influence on the particle velocity variances. For simplicity and definiteness, we shall consider the velocity variances only in a vertical suspension flow (or in a fluidized bed). Vectors S, B and C defined by equation (74) are collinear in such a flow, and it is easy to understand that 1) only diagonal components of the fluid and particle velocity variance tensors differ from zero, and 2) these tensors are axially symmetric with respect
1287 to the vertical axis. Thus, only diagonal components of the corresponding spectral density tensors are actually necessary to evaluate the vertical and horizontal velocity variances. Expressions for these diagonal components follow from equation (73). Under the conditions enumerated above, these expressions look as follows:
{ (-~14(1-~u ) 2k?
tlJvl'vl((~
2) $2
(76)
q2 qbC + sB - ~bC + s B + i -u~b
qJ~,~(co.k)
Wv2,v2(co, k) ~ (1 + fl)2 Ww2,w2(Co,k )
Wwl,wl(CO,k) ~ f12r
q20C-
B + l-~b
1 +q2)
qbC+s B + i , 0
q~,r
~w2,w2(co,k)~ f1202(k~+q2 1 ) q2 r
B+
{(/2 s2 1_uo
(77) kl2
k2 )
where
a fl=~=cflqk,
2 cfl=-~Ca,
s = ~ fl~ , l+fl
fl E q = - qbpp o~pf l+fl u
(78)
Equations (76) and (77) are formulated in a coordinate system which first coordinate axis is directed along u, its two other orthogonal axes being arbitrarily chosen in a plane normal to u. Expressions for the spectral density tensor components that correspond to the third coordinate axis are to be obtained
1288
from those corresponding to the second coordinate axis, if the third wave-number vector component, k3, is substituted for the second one, k2.
6.3 Experimental Verification If the vertical suspension flow under study is uniform, then u = u, - -(K-
1) (1 - q6)z" m(r
1 - # Vt
(79)
where Vt is the terminal velocity of a single particle falling in an unbounded fluid. Using this expression in equation (74) yields
dlnM
,-
0 u,
,
c=
1-#
u,
(80)
#(1-#)
Equation (80) enables us to substantially simplify expressions (76) and (77). To make further calculation of the fluid and particle velocity variances even simpler, we assume that a characteristic scale of k is much smaller than q. If we evaluate this scale as ck#a, in accordance with equation (65), then this condition of q being relatively large can be roughly formulated as follows [56]" tr - 1 (1 - r ~Fr<
lo
r
Fr-
ga 3 2 vf
(81)
When formulating this inequality, the formula for relative suspension viscosity from equation (4) and equations (78), (79) have been used. Condition (81) is always satisfied if suspended particles are fine enough, so that their Froude number, Fr, is small. However, inequality (81) becomes invalid, and it ultimately transforms into an inverse strong inequality, in the dilute limit, as # approaches zero. If inequality (81) holds true, the terms proportional to k~2 in the denominator and numerator of equations (76) and (77) can be ignored compared to the terms that are proportional to q2. This simplifies the spectral density expressions identified in equations (76) and (77) to quite a considerable extent, so that these expressions can easily be integrated over the wave-number space. In particular, by making use also of equation (80), we obtain from equation (77) the following simple approximate formulae for the diagonal components of the particle velocity variance tensor in a uniform vertical flow [56]:
1289
'
(
/ u_, ) 2
Wwl,wl(CO,k) ~ ]320 2 1- y ~-~-j 1 r
%2,wz(co,k) ~r162
~2 )
qJ~,~(co,k)
i - r %,q~(co,k),
ca~ 2 y = ~ l+c,ar
(82)
The wanted particle velocity component variances are to be calculated by integrating over co and k the corresponding expressions for the spectral density tensor components as defined by equation (82). As has been already pointed out, the integral of the concentrational fluctuation spectral density over co is assumed to depend only on modulus k of wave-number vector k. The coefficients at this spectral density in expressions (82) involve only angular coordinates in the wave-number space, but they are dependent on neither k, nor co. Furthermore, the integration of the concentrational fluctuation spectral density over k and co gives variance (75) of these fluctuations. Keeping in mind these simple considerations, and also making allowance for equations (4) and (79), we obtain 1/2
vt
~Cw 1 - 3 + 5
(1-~) s/2 1- ~
(83) vt
~ Cw ~
1-
1
cw = cp r~-~
According to this equation, the ratio between lateral and longitudinal RMS particle fluctuation velocities is equal to
iw+w211'2 iw ll,2 ( (wf21"2
_
2y2 (wf2) 1/2 -_ 15-107'+3y 2
11,2 (84)
In conformity with the formulae in equation (83), both longitudinal and lateral RMS velocity components tend to finite limits as ~b comes to zero. Thus, the pseudoturbulent particle velocity variances do not diverge in the dilute limit. However, the adequacy of these formulae is conditioned by inequality (81) being valid, so that they cease to be true in the dilute limit. It can be shown that these formulae give way in the dilute limit to other relations for the said RMS velocity components [56]. The last relations prove both lateral and longitudinal RMS
1290
velocities to be proportional to r 2, their ratio 6~ tending to a constant value of 0.067 as r approaches zero. Similarly, the coefficient of particle pseudoturbulent self-diffusion can be proven to vary proportionally to r at low ~. Equations (83) and (84) provide an opportunity to check theoretical conclusions by comparing them with available experimental evidence. Such a comparison with data obtained for sedimenting suspensions in [53] is illustrated by Figure 10 and Figure 11. The agreement between our model and experiments as evidenced by these figures seems satisfactory, in spite of the fact that specific values have been used for the adjustable factors that appear in theoretical equations (83) and (84). The deviation of the theoretical curves from experimental dots at low concentrations is quite understandable because these equations fail to be valid in the dilute limit, and they have to be replaced by other equations, as has been already pointed out. As has been shown in [56], both longitudinal and lateral coefficients of particle pseudoturbulent self-diffusion as evaluated on the basis of the model developed also agree with the experimental data of [52,53], and this fact lends an additional support to the model developed for pseudoturbulent fluctuations in eollisionless suspensions. 0.8 -..\ . . . . . . . . . . . 0.7 0.6 0.5
9
Cq
0.4
~.
I
o.3 0.2 0.1 0 .
0
.
.
.
.
.
0.1
.
.
.
.
0.2
.
.
.
.
.
0.3
.
.
.
.
0.4
.
.
.
0.5
Figure 10. The longitudinal (vertical) particle velocity fluctuation scaled with particle terminal velocity according to equation (80) at ~ = 0.6 and cp = 0.6; the upper and lower curves correspond to c:v~/2 = 1.2 and 1.4, respectively; dots, experiment in [53]. 6.4 Pseudoturbulent Stresses
As cannot Rather flows.
follows from our model, the chaotic particle pseudoturbulent motion be viewed as statistically independent fluctuations of individual particles. it is similar to turbulent fluctuations of fluid moles in one-phase turbulent The same statement holds true for the pseudoturbulent motion of the
1291
ambient fluid as well. For this reason, equation (42), that has relevance to fluctuations of individual statistically independent particles which do not collide but interact only with the fluctuating ambient fluid, certainly cannot be applied to the description of stresses that originate due to pseudoturbulent fluctuations. Thus, an alternative way to describe these stresses has to be put forward.
!.8
I hl.6
1.4
1.2 0
0.i
0.2
0.3
0.4
0.5
0
Figure 11. The ratio between longitudinal and lateral standard particle velocity deviations as a function of suspension concentration; curves correspond, in the descending order, to equation (81) with c~ = 0.4, 0.6 and 0.8; dots, experimental data from [53 ]. Such a method can be developed from considerations that are analogous to those used by Roco [59] in his treatment of two-phase turbulent flows. According to this method, conservation equations (1) - (3) are to be employed again to describe the averaged fields of smoothed flow variables pertaining to the interpenetrating continua that model the suspension phases. After that, these fields are supposed to fluctuate in just the same way as the fluid velocity and pressure fields fluctuate in an one-phase turbulent flow, and a new, secondary, averaging procedure is applied to average these equation with respect to such quasi-turbulent fluctuations. This second averaging procedure is quite similar to that which is commonly used while dealing with one-phase turbulent flows and that leads to the appearance of Reynolds stresses in the newly averaged equations. If this procedure is adopted, the wanted expressions for pseudoturbulent stresses should not differ from those for the Reynolds stresses in turbulent flows. As a result of applying this averaging procedure to equations (1) - (3), we arrive at additional Reynolds-like stresses a
-p<
-
s,
=
(85)
1292 that are supposed to act in flow of both continuous and dispersed phases due to their pseudoturbulent fluctuations. The averages that appear in equation (85) can be calculated from the model of pseudoturbulence as developed earlier in this section. In particular, while dealing with equations (76) and (77), we can use precisely the same assumptions that have led to formulae (83). In this case, we shall obtain, atter a manipulation, expressions for normal pseudoturbulent stresses that act in a nonuniform vertical suspension flow. We have for the continuos phase
(1+ Cfl~)2 i1 _ ~m_m/IV(0)+ V(1) z/
~ f = - 15 c fl2CN(1 - qk)
u
(86)
u,
and for the dispersed phase
+ 15 cfl 2 c N
--+
pp
(87)
U,
(1 - ~b)2
where j = 1, 2, 3, new coefficients are introduced that are functions of
W l ( 0 ) = 8 - 4 s + 3 s 2, V/(0) = (1 + s) 2 ,
Wi(k)=V ( k ) ,
Wl(1)=2s(-2+3s)(h+l), V(1) = 2s(1 + s)(h + 1),
i=2,3,
k=0,1,2,
Wl(2)=3s2(h+l) 2
V(2) = s 2 (h + 1)2
(88)
h=(1-~b) aln------M-M
a0 and expressions for B and C resulting from equation (74) have been employed. The pseudoturbulent normal stresses defined in equations (86) - (88) represent fimetions of two flow variables: local volume concentration ~ and local mean fluid slip velocity u which are in general mutually independent. In uniform flows, these variables are related to each other by equation (79), so that the normal stresses are completely determined by a single flow variable. As follows from equations (86) - (88), normal stresses that act in a uniform flow in either continuous or dispersed phase of a dilute suspension and that are attributed
1293
to the longitudinal (vertical) direction are eight times as large as corresponding normal stresses attributed to any of the lateral (horizontal) directions. It is worth noting in conclusion to this section that there exists an alternative model of pseudoturbulent fluctuations [60]. In contrast to the model as presented above, this alternative model implies that particles that undergo pseudoturbulent fluctuations may be assumed to behave themselves as approximately statistically independent entities. Such an assumption contradicts the widespread notion that particles fluctuate as members of groups composed of closely correlated particles. However, this model findings may also be supported by some experimental evidence [60]. It seems just possible that the overall pseudoturbulent fluctuating motion might involve two different constituents that have different length scales. According to such a scheme, approximately independent fluctuations of particles, presumably due to the nonlinearity of the dependence of the interphase interaction force on local concentration, are superimposed on fluctuations of groups containing many particles, presumably owing their origin to the interaction of the gravity field with density fluctuations that are known to be a primary cause of the divergences discussed in [49-51, 54, 55], and also earlier in this paper. Without doubt, this important issue deserves the most close attention in future theoretical studies. 7. CONCLUSIONS The two central notions advanced in the present Chapter are actually rather trivial. The first notion represents a recognition of the simple fact that even simplest suspension flows are subject to the stratification caused by forces exerted on suspended particles in the direction normal to the dispersed phase streamlines. If those forces are not opposed by dispersed phase stresses, this stratification ultimately leads to the separation of the whole flow domain into regions containing close-packed particles or altogether devoid of particles. Consequently, steady suspension flows cannot be adequately and effectively described with the help of existing conventional fluid dynamic models, if these models do not allow for concentration-dependent stresses that are specific to the dispersed phase and that condition this phase stratification. The second notion intimates that the particulate stresses, which are capable of hampering the flow stratification in unsteady flows and of bringing it to an end in steady flows, are due to random fluctuations of the suspended particles. The action of the particulate stresses results in the establishment of non-trivial particle distributions that play a paramount role in evolving the fields of all the flow mean variables and in making up the flow hydraulic characteristics.
1294
Apart from turbulent fluctuations of suspended particles which have not been considered in this Chapter at all, fluctuations of three distinct types that owe their origin to different physical mechanisms have been distinguished. These are the thermal particle fluctuations, and also two types of hydrodynamically-induced fluctuations: the shear-reduced and pseudoturbulent ones. The fluctuations of all the types stimulate particle migration which in arm produces a net diffusion flux down a particle concentration gradient. This flux partially compensate for the convective flux of the particles caused by the forces directed normally to the dispersed phase flow streamlines. As a result of the combined effect of these fluxes, a particle distribution originates that, in steady flows, correspond to dynamic equilibrium attained between convection and diffusion of particles. When dealing with a suspension flow, simultaneous allowance for the relevant fluctuations belonging to different types seems imperative. First, the coupling of different fluctuations results in the occurrence of non-monotonous dependencies of meaningful flow characteristics on parameters, such as the particle size, which can significantly influence various technological processes. This effect of the coupling is exemplified by a uniform shear layer flow that has been considered at some length in [38]. A convenient measure of the ability of uniform horizontal steady shear flow of a given shear rate to suspend particles is the total mass (or volume) of suspended particles per unit area of the flow bottom plate. This total mass has been proven to be a non-monotonous function of particle size that has a minimum at a certain size value, all other physical parameters being presumed fixed. For smaller particles, the flow suspending ability increases owing to the corresponding enhancement of thermal fluctuations. For larger particles, this ability also increases, but due to the augmentation of shear-induced fluctuations. This conclusion has an immediate bearing on manifold field-flow fractionation processes in practice. A similar non-monotonous dependence on particle size can be proven to be specific to fluidized beds and quiescent sedimenting suspensions of fine particles. The total mass of particles that form an upper part of a fluidized bed and that is situated above the approximately uniform bulk region of the bed has a minimum for particles of a certain radius. This minimum has precisely the same physical origin as that for the total mass of particles suspended by a uniform shear flow, except for the fact that the role of shear-induced fluctuations is now played by pseudoturbulent fluctuations. Secondly, failure to simultaneously account for fluctuations of different origin can lead to unwarrantable conclusions that may be erroneous not only quantitatively, but also qualitatively. A convincing example is provided by a pressure-driven one-dimensional steady suspension flow in a channel. Even if the suspension is non-Brownian, neutrally buoyant and finely dispersed, allowance
1295
for the stresses due to both shear-induced and thermal fluctuations is vitally important. If only shear-induced fluctuations were allowed for, we would come with a conclusion that a close-packed core region always obtains in the flow, no matter how low is the particle concentration averaged over the flow cross-section [36,38]. Indeed, in such a case, the lateral particulate stress disappears at the channel central axis (or plane) where the mean shear rate turns to zero. Hence it follows that this lateral stress cannot be maintained uniform over the crosssection, if the suspension concentration at the central axis (plane) is less than its value associated with the close-packed state. In the opposite case where only pseudoturbulem fluctuations were taken into account, we would conclude, also erroneously, that the particles are uniformly distributed in each cross-section. It is clearly the conditions of suspension concentration and fluid slip velocity being invariable throughout any crosssection that provides for the required tmifonmty of the lateral pseudoturbulent stress. If the concentration does not vary in a cross-section, the fluid slip velocity is also uniform. (Indeed, relative fluid flow is caused by the very same longitudinal pressure gradient that produces the flow. This pressure gradient is well-known to be uniform in a steady one-dimensional channel flow. Thus, given that the pressure gradient is fixed, the distribution of fluid slip velocity over flow cross-sections actually a function of suspension concentration alone.) Simultaneous allowance for fluctuations of different types becomes even more as far as general three-dimensional unsteady flows are concerned. In particular, this is due to the fact that the corresponding particulate stresses influence not only particle distributions as attained in various steady flows, but also the rate with which these stationary distributions, and also the steady flows, establish themselves. Besides, these stresses sometimes play a fundamental role in making for hydrodynamic stability of steady flows [46, 48, 58]. It is expedient to emphasize in conclusion that much remains to be done in order to develop a completely reliable, comprehensive and workable theory to treat fine suspension flows. The point is that the tentative model as developed and presented in this Chapter is not free of numerous semi-empirical assumptions the validity of which must be examined in detail. To provide for such an examination, tangible subsidiary models must be further elaborated and made more precise for fluctuations of different origin, and also for stresses induced by these fluctuations. Unavoidably, this task requires very serious efforts to be applied in a specific research area bordering on mechanics of multiphase flow, rheology and statistical physics. However, such efforts seem to be fairly justified and worth being undertaken in view of important practical issues that are likely to come about in connection with munerous branches of technology.
1296
REFERENCES
1. Y.A. Buyevich, Arch. Mech. 42 (1990) 429. 2. P.G. S ~ a n , J. Fluid Mech. 22 (1965) 385; and 31 (1967) 624. 3. M. Ishii, Thermo-Fluid Dynamic Theory Two-Phase Flow (Eyrolles, Paris, 1975). 4. D.A. Drew, Ann. Rev. Fluid Mech. 15 (1983) 261. 5. R.I. Nigmatulin, R. T. Lahey and D. A. Drew, Chem. Engng Comm. 141142 (1996) 287. 6. Y.A. Buyevich and I. N. Shchelchkova, Progr. Aerospace Sci. 18 (1978) 121. 7. Y. A. Buyevich and T. G. Theofanous, ASME Winter Meeting, Dallas, 1997. 8. G.K. Batchelor and J. T. Green, J. Fluid Mech. 56 (1972) 401. 9. G.K. Batchelor, J. Fluid Mech. 83 (1977), 97. 10. Y.A. Buyevich, Chem. Engng Sci., 50 (1995) 641. 11. L.D. Landau and E. M. Lifshitz, Fluid Dynamics (Pergamon Press, Oxford, 1959). 12. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge Univ. Press, Cambridge, 1967). 13. N. Zuber, Chem. Engng Sci. 19 (1964) 897. 14. L. van Wijngaarden, J. Fluid Mech. 77 (1976) 27. 15. J.B.W. Kok, Physica A 148 (1988) 240. 16. A. Biesheuvel and S. Spoelstra, Int. J. Multiphase Flow 15 (1989) 911. 17. B.U. Felderhof, J. Fluid Mech. 225 (1991) 177. 18. D.Z. Zhang and A. Prosperetti, J. Fluid Mech. 267 (1994) 185. 19. R.-H. Jean, and L.-S. Fan, Powder Techn. 72 (1992) 201. 20. G. Astarita, Chem. Engng Sci. 48 (1993) 3438. 21. Y.A. Buyevich, ASME AMD-Vol. 217 (1996) 161. 22. F. Feuillebois, J. Fluid Mech. 139 (1984) 145. 23. A.D. Maude, J. Fluid Mech. 7 (1960) 230. 24. B.S. Yendler, Inzh.- Fiz. Zh. 37 (1979) 110. 25. J.F. Brady, J. Chem. Phys. 98 (1993) 3335. 26. J.F. Brady, J. Chem. Phys. 99 (1993) 567. 27. G.K. Batchelor, J. Fluid Mech. 74 (1976) 1. 28. J.-Z. Xue, E. Herbolzheimer, M. A. Rutgers, W. B. Russel and P. M. Chaikin, Phys. Rev. Letts 69 (1992) 1715. 29. N.F. Carnahan and K. E. Starling, J. Chem. Phys. 51 (1969) 635. 30. D.L. Koch, Phys. Fluids A 2 (1991) 1711.
1297
31. P. M. V. Resibois and M. de Leneer, Classical Kinetic Theory of Fluids (Wiley-Interscience, New York, 1977). 32. G.K. Batchelor, J. Fluid Mech. 193 (1988) 75. 33. E.C. Eckstein, D. G. Bailey and A. H. Shapiro, J. Fluid Mech. 79 (1977) 191. 34. D. Leighton and A. Acrivos, J. Fluid Mech. 177 (1987) 109. 35. F. Gadala-Maria and A. Acrivos, J. Rheol. 24 (1980) 799. 36. R. J. Phillips, R. C. Armstrong, R. A. Brown, A. L. Graham and J. R. Abbot, Phys. Fluids A 4 (1992) 30. 37. P.R. Nott and J. F. Brady, J. Fluid Mech. 275 (1994) 157. 38. Y.A. Buyevich, Chem. Engng Sci. 51 (1996) 635. 39. I.M. Krieger, Adv. Colloid Interface Sci. 3 (1972) 111. 40. W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions (Cambridge Univ. Press, Cambridge, 1989). 41. Y. Buyevich and I. N. Shchelchkova, Inzh.- Fiz. Zh. 33 (1977) 872. 42. G.K. Batchelor, J. Fluid Mech. 131 (1983) 155. 43. Y.A. Buyevich and A. O. Ivanov, Physica A 192 (1993) 375. 44. S. V. Bushrnanova, Y. A. Buyevich and A. O. Ivanov, Physica A 202 (1994) 175. 45. Y.A. Buyevich, J. Fluid Mech. 49 (1971) 489. 46. R. Jackson, Trans. Instn Chem. Engrs 41 (1963) 13. 47. D. Gidaspow, Multiphase Flow and Fluidization (Academic Press, Boston, 1994). 48. Y. A. Buyevich and S. K. Kapbasov, in: Multiphase Reactor and Polymerization System Hydrodynamics (Gulf, Houston, 1996), p. 119. 49. R.E. Caflish and J. H. C. Luke, Phys. Fluids 28 (1985) 759. 50. E.J. Hinch, in: Disorder and Mixing (Kluwer, Boston, 1985), p. 153. 51. A.J.C. Ladd, Phys. Fluids A 5 (1993) 299. 52. H. Nieolai and E. Guazelli, Phys. Fluids A 7 (1995) 3. 53. H. Nicolai, B. Herzshafi, E. J. Hinch, L. Oger and E. Guazelli, Phys. Fluids A 7 (1995) 12. 54. D.L. Koch and E. S. G. Shaqfeh, J. Fluid Mech. 224 (1991) 275. 55. A.J.C. Ladd, Phys. Fluids A 9 (1997) 491. 56. Y.A. Buyevich, Fluid Mech. Res. 22 (1995) 41. 57. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, vol. 2 (MIT Press, Cambridge, MA, 1971). 58. Y.A. Buyevich, Chem. Engng Sei. 26 (1971) 1195. 59. M. Roco, in: Encyclopedia of Fluid Mechanics, Vol. 10 (Gulf Publ. Co., Houston. 1991), p. 1. 60. Y.A. Buyevieh, ASME FED-Vol. 231, MD-Vol. 66 (1995) 107.
1299
RHEOLOGICAL SUSPENSIONS
PROPERTIES
OF CONCENTRATED
P.J. Carreau, P.A. Lavoie, and F. Yziquel Center for Applied Research on Polymers, CRASP, Department of Chemical Engineering, Ecole Polytechnique, Montreal, QC H3C 3A 7, CANADA
1. I N T R O D U C T I O N Concentrated suspensions and multiphase polymeric systems are encountered in many industrial sectors. Paints, foodstuffs, pulp and paper, mineral slurries, filled elastomers and reinforced plastics illustrate very well a large sector of applications. The processing/morphology/property relationships in suspensions and in immiscible blends remain poorly understood and a given processing strategy may result in high value-added products. The dispersion of fillers, the orientation of fibers as well as the morphology of blends are strongly dependent on the rheological properties of the components. Rheological properties are needed to understand the phenomena encountered and changes occurring during processing. Rheological data are also needed to assess constitutive equations required for designing equipment and predicting changes under processing. Finally, rheological methods could be powerful tools to establish relationships between structure, formulation, processing and properties. Most industrial suspensions show shear-thinning and thixotropic behavior. Highly concentrated suspensions do not exhibit the low shear (Newtonian) plateau depicted by homogenous polymer solutions or melts. The unbounded viscosity or solid-like behavior at low shear rates or frequencies is the result of particle-particle interactions and it is referred to as a yield stress. The notion of yield stress is quite controversial and obtaining a significant value is very difficult, mostly for high viscosity matrices such as polymer melts for which the high values of stresses mask
1300
the weaker contribution of the particle-particle interactions. Other physical aspects are related to the degree of solids dispersion in the suspending fluid or matrix. For highly concentrated suspensions, the solids are not, in general, evenly distributed and aggregates and flocs may be present, forming a structure which may change with time due to forces acting on the system. A second aspect is related to the viscoelastic nature of suspensions. For concentrated suspensions in Newtonian fluids, viscoelastic effects have been observed only when yield stresses are detected. Fillers in viscoelastic polymers are believed to reduce the elasticity of polymers [ 1], but this remains unclear as the definition of elastic depends on the experiments used to obtain the appropriate data, for example primary normal stress differences in simple shear flow or elastic modulus in small amplitude oscillatory shear flow. Measurements in the linear viscoelastic domain, such as small amplitude oscillatory shear flow, can yield very useful information on the suspension structure and interactions between the various components. For small enough strain, the structure of multiphase polymeric systems is expected not to change and the linear viscoelastic functions (the storage modulus G' and the loss modulus G") are enough to describe the equilibrium properties. Results of measurements, however, in the so-called non linear viscoelastic domain, although very complex, are essential for understanding and predicting morphology changes during processing. 1.1 Scope In this chapter, we review the rheological behavior of multiphase systems (suspensions in Newtonian fluids and polymers filled with or non interactive particles). First, we present some basic definitions and concepts useful for the understanding of the rheological properties of concentrated suspensions. Then, we summarize problems encountered in rheometry, and review the main rheological models proposed in the literature to describe the properties of concentrated suspensions. Finally, we focus on non linear effects observed for concentrated suspensions in small strain experiments using data of our laboratory, and conclude with challenges and trends for future research.
1.2 Forces acting on particles in suspensions Various forces affect the rheological behavior of concentrated suspensions, especially in the case of colloidal (submicron) particles for which Brownian forces
1301 and particle-particle interactions play a major role. The internal forces can be subdivided in hydrodynamic and non hydrodynamic forces. The hydrodynamic forces result from the relative motion of the suspending medium with respect to the particles. These are the only forces considered in the Einstein [2] analysis of the viscosity of a dilute suspension of rigid spheres in a Newtonian fluid, which led to the well known result: ]]r
-
1 + 2.5ff
-
(1.1)
11 O
here fir, rl~ and I"1oare the reduced viscosity, the suspension and the fluid viscosity respectively. The hydrodynamic forces are responsible for migration of particles, alignment or orientation in the case of non spherical particles and finally structure break-down in the cases of flocs and aggregates. The non hydrodynamic internal forces consist namely of the Brownian forces, responsible for the internal motion of the particles and diffusion, and forces arising from physical and chemical interactions. These forces can affect the internal structure of suspensions of colloidal particles, for concentrations well below that of the maximum packing. Brownian motion allows low range attractive forces to promote floc building, network and weak gel formation. Such attractive forces are characterized by the Hamaker constant defined by the Israelachvili equation as reported by Tsai and Ghazimorad [3]:
A 3 (c, c3/ -
+
4kT
el +e3
+
3hVe , 2 23
(1.2)
16v/-2(n?+n2)
where k is the Boltzmann constant, T is the absolute temperature, c~ et c 3 are the dielectric constants of the particles and the fluid respectively, n/and n 3 are the corresponding refractive indices, h is the Planck constant and v~ is the absorption frequency. For a given concentration, floc formation increases with increasing value of the Hamaker constant [4]. Flocs can be destroyed under shear stresses, increasing the shear-thinning behavior. The power-law index has been shown to decrease with solids concentration. The results of Tsai and Zammouri [4] are in agreement with those of Gadala-Maria and Acrivos [5], who attributed the increasing shear thinning to a reduction of the distance between particles with
1302
increasing solids concentration. Electrostatic repulsion can occur when particles are superficially charged. These long range interactive forces are characterized by the zeta potential (~) obtained, for example, from electrophoretic mobility. In ionic fluids, a double ionic layer masks the repulsive field. Moreover,when the surface is saturated the global charge is zero and the repulsion range is reduced to the thickness of the double ionic layer, l D (Debye length). The interesting results obtained by Krieger and Eguiluz [6] show a drastic reduction of the viscosity for a 40% volume suspension of 0.11 ~tm polystyrene beads with increasing ionic content (HC1). These results are presented in Section 3. In pure water, the suspension shows a yield stress whereas at ionic saturation, Newtonian behavior is observed. Leong and Boger [7] have studied the influence of the surface chemistry of different coals. The addition of electrolytes was shown to increase the suspension viscosity until a yield stress appears, in contrast to the findings of Krieger and Eguiluz [6]. Tsai and Viers [8] have investigated the effect of polarity of non-aqueous solvents on the rheological properties of graphite and polystyrene suspensions. The effect was found to be important for the graphite suspensions, but negligible for the polystyrene systems. This was explained by the large differences in the Hamaker constant which was found to be ten times larger for graphite particles than for polystyrene. Tsai and Ghazimorad [3] have correlated the power-law index of the shear viscosity vs. shear rate and the Hamaker constant for systems of various zeta potentials. Newtonian behavior was observed when the Hamaker constant was zero and the zeta potential negligible. Suspensions with significantly non-zero zeta potential depicted a Newtonian behavior for the higher Hamaker constant values. That is: Newtonian behavior results from an equilibrium between attractive and repulsive forces. Repulsive forces can be induced by polymer stabilization with polymer chains adsorbed on particle surface, which interact to form a network stabilized by steric repulsion. This is comparable to electrostatic repulsion. Mewis et al.[9] have examined the rheological properties of suspensions of PMMA particles stabilized with polyhydroxystearic acids grafted on surface. Their main conclusion was that the deformation of polymeric layers during flow affects the rheology of the suspensions. As expected, the effect was more important for smaller particles with a smaller ratio of particle diameter to stabilization layer. The critical ratio below which hard sphere assumption is no longer valid is somewhere between 5 and 30. For a critical ratio of five, the maximum packing was calculated
1303
via the Krieger-Dougherty model to be 0.72 and 0.96 using the values for the zero shear viscosity, rio, and the high shear rate limiting viscosity, rl=, respectively. The corresponding values for a suspension of spheres with a ratio 60 were 0.49 and 0.62. External forces may affect the behavior of suspensions. Buoyancy effects become important when the particle and the fluid densities are significantly different. These effects are responsible for sedimentation or flotation, affecting the stability of the suspensions and resulting in concentration profiles. As illustrated in the next section, apparent thixotropy can be related to sedimentation. Other external forces such as that due to an electric field can drastically modify the rheological properties of suspensions. Table 1.1 summarizes the main forces acting on colloidal particles and key expressions for calculating their order of magnitude are presented. Table 1.1 Forces acting on colloidal suspensions (Adapted from Russel et al.[ 10])
Force
Order of magnitude
Brownian
kT dP
van der Waals
A/dp
Definitions k: Boltzmann constant T: temperature [K] dp: particle diameter A" Hamaker constant
Viscous
rl" fluid viscosity U: relative velocity of particles
Inertial
pp" density of particles
Gravitational
dp3Apg
A 9" density difference between fluid and particles g: gravitational acceleration
1304
1.3 Yield stress
The yield stress is a useful rheological parameter to characterize particleparticle interactions in suspensions. Classically, the yield stress (Oo) is the stress value below which a material cannot deform. The simplest model to describe fluids exhibiting a yield stress is the Bingham model [ 11 ]. Other well known empirical models are the Casson equation [ 12], the Herschell-Bulkley model [ 13] for powerlaw fluids, and the equation of Poslinski et al.[ 14] for Carreau fluids. A better definition is to consider two distinct deformation modes, as proposed by Oldroy [ 15]" first an elastic-solid behavior for stresses below the yield value, then followed by a fluid behavior. Yoshimura and Prud'homme [ 16] and Doraiswamy et al.[ 17] used the same concept. For example, the Herschel-Bulkley model can be written as:
o :
- G ov,
o = - (
o
[o1 < o ~ : G ovo
o + m I~t[(~-l)) ~t,
(1.3)
Iol > o o
where Go is an elastic modulus. In proposing this model, Doraiswamy et al.[ 17] assumed that the structure was not modified at low strain (ITI -< To). The experimental results of Gadala-Maria and Acrivos [5] and our results shown in Section 4 do not verify this assumption. G ' and G "data are clearly non-linear functions of strain. The concept of s t r a i n - i n d u c e d structure can partially explain yield stresses observed by Husband and Askel [ 18] for non colloidal dispersions in polymeric matrices. Finally, we stress that if many experimental methods (mostly based on extrapolation of shear rate data to zero shear rate) have been proposed to determine the yield stress, the determined value depends strongly on the rheological method or model used [19]. Difficulties in measurements as discussed below may be other sources of errors. 2. D I F F I C U L T I E S
AND MEASUREMENT
PROBLEMS
Many difficulties arise in the determination of the rheological behavior of non homogeneous media. The following criteria are useful in assessing potential difficulties. 2.1 Basic criteria
A first simple criterion is related to the dimension of the suspended particles
1305
with respect to the characteristic dimension of the measuring element (in cone-andplate geometry, gaps of less than 1O0 gm are common)" dp h
<< 1
(2.1)
dp
where is a characteristic dimension of the particles and h is the gap of the measuring device. Obviously it is not always possible to respect this criterion and systematic experimental errors could result. Inertial and gravitational effects must be negligible. The criteria for both effects to be negligible can be written as"
4vp
<< 1
(2.2)
and: texp h
d2plPP-Pmlg <, 1
(2.3)
1]
The first criterion (Equation 2.2) just expresses that the Reynolds number based on the particle and the average velocity of the fluid, V, has to be very small (creeping flow regime). The second one relates the experimental time, t~• to a characteristic time of gravitational or buoyancy effects. With large size particles, other effects are to be expected such as due to a depletion layer at the wall and/or orientation along the wall surface in the case of non spherical particles resulting in lubrication effects or apparent slip. 2.2 Effects of dispersion During measurement or processing, the degree of dispersion of the solids or the structure due to particle-particle interactions may be considerably modified with time. An example of the strong influence of the mixing history of the fillers on the complex viscosity of a suspension of ruffle (TiO2) particles in a low molecular weight polybutene (liquid at room temperature) is illustrated in Figure 2.1. The untreated rutile particles of average diameter equal to 0.6 gm were first mixed to the polybutene using a Brabender plasticorder until an equilibrium torque was reached. Then a second operation was applied to some samples, which consisted of mixing by hand using a thin spatula up to five hours. The behavior of the
1306
complex viscosity curve and its value are drastically changed following the second step mixing. The power-law behavior with a slope close to - 1 for this 31 vol % suspension after mixing using the Brabender,is indicative of strong interactions 10 7
~
'
~
10 6 c~
10 s
..x-
10 4
alter 1st mixing
" O " after 2"d mixing
10 3
10 2
Maron -Pierce ~m= 0.68 . . . . . . . .
i
lO-a
i
i
10 -2
i IiiIll
10 -1
t
..I i
i iiilll
i
i
i ii1'11
10 ~
. . . . . . . .
i
. . . . . . .
10 2
101
10 3
o [s-q Figure 2.1
Complex viscosity for a 31 vol % TiO2 in polybutene at 25 o C (From Carreau et al. [20]).
100
9
9
9
.
. . . . i
II~
9
'
9
9
9
..=.|
9
9
.
9
. . . . b
9 Manual I:I Ultrasonic
r'--"l r~
10-1
10-2 lO-I
10 ~
101
10 2
10 3
[s-]l Figure 2.2
Shear viscosity of 6 mass % suspension of fumed silica A200 particles of diameter equal to 12 nm after a manual mixing and after ultrasonic dispersion (Unpublished data).
1307 between aggregates of rutile particles. The second mixing step broke down a large portion of the aggregates and the behavior approaches that of a suspension made of non imeractive particles. The complex viscosity remains, however, considerably larger than predicted by a Maron-Pierce equation (Equation 3.1 with p = 2 and assuming ~m = 0.68). Another drastic effect of dispersion is illustrated in Figure 2.2 for a low concentration suspension of fumed silica particles in water. These silica particles have silanol groups at the surface which interact via water through hydrogen bonding, forming very large aggregates and possibly a network. Following a preparation via a manual mixing, the viscosity of an aqueous suspension containing 6 mass % of A200 (Degussa) particles (diameter of 12 nm) is shown in the figure to be highly shear-thinning, typical of a gel-type structure which is broken down under shear flow. However, the structure can be largely broken down prior the rheological measurements using an ultrasonic vibrator. The values for the suspension viscosity are drastically decreased and shear thinning is now very light. 2.3 Effects of adsorption or changes in ionic concentration For molten polymer matrices, degradation, reentanglement and r are frequently encountered. For suspensions in polymer solutions, the main problems are associated with mechanical degradation, solvent evaporation and changes in solubility of the polymer, i.e., shear induced crystallization and adsorption of the polymer on the filler surface. Spectacular effects of polymer adsorption reported by Otsubo [21 ] are shown in Figure 2.3 for suspensions of fumed silica particles in a polyacrylamide solution. At the lower solids concentrations (up to 5 mass %) the shear viscosities of the suspensions are shown to be smaller than that of the suspending solution (0 % solids). The hydroxyl groups at the surface of the fumed silica particles favor the adsorption of the polyacrylamide chains. Otsubo has estimated that the adsorption layer is of the order of 8 to 10 nm, which is comparable to the size of the silica particles. These specular effects could possibly be caused by other factors such as changes in the ionic content (an example of such effects is illustrated next). At high solids concentration, the effect of dilution due polymer adsorption is overcome by the effect of solids loading and the viscosity of the suspensions becomes considerably larger than that of the suspending fluid. The behavior becomes quite complex. The rapidly increasing viscosity with decreasing shear rate at low shear rates is indicative of a gel-type behavior and the shear thickening at high shear rates suggests flow induced structure.
1308
103
13%
~_~ 10 2
10 7,~
~'1 0]
2"~ 3-~
100
9
10 -3
10 -2
,
I
10 -!
10 0
I
10 l
10 2
[S "1]
Figure 2.3
Shear viscosity of suspensions of fumed silica particles in a 1.5 % mass polyacrylamide (PAA) solution in glycerine. The average particle size is 20 nm and the weight average molecular weight of the PAA is 2.1 x 106 kg/kmol. The particle concentrations are in mass percent (Adapted from Otsubo [21 ]).
Figure 2.4 reports simple shear viscosity data for suspensions of glass beads (average diameter of 49 ~tm) in a 1 mass % aqueous solution of gellan (a polysaccharide of molecular a weight of approximately 10~ kg/kmol). The unfilled gellan solution is shown to behave as an homogeneous polymer solution, describing a Newtonian plateau at low shear rate followed by a typical power-law behavior. Adding 4.76 % per volume of glass beads did not change much the rheological properties. The viscosity in the plateau region for the 4.76 % suspension is, however, slightly lower than that of the suspending solution. Adding more glass beads resulted finally in a large increase of the viscosity in the plateau region, but strangely enough the viscosity is shown to be unaffected by the solids content in the power-law region. To discriminate between the effects of polymer adsorption at the glass bead surface (believed to be negligible here) and effects of ionic content, we did measure the pH of the solution. It was found to increase from 6.79 for the unfilled solution to 11.53 for the 37.5 % suspension. Obviously, the chemicals used for the surface treatment of the glass beads interact strongly and change the polymer conformation in solution. Similar effects have been reported previously by Carreau [22]. Figure 2.5 shows that the viscosity of the unfilled aqueous solution of gellan is considerably reduced by increasing the pH of the solution from 6.8 to 11.4.
1309
7
10 3
~ ....
m
........
m
........
*~, **, "*._ 9 9 ~ - A 9149149149
10 2
m
. . . . . . .
9 [] * 9 9
0% glass beads 6.6% 14.6% 28.5% 37.5%
"m
........
u
........
m
pH = 6.8 pH=10.5 pH = 10.9 pH=11.3 pH=ll.5
101 100 10-]
........
m
........
10 "2
10-3
n
........
m
10]
m
........
10 ~
m
........
101
.
10 2
. . mi~mmm]
10 3
[s,] Figure 2.4
Simple shear viscosity of glass bead suspensions in a 1.0 mass % gellan solution in water (Unpublished data).
O001~ 9
. . . . . . . .
101
I
. . . . . . . .
I
. . . . . . . .
mmlmmmmmm m
I
9 []
r
9
10 0 9 9
9
pH = 6 . 8 pH =11.4
9 9 mo 9 9
lO-1 9 ..mm! . . . . . . . .
10 -2
I
. . . . . . . .
1O 1
I
I
10 ~
I
......
I
101
. . . . . . . .
I
10 2
10 3
~/ [S -1]
Figure 2.5
Simple shear viscosity of unfilled aqueous 1 mass % gellan solutions for pH values of 6.79 and 11.4 (Unpublished data).
2.4 Buoyant
or gravitational effects
Buoyant or gravitational effects could lead to results which resemble typical shear-thinning or thixotropic effects. This is demonstrated in Figure 2.6 taken from Acrivos et a1.[23] for two types of suspensions. The first one consisted of polymethy methacrylate particles in a mixture of glycerine and water. The particle
1310
diameters ranged from 125 to 150 ~tm and the density differences between that of the particles and fluid were 0.03 and 0.04 g/mL. The second one consisted of acrylic particles (diameter between 75 and 106 pm) in a Dow Coming fluid with a density difference of 0.07 g/mL. A double gap Couette geometry was used for measuring the viscosity. In the figure, the reduced viscosity as a function of a reduced shear rate in steady-state conditions is reported for four different solids volume fractions. The reduced shear rate is based on the fluid viscosity and gives an order of magnitude of the ratio of viscous to buoyancy forces; ho is the height of fluid in the Couette geometry. The apparent shear-thinning effects observed in Figure 2.6 have nothing to do with shear-thinning properties of the suspensions. Initially due to the small differences in density, sedimentation proceeds leading to higher stresses in the lower part of the Couette geometry. But at higher stresses particles are re-suspended by shear-induced diffusion and the particle concentration reaches an equilibrium profile, which is a function of the applied shear rate. The solid lines in the figure are the predictions of the model developed by Acrivos et a1.[23] to account for sedimentation and shear-induced effects.
102
a,
r
050
0.40
~- 101
0.30
10~ [_.,.,., 10 -3
10 -2
10 -1
100
101
102
9rl~' 2h0gAp
Figure 2.6
Apparent reduced shear viscosity for suspensions of non colloidal particles in Newtonian fluids, due to density differences between the particles and the suspending fluid. The open symbols are for the first suspensions; the close symbols are for the second type of suspensions (Adapted from Acrivos et al. [23]).
1311
These simple results stress how careful one has to be in interpreting unusual phenomena. Many other difficulties may be encountered, especially when using polymer matrices. Other major problems are related to polymer degradation when experimenting with polymer melts at high temperature. In the case of polymer solutions, solvent evaporation could be a major nuisance as indicated by Acrivos et a1.[23] when long experimental times are required to reach, for example, steady conditions. For highly filled systems, it may just be impossible to make reasonable measurements especially at high shear rates due to wall effects, fracture in the suspensions, and other complications. 3. M O D E L I N G In this section we discuss key results of the literature illustrating various behaviors observed for non interactive and interactive particles in Newtonian and non-Newtonian matrices. Useful empirical viscosity expressions to describe properties of concentrated suspensions, as well as recent rheological models for characterizing thixotropic effects, are presented. 3.1 Hard sphere suspensions in Newtonian fluids Following the theoretical analysis of Einstein [2] for creeping flow of very dilute suspensions of spheres, various phenomenological models have been proposed to predict the rheological behavior of non dilute suspensions of hard spheres in Newtonian fluids. In the hard sphere concept, the spheres are assumed to be ideal spherical particles with a Dirac interaction potential. These assumptions lead to only hydrodynamic and Brownian forces. The Einstein theory neglects the effects of other particles (particle-particle or/and hydrodynamic interactions). Interactions with neighboring particles is accounted for in a first approximation by a quadratic term with respect to the solids fraction, as proposed by Batchelor [24]. Krieger and Dougherty [25] modified the Mooney [26] model to account differently for crowding effect. The relative viscosity rl~ as a function of the volume fraction ff is then given by the following simple expression:
(l m)
-p
(3.1)
1312
where p = (l)m[]]], (])m is the maximum packing fraction and [11] the imrinsic viscosity. The Krieger-Dougherty relation reduces to the Maron and Pierce [27] expression for p = 2, which corresponds to the theoretical case of spherical particles with [11] equal to 5/2 and ~ = 0.80. However, (I)m is usually kept as a fitting parameter. The viscosity models for concentrated suspensions are usually expressed as a function of the maximum packing fraction ~ , which is a function of shape and size distribution of the particles. The values reported in the literature range from 0.18 for carbon fibers with shape factor L/dp = 27 to 0.71 for uniform hard spheres. Chang and Powell [28] have investigated the effect of size distribution using particles of two well-defined sizes. The results were reported in terms of two parameters K, the diameter ratio of the small particles, and r the volumetric proportion of small particles. Their results and those of Storms et al. [29], Poslinski et al. [ 14] and Shapiro et al. [30] show that for a fixed volume fraction, the relative viscosity decreases as K decreases and a minimum in viscosity is observed around = 0.25. The viscosity reduction could be quite spectacular as shown by Poslinski et al. [14].
I
103[ .... ~t_,, 0
102 10 1
10~
Figure 3.1
I
I
I
0.1
0.2
0.3
I
0.4
I
i
0.5
0.6
Relative viscosity versus dispersed phase volume fraction for latex and silica suspensions (combined data of Krieger (1972) and de Kruif et al. [31]). The solid lines are the best fits using the KriegerDougherty equation (3.1) (Adapted from Barnes et al. [33]).
1313
The value of the maximum packing fraction also depends on the arrangement of particles. The maximum packing value for monodisperse particles ranges from 0.52 for cubic arrangement to 0.74 for a hexagonal close packed one. The maximum packing organization varies with flow: de Kruif et al. [31 ] and Krieger [32] have shown that the maximum packing fraction and the intrinsic viscosity [1]] vary with shear rate. Figure 3.1 reports the reduced viscosity as a function of the volume fraction for the combined experimental data of de Kruif et al. [31 ] for silica suspensions and of Krieger [32] for latex suspensions. The best fit of the KriegerDougherty equation allows to obtain at high shear rates ~m== 0.71 and [rl]= = 2.71 and at low shear rates d~mo = 0.63 and [1]]o = 3.13. Since only the hydrodynamic and the Brownian forces govern the behavior of suspensions of colloidal hard spheres, the rheological properties can be scaled as a function of the radius of the particles, a, the Boltzmann constant, k, and the temperature T. Krieger [32] proposed a semi-empirical relation based on the CrossWilliamson [34] model to describe the shear dependence for the viscosity of concentrated suspensions viscosity. The viscosity is defined by the following relation: rl -rl~
1
11o-r1=
1 +o/o c
(3.2)
26 O BZOH
22-
-"~
9 m-CRESOL "
~"18
14 10
'
]0-2
,
]0-I
]0 0
,
]01
O"r
Figure 3.2
Reduced viscosity versus reduced shear stress for 50 vol %
monodispersions of polystyrene latex particles in three different media (Adapted from Krieger [32]).
1314
This equation describes shear-thinning effects from a zero shear rio to a high shear limit rl=; Oc is a critical reduced stress for which the reduced viscosity, (rl - rl=)/(rlo- rl=) - 1/2; %(- oa3/kT) is a reduced shear stress. Figure 3.2 reports the reduced viscosity as a function of the reduced stress for suspensions containing particles of two different diameters dispersed in two different media. A unique curve is obtained, independent of particle sizes and of nature of the suspending medium. I
I
I
I
101
rim
-~--O O
l0 0
.
O
o /
10 -l
a3G ' kT
[]
[]
10 -2
10 -3
m
10 "4
10-1
I
I
I
10 0
101
10 2
10 3
a20~ Do Figure 3.3
Dimensionless elastic modulus and reduced dynamic viscosity versus dimensionless frequency for a 46 vol % suspension of silica spheres in cyclohexane- ffl a = 28 • 2 nm; (3 a - 76 + 2 nm. The solid lines are the predictions using the Jeffreys model (Adapted from Mellema et al. [35]).
Mellema et al. [35] have investigated the linear viscoelastic properties of suspensions of silica spheres of two different radius values (28 and 76 nm) dispersed in cyclohexane. They obtained master curves for a dimensionless elastic
1315
modulus G 7cT/a 3 and the reduced dynamic viscosity rl'lrlm as a function of a dimensionless frequency eoa2/Do where Do is a characteristic diffusivity. The results are illustrated in Figure 3.3. The dimensionless elastic modulus is shown to be a quadratic function of the frequency at low frequencies and reaches a plateau at high frequencies. The reduced dynamic viscosity depicts slightly shear-thinning effects between a low frequency plateau to a limiting high frequency value, which corresponds to a high frequency viscosity larger than that of the suspending fluid. A very similar behavior was observed for the steady shear viscosity. The data are fairly well described by a simple Jeffreys model. The electrostatic and sterically stabilized particles are assumed as a first approximation to form a rigid stabilized layer. The scaling rules for hard spheres suspensions, as Equation 3.1 or 3.2, can be used provided the thickness of the stabilized layer is taken into account in calculating the effective radius of the particles. In electrostatic stabilization, the thickness layer, the so-called Debye length, is a function of the electrolyte concentration and of the medium. In the case of steric stabilization, one must include the polymer layer thickness which depends strongly on the affinity between the polymer and the suspending medium, as characterized by the Flory-Huggins parameter X- This approximation is valid when the compressibility of the layer is negligible, that is: for a thickness layer smaller than the core particle radius and for moderately concentrated suspensions. Frith et a1.[36] proposed the following modified Krieger equation to predict the viscosity of soft sphere suspensions: rl - rloo
1
(3.3) 13o - rloo
1
+(Or/Oc )m
where m is a material parameter. Good agreement between the model predictions and the experimental data was obtained for suspensions of PMMA latex (sterically stabilized) particles in Exol. When the interparticle interactions become dominant, the rheological behavior of suspensions deviates from the behavior of hard sphere suspensions. Krieger and Eguiluz [6] observed that electrostatic repulsion between particles can drastically modify the suspension viscosity at low shear rates and can eventually induce a yield stress. Figure 3.4 reports the reduced viscosity as a function of the reduced shear stress for suspensions of PVC latex particles dispersed in water containing HC1 at different molar concentrations. The electrolyte had been added to suppress electrostatic repulsion between PVC latex particles. The added counterions screen
1316
off the charges on the particle surface. When electrostatic repulsion is dominant, an agglomeration of particles is induced. At low shear rates, the viscous forces are not sufficiently large to break down the structure, responsible for the rapidly increasing viscosity with decreasing shear rate and for the existence of a yield stress. At high shear rates, the flow is controlled by the hydrodynamic forces and the shear viscosity becomes approximately independent of the electrolyte concentration. 10 6
O 1.9xl04 M HCI ~
deionized
O
ut.I O ~ . - -
D ~o
10 4 s..
9
10 2
1.9x10"3M HCI
9.4xl(I2M HCI 9
A~ m . $ gA~llLemm~
O
%Oo
1.9xl0"2M HC! ' ~
10 ~ 10 .2
I
10 o
10 2
O'r
Figure 3.4
Reduced viscosity versus reduced shear rate for 40 vol % suspensions of PVC latex particles in water containing HC1 at various molar concentrations (Adapted from Krieger and Eguiluz [6]).
3.2 Suspensions in non-Newtonian fluids
In industrial applications, non-Newtonian polymer solutions are frequently used as the suspending fluids. Generally, adding particles to a shear-thinning fluid does not alter the general shape of the viscosity curve. The viscosity curve is shifted vertically upward and the onset of shear thinning occurs at smaller shear rates with increasing solids loading, as observed by Nicodemo et al. [37]. The power- law index does not significantly change with solids concentration [38]. For high solids concentrations or low viscosity matrices, particle-particle interactions may become significant resulting in an apparent yield stress. Such yield stresses have been observed by Lobe and White [39] and Tanaka and White [40] for polystyrene melts filled with carbon black, titanium dioxide and calcium carbonate. Poslinski et al. [ 14] proposed a modified Carreau model to predict the rheological behavior of filled polymers:
1317
0o
rl -
1] 0
IYI + (1 +(t,~/)2) (1-")/2
(3.4)
where the zero shear viscosity and the characteristic time are described by MaronPierce expressions" -
-
(3.5)
and -2
(3.6) rim and tlm are respectively the viscosity and the characteristic time of the polymeric matrix. Poslinski et al. [ 14] obtained good agreement between the proposed model and the experimental data for glass spheres suspended in thermoplastic matrix, as shown in Figure 3.5. In this case, the glass spheres are not sufficiently interactive compared to the viscous forces and no apparent yield stress is observed. The rheological behavior of the suspensions is very similar to that of the unfilled polymer. The zero shear viscosity increases and shear thinning occurs at smaller shear rates as the glass spheres volume fraction increases.
lO
5
9 0% O13% m26% t~35% A46 % ~x60 %
104 r~
10 3
~" 102 101 10~ 10-2
10-1
10o
101
102
10 3
10 4
"~ [S "l ]
Figure 3.5
Shear viscosity of glass spheres at various volume fractions, dispersed in a thermoplastic polymer at 150~ (Adapted from Poslinski et al. [14]).
1318 Doraiswamy et al. [17] proposed an extension of the Cox-Merz rule for concentrated suspensions in polymer exhibiting a yield stress. They have shown that the steady shear viscosity and the complex dynamic viscosity of a concentrated suspension of silicon particles in polyethylene can be superimposed if the complex viscosity is reported as a function of the strain rate amplitude coy~ for oscillatory shear data obtained in the non linear domain. 3.3 Time
dependent
and
structural
models
Most industrial suspensions consist of interactive particles: paints, coating colors, inks,... The interactions between particles are responsible for the formation of a structure which is time and shear dependent. Adequate experimental characterization and modeling of the rheological properties of suspensions, especially for interactive particles, are very difficult tasks. The reader can find a rather complete review of thixotropic phenomena in [41 ]. Here, we summarize the main rheological models which are capable of describing yield stresses and thixotropic behavior and present our most recent ideas on the subject. The rheological behavior of suspensions made up of interactive particles is governed by a competition between micro-structure break-down due to flow and build-up due to collisions between particles induced by Brownian motion and flow. Hence, the evolution of the micro-structure is generally described by a kinetic equation of the following form [41 ]:
- a(1-~)b-c~'~a at
(3.7)
where ~ is the structural function and a, b, c and d are constants. To describe thixotropy, Quemada [42] used a Krieger-Dougherty expression:
l"lr (1 -
-
(D /-2 ~)m(~)
(3.8)
where the maximum packing factor is given by the following function of the structural parameter: ~),,,(r
_
1
( 1/~mo - 1/~,,,~,)+ 1/~,,,oo
(3.9)
1319
Here the maximum packing factor varies from (~moat low shear rates to (I)mooat high shear rates. The structural function ~ is obtained by solving the following kinetic equation:
0t
ta
(3.10) td
where t~ and td are characteristic times associated with the aggregation and desegregation process respectively. The ratio t~/td is assumed to be a function of the shear rate as follows: a
t
_
(3.11)
ta
where p ( > 0) is an empirical parameter and tc is a characteristic time.The following expression is obtained for the steady shear viscosity:
1 +(tc~)P ] (3.12)
where 1]o and ~1=are the limiting viscosity at low and high shear rate respectively. Good agreement was obtained between the proposed model and experimental data for blood and latex and silica particle suspensions. Quemada [43] also proposed the following Jeffreys-type model to describe the non linear viscoelastic behavior of concentrated suspensions" 1
Oo
G(~) at
+ ~ 1o
rl(~)
- - ~t - ~,(~) O~t
Ot
(3.13)
where G is a modulus and ~ a relaxation time, both functions of the structural parameter, ~. De Kee et al. [44] have described the viscosity of yogurt and mayonnaise using a model based on a structure kinetics. The structural function is obtained by solving the following equation"
1320
at
(3.14)
-C [~ l d( ~ - ~equiZ) ~
The shear viscosity is then given by the relation" O - ~(•
- "~E'qpe
-tp ~
)
(3.15)
Leonov [45] has developed a model to describe yield stress and thixotropic phenomena in filled polymers. He assumed that the total stress contains two main contributions: a viscoelastic contribution from particle interactions and a viscous one from the suspending medium" o - op + o m
(3.16)
The total stress tensor consists of a contribution due to particle-particle interactions, Op, and that of the suspending polymer matrix, o~, considered to be filled with inert particles. The Leonov model developed for homogeneous polymers is used to describe this last contribution. A kinetic equation is proposed to describe floc rupture and build-up in terms of characteristic times. Floc break-down is related to a critical elastic deformation. Leonov have shown that his model can describe thixotropic effects in stress growth and relaxation experiments, but it has been assessed using only steady shear (viscosity and primary normal stress difference) and and elongational viscosity data for filled polymer melts. Good agreement has been shown, as evidenced here for the steady shear viscosity of two filled systems. Figure 3.6 compares the model predictions to the steady shear viscosity as a function of shear rate for carbon black particles in polystyrene at various volume fractions (data of Tanaka and White [40]). In Figure 3.7, the corresponding data for a molten polystyrene filled with calcium carbonate particles of different sizes (data of Suetsugu and White [46]) are compared to the model predictions. Figure 3.7 shows that the Leonov model is able of predicting the drastic increases of the shear viscosity with decreasing size of the particles at constant solids volume fraction, ~ = 0.3 (comparable results have been reported for the effective elongational viscosity [45]). However, these data do not allow to test the model ability to describe time-dependent or thixotropic effects.
1321 106 PS/CB ~10
5
20% 10% 104 10 -2
Figure 3.6
10-1 'y [S-1]
......
10o
The predictions of the Leonov model for the steady shear viscosity compared with the data of Tanaka and White [40] for polystyrenecarbon black (dp ~ 0.045 pro) compounds at various volume fractions (Adapted from Leonov [45]). 107 PS/CaC03 [ 106
1 <--0.07 p m
z~ ~ - 10 5
9
_ 9
104 10-2
-~'a~~
-~-"
9
<--0.5 p m
- " " ' ~ . - - - - . ~ <-- 3
I 10-1
lam'
10o
? [s-l] Figure 3.7
The predictions of the Leonov model for the steady shear viscosity compared with the data of Suetsugu and White [46] for polystyrenecalcium carbonate compounds with various particle sizes at constant solids volume fraction, ~ - 0.3 (Adapted from Leonov [45]).
1322
Coussot et al. [47] proposed a model similar to that of Leonov to describe the rheological properties of a granular grease and a clay platelet suspension. The viscoelastic contribution is given by a modified Maxwell equation: 1 0o
1
P + ~ O
-
-
?
(3.17)
q(~) p
G O at
where Go is a constant elastic modulus and rl(~) is a viscosity function for the viscous contribution of the particles, which depends on the structural factor ~, determined from the following kinetic equation:
X--~- =
(1
-E.j)--yc ---0
(3.18)
with n T](~)
-
(3.19)
T}p
(1 - ~ ) - ' - 1
X is a kinetic parameter, yc a critical strain, 0 characteristic time, TIpa characteristic viscosity and n an empirical parameter. Yziquel et al. [48] proposed a different kinetic network model based on ideas ofMarmcci et a1.[49] and Coussot et al. [47] to describe the non linear behavior of fumed silica suspensions observed during oscillatory measurements. The stress is described by a modified upper convected Jeffreys model with a single relaxation time" 6 6t G(~)
--
+ n(~)
(3.20)
~r - q~.
I + n(~)
~
G(~)
with
G(r
= Gor + G=
(3.21)
where Tl= and G= are respectively the viscosity and the elastic modulus for the
1323 suspension with a destroyed structure and Go + G~ is the equilibrium value of the elastic modulus of the suspension; 6 /6t is the upper convected derivative. Contrary to Coussot et al., we assume that the modulus depends on a structural parameter as proposed by Marrucci et al. [49] and Quemada [43]; rl(~) is the structure-dependent viscosity defined by the relation: r I(~) -
1]~
(3.22)
1"1ois a characteristic viscosity and fl~) an empirical structural function chosen to obtain solid-like behavior at very small strains: 1
=
/ (1-n)/2 1
(3.23)
I
where ~, the structural parameter is related to the number of interparticle bonds and ranges from 0 and 1. It is obtained by solving the following kinetic equation:
~'o0~
= (1-~)
- k2
z
5-
(3.24)
o Yc with = 0
Go
rio +G
(3.25) oo
k I and ke are kinetic constants for the thermal build-up of the suspension microstructure and for shear induced break-down respectively, )~ois a characteristic relaxation time and I/, is the second invariant of the rate-of-strain tensor. Figures 3.8 and 3.9 compare the experimental data for the elastic and viscous moduli versus strain amplitude obtained for suspensions of fumed silica particles in paraffin oil and the values predicted by the Yziquel et al. [48] model. Two types of particles were used: Aerosil A200 (Degussa) particles have a diameter of 12 nm and are hydrophilic with surface silanol groups that can participate in hydrogen bonding. Aerosil R974 (Degussa) particles have also a diameter of 12 nm but are
1324
hydrophobic: a significant fraction of silanol groups have been substituted by methyl groups. Since the paraffin oil has no hydrogen bonding capacity, the particles can only interact between themselves. These interactions are stronger for the hydrophilic particles than for the hydrophobic fumed silica which has less silanol groups on their surface. The elastic modulus is shown to decrease rapidly above a strain exceeding 0.03 for A200 particles (Figure 3.8) and 0.008 for R974 particles (Figure 3.9) whereas the loss modulus increases markedly with strain amplitude, but the increase is less important for the R974 suspensions. For these suspensions, elastic effects dominate viscous ones, but the A200 suspensions have the highest elastic modulus (stronger network due to the hydrogen bonding). The moduli for these suspensions were found to be almost independent of the frequency [48]. Note that the moduli reported in the non linear domain are apparent values since the output signal is no longer a perfect sine wave. The model accurately predicts the elastic modulus decreases with strain as well as the initial increases of the loss modulus with strain. However, the model predicts subsequent decreases of the loss modulus at a strain that is smaller than experimentally observed. The model parameters used to fit the data are listed in Table 3.1 (the parameter k~ was found to be a non significant role on the predictions). As far as we are aware, no other models in the literature are able of quantitatively predicting the initial increase of
105
[
-
__
G' / ' - - - - - -
_----__ ...... i "~~"~".,.
---_-_- ~-~ i .....
1 04 9
7.0%
"
9
8.2%
Q~
"
10.0%
~.~" 1 0 3
"
11.5%
----- m o d e l
,,-" "
,,,,k,n~.~ . . "
9
""
-
102 10 .3
Figure 3.8
10 .2
~,o
1 o-1
Elastic and loss moduli as a function of stain amplitude for suspensions of A200 fumed silica particles in paraffin oil at different solids loadings. The solid lines are the Yziquel et al. model predictions. (Adapted from Yziquel et al. [48]).
1325
105 . . . . . . . . . . . . . . . . . . . . . . .
G
?
G
??
104
103 .
.
.
.
|
.
.
.
.
.
.
.
.
!
10.3
.
.
.
.
.
.
10 -2
.
.
10-1
3,o
Elastic and loss moduli as a function of stain amplitude for suspensions ofR974 fumed silica particles in paraffin oil at different solids loadings. The solid lines are the Yziquel et al. model
Figure 3.9
predictions. (Adapted from Yziquel et al. [48]). M o d e l parameters used for describing the data o f Figures 3.8 and 3.9
Table 3.1
Suspensions in naraffin oil
Go + G= .
(kPa~ .... ,
,,
,
~.o
Y C
nJ(Go + G3
k, / kl
n
(S)
(s) . . . .
7.0 % A 200
15.6
1.19
0.0320
0.0030
0.0085
8.2 % A 200
31.0
1.27
0.0320
0.0025
0.0075
10.0 % A 200
64.2
1.83
0.0320
0.0020
0.0041
11.5 % A 200
140
2.07
0.0320
0.0020
0.0020
8.2 % R974
4.70
0.72
0.0090
0.0620
0.0270
0.08
10.0 % R974
11.5
0.72
0.0070
0.0620
0.0270
0.08
11.5 % R974
19.0
0.72
0.0066
0.0620
0.0270
0.08
13.5 % R974
32.5
0.72
0.0060
0.0620
0.0270
0.08
0.0620
0.0270
0.08
,,
,
,
,,,
,
14.2 % R974
51.4
0.72
0.0057
1326
the loss modulus with strain. For example, the Coussot model [47] can describe as well the decrease of the elastic modulus with strain reported here, but the predictions for the loss modulus are considerably higher than the experimental values [48]. As mentioned by Leonov [45], the main advantage of these approaches is that no yield criterion is necessary to describe the transition from solid-like to liquidlike behavior as in the Oldroyd model, discussed in Section 1.
3.4 Shear thickening Shear thickening is defined as the increase of viscosity with shear rate. Most concentrated suspensions exhibit shear thickening at high shear rates, but the onset and the importance of shear thickening depend on the volume fraction, particle size distribution and viscosity of the suspending fluid. The increase of viscosity is attributed to a transition from a two dimensional layered arrangement of particles to a random three dimensional form as confirmed by Hoffman [50] using light diffraction and more recemly by Boersma et al.[51] using neutron scattering. According to Laun et al. [52], this order-disorder transition cannot explain all cases of shear thickening. Brady and Bossis [53] proposed that hydrodynamic clustering may even explain shear thickening. Shear thickening appears generally for volume fractions of solids larger than 50 % and the onset (shear rate at which shear thickening appears) decreases as the volume fraction increases. The increase in viscosity depends strongly on the shape and size of particles. According to Frith et al. [54], the data are not sufficiently accurate to affirm that the onset of shear thickening is a quadratic or a cubic function of the particle radius, but it seems that a cubic function is more appropriate for hard sphere suspensions and a quadratic function for soft spheres. Collins et al. [55] have noted that a large distribution of particle sizes can suppress shear thickening. The shape of particles is also quite important and particle anisotropy tends to produce shear-thickening effects at a lower shear rate and smaller solids fraction. Shear thickening in suspensions of aggregated particles depends strongly on the shear history. An example is given in Figure 3.10 which reports the viscosity of a 8 mass % suspension of fumed A300 (Degussa) silica particles in polypropylene glycol for different pre-shearing histories. The fumed silica are colloidal particles (diameter of 7 nm) which could form aggregates through hydrogen bonding via a polar solvent such as polypropylene glycol. Each curve
1327 presented in the figure was obtained using a controlled stress rheometer in the viscosity mode and steady or pseudo steady viscosity is reported as a function of the measured shear rate. A pre-shearing at different stress levels, but above the maximum stress obtained in the viscosity mode, was applied to the sample sufficiently long to reach steady states and erase previous shearing effects. Then the subsequent viscosity curve was obtained. We observe in Figure 3.10 that the low shear rate viscosity, the onset and the importance of the shear thickening decrease with pre-shearing effect. At low shear rates, the behavior is slightly shearthinning. Above a critical shear rate, the viscosity rapidly increases due to formation of aggregates. Under higher and higher stresses, the hydrodynamic forces break down the aggregates and the hydrodynamic radius reduces with the increasing stress level. Therefore, the viscosity is found to be considerably reduced and the onset of the shear thickening occurs at a larger shear rate. Finally, we observe that above a critical shear rate, the viscosity begins to decrease. This subsequent shear thinning is attributed to a further reduction of the aggregate sizes at higher shear rates.
101
.! , ,
9
9
aiiOOOPa
c~
.9 YYvV9 9
Wv ~Vv v V V v V V V v v v v v v v v v v v v W W Vv
'
l0 0
! .... ~
10 .2
........
~
10 -1
~ 9 ..~,d
~ ~ ......
10 ~
~
101
.
. ~ .... ~
102
........
I
103
Is-'] Figure 3.10 Pre-shearing effects on the steady shear viscosity for a 8 mass % suspension of fumed silica (A300) particles in polypropylene glycol (Unpublished data). Few models are available in the literature to describe shear thickening. A flexible one is the viscoelastic model proposed by De Kee and Chan Man Fong [56], which can predict shear thinning, shear thickening and thixotropy or rheopexy
1328
depending on the choice of model parameters. The following viscosity expression is obtained:
rI
_
rl~
(1
+
(bf~ - cfz) e
-IX o l
)
(3.26)
tXo
with r162 = k ( 1
(3.27)
+ bf~ - cf2 )
where rio, kc, b, c are contants and f~ and fe are functions of the shear rate. For (b fl - cf2 ) > 0, the model predicts thixotropy and for (b f / - c f2) < 0 rheopexy is predicted. For steady state, the viscosity becomes"
n(?) -
(3.28) tz
o
Shear thinning or shear thicknening is predicted if ( b f ( - c f 2 / ) > O or < 0, where fl/, f2/ are the derivatives with respect to the shear rate off1 and f2 respectively. Although the De Kee - Chan Man Fong viscoelastic model appears to be quite flexible, it cannot describe the pre-shearing effects reported in Figure 3.10. 4. L O W S T R A I N H A R D E N I N G
PHENOMENON
In this section, we highlight non linear effects encountered at relatively low strain with concentrated model suspensions and suspensions of industrial interest (coating colors used in the paper industry). The model suspensions discussed here consist of non colloidal particles suspended in a viscous matrix, a polybutene (PB, Indopol H100 of Stanchem), of low molecular weight, density equal to 890 kg / m 3 and viscosity of 24.5 Pa.s at 25 ~ This material is viscous enough so that the rheological properties are easily measurable, but not too much in order to highlight mechanical interactions between the particles. The industrial suspensions consist of coating colors used in the paper industry. Kaolin particles used for these suspensions are smaller, in the form of hexagonal platelets and show strong ionic interactions. Because of these strong particle-particle interactions, elastic effects are much more important, but some of the characteristics observed with the model suspensions are still detectable.
1329
4.1 Suspensions PVC particles in polybutene The non interactive PVC (polyvinyl chloride) particles were suspended in a nonpolar viscous matrix PB. The powder was an industrial non formulated PVC, with an average diameter of 12 ~tm and 50 % of the particles being smaller than 10 ~tm. A second maximum could be observed at 3 ~tm on the particle size distribution. Thus, the particle size distribution could be defined as just above the upper limit of the colloidal domain. Their shape was slightly ellipsoidal with a small to large axis ratio of 0.6. Suspensions at six different volumetric fractions (~ = 0.09, 0.18, 0.28, 0.37 and 0.47) were investigated, using a Weissenberg rheogoniometer equipped with parallel plates of 1.2 mm gap, and a Bohlin VOR with a Couette geometry. The results presented here are extracted and adapted from a previous publication on theological properties of filled polymers by Carreau et al. [20]. The specific viscosity as a function of the shear rate for the PVC suspensions is reported in Figure 4.1. For suspensions containing less than 30 vol % solids, the data fall approximately on a unique curve. The horizontal solid line corresponds to the theoretical value of 2.5 obtained from the Einstein analysis (Equation 1.1). This value represents fairly well the data for the most dilute suspensions, mainly at high shear rates. Using a maximum packing factor of 0.68, the Maron-Pierce equation (3.1) is shown to describe well the data for the 0.38 and 0.47 vol % suspension respectively at high shear rates. The shear-thinning effect obviously cannot be described by this simple relation, nor can be related to settling effects as reported by Acrivos et al. [23] and shown in Figure 2.6. These particles are non interactive, large enough so that the viscosity decrease with shear rate could not be associated with floc size variation nor to changes in the maximum packing fraction. The shape factor of 1.6 could explain the slight shear thinning observed for the lower concentrations but not the drastic effect observed for the 0.47 vol % suspension. For this suspension, the viscosity becomes unbounded as the shear rate goes to zero, indicating the presence of a yield stress and gel-type behavior, normally observed for particles having strong physical or chemical interactions. However, it is not the case here, but the yield stress is believed to be due to particle-particle steric-type interactions, as discussed below. As for time dependent models presented in Section 3.3, a structural parameter can be defined and included in a kinetic equation to follow the particles organization. With this approach, shear-thinning behavior (under steady state) is essentially based on effective volume fraction. In other words, when structural equilibrium is reached at a given shear rate, the structure is stable and can be modeled using the maximum packing fraction as a single parameter. As shown in Figure 4.2, the relative viscosity of each suspension is plotted against the volume fraction for four steady shear rates" 0.156, 1.56, 15.6 and 39.3 s~. The solid lines
1330
100 90 il 9 9
.
.
.
.
.
80 70 ~ - 60 ,
.
.
.
,
,i.
i..m
.i..,....,
0681
Einstein's equation -Maron-Pierce(O., = . ) [] 9% PVC / PB 1 A 18%PVC/PB "~ Q 28% PVC / PB --I
_
50
....
"9
9V 437 % P V C / P B
~o 1d
mmmmmmmmImm~m_m m
40 ~=47 30 20 ~ ~_ V _ _" _ ~ ...~]E _ l"Itl?_'~..ll'..~ llt .ay_,~. , _ ~ _ 3-7~176~ 10 o ...... , . . . . . . . . , 9 . 7 . . . . . , , ...7...# 10 -2
10-t
10 o
10 2
101
'~[S "1]
Figure 4.1
Specific viscosity for five different volume fractions of PVC/PB suspensions vs. shear rate. The horizontal solid line at the value of 2.5 represents the Einstein result. The horizontal dashed lines are the results calculated via the Maron-Pierce equation using a maximum packing factor of 0.68 (Adapted from Carreau et al. [20]).
35 30 25 ~. 20
O
0 . 1 5 6 s -1
121 A
1.56 s -! 15.6 s-I
V
39.3 s "j
15 10 5 0 0
10
20
30
40
50
60
r Figure 4.2
Relative viscosity of PVC/PB suspensions for four different shear rates as a function of the volume fraction. The solid lines represent the Maron-Pierce model predictions using a maximum packing fraction of 0.575, 0.629, 0.705 and 0.756 for the lowest to the highest shear rate (Adapted from Carreau et al. [20]).
1331
represent the Maron-Pierce model predictions using the maximum packing factor as an adjustable parameter. For each shear rate, the optimal values of ~)m equal to 0.575, 0.629, 0.705, 0.756 for each shear rate respectively. Hence, the steady state viscosity of this PVC/PB system can be simply modeled by this Newtonian empirical model using an adjustable maximum packing fraction or effective volume fractions.
"~n'
......
'
'
! 9%
"<) . . . .4.5%r . PVC/PB
0.1
18% 28% 37%
.~, ,~
,=.. I " ~ ~ 0.0
" 9.
10~
Ha 47%
~ "
"
"
_
,
"" ' ' I
I
101
I,
.
,
,
, , , . I
10 2
I
I
I
9
" ' ' "
10 3
[s-q Figure 4.3 Characteristic elastic time (~. - Nl/rlm?2) as a function of shear rate for PVC/PB suspensions at six different concentrations (From Carreau et al. [20]. Although the Polybutene matrix was inelastic in the measured shear rate domain, significant primary normal stress differences, N1, w e r e measured for the PVC/PB suspensions and increased rapidly with solids concentration. The results are reported in Figure 4.3 in terms of a characteristic elastic time, ~. = N~/1]m:~2 , a s a function of shear rate. For concentrations lower than 0.28, the elastic time is not significantly different from zero. On the other hand, for higher concentrations the elastic time is large at low shear rates and increases with concentration. However, drops rapidly with increasing shear rate, as observed before for the specific viscosity. Because of the nature of these suspensions, the elastic properties have to be related to particle-particle steric interactions; the particles are large and not affected by the Brownian motion and the matrix is purely viscous. The 47 vol % suspension depicts the larger elastic characteristic and the stronger shear-thinning behavior. To prove that shear thinning of such concentrated suspensions of non colloidal particles is not due to hydrodynamic effects, we present in the next three figures oscillatory data for the 47 vol % suspension. All experimems ,sere carried out using a controlled stress rheometer (CSM of Bohlin) equipped with a 25 mm
1332
parallel plate geometry. As illustrated in Figure 4.4 for a frequency of 0.628 rad/s, no viscoelastic linear domain could be detected. At low deformation, G 'and G "decrease with increasing strain, similarly to a viscoplastic material. The loss (viscous) modulus is much larger than the storage (elastic) modulus, as expected for non-interactive particles in a viscous matrix. At a strain equal to 0.25, G 'starts to increase rapidly followed by a similar, but more modest, increase of G '; at a deformation of about 0.5. Then at larger strain values, both moduli are shown to decrease. Obviously, for this system no linear or close to linear domain is depicted. Similar phenomena could be observed for concentrated suspensions made of different types of solids. Figures 4.5 and 4.6 present two more examples. .
.
.
.
.
.
9" I
.
.
.
.
.
.
.
1
. . . . . . .
1
.
.
.
.
.
.
.
1
10 3
.
.
=
.
.
.
.
"1
28
. . . . . . . .
S-I.,
02 ro
~,.~" 10 l
10 ~ 10 .4
10 .3
10 .2
10 1
-
-~
10 ~
101
102
yo Figure 4.4 Elastic and loss moduli at 0.628 rad/s vs. strain amplitude for a 47 vol % PVC/PB suspension (From Carreau et al. [20]). Figures 4.5 and 4.6 report the elastic and loss moduli as functions of the strain amplitude at ~ = 0.628 s~. The results of Figure 4.5 were obtained for 47 vol % dispersion of coarse polypropylene (PP) particles in the same polybutene. The average diameter of these PP particles was 300 lam and were spherical. To respect criterion 2.1, a gap of 3.0 mm was used. The particles and the fluid had almost the same density and measurement was not affected by settling, at least in the time frame of the experiments. Obviously, particles of this size are not affected by any type of colloidal forces. In this figure, a viscoelastic linear domain is observed until the strain amplitude reaches 10%, then a large and rapid increase of G "is observed, from a value of 13 to 160 Pa. Also G "is seen to start to increase at a larger strain
1333 103
a
a
I
a
9
!
a
l
a
i
e
9
I
i
| ,,,
ff
G 102
101
.
i
I
I
I
I
I
10 "1
Figure 4.5
I
I
I
I
i
i
i
10 ~
?o
Elastic and loss moduli at 0.628 rad/s vs. strain amplitude for a 47 vol % PP/PB suspension (Unpublished data [57]).
10 2
.
.
.
.
.
.
.
.
.
.
.
.
.a
.
G" lO1
10 o 10-1 Figure 4.6
O
100
Elastic and loss moduli at 0.628 rad/s vs. strain amplitude for a 24 vol % GB/PB suspension (Unpublished data [57]).
amplitude (~ 0.15) and increases from 230 to 370 Pa at a strain amplitude of 0.7, but the maximum is not reached. The results of Figure 4.6 were obtained for a 24 vol % of spherical glass beads (GB) of average diameter equal to 10 gm in the same PB. Here again for the time frame of the experiments, the effects of settling are believed to be negligible. The general aspect of the curves remains the same. For the lower strain amplitude a viscoelastic linear domain is observed, followed by slight decreases of the moduli up to a deformation of about 0.7, which is the critical strain at which the structure
1334 induction phenomenon is apparent. The strain hardening observed in this figure as well as the previous ones is believed to be caused by particle-particle interactions due to crowding or steric effects. From our observations, the critical strain is shifted to higher strain values for lower solids concentrations. We assume that this critical strain scales with the distance between particles or eventually with the radial particle concentration distribution function. The behavior presented here seems similar to the hardening behavior observed by Gadala-Maria and Acrivos [5]. The results of G 'and G "obtained from oscillatory measurements are based on the first harmonic of the measured strain signal and do not represent the entire signal. However, the increases with strain observed for each suspension are not an artifact due to instrumentation. At the critical and above the strain for which G ' reaches a minimum, the measured (output) strain was slightly skewed compared to the imposed sinusoidal stress (input signal of the controlled stress rheometer). During the tests, the edge of the sample was marked with a tracer and was videotaped using a camera equipped with a macro lens. No slippage at the wall or fracture in the sample could be detected. Figure 4.7 reports typical normalized signals for the suspensions which are discussed in Figures 4.5 and 4.6. The strain response for the Newtonian matrix (PB) is also presemed as a reference. The two solid lines represents the normalized stress signal (input) and the normalized amplitude strain response (output) respectively for the Newtonian PB, at 0.628 rad/s. The dot-dashed line represents the strain signal obtained for the 47 vol % PP/PB suspension and the double dot-dashed line corresponds to the 24 volume %
1
0
-1
0 Figure 4.7
2
4
t[s]
6
8
10
Normalized strain and stress signal obtained at a frequency of 0.628 rad/s vs. time for a 47 vol % PP/PB, and a 24 vol % glass beads/PB suspension. Also shown is the normalized strain output for the Newtonian PB matrix (Unpublished data [57]).
1335
GB/PB suspensions. In the strain-hardening domain, the deformation of the signal increases with volume concentration. For both suspensions, no deformation of the signal could be depicted at very low deformation. Figure 4.8 shows the influence of frequency on strain hardening for the 47 vol % suspension of PVC particles in PB, in terms of the complex modulus as a function of the measured strain amplitude. Strain hardening occurs approximately at the same strain, independently of the frequency, but because of instrument limitations it could not be observed at the largest frequency (63 rad/s). This finding supports our assumption that this strain hardening is due to steric or crowding effects and not controlled by stress. However, the magnitudes of the energy dissipated and of the strain hardening depend on the frequency. The same oscillatory shear data are re-plotted in Figure 4.9 in terms of the stress amplitude as a function of the strain rate amplitude, ~,~ and compared to the steady shear stress vs. shear rate data. The low frequency data reveal a very small elastic yield stress, ao, of approximately 1 Pa. The strain-hardening effect for the three lowest frequencies is quite visible and it is interesting to note that the stress amplitude at low frequencies approaches the steady shear stress. At high frequencies, the curves appear to collapse into a single curve similar to the steady shear curve, but much lower. Hence, because of strain hardening, the resistance to flow of concentrated suspensions is considerably greater under steady shear conditions.
9
104
"
~"
" ~ 1
~
------i
. . . . . . . .
i
~
. . . . . . .
i
. . . . . . . .
i
. . . . . . . .
i
~
~~i,t,i~,ati,,,,i.m~
63 s'l
103
6.3 s"l
ak
V
~
.
6
3
s-~
102 V V
101 10 -4
V V ~ . 0 6 3 ........
I
10 .3
. . . .
I ...I
10 -2
........
I
10-1
.
." .**.".'...I
10 ~
........
I
101
s-~ .......
102
0
Figure 4.8
Complex modulus vs. strain amplitude at different frequencies (0.0628, 0.628, 6.28 and 62.8 s~) for the 47 vol % PVC/PB suspension (From Carreau et al. [20].
1336 10 4 10 3
o 102 D 10~ 10o 10-4
constant shear and model
Jy
0
6 3 s -1
c o [] A V
6.3 s -1 0.63 s 1 0.063 s-1
....................................
l 0 "3
10 .2
10 -1
10 ~
101
10 2
~1, (1)y~ "1]
Figure 4.9
Steady shear stress and shear stress amplitude as functions of shear rate and strain rate amplitude for the 47 vol % PVC/PB suspension (From Carreau et al. [20].
Creep experiments were also carried out to confirm the strain-hardening phenomenon. The transient shear viscosity obtained in creep and the complex viscosity (at 0.628 rad/s) are plotted as functions of the strain and the strain amplitude respectively in Figure 4.10 for the same 47 vol % PVC/PB suspension. Inertial effects in creep were negligible and the viscosity obtained is the imposed stress divided by the instantaneous shear rate calculated for each experimental point. The viscosity functions, 11 and r I*, show two different distinct pattems. The complex viscosity, rl*, decreases initially with strain as shown before for the complex or the elastic and loss moduli. The creep viscosity, rl, obtained at constant shear stress slightly increases in the beginning because of a shear rate decrease with increasing strain. The interest of Figure 4.10 is, however, to show that the strainhardening phenomenon is observed at about the same critical strain for the two different experiments. This is another confirmation of our assumption that this strain hardening is due to steric effects. This can be seen as a structural barrier, large enough to increase the resistance to flow and to be responsible for an effective yield stress in non colloidal suspensions. From a rheological point of view, this limiting case is similar to a shear-thickening effect with a limiting infinite viscosity.
1337
[] r I , t ~ = 2 5 P a O rl, cr = 10Pa ~x rl*, 03 = 0.63 s-1
10 3
102
10 -3
10 -z
10-1
10~
101
~/, yo
Figure 4.10 Complex viscosity and transient shear viscosity in creep as functions of strain amplitude and strain respectively for the 47 vol % PVC/PB suspension (From Carreau et al. [20]). These experimental observations allow us to make the assumption that the strain hardening effects are included in the steady rheological properties. This contribution for non colloidal particles could be significant compared to hydrodynamic and other forces. As shown in Figure 4.9, the steady shear viscosity of concentrated suspensions in a Newtonian matrix can be modeled by a MaronPierce equation adding the contributions for the elastic yield stress, o o, and the supplementary strain induced stress, Ooi. The induced yield stress was found, from oscillatory shear experiments for the 47 vol % PVC/PB suspension, to be proportional to the frequency at a power 0.65. Hence, we have proposed the following extension of the Maron-Pierce model: o
-
-
G oY ,
o
-
9
(o ~ +
1o1
_< o 0 s)
-
~]m(1
(4.1)
GoY ~ -
(~m
)-2~t,
Iol > Oo
(4.2)
Using o ~ = 1 Pa, D= 188 Pa.s 0.65, s = 0.65 and (l)m-- 0.74, an excellent fit is obtained for the shear viscosity data of the 47 vol % PVC/PB suspension, as illustrated by the solid line in Figure 4.9. 4.2 C o a t i n g c o l o r s
Many industrial processes are transient from a material point of view. Although
1338
the flow rate or deformation rate could be quite large, the total deformation experienced by the material may remain relatively small. In general, restricting the flow analysis of industrial processes by using time independent or steady state viscosity could lead to a poor understanding of the phenomena envolved and incorrect modeling. One example of such industrial applications is paper coating, where a coating color is applied on the paper sheet and excess removed by a thin trailing blade. The gap between the paper sheet and the blade is in the order of few microns. As the blade is fixed and the speed of the paper sheet is in the range of 100 km/h, the calculated apparent deformation rate is of the order of 106 s~, but the corresponding deformation or strain for the flow under the blade is less than 10. Coating colors are concentrated colloidal suspensions. A typical formulation consists of water, mineral pigments as kaolin and/or grinded calcium carbonate, latex particles and/or starch used as binder and of different other additives playing a role on the rheological properties. Brownian and other colloidal forces affect the rheological behavior of this type of suspension. The results presented in this section are extracted and adapted from a publication on rheological properties of coating colors by Lavoie et al. [58]. The coating colors, used in this work, consisted of 100 parts by weight of pigments and 11 parts by weight of styrene butadiene carboxylated latex particles of diameter equal to 0.13 lam. The pigments were a standard #2 US kaolin and a standard delaminated US kaolin. The pH of the colors was adjusted to 8.0. The Brookfield viscosity of each color was then adjusted to 1 Pa.s (1000 cP) at 100 rpm by adding a "high viscosity" type CMC as dry powder. The coating colors containing the standard #2 kaolin were prepared at total solids of 67 and 63 wt.%, and the coating colors made up of the standard delaminated kaolin were prepared at 66, 64 and 62 wt. % solids. To emphasize the structure induction (strain hardening) phenomenon presented in the previous section, results obtained on two kaolin slurries at 68 and 71 wt. % and those of an industrial coating color of unknown formation are also presented. The rheological properties of the colors were determined using a strain-controlled Bohlin VOR and a stresscontrolled Bohlin CVO rheometer. A constant temperature of 20 ~ and a bob-andcup geometry (25 mm diameter and 1 mm gap) was used at all times. Steady shear viscosity, transient viscosity (imposed shear rate) as well as strain and frequency sweep measurements were conducted. At low strain or low stress, the rheological behavior of a coating color suspension is dictated by particle-particle (colloidal) interactions, it is strongly related to the formulation. Figure 4.11 presents the complex modulus as a function of the shear strain, for the coating color made with the delaminated US kaolin containing 66 wt. % solids and for the industrial coating color. A linear viscoelastic domain is observed for the complex modulus, characteristic of solid-like
1339
viscoelastic behavior. The limit of the viscoelastic domain corresponds to the yield strain, Yo, which is related to the yield stress by Oo/Go. The rapid decrease of the complex modulus corresponds to plastic flow behavior, and the strain hardening, as discussed in the previous section, appears at a critical deformation. 1~ ." . . . . . . . . ! -
. . . . . . ii' ~ . i o ~ & ~
[] ~
"
[]
~
/
o
L
10 ! =--
.
S
000
del.cdcr(0.lI-lz)
9
Plastic Flow
Msmelmtic 100
.......
0 ~~cacr(U-~)
~
'
Dxr~ .
10 -3
.
.
.
.
.
.
.
.
.
.
.
.
.
10-2
.
.
I
!
i
|
|
10-!
. . . .
10 ~
yo
10 !
Figure 4.11 Complex modulus as a function of strain amplitude for an industrial and a 66 wt. % delaminated kaolin (model) color (From Lavoie et al.
[58]). The oscillatory data for the industrial coating color, in terms of the amplitude of the complex stress (o ~ G*y ~ as a function of the strain rate amplitude (y~ is presented on Figure 4.11. Also shown are the steady shear stress vs. shear rate data. As for the 47 vol % PVC/PB suspension, strain-hardening effects appear for the 101
Model (5" Steady Shear
10 ~ o
~
10 -! cyo 10
-2
10 -3
10-2
10-1
10 0
% 7~
[] A V 101
co 31.4 s -1 6.28 s -1 0.628 s - ! 10 2
10 3
[s-q
Figure 4.12 Steady shear stress and shear stress amplitude as functions of shear rate and strain rate amplitude respectively for the same industrial coating color as in Figure 4.11 (From Lavoie et al. [58]).
1340
three frequencies, and the stress amplitude at low frequencies approaches the steady shear stress. The resistance to flow in steady shear situation is larger than that in high frequency small amplitude shear case because of the stain induced structure. The induction structure effect for the 68 wt. % standard American delaminated kaolin suspension is presented in Figure 4.13. The complex viscosity (1"1")at o equal to 0.628 rad/s and the transient viscosity (1]) in creep are presented as functions of the strain amplitude (yo) and shear strain (y) respectively. These rheological properties reflect strong particle-particle interactions. No linear viscoelastic domain is shown and at low strain the elastic contribution is 5 times larger than the viscous one, characteristic of a solid-like structure. At a strain amplitude of 1.2, the structure induction phenomenon from the complex viscosity becomes apparent. The creep experiments were conducted at three different shear stresses, one lower than the yield stress (0.5 Pa), a second just above the yield stress (1 Pa) and finally at a stress of 5 Pa for which the slurry behaves as a fluid. These curves confirm the different flow regimes observed in oscillation. For a shear stress of 0.5 Pa, the slurry behaves like a solid. The stress is large enough to deform the material but not enough to make it flow. The large increase in the viscosity is the reflect of the structure break-down under strain, as discussed in Section 3 in terms of the loss modulus for fumed silica suspensions (see Figures 3.8 and 3.9). For the second experiment (at 1 Pa) a large increase for the creep viscosity is also observed, but at larger strains, in the domain of the induced structure phenomenon. After reaching a maximum, the viscosity goes down as the material is beginning to flow. Finally, the experiment at 5 Pa reveals the same type of behavior, but the strain hardening effects are attenuated because the hydrodynamic forces dominate. Comparable behavior would be observed for different types of concentrated suspensions, including industrial coating colors. During the paper coating process, coating color experiences a wide range of deformation rates, the highest being in the nip of a roll or under a blade. However, at high deformation rates, contact times are very short and, in these conditions, the total deformation remains very small. Hence, the low strain behavior is as much important as the steady flow properties for applications such as blade coating. We stress that the coating process cannot be fully characterized using only steady state rheological parameters. The magnitude of the deformation for the transient experiments used to generate the data of Figure 4.13 is comparable to the total deformation experienced in coating for the flow under the blade. The rheological behavior under the blade is essentially that of a thixotropic material, which is time dependent.
1341
|01 o" = 0.5 Pa
W 10~
r
q lO-1 10 -2
Pa
O- = 5 p a 10 1
10 ~
101
10 2
T,T ~
Figure 4.13 Complex viscosity (60 = 0.628 rad/s) and creep viscosity as functions of strain amplitude and strain respectively for a 68 wt. % delaminated clay slurry (Adapted from Lavoie et al. [58]).
5. C O N C L U D I N G R E M A R K S In this chapter, we have illustrated and discussed various aspects related to the rheological behavior of concentrated suspensions. Concentrated colloidal suspensions are characterized by yield stresses, shear thinning and thixotropic behavior, and for small amplitude shear flow, the elastic moduli could be significantly larger than the loss moduli. Obviously, colloidal forces and particleparticle interactions play a major role on the properties. Changes in ionic strength of the suspending fluid could lead to drastic differences in the viscosity and the shear-thinning characteristic of colloidal suspensions. Suspensions of non colloidal particles in Newtonian fluids are expected to be simpler to describe. This is done using the concept of hard spheres and the increase of viscosity can be accounted for by using the Krieger-Dougherty, Maron-Pierce or other equivalent expressions, in terms of a maximum packing factor. For suspensions in non-Newtonian fluids, the behavior is similar to that of the suspending fluid, provided particle-particle interactions make negligible contributions to the stresses. The viscosity can then be described combining viscosity models for polymers and Maron-Pierce expressions for the filler effects. We have shown that many experimental problems could lead to erroneous results or results which lead to wrong interpretations. The dispersion of solids especially in the case of colloidal particles could result in totally different rheological properties. For example, an apparent yield stress and shear thinning may be caused
1342
by aggregates which could be broken down by stronger shearing. Buoyant or gravitational effects can lead to time dependent and shear-thinning properties. The interesting results of Acrivos et al. [23] reporte d in Figure 2.6 stress how carefully experiments have to be designed and conducted if one wishes to obtain the true rheological properties of suspensions. Many other sources of difficulty have been identified in Section 2 and this implies that considerably more experimental effort is needed to elucidate the phenomena responsible for the rheological behavior of concentrated suspensions. Badly needed are transient data which will allow us to better characterize thixotropy, determine the key parameters which control thixotropy, and assess constitutive relations proposed to describe their behavior. Some of the recently proposed rheological models discussed in Section 3 appear to be quite promising with respect to accounting for thixotropic as well as non linear and shear-thinning effects. A most interesting feature of these models is that they are capable of describing a smooth transition from solid-like viscoelastic behavior to a liquid-like viscoelastic one. However, the specific forms for the kinetic equations proposed to describe the time evolution of the structure need to be assessed using more transient data, such as stress growth and large amplitude oscillatory shear data. For example, in the rheological models of De Kee and Chan Man Fong [56] and ofYziquel et al. [48], the structure is assumed to be dependent on the second invariant of the rate-of-strain tensor. In light of our results on strain hardening discussed in Section 4, we believe that invariants of the strain tensor need also to be considered, at least to account for the crowding effects in highly concentrated suspensions. The rheological behavior of a concentrated suspension is the results of various contributions which are very difficult to measure and accurately describe. When dealing with industrial suspensions, other complications may arise from uncontrollable parameters, such as purity and technical specifications of the various components, etc... Suspensions are usually considered as homogeneous materials (a continuum) and the solid particles are assumed to move as the surrounding fluid (the so-called affine assumption). Close to maximum packing, the continuum hypothesis is no longer valid as the motion of a particle is restricted by the surrounding particles. We have shown, using many different concentrated suspensions, that such steric effects lead to strain-hardening phenomena for strain values of unity or less depending on the distance between surrounding particles. This strain hardening has been observed in different deformation modes such as oscillatory, creep or shear stress growth experiments. This is responsible for the non linear behavior at very low strains of concentrated suspensions of so-called non interactive particles, and apparent yield stresses, shear-thinning effects and primary normal stresses detected for non colloidal suspensions in inelastic fluids. This
1343
strain hardening appears to be an important rheological characteristics of industrial suspensions in high strain rate but small strain processes such as coating colors in blade coating. The maximum stress for various coating colors in stress growth experiments have been successfully correlated to their runnability in coating [58]. We believe that the continuum concept is no longer valid for highly concentrated suspensions. Hence, in modeling the rheological properties of such materials, one needs to relax the assumption of affine deformation embedded, for example, in the model of Yziquel et al. [48]. This can be done empirically using the GordonSchowalter [59] derivative with a slip parameter instead of the usual upper convected derivative. A better and probably more successful route is to use computer (direct) simulations of the dynamics of particles injected in a suspending fluid. Such (2-D) computer simulations have been carried out for a few hundred long rigid fibers [60] and the qualitative results were shown to be of interest. The 2-D simulations of large size spherical particles, for a number large enough to account for situations close to maximum packing, should reveal realistic results that explain the strain-hardening effects discussed in Section 4. We intend to report such results in a near future and eventually to extend the simulations to 3-D. REFERENCES ~
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Metzner, A.B., J. Rheol., 29 (1985) 739. Einstein, A., Ann. Physick, 19 (1906) 289; 34 (1911) 591. Tsai, S.C., K. Ghazimorad, J. Rheol., 34 (1990) 1327. Tsai, S.C., K. Zammouri, J. Rheol., 32 (1988) 737. Gadala-Maria, F. and A. Acrivos, J. Rheol., 24 (1980) 799. Krieger I.M. and M. Eguiluz, Trans. Soc. RheoI., 20 (1976) 29. Leong, Y.K. and D.V. Boger, J. Coll. Int. Sci., 136 (1990) 249. Tsai, S.C. and B.M. Viers, J. Rheol., 31 (1987) 483. Mewis, J., J.W. Frith., T.A. Strivens, and W.B. Russel, AIChE J., 35 (1989) 415. Russel, W.B., D.A. Saville, and W.R. Schowalter, Colloidal dispersions, Cambridge University Press, Cambridge, MA (1989). Bingham, E.C., Fluidity and Plasticity, McGraw-Hill, New York, NY (1922). Casson, N, in Rheology of Dispersed Systems, edited by C.C. Mill, Pergamon Press, New York, NY (1959). Herschel, W.H. and R. Bulkey, Kolloid-Z., 34 (1926) 291. Poslinski, A.J., M.E. Ryan, R.K. Gupta, S.G. Seshadri, and F.J. Frechette, J. Rheol., 32 (1988) 703; and 751. Oldroyd, J.G., Proc. Cambridge Philos. Soc., 43 (1947) 100.
1344 16. 17. 18. 19. 20.
21. 22.
23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.
Yoshimura, A.S., and R.K. Prud'homme, Rheol. Acta, 26 (1987) 428. Doraiswamy, D., A.N. Mujumbar, I. Tsao, A.N. Beris, S.C. Danford, and A.B. Metzner, J. Rheol., 35 (1991) 647. Husband, D.M, N. Askel, and W. Gleissle, J. Rheol., 37 (1993) 215. De Kee, D., and C.J. Duming, in Polymer Rheology and Proceeding, edited by A. Collyer and L.A. Utracki, Elsevier, New York, NY (1990). Carreau, P.J., P.A. Lavoie, and M. Bagassi, in Macromolecular Symposia, edited by E. Papirer and J.P. Trotignon, Huthig and Wepf Verlag, Oxford, CT, 108 (1996) 111. Otsubo, Y., J. Coll. Int. Sci., 112 (1986) 380. Carreau, P.J., in Transport Processes in Bubbles, Drops and Particles, edited by R.P. Chhabra and D. De Kee, Hemisphere Publishing Corporation, Washington, DC (1992) 165. Acrivos, A., X. Fan, and R. Mauri, J. Rheol., 38 (1994) 1285. Batchelor, G.K., J. Fluid Mech., 83 (1947) 97. Krieger, I.M. and I.J. Dougerthy, Trans. Soc. Rheol., 3 (1959) 137. Mooney, M., J. Coll. Sci., 6 (1951) 162. Maron, S.H. and P.E. Pierce, J. Coll. Sci, 11 (1956) 80. Chang, C. and R.L. Powell, J. Rheol., 38 (1994) 85. Storms, R.F., B.V. Ramarao, and R.H. Weiland, Powder Tech., 63 (1990) 247. Shapiro, A.P. and R.F. Probstein, Phys. Rev. Lett., 68 (1992) 1422. de Kruif, C.G., E.M.F. van Iersel, A. Vrij, and W.B. Russel, J. Chem. Phys., 83 (1985) 4717. Krieger, I.M., Adv. Coll. Int. Sci., 3 (1972) 111. Barnes, H.A., J.F. Hutton, and K. Walters, An Introduction to Rheology, Elsevier, New York, NY (1989). Cross, M.M., J. Coll. Sci., 20 (1965) 417. Mellema, J., C.G. de Kruif, C. Blom, and A. Vrij., Rheol. Acta, 26 (1987) 40. Frith, W.J., J. Mewis, and T.A. Strivens, Powder Tech., 51, 27 (1987). Nicodemo, L., L. Nicolais, and R.F. Landel, Chem. Eng Sci., 29 (1974) 729. Mewis, J. and R. de Bleyser, Rheol. Acta, 14 (1975) 721. Lobe, V.M. and J.L. White, Polym. Eng. Sci., 19 (1979) 617. Tanaka, H. and J.L. White, Polym. Eng. Sci., 20 (1980) 949. Barnes, H.A., J. Non-Newtonian Fluid Mech.,70 (1997) 1. Quemada, D., Rheol. Acta, 17 (1978) 632. Quemada, D., Rev. G6n. Therm. 279 (1985) 174. De Kee, D., R.K. Code, and G. Turcotte, J. Rheol., 27 (1983) 581.
1345
45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
Leonov, A.I., J. Rheol., 34 (1990) 1039. Suetsugu, Y. and J.L. White, J. Appl. Polym. Sci., 28 (1983) 121. Coussot, P., A.I. Leonov, and J.M. Piau, J. Non-Newtonian Fluid Mech., 46 (1993) 94. Yziquel, F., P.J. Carreau, and P.J.Tanguy, submitted to J. Rheol. (1997). Marrucci., G., G. Titomanlio, and G.C. Sarti, Rheol. Acta, 12 (1973) 269. Hoffman, R.L., Trans. Soc. Rheol., 16 (1972) 155. Boersma, W.H., J. Laven, and H.N. Stein, AIChE J., 36 (1990) 321. Laun, H.M., R. Bung, S. Hess, W. Loose, O. Hess, K. Hahn, E. H~idicke, R. Higmann, F. Schmidt, and P. Lindner, J. Rheol., 36 (1992) 743. Brady, J.F. and G. Bossis, Ann. Rev. Fluid Mech., 20 (1988) 111. Frith, W.J., P. d'Haene, R. Buscall, and J. Mewis, J. Rheol., 40 (1996) 531. Collins, E.A., J. Coll. Sci., 71 (1979) 21. De Kee, D. and C.F. Chan Man Fong, Polym. Eng. Sci., 35 (1994) 438. Cotton, F., M.A.Sc. Thesis, in preparation, Ecole Polytechnique, Montr6al, QC, Canada (1997). Lavoie, P.A., P.J. Carreau, and T. Ghosh, Tappi J., in press (1997). Gordon, R.J. and W.R. Schowalter, Trans. Soc. Rheol., 16 (1972) 79. Thomasset, J., M. Grmela and P.J. Carreau, J. Non-Newtonian Fluid Mech., in press (1997).
1347
MODELLING THE FLOW NARROW GAPS
OF FIBER SUSPENSIONS
IN
F. Dupret and V. Verleye ~
CESAME, Unitd de Mdcanique Appliqude, Universitd catholique de Louvain, avenue G. Lemaftre 4-6, B-1348 Louvain-la-Neuve, Belgium Tel "32 (0) 10 472350, E-mail "[email protected] a current address 9 TECHSPACE AERO, route de Liers 121, B-4041 Milmort, Belgium
1. INTRODUCTION A sustained industrial interest has been focusing on fiber-reinforced polymers, which combine enhanced stiffness with all the attractive properties of thermoplastic or thermoset materials [1-4]. Thin composite parts are often mass-produced by injection or compression molding, which result in a reduced cycle time and automatic processing. However, the thermo-mechanical properties of the endproduct are highly dependent on the fiber orientation distribution [5-7], which is itself governed by the flow of the compound during processing [1-3,6,8,9]. In particular, obtaining random orientation (which is convenient in most practical cases) is almost impossible in view of the strong aligning effect of shear and extensional flows [10-13]. This is where numerical simulation can play a key role, the ultimate aim being to control the flow during filling by designing the delivery system (or selecting the initial load location) and introducing appropriate stiffeners, in order to obtain the desired fiber orientation in the final part. The physical behavior of fiber suspensions is very complex in view of the effect of fiber-fiber, fiber-flow and fiber-wall interactions. The orientation field is indeed govemed by the ensemble averaged motion of the compound, but in tum affects the rheology of the mixture, while the walls play an essential role in narrow gap flows. Several material and geometrical parameters must be considered in order to understand and classify these interactions.
1348 9 The effect of the solvent rheology is certainly essential [14,15]. However, the available fiber motion models are basically limited to Newtonian solvents. 9 The fibers are first of all characterized by their aspect ratio A r - I/d, where l and d are the fiber length and diameter [10-12]. In addition, the fiber thermo-mechanical properties (including the fiber flexibility [16-17]) can play a non-negligible role. Most theories are, however, restricted to rigid fibers. 9 The volume fraction of the fibers (r directly affects the mixture rheology [18]. General suspensions are classified as dilute
(O(Ar-2)
or concentrated
(~O
(O(Ar-1)<~)), depending on
whether the particles do not interact, have only hydrodynamic interactions, or interact through both hydrodynamic and direct kinematic effects [3,19-24]. Whereas the dilute and semi-dilute theories have been extensively investigated [12,21-30], the much more frequent case of concentrated suspensions remains limited to phenomenological models [1-3,6,20,31-33] and a theory in its beginning stages [34-37]. 9 In narrow gaps of thickness 2h, fiber out-of-plane rotation is basically governed by the ratio I/h (narrow meaning that e-h/L<
1349
42,53-56] in order to predict the flow-induced orientation field in short or long fiber injection or compression molded suspensions will be analyzed. An importam part of this work was reviewed by Crochet et al. [9]; subsequent developments were basically devoted to developing and validating the natural closure approximation [4,40-42,56,57] and to using the simulation results to predict the thermo-mechanical properties of the molded part [5-7]. The flow numerical model will only be summarized, since it is detailed in the companion chapter "Industrial Flows" Polymer Processing and Rheology" Modelling and Simulation of Injection Molding", that will be referred to as "Chapter IM" in the sequel. The plan of the chapter is as follows : Section 2 is devoted to the general physics of fiber suspensions, considering the effect of the flow on the orientation field, and the fiber-fiber and fiber-flow interactions; the entire Section 3 focuses on the closure approximation problem; Section 4 investigates the particular theory and numerical modelling of fiber suspension flows in narrow gaps; finally, some simulation results and experimental validations are presented in Section 5.
2. THE PHYSICS OF F L O W I N G SUSPENSIONS 2.1 The motion of a single fiber in a liquid The starting point of fiber orientation modelling was the work of Jeffery [10], who analyzed the motion of a rigid infinitely small particle in an incompressible Newtonian solvent, with no external forces or torques nor Brownian motion. From these hypotheses, it results that the flow around the particle is of a creeping type and that, at the particle scale, the asymptotic velocity gradient at infinity is uniform at all time. In addition, as the particle (or fiber) was assumed to be an axisymmetric ellipsoid (Figure 1), Jeffery could prove that any vector Pi of constant length and parallel to the fiber axis satisfies the evolution equation []
(1)
Pi - - ~" dkl PiPkPl (PmPm) -1 , []
where ~, = (ArL 1)/(Ar 2+ 1) is the fiber shape factor and
Pi
denotes a mixed
convected derivative of Pi" a
OPi
Pi = Dt
1 (1 + ~) ~ 1 ~j 2 ~ j j P J + -2 ( 1 - A ) -~ii P J "
(2)
The Einstein summation convention over repeated indices is used and t , x i and stand for time, Cartesian coordinates and velocity components, while the Vi
1350
l
e1
Figure 1. Sketch of an ellipsoidal fiber and unit vector representing the fiber orientation.
+&jl&i)
DPilDt-OPilOt+vjOPilOx2 and d i j - ~ 2 ( & i l & j denote the material derivative of Pi and the rate of strain tensor, respectively.
expressions
Equation (1) deserves a short comment. In a general flow, the evolution equation of an infinitesimal material segment d x i is dxi
-
--(dxi
Dt
) -
dxj
-- 0
,
while an infinitesimal material surface
(3)
dS
normal to the unit direction
ni
obeys, in incompressible flow, the equation dS i -
with
( dSi ) + dS i - n i d S .
- dx,/lldxll
or
dS j - 0
The unit vectors
da,/lldSll)
(4)
,
Pi
parallel
to
dx i
or
dS i
(with
satisfy (1) with ~ - 1 or - 1, respectively, corre-
sponding to infinitely slender or flat particles. Hence, while the left-hand side of (1) is a mixture between upper and lower convected derivatives, due to the effect of fiber aspect ratio, the right-hand side arises from using normalized vectors Pi (instead of deforming vectors such as dxi o r d S i ) t o represent the fiber orientation. Observe also from (1) that % t (PiPi)-0. The presence of a normalization term in the right-hand side of (1) causes significant modelling difficulties via the induced non-linearities. The Jeffery equation has been extensively studied (cf. [8,58-60]). Bretherton [11] developed the general solution in simple shear flow, and proved that the motion of the axis of most rigid bodies of revolution is identical with that of some ellipsoid. The creeping motion of infinitely small particles in a Newtonian solvent was further investigated by Batchelor [19,61], who developed the slender
1351
body theory, and Brenner (cf. the references in [12]), who highlighted the intrinsic tensors governing the motion of arbitrary particles under external forces and torques. Vincent and Agassant [13] validated the Jeffery model by means of extensional flow experiments, while Givler et al. [62, 63] and Henry de Frahan et al. [39] developed numerical tools to solve (1) for plane and multi-faceted thin shapes. From these investigations, it is worth recalling that, in extensional flow, the fibers quickly align with the principal stretching direction. In shear flow, each fiber rotates along a given "Jeffery orbit" at a non-uniform rate with a period
T-2zr (Ar + Ar-1)/y
(where
~ ' , a s defined
from )?2 2dijdij,
denotes the
shear rate); for slender fibers (Ar >> 1), rapid tumbling motions separate long periods of nearly in-plane orientation; for inf'mitely slender fibers ( Ar = oo ), each in-plane orientation is an unstable equilibrium position and the motion is no longer periodic (see also the stability analyses of MacMillan [64] and Szeri et al. [60], and the investigation of the 3D transition from in-plane to uni-directional orientation in shear flow in [9]). The Jeffery model remains useful to evaluate closure approximations or other effects by averaging the results obtained with a large set of fibers [65-67]. Ingber and Mondy [59] have shown from exact flow calculations that Jeffery's theory is robust for rigid fibers, even when weak hydrodynamic interactions with other fibers or the walls are considered. Similar conclusions can be drawn for semidilute suspensions from the analysis of Koch and Shaqfeh [23] and the experiments of Stover et al. [68]. The theory should be further elaborated in order to take the solvent viscoelasticity into account, since experiments indicate that various behaviors exist, which often disagree with theoretical predictions (cf. Feng and Joseph [14], Iso et al. [15] and the literature therein).
2.2 Distribution functions; fiber-fiber interaction The Jeffery model is unable to take into account fiber-fiber and fiber-flow interactions (this being necessary with semi-dilute or concentrated suspensions). Moreover, orientation must be treated as a random phenomenon, since fibers are never perfectly aligned in the initial state or in the inlet flow sections. Distribution functions must therefore be introduced [20-22,27,30,58,68-72]. The distribution function ~?(p, x,t) is the orientation probability density as a function of p , x and t. Hence, dP - ~?(p,x,t)dp is the elementary probability of finding a fiber whose orientation is located in the infinitesimal zone dp surrounding p. Note that p is a unit vector, and dp is an element of area of the unit sphere U . In the 2D case (which applies in the molding of concentrated suspensions of
1352 long fibers),
dp is an element of length of the unit circle (which will also be
denoted by U ). The distribution function is normalized by the condition ~u ~ dp - 1, while the symmetry condition W(p) - q ' ( - p ) is introduced to remove the indeterminacy between fibers oriented in the directions p and - p . The evolution equation of W has the form [1,73] : ,
(5)
o%t
-
where
O~/03Pi denotes the constrained divergence operator on U , while Pi
represents the material derivative of Pi expressed from a constitutive equation as a function of p, Vv and W. The operator tg/Opi is constrained because
Ilpll- . is no problem because Pi is itself constrained to be orthogonal to Pi when lid- 1, and it is sufficient to replace everywhere tg/o3Pi by (t)/o3Pi- PiPj t~/Opj) to obtain (Pl,P2,P3)
are not independent variables since
non-constrained equations. The constitutive equation proposed by Folgar and Tucker [20] was based on adding a rotary diffusion term to the material derivative Dpi/Dt provided by Jeffery's equation (1), in order to represent the geometrical and random effect of fiber-fiber interaction : ~ln~I/ 1 o~vi 1 ~j Pi -- -~(1 + / ~ ) ~ j pj ---~(1--/~)~/pj -/], dkt PiPkPt - C 1 9 7 ~ oqPi
(6)
where & is the equivalent fiber shape factor of the fiber (taking its real shape into account) and C I is the (phenomenological) interaction coefficient. The last term of (6) tends to randomize the fiber orientation by flattening the distribution function. Due to the factor ~ , the final fiber orientation does not depend on the global magnitude of the velocity field, but only on its spatial distribution. More precisely, when velocities are multiplied by an arbitrary uniform positive scale factor F(t), the final orientation field is not changed provided the time scale is adapted, because all the terms on the right-hand side of (6) are multiplied by F (t). On the other hand, reverse motion does not bring back the initial orientation (as the Jeffery model does), because ~' does not change sign with the velocity direction. All this is in agreement with the physics of fiber-fiber interactions of purely hydrodynamic or kinematic origin. On the contrary, in the case of Brownian interactions, D t =Ct~, (and not C t ) is a material function of temperature, volume fraction and orientation distribution [58,69,70]. It should be noted
1353
that Hinch and Leal [69a] def'me the Brownian effect as strong, weak, or very weak, according to whether
(Ar + Ar -])-3 3.01 .
>> D , / y .
DI/~/>> 1,
1 >>
DI/~z >> (Ar + Ar -1)-3,
The same scaling could thus be considered for
0 Experimentali Approximated Fit
or
C I .
(a)
2.5 C I =0 . 0 0 4 4
=.9 2.0
_
Aspect ratio
=83 g ].5
Volume fraction =0.013
o'1
?:51.o
O.5 0
(3
-rr/2
3.0
0 Orientation Angle, c~
77"/2
' ( y : 1.56) 0 Data Points
'
(b) . . . .
- - Approximated Fit
2.5
2.5
g~o
~'2.0
(c)
LI.
t.l.
~0
0 Data Points (7': 1.56) Initial Orientation (Exp.) Approximated Fit
1.5
C
.91,5 C I: 0.004
e~
.o
-~. =ul
~1,0
i:51.0 -
-.,,
0.5
0.5
0
-
CI: aO15
-
1~~
o_ D o o o 0 -~'/2
i 0 Orientation Angle, ~)
0 71"12
-71"12
I 0 Orientation Angle,
~'/2
Figure 2. Experimental validation of the distribution function in shear flow : (a) steady distribution and best fitted theoretical result; (b) transient data and theoretical predictions with the same parameters as in case (a); (c) same situation as in case (b), with C / - 0.015. (From [20]).
1354
The most significant consequence of (6) is that, in any steady homogeneous flow, an asymptotic W exists that can easily be determined by letting DW/Dt vanish in (5). According to Folgar and Tucker [20], this steady state is more aligned in elongational than in shear flow (in agreement with experimental observations in molding processes). Figure 2(a) shows a typical comparison between experimental orientation measurements and best fitted theoretical predictions in steady shear flow. Although the agreement is excellent, further comparisons in transient shear flow with the same value of CI (Figures 2(b) and 2(c)) indicate that interaction is more important when the fibers are in a random state than when alignment is high. It should be emphasized that the Folgar-Tucker model of fiberfiber interaction is phenomenological and does not rest on a kinetic theory. Experimental measurements of C/ are described in [1,20,74] (see also [8]), while the numerical simulation of the motion of many simplified rod-like particles by Yamane et al. [66] provides estimates of CI whose order of magnitude agrees with the experimental values. A correlation between CI and the average interfiber spacing in concentrated suspensions is proposed in [34,35]. However, the only comprehensive theory available concerns semi-dilute suspensions [22-24,2830,68,71,72,75]. Very briefly, the principal concept of this theory is the screening length (Batchelor [ 19]), which scales the short-range screening of hydrodynamic fiber-fiber interactions and is of the order o[d/2 (r
111~-1)1/2] (it is thus
close to the average inter-fiber spacing) [22,23,28,75]. The interaction model of Fredrickson and Shaqfeh [28] and Shaqfeh and Koch [22,23] is based on using multiple scattering expansions of the average Green's function in the LaplaceFourier space. Orientational and translational di~sivities, and drift velocities can be described by this approach [22,71,72], which was validated by the experiments of Stover et al. [68] and the multi-particle numerical simulations of Claeys and Brady [76] and Mackaplow et al. [29,30] (note that both the dilute/semidilute transition and the semi-dilute regime are very well predicted by a dilute model with two-body interactions [28b,29,30,75]). In spite of these significant developments, the resulting models are extremely complex and do not reach the concentrated regime (except in recent work by Sundararajakumar and Koch [37]). They thus remain basically useless for the practical simulation of molding processes. Further research is therefore needed. More complex constitutive laws than (6) can be imagined, since interaction depends on the rate of strain and is different when the fibers are aligned and when orientation is random. Anisotropic effects could be considered by replacing in (6) the orientation rotary diffusion flux Cti --CtYc)W/OPi by an expression of the form ~Oli - - K i j ~OtlJ/6~Pj, with a general positive definite interaction ten-
1355
sor Kij. The latter can be developed as a sum involving the 9 base syrrmaetric tensors constructed from pi and dii (with coefficients depending on their invariants and W ), and must respect invariance of fiber orientation with respect to time scale changes. Hence, the general theory is very complicated.
FiA Figure 3. Conservation of the fibers from isotropic to actual configuration. A particular class of distribution functions introduced by Dinh and Armstrong [21] exhibits properties of key importance for the closure problem. This class is obtained by solving the evolution equation (5) with infinitely slender fibers (~ = 1) and a vanishing CI, starting from the isotropic orientation state W0,
~o - / J r
for 3D orientation,
~o - l~2jr for 2D orientation.
(7)
A simple analysis shows that Wc is governed by the deformation gradient Fia from initial to actual state (FiA - Ox i/OxOA, with capital letters indicating the reference configuration). Indeed, the relation between initial and final orientations (pO and Pi ) is directly obtained by solving (1) for there is no fiber-fiber interaction. Moreover, infinitely slender fibers obey the same equation as normalized infinitesimal vectors, and thus, from the relation dx i - giA dx 0 , it is easy to show that Pi-FiA pO
i..o]l-1
and
p~
Pi
]-l.pl1-1.
On the other hand, as
C t = 0, the distribution function evolution equation can be integrated with respect to time (Figure 3) :
Wo dpo = Wc (p,x,t) dp ,
(8)
where qJ0 is given by (7), while dpo and dp denote associated infmdtesimal elements of area (or length) of U . Therefore, dPo/dp is the Jacobian of pO expressed in terms of pi. After some calculations, this provides from (7) the distribution functions one is looking for :
1356
{
WC ( P) - l~4zr(Bij-lpi Pj )-3/2
in 3 D orientation,
q*c (P) - /l~2jr (Bij-lpi Pj ) -1
in 2 D orientation,
(9)
where B 6 - Fia FjA is the Finger strain tensor from initial to actual state. It is clear that W c
has 5 degrees of freedom in the 3D case and 2 in the 2D
case, since Bij is symmetric and its determinant is 1 due to the suspension incompressibility. When the coefficients Bij are arbitrary numbers constrained by these requirements, the distributions
Wc(p)
form a class called by Verleye et
al. [40-42,56] the class of canonical distribution functions, since they considered that it is the simplest class of distribution functions and used it to build the natural closure approximation by non-linear projection. This class was also used by Szeri and Lin [46] to approximate the solution of the fiber orientation problem (with Brownian motion) on the basis of the double-Lagrangian technique of Szeri and Leal [44]. Lipscomb et al. [27] proved that the same class is generated by the motion of finite aspect ratio fibers without fiber-fiber interaction, since an equivalent strain tensor Fi,~q can be defmed from (2) by the relation Dt
=
]
+ (~ - 1)dij Ffq
"
(10)
Both 2D and 3D analytical solutions for W without fiber-fiber interaction were investigated by Altan et al. [73]. See also numerical solutions in [77]. 2.3 Orientation tensors Although W provides a complete description of the orientation state in the suspension, its use is cumbersome since W ( p , x , t ) has 6 independent variables in
the 3D case (and 4 in the 2D case). This is why Hinch and Leal [26b], Lipscomb et al. [47] and Advani and Tucker [43a], followed by many other authors [2,4,9, 27,39-41,43b,45,48-50,53,65,73,78-88], developed the use of orientation tensors (which are successive and more and more detailed summaries of W). The n thorder orientation tensor a n is obtained by averaging the n th dyadic power p| of p : an - ~W(p,x,t)p|
.
(11)
u From the normalization and synnnetry properties of W, it is clear that a 0 is the constant 1 and that all odd-order orientation tensors vanish.
We will restrict
1357
ourselves to the 2 nd-, 4 th- and 6 th- order tensors aij, aijkl and aijklmn, with
aij - ~W piPjdp ,
aijkl
~W piPjpkPldP ,
--
u
aijklmn
=
.-..
(12)
u
The basic properties of orientation tensors are full symmetry and normalization [43a]. It is convenient to introduce the symmetrization operator S as follows : for a genetic tensor b, S(b) is the average of all the transposes of b. Hence,
and
S (bij)
S (bijkl)
are defined by ,
S ( b ij ) - /~2 ( b ij + a f t )
(13)
S(b(ikl) = /~24(bijkl +bijlk +bikjl +bjikl +...)
(24terms),
(14)
S(bijklmn ) . The full symmetry
with a similar expression (with 720 terms) for
property, which is a direct consequence of (12), writes as : aij - S ( a i j ) ,
aijkl - S ( a i j k l ) ,
aijklmn = S(aijklmn )
.
(15)
On the other hand, the normalization condition results from the fact that Pi is a unit vector. Hence, from (12), aii -
1,
aijkk
-- aij ,
aijklmm -- aijkl
9
(16)
This shows that all the informations provided by aij are contained in aiju , and the same for aijkt and aijklmn. Nevertheless, the deviatoric parts of the orientation tensors can be defined by the expressions [43a]
I,ta ij d aijkl
- a ij -
_iS. . N tj ,
aijkt
--
(17) 3
6------~-S ) "kS(~ij(~kl), 4 + N (Sijakt (4 + N)(2 + N)
and so on with N = 3 or 2 depending on the problem dimension, in such a way that their traces vanish : d
d
aii - O ,
aijkk -- O ,
The tensors
d aij,
d aijkt,
d
aijklmm -- 0
(18)
.
d aijklmn ... are independent and
W
can be recovered
from the expansion [89]
W(p) - A o +
A1
d )d a(i ( P i P j
d
+ A 2 aijkl(piPjPkP
l
)d
+ ...
,
(I9)
1358
where ( p i p j ) cl , ( p i P j P k P t ) d ... are the deviatoric parts of P i P j , PiPjPkPt ... d (defined in the same way as aij,
d aijkl ... in (17)), while
A0,
A1, A2 ... are
coefficients which depend on N (see [43a]). It should be pointed out that the approximations obtained by limiting the number of terms in (19) are not very accurate since the expansion is linear in the powers of Pi (see [90]). Once the orientation tensors are defined, it is very easy to establish their evolution equations by orientation averaging. According to Advarli and Tucker [43a], this is performed by multiplying (6) by the successive odd dyadic powers of Pi and integrating the result over U . After some calculations involving careful integrations by parts since c)/Op i is constrained, this provides the equations ij
- -2~aijktd~a - 2 C 1 ~ ' N a q
,
( [aokt
- - 4 X aokt,,, , dmn - 4 C r y
d 3
N
( N + 2)a01 a + ---S(a~6kl +4 )N I-1
(20)
)
"
rl
The mixed convected derivatives a ij and aokt are defined in the same way as rl
P i in equation (2)" []
V
ijkl
--
A
v a ~2(1 + Z) aijkt+/~2 (1 - X)aiykt,
V
V
A
(21)
A
where a ij , a ijkt, a q and a ijkt denote the upper- and lower-convected derivatives of aq and aqk l . In particular, [] Daij aij = Dt
l(l+,~)(O~ia Ova) 1 (~. oars) 2 ~.Ox k kj + aik + -~(1 - ~) -'--t akj + aik . (22)
The 1~t and 2 "a equations (20) are embedded, in the sense that the trace of the 2 nd equation is exactly the 1st equation, whose trace is itself vanishing. Hence, only one equation need be considered in a given model. Most authors select aij as the basic variable (with 5 unknowns in 3D problems) [2,9,26b,27,39-41,43,4549,53,78,79,81 ], while Altan et al. [65,80,82,83,86,87] investigated the 2 "a option (i.e., to solve for aijkt, with 14 unknowns in the 3D case), in view of the direct effect of aijkt on the thermo-mechanical properties of the product [7].
1359
Solving the problem in terms of orientation tensors has the drawback that the evolution equation of the 2nO-order orientation tensor involves the 4th-order tensor, whose equation itself involves the 6th-order tensor, etc. To circumvent this difficulty, a closure approximation expressing aijkl algebraically in terms of aij (or aijklmn in terms of aijkt ) is required. In view of its importance, this issue is ex-
amined in a separate section.
2.4 Rheological aspects; fiber-flow interaction Detailed analysis of the theology of fiber suspensions is not the object of this chapter, since basically orientation-flow decoupled simulation models are considered. Nevertheless, the relevant literature will be reviewed in view of its importance for future developments. Following Prager [91], Batchelor [25], Brenner [12], Hinch and Leal [26b], Lipscomb et al. [27] and Ausias et al [92], the extrastress zij can be expressed for a Newtonian solvent in the generic form [2] 9 ~'ij -- 21"ld ij + 2 Tl ~) [ A a ijkl d kl + B ( d ik a kj + a ik d kj ) + C d ij + 2 O a ij ] ,
(23)
where 7"/ is the solvent viscosity, while A, B, C and D are coefficients which primarily depend upon the suspension concentration and the aspect ratio and degree of alignment of the fibers, but might also be functions of the invariants of ai# l and dij [93]. The dilute theory is well-established [10,12,25,27,58,61,69, 94] while, starting from the pioneer work of Batchelor [19] and Dinh and Annstrong [21], the semi-dilute theory has been developed by Shaqfeh, Koch, Fredrickson and co-workers [28-30,37,75], and validated by means of experiments [68,92,95] and numerical simulations [29,30,37]. Mass, momentum and heat transfer have been investigated. It is worth noting that, when A is put in the form A - 1 / 2 ar 2 [ln(X/d)] -1 , the length X
(which is thereby defined) scales
as the fiber length l for dilute suspensions, and as the screening length for semidilute suspensions. Hence, as explained in Section 2.2, X is in this latter case of the order of magnitude of the average inter-fiber spacing (and not the average shortest distance between two fibers as in [21]), for any orientation distribution. The coefficient B is negligible for slender fibers [27], while C = 2 and D vanishes in the absence of Brownian motion. No complete theory exists for concentrated suspensions. Semi-empirical values of A, B, C and D are proposed in [96]. Phenomenological models combining extensional and shear viscosities were developed by Gibson and co-workers [6, 31-33] (see also [97]), while Toll and M~nson [36] and Sundararajakumar and
1360
Koch [37] investigated the effect of non-hydrodynamic mechanical fiber-fiber contacts. Dynamic simulations of suspensions of rigid or flexible fibers were performed by Yamamoto et al. [17b] by modelling each fiber as an array of spheres. Another important issue concerns the coupled numerical solution of the orientation and flow problems, which was investigated in planar or axisymmetric 2D situations by Lipscomb et al. [47], Papanastasiou and Alexandrou [77], Rosenberg et al. [48], Ranganathan and Advani [35], and Azaiez et al. [51] (see also [82,83, 86]). The complex numerical effect of the wall no-slip condition, which causes periodic fiber tumbling and generates very sharp orientation boundary layers that cannot be resolved by mesh refinement, is discussed in [47,48]. Recent work by Kabanemi et al. [52] indicates that coupling the flow and orientation predictions becomes possible in 3D free-boundary flows by using the VOF method. The coupled models adapted to narrow gap flows are reviewed in Section 4. Finally, it should be noted that several authors are currently investigating the combination of viscoelastic and fiber-induced stresses, without however taking into account the local effect of viscoelasticity on the fiber motion (cf. [51,98]).
3. THE CLOSURE PROBLEM
The use of orientation tensors to simulate fiber suspension flows represems the best modelling methodology as long as an appropriate closure approximation is developed (Hand [93], Hinch and Leal [26b], Advani and Tucker [43]). This issue brings about the question of whether the (2n+2)th-order orientation tensor can be expressed as a given function of the (2n)th-order tensor in such a way that the predicted values of the 2nth-order tensor obtained by solving the associated momem equations provide good results "in most circumstances of practical interest" This question expresses the situation well. Exactly in the same way that orientation tensors are summaries of the distribution function, the moment equations completed by a closure approximation represent summaries of the distribution function evolution equation, and summaries never provide full information. Hence, the closure problem is not precisely posed in a mathematical sense, because this would necessitate to define the probability of the different states of the system (each being represemed by a given distribution function), which is impossible for there is a lack of sufficiem knowledge about the problem physics and also because of the tremendous difficulty of the mathematical problem. Therefore, heuristic methods, which combine theoretical and physical considerations with numerical and real experimems, remain the only way to tackle the question.
1361
3.1 Basic concepts Some basic theory is necessary. Details can be found in [43]. For the sake of simplicity, only the 4th-order tensor closure will be investigated. Both the 2D and 3D problems will be addressed in view of their technological importance. According to the Cayley-Hamilton theorem, any 3D closure approximation can be expressed as a sum of products of the 27 base tensors ((~ij(~kt, ~ik~jt ,... t$ijakt .... aijakl .... t$ija~amt , ... a i m a m j a ~ a n t , ...) by scalar functions of the invar-
iants of aij, P - det (aij)
,
-1
D - P aii
.
(24)
In the 2D case, the sum is limited to 12 terms and a single invariant needs to be considered: O - det (aij)
(25)
.
Note that symbols are selected in such a way that equations (24) and (25) match when a 3D orientation state tends to becoming 2D. Optimally, closure approximations should satisfy the f u l l normalization and symmetry conditions (16.2) and (15.2). However, this is not a constraint [26b,43b] and it is sufficient to require p a r t i a l normalization and symmetry : aiikl -- 1///N (aiijj - 1)t$kl -- akl ,
(26)
aijkl - ajikl -- aklij ,
(27)
since only the product aijkl dkl
plays a role in the moment equations (20), while
partial symmetry guarantees that aijkl has the symmetries of an elasticity tensor and can be used for thermo-mechanical property predictions [7]. To achieve partial symmetry, the number of base tensors in the expression of Ctijkl reduces to 12 in the 3D case, and 6 in the 2D case, while full symmetry is obtained when aijkl is expressed in the form a ijkl -- [~1 S ( ~ij ~kl ) + [~2 S ( ~ij a kl ) + ff~3 S ( a ij a kt ) + + r4 S(t$ijakmaml) + r5 S(aijalonaml) + r6 S(aimamjaknanl)
or
,
(29)
aijkl -- 131S(tSijgkl)+ 132 S(~ijakl ) + 133 S(aijakl ) ,
depending on whether the 3D or 2D case is considered. The coefficients or
fll-3
are functions of the invariants
P
and
D , or
(28)
fll-6
D , respectively.
1362 Normalization imposes the conditions
I101!1I1 I12o 4r 0 0
7 0
132 = 133
or
I80
0 7
1-6D 5
]Iil
4(P-D) 2(3- 4D)_]
+
(30)
,
[-4D
depending on the problem dimension. The first closure approximations that were successively proposed by various research groups are listed below. All these formulae, which are discussed in [43b], satisfy full normalization. 9 The linear closure (Hand [93]) writes as
aii~t =
3 6 (4 + N)(2 + N) S (6iy6 ~ ) + 4 + N S (aij6~t ) ,
in such a way that
(32)
a~.kt- 0. This fully symmetric closure approximation is
exact when the fibers are in a random state. It can lead to unacceptable physical inconsistencies both in shear and elongational flows [4,40,43b]. 9 Two composite closure approximations (named HL1 and HL2 in [43b]) were developed by Hinch and Leal [26], in order to fit properly both in the strong and weak Brownian motion limits. These closures behave well in some cases (especially HL1), but have strong deficiencies (with possible pathological behavior) in other cases [43b,44b,45,99]. 9 The quadratic closure aijkt- aijakl was proposed by Hinch and Leal [26b], Doi [70] and Lipscomb et al [27,47], and has only partial symmetry. It is exact when the fibers are perfectly aligned, but can provide poor (and even nonphysical) results in transient flows or with a non-vanishing C t [4,40,43b,44b, 73,84]. 9 The hybrid closure approximation was further developed by Advani and Tucker [43] in order to remove the problems associated with the linear and quadratic closures. In its best formulation [43b], it writes as
aijkl -- f , where f ' -
I
6
1
3 + N) S(t~ijt~kl)+ 4 + N S(aijt~kl ) + ( 1 - f ' ) (4 + N)(2
aijakl
1
(33)
N N det (aij) . The hybrid closure is exact when the fibers are in a
random state or perfectly aligned, and never behaves badly (except that it
1363
accelerates the orientation transients) [4,43b,73]. Moreover, it can be very easily implemented. 9 Quadratic and hybrid closure approximations for aijklmn in terms of aij~l are investigated in [35,65,67,80,82,83,86,87]. In spite of these results, several groups pursued their investigations in parallel since the problem had not received a really satisfactory solution. In the sequel, we will focus on the natural closure approximation [4,40-42,57], while the principal aspects of the best fitted orthotropic closure approximation [45] and the double-Lagrangian technique [44,46] will be summarized. ,r/
1
I 0.8
--,,--- SimpleShear
.-.
BiaxialElong.
-E?r- Shear/StretchA
o L UniaxialElong.
- O - Shear/StretchB
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
Figure 4. Left : space of possible orientation states in principal values and fitting domain. Right: points generated by calculation of the distribution function and used for the orthotropic fitted closure. (From [45]). 3 . 2 0 r t h o t r o p i c closures The term "orthotropic" closure was introduced by Cintra and Tucker [45] to denote a 3D closure approximation of aijkl that has the same principal axes as aij. Since aiykl must take all its directional information from aij, any closure approximation must be orthotropic. Hence, the closure relation expressing aijkt in terms of aij can be considerably simplified. Letting ~, 7/ and ~" denote the principal values of aij, it is easy to show by s ~ e t r y considerations that, for any orthotropic closure satisfying the normalization and partial symmetry conditions (16.2) and (27), there are, in principal axes, exactly 6 independent components aij~l. This number reduces to 3 when full symmetry (equation (15.2)) is required. Cintra and Tucker [45] selected allll, (/2222 and a3333 to provide these basic
1364
1.0
/
all
I-/
\\'.
'"
"
';i
---
,,,
.
[ an
.
.
/ 0.8 rl._
,--...-.- DFC - ORF
o.~
....
. ........... "~
..-".........
!
i~I
.<,
i
L .... "'"*"" ..~/. ~.~
I
................
0.2
0
............. 5
10
~"" 20
15
o.o 25
30
o
5
15
lO
Gt = reduced time
L~. . . . . 25
20
30
Gt = reduced time 0.3 az3
a12
-----
, DFC
......
/ ".. i "',,...
oaJ=
0.2
~,
,~" ...........
~x.~:
"-,~,..,
------
HL1
....
H,~
t " " ~.'K
.................................\
! "
i
. . . .
5
. . . . .L1 i
. . . .
HL2
'.~~\ ................HY8
0.1
~%. 0.0
. . . .
NAT
____"
............... . , ,
X.
DFC ORF
9
10
. . . .
! '
15
. . . . . . . .
! '
20
. . . . . . . .
! *
. . . . .
25
.
.
\./
0.1
. . . . .
30
0
'
. . . .
5
Gt = reduced time
'
10
. . . . .
*
. . . .
15
'
20
. . . .
*
25
. . . .
30
Gt = reduced time
Figure 5. Tensor components obtained by means of the best fitted orthotropic closure. The flow starts with simple shear, switches to shearing/stretching flow, and finally shearing continues with a strong stretching in the 3 rd direction (DFC" integration of distribution function; ORF" orthotropic fitted closure; NAT" natural closure approximation, 1 st version; HL1 and HL2" Hinch and Leal's 1 st and 2 no composite closures; HYB 9 hybrid closure). (From [45]). expressions. As ~, 7/ and ~" are linked by the condition (16.1), ~+r/+~"-1
(O<~,q,~'
,
(34)
two independent variables (~ and O ) can be chosen and, due to symmetry, it is sufficient to consider the domain
1 > ~ > 7/> ~" > 0
(Figure 4). Hence, 3 func-
tions (a1111, a2222 and a3333 ) m u s t be evaluated in this region. In order to obtain a fitted orthotropic closure, the 1st step consists in selecting an appropriate finite dimensional space for the functions a~l~l, a2222 and a3333.
1365
Quadratic polynomials in ~ and 7/ were selected in [45]. The 2 nd step consists in choosing a set of flows that generate a wide variety of orientation states. For that purpose, the simple shear, uniaxial and biaxial elongational flows, together with two sheafing/stretching flows were selected in [45] (Figure 4). In the 3 rd step, the polynomial coefficients of a l l l l , a2222 and a3333 are determined by least square best fit, with the exact values obtained by integrating the evolution equation (5). Impressive results were obtained by this technique (Figure 5), except for very low values of the interaction coefficient C I . In conclusion, the best fitted orthotropic closure approximations can easily be tuned to general or particular classes of flows as long as the fitting objectives are clear. Indeed, there is no unique solution to the closure problem and, if a general approximation is desired, there is a limit accuracy that cannot be improved on. Nevertheless, the formula proposed by Cintra and Tucker [45] today certainly represents the best choice in solving a very broad class of problems in view of its accuracy and ease of implementation. Further investigations could bear on selecting an optimal approximation space for allll, a2222 and a3333. A related question is to know whether constraints should be introduced in order to avoid discontinuities and derivative discontinuities when the solution is extended to the whole triangular domain of orientation states (Figure 4). 3.3 Natural closures The natural closure approximation of Verleye et al. [4,40-42,56,57] is defined by considering the set of canonical distribution functions introduced in Section 2.2. As this class has 5 degrees of freedom in the 3D case and 2 degrees of freedom in the 2D case, while the positive semi-definite 2nO-order orientation tensor itself has 5 or 2 degrees of freedom, respectively, since it is synunetric and its trace is l, there is a one-to-one correspondence between the set of 2nd-order orientation tensors and the subset of canonical distribution functions. The natural closure approximation is generated by non-linear projection onto this subset. More precisely, the relation (12.1), which links aij to ~ , can be inverted and
thus aij~t, which is a tensor functional of q~, can be expressed in terms of aij, and this in principle provides the natural closure. The existence of this theoretical closure was demonstrated by Lipsc0mb et al. [27]. However, no attempt was made by these authors to directly investigate and exploit the resulting relation between aijk! and aij. In theory, the natural closure approximation is completely specified by the above definition. However, direct explicit calculation of aij~:t as a function of
1366
aij is only possible (to date) in the 2D case [40]. Tuned numerical methods are therefore required in the 3D case to obtain an accurate approximation of this relation. Hence, two levels of approximation must be considered, viz., the theoretical natural closure, which provides an approximation of the 4th-order orientation tensor aijkl, and its numerical approximation (which is thus an approximation of an approximation). To avoid any misunderstanding in the sequel, the term "natural closure" will concem the theoretical closure, while the term "natural closure approximation" will be used for the numerical approximation of this theoretical closure only. It should be emphasized that the natural closure concept is based on establishing a relationship between aij and aijkl by neglecting the effect of fiber-fiber interaction and starting from isotropic orientation in a hypothetical flow. These assumptions are only necessary to define the closure, which is further used in situations where they are no longer valid. There is some physical evidence, which is corroborated by the investigations of Verleye et al. [4,40-42,56,57] and Cintra and Tucker [45] (see Figure 5), that the natural closure behaves well in many circumstances, especially for molding problems. However, this cannot be always true, and in particular the natural closure should not be used when two (or more) preferential orientations of the fibers are present. These states are highly unlikely in practical situations.
3.3.1 The 2D natural closure In 2D problems [40], the full symmetry and normalization conditions impose to
,5
1.4
i
i
i
i
a,,
1.3 1.2 1.1 1.0 -
,I
i
i
i
Linear closure ........ Quadratic closure . . . . . . . Hybrid closure Natural closure .'" Distributionfunction
. "
.
.....:.,'-- --
0.9
---"_-z_-':"-". . . . . . . .
"
. " " / . 9' " " - ~ ~ ~ ~ ' ~
08
0.7
~
ot"
0.6 ,,-'~E , , , ,Reducedtime ~: = ~/~ t 0.5 , , , 0 1 2 3 4 5 6 7 8 9 Figure 6. Prediction of 2D fiber orientation in elongational flow. (From [40]).
1367 write the natural closure in the form
aijkl - ~8 ((l + 4D)~3 -1)S(6ij61d) - (~3 -1)S(aij6kl ) + ~3 S(aijalcl ) ,
(35)
which results from (29) and (31). On the other hand, the deformation gradient F~A from fictive initial to actual time can be written in principal axes. Hence,
[FiA]_
[ J F
0
0
F -1
'
(36)
where F is a non-vanishing positive constant. Finally, the current orientation vector Pi can conveniently be expressed in polar form. Therefore, combining (36), (10) and (9.2) with the definition (12), the components aij ( - ~, 71 in principal axes) and aiju are easily calculated as a function ofF: -
,
F2+l
(37)
1
[a1111 a~122]_ 1 a1122 a2222 2(F 2 + 1)2
and thus
[F2(2F2 + l) F2
F2 1 2+ F2
[ allll a1122]- 1 I~(l+~) a1122 a2222 2 ~r/
'
~F] 1 77(1 + 7/) "
(38) (39)
Expressing (35) in principal axes and comparing the result with (39) shows that /33 - 1, from which the 2D natural closure is obtained" aij t
-
S(a j
t)
9
(40)
The 2D natural closure performs very well when compared with other 2D closures [4,40,51]. Figure 6 illustrates this behavior in simple elongational flow (v 1 = ~ x 1, v2 = 0 , v3 = - ~ x 3 ) , starting from isotropic 2D orientation (all =a22 = 0 5 , a12 = 0 ) and using the value C I =0.01. Impressive experimental validations have been obtained in the prediction of Bulk Molding Compound (BMC) injection molding with the 2D natural closure (see Section 5).
3.3.2 The 3D natural closure When the fibers can rotate in all directions, a 3D closure is needed. Unfortunately, the problem of determining accurate approximations for the coefficients
1368 fll to t6 in (28) in terms of P and D turns out to be extremely difficult in view of the presence of complex singularities in several expressions. Therefore, many different formulations were investigated until the present theory was elaborated. To summarize this work, let us first establish the following non-linear partial differential equation-
aijkq (~pm + a p#a (~im + a ipgq 3jm -- a pjmq tSik - a ipmq r +2
-- aijmp t~qk +
a ijkp 3 a ijmq 3 a ijmq Oa ijkp -- 0 Oars arsmq - 2 Oars arskp + 2 Oaks aps - 20ams aqs ,
( 41 )
which is a direct (but not obvious) consequence of equations (20.1) and (22) and the definition of the natural closure. Indeed, from the identity D F i A / D t - ~ V i / ~ X j FjA, and taking the conditions ~ - 1 and C t - 0 into account, (20.1) and (22) can easily be combined as follows in differential form :
daij - dFiA FAT a kj - aik FAT dFjA
- -aijkl (dFkA FAll + FAT dFia ) ,
(42)
where aijkt is a function of aij, which is itself a function of Fia. The key point is to observe that the differential daij in (42) must be exact when aijkl depends on
aij
through the natural closure.
o32aij/CgFkloqFmn and o32aij/cgFmn3Fkt
More precisely, the derivatives
provided by (42) must be equal for any
value of FiA when aijkt is defined by this closure. After quite long calculations (which must be performed with care for aij is constrained by (15.1) and (16.1)), this condition provides equation (41), which is the comerstone of the numerical theory of the natural closure approximation. Also, (41) can be extended to higher-order closures and, in particular, it could help in defining a closure approximation for the 6th-order orientation tensor in terms of aijkl (by extending the space of canonical distribution functions, since 14 degrees of freedom are required). This theory is under investigation. At this stage, since the invariance properties of indicial tensor notation have been used to provide (41), the calculations can be pursued with the tensors aij
and Fim in diagonal form. The following notations are used (Figure 7) : (a11,a22,a33) = (~,rl,()
,
(a2233,a1133,a1122) = ( X , Y , Z )
,
(43)
while, from (24), the invariants P and D read as (44)
1369
r/=l
~-1/8 m
r
r/=O
~=1
Figure 7. Left'triangular domain T of eigenvalues of aij. Right" isovalues of
Z-a1122 in T (the maximum Zmax is 1/8). (From [4]). Moreover, the normalization condition (16.2) becomes a1111 + Y + Z - ~ ,
a2222 + X + Z - 77 ,
a3333 + X + Y - ~" ,
and a l l l l , a2222 and a3333 can be eliminated in terms of X , Y, Z, and ~" (the natural closure is a particular orthotropic closure [45]). Furthermore, from (28) and (34), the expression of Z writes as
(45) ~, 77
Z - 112/31 + (~ + r/)/32 + 2~r//33 + (~2 + r/2)f14 + (~ + r/)~r//35 + 2~2r/Zf16], (46) 6 with similar formulae for X and Y. Combining these relations with (44) provides (X + Y+ Z), (X~ + Yr/+ Z~') and (X~ 2 -k-Y/72+Z~ 2) in terms of P , D and 131 to 136. After some calculations, and with use of (30), these relations can be cast in the form : 1A0 2
-
7
1 4
0
0
--10
/~6
-4P
6P
2D
-2
10
0
6P
1-4D
-1
-2D
1
7
o
2D
-1
-3
(D-15P)
-2
21
-35
0
1 X+Y+Z +
+
X~ 2 + g r / 2 + Z ~ . 2
, (47)
1370
with
A - - 2 7 P 2 + 18PD
+ D2
-4D 3 -4P
.
(48)
Equation (47) proves extremely useful in evaluating the natural closure. Finally, equation (41) can be transformed after very long calculations (involving trace operations) into the following set of equations"
+
[ (4Z + X - r/)(2Z- ~) 1
+ (rt- r162 r L_(4 z + Y - , ~ ) ( 2 Z - r l ) _ ] X - Z)(2Y - ~)(7/- ~') 1 - 0 (Y - Z)(Z X - rl)(r - ~) J
(49)
with t5 - (~ - 7"/)(7/- ( ) ( ( - ~), and A - ~ 2 . Similar pairs of equations govem the derivatives of X and Y. It can be proved that no other independent relationship than these 6 equations can be derived from (41). It should also be observed from (34) that ~, 7/ and ( are are not independent variables and only the constrained partial derivation operators (tg/0~- 0/tg(), ( 0 / 0 r / - 0/0~) and ( 0 / 0 ~ - 0/0r/) can provide meaningful results. Indeed, when a given function of and r/ (or ~ and ( , or r/ and ( ) is regularly extended to a function of ~, 7/ and ( (considered as independent variables), each of these operators provides a unique derivative along the subset of admissible values of (~, r/, ~'), as defined by (34), whatever extension is performed. 3.3.3 Numerical approximations o f the 3D natural closure In the 1 st strategy selected by Verleye et al. [41,56] to obtain a closed approxi-
mation of the natural closure, rational expressions were determined to accurately approximate the coefficients fll to f16- For that purpose, the canonical distribution function q'c was integrated numerically for many values of the cumulative deformation gradient FiA, thereby providing a set of pairs (aij, aijkt) from which a rational approximation of fll to f16 respecting (30) was further obtained by best fit. The drawback of this method (which nonetheless produced good results) is that the error decreases only slowly when the degrees of the numerator
1371
0.14
9
.
.
.
.
.
.
.
Analytical Z ~ Approximate Z - -
0.12
/
0.1
~,,
/
Anal.ytical Y - -
0.08 0.06 0.04 0.02 0.0 -0.02 0.0 0.5 -
i
i
i
i
i
i
i
0.1
0.2
0.3
0.4
0.5
0.6
0.7
i
0.8
0.9
1.0
Analytical dY/d~ Approximate dY/dr
0.4 0.3 0.2
Analytical dZ/dr Approximate dZ/d~ - -
0.1 0.0 -0.1 -0.2 ,, -0.3 Figure 8.
Comparison between the theoretical and approximated values of and Y - a l l 3 3 (top), and dZ/d~ and dY/d~ (bottom), along
Z -- a1122 - 0 . The 2 "d strategy was used. (From [4]).
and the denominator of the approximants are increased (in [41,56], these degrees were 8 and 1). This results from the singularity of the natural closure along the boundary 3/" of the triangular domain T of admissible (~,r/,~') (Figure 7). Altemative approaches, taking into account the theory of the natural closure, were thus investigated. In the 2 "a strategy [4], the following decomposition was introduced" Z-/~2~r/+PZ where
Z
,
is analytical and
(50) PZ
tends towards 0 when
This form is justified by the fact that
~r//2
P
tends towards 0.
reduces to the 2D natural closure
1372 A
(40) when P - 0 . An approximation of Z was obtained from 3 basic properties of the natural closure. First, numerical experiments showed (and this should be confirmed with the help of (49)) that 2-
3/10 ,
(51)
when ~ r / - ~,2 or, equivalently, when p _ D3
_
~3
D - ~ .
,
(52)
Secondly, the following derivatives of X , Y and Z (49) along the boundary aT when ~" - 0 :
ae, g-r z - - ~ n ,
ar ac X -
a a)z_l~r an ar
a~
(aa~ ara] x - a(an
~)
can be calculated from
y =
1 2 1
(53)
2
with similar equations when ~ = 0 or 7/= 0. Thirdly, the axisymmetric natural closure [4] (Figure 8) is easy to calculate : 9 either by solving the differential equation provided by (49) along the bisectors of the triangular domain T :
(1-3~)(8Z-(1-~))dZ/d~+40Z 2 + Z ( 8 ~ - 7 ) +
1/4 (1-~')2 - 0 ;
(54)
9 or in parametric form from (42) : _
_ (4 cosh 2 S - - 1) cosh s coshs
,6s s
S ~
sinh3s
~
s+
cosh 2s (2 cosh 2s + 1)
(55)
16sinh4s
if
"
sinh2s
o
(~<s
(4COS2S- 1) COSS COS2S(2COS2S + 1) . . . . . . . . . s-k16sin 5s 16sin 4s coss 1 _:-~_3-s+~2-, if )~<~'_<1 sin s sin s In both cases, (34) provides the relation ~ - 7 / - ~ 2 proved that X - Y - ~
(1 -
u
8Z).
(56) (O<s_<~).
( 1 - ~'), while it can also be
1373
The approximation (X ap, I?ap, Zap ) tested in the
Zap
lnD 1+/7 lnP
2 nd
strategy was of the form
+(~
I l)I ll
~
with symmetricexpressions for
(57)
In
f~ap
and I~ap. In (57), P(v)
and G(v)
are
polynomials in 77= lnD/lnP, which vanish when "c = 0 and are selected in order (i) to respect condition (51) when equation (52) holds :
and (ii) to provide a very close approximation of the axisymmetric natural closure when ~ - U . This was achieved by constraining X ap, Yap and Zap to present the same expansions (up to a given order) as )(, I7 and Z around ~ - 0, = 1/3 and ~=1. In particular, complex calculations around ~ = 0 produce the following expansions : Z=
1
( 1+
8
4
+...,
X -Y-
1+
+...,
(59)
which provided conditions for the selection of polynomial coefficients in (57). Equations (59) illustrate the singularity exhibited by the natural closure in the case of a nearly 2D orientation. When the expressions of f~ap, Yap and Zap are selected, it becomes easy to calculate
Xap + Yap + Zap, ~ Xap + 71Yap + ~ Zap
and
~2 Xap + 02 Yap + ~2 Zap
in terms of P and D from (44) and (50), and thus, in principle, to obtain approximate formulae for the coefficients fll to [36 by means of (47) and (30). However, a significant difficulty appears at this stage, since the determinant A defined by (48) vanishes identically along the bisectors of the domain T. Therefore, the equations cannot be solved in the axisymmetric case. This problem is worse than it seems, since the fight-hand side of (47), which vanishes identically when the exact natural closure is used and A - 0, never did vanish when an approximate closure was introduced by means of the 2 nd strategy (a simple formula satisfying this condition was impossible to obtain). Hence, artificial singularities were introduced for nearly axisymmetric orientation states. To circumvent this difficulty, either the rational approximants determined by means of the 1st strategy or the expression (57) were used for low values of A.
1374
Figure 9. Isovalues of
(Zap -Z)/Zma x in % (2 ~d strategy). (From [4]).
Whereas very good results were obtained by using this 2 nd method [4], the solution adopted to remove the artificial singularities was not completely satisfactory, since a closed analytical expression was expected for the closure approximation. Moreover, since in molding processes the fibers are most often strongly aligned by shear and elongational effects, the vicinities of the comers of the domain T must be given particular attention. In particular, overlapping (and thus non-regular) solutions must be avoided. This is the reason why a 3 rd strategy, which is still under development [42], was investigated. Recent work also suggests that non-linear algebraic relations linking fll to /56 can be obtained by equaling the cross 2 "d derivatives of Z (obtained from (49)). Hence, a very high accuracy should eventually be reached. The results obtained by using the 1~t strategy (with rational approximants) were compared by Cintra and Tucker [45] to their best fitted orthotropic closure (see Figure 5). They showed that this natural closure approximation generally behaves very well, whereas slightly better results are obtained with the optimal orthotropic closure. We believe (but this needs confirmation) that this can be related to the lower quality of the 1~t version of the natural closure approximation in the vicinity of the boundary (and especially the comers) of the domain T. Further numerical experiments are needed. On the other hand, the natural closure approximation does not oscillate in the same way as the best fitted orthotropic closure for low values of C I [45,88]. The isovalues of the approximate solution Zap obtained with the 2 nd strategy are represented in Figure 7 along T. The good quality of this approximation is illustrated by Figure 8, which depicts the values of X, Z, X ap, Zap and their
1375
derivatives along the bisector ~ = 7/, and by Figure 9, where the relative error
(Zap- Z)/Zma x (in %)is represented over T. Near to the comers, the error is more important because another approximation was used. 3.4 D o u b l e - L a g r a n g i a n technique; deformation tensor model
In order to avoid the non-physical behaviors exhibited by the quadratic and HL1 closures, Szeri and Leal [44] elaborated the "double-Lagrangian" computational technique, which consists in solving microdynamical equations of the form (6) in a space that is Lagrangian for both the coordinate (x ~ and orientation (pO) variables. This method requires to express W in the Lagrangian form [44a]
~F(x ~ p~ t) - v~o /det(0pi/oap 0) ,
(60)
where 0pi/Op ~ (x ~ p0, t) represents the Jacobian matrix of the solution of (6), while q'0 - W ( x~ p0,0). Hence, a system of ordinary differential equations is found, which can be solved numerically by generating a large number of Lagrangian fibers. Details are in reference [44b]. This serf-adaptive scheme is of particular interest for those problems where a good closure approximation is not available, or when steep orientation gradients are present in the boundary layers. In a subsequent work, Szeri and Lin [46] developed an accurate method to solve the fiber orientation problem by tracking an equivalent deformation gradient tensor FleAq (x,t) obeying the equation
Dt
-
where D/
-~j + ( ~ - l)dij + 3hDI
1
(BiJq(-1)-3B~c(-1) ~iJ) F ~
,
(61)
is Brownian diffusivity, while Bi~q is defined by Bi~q-FieAq Ff~,
and h-h(Feq(T).Feq)
is an average value of p~176
which is designed to match selected asymptotic states. The tensors
aijkl, which are recovered from the relation Pi
--
(pOEU), aij and
Fleaq pOA[~'P0[I-1 by exact or
Gaussian quadrature, are thus linked by the theoretical natural closure [4,40-42, 56]. This method extends the theory of Lipscomb et al. [27] (equation (10)) to the case D I ~ 0 (see also [77,94,100,101]). First tests [46] indicate that the "deformation tensor model" provides slightly less accurate results in shear flow than the orthotropic fitted closure [45] or the natural closure. Additional experimental comparisons are necessary.
1376
4. A P P L I C A T I O N TO SUSPENSION F L O W S IN N A R R O W GAPS Investigating the flow of fiber suspensions in narrow gaps is of particular interest in view of the model simplifications resulting from this dimensional hypothesis ( e - h/L << 1 ), and the direct applicability of the model to injection and compression molding simulation [ 1-9,38-42,49,50,53-56,74,78-81,85,87,88,102104]. It must, however, be noted that the gap between model and actual process remains quite wide, especially concerning the effect of the polymer rheology and the fiber concentration and flexibility on the flow and fiber orientation. Hence, only rather idealized simulation tools are currently available. In an important paper, Tucker [2] identified 4 suspension flow regimes in narrow gaps, considering rigid slender particles and an incompressible Newtonian solvent. His classification was based on defining 2 dimensionless numbers, namely, the particle n u m b e r N p ( w h i c h scales the macroscopic extra-stresses caused by the particle resistance to flow stretching effects) and the order of magnitude of out-of-plane orientation 6. Neglecting the Brownian effect and recasting equation (23) in the form
~ij
--
27/(1 + CO) [dij + Np aijkl d u + N s (dikakj + aikdkj )] ,
(62)
provides a convenient way to define and understand the role of N o (the shear number N s is negligible for slender particles). Tucker [2] proved that Pi, aij, aiju ... are all of the order O(6S), where s is the number of out-of-plane components among the indices i , j , k .... while, from (20.1), 6 - m a x ( e , C ~ i 3 ) . (Note that aiju is scaled in [2] from the definition (12.2), without considering the effect of the closure approximation. First inspection reveals that classical closures such as the quadratic, hybrid ... closures, but not the linear closure, are consistent with this scaling.) The flow regimes identified by Tucker [2] are : 9 I. The decoupled lubrication regime ( N p S 2 << 1, S > e), which allows to separate the flow and orientation calculations since the gapwise shear stresses are governed by the solvent viscosity (without orientation effect) and dominate the in-plane momentum balance. . II. The coupled lubrication regime (Np 62 > _ 1 , e << 6 << 1), where the gapwise shear stresses dominate the in-plane balance, with orientation effects, ~'3a - 2r/(1 + CO)[d3a + 2Np a3ap3 dp3 ]
(63)
1377
(Greek indices, from 1 to 2, indicate the in-plane directions). 9 lIl. The general narrow gap regime ( N p S 2 _>1, S = E ), where few simplifications are allowed. 9 IV. The plug flow regime with shear boundary layers (NpC 2 >> 1, ~ << c), where the velocity profile is flat across the gap and generates two wall lubricating layers of thickness [3-O(LNpl/2), while both in-plane (fiber-induced) stretching stresses and wall shear stresses are significant. A different approach was selected by Barone and Caulk [38] who, considering the compression molding of Sheet Molding Compound (SMC), assumed a priori a plug flow in the cavity, with in-plane stretching stresses and wall friction layers. When in-plane orientation is random, this model is equivalent to the regime IV of Tucker [2]. However, while the latter was rigorously derived for rigid fibers, the former is phenomenological and can be considered whenever the suspension is concentrated and the fiber length prevents out-of-plane tumbling. With long flexible fibers, the lubricating layers (of thickness 13) must be considered as regions where less fibers are present (due to the wall effect) and whose rheology is governed by the solvent viscosity, while the plug flow results from kinematic fiberfiber interactions. The applicability of this model depends on the solvent rheology and the real size of the fibers, which can be affected by the gates, the walls ... Experiments [33,102,105] indicate that injection molded long-fiber composites have a significantly thicker core and a thinner shear layer than short-fiber parts. Finally, it should be noted from (62) that the ratio between the powers dissipated by out-of-plane shear stresses and in-plane stretching stresses is of the order O(L2fl-lh-lNpl). The latter can thus be neglected in the in-plane momentum balance when L is large, since the order of magnitude of ]3 does not increase with L for flexible fibers as in regime IV. This justifies the thin charge approximation of Barone and Caulk [38]. The remainder of this section will be devoted to decoupled models. The coupled lubrication regime II of Tucker [2] was investigated by Chung and Kwon [49, 50], who extended the Hele Shaw model to that case (see Section 5). 4.1 Decoupled mathematical and numerical models Two problems are posed, considering a generalized Newtonian rheology and an incompressible suspension. 9 P1 : decoupled lubrication flow of a short-fiber suspension [2-4,9,41,42,49,50, 56,80,81,87,88,104]. The Hele Shaw approximation is used and orientation is 3D (using a 2D model produces erroneous results; if simplifications are
1378
required, imposing constant out-of-plane fiber orientation components is possible, but their selection is critical [88]). This model applies rigorously when the suspension is dilute. 9 P2 9plug flow of a long-fiber suspension with shear layers [2-7,9,38,40,78,79, 85,103]. The thin charge model of Barone and Caulk [38] is used and the fiber orientation field is 2D and decoupled. This regime occurs when the suspension is concentrated and the in-plane stretching stresses can be neglected with respect to friction in the skin layers. For both problems, the kinematics are governed by the pressure gradient and the fluidity. Letting v-~ denote the gap-averaged in-plane velocities, the volume flow rate can be written in the form 2hV a - - S Op/Ox a
(64)
,
where the fluidity S is calculated differently in P1 and P2. The pressure field is governed by mass conservation"
&
+ ~
3x~
S
-
0
(65)
where 3h/oat is included to provide a single model for injection and compression molding. The Hele Shaw approximation, which is used to model problem P1, has been presented in Chapter IM. Some points need to be recalled : 9 The pressure field is approximately constant across the gap. 9 Assuming a power-law viscosity 77- moe b p - a T ~ n - 1 , where T denotes temperature while n, m0, a and b are material constants, the shear rate can be approximated from the relation ~2 _ (oava/o3z)2 (with z - x3). 9 The fluidity is given by the expression S - 2h
[[Op/&,~l[~/n-~e-bp/" m~ ~/"~
(66)
,
where ~ denotes the gap-averaged "profile function" 09, which is proportional to the velocity profile in the gap 9 -
jz (e- z0) le- z0I1/n-1
(67)
eaT(~)/nd2
with the wall no-slip condition o g ( - h , T ( . ) ) - 0 .
In (67),
z0 stands for the
level at which ~ ' - 0. The profile function depends upon the temperature profile T(.) only. Its average is calculated by integration by parts 9
1379
rhl
j_hlz -- Z01Vn+l
2h~-
eaT(z)/nd z
(68)
.
In-plane velocity components can be recovered from the relation"
--
va -
3x a
~
e-bp/nmoVn(o(z,T(.))
Oxt~
.
(69)
In problem P2, the gap-integrated in-plane momentum equations read as - 2 h O p / O x a + "c3a ( z - h ) - "c3a ( z - - h )
- 0
(70)
.
Letting 13+ and 13- denote the upper and lower lubricating layer thicknesses (with ~ - / h ,
{
~+/h
~v~l~z
- - ~1~+
~v=/az
-
<< 1), the velocity gradient and shear rate in the skins are"
+ e -I1~=11/~
,
i, - I I ~ l l / ~
~-~/~,
, (-h_
where v~ stands for the constant in-plane core velocity. Furthermore, assuming a power-law skin viscosity, and using (70) and (71) with the approximation "r3a - 7"10v a I O z , the core velocity norm I1~11 ca. be expressed in the form
II~ll-
][Oqp 1In le_aT+ e_aT-I-Vn (2h)Vn -~a moVne-bp/n -~n + ~~-n '
(72)
where the temperatures T § and T- in the upper and lower skin layers are supposed to be constant (and this is a restrictive assumption). Introducing (71) and (72) into (70), the core velocity is therefore approximated by the expression
V'-'a= ~ t I1~11 ]I~
(2h)l/n mo-1/ne-bp/nle-aT+ ~+~+n e-aT- I
'
(73)
which is of the form (64), with
s
n-1(2h)l/n+lmo-Vne-bp/n~+~-~~+n /ea+ /
I1~11
(74)
From this discussion, it results that the same equations (64) and (65) will be used to predict the kinematics in P1 and P2. Thermal effects can play a nonnegligible role. The effect of chemical reactions could be easily considered (cf. also [ 104]). The solution of these flow problems was investigated in Chapter IM,
1380
to which we refer the reader for any details. Once the flow is known, fiber orientation must be calculated. It is sufficient to consider the 2~ tensor aij, whose evolution is governed by (20.1). In the framework of the finite element technique used in the MOLDSYS software (see Chapter IM, Section 4), aij is a 2D or 3D additional field, which is integrated in a similar manner to T. Details will not be repeated and we will focus on some particularities of the algorithm related to the tensorial nature of equation (20.1). To cast the equations into a weak f o r m , the residual of (20.1) is multiplied by an arbitrary tensor test-function aij (defined over the filled zone f~(m) of the midsurface) and the result is integrated over f~(m) : r-I d
"
(75)
(aij + 2~,aijud u + 2Ct ~' N aij ) a(i dS - 0 . ~(m)
Recalling that aij is constrained by equations (15.1) and (16.1), while aij and dij
must have an equal number of degrees of freedom in order to produce an
equal number of discrete unknowns and equations, d o is itself constrained 9 aij -- d j i ,
(76)
aii - 0 .
On the other hand, when the average mass equation (65) is put into a weak form, integration by parts is carried out o v e r ~,-~(m) and the weak solution belongs to the Sobolev space H l ( f ~ ( m ) ) .
p(xu,t)
The velocities provided by
(69) or (73) belong to L2(f~ (m)) and theft in-plane derivatives are measures [106]. Hence, whereas (75) does not violate any distribution product rule when aij and dij themselves belong to H l(~(m)), integrating this equation is difficult and much better results are obtained by integrating by parts every term involving velocity in-plane derivatives (Verleye [4], Crochet et al. [9]). This operation is delicate in 3D problems when 2h is not a constant over f~(m), since the different fluid layers are no longer parallel to the midsurface (parallel layers are defined by z - c o n s t a n t , while effective layers obey the equation z/h - constant ). To solve this problem, let f denote a generic quantity and de-
free a*/
ana a*I /ax
by fixing z/h
(instead of z ) when derivation is
performed"
a*I &
oy &
ah oy h&&'
cg*f _ oaf t- zOhOf . 0x a oaxa h 0Xa 0z _
(77)
1381
Splitting the out-of-plane velocity component w as follows : W
-
-
-
+w
+V a
-
h
,
(78)
it results from mass conservation and the wall no-slip condition that 9
3"
3"
W ---~~hd2+<Sz
h
zV~d2
(79)
.
Combining (77), (78) and (79) shows that
Df _- c)*f + v a c)*.__._f_+f w* -oaf Dt Ot oaxa cgz '
(80)
which is a convenient form because in-plane advection is considered along effective layers. (In practice, due to numerical stability concerns, w is often approximated by setting w* to zero in (78), which corresponds to imposing a locally convergent, divergent or parallel velocity profile; see [107] and Chapter IM, Section 3.) Hence, in order to integrate by parts f Og/oaxa (where g is a 2 nd genetic quantity), the following formula must be used :
[.fOg
~2(m)
--
S (
~-~( m )
a*f ~
0~
g+f
z Oh Og h O~Xo~ ~ Z
) dS + S f g nads '
(81)
~-~(m )
where n a denotes the in-plane outer normal along o ~ (m) . In the fight-hand side of (81), the 1st term is easy to calculate in a layer-by-layer procedure, while the 2 nd term is indispensable. To complete the weak formulation of (20.1), boundary conditions must be treated. Along the gates, aij is specified by means of essential conditions. Along the side walls, no treatment is mandatory because the characteristics of (20.1) are tangent to the side walls. However, the normal in-plane velocity may be usefully set to 0. The conditions to be applied at the flow front and along the singular lines of the midsurface (namely, the edges, abrupt changes of thickness, bifurcations ... of the part) are examined in a separate section. After the evolution equation of aij has been put into a weak form and the boundary conditions have been treated, the semi-discrete scheme is built up in two steps. First, aij is approximated by the discrete sum
aij - ~.~ r m
(xo~) aij (m) (z,t) ,
(82)
1382
where aij (m) (z,t)
and qb(m) (xa)
denote the nodal values of aij and the shape
functions. The tensor a ijkt is discretized in the same way"
aijkl -- Z ~)(m) (Xot ) aijkl (m) (z,t) m
,
(83)
without loss of accuracy [9] (evaluating the closure approximation of aijkl at the sole mesh nodes is very efficient). Secondly, in order to obtain an equal number of equations and unknowns, the test-functions aij are also discretized taking constraints (76) into account"
[
~n)
0
0
0
,
o
o -r o
0
0
^
o 0
8(n) 0 o ; o
-r
0 0 ~(n) 0 o 0 ~,., o o
o
-~n)'
~~
0
0
0 (84)
~(n,
0
'
for a 3D or 2D problem, respectively. In (84) and (85), and (82) and (83), the ^
functions r
and q~(l) differ, due to the effect of applying a modified SUPG
technique (cf. Chapter IM, Section 4).
4.2 Effect of the fountain flow; abrupt changes of thickness and bifurcations In 3D Hele Shaw flows, the fountain effect results from the deflection of the core fluid towards the walls of the narrow cavity when it approaches the flow front, as a consequence of the fluid-wall no-slip condition. Modelling this effect is an important issue, since appropriate temperature and orientation conditions must be imposed at the front to obtain correct simulation results. Both the flow and its influence on the fiber orientation must be considered. The effect of the fountain flow on the kinematics has been thoroughly investigated in Chapter IM and only the principal results will be stunmarized. Basically, the Hele Shaw approximation generates a singular perturbation problem [108], whose outer zone comprises most of the flow domain and is governed by the
1383
\\\\\\\\\N\\~z\\\\\\
\\\\N\\ . . . .
j--) __
dZl~z l_l
! i i I
dx!2 \\\\'N\\\\\\\\ ' 7 ...... a 2: . . . . . . . . . . . . . . . . .
dZl~z I
_~
///////
,I
z z~ ~ ~ - ~
dz 2
t11111111111'1/
x1
Figure 10. Simplified models for the effect of fountain flow (left), and for the flow across an abrupt change of thickness (fight). lubrication approximation, whilst the inner front zone has a characteristic dimension of the same order of magnitude as the gap thickness. Matching the inner and outer zones imposes to select precise front boundary conditions. The fountain flow thermal model developed by Dupret and Vanderschuren [107] was thus based on the theory of matched asymptotic expansions. The following points should be stressed : 9 The Hele Shaw approximation governs the whole flow domain. 9 The front region is infinitely thin and consists of straight segments perpendicular to the midsurface. 9 Heat diffusion and viscous heating are negligible in the front zone. The core material points entering the front at a given level leave the front at another level while keeping the same temperature. This model provides a front thermal boundary condition for each level in the gap where the material points move slower than the front (according to the Hele Shaw model). This agrees with the fact that the characteristics of the simplified energy equation enter the flow domain in these layers [9]. Once the flow is modelled, its effect on the fiber orientation can be investigated. This problem is complicated, since orientation is governed by the motion, rotation and deformation of the elementary material volumes that cross the front region. Ideally, the effect of both the inner and outer asymptotic flows should be taken into account. However, calculating the inner flow means solving a free boundary problem at each time and position along the front, whereas a unique inner solution can be determined once and for all only if the viscosity is a constant. Approximating the viscosity by a constant in the inner zone and predicting the fiber orientation by matching techniques thus represents an attractive solution, which has not yet been implemented. The method developed by Verleye and Dupret [4,9,40,53] (which is very close to the model of Bay and Tucker [81 ]) extends the thermal model of [ 107]. Shear
1384 deformation is neglected when the suspension crosses the front region. Consider in Figure 10 an elementary parallelipiped entering the front at level z 1 and leaving it at level z 2 , and let (dx 1,dy 1,dz l) and (dx 2 ,dy E ,dz a) stand for the initial and final side lengths of this material volume (which remains a parallelepiped since shear deformation is neglected). The deformation gradient F~/(l-a) from z I to z a can be expressed in terms of the profile function to and its average defined by (67) and (68). Indeed, from mass conservation, the relation linking the associated pairs
(Zl, z2 ) is
Sz 2 ( ~ - (3.)(Z))dz - O, from which it results 1
that (~-(.o(z2))dz 2 - ( c o ( z 1 ) - m---)dZl, while mass conservation in the 2D front flow also produces the relation dx 2 dz 2 - dx 1 dz 1. Hence, -- ~" -- ('O(Z2)
r__,lF,~l_2)/ _ L
-J
0
0
(.O(Zl) - ~ 0
1
0
0
0
- o~(zl)-~
(86)
j
~" _
(.D(Z 2 )
where the negative signs account for the 180 ~ rotation. In order to approximate the relation relating a/~2) to a hi) , Verleye and Dupret [40] analyzed in a 1~t step the effect of the fountain flow in a theoretical problem where in-plane orientation was assumed, while the evolution equation of aij was supposed to be governed by the 2D natural closure (see Section 3.3). In that case, the squares of a~1) and a~2) can be expressed as functions of the deformation gradients F/(a1~ and F/(A2) from fictive initial to actual time"
amnUnm [~am(2)...(2)] Unm
a/(]) a , ( 1 ) - ( F f f ~ ) F i ~ ~:J -
a[~ ) akj
~'kB r kB
Fi(A1) F)A ) , --1
)
(87)
F,'2
since this latter relation is valid in principal axes (from (36) and (37)) and satisfies frame indifference (because Fi(A1) F),~) and Fi(A2) F),~ ) are the Finger strain tensors from fictive initial to actual state). Therefore, assuming that fibers are slender and that fiber-fiber interaction is negligible during the short time the fibers undergo fountain flow, the relative deformation gradient
F/(2) and
F(1-2) F ( ~ ) - F i (2) F/(A1) by the relation "ij
Fij (1-2)
Eliminating
is related to
F/(2)
thereby
1385
from (87.2), and then FiCA 1) from the result by means of (87.1) provides after some calculations the relation
(F(1-Z).a(1).a(1).F (1-2) T)l/2
a (z) =
tr[F
]
(88)
where the conditions (15.1) and (16.1) have been taken into account. The fountain flow model of [40,53] consists in using the jump condition (88) as an orientation front condition for the layers moving slower than the front. Whereas quite restrictive assumptions were introduced to establish (88), this model can efficiently be used in more general situations, in particular when the fibers are not parallel to the midsurface or when fiber-fiber interaction is not negligible. The above model can also be used in the presence of other singular regions, such as abrupt changes of thickness or midsurface bifurcations (Figure 10). Considering the theory of singular regions developed in Chapter IM, this methodology provides a convenient way to treat all the boundary and jmnp conditions associated with the prediction of fiber orientation in Hele Shaw flows. On the other hand, there is, in principle, no fountain effect when the 2D plug flow model applies. Nonetheless, bifurcations and abrupt changes of thickness can be treated by using the above orientation jump model. As an example, the deformation gradient tensor
/]7(1-2)/ -t'ij
_
Fij(1-2) across an abrupt change of thickness is
0
0 1
0 0
0
0 h2/h 1
(89)
where 2h 1 and 2h 2 denote the upstream and downstream thicknesses. The ex-
Fij (1-2) provided by (89) can be introduced into (88) for calculating the jump of aij. Applying this model to long-fiber suspensions however requires
pression of
some caution. In particular, experiments indicate that the velocity profile is not completely flat in some circumstances [67,102,105], with a consequently low, but non-vanishing fountain effect (in which case the fibers in the front zone can be bent across the whole gap [33,105]), while a plug flow is clear in other cases [38]. The model to be selected therefore depends on the material (SMC, BMC, GMT ...) and the processing conditions. Additional modelling effort is needed.
1386 5. E X A M P L E S O F N U M E R I C A L TAL VALIDATION
SIMULATIONS
AND EXPERIMEN-
Few publications report quantitative data describing the orientation field of injection molded fiber reinforced composites and, in addition, most results are presented in such a way that it is difficult to compare the observations with theoretical predictions and to draw general conclusions. Most often, the publications focus on the effect of the orientation state on the suspension rheology or the Z
I
Num. a~ m Exp. al,
"
r
0.8
Point
A
0.2
2h = 3.18 mm
0.0 -0.2 -0.4 -1.0 (b)
.
.
.
.
.
/
.
Point
-0.6
, 1.2 (C) Num. a t , - Exp. a,, ~-' t 1.0
m
B
t tt
-0.2
0.2
0.6 |
. . . . .
i
1.0 |
Num. a , l - Exp. ajj ~-~
Point
C
-~t
.]t
0.8
t
t
0.6 0.4 0.2
~r 9
' 9
9
9
9
9
9
-
. E x p_'a,3~--f _
tl
i , . I .
, ,,
0.0
taF
Num.
-0.2 i
- 1.0
i
-0.6
i
i
-0.2
i
|
0.2
I
I
0.6
I
-0.4 1.0
- 1.0
i
i
-0.6
i
i
-0.2
I
I
I
0.2
I
0.6
i
1.0
z~ z~ Figure 11. Filling of a short-fiber reinforced nylon disk" sketch of the part, and numerical and experimental profiles of a l l ( = arr ) and a13 ( - arz ) across the gap 9 (a) r - 19.7 m m ; (b) r - 36.2 m m ; (c) r - 47.5 m m . (From [4]).
1387
1.2 W = 25.4 m m
.....
i
9
i
i
'
(a)
"
i
'
Num. a , l - Exp. al~ ,*--,
1.0 0.8
L = 20_3._2 m m C
B
0.6
t
0.4 .......... Xl
Num. a~3m Exp. a i 3 ~..-,
0.2
2h = 3.18 m
0.0 ~ -0.2 -0.4 0.0
(b)
Nurfi. a,, Exp. a l l t "
Point
t
I
I
I
I
0.2
0.4
0.6
0.8
(C)
0.0
|
t"
i
B
"
9
9
!
0.2
0.4
1.0
um. a l l w Exp. a~l ,*--,
Num. a ~ 3 m Exp. al3 ,.,---,
9
,,,
, 0.6
"
0.8
I
L_
Num.
Exp.
\'
~f~----'
I
'
!
I
9
al3_. a ~3 ,-,-.-,
I___
-0.2 ~ - - - - - - - - ~ - - - - - - - 1.0 0.0 0.2 0.4 0.6 0.8
z~
1.0
z~
Figure 12 Filling of a short-fiber reinforced nylon strip" sketch of the part, and numerical and experimental profiles of al 1 (-- axx ) and a13 (-- axz ) across the gap 9 (a) x - 9.1 mm; (b) x - 54.1 mm; (c) x - 167.0 mm. (From [4]). thermo-mechanical properties of the product. However, since the physical behavior of the solid part can be quite well predicted by means of homogenization methods based on the orientation field description (Lielens et al. [7]), it is of prime interest to validate the orientation prediction models.
5.1 Filling of short-fiber reinforced parts In this section, we analyze the filling of a center-gated disk (Figure 11) and a film-gated strip (Figure 12), using experimental results provided by Davis [109] and Bay and Tucker [81 ]. The reader is referred to these publications for further
1388
detail as to the experimental conditions and thermo-physical data. The injected material is nylon 6/6 reinforced with glass fibers (~ = 43 % weight ratio, average l = 210pro, average d = 11 pm). It should be noted that a rather complex analysis is required to characterize 3D non-planar orientation fields. The filling of a center-gated disk represents a useful benchmark, since it is commonly encountered in mold filling operations. As the flow decelerates radially, the in-plane extensional velocity gradient tends to align the fibers perpendicular to the flow direction in the core of the cavity. Across the thickness, however, shearing competes with extensional effects and tends to align the particles with the flow in the shell layers. Other factors that complicate this analysis are the radial dependency of the velocity gradient, and the fountain flow effect (which generates the skin layers). This problem was solved by Bay and Tucker [81] and Verleye [4,42], using the non-isothermal generalized Newtonian Hele Shaw flow model with exponential and power-law dependence of the viscosity upon temperature and shear rate, respectively. Fountain flow was taken into account, while the packing stage was not considered. Decoupled orientation predictions were performed using either the hybrid [81] or the natural [4,42] closure approximation, and an interaction coefficient C~ = 0.002. Altan and Rao [87] also studied this problem by means of steady Newtonian flow simulation and decoupled prediction of the distribution function. The interaction coefficient was adapted to achieve best fit with experimental data. The numerical predictions of Bay and Tucker [81b] look very well, but do not display satisfactory quantitative agreement with the experimental results throughout most of the disk. In particular, far from the inlet gate, the shape and thickness of the core region and its dependence upon the radial distance are not correctly predicted. The small out-of-plane fiber orientation is overpredicted and the coreshell transition is placed too close to the mid-plane (see Figure 8 in [8 l b]). This discrepancy results from using the hybrid closure approximation, as confirmed in [4,42,87], since the exact transient or steady-state orientation fields cannot be obtained by this way [87]. Much better agreement could certainly be obtained from the best fitted orthotropic closure approximation [45]. The orientation results obtained by Altan et al. [87] are in slightly better agreement with reality. Unforttmately, their simulations were performed without considering the moving front effect, and the orientation structure is not accurately described in the regions affected by the fountain flow. The natural closure approximation behaves much better and allowed Verleye [4, 42] to obtain very good agreement between experiments and numerical predic-
1389
tions, except near to the gate. In Figure 11, the thickness of the core region and its evolution are quite well predicted. Furthermore, the correct fountain flow and frozen layer treatments induce very good prediction of the orientation in the skin and shell layers (whose mastery is essential for realistic prediction of the product thermo-mechanical properties [7]). Around the gate, orientation was never predicted accurately since, as the Hele Shaw model is no longer valid in its vicinity, the temperature, pressure and orientation fields were approximated as inlet boundary conditions into the cavity. Fortunately, the effects of shearing and elongational flow rapidly erase the discrepancies downstream from the gate. Bay and Tucker [81] also investigated the injection of a film-gated strip (Figure 12). The general orientation behavior was well captured by the simulations, and some orientation components were correctly predicted, while the correlation between experiments and simulation was of lower quality for the other components (see Figure 5 in [8 lb]). The origin of this discrepancy must again be found in the use of the hybrid closure. For the same problem, Verleye [4] obtained excellent agreement by using the natural closure approximation, the only discrepancies being observed near the gate (in point A of Figure 12). It should be noted that the layered structure obtained by simulation and experiments is similar to that of the center-gated disk, except that there is no longer a skin layer. Along the mid-plane, the core orientation is not significantly modified after the fibers exit the gate. Near the walls, in the shell layers, shearing aligns the fibers in the flow direction. The thickness of the different layers changes along the flow, this change being a function of the flow rate, the gap thickness and the polymer material properties. Near to the cavity extremity, Bay and Tucker [81b] predict a core layer thinner than the experimental one, and this might be related to the fountain flow model implemented. However, the effect of the packing stage, and possibly the in-plane viscosity, which are not considered in [4,42,81 ], should be analyzed before drawing final conclusions. The parameter analysis of Bay and Tucker [8 lb], confirmed by Verleye [4], provides key information, for various shapes, about the influence of the injection and material parameters on the orientation field. The process parameters that significantly affect fiber orientation are identified, together with the physics to be included in the model for good orientation prediction. The influence of the wall temperature proves to be limited [49,81b,104], although it has some impact that will not be detailed. The inlet temperature has a reduced effect because it can only vary over a narrow range. The effect of the flow rate is much more significant, since it govems the thickness and orientation of the skin layer, which could even disappear in very fast filling processes [81b,110,111]. This last conclusion is directly drawn by comparing the fast filling of the film-gated strip and the slow
1390
filling of the center-gated disk. Conversely, the transition between the core and shell layers is less stiff in the case of fast filling than for long injection times. Bay and Tucker [81b] finally observed that the sensitivity of fiber orientation to the velocity profile and the polymer rheology is high, basically since the orientation profile cannot be accurately predicted if the velocity profile is inaccurate. With a view to investigating the role of the suspension rheology, Chung and Kwon [49,50] developed a simulation program to predict the injection of shortfiber reinforced thermoplastics in arbitrary 3D mold cavities. Their lubrication flow model was based on combining the Hele Shaw approximation with the DinhArmstrong model, in order to couple the velocity and orientation predictions. The hybrid closure approximation was used. The filling of a tension test specimen and a center-gated disk was analyzed. The presence of different layers across the thickness was well predicted, and the same conclusions as Bay and Tucker [8 l b] were obtained from parametric analysis.
5.2 Filling of long-fiber reinforced parts Jackson et al. [74] analyzed the fiber orientation in thin compression molded parts by integrating the distribution function evolution equation with a non-vanishing interaction coefficient. Very good qualitative correspondence between predictions and experiments was obtained by assuming a plug flow across the thickness (without fountain flow effect) and considering a 2D orientation field. Later, Reifschneider and Akay [85] studied the compression molding of a truck head with multiple charges, whose thermo-mechanical properties were predicted by means of the Halpin-Tsai and Scharpery equations. Verleye [4] analyzed the filling of a BMC box (Figure 13). Although the polymer was a thermosetting resin, curing was neglected since it takes place mainly after filling. The coupling between flow and orientation was also neglected (whereas it is known that in-plane viscosity can play a significant role in some cases). Experimental data were provided by the Owens Coming company. The simulation was performed by adapting the power index and using the 2D natural closure. As a result, the relative orientation state was very well predicted, in particular near to the gate, the welding line and the last flow front (where agreement is difficult to achieve), while the pressure drop was less accurately estimated.
Figure 13 (next page). Filling of a thermoset box: (a) finite element mesh; (b) successive fronts; (c, d) predicted fiber orientation from 2 viewpoints; (e) experimental orientation. (From [4]).
1391
1392
Figure 14 (next page). Compression molding of a container" (a) initial load and successive flow fronts; (b) final fiber orientation field; (c, d) comparison between the mechanical effect of the flow-induced orientation (c) and a reference case with planar isotropic orientation (d) (isovalues indicate the von Mises stress, and w is the camber); (e) example of temporary mesh; (f) transient orientation field during filling. (From [7]).
5.3 Calculation of thermo-mechanical properties This section will be closed by recalling that the ultimate aim of fiber orientation prediction in molding or other forming processes is to correlate the thermo-mechanical properties of the product with the final orientation distribution in the part and the processing conditions. A single example, from the work of Lielens et al. [7,112], will be considered to illustrate this issue, which is beyond the scope of the chapter. See also [5,6]. Using the predicted fiber orientation distribution represented by the 2nd-order tensor aij, the authors developed a homogenization theory in order to evaluate the stiffness, thermal stress and thermal conductivity in multiphase composite parts with inclusions of arbitrary orientations. It is worth noting that orientation averaging over the aggregates was directly performed by using the aij and aijkl tensors, the latter being estimated by means of the natural closure approximation. The simulation results, which concern the compression molding of a 5mm thick container, are depicted in Figure 14. In view of the part's symmetry, only one quarter of the geometry is represented. In this problem, the abrupt contraction located at the border of the upper horizontal face, and whose thickness ratio changes when compression progresses, has a strong impact on the downstream orientation distribution, which in tum direcly affects the thermo-mechanical properties in the molded part. Using these results, it was possible to calculate the reaction of the container when clamped on its lower horizontal face and loaded with an internal pressure of 1 bar [6,7,112]. The computation software SAMCEF was used for that purpose. 6. CONCLUDING REMARKS AND A C K N O W L E D G M E N T S This chapter was first devoted to analysing the general behavior of fiber suspension flows, with a particular attention focusing on the decoupled prediction of the fiber orientation evolution in a Newtonian solvent. Since orientation
1393
1394
tensors represent the best approach to model fiber orientation, the closure approximation problem has been investigated, and the theory of the natural closure approximation has been thoroughly developed. The second objective of the chapter was to review the models and numerical algorithms that are available to predict the fiber orientation distribution in injection and compression molded parts. The particular techniques implemented in the MOLDSYS software in order to perform decoupled orientation predictions have been detailed (the flow problem is addressed in a companion chapter). Several examples, including successful experimental validations, demonstrate the quality of the models and numerical methods that have been elaborated Besides the authors of this chapter, several people participated in this research, including Marcel Crochet, Alain Couniot, Laurent Dewez, Hubert Henry de Frahan, Bernard Languillier, Gregory Lielens, Pascal Pirotte and Luc Vanderschuren, whom the authors wish to thank here for their contributions. The work was carded out within the framework of the "Multimat6riaux" project of the Walloon Region of Belgium, the program of Intenmiversity Attraction Poles initiated by the Belgian state and the COST 512 European project, and in collaboration with the Shell Research and Technology Center in Amsterdam (the Netherlands) and the Owens Coming company in Battice (Belgium). Grants from the IRSIA (Belgium) are acknowledged. The authors wish to thank Natasha Van Rutten for the drawing of several figures, and Jacques G6rard, Emile Franck and Jacques Bleyfuesz for their help and advice in performing the molding experiments. The efficient page setting work of Victor Vermeulen was also appreciated. REFERENCES 1. W.C. Jackson, F. Folgar and C.L. Tucker 1II, in C.D. Han (ed.), Polymer Blends and Composites in Multiphase Systems, American Chemical Soc., Washington (1984) 279. 2. C.L. Tucker lJ/, J. Non-Newtonian Fluid Mech., 39 (1991) 239. 3. C.L. Tucker Ill and S.G. Advani, in : S.G. Advani (ed.), Flow and Rheology in Polymer Composites Manufacturing, Elsevier, Amsterdam (1994) 147. 4. V. Verleye, Ph.D. Thesis, Universit6 catholique de Louvain, 1995. 5. V. Verleye, G. Lielens, P. Pirotte, F. Dupret and R. Keunings, Proc. Numiform'95, Balkema, Rotterdam (1995) 1213. 6. A. Couniot, L. Dewez, F. Dupret, R. Keunings, G. Lielens, P. Pirotte, J. Bland, A.G. Gibson, G. Kotsikos, R. Gebart, J. Krispinsson, F. Vahlund, S. Toll, J.-A.E. M~ason and C. Servais, Proc. MMSP'96 General COST 512 Workshop, Davos, Switzerland, European Commission (1996) 99.
1395 7. G. Lielens, P. Pirotte, A. Couniot, F. Dupret and R. Keunings, Composites Part A, 29A (1998) 63. 8. M.C. Altan, J. Thermoplastic Composite Mat., 3 (1990) 275. 9. M.J. Crochet, F. Dupret and V. Verleye, in : S.G. Advani (ed.), Flow and Rheology in Polymer Composites Manufacturing, Elsevier, Amsterdam (1994) 415. 10. G.B. Jeffery, Proc. Roy. Soc., A102 (1922) 161. 11. F.P. Bretherton, J. Fluid Mech., 14 (1962) 284. 12. H. Brenner, Int. J. Multiphase Flow, 1 (1974) 195. 13. M. Vincent and J.F. Agassant, Rheol. Acta, 24 (1985) 603. 14. J. Feng and D.D. Joseph, J. Fluid Mech., 324 (1996) 199. 15. Y. Iso, D.L. Koch and C1. Cohen, J. Non-Newtonian Fluid Mech., 62 (1996) 115; 135. 16. S. Yamamoto and T. Matsuoka, J. Chem. Phys., 98 (1993) 644; 100 (1994) 3317. 17. S. Yamamoto and T. Matsuoka, Polym. Eng. Sci., 35 (1995) 1022; 36 (1996) 2396. 18. A.B. Metzner, J. Rheol., 29 (1985) 739. 19. G.K. Batchelor, J. Fluid Mech., 46 (1971) 813. 20. F. Folgar and C.L. Tucker III, J. Reinforced Plastics and Composites, 3 (1984) 98. 21. S.M. Dinh and R.C. Armstrong, J. Rheol., 28 (1984) 207. 22. E.S.G. Shaqfeh and D.L. Koch, Phys. Fluids A, 2 (1990) 1077. 23. D.L. Koch and E.S.G. Shaqfeh, Phys. Fluids, 2 (1990) 2093. 24. D.L. Koch, Phys. Fluids, 7 (1995) 2086. 25. G.K. Batchelor, J. Fluid Mech., 41 (1970) 545. 26. E.J. Hinch and L.G. Leal, J. Fluid Mech., 71 (1975) 481; 76 (1976) 187. 27. G.G. Lipscomb II, M.M. Denn, D.U. Hur and D.V. Boger, J. Non-Newtonian Fluid Mech., 26 (1988) 297. 28. G.H. Fredrickson and E.S.G. Shaqfeh, Phys.Fluids A, 1 (1989) 3; 2 (1990) 7. 29. M.B. Mackaplow, E.S.G. Shaqfeh and R.L. Schiek, Proc. R. Soc. Lond. A, 447 (1994) 77. 30. M.B. Mackaplow and E.S.G. Shaqfeh, J. Fluid Mech., 329 (1996) 155. 31. A.G. Gibson and G.A. Williamson, Polym. Eng. Sci., 25 (1985) 968, 32. A.G. Gibson, Composites, 20 (1989) 57. 33. A.N. McClelland and A.G. Gibson, Composites Manufact., 1 (1990) 15. 34. S. Ranganathan and S.G. Advani, J. Rheol., 35 (1991) 1499. 35. S. Ranganathan and S.G. Advani, J. Non-Newtonian Fluid Mech., 47 (1993) 107.
1396 36. S. Toll and J.-A.E. M~-ason, J. Rheol. 38 (1994) 985. 37. R.R. Sundararajakumar and D.L. Koch, J. Non-Newtonian Fluid Mech., 73 (1997) 205. 38. M.R. Barone and D.A. Caulk, J. Appl. Mech., 53 (1986) 361. 39. H. Henry de Frahan, V. Verleye, F. Dupret and M.J. Crochet, Polym. Eng. Sci., 32 (1992) 254. 40. V. Verleye and F. Dupret, Proc. ASME'93, New Orleans, Louisiana, AMDVol. 175 (1993) 139. 41. V. Verleye, A. Couniot and F. Dupret, Proc. ASME'94, Chicago, Illinois, MD-Vol. 49, HTD-Vol. 283 (1994) 265. 42. F Dupret, V. Verleye and B. Languillier, Proc. ASME'97, Dallas, Texas, FED-Vol. 243, MD-Vol. 78 (1997) 79. 43. S.G. Advani and C.L. Tucker 111,J. Rheol., 31 (1987) 751; 34 (1990) 367. 44. A.J. Szeri and L.G. Leal, J. Fluid Mech., 242 (1992) 549; 262 (1994) 171. 45. J.S. Cintra and C.L. Tucker III, J. Rheol., 39 (1995) 1095. 46. A.J. Szeri and D.J. Lin, J. Non-Newtonian Fluid Mech., 64 (1996) 43. 47. G.G. Lipscomb II, R. Keunings, G. Marucci and M.M. Denn, Proc. I X th Int. Congress on Rheology, Acapulco, Mexico, Elsevier, 2 (1984) 497. 48. J. Rosenberg, M. Denn and R. Keunings, J. Non-Newtonian Fluid Mech., 37 (1990) 317. 49. S.T. Chung and T.H. Kwon, Polym. Eng. Sci., 35 (1995) 604. 50. S.T. Chung and T.H. Kwon, Polym. Composites, 17 (1996) 859. 51. J. Azaiez, R. Gu6nette and A. Ait-Kadi, J. Non-Newtonian Fluid Mech., 73 (1997) 289. 52. K.K Kabanemi, J.F. H6tu and A. Garcia-Rejon, Intern. Polym. Proc. 12 (1997) 182. 53. V. Verleye and F. Dupret, Proc. Numiform'92, Balkema, Rotterdam, (1992) 389. 54. V. Verleye, A. Couniot and F. Dupret, Proc. ICCM/9 (9th Int. Conf. on Composite Materials), Madrid, Spain (1993) 642. 55. A. Couniot, V. Verleye and F. Dupret, Proc. FPCM'94 (Flow Processes in Composite Materials), Galway, Ireland (1994) 227. 56. V. Verleye, A. Couniot and F. Dupret, Proc. CADCOMP'94 (CAD in Composite Material Tech.), Southhampton, United Kingdom (1994) 303. 57. F. Dupret and V. Verleye, in preparation (1998). 58. L.G. Leal and E.J. Hinch, J. Fluid Mech., 46 (1971) 685. 59. M.S. Ingber and L.A. Mondy, J. Rheol., 38 (1994) 1829. 60. A.J. Szeri, W.J. Milliken and L.G. Leal, J. Fluid Mech., 237 (1992) 33. 61. G.K. Batchelor, J. Fluid Mech., 44 (1970) 419.
1397 62. R.C. Givler, in M.M. Chen, J. Mazumder and C.L. Tucker (eds.), Transport Phenomena in Materials Processing, ASME, New York (1983) 99. 63. R.C. Givler, M.J. Crochet and R.B. Pipes, J. Composite Mat., 17 (1983) 330. 64. E.H. MacMillan, J. Rheol., 33 (1989) 1071. 65. B.N. Rao, S. Akbar and M.C. Altan, J. Thermoplastic Composite Mat., 4 (1991) 311. 66. Y. Yamane, Y. Kaneda and M. Doi, J. Physical Soc. Japan, 64 (1995) 3265. 67. K.A. Ericsson, S. Toll and J.-A.E. M~nson, J. Rheol. 41 (1997) 491. 68. C.A. Stover, D.L. Koch and C1. Cohen, J. Fluid Mech., 238 (1992) 277. 69. E.J. Hinch and L.G. Leal, J. Fluid Mech., 52 (1972) 683; 57 (1973) 753. 70. M. Doi, J. Polymer Sci. : Polymer Physics Ed., 19 (1981) 229. 71. M. Rahnama, D.L. Koch, Y. Iso and C1. Cohen, Phys. Fluids, 5 (1993) 849. 72. M. Rahnama, D.L. Koch and E.S.Go Shaqfeh, Phys.Fluids, 7 (1995) 487. 73. M.C. Altan, S.G. Advani, S.I. Giigeri and R.B. Pipes, J. Rheol., 33 (1989) 1129. 74. W.C. Jackson, S.G. Advani and C.L. Tucker, J. Composite Materials, 20 (1986) 539. 75. E.S.G. Shaqfeh, Phys. Fluids, 31 (1988) 2405. 76. I.L. Claeys & J.F. Brady, J. Fluid Mech., 251 (1993) 411; 443; 479. 77. T.C. Papanastasiou and A.N. Alexandrou, J. Non-Newtonian Fluid Mech., 25 (1987) 313. 78. C.L. Tucker III, in : T.G. Gutowski (ed.), The Manufacturing Science of Composites, ASME, New York (1988) 95. 79. S.G. Advani and C.L. Tucker, Polym. Composites, 11 (1990) 164. 80. M.C. Altan, S. Subbiah, S.I. Gtigeri and R.B. Pipes, Polym. Eng. Sci., 30 (1990) 848. 81. R.S. Bay and C.L. Tucker III, Polym. Composites, 13 (1992) 317; 332. 82. M.C. Altan and B.N. Rao, Proc. ASME'92, Anaheim, California, AMD-Vol. 153, FED-Vol. 141 (1992) 93. 83. M.C. Altan, S.I. Giigeri and R.B. Pipes, J. Non-Newtonian Fluid Mech., 42 (1992) 65. 84. M.C. Altan and L. Tang, Rheol. Acta, 32 (1993) 227. 85. L.G. Reifschneider and H.U. Akay, Polym. Composites, 15 (1994) 261. 86. L. Tang and M.C. Altan, J. Non-Newtonian Fluid Mech., 56 (1995) 183. 87. M.C. Altan and B.N. Rao, J. Rheol., 39 (1995) 581. 88. B.E. VerWeyst, C.L. Tucker m and P.H. Foss, Intern. Polym. Proc. 12 (1997) 238. 89. E.T. Onat and F.A. Leckie, J. Appl. Mech., 55 (1988) 1. 90. C.V. Chaubal and L.G. Leal, J. Rheol., 42 (1998) 177.
1398
91. S. Prager, Trans. Soc. Rheology, 1 (1957) 53. 92. G. Ausias, J.F. Agassant, M. Vincent, P.G. Lafleur, P.A. Lavoie and P.J. Carreau, J. Rheol., 36 (1992) 525. 93. G.L. Hand, J. Fluid Mech., 13 (1962) 33-46. 94. B.N. Rao, L. Tang and M.C. Altan, J. Rheol., 38 (1994) 1335. 95. M.A. Bibbo, S.M. Dinh and R.C. Armstrong, J. Rheol., 29 (1985) 905. 96. N. Phan-Thien and A.L. Graham, Rheol. Acta, 30 (1991 ) 44. 97. R.B. Pipes, J.W.S. Hearle, A.J. Beaussart, A.M. Sastry and R.K. Okine, J. Composite Materials, 25 (1991) 1204; 1379. 98. A. Ramazani, A. Ait-Kadi and M. Grmela, J. Non-Newtonian Fluid Mech., 73 (1997) 241. 99. C.V. Chaubal, L.G. Leal and G.H. Fredrickson, J. Rheol., 39 (1995) 73. 100. P.K. Currie, J. Non-Newtonian Fluid Mech., 11 (1982) 53. 101. A.J. Szeri and L.G. Leal, J. Fluid Mech., 250 (1993) 143. 102. R. Blanc, S. Philipon, M. Vincent and J.F. Agassant, Intern. Polym. Proc., 2 (1987) 21. 103. T. Matsuoka, J.-I. Takabatake, Y. Inoue and H. Takahashi, Polym. Eng. Sci., 30 (1990) 957. 104. B. Friedrichs, S.I. Gii~eri, S. Subbiah and M.C. Altan, J. Mat. Process. & Manufact. Sci., 1 (1993) 331. 105. S. Toll & P.-O. Andersson, Polym Composites, 14 (1993) 116. 106. L. Schwartz, "Th6orie des distributions", Hermann, Paris (1966). 107. F. Dupret and L. Vanderschuren, AIChE Journal, 34 (1988) 1959. 108. M. Van Dyke, Applied Mathematics and Mechanics, Vol. 8, Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964. 109. R.B. Davis, M.Sc. Thesis, University of Illinois, Urbana, 1988. 110. S. Kenig, Polymer Composites, 7 (1986) 50. 111. P.F. Bright, R.J. Crowson and M.J. Folkes, J. Mat. Sci., 13 (1978) 2497. 112. G. Lielens, Ph.D. Thesis, Universit6 catholique de Louvain, 1998.
1399
R E C E N T A D V A N C E S IN T H E R H E O L O G Y MATERIALS
OF FLUIDIZED
S. I. Bakhtiyarov and R. A. Overfelt Space Power Institute, 231 Leach Center, Auburn University, Auburn, AL 36849-5320, USA 1. INTRODUCTION Flows of particulate materials are encountered in numerous commercial industries including metallurgical, chemical, agricultural, pharmaceutical, plastics, and food processing. Extensive applied research in these areas over the years has developed a large database of empirical correlations that are used by designers to engineer practical reactors and devices. However, the empirical nature of the correlations often limits the applicability of the data to specific situations. The successful application of computational techniques to accurately simulate single-phase flows in complex flow geometries makes their application to multiphase flows also highly desirable. Although rigorous two-phase flow algorithms have been demonstrated for simple geometries, excessive computational times preclude their widespread application to complex industrial problems. Single phase algorithms must be modified to enable routine design calculations. Central to the successful exploitation of single phase codes is the measurement of appropriate transport properties (e. g., viscosity) that represent the complex rheological behavior of these multi-phase fluids. In this review we consider the works related to achieving an understanding of the hydrodynamics of aggregative fluidization (gas-solid beds) with special emphasis on rheological aspects of the problem. The review part of this work covers the literature that was published in 1985 or later (also unpublished) and a few earlier materials concerning the rheology of fluidized beds which were out of the scope of previous major review works [ 1, 2].
1400
2. FLOW REGIMES IN FLUIDIZED BEDS
In both particulate (fluidization with a liquid) and aggregative fluidizations there are several hydrodynamic flow regimes primarily related to the characteristics of the solid phase, the fluid phase or the design of the equipment. To understand the flow regimes in aggregative fluidization Grace [3] applied the analogy with gas-liquid two phase flow. The following flow regimes have been identified 9 delayed bubbling regime, 9 bubbling regime, 9 slugging regime, 9 turbulent regime, 9 fast fluidization, and 9 pneumatic conveying. Bi and Grace [4] presented flow regime maps for gas-solid fluidized beds (fixed bed, bubbling regime, slugging regime and turbulent regime) and gassolids upward transport (dilute-phase flow, fast fluidization or turbulent flow, slug/bubbly flow, bubble-free dense-phase flow and packed bed flow). The flow regime maps are shown with Re / Ar u3 plotted against Ar u3. Practical flow regime maps are presented as ~ plotted against Qp [ pp. In delayed bubbling regime (fixed bed) the particles contained in the bed are motionless and are supported by contact with each other. The pressure drop flowrate diagram obtained by Bakhtiyarov and Overfelt [5] for fluidization of silica sand by air is a good indication of flow regimes in aggregative fluidization (Figure 1). At low air flow rates the pressure drop increases with air flow rate. This branch of the curve pertains to air flow through the fixed bed. At a certain air flow rate the pressure drop decreases rather suddenly, which is obviously essential for predicting the onset of bubbling regime. The value of the pressure drop at bubbling onset significantly depends on the initial fixed bed height in the fluidization chamber. As the air velocity continues to increase the pressure drop increases slowly but cannot reach the pressure drop at onset of channeling. Qualitatively these curves are in good agreement with the data of earlier experiments [6]. There are several approaches to predict the superficial velocity at minimum fluidization; however, most equations are of the form [7] Remf = (C12 + C2 Ar) 0"5 - C l , where
Remf = pg dp Umf / ~g and Ar = pg (pp - pg) g dp3 ] ~j,g2 .
(1)
1401
Most experimental studies of bubbling phenomena are interested in bubble size, velocity, shape and flow patterns. Jackson [8] concluded that the theory of stability of uniform fluidized beds reveals the basic instability without which the process of bubble growth cannot begin. Many researcher confirmed that the bubbles in fluidized beds can be developed from a small perturbation. Experimental data show that increasing pressure causes bubbles of smaller sizes resulting in smoother fluidization [9, 10]. Based on their experimental data with fluidized cracking catalyst and polypropylene powder, Piepers and Rietema assume that the elasticity modulus is a significant property of the homogeneous and heterogeneous fluidized beds [10]. Gautam, Jurewicz and Kale [11] applied laser Doppler anemometry to non-intrusively measure the through-flow component in single isolated bubbles in two-dimensional fluidized beds. They observed that the bubble velocity increases with distance from the air distributor and the through-flow inside the bubble (at a given height above the air distributor) is constant. Recirculation zones of gas were not observed inside the bubbles. Gera and Gautam [12] reported that the bubble aspect ratio plays an important role in predicting an accurate gas flow through the bubble. They also observed that the through-flow velocity component at the nose of an elongated bubble is higher than that in a flattened bubble. Utilizing a newly developed capacitance imaging system, Halow et al. [13] reported that the aspect ratio of bubbles appears to increase linearly with bubble diameter up to a bubble to tube diameter ratio of about 0.5, and remains constant up to a db/ d of 0.7 - 0.9, then increases rapidly as db --- d. Using X-rays, Yates, Cheesman and Sergeev [ 14] examined air-fluidized beds (solid particles of A and B groups of the Geldart classification) and found that the rising bubbles are surrounded by an expanded "shell" of particles in which the void factor is significantly in excess of that of the emulsion phase remote from the bubbles. Based on Lavrentyev's turbulent-eddy flow model, Bakhtiyarov and Overfelt [5] predicted the friction coefficient during the bubbling regime = 12 c / Re [1 - ot2(3 - 2o0 + 6o~2(1 - o~) n/(n+l)],
(2)
where c, n and ~ are dimensionless parameters depending on particle and gas properties and the design of the fluidization chamber. The experimental results, obtained employing combined capacitance-pressure probes on the apparatus shown in Figure 2 with silica sand fluidized by air [5, 15], confirmed the predictions of equation (2) (Figure 3). As seen from the comparison of analytically predicted and experimentally obtained data, there is good agreement at a wide range of Reynolds number values.
1402
Figure 1. Variation of the pressure drop as a function of the average air velocity and height of the fixed silica sand bed [5]. Slugging is a flow regime in which gas bubbles increase to the diameter of the fluidization chamber. The slugs of solid particles will move upward in a pistonlike manner, reach a certain height, and then rain through the gas phase in the form of aggregates or as individual particles. In group A of HovmandDavidson's [ 16] classification, air slugs rise through the bed of particles which rain down through the void to allow its upward motion. In group B, the upward movement of the interface is slow and is caused by particles raining down uniformly through the disperse regions. A slug flow of group A is analogous to the flow of the two-phase (gas-liquid) system. Therefore the group A slug flow has been studied thoroughly. A slug flow of group B occurs at certain values of the bed height to diameter ratio (H/D) and the superficial velocity of gas. To predict the slug flow regime some investigators introduce an "equivalent bubble diameter" which is about 1/3 - 1/2 of the bed diameter. The effect of slugging regime on the energy loss in fluidized beds has been investigated analytically and experimentally by Bakhtiyarov and Overfelt [15]. Based on Meshchersky's model of motion of a body having variable mass and the B lake-Kozeny-Carman equation of porous media flow, the following expression for friction coefficient as a function of the Reynolds number, void factor, gas velocity, sizes of the slug and fluidization chamber has been obtained
1403
Figure 2. Apparatus for study of hydrodynamics of fluidized bed [5, 15]. 10000
!_
o C"
h=7cm h=18.4cm
A
h=33.65cm
- - " ' e q u a t i o n (2)
0 t~ lie
o
0 o
1000
,m
~0 !._ lie
100 10
100
Reynolds Number
Figure 3. Friction factor as a function of Reynolds number in bubbling regime of fluidized silica sand bed [5].
X = 321/{Re [1- [~4 -I- (1- 1~2)2 / In 13 + 8 e3 132/ 5 R z S 2 (1- e)2]}, where ~ is a design parameter of the fluidization chamber.
(3)
1404
Also experiments were run in the fluidization chamber (Figure 2) with silica sand at different values of the fixed bed height, pressure and air flow rate [15]. The study of the slug motion dynamics demonstrated that both the drag coefficient and the resistance factor decrease with increasing the Reynolds number and the porosity of slug. At a given Reynolds number, the values of the friction coefficient and the resistance factor are higher than for fluidized bed with large value of the fixed bed height (Figure 4). There is a qualitative agreement between the experimental data and predictions, with better agreement at lower values of Reynolds number. One of the major complications in the fluidization of materials is a tendency towards radial segregation in the bed. Segregation is a state of the fluidized bed in which a larger proportion of solid particles is found in one region than in another one. In many experimental studies of gas-solid particles two-phase flow the radial segregation phenomenon has been observed, where a fluidized bed comprises a rapidly rising core with a low concentration of solids and an annulus region near the column wall where the solids concentration is higher. The research literature related to segregation phenomena and published before 1985 was reviewed by Nienow and Chiba [17]. Shape factor, density, moisture content, size and size distribution of the solids are the main factors that affect segregation tendencies. Nienow, Naimer and Chiba, [18] concluded that the distributor design has a significant effect on segregation tendencies. According to their results, standpipe and perforated plate distributors provide better mixing than a porous plate at the same superficial gas velocity. To predict the presence of segregation during fluidized bed operations Delebarre, Pavinato and Leroy [19] introduced an index based on pressure losses along the fluidized column. Introducing suspended particle distribution coefficients, Hong and Tomita [20] presented a model for high density gas-solid stratified pipe flow, in which the particle-particle interactions between the suspension and the sliding bed were taken into account. They found that particles begin to drop out of the gas phase at the saltation point. The impact of the inlet configuration on the rate of segregation of particles to the wall and the internal recirculation during the steady, developing flow of gas-solid suspensions in a vertical tube were investigated numerically by Pita and Sundaresan [21]. Examining the three inlet configuration (uniform, core-annulus and circumferential), these authors conclude that a circumferential injection of gas has a favorable effect on the flow in the sense that it can decrease the extent of internal recirculation, and hence, the segregation of particles to the wall. Dasgupta, Jackson and Sundaresan [22] analyzed the time-smoothed equations for the motion of dense
1405
suspensions to demonstrate the impact of density fluctuations on the occurrence of segregation. It was shown that the solid particles will congregate in regions where the kinetic energy of fluctuations associated with the particles is small. 100
9
0
h=18.4cm
[]
h=33.65cm
!....
o .0
10
.
.
t~
tl. ~:
.
h=18.4cm (eqn.3)
9
h=33.65cm
,,i.I 0
= 9
,,_
(eqn.3)
"~
1
N 0.1
. . . . . . . . 10
~
~-
100
. . . . . . 1000
Reynolds Number
Figure 4. Variation of friction factor with Reynolds number at slug flow regime
[15]. Rhodes, Zhou and Benkreira developed semi-empirical models circular tube. According to the solids is dependent on the average
[23] and Bakhtiyarov and Overfelt [24] for radial segregation of solid particles in a models proposed, the radial distribution of solids concentration as
(1 - 8) / (1 - 8w) = C1 (r/R) n,
(4)
(r/R) n,
(5)
(1 - ~) / (1 - ~av) = C2
where C~, C 2 and n are constants (C~ = 1, [14, 15]; C2 = 2, [14, 15]; n = 2, [14]; n = 3, [15]). In Figures 5 and 6 the radial solids concentration profiles predicted by theoretical analysis [23, 24] are compared with experimental data of different researchers [24-29]. As seen from these figures, there are qualitative and quantitative agreements between the experimental data and the theoretical predictions. Spouting is a state of fluidized bed in which a fluid jet at high rate pierces through the bed and a central channel is formed. The spouted bed is limited to relatively large solid particles and mainly is affected by construction of the
1406
9 A
.=
2.5
I ~o !
2 1.5
A
r!
1
qlI
~"
0.5 "~"
U=3.8m/s [27]
[]
U=4.0m/s [28]
9
U=3.3m/s [26]
A
U=O.2m/s [25]
0
U=1.3m/s [24]
0
U=l.7m/s [24]
[]
U=2.3m/s [24] --"theory [23] theory [24]
0
0.2
0.4
0.6
0.8
1
rlR
Figure 5. Radial profile of solids concentration. fluidization chamber. Bakhtiyarov and Overfelt [30] reported the results of the theoretical and experimental analyses of the spouted bed concerning the influence of the design parameters of the fluidization chamber on the hydrodynamic characteristics of the fluidized bed. The friction factor was expressed as a function of Reynolds number, as following ~, = 8 13/ k Re,
(6)
where ~ is a design parameter of fluidization chamber and k is a function of the permeability of the spouted bed. The experimental data obtained with silica sand on the apparatus described in Figure 2 compare well with the analytical predictions (Figure 7). In the past years experimental and theoretical studies of the gas-solid turbulent flow have attracted considerable interest. Gidaspow et al. [31] and Miller and Gidaspow [32] measured the particle average velocity and concentration of gassolid flow in a vertical tube. It was shown that the main differences between dilute and dense gas-solid turbulent flow are the mechanism of exchange of momentum and fluctuation kinetic energy between the particulate and the fluid phases. To establish a thermo-mechanical formulation for turbulent gas-solid flows, Ahmadi and Ma [33] utilized a phasic mass-weighted averaging technique. The velocity, void factor and fluctuation kinetic energy of different phases were predicted. This model was used by Cao and Ahmadi [34] to analyze the steady fully-developed, dilute and dense, gas-particle turbulent flow between two parallel plates. The model predictions for the particulate and gas phases were compared with the experimental data of Miller and Gidaspow [32] and Tsuji, Morikawa and Shiomi [35] and good agreement was found.
1407
Introducing a new mean drag law for the solid particles, Singh and Joseph [36] proposed one dimensional model of fluidized beds where the fluid and solid momentum equations are decoupled. The linear stability analysis of the model shows that uniform fluidization is unstable even when the force acting on a solid particle is assumed to depend on the area fraction. The results of this analysis agree with the experimental results. A theoretical model for the fully developed flow of gas-solid suspensions in a vertical tube was developed by Sinclair and Jackson [37]. Over the whole range of co-current and counter-current flows, they predicted the relation between gas pressure gradient and the flow rates of the two phases. A drag force between the gas and the solid particles is a function of their relative velocity. The existence of the mutual interactions between the solid particles through inelastic collisions has been reported. The model has been expanded by Pita and Sundaresan [38] for gas-solid suspensions flow in vertical tubes of different diameters. The flow of gas-solid suspensions in ducts of arbitrary inclination was considered by Ocone,Sundaresan and Jackson [39].These authors conclude that, as a result of the compaction due to gravity, it is necessary to take into account forces transmitted between particles at points of sustained, rolling and sliding contact. Tu and Fletcher [40] reported the results of numerical computations and comparisons with the LDV measurements of earlier experiments for turbulent gas-solid particle flow in a 90 ~ square-sectioned bend. The comparison of the mean velocity profiles of both the gas and particulate phases gave good agreement with the LDV measurements. Authors predicted the localized high particulate concentration near the outer curve of the bend that occurs at large Stokes numbers. 3. R H E O L O G Y OF FLUIDIZED MATERIALS There are several different experimental approaches to the study of the transport properties of fluidized materials. The problem of the rheologist is the interpretation of the flow behavior of a fluidized material in terms of its physical and chemical properties and its state of fluidization. The first measurements of an "apparent viscosity" of a fluidized material were reported by Matheson, Herbst and Holt [41]. They used a modified Stormer viscometer with a paddle 1.9 cm wide and 3.81 cm high. It was immersed in an aerated bed of solids 7.62 cm deep. A viscosity was obtained at a sensor speed of 200 rpm by comparing a liquid of known viscosity. They reported that the fluidized bed
1408
3 > 2.5 I r Q,)
xI
2
i qlm
1.5 A
I"
0.5 0 0
0.2
0.4
0.6
0.8
1
rlR F i g u r e 6. Radial Profile of solids concentration.
[29]" 9 - dp = 0.054 m m , pp =
930 k g / m 3, U = 3.58 m/s; II - dp = 0.075 mm, pp = 608 k g / m 3, U = 3.58 m/s; ~ dp = 0.043 mm, pp = 2003 k g / m 3, U = 3.58 m/s; A - dp= 0.054 m m , pp = 930 k g / m 3, U = 1.40 m/s; "ff - dp = 0.054 mm, pp = 930 k g / m 3, U = 3.15 m/s; 9 - dp = 0.054 m m , pp = 930 k g / m 3, U = 2.8 m/s; O - dp = 0.054 mm, pp = 930 k g / m 3, U = 2.16 m/s; X - dp = 0.075 mm, pp = 608 k g / m 3, U = 2.16 m/s; [24]" dp = 0.32 mm, pp = 2593 k g / m 3, A - U = 1.25 m/s, ~ - U = 1.7 m/s, .I, - U = 2.25 m/s; theories" m m [23], ~ [24]. 1000
It.
100
10
i
100
i
*
*
Reynolds Number
.
*
9
.
1000
F i g u r e 7. Variation of friction factor with R e y n o l d s n u m b e r at spouted flow r e g i m e [30].
1409 was significantly less viscous as the gas flowrate was increased and particle size was decreased. The addition of particles of large size had little effect on the viscosity, but the addition of negligible amounts of fine particles significantly lessened the viscosity. Trawinski [42] suggested an interesting model to explain this phenomena. According to his model, fine particles act as "ball bearings" between moving surfaces. Later, Kramers [43], Furukawa and Ohmae [44], Shuster and Haas [45] and Grace [46] reported viscosity data also using Stormer viscometers obtained with fluidized beds. The viscosity data actually reported by a Stormer viscometer are the weights required to spin the immersed paddle at a certain angular velocity. A different approach using a falling sphere was applied by Peters and Schmidt [47] and Trawinski [42]. Applying the hole of the liquid state, Trawinski [42] also analytically described the fluidized bed. The following formula has been proposed to predict the effective viscosity of the fluidized system
'1]a=[TIg+64(1-Emf)Pp(pp-pg)gdp 3] [U/(Umf E m f ) - K ( U [ U m f ) ~
0"5 -
1 ].
Viscometers based on Hagen-Poiseuille laminar flow have also been used, among the first by Siemes [48] and Siemes and Hellmer [49]. The main advantages of this method are that (a) the flow process has no effect on the state of the fluidized material and (b) the fluidized material flows in inclined channels like a Newtonian liquid. However, Wu et al. [50] observed an interesting effect in sand flow through orifices. At small particle diameters (0.01 to 0.1 mm) the flow of solids is steady in the narrow range of the orifice diameter to particle diameter ratio (2 to 12). At larger values of this ratio, the oscillatory flow of sand through the orifice has been observed. Using a twophase fluid flow model based on kinetic theory for granular flow, Manger et al. [51 ] developed a numerical model of the oscillatory flow of solid particles. The simulations show qualitative agreement with the experimental data of Wu et al.
[50]. Based on the behavior of gas bubbles (shape, pressure drop, size, etc.) some researchers attempted to predict the apparent viscosity of fluidized materials. The method based on shape of bubbles in fluidized beds to predict the viscosity was proposed by Grace [46], Davies and Taylor [52] and Mendelson [53]. It was found that the bubble shape factor is a function of the Reynolds, Weber and EtStvos numbers. Using the "included angle" (0) (angle of supporting surface from horizontal) Grace [46] proposed the following empirical formula to compute the apparent viscosity Tla = 0.0435 dB UB pp (1 - a m f ) exp (0.004 0).
1410
Estimated values of the apparent viscosity were found to coincide with those obtained in [46, 54]. Murray [55] predicted the apparent viscosity of a fluidized materials by estimating the friction factor for rising bubbles. A method based on pressure measurements as a bubble rose through a fluidized bed was reported by Stewart [56]. Row and Partridge [57] estimated the apparent viscosity of a fluidized bed through the bubble size measurements. According to Sch~gerl, Merz and Fetting [54] only the viscosity data measured by a torsional pendulum oscillating at low amplitudes or by Couette viscometer at low angular velocities of the inner cylinder can be considered as quantitative. Capillary tube viscometers are preferred when the data are to be used for pipe flow problems, and rotational viscometers, which subject the material under the test to a precise and uniform rate of shear, have definite advantages in the analysis of complex systems [58]. Simple mathematical models cannot at present describe the general flow behavior of fluidized materials. The apparent viscosity of the fluidized material is a multiparametric function and is dependent on the physical and chemical properties of both the solids and the aerating fluid. Fluidized materials are particularly complex and, if numerical simulations of the behavior are to be reliable, it is critical that the measured values be consistent regardless of the measurement techniques applied. Bakhtiyarov, Overfelt and Reddy [58] presented the results of the experimental study of the "apparent viscosity" of fluidized silica sand utilizing both Poiseuille flows (capillary viscometer) and Couette flows (rotational viscometer). Viscometric measurements by capillary tube viscometers were run in the system shown schematically in Figure 8. The system consists of a clear acrylic cylindrical chamber 457.2 mm long and 69.85 mm inside diameter which is sealed at the top and bottom. A funnel with an angle of approach 57030 , was attached at the bottom of the cylinder. Precision-bore copper capillary tubes of 4.7625 mm and 7.9375 mm inside diameters and five different lengths (L/D = 15.328, 30.656, 61.333, 122.667 and 245.334) were screwed into the funnel exit. Air at a carefully controlled constant pressure was admitted through the top plug of the chamber. The rate of flow of solid particles is typically measured by collecting a sample over a measured time and determining its mass. The experiments were run using both non-coated capillary robes and capillary tubes coated (to prevent wall slippage) by sand of the same particle diameter of sample to be tested. Water and 50/50 glycerol/water mixture were used as calibrating liquids.
1411
One of the main difficulties in capillary tube viscometry is in accurately determining the appropriate pressure drops. The corrections for the head of sample over the tube, for kinetic energy effects, and for entrance losses are required. The first correction is straightforward. The other two corrections can be respectively estimated by (1) repeating experiments in capillary tubes of different lengths and extrapolating the overall pressure drop to zero length and (2) experimental calibration with Newtonian fluids of known viscosity and density.
Figure 8. Capillary tube viscometer used in [58]. A computer controlled Brookfield HADV-II+ rotational viscometer was also used in the experimental program [58]. This device measures the torque required to rotate a spindle immersed in a fluid. For a given viscosity, the viscous drag, or resistance to flow, is proportional to the spindle's speed of rotation and is related to the spindle's geometry. Measurements made using the same spindle at different speeds are used to detect and evaluate the rheological properties of the test material. Viscosity measurements were made at spindle angular velocities in the range of 1 to 100 rpm. The viscometer was calibrated by using 50/50 and 99/1 glycerol/water mixtures and 3.5% polyacrylamide solution. A round acrylic tube with internal diameter of 58 mm and 227 mm length was used as a fluidization chamber (Figure 9). Pressure taps were provided along the column. The fluidization chamber was arranged for use with air as the operating fluid. Compressed, dried and pre-filtered air under the pressure up to 840 kPa was supplied to the bottom of the fluidization chamber.
1412
Air pressure and flowrate were measured by computer controlled pressure transducer and rotameter, respectively. A polypropylene porous plate with 0.250 mm pore size and 3.175 mm thick was used as an air distributor. All measurements were made with silica sand of average particle density 2.593 g/cm 3. The distributions of particle diameters used were as follows: less than 0.212 mm, from 0.212 mm to 0.425 mm, and from 0.425 mm to 0.710 mm. To prevent wall slippage effects, the surface of the spindle was coated by sand of the same particle diameter of sample to be tested.
Figure 9. Apparatus for fluidized sand viscosity measurements by rotational viscometer [58]. The apparent shear strain rate and the apparent shear stress at the capillary wall are defined by Ta=4 Q / p r ~ R 3 = 4 t ~ / R ,
(7)
Xa= AP R / 2 L .
(8)
The apparent viscosity is defined by the ratio of Xa over ~/a" Tla - 'l;a / ~/a 9
(9)
In the case of sand flow, the capillary viscometer technique determines the apparent viscosity as the averaged value of all inner local viscosity. The value of the apparent viscosity is obviously a function of the shear strain rate q(a-
1413
Figure 10 shows the results obtained using the Brookfield viscometer for silica sand (0.212 mm < dp < 0.425 mm) at different bed void factors. As either the shear rate or the bed void factor (air flow rate) increases, the apparent viscosity decreases. The apparent viscosity is most sensitive to shear rate and voidage effects near incipient fluidization. Thereafter, the rate of decrease lessens. However, the measured viscosity was only slightly affected by changes in particle diameter for the diameters investigated. The apparent viscosity can be correlated with the apparent shear strain rate in the usual way as [58] 'rla = K~~an-1 .
(10)
where ~ is a measure of consistency of the fluidized bed. Lower void factor of the fluidized material yields larger values of consistency, n is a measurement of the fluidization behavior of the fluidized bed and is a function of the design and operating characteristics of the fluidized bed. Fluidized beds can exhibit behavior that is almost Newtonian (n = 1) or can display several anomalous behaviors. The greater the divergence of n from unity, the more anomalous is the fluidized bed. Figure 11 relates the apparent viscosity of fluidized silica sand measured by SchOgerl, Merz and Fetting [54] and Bakhtiyarov, Overfelt and Reddy [58]. From these data one can recognize that at low air flow rates (or low void factor) the apparent viscosity increases. Data obtained with the capillary viscometer [58] show that the mass flow rate of solids does not depend on the overall pressure drop (column height) at certain drained angles of repose. Gregory [59] also observed that in stick-and slip flow of solid particles no external application of pressure is required to push the solids through the pipe, the force exerted by gravity being all that is demanded. For this type of flow he proposed the following formula Qp = kl d z'5 ,
(11)
where kl is a coefficient which depends on a tube length, particle diameter, shape factor, etc. Rausch investigated gravity flow of a wide variety of solids in 76.2 mm and 203.2 mm diameter tubes [6]. He correlated all data by the following equation Qp = f c Cw (d]dp) 2"7 g0.5 Pb dp 2"5 [ (tan 13)0.5 ,
(12)
1414
Figure 10. Apparent viscosity of fluidized sand bed as a function of the shear rate and void factor [58].
0
10000 W
o
a.
50 rpm [58]
[] 100 rpm [58]
E
[]
0
~, [ 5 4 ] o
0 o
(/)
oo
1000
[] []
/1 []
/1
eG)
12, < 100
'
~o
'
' ' ''"~
'
~oo
'
. . . . . . . . . . . . . . .
~ooo
~oooo
Air Flowrate, cm31s Figure 11. Apparent viscosity of fluidized sand as a function of air flowrate.
1415
where c is a correction factor pertaining to the angle of approach in the hopper bottom; Cw is a wall-effect correction factor; dp = (d~ d2)~ , d~ and d2 are adjacent sieve openings; f is a correction factor for tube length to diameter ratio, introduced in [58]. Variation of the Gregory coefficient (k~) and the factor (f) versus the capillary tube length to diameter ratio (L/d) are shown in Figure 12 [58]. Both correction factors appear to be approaching a limiting values as L/d increases. 1.2
~
0
.= 0.8 ll_ .=_o 0.6
...__.o...-.-=-.---
0.4 0
0.2
0
50
100
150
200
250
Lid
Figure 12. Correction factors as functions of the capillary tube length to diameter ratio [58]. The apparent viscosity of sand particles (0.425 mm < dp < 0.710 mm) obtained on both the rotational and the capillary viscometers as function of the apparent shear rate is shown for comparison in Figure 13. Regardless of the method of measurement, there is a general consistency in the data obtained with the Brookfield viscometer (low shear rates) and capillary viscometer (high shear rates). As seen from this figure, the apparent viscosity can be satisfactorily correlated with the apparent shear rate by the following empirical power law equation: 'qa = K~ ~r
.
(13)
A non-continuum approach to describe particle-gas suspension flow was presented by Cundall and Strack [60]. The method was based on the use of a numerical scheme in which the interactions of the particles are monitored contact-by-contact and the motion of the particles simulated particle-by-particle. Aizawa, Tamura and Kihara [61] calculated the particle-to-particle interaction
1416
forces by assuming that the interaction force between two solid particles is equal to the overlapped volume force from the elastic contact between the particles. To stabilize the numerical solution, they introduced a friction-like energy-dissipation mechanism in the contact algorithm. 1.00E+07 w m
a,, E
:r w o o w
",o,,
1.00E+06
',o,,
1.00E+05
~,,,,,,,
,,,,=
O
Brookfield [58]
[]
L/D=245.33 [58]
o
L/D=122.67 [58]
[]
L/D=61.33 [58]
9
L/D=30.67 [58]
9
L/D=15.33 [58]
-'-- - - E q n . (13)[58]
r., -9t~ 1.00E+04 o,, D, J
1.00E+03 0.1
a
1
,
,
|||,=!
9
10
|
J
**1.,|
|
100
9
9 , ~ , 1 m
1000
Apparent Shear Rate, 1Is
Figure 13. Apparent viscosity of sand bed as a function of the apparent shear rate [58]. Although these rigorous many-body algorithms have been demonstrated for simple geometries, excessive computational times preclude their widespread application to complex industrial problems. Continuum algorithms must be modified to enable design calculations. Central to the successful exploitation of such an approach is the measurement of appropriate transport properties (i.e., viscosity) that represent the complex rheological behavior of these multi-phase fluids. Therefore, continuum theories have been developed to describe the dynamics of particle-gas suspension flow. These theories develop hydrodynamic models based on the Navier-Stokes equations for multiphase flow. Using fluid mechanics principles, Anderson and Jackson [62] proposed a two-phase flow model. Pritchett, Black and Grag [63] applied a two-fluid model to solve the problem numerically. Gidaspow [64] presented the hydrodynamic model describing gas and solid phases in terms of separated conservation equations. Tsuo and Gidaspow [65] generalized the Navier-Stokes equations for two fluids to predict flow regimes in circulating fluidized beds. Using a two-phase 2-D computational fluid dynamics model, Samuelsberg and
1417
Hjertager [66] computed axial solid velocity, void factor and solid shear viscosity in the riser of a circulating fluidized bed. The radial profile of solid shear viscosity computed by the turbulent kinetic energy model was lower in the core than that found experimentally, but with a linear function of solid volume fraction in the measurement, the computed profile agrees well with experiments. Delassade [67] considered the fluidized bed as a mixture of a micro-stretch fluid representing the particulate phase, and a Newtonian fluid. The effect of particle inertia and other consequences of non-negligible spatial gradients of velocities, concentration and void factor on vertically propagating concentration waves of small amplitude has been analyzed. Scht~gefl [68] carried out the flow equation of fluidized systems from the shear diagrams using the method of Pawlowski [69]. According to his results, the apparent viscosity of the particle-gas suspension is independent of the flow rate at all particle diameters. However, although the viscosity is independent of the particle diameter at particle diameters dp > 0.100 mm, the viscosity decreases with decreasing particle diameter at particle diameters dp < 0.100 mm. Reviewing a number of expressions proposed for correlating fluidized bed viscosity with overall bed void factor, Johnson [70] proposed that the fluidized bed viscosity is related to bed voidage by Tla "-'Fig [1 + 0.5 (1 - e)][(1 - ~) / ~]9 / ~4.
(14)
Simulations made by equation (14) show a dramatic decrease in bed viscosity as the void factor increases. A kinetic theory based on a moment expansion was developed by Tsao and Koch [71], who numerically predicted the behavior of a dilute fluidized beds subjected to a simple shear flow. The results of simulations show that as the Stokes number is increased, the velocity distribution approaches the Maxwellian and the rheology becomes more Newtonian. At small Reynolds numbers the non-linearity of the drag increases the non-Newtonian behavior of the bed. Hydrodynamics and erosion modeling of fluidized beds have been introduced by Li and Zakkay [72]. Particle velocity fields and fluidized bed dynamics (bubble formation and motion, bed expansion and collapse) were analyzed. Utilizing the Finnie erosion model, these authors show that the distance from the air distributor to the tubes, tube size, tube orientation and operating pressure have significant effect on bed dynamics and tube erosion. However, in order to accurately analyze the kinetic and transport processes in these complex gas-particle systems, it is critical to determine a rheological
1418
model to describe their flow behavior. Since the dynamics of gas-solid suspension flow is a multiparametric process as noted above, it is convenient to develop the theological model on the basis of a semi-empirical correlation for fluidized beds. Bakhtiyarov and Overfelt [73] developed a rheological model of gas-solid suspension flow based on Ergun's semi-empirical correlation [74]. Introducing two components of the apparent viscosity, the model predicts that the flow curves of gas-solid suspensions depend on the gas properties, particle diameter and sphericity, void factor and tube diameter. The following form of the theological model has been developed (15)
q;a = [Tla.sh. + Tla.k.(~a) ] ~ a ,
where two components of the apparent viscosity as apparent shear viscosity ~a.sh. and apparent kinetic viscosity ~a.k.(~a), defined as 1]a.sh. = 75 (1
- I~) 2
~l.gd 2 / 16 e 3 (0 dp)2 ,
Tla.k"---- 7 (1 - e) Og d 3 "Ya[ 1024
1~3 ~ dp,
(16) (17)
where a sphericity of particles r is defined as a ratio of the surface areas of a sphere and the particle having identical volumes: t~ = S~ / Sp. According to formula (15), the net apparent viscosity is the sum of the two components: the first is due to shear viscous pressure drop and the second due to the kinetic pressure losses. Therefore, it is of interest to establish the contribution of each viscosity component to the total apparent viscosity. The viscosity ratio ~ = 1"la.k./Tla.sh.as a function of the apparent shear rate and particle diameter is shown in Figure 14. As seen from this figure, the viscosity ratio increases as both the shear rate and the particle diameter increase. Figures 15 and 16 show the variation of the apparent shear stress and the apparent viscosity, respectively, with the apparent shear rate obtained experimentally for silica sand of different particle diameters and over-all voidage fraction measured by procedures described in [73]. The predictions of the rheological model using equation (15) [73] are also shown in Figures 15 and 16. As seen from these figures, there is a good agreement between the model and the experimental data. Figures 15 and 16 show the variation of the apparent shear stress and the apparent viscosity, respectively, with the apparent shear rate obtained experimentally for silica sand of different particle diameters and over-all
1419
voidage fraction measured by procedures described in [73]. The predictions of the rheological model using equation (15) [73] are also shown in Figures 15 and 16. As seen from these figures, there is a good agreement between the model and the experimental data.
Figure 14. Viscosity ratio ~ = Tla.k.]Tla.sh.as a function of the apparent shear rate and particle diameter [73]. An apparent viscosity model based on the integral characteristics of the radial segregated fluidized beds has been developed by Bakhtiyarov and Overfelt [75]. Assuming that (i) the flow is laminar and steady on either side of the interface between core and annular space, (ii) the interface is stable and smooth, (iii) the pressure gradient is constant, (iv) end effects and a mass transfer between core and annular flows are negligible, it has been determined that the apparent viscosity of the segregated fluidized bed is a function of the viscosity and density ratios of the core and annular flows. This embodies the effects of the grain size and shape factor of particles, rheological characteristics and relative velocity of the fluidizing fluid giving l q s e g ~-
[Xo3 q- p / 11 -
Xo 3
(3 - p) / 211 - 3 Xo (p - 1) / 2111 / 11,
(18)
1420
where 1"1 and p are viscosity and density ratios of the core and annular flows, respectively; and Xo is a dimensionless radius of the core. 50
m
Ix. 40 W G) L_
30 L_ m 0
I
... 20
I
I
G} t_ m Q.
a. lO
0
500
1000
1500
2000
2500
3000
Apparent Shear Rate, 1/s
Figure 15. Apparent shear stress of air-sand suspension as a function of the apparent shear rate [73]" r-! _ dp = 0.653 mm, ~ = 0.55; + - dp = 0.505 mm, ~ = 0.64; O - dp = 0.357 mm, ~ = 0.74; A - dp = 0.252 mm, e = 0.80; 9 - dp = 0.212 mm, e = 0.86. Figure 17 shows the variation of the dimensionless apparent viscosity (1/Tlseg) with the parameter rl, a ratio of the viscosities in core and annulus regions, and with the dimensionless radius of the core (Xo). The radius of the core plays a prevailing part in determining the apparent viscosity of the system. That is especially conspicuous at high values of the parameter 1"1. The apparent viscosity of segregated bed (1/rl~g) also shows strong subordination on the parameter 1"1 which has an increasingly larger effect on 1/l"lseg with increasing values of Xo. Theoretical predictions for the dimensionless apparent viscosity (1/Tlseg) are shown in Figure 18 together with experimental results obtained in [24] for silica sand with grain size 0.212 mm to 0.425 mm and the average particle density 2.593 g/cm 3 fluidized by air. The apparent viscosities and average densities of the segregated fluidized beds in the core and annulus regions have been determined by using the procedure described in [24]. As seen from the figure,
1421
the apparent viscosity of the segregated fluidized bed is reasonably well predicted at all values of the viscosity ratio rl. 16
w == 12 o. E
u
8
,,~ -_
13... " ~
F.
<
.....
I:_,. .- - , ' - " " "
0
ii
ii
*
500
~ l
'
"
*
II
I
1000
-'~
*
'
*
I
.6 ~'' .'~;~
9
"
/
=_.,. . . . . . . . .
I
1500
I
I
I
I
I
/
I
I
2000
I
'
i
2500
.
,
,
,
3000
Apparent Shear Rate, 1Is
Figure 16. Apparent viscosity of air-sand suspension as a function of the apparent shear rate [73]: I'1 - d p = 0.653 mm, e = 0.55; + - d p - 0 . 5 0 5 m m , E = 0.64; O - d p = 0.357 mm, e = 0.74; A - d p = 0.252 mm, e = 0.80; @ - d p - 0.212 mm, ~ = 0.86. The hydrodynamic behavior of fluidized beds in the presence of an external physical fields such as magnetic, electric, acoustic or gravity are of both methodological and practical interests. Sergeev and Dobritsyn [76] considered the propagation of solid concentration disturbances in fluidized beds in an external magnetic field. Using the Korteweg-de Vries-Burgers equation for the departure of solid concentration from the uniform state, authors predicted the structure of the wavefront and the thickness of the front as a function of magnetic and other physical parameters of the bed. A general approach for evaluating the sedimentation velocity and drag force of charged particles in bounded system for small particle Peclet numbers and small particle-surface potentials has been described by Pujar and Zydney [77]. The viscous force that arises from the alteration in the velocity profiles associated with the interaction between the electric field and the fluid was evaluated using a generalized form of the Lorenz reciprocal theorem.
1422
Figure 17. Variation of the dimensionless apparent viscosity (1/~t) with the parameter 1"!, a ratio of the viscosities in core and annulus regions, and with the dimensionless radius of the core (x0) [75]. The effect of the microgravity on the "apparent viscosity" and onset of fluidization of the fluidized solid particles, which is a two-phase flow system free from surface tension, has not received much attention in the past mainly due to the complex rheological behavior of this system. Bakhfiyarov and Overfelt [78] first reported the results of the low gravity experiments developed and conducted on board the NASA/KC-135 aircraft which were specifically aimed to understand the effect of the gravitational forces on the "apparent viscosity" of the fluidized materials. Short (-- 20 s) duration of microgravity (-- 0.1 m/s 2) are created aboard NASA's Zero-G KC-135 aircraft by a series of parabolic trajectories. Each microgravity period followed a-- 50 s period of 1.8g gravity during the flight maneuvers. The aircraft flew -- 40 parabolas on each flight which allowed for repeating the tests several times and for changing the flow regimes. The KC135 aircraft climbs and descends between altitudes of 7.6 and 10.6 km. The gravitational acceleration was measured by a computer operated capacitive three axis accelerometer which enabled recording typical gravitational
1423
acceleration cycles in all three directions for each parabola during the flight maneuvers of the KC- 135. 250
200
150 =I.
100
50
0 0
20
40
60
80
100
Figure 18. Comparison of the experimental data [24] and theory [75] for the dimensionless apparent viscosity as a function of the parameter 11" x0 = 0.5, !"! u0= 1.25 m/s, A - u0= 1.70 m/s, O - u0= 2.25 m/s. The experimental set up, presented in Figure 19, was specifically designed and built with the safety and portability requirements of the flight program. A computer controlled Brookfield HADV-II+ rotational viscometer was used in both the ground based proof and the flight experimental programs. The advantage of the measurements in reduced or increased gravity conditions is the possibility to change the buoyant weight of the system at the same fixed bed height and voidage. Figures 20 and 21 show the variations of the expanded bed height to the fixed bed height ratio and the overall voidage, respectively, with the air velocity at three different values of the gravitational acceleration during the fluidization of the silica sand particles tested (dp < 0.212 mm and 0.600 mm < dp < 0.710 mm) where data pertained to the g = 9.81 m/s 2 were obtained in the ground based experiments [78]. As seen from these figures, the particle diameter plays a predominant role in expansion of the bed and the void factor. The bed expansion and the voidage also show strong dependence on the gravitational acceleration which has an increasingly larger effect on bed expansion and the void factor with increasing values of the air velocity. Figure 22 shows the gravity effects on the air velocity required for
1424
the incipient fluidization for two solid particle sizes [78]. The experimental data show that increasing the gravitational acceleration causes suppression of the bed and increases the amount of gas necessary to initiate fluidization.
Figure 19. Schematic diagram of experimental apparatus designed for flight experiments [78]. As we mentioned earlier, the "apparent viscosity" of fluidized systems strongly depends on the overall void factor of the bed. Since the incipient fluidization and the voidage are functions of the gravitational forces one would expect an effect of the gravitational forces on the "apparent viscosity" of fluidized beds. The results of the "apparent viscosity" measurements [78] for the fine (dp < 0.212 mm) and for the coarse (0.600 mm < dp < 0.710 mm) silica sand particles at different spindle angular velocities and gravity levels are shown in Figures 23 and 24, respectively. As seen from these figures, a predominant influence of the gravity level on the "apparent viscosity" is particularly noticeable at small shear rates (angular velocities of the spindle). One is drawn to the inescapable conclusion that at high shear rates the influence of the centrifugal forces and the related phenomena (radial segregation, turbulence, etc.) on the measured "apparent viscosity" become more significant. This conclusion can be succinctly demonstrated in Figures 25 and 26 [78], which show the variation of the "apparent viscosity" of the fluidized silica sand particles (dp < 0.212 mm and 0.600 mm < d p < 0.710 mm, respectively) with imposed gravitational
1425
acceleration at a wide range of the spindle angular velocity changes (10 to 100
rpm). Recently, experimental results obtained by Hunt, Hsiau and Hong [79] show that the expansion of the bed increases significantly beyond a critical frequency (10 Hz), and that the expansion does not depend on the amplitude of the vibration. The effect of the vibrational amplitude may be significant if the ratio of vibrational amplitude to particle diameter is significantly larger or smaller than unity. Therefore, a more detailed study of the fluidized beds in microgravity should include the effects of small oscillations in acceleration under the reduced gravity conditions, called g-jitter. 0.4 "o m ._x u.
0.3
O.o
rfl
~
"o
t
m
/,.;"
0.I
I
m
I
-031
A
I
0.1
1
10
Air Velocity, m l s
Figure 20. Variation of the expanded bed height to the fixed bed height ratio with imposed air velocity (dp< 0.212 mm: 9 - 9.81 m/s 2, I - 11.772 m/s 2, A 17.658 m/s2; 0.600 mm < dp< 0.710 ram: -~- - 9.81 m/s 2, r-1 _ 11.772 m/s 2, A 17.658 m/s 2) [781. 4.
CONCLUSIONS
Hydrodynamics of gas-solid fluidized beds cuts across many disciplines and is of significance to many industrial processes. During the last decade substantial progress has been made in this area. Appealing fundamental and complex practical problems have challenged scientists and engineers to create new theoretical models and unique experimental techniques for better understanding
1426
0.56
9 /
I,,l
0.52
I:t
0
01 0.48 o >
..;"
0.44
0.4
.
9 9 9
I'''l
0.01
I
l
9 ..,,.l
.
0.1
.
.
9 ,.,.
1 Air Velocity,
10
m/s
Figure 21 Variation of the void factor with imposed air velocity [78] (dp< 0.212 mm: 9 - 9.81 m/s 2, i - 11.772 m/s 2, A - 17.658 m/s2; 0.600 m m < dp< 0.710 mm: + - 9.81 rn/s 2, r-I _ 11.772 m/s 2, A - 17.658 m/s2).
o
0.8 o
E
0.6
0 ~
D4
O
_c
0.4
-~ m
8
!
i
12
16
Gravitational
Acceleration,
20 m/s 2
Figure 22. Variation of the incipient fluidization air velocity with imposed gravitational acceleration ( i - dp<0.212 mm; r'i _ 0.60 mm< dp< 0.71 mm) [78].
1427
250
m 200
., 0o ~
"ILZ~:"
100 I~
E
50
<
~ ~"'~o " ~ ' " -
XT
9I , . ' ' ~ - . .
I " ~-.
"
0
0
20
40
60
80
100
Spindle Angular Velocity, RPM
Figure 23. Variation of the "apparent viscosity" of fluidized silica sand particles with imposed spindle angular velocity (dp< 0.212 mm: -~ - 0.981 rn/s 2, II - 9.810 m/s 2, A - 11.772 m/s 2, O - 17.658 m/s 2) [78].
600
w te 500 Ix >; ul 400 o u .w_ 300 > r
Vi/x i
200
i
D. o. 100 <
~Q
_ 0
II
20
40
"lOm
D~
-~" :." ~.-.:.-..~ 60
80
100
Spindle Angular Velocity, RPM
Figure 24. Variation of the "apparent viscosity" of fluidized silica sand particles with imposed spindle angular velocity (0.600 mm < dp< 0.710 mm: -~ 0.981 m/s 2, II - 9.810 m/s 2, A - 11.772 m/s 2, O - 17.658 m/s 2) [78].
1428
250
ca 200 a.
100
L ~ ~
,~ so
~
li.~-
- "~" " "
""
"'''2"'-'''-
0 0
5
10
15
20
G r a v i t a t i o n a l A c c e l e r a t i o n , nl/S 2
Figure
25. V a r i a t i o n o f the " a p p a r e n t v i s c o s i t y " o f f l u i d i z e d
silica sand
p a r t i c l e s w i t h i m p o s e d g r a v i t a t i o n a l a c c e l e r a t i o n (dp< 0 . 2 1 2 m m : +
- co = 10
r p m , r-I _ ~ = 30 r p m , A - o~ = 6 0 r p m , O - co = 100 r p m ) [78]. 600 ca 500 (g a. _~ 400 (n
o o
300
r~
9 2oo D,, ,< 100 ---I 0
5
10
.....
x ..... 15
20
Gravitational Acceleration, m/s 2
Figure
26. V a r i a t i o n o f the " a p p a r e n t v i s c o s i t y " o f f l u i d i z e d
particles with imposed gravitational acceleration (0.600 mm
silica sand
< dp< 0 . 7 1 0 m m : +
- CO= 10 r p m , r-I _ co = 12 r p m , A - o~ = 20 r p m , I - co = 30 r p m , 9 - o~ = 5 0 r p m , O - co = 6 0 r p m , & - t~ = 100 r p m ) [78].
1429
of fluidization phenomena. The heterogeneity of the fluidization still remains a daunting feature of gas-solid systems. The present challenge is to understand the physical mechanisms of heterogeneity and to find effective ways to prevent this undesirable phenomenon. It is hoped that in the next years efforts will be made to advance the computer-based hydrodynamic and rheological models of fluidization. Systematic study is needed to establish the influence of external physical fields (e. g. magnetic, electrical, acoustic, gravity, centrifugal) applied individually or in combination on behavior and rheological parameters of fluidization. REFERENCES
1. J. F. Davidson, R. Clift and D. Harrison, Fluidization, 2 nd edn., Academic Press, London, 1985. 2. D. Geldart, Gas Fluidization Technology, Wiley, Chichester, UK, 1986. 3. J. R. Grace, Can. J. Chem. Eng., 64 (1986) 353. 4. H. T. Bi and J. R. Grace, Int. J. Multiphase Flow, 21 (1995) 1229. 5. S. I. Bakhtiyarov and R. A. Overfelt, 31 st ASME National Heat Transfer Conference Proceedings, 1 (1996) 239. 6. M. Leva, Fluidization, Mc. Graw-Hill Book Co., New York, 1959. 7. K. S. Lim, J. X. Zhu and J. R. Grace, Int. J. Multiphase Flow, 21 (1995) 141. 8. R. Jackson, A. I. Ch.E. Symposium Series, 90 (1994) 301. 9. M. C~rsky, M. Hartman, B. Ilyenko and K. E. Makhorin, Powder Technology, 61 (1990) 251. 10. H. W. Piepers and K. Rietema, in Fluidization VI, Edited by Grace, J. R., Shemilt, L. W. and Bergougnou, M. A., Engineering Foundation, New York, 1989, 203. 11. M. Gautam, J. T. Jurewicz and S. R. Kale, J. Fluids Eng., 116 (1994) 605. 12. D. Gera and M. Gautam, J. Fluids Eng., 117 (1995) 319. 13. J. S. Halow, G. E. Fasching, P. Nicoletti and J. L. Spenik, Chem. Eng. Sci., 48 (1993) 643. 14. J. G. Yates, D. J. Cheesman and Y. A. Sergeev, Chem. Eng. Sci., 49 (1994) 1885. 15. S. I. Bakhtiyarov and R. A. Overfelt, IEEE 31 st Intersociety Energy Conversion Engineering Conference Proceedings, 2 (1996) 793. 16. S. Hovmand and J. F. Davidson, Fluidization, Edited by Davidson J. F. and Harrison, D., Academic Press Publishing Corp., New York, 1971, 193. 17. A. W. Nienow and T. Chiba, Fluidization, Edited by J. F. Davidson, R. Clift and D. Harrison, 2nd edn., Academic Press, London, 1985, 357.
1430
18. A. W. Nienow, N. S. Naimer and T. Chiba, Chem. Eng. Commun., 62 (1987) 53. 19. A. B. Delebarre, A. Pavinato and J. C. Leroy, Powder Technology, 80 (1994) 227. 20. J. Hong and Y. Tomita, Int. J. Multiphase Flow, 21 (1995) 649. 21. J. A. Pita and S. Sundaresan, A. I. Ch. E. Joumal, 39 (1993) 541. 22. S. Dasgupta, R. Jackson and S. Sundaresan, A. I. Ch. E. Joumal, 40 (1994) 215. 23. M. Rhodes, S. Zhou and H. Benkreira, A. I. Ch. E. Journal, 38 (1992) 1913. 24. S. I. Bakhtiyarov and R. A. Overfelt, Powder Technology, submitted (1997). 25. A. Berker and T. J. Tulig, Chem. Eng. Sci., 41 (1986) 821. 26. E. U. Hartge, D. Rensner and J. Werther, Circulating Fluidized Bed Technology (II), Basu and Large, eds., Pergamon Press, 1988, 165. 27. B. Herb, K. Tuzla and J. C. Chen, Distribution of Solid Concentration in CFB, in Fluidization VI, Edited by Grace, J. R., Shemilt, L. W. and Bergougnou, M. A., Engineering Foundation, New York, 1989, 65. 28. H. Mineo, High Velocity Circulating Fluidized Beds, Ph. D. Thesis, University of Tokyo, 1989. 29. Y. Tung, Z. Zhang, Z. Wang, X. Lui and Z. Qin, Eng. Chem. Met. (China), 10 (1989) 17. 30. S. I. Bakhtiyarov and R. A. Overfelt, Study of Spouted Flow in Fluidized Bed, Journal of Powder Technology, submitted. 31. D. Gidaspow, R. Bezdaruah, A. Miller and U. Jayaswal, Dense Transport and Fluidization of Solids in Gas or Liquid Using Kinetic Theory, Joint DOE/NSF Workshop on Flow of Particulates and Fluids, Worcester, MA. 32. A. Miller and D. Gidaspow, A. I. Ch.E. J., 38 (1992) 1801. 33. G. Ahmadi and D. Ma, Int. J. Multiphase Flow, 16 (1990) 323. 34. J. Cao and G. Ahmadi, Int. J. Multiphase Flow, 21 (1995) 1203. 35. Y. Tsuji, Y. Morikawa and H. Shiomi, J. Fluid Mech., 139 (1984) 417. 36. P. Singh and D. D. Joseph, Int. J. Multiphase Flow, 21 (1995) 1. 37. J. L. Sinclair and R. Jackson, A. I. Ch. E. Journal, 35 (1989) 1473. 38. J. A. Pita and S. Sundaresan, A. I. Ch.E. Joumal, 37 (1991) 1009. 39. R. Ocone, S. Sundaresan and R. Jackson, A. I. Ch. E. Joumal, 39 (1993) 1261. 40. J. Y. Tu and C. A. J. Fletcher, A. I. Ch.E. Joumal, 41 (1995) 2187. 41. G. L. Matheson, W. A. Herbst and P. H. Holt, Ind. and Eng. Chem., 41 (1949) 1099. 42. H. Trawinski, Chemie. Ing. Techn., 25 (1953) 201. 43. H. Kramers, Chem. Eng. Sci., 1 (1951) 35
1431
44. 45. 46. 47. 48. 49. 50.
I. Furukawa and T. Ohmae, Ind. and Eng. Chem., 50 (1958) 821. W. W. Shuster and F. C. Haas, J. Chem. Eng. Data, 5 (1960) 525. J. R. Grace, Can. Joum. Chem. Eng., 48 (1970) 30. K. Peters and A. Schmidt, (3st. Chem. Ztg., 54 (1953) 253. W. Siemes, Chem. Ing. Techn., 24 (1959) 82. W. Siemes and L. Hellmer, Chem. Eng. Sci., 17 (1962) 555. X. L. Wu, K. J. Maloy, A. Hansen, M. Ammi and D. B ideau, Phys. Rev. Lett., 71 (1993) 1363. 51. E. Manger, T. Solberg, B. H. Hjertager and D. Vareide, Int. J. Multiphase Flow, 21 (1995) 561. 52. R. M. Davies and Sir G. I. Taylor, Proc. Roy. Soc., A 200 (1950) 375. 53. H. D. Mendelson, A. I. Ch. E. J., 13 (1967) 250. 54. K. Scht~gefl, M. Merz and F. Fetting, Chem. Eng. Sci., 15 (1961) 1. 55. J. D. Murray, Rheologica Acta, 6 (1967) 27. 56. P. S. B. Stewart, Trans. Inst. Chem. Eng., London, 46 (1968) 80. 57. P. N. Rowe and B. A. Partridge, J. Fluid Mech., 23 (1965) 583. 58. S. I. Bakhtiyarov, R. A. Overfelt and S. Reddy, ASME International Mechanical Engineering Congress, Proceedings, AMD-217 (1996) 243. 59. S. A. Gregory, J. Applied Chemistry (London), 2 (1952) 1. 60. P. A. Cundall and O. D. L. Strack, G6otechnique, 29 (1979) 47. 61. T. Aizawa, S. Tamura and J. Kihara, ASME Proceedings, PED-61 (1992) 31. 62. T. B. Anderson and R. Jackson, Ind. Eng. Chem. Fund., 6 (1967) 527. 63. J. W. Pritchett, T. P. Black and S. K. Grag, A. I. Ch.E. Joumal, 74 (1978) 134. 64. D. Gidaspow, App. Mech. Rev., 39 (1986) 1. 65. Y. P. Tsuo and D. Gidaspow, A. I. Ch. E. Joumal, 36 (1990) 885. 66. A. Samuelsberg and B. H. Hjertager, A. I. Ch. E. Joumal, 42 (1996) 1536. 67. X. A. Delassade, ASME Proceedings, AMD, 217 (1996) 251. 68. K. Sch~gefl, in "Fluidization", J. F. Davidson and D. Harrison, ed., Academic Press Publishing Co., London-New York (1971) 261. 69. J. Pawlowski, Kolloidzschr, 30 (1953) 129. 70. E. Johnson, Institute of Gas Engineers (London), 378 (1950) 179. 71. H.-K. Tsao and D. L. Koch, J. Fluid Mech., 296 (1995) 211. 72. C. Li and V. Zakkay, Trans. of ASME, 116 (1884) 746. 73. S. I. Bakhtiyarov and R. A. Overfelt, ASME FED SM, Proceedings, Vancouver, Canada (1997) $245. 74. S. Ergun, Chem. Eng. Progress, 48 (1952) 89.
1432
75. S. I. Bakhtiyarov and R. A. Overfelt, Proceedings of Third International Conference on Multiphase Flow, Lyon, France (submitted). 76. Yu. A. Sergeev and D. A. Dobritsyn, Int. J. Multiphase Flow, 21 (1995) 75. 77. N. S. Pujar and A. L. Zydney, A. I. Ch.E. Journal, 42 (1996) 2101. 78. S. I. Bakhtiyarov and R. A. Overfelt, IEEE 32 ~d Intersociety Energy Conversion Engineering Conference Proceedings, 2 (1997) 1439. 79. M. L. Hunt, S. S. Hsiau and K. T. Hong, J. Fluids Engineering, 116 (1994) 785. NOMENCLATURE Ar d g K L P Q R Re S U 13 e t~ ), ~, 1"1 0 p x
Archimedes number diameter gravity expansion coefficient length pressure (AP is a pressure drop) mass flow rate radius of fluidization chamber or tube Reynolds number specific surface area (surface area per unit volume of medium) superficial velocity angle of repose porosity sphericity of solid particles shear strain rate consistency friction factor dynamic viscosity structural viscosity included angle density shear stress velocity viscosity ratio
Subscripts a
apparent
1433
av
b B
g k mf P S
seg sh W
average value bulk bubble gas kinetic minimum fluidization particle sphere segregated fluidized bed shear wall
1435
H E A T AND M A S S T R A N S F E R IN R H E O L O G I C A L L Y COMPLEX SYSTEMS R.P. Chhabra
Department of Chemical Engineering, hTdian hTstJtute q/Technology Kanpur, h~dia 208016
1. INTRODUCTION During the last four to five decades, considerable attention has been accorded to the fluid mechanics of theologically complex materials and therefore significant advances have been made in developing better insights into the underlying physical processes. In contrast to this, the transport of heat and mass in nonNewtonian materials have received much less attention. It is readily agreed that most unit and processing operations encountered in the handling and processing of non-Newtonian materials entail temperature and/or concentration gradients within the fluid medium, thereby resulting in the net transport of heat (or mass) from one region to another. For instance, in many industrial applications, process streams need to be heated or cooled and a wide range of equipment may be utilized for this purpose ,e.g., double pipe or tubular heat excl~angers or stirred tanks fitted with cooling coils or steam jackets. Similarly, inter-phase mass transfer between a non-Newtonian medium and a particulate please is frequently encountered in fixed, fluidized bed and three phase reactors used to carry out a range of polymerisation and biochemical reactions. Further examples are found in devolatilization of thin films, de-gassing of molten polymers, drying and aseptic processing of liquid food stuffs, concentration of fruit juices, oxygenation of blood, etc. Solnetimes, heat is generated in the process itself, sucl~ as in extrusion of polymers and foodstuffs. It may too be necessary to reduce the rate of heat loss from a vessel or ensure that heat is removed at a sufficient rate in equipment such as screw conveyors. In most applications, it is the rate of heat transfer within the process equiplnent which is of plincipal interest, though with thennally sensitive materials (such as food stuffs, fermentation broths, etc.), the telnperature profiles must be known and maximum permissible temperatures must
1436
not be exceeded. Obviously, the rate of heat (or mass) transfer and the temperature distribution in a given application are strongly dependent upon the geometry of flow, kinematic conditions and the physical properties of the fluid. Indeed this interplay between these factors is fi~rther accentuated in the case of non-Newtonian fluids due to their non-linear flow behaviour. This chapter aims to provide a state of the art review of the currently available body of infonnation in this field. 2. SCOPE
In this chapler, consideration is given to lhe phenomena of heat and mass transport in processes involving rheologically complex materials, especially with regard to the implications of shear dependent viscosity and viscoelasticity of materials. As suggested earlier, the highly non-linear and strongly coupled nature of the field equations preclude the possibility of analytical results and therefore usually numerical solutions are sought, even for as simple a situation as that of laminar flow of a purely viscous fluid in a circular tube. In this chapter, the main emphasis is on the presentation of the results and the underlying physical processes rather lhan on the numerical techniques p e r se. Clearly, it is also not feasible to include here all possible geometries encountered in engineering applications. Instead this review mainly addresses the phenomena of heat and mass transfer in internal (conduit) flows and in boundary layer flows, followed by a short section on heat transfer in geometries of practical interest. First of all, however, the thenno-physical properties of the commonly used non-Newtonian materials will be described. 3. THERMO-PItYSICAL PROPERTIES In addition to tile flow characteristics (viscous and viscoelastic), the other important physical properties of non-Newtonian substances are thermal conductivity, density, specific heat, surface tension, coefficient of thermal expansion, solubility and molecular diffusion coefficients. While the first three enter into virtually all heat transfer calculalions, surface tension exe~ls a strong influence on boiling heat transfer and bubble dynamics in non-Newtonian fluids. Likewise, the coefficient of thennal expansion is impo~lant in heat transfer by free convection. Finally, the last two properties, namely, solubility and molecular diffusivity play central roles in mass transfer processes. Admittedly, only very limited measurements of physical properties have been made, but for dilute and moderately concentrated aqueous solutions of commonly used polymers including carboxymethyl cellulose (Hercules), polyethylene oxide
1437
(Dow), carbopol (Hercules), polyacrylamide (Dow), etc., density, specific Ileal, thermal conductivity, coefficient of thermal expansion and surface tension differ from tile values for water by no more than 5-10% [1-8]. The thennal conductivily and molecular diffusivity may be expected to be shear rate dependen! for both of these and the viscosity are dependent on structure. Although recent measurements [9] on aqueous carbopol solutions confinn this expectatir :l for thennal conductivity, the effect, however, is small [10]. For engineering design calculations, there will be little enor in assuming that all the above physical properties, except molecular diffusivity, of aqueous polymer solutions are equal to the values for water at the same temperature. The available literature on the values of the molecular diffusivity into non-Newtonian polymer solutions is much more controversial and inconclusive, as revealed by the available reviews [11,12]. For instance, from the data reported for gases and neutral solutes diffusing in non-Newtonian polymer solutions, all one can conclude is that there is some effect of polymer concentration on diffusivity. More authors report that diffusivity increases with increasing polymer concentration (increasing consistency), eg., diffusivity values as large as 2.5 times the value in the virgin solvent (zero polymer concentration) have been reported for CO= diffusing in solutions of cellulose polymers. The effect is not as great for solutions of polymers having simpler structure with no branching, etc. All in all, the values ranging from --- 10 -~= to ~ 10~ m=/s have been reported for CO2, SO=, C2H2 and O= diffusing in aqueous solutions of poly ethlyene glycol (PEG), carbopol, polyethylene oxide (PEO), hydroxyethyl cellulose (HEC), etc. and of polystyrene in toluene. Similarly, the corresponding values for solid solutes inferred from the rates of dissolution from inclined surfaces and rotating disks range fiom --- 3.4 x 10-12 to --- 2.5 x 10-11m2/s for 13-naphthol, benzoic acid and oxalic acid in aqueous solutions of PEO and CMC. Little is known about the role of viscoelasticity on diffusion coefficient [13]. Much confusion, however, exists regardi~.~ the influence of shear rate on molecular diffusion. For instance, Wasan et al. [14] used a wetted wall column for measuring the diffusivity of oxygen in a series of polymer solutions and reported a rather steep rise in the value of diffusivity with increasing shear rate. On the other hand, many other workers [15,16] reported very weak dependence on shear rate. Likewise, most of the cunently available data pertain to room or near room temperatures and hence little is known about the temperature dependence of the diffusivity in polymer solutions. Some data of these properties for polymer melts are also available [17-19]. Admittedly, some attempts have been made at developing semi-theoretical expressions for the prediction of thennal conductivity [20], diffusion [21,22], solubility [23], etc., these are not yet refined to the extent of being completely predictive. Besides, the values of thenno-physical properties in these systems
1438
seem to be strongly influenced by the detailed molecular structure (molecular weight distribution, method of preparation, etc.) and therefore extrapolation from one system to another, even under nominally identical conditions, can lead to siD~ificant errors. For industrially important particulate slurries and pastes exhibiting strong nonNewtonian behaviour, the thenno-physical properties (density, specific heat and thennal conductivity) can deviate significantly from those of its constituents. Early measurements [24] on the aqueous suspensions of powdered copper, graphite, aluminium and glass beads suggest that both the density, p, and the heat capacity, Cp, can be approximated by the weighted average of tl~e individual constituents, i.e., Psus
=
~Ps +
(l
--
-
~)PL
+
(]
-
(1) (2)
)CpL
where ~ is the volume fraction of the solids, and the subscripts L, s and sus refer to the values for the liquid, the solid and the suspension respectively. The thennal conductivity, k, of these systems, on the other hand, seems generally to be well correlated by the following expression [24-26]: 1 + 0.5(ks/kL)-~b(l-(ks/kL) ) k sus = k L
1
+
0.5
(k S / k L ) ~+b ( 1
-
(k S /kL) )
(3)
Thennal conductivities of suspensions up to 60% (by weight) in water and other suspending media are well approximated by equation (3). It can readily be seen that even for a suspension of highly conducting particles (ks/kL ~ ~), the thermal conductivity of a suspension can be increased only by a fewfolds. Furtherlnore, the corresponding increase in viscosity from such addition would more than offset the effects of increase in thermal conductivity on the rate of heat transfer. For suspensions of mixed size particles, the following expression due to Bmggemann [27] is found to be satisfactory:
(ksus/ks)- 1 = (k~usl"~ (i- r
(k /ks)-1
Lk J
(4)
1439
The scant experimental data [28] for suspensions (~ _<0.3) of alumina (0.5 - 0.8 l.tm) particles in a paraffin hydrocarbon are in line with the predictions of equation (4). An exhaustive review on the thennal conductivity of structured media including polymer solutions, filled and unfilled polymer melts, suspensions and food stuffs has been published by Durra and Mashelkar [29]. Of all the physico-chemical properties, it is the rheology which shows the strongest temperature dependence. For instance, the decrease is apparent viscosity at a constant shear rate is well represented by the usual Arrhel~ius type exponential expression; both the pre-exponential factor and the activation energy are generally shear rate dependent. It is thus customary to denote the temperature dependence using rheological constants such as the power law consistency index and the flow behaviour index. It is now reasonably established that the power law flow behaviour index, n, of suspensions, polymer melts and solutions is nearly independent of temperature, at least over a 40-50~ temperature interval whereas the consistency index follows the exponential dependence on temperature, i.e., m = mo exp (E/RT)
(5)
where mo and E are evaluated using experimental results in the temperature range of interest. Similarly, for Bingham plastic fluids, both the plastic viscosity and the yield stress decrease with temperature in a similar fashion but with different values of the pre-exponential factors and the activation energies. Temperature dependencies of the other rheological characteristics such as the first and second nonnal stress differences, extensional viscosity, storage modulus, compliance, etc., though studied less extensively, have been discussed by Ferry [30]. 4. HEAT TRANSFER IN DUCT FLOWS The study of flow and heat transfer to non-Newtonian fluids in ducts is of both theoretical and practical importance, owing to its wide ranging applications in a spectrum of industries including food, chemical and polymer processing. Though process streams are heated or cooled in a wide variety of geometries, most studies to date have employed circular tubes and occasionally triangular, square, and elliptic tubes have also been used. Furthennore, owing to generally high consistencies of polymeric and other non-Newtonian systems, laminar flow is encountered much more frequently than in the case of Newtonian fluids. Hence, the rates of heat transfer are inherently lower under these conditions than those achievable under turbulent conditions. The rate of heat transfer is further influenced by the type of boundary conditions imposed, e.g., constant
1440 temperature or constant heal flux. Additional complications arise from the viscous dissipation effects which intensified the coupling between the momentum and energy balance equations.Thus, many possibilities exist in studying heat transfer with non-Newtonian fluids in duct flows. This section is primarily concerned with the current state of the art on l~eat transfer to non-Newtonian (purely viscous) fluids flowing in circular tubes and ducts of regular crosssections, though frequently reference is made to other important studies and reviews available in the literature on this subject [1,2,31-34]. In addition to hydrodynamic boundary layers, lhermal boundary layers are also present when a fluid entering a duct is at a temperature different fiom that of the duct walls. Thus, a thennal entrance region exists, and the duct can be divided into two regions: the thennal entrance region and the fully developed thermal region. While the notion of fully developed flow is easy to visualise at points remote from the entrance of tube, the fi~lly developed temperature profile needs fi~rther elaboration. Following Kays [35], the tenn implies that there exists, under certain conditions, a generalised temperature profile which is independent of the axial coordinate, i.e.
0z
T,,, -
T b
where T = T(r, z) is the local fluid temperature, T,,. is tile duct wall temperature and Tb is the bulk fluid or mixing cup temperature. Needless to say that the fully developed temperature distribution implies fidly developed flow. From engineering applications standpoint, the central problem in duct heat transfer is to predict either the rate of heat transfer, q, fiom the duct walls to the fluid for a known temperature difference, or the fluid-to-wall temperature difference for a given rate of heat flow. In either case, it is convenient to introduce a heat transfer coefficient, h: q = h(Tw- Tb)
(7)
From the scaling of the field equations and the relevant boundary conditions, it can readily be shown that for flow in a tube, the dilnensionless heat transfer coefficient is a fimction of the Reynolds and Prandtl numbers, i.e., Nu =
hD = f(Re, Pr) k
(8)
1441
where Nu is the Nusselt number, Re, the Reynolds number, and Pr is the Prandtl number. Thus, the central objective of all theoretical and experimental investigations is to establish this fimctional relationship.lt is convenienl to begin with the discussion of heat transfer in circular tubes, followed by a similar treatment for non-circular ducts. 5. H E A T T R A N S F E R IN C I R C U L A R T U B E S 5.1 L a m i n a r Flow
5. l. l Fully Developed and Constant Physical Properties Case The first generation of analyses dealt with the thennally fiflly developed conditions and for constant physical properties, with no viscous dissipation. For the constant heat flux conditions at the tube wall, the asymptotic Nusselt number for power law fluids is simply given by [36]: Nu =
8(5n + l)(3n + 1)
(9)
(31n 2 + 12n + 1)
It is not difficult to show that another form of tllis expression is possible as [37]" 3n + 1~ ~/3
Nu = 4.36 ~ , ~ j
(1o)
In another analytical study, Skelland [38] extended the Newtonian solution developed by Rohsenow and Choi [39] to power law liquids as Nu =
n+l
/
-2
1/8)+
3n + 1
(11) 15n: T 8n + 1
Yet another version was presented by hunan [40]. All these expressions reduce to file generally accepted value of Nu = 48/11 for n = 1 and for the plug flow condition at n = 0, Nu = 8. Analogous results for the Bingham plastic fluids have been presented by Grigull [37] and by Matsuhisa and Bird [41]. For tile case of constant wall temperature, the Nusselt number in the therlnally fiflly developed region is given by: 2
Nu = /31
(12)
1442
where 13~ is the lowest eigen value for the boundary value problem governed by the equation: 1 d dZ, l 3n+ 1 ~- d---~ F,-~j+[3~ n + 1
1
-~,"JZ,
: 0
with Zi (1)= 0 and Zi (0) = 0 where ~, = r/R. The resulting values for the Nusselt number for n = 1, 0.5, 0.333 as calculated by Lyche and Bird [42] are 3.657, 3.95 and 4.175 respectively. The inclusion of viscous dissipation term in the energy equation produces a dramatic effect on the heat transfer coefficient. Using an elegant analytical scheme, Sestak and Charles [43] obtained a closed fonn expression for the asymptotic Nusselt number for the constant wall flux condition: Nu = [ 3 InS + 43n2 + 13n + 1
L8(5n + l)(3n + l)(n+ 1)
-1
n(6n + 2)" Br 8n,,I
(14)
When the Brinkman number, Br = 0, equation (14) almost coincides with equation (11). An inspection of equation (14) suggests that the negative values of Brinkman number (i.e. heating) drive the value of Nu towards zero. Under these conditions, the temperature difference, AT, between the fluid in the wall region and the wall is reduced. For a specified value of n, there will be a value of the Brinkman number for which AT = 0.
5.1.2 Effect of Temperature Dependent Viscosity In contrast to these constant properties solutions, Joshi and Bergles [44] employed a numerical scheme to account for the temperature dependence of the power law viscosity (m = mo exp(-bY)) on the value of Nusselt number under constant wall heat flux condition. Their results indicate that the Nusselt number is nearly independent of the ratio (lUb/la,,.) but strongly dependent on the parameter (Rb q,,./k) = ~. Their numerical results are well represented by the expression: NU~p _ 1 + (0.124 - 0.0542n) W - (0.01013 - 0.0068n) ~2 Nu
(15)
where the subscript "vp" refers to the temperature dependent viscosity value. Equation (15) is applicable for tit < 6. In qualitative tenns, the overall effect of temperature-dependent power law consistency index is to enhance the rate of heat transfer for heating and to decrease it in the case of cooling of a fluid.
1443
5.1.3 Thermally Developing Heat Tran,~fer The next generation of solutions dealt with the analysis of heat transfer in the so called thermal entrance region where the velocity profile is fiflly developed. The physical properties are assumed to be temperature independent. For the constant wall heat flux condition, the Nusselt number is given by: Nuz = 1.41 5]/3Gz 1/3
Gz > 25 :t
(16)
The conesponding expression for the case of constant wall temperature is given by: Nuz = 1.16 61/3 Gzl/3
G,. > 33 rt
(17)
where 8 = ((3n + 1) / 4n). It is interesting to note that the shear thinning effects are completely accounted for via the factor 8. All the aforementioned treatments also neglect the axial conduction term in the energy equation. A recent study [45] suggests this to be a reasonable approximation provided Pe > 1000 for the constant wall temperature condition and Pe > 100 for the constant heat flux condition. The role of the non-unifonn inlet temperature profile for power law fluids has been investigated by Tonini and Lemcoff [46]. They found that at high values of the Graetz number, Gz, the lowest temperature gradient occurred at the wall and hence the lowest Nusselt number corresponds to the unifonn initial temperature profile while at small values of Gz, the reverse is true. Likewise, Faghri and Welty [47] investigated the influence of circumferentially varying heat flux. The thennally developing flow and heat transfer with the Bingham plastic model has received much less attention. The earliest analysis for a constant wall temperature and constant physical properties is due to Pigford [48]. Under these conditions, the mean Nusselt number is given by: Nu =
1.75 51/3 Gz 1/3
(18)
1-1"1
where 8
111z 3 rl+-g.~
4
and
q
=
"toB / z,,,
These results have been substantiated subsequently by numerical solutions [49,50] whereas the effect of axial conduction has been shown to be negligible
1444
for Pe > 1000 [51]. Moudachirou et al. [52] have studied the flow of HerschelBulkley fluids in tubes with constant heat flux at the tube wall and presented correlations for pressure drop and Nusselt number. As mentioned previously, for most non-Newtonian fluids, the consistency index is strongly temperature dependent. Therefore, any serious analysis must take into account at least the temperature dependence of consistency index. A common practice in Newtonian flow studies is to account for the temperature dependent viscosity via the Sieder-Tate type empirical correlation, namely, (~tb/~t,,,)~ In the initial studies, this approach was assumed to be applicable for non-Newtonian fluids also [53]. Subsequent extensive numerical computations, however, clearly bring out the deficiencies of this empirical approach [54]. Many forms of telnperature dependence of the non-Newtonian characteristics have been used in the literature. For instance, in the early numerical studies [54-56], the shear stress itself was written as a fimction of temperature as:
Zrz=ln
-
exp
(19)
where E is the activation energy of viscous flow. With this fonn of temperature dependence, Christiansen and Craig [54] solved the coupled field equations numerically for different values of n and tlle parameter ~(E)= (E/R) (1/To -I/T,,.) where To is the unifonn temperature of the fluid at the inlet. These calculations show that for n = I and ~(E) = 2, the Sieder-Tate equation overpredicts the Nusselt number by 10% at Gz = 50 while it underpredicts by 11% at Gz = 5000. This divergence rises with increasing values of ~I~(E). Positive values of ~I~(E) implies that the fluid is being heated, i.e., T,,, > To. Subsequently, these authors have obtained similar results for the cooling of fluids, i.e., ~II(E) < 0 [57]. Figure 1 shows representative results illustrating the interplay between various factors. On the other hand, Forrest and Wilkinson [58,59] denoted the telnperature dependence of the apparent viscosity as:
la= [1 + flw (T - To)]"
(20)
where jao is the apparent viscosity at the fluid inlet temperature To and [3w is the temperature-viscosity coefficient to be evaluated using experimental data. Figures 2a and 2b show their representative results for tile constant wall temperature and the constant heat flux boundary conditions respectively when the viscous dissipation effects are negligible. Broadly, the telnperature dependent
1445
consistency index facilitates the heating of fluids whereas the cooling is impeded. The effect is, however, very weak for the constant wall flux conditions. 10 3
10 v(E)= L.
-3 x - 2 . -1
0
1 2
10 2
F 101
10 ~
...... 10 ~
I
,,
lO l
I
,
~
10 2
Graetz
Number,
10 3
I 10 4
,
lO s
Gz
Figure 1 .Nusselt number as a fimction of Graetz number and ~I~(E)for n = 0.3 and (E/RTw) = 10. By far the most popular form of the viscosity-temperature relationship is the exponential form, i.e., g =mo exp (- bT). This form has been found convenient from a numerical standpoint as well as to approximate the experimental rheological data. Consequently, it has been used widely to mimic the role of temperature dependent viscosity on heat transfer [44,60-62]. Kwant et al. [60] presented extensive numerical predictions of local and average Nusselt number distribution and of the pressure loss trader non-isothermal conditions. The ratio of the true pressure drop evaluated at the tube wall temperature to that under isothennal conditions is given by: (- Ap/L)w (- ap /
_ [ _ ~ 1 - ~'~b(T' v~ -
(21)
Equation (21) is applicable in the range 0.2 < _ n < _ 1 and 0.001 _<X * < 0.4. For heating of fluids, i.e., b(Tw- To) >_ 0, O~'k = 20tk n/(n + ]) with Otg values given in Table 1 and for cooling, Ot'k takes on constant values as:
1446
"
i
10 3
I
13w=0
'
!
13w= 1 0 , ~ =
1.2
_
n = 0.5
101 13w = 1 0 , 4 ) =
0.91
10 ~ 10 2 13w = 0
.,.
Nu
13w= 1 0 , V = ' 0 " 1 I
10 ~ 10 ~
101
....
I
I
10 2
10 3
1 10 4
Gz
Figure 2.
(/,'k
Mean and local Nusselt number for heating and cooling of power law fluids with temperature dependent consistency index (11 = 0.5).
=0.125 =0.135
(22a)
- 2.5 < b ( T , , - To) < 0
(22b)
b (T,,. - To) < 2.5
Table 1 Values of a'k for heating [60]. b(Tw- To) 1 2 3
n~ 1 0.105 0.1 0.95
0.75 0.115 0.107 0.101
0.5 0.129 0.105 0.086
0.33 0.140 0.094 0.066
0.2 0.141 0.090 0.062
1447
In addition to the extensive numerical results, Kwant et al.[60] argued that when the consistency shows strong temperature dependence, the only significant parameter governing the rate of heat transfer is the shear rate at the wall. Based on this premise, Kwant et al. [60] reconciled their results for the constant wall temperature condition as: Nu
vp
Nu
- 1 + 0.271 In ~o + 0.023 (In ~o)=
where r = [1 " or' k b ( T w - T o ) ] l/n (X*/6) - ot 'k b(Tw " T o ) / n
(23) (24)
for 0.001 < X* < 0.4 and 0.2 ___n < 1. Similar expressions, though somewhat more involved, have also been presented by Joshi and Bergles [44]. The effect of viscous dissipation on entrance heat transfer has been examined by numerous investigators [58,59,63-73]. Since detailed discussions are available elsewhere [33,36,69,74], only the salient features are recapitulated here. In practice, a fully developed entrance flow condition is accomplished by preceding the healed section with another long section. Under the conditions of significant viscous dissipation (large Brinkman numbers), Gill [64] argued that the establishment of fully developed flow is not feasible under the constant wall temperature condition and therefore a plausible boundary condition can be provided by the solution of energy equation with viscous dissipation in an infinitely long isothennal section. On the other hand, Forrest and Wilkinson [58,59] solved the fifll energy equation with llle viscous dissipation as well as internal source tenns. Figure 3 shows typical results elucidating the effect of viscous dissipation on the mean Nusselt number. Qualitatively, the study of Fo~Test and Wilkinson [58,59] shows that even though the mean temperature of the fluid may be lower than that of the wall, for certain values of Brinkman number there exists the possibility of a fluid layer in the wall region with an average temperature higher than that at the wall thereby resulting in heat transfer from the fluid to the wall. Under these conditions, the local Nusselt number would obviously be negative. Some idea about the role of viscous dissipation can also be gauged from approximate solutions [74]. For instance, for the Poiseuille flow of constant properties power law fluids, in the so called equilibrium regime, the temperature of the fluid becomes independent of the axial coordinate z and is a function of r
1448
alone. Under these conditions, the maximum temperature rise occurs at the center of the tube which is given by:
10 3
!
i
ia
!
!
i
!
Br=l,~=l.2
B r = 1 0 , ~ = 1.2 Br=0 Nu
i
~
B r = 10, ~ =0.91
h
/
101
/
I
/
B r = 1 , ~ = 0.91 I
t l
10 ~ 10 ~
101
10 3
lO s
Gz Figure 3. Mean Nusselt number as a fimction of Graetz number with si/:,mificant viscous dissipation effects (n - 0.5). nt
,ST ] ma• -- k
n 3n + i)'-"
(25)
v"+' R"-!
Figure 4 shows the value of ATmax as a function of n for a polymer (m = 104 Pa.sn; k = 0.2 W/In~ flowing at an average velocity of 0.2 m/s in a capillary of 5 ~run radius. Obviously, as n decreases, the apparent viscosity of polymer decreases, and ATm,~,,drops. The rather large values of ATma~ shown here for n = 1 are not realistic. In the so-called transition region, the fluid temperature depends on both the radial as well as the axial positions, and under these conditions, an approximate expression for the temperature rise is given by:
Tb(z)-Tlr-o
411+1 \5n +
I t 2 /Sn+,ll3n+,// l
ATm,~• 1 - exp -
Gzz
n
4,1 + 1
(26)
where ATmaxis given by equation (25). The effect of free convection deserves particular attention for two reasons first, enhancements of up to 200-300% in heat transfer have been documented in the
1449
2000
g al
l::
.<3
1000
0 0
0.5
n
1.0
u162
Figure 4. Maximum temperature rise due to viscous heating for a power law fluid in thennal equilibrium regime. literature with natural convection being present in experimental studies [75]; Secondly and more significantly, velocity profile distortions caused by free convection are known to induce flow instability even at low Reynolds numbers thereby resulting in underestimation of the rate of heat transfer. Scheele and Greene [76] suggested that flow transition and/or reversal will occur in vertical upward flow at Gr'/Re' > ~ 52 while the corresponding value is 33 for downward flow. Transition may occur even at lower values of Gr'/Re' in shearthinning fluids due to greater distortions of the velocity profiles [77]. Figure 5 shows tlle critical values of Gr'/Re' at which the maximum in velocity occurs at off-center for upflow heating and at which the wall velocity gradient becomes zero in downflow heating. Both Gori [78,79] and Mamer and McMillen [80] have carried out detailed theoretical analyses to highlight the role of fiee convection on heat transfer in power law fluids and the resulting correlations for the Nusselt number tend to be rather cumbersome. Analogous treatments for thermally developing flows for the other generalised Newtonian fluid models are also available. For instance, heat transfer to the Ellis model fluids has been investigated by Matsuhisa and Bird [41] and by Gee and Lyon [81]. Similarly, heat transfer to Bingham and yield-pseudoplastic model fluids has been analyzed among others by Wissler and Schechter [49], Vradis et al. [73], Hirai [82], Schechter and Wissler [83], Henning and Yang [84] and
1450
Dakshina Murty [85]. Schenk and van Laar [86] investigated tl~e behaviour of Prandtl-Eyring model fluids. 60
120
i -
40
80
20
40
i
i
9
i
1
i|
I
Gr' / Re'
oi 0
'
~, I,,
,I
1
~ 2
[
O! 0
,
l
,
1
.
.I
2
Figure 5. Dependence of Gr'/Re' on power law index at which maximum velocity moves off-centre for upflow heating (left figure) and at which wall velocity becomes zero for downflow heating (right figure). 5. I. 4 Simultaneously Developing Flows" When the Prandtl number is smaller than unity, the temperature profile develops more rapidly than the velocity profile. This regime of heat transfer has received much less atlention. Besides, owing to generally high consistency of nonNewtonian materials, the corresponding Prandtl numbers are high and therefore many authors have questioned the relevance of this regime to the processing of non-Newtonian materials [33]. McKillop [87] extended the method of Atkinson and Goldstein [88] to analyze the heat transfer to power law fluids in the entrance region of a tube. Remote from the entrance, a perturbation solution was sought and matched with that near the entrance. Table 2 gives a summary of their results for three values of the Prandtl number and n = 0.5. Subsequently, McKillop et al [89] have elucidated the effect of temperature dependent viscosity. The entrance region heat transfer to Bingham plastics has been analyzed by Samant and Marner [90] while Lin and Shah [91] and Victor and Shah [92] have performed similar analysis for Herschel-Bulkley model fluids, though all are based on the assumption of the constant physical properties.
1451
5.1.5 Experimental Studies Laminar Regime
Numerous experimental studies oll heat transfer to power law fluids in circular tubes have been reported in the literature [48,53,87,93-101 ].Detailed discussions regarding their reliability and range of applicability have been provided by Cho and Hartnell [2] and by Lawal and Mujunldar [33]. However, the salient featllres of lhe experimental studies are re-capitulated here. In the filly developed flow regime, the maximum or minimum temperature, depending upon healing or cooling, occurs at the centre of the tube and the magnitude of this temperature is important while handling temperature sensitive materials (e.g., food, fennenlaiton broths etc.). Chann [102] measured the centre-line temperatures of banana pure6, apple sauce and ammonium alignate (all power law fluids) during heating or cooling in a straight tube of constant wall temperature. Except for the ammonium alignate solution, fully developed velocity and temperature profiles were not realised owing to the finite length (3.8 m) of the experimental section. However, the resulting values of the Nusselt number were found to be substantially gneater than the predicted asymptotic values of the Nusselt number thereby suggesting the presence of fiee convection. Matthys and Sabersky [103] studied the flow and heat transfer characteristics of tomato pure6 in circular tubes but no con'elations were presented. In the thermal entrance region, scores of conelations are available with some of these built in corrections for natural convection effects. Table 3 gives a selection of widely used conelations available in the literature. While some of these [53,62,93,94,98] account for the temperature dependence of power law consistency, only three of them take into consideration the fiee convection effects; the latter tend to be more important for relatively less viscous fluids. For relatively small fluid bulk-to-wall temperature difference, the available experimental results [104] are in good correspondence with the predictions of Bird [105] for constant wall flux condition, as shown in Figure 6 for a carpobol solution (n = 0.73). Note that the lilniting Nusselt number, equation (9), is also seen to be approached for diminishing values of the Graetz number. Similar comparisons are obtained with the data of Mizushina et a1.[94]; Mahalingam el al. [62]; Bassett and Welty [96]; Scirocco et al. [97] and Deshpande and Bishop [98]. Finally, the scant experimental results for viscoelastic fluids [95,104] suggest that the value of the Nusselt number is not altered appreciably by the viscoelasticity of the fluid.
1452
Table 2 Local and mean Nusselt numbers for simultaneously developing flow of a powerlaw fluid ofn = 0.5 in a circular pipe Constant heat flux
Constant wall temperature x*
Nu
0.008 0.019 0.059 0.099 0.15 0.20 0.30
10.05 7.95 5.48 4.61 4.18 4.04 3.96
0.005 0.010 0.020 0.060 0.10 0.15 0.30
11.32 8.68 6.64 4.78 4.30 4.08 3.95
0.001 0.0195 0.0595 0.0995 0.1495 0.1995 0.2995
8.01 6.41 4.75 4.29 4.08 3.99 3.95
Nti Pr = 1 18.27 12.98 8.57 7.13 6.20 5.67 5.11 Pr = 10 16.55 13.31 10.45 7.13 6.08 5.45 4.73 Pr = 100 11.08 9.03 6.56 5.73 5.20 4.91 4.59
Nu
Nu
15.43 10.77 7.15 5.97 5.31 5.02 4.82
27.93 19.29 12.02 9.79 8.39 7.57 6.68
14.55 11.22 8.52 6.06 5.40 5.06 4.79
24.52 18.83 14.32 9.43 7.93 7.03 5.97
10.26 8.16 6.00 5.38 5.05 4.90 4.79
15.59 12.23 8.57 7.39 6.66 6.23 5.76
From the foregoing brief account, it can thus be concluded that for tile constant wall flux condition, the analytical predictions of Bird [105] provide good estimates of the Nusselt number for viscous and viscoelastic systems for small values of AT, the correlation of Mahalingam et al. [62] and Deshpande and Bishop [98] might be the best ones to use in design calculations.
1453 Turbulent Regime Despite the fact that turbulent flow is encountered much less frequently with non-Newtonian systems (except with the so called drag reducing dilute polymer
Table 3 Experimental correlations for laminar heat transfer
Constant wall heat flux
Mizushina et al. [94]
Nuz = 1.4 ~51/3Gz~/3 (~-~w):' '~176
Bassett and Welty [96]
Nuz = 1.85 Gz 1/3- 0.03 6
Mahalingam et al. [62]
Nuz = 1.46 81/3
[Gz + 0.0083 (GrPr)~] ''~
Mehta et al. [ 106]
Nuz = 1.873 Gz 1/3 + 0.87
Deshpande & Bishop [98]
Nuz = 1.41 81/~
IYIb
ln b
o3,:,,,o
Constant wall temperature (mean Nusselt number) Metzner et al. [53]
w-w10.14 Nu = 1.75 81/3 Gz1/3 I'~mb
Oliver & Jenson [93]
Nu = 1.75 [Gz + 0.0083 (GrPr)3(.4]1,3 mb
I'~.1014
1454
60 1 t
! Nuz
!
,
!
43 < Re < 1780
I I
I
I
'
i I
I
, 105 < Pr < 231
"1
B i r d 11051
n=0.73
.
,
_
10 I-
I Equation(9) I I
2_ 101
I
I I 102
!
I
~ II 103
I
I
I l 104
Graetz Number ,Gz
Figure 6. Typical comparison between predictions and experiments for laminar heat transfer in a circular tube (n = 0.73). solutions and low concentration particulate suspensions), considerable effort has been expended in investigating heat transfer in turbulent regime [2]. It is readily acknowledged that much longer thennal entrance lengths (up to 400-500 D) are needed under turbulent conditions as compared to the corresponding 10-15 D required in the laminar regime. This fact alone raises questions about the utility of some of the early data on heat transfer in the turbulent region. Metzner and Friend [107] extended the Reichardt's framework of analogy between heat and lnomentum transfer to power law fluids as: St =
f /2 1.2 + l l . 8 4 f / 2 ( P r -
(27) 1) Pr ''~
where the fi-iction factor, f, fimction of the generalised Reynolds number (pV 2-" D"~/8"-~m 6n) and the power law index, n, is given by the following equation[ 108] (l/f) ~ = 4 n ~ log (Re f(2-,)/2) _ 0.4 n "12 (28) Equation (27) is restricted to the condition (Pr Re2)f > 5 x 105. Preliminary comparisons showed equation (27) to be adequate. Since this pioneering study, many workers have reported new experimental data and/or analysis for turbulent heat transfer to non-Newtonian systems, see [2] for an exhaustive compilation. Based oll a critical evaluation of most previous data oll heat transfer for purely viscous fluids, Yoo [109] put forward the following empirical correlation:
1455
St Pr,~2/3 = 0.0152 Re~"~
(29)
Equation (29) is based on data extending over the ranges 0.2 < n < 0.9 and 3000 _
In contrast to the volulninous literature on heat transfer in circular tubes, the corresponding literature for non-circular conduits is much less extensive. Some of the comlnonly studied configurations include parallel plates or slits, square and rectangular ducts, concentric cylindrical annuli, triangular and trapezoidal ducts [33,34]. While some of these shapes are used in extrusion, others find applications in heat exchangers. One distinguishing feature of heat transfer in non-circular ducts is the variety of boundary conditions in addition to those of isothennal walls or constant wall flux. Since extensive compilations and discussions thereof have been provided recently by Lawal and Mujulndar [33] and Hartnett and Kostic [34], only the key points are su~mnarised here. 6.1 Parallel Plates For the fiflly developed flow, constant properties and in lhe absence of natural convection effects, the asymptotic value of the Nusselt number for the constant
1456
wall flux condition is given by [116]:
~( it
n
5n +
2(n+ 1) NH
=
n
-!
,In +
(30)
2n+l
The corresponding expression with viscous dissipation is:
[~ (32n'-
+ 17n + 2)
Nu =
(4n + 1)(5n + 2)
(4n + l)(5n + 2)
(31)
Payvar [116] also presented an analogous expression for Binghaln plastic fluids. On the other hand, in the hydrodynamically and thennally developing flow regime, when the classical separation of variables method is applied to the Graetz problem for power law fluids, a series solution is possible only for certain values of n. Tien [117] obviated fills difficulty by using the approximate velocity distributions deduced from file variational principle. Richardson [66] investigated fl~e similar problem wifl~ constant thennophysical properties but included viscous dissipation effects. His solution, however, converges only for large values of Gz and is fires restricted to very close to fl~e entrance. Numerous other studies of laminar heat transfer in slit geometries have been reported but all have presented fl~eir results in gn'aphical form and therefore reference must be made to original papers [118-123]. However, Kwant and Van Ravenstein [123] extended flleir previous study to channel flows and presented fl~e following expressions for pressure loss and mean Nusselt number: !
(- Ap / L)isothemlaI for 0 . 2 <_ n <_1
;
0.001_<
X*
_<0.9
(32)
For heating, i.e., b(Tw - To) _> 0, C~'k = 2 ak n/(n + 1) where ak is tabulated in Table 4 while for cooling C~ ~
k
-~
0.11
- 2 _< b ( T , , , -
To) < 0
1457 = 0.116
b (T,,,- To) < - 2
The mean Nusselt number with temperature dependent viscosity is given by:
Ntl] vp Nu
(33)
= 1 + 0.238 In~b, + 0.0224 (ln ~bo)-"
where ~, = 1 - Ct'k b (Tw- To) TM (X*/1.8) - Ct'k b(Tw- To)/n Table 4 ~k as a fimction of b(Tw- To) and n in a parallel plate geometry b(Tw- To ) 1 2 3
n --~ 1 0.105 0.10 0.095
.....
.
.
,,
.
.
0.75 0.109 0.102 0.096 .
.
0.50 0.123 0.100 0.082
0.33 0.134 0.88 0.061
0.2 0.132 0.070 0.049
Note the increasing effect of b(T,,,- To) with an increasing degree of shear thinning behaviour. Aside from these studies, Lin and Shah [91] and Shulman and Zaltsgendler [124] have studied heat transfer to viscoplastic fluids whereas Yau and Tien [125] studied developing flow heat transfer in power law fluids.
6.2 Rectangular and Square Ducts The geometries considered so far, namely, circular pipes and parallel plates are not only characterized by a high degree of symmetry but there is only one nonzero component of velocity. This attribute results in appreciable simplification in both the field equations and the non-Newlonian constitutive equations. In turn, the shapes like square and rectangular, though still possess symmetry, are much more difficult to analyze owing to the 2-D flow. Furtherlnore, additional possibilities also exist in terms of boundary conditions. For instance, one also needs to specify the condition at the duct wall in a circumferential direction, a common variant being constant heat flux in both flow and circumferential directions (H2) or constant heat flux in the flow direction and constant temperature in the circumferential direction (HI) or constant temperature in both directions (T). For fully developed flow and constant physical properties, the results of Chandraputla and Sastri [126] for a square duct are shown in Table 5.
1458
Table 5 Fully developed laminar flow in a square duct n
NLI1-
NLIIII
Nu112
1 0.9 0.8 0.7 0.6
2.975 2.997 3.030 3.070 3.120 3.184
3.612 3.648 3.689 3.741 3.804 3.889
3.O95 3.106 3.135 3.171 3.216 3.274
0.5
, ,
....
Note that shear thinning behaviour only marginally enhances the value of the Nusselt number. Chandraputla and Sastri [126] also computed the values of the Nusselt number as a fimction of the Graetz number and power law index for the thennally developing flow in square ducts. Finally, they [127,128] also tackled the problem of heat transfer in simultaneously developing flows. Representative results are shown in Figure 7 for the isothennal wall boundary condition. Lawal and Mujumdar [129] have also studied heat transfer to power law fluids in square ducts under silnultaneously developing conditions. Analogous results for rectangular ducts are fi~rther compounded due to the additional geometric factor, namely the aspect ratio [34]. Few numerical studies [130-134] and experimental data [135-137] on laminar heat transfer to purely viscous and viscoelastic polymer solutions are available. Most theoretical analyses assume constant physical properties; however, a notable exception being a recent study of Shin and Cho [138] for a 21 rectangular duct. The effect of viscoelasticity on heat transfer has been found to be small in rectangular ducts [139]. Limited data on turbulent heat transfer suggest the j factor to be smaller for viscoelastic fluids in rectangular ducts [140]. 6.3 Triangular Ducts The fidly developed laminar heat transfer to power law fluids in isosceles triangular ducts has been numerically solved by Cheng [141] for the boundary conditions H1 and H2; the resulting mean values of the Nusselt number are summarised in Table 6 for both shear thinning and shear thickening materials. Here also, both shearthinning and shear-thickening behaviours seem to have little influence on the value of the Nusselt number. The effect of temperature dependent viscosity on the laminar heat transfer to power law fluids in equi-lateral triangular ducts has been investigated by Lawal [142] in the thennal entrance region. He reported detailed temperature profiles at
1459
various axial locations thereby showing the gradual development of temperature distribution. The role of free convection was also discussed, especially in relation to the distortion of velocity profiles thereby yielding higher rates of heat transfer. 12 n = 0.5
Nu
Pr'= 0
8
O0
10
20 Graetz
50 Number
100
200
, Gz
Figure 7. Mean Nusselt number as a fimction of Prandtl and Graetz numbers for simultaneously developing profiles for isothermal condition. Subsequently, Lawal and Mujumdar [143] and Etemad et al. [144] respectively have elucidated the role of viscous dissipation and simultaneously developing flow on the laminar heat transfer to power law fluids in equi-lateral triangular ducts. Lawal [142] also paralleled a similar study for trapezoidal and pentagonal ducts as encountered in the extrusion of polymers and food stuffs. 6.40iher
Geomeiries
In addition to the aforementioned ideal flow geometries, industrial processes entail a great variety of complex shapes of flow passages including helical, double-sine [145], tapered tubes and these are not readily amenable to rigorous analysis. For instance, Jarzebski and Wilkinson [146] solved the laminar 11o11isothennal developing flow, lemperature dependent consistency power law fluid flow in a slightly tapered tube to simulate the flow in extrusion dies and they developed a predictive expression for pressure drop. Similarly, sudden contraction also represents a geometry of considerable theoretical and pragmatic significance in polymer processing. Christiansen and Kelsey [147] analyzed the
1460
Table 6 Fully developed laminar flow Nusselt numbers for isosceles triangular ducts Apex angle (deg)
n
NuHj
Null2
1.2 1.0 0.6
2.374 2.391 2.434
0.0756
1.2" 1.0 0.6
3106 3.101 3.250
1.882 1.891 1.933
1.2 1.0 0.6
2.946 2.974 3.098
,
10
0"
90
0.0792 0.0846 . . . .
..
,.
_
1.350 1.351 1.356
flow and heat transfer characteristics of Powell-Eyring model fluids with temperature dependent viscosity. Likewise, Halrnnad and Vradis [72] have studied the flow and thennal behaviour of Bingham plastic fluids. Rao [148] has, on the other hand, reported scant data or turbulent heat transfer to viscoelastic fluids in helical tubes and reported 10-15% enhancement in heat transfer. Turbulent heat transfer data with drag reducing separan solutions in sudden expansions has been reported by Pak et al. [149]. Mass transfer to pseudoplastic fluids in spiral flow has been studied by Wronski and Jastrzebski [150] and they showed that with a suitable choice of effective viscosity, it was possible to use the Newtonian correlation for power law fluids also. It is unlikely that theoretical and/or experimental results of heat transfer would ever become available for all possible geometries and conditions. In this context, the following correction factor, due to Cheng [141 ], is of considerable significance: NUpo,,o~,~,, = [ a + b n ] ''~ Nu~,,.,..... (a + b)n
(34)
where a and b are purely geolnetric factors introduced by Kozicki and Tiu [151] and are reproduced here in Table 7. The advantage of this simple approach which is restricted to laminar flow is obvious. Dunwoody and Hamill [152,153] predicted enhancement in heat transfer for 3rd grade Rivlin-Ericksen fluids in rectilinear flow in rectangular channels. Similarly,
1461 Table 7 Duct Flows: Geometric Constants
Geometry
CONCENTRIC ANNULI
0t* = inner radius/outer radius
RECTANGULAR DUCT a * = height / width
ELLIPTICAL DUCT 0t* = minor axis / major axis
ISOSCELES TRIANGULAR Apex angle = 2
REGULAR POLYGON with N sides
(x*
a
b
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.25 0.50 0.75 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2~(deg) I0 20 40 60 80 90 N 4 5 6 8
0.4455 0.4693 0.4817 0.4890 0.4935 0.4965 0.4983 0.4992 0.4997 0.5000 0.5000 0.3212 0.2440 0.2178 0.2121 0.3084 0.3018 0.2907 0.2796 0.2702 0.2629 0.2575 0.2538 0.2515 0.2504 0.2500
0.9510 0.9739 0.9847 0.9911 0.9946 0.9972 0.9987 0.9994 1.0000 1.0000 1.0000 0.8482 0.7276 0.6866 0.8766 0.9253 0.9053 0.8720 0.8389 0.8107 0.7886 0.7725 0.7614 0.7546 0.7510 0.7500
0.1547 0.1693 0.1840 0.1875 0.1849 0.1830
0.6278 0.6332 0.6422 0.6462 0.6438 0.6395
0.2121 0.2245 0.2316 0.2391
0.6771 0.6966 0.7092 0.7241
, secondary flow patterns in flattened tubes and coils have been shown to yield higher heat transfer to non-Newtonian fluids [154-156]. Such enhancements ill
1462
heal transfer have been observed experimentally [157]. The scant results point to fi~rlher increases in heat transfer due to free convection in non-circular conduits for purely viscous and viscoelastic fluids [158-160]. This section is concluded by noting that the highest value of the Nusselt number occurs in circular tubes and then progressively decreasing as the cross-section of flow changes to pentagon, square, trapezoidal and triangular. This trend is seen for both shear thinning and shear thickening materials. Furthermore, if the various geometries are ranked on the basis of the ratio of heat transfer to pressure drop, a large value of this ratio is desired. In this ranking, for n < 1, the circular duct is evidently superior to other shapes while short square tubes are to be preferred for shear thickening materials [33]. 7. HEAT AND MASS TRANSFER IN BOUNDARY LAYERS
Considerable research effort has been expended in elucidating the role of nonlinear flow characteristics on the rates of heat and mass lransfer in boundary layers over submerged objects. Though theoretical results are available for highly idealised shapes such as plates, cylinders and spheres, often this treatment serves as a useful starting point for more complex situations encotmtered in engineering applications. The available voluminous literature on both forced and flee convection has been reviewed by several authors [161-165], only key findings are summarised here. At the outset, it is instructive to point out l~re that the flow of shear lhinning (usually modelled as power law fluids) and viscoplastic media have been studied most extensively, followed by the flow of viscoelastic fluids. Furthennore, though it is realised that the momentum boundary-layer analysis for nonNewtonian fluid is gennane to the study of the convective heat and mass transfer in these systems, these are not included here for the sake of brevity. However, attention is drawn to the pioneering efforts of Schowalter (1960); Bizzell and Slattery (1962); Na and Hansen (1966) ; Lee and Ames (1966); Chen and Radulovic (1973), etc. for power law liquids and of others [171-173]for viscoelastic boundary layers. A cursory examination of the available reviews reveals that thermal convection has been investigated more extensively than the analogous mass transfer process. The three geometries that have been accorded the greatest amount of attention are spheres, cylinders and plates, although some authors [174-178] have attempted to develop general framework for 2-D boundary layer flows. The ensuing discussion is presented in tlu-ee sub-sections, namely, spheres, cylinders and plates, with fiwther classification depending upon the mechanism of transport, e.g., free, forced or mixed convection.
1463 7.1 Spheres 7.1.1 Free Convection
The general field equations describing fiee convective heat/mass transport in external flows and the associated difficulties, due to non-linear viscosity models, in seeking rigorous solutions have been alluded to by some authors[31,161], Similarity solutions are not feasible even for the simplest non-Newtonian viscosity model, namely, power law model [166,179]. Acrivos [179] employed a Mangler-type transformation to derive the following expression for the mean Nusselt number for the constant temperature condition in the laminar boundary layer region: |
Nu = ~hR = C1(ll) G r o ~
B
Pro3n+,
(35)
k
where cl(n) is a function of n. In another study, Stewart [180] also obtained approximate solutions for laminar boundary layer equations for a range of shapes including a sphere and the resulting expression for lnean Nusselt number is: 1
Nu = c, (n) (Gr, Pr,) ~~
(36)
where c2(n) is another fimction of n. Tile values of both cl(n) and c2(n) are available in respective papers. Equation (36) was stated to be particularly appropriate for high Prandtl numbers, as is generally the case with nonNewtonian fluids. The theoretical developments in this field have been matched by excellent experimental studies. Both Liew and Adehnan [181] and Amato and Tien [182] measured the temperature distributions and the mean Nusselt numbers for electrically heated copper spheres immersed in polymer solutions. In the range 0.66 _
(37a)
and Nu = 0.49 f2
(37b)
1464 |
!1
for 10 < s < 40 where s = G r ~ Pro~~ . Equation (37) was stated to correlate their results with a mean error of 8%. Funhennore, the numerical coefficient of 0.49 in equation (37b) is remarkably close to the theoretical value of 0.468 predicted by Acrivos [179]. Further examination of the data of Amato and Tien [182] suggests that the ratio of the thermal and momentum boundary layer thicknesses is of similar magnitude for water and polymer solutions, except near the stagnation points. Alhamdan and Sastry [183] used a musluoom shaped aluminium panicle to measure the heat transfer coefficients under natural convection regime in the aqueous carboxymethyl cellulose solutions and developed empirical correlations in terms of the Rayleigh number. They estimated the apparent viscosity based on the actual fiee convection velocity measured by tracer particles. This severely limits the utility of their correlations because neither such velocities can be measured with great accuracy nor can these be estimated independently. Awuah et al. [184] while attempting to measure force convective heat transfer from a sphere to aqueous CMC solutions reported that the free convection was important in their study, and successfiflly correlated their data in tenns of the Rayleigh number with an index o f - 0.1 - 0.12 which is rather small compared with the usual 1/4 power for Newtonian systems. The analogous mass transfer data for flee convection fiom spheres has been reported by Lee and Donatelli [185] in the range (0.58 < n < 1). By invoking the usual analogy between heat and mass transfer, they adapted equations (35) and (36) by noting equality of the Nusselt and Shel~,ood numbers, Prandtl and Schmidt numbers. But they found the following correlation [186] to represent their data better than the mass transfer analogues of equations (35) and (36), especially with decreasing values of Gr,n: i
Shl =
k d
= 2 +
c,' (Qr,,,Sc,)s"-' -
4
(38)
1 + (0.43/ Sr ~ Witll the diminishing value of Grin, one would intuitively expect the Sherwood number to approach the pure diffi~sion limiting value of 2. However, due to the thin boundary layer assumption inherent in the aforementioned dleories, neither equation (35) nor (36) approaches fills limit whereas equation (38) does. Widl fl~e generally large values of die Schmidt number for non-Newtonian fluids, Lee and Donatelli [185] simplified equation (38) as: i
Shi = 2 + c2'(n) (GrlmSc,)~"*'
(39)
1465
and they found tl~at c2'(l) = 0.59 provided satisfactory conelation of their mass transfer data in the range 0.58 _
Westerberg and Finlayson [187] numerically solved the governing equations for heat transfer from a sphere to a typical Nylon-6 melt, and elucidated tlle effect of various kinematic and physical parameters. Qualitatively, the Nusselt number for a single sphere is influenced, in decreasing order, by the value of the Peclet number, viscous dissipation, extent of shear thinning, temperature dependence of viscosity, viscoelasticity, and the telnperature dependent thermal conductivity. The observation that the fluid viscoelasticity plays little role in determining tl~e value of the Nusselt number is consistent with other works for heat transfer [188] and for mass transfer [189,190]. Morris [191 ] studied the relative importance of the temperature and shear rate dependence of viscosity on the flow field produced by an isothermal sphere moving in Newtonian and power law fluids. In a series of papers, Nakayama and co-workers [176-178] have developed an integral method for analyzing the forced thennal convection in laminar boundary layers over axisymmetric bodies immersed in power law liquids. By a suitable choice of shape factors, the results can be obtained for spheres, cylinders, ellipsoidal shapes, though the validity of this approach has been questioned for highly shear thinning fluids (n < --- 0.3) [192]. Mixed convection heat transfer from spheres to power law fluids has also been analyzed [193,194]. In the creeping regime, the mass transfer between a sphere and power law fluids has been investigated by Kawase and Ulbrecht [195]. Owing to the large values of Schmidt numbers, the fonnalism introduced by Lochiel and Calderbank [196] is applicable for non-Newtonian systems. Thus, the Sherwood number, Sh, is given by:
where V0* is the tangential velocity on the surface of the sphere. Kawase and Ulbrecht [195] used the approximate stream fimctions available in the literature to deduce the value of the Sherwood number. The predicted 50-70% enhancement in mass transfer due to shear thinning viscosity (n --- 0.7) is in line with the scant experimental results [197,198].Beyond the creeping flow regime, little theoretical information is available on heat and mass transfer beween a non-Newtonian fluid and spherical particles. Based on the penetration model [199,200], Kawase and Ulbrecht [201 ]) derived the following formula for a sphere:
1466
1,-A
Shl =
"~"
n-*2
1
= A(n) Re,,3~.,,> Sc ~
(41)
DAB
where the values of A(n) are available in their paper; A(n=l) = 0.85. The scant experimental results on mass transfer from spheres [202] appear to be within +30% of the predictions of equation (41) [165].Likewise, admittedly numerous experimental studies on forced convection heat transfer between a sphere and inelastic power law type polymer solutions have been reported but sufficient details are not available to recalculate the experimental results in the form needed by theoretical analyses mentioned in the preceding section. For instance in some studies [203,204] particles were held stationary in moving polymer solutions while in olher stLidies [205] particles were being conveyed in moving liquid streams. Similarly, significant uncertainty also exists with regard to the effect of particle-to-tube diameter ratio in exerimental results. Clearly there are some variables which are not being duly accounted for. Similarly, based on their experimental data, Yamanaka and Mitsuishi [206] put forward the following semi-empirical correlation for mixed convection from a sphere to power law fluids: Nu
=
hd k
2
!
3
= 2 + { (0.866o- Pe~ - 0.553o-- 0.341) -~ !
I
n
3
2 [ l l l b ] - -3n-,-!
+ (0.44 Gr~ <~ pr~"")2 }~
(42)
\ 111.
where
cr = -2.475n 3+6.74n 2-7.67n+4.74.
Similar empirical conelations have been put fo~vard by many other investigators, e.g., see [204,207,208] but all these are too tentative to be included here but these have been reviewed recently [209,210].However, Ghosh et al. [165,211] have developed a unified correlation for heat and mass transfer fiom spheres to power law fluids by invoking the usual heat-mass transfer analogy, i.e., Pr ~ Sc and indeed this approach does reconcile most of the data on heat and mass transfer available in the literature. Their correlation is given by: Y
=
1.428 Rep !/3
Rep<4
(43a)
Rep 1/2
Rep < 4
(43b)
1467 1
where for heat transfer, Y = (Nul - 2)
m,
P r ? and Y = (Shl - 2) Scp-~/3 for
1-n b ,,/
P
mass transfer. Figure 8 shows the overall correlation. Furthermore, equation (43) also seems to correlate the data for short (L/D--- 1) cylindrical pellets provided the equal volume sphere diameter is used.
10 2
Equation (43a)
lO~ Equation (43b)
i
10 -2 10. 3 L 10 -7
I
I 10 -5
|
I 10 -3
I
I I 0 -1
Reynolds Number,
_
!
!
101
10 3
Rep
Figure 8. Overall correlation for heat and mass transfer from a sphere (0.32 _< n <_ 0.93). Mass transfer fiom spheres suspended in pseudoplastic solutions in agitated vessels has been investigated [212,213] and the corresponding Newtonian fonnulae provide satisfactory representation when the effective viscosity is evaluated as m (4nN/n) nl. 7.2 C y l i n d e r s
7.2.1 Free Convection The analysis of Acrivos [179] can easily be adapted for free convection from a horizontal cylinder and the resulting expression is of a form similar to that for a sphere, i.e. equation (35). However, Ng and Hartnett [214] have re-cast equation (35) in tenns of the Rayleigh number as follows: 1
Nu = c(n) Ra 3~
(44)
1468
The thin boundary-layer assumption inherent in the analysis of Acrivos [179] !
now becomes Ra ~3~ > 1. Since c(n) is a weak fimction of n, the Nusselt number rises with the decreasing value of n at a fixed value of Ra. Analogous results for constant properties Ellis model fluid are also available [215]. The available limited experimental results for shear thinning polymer solutions [216] are in fair agreement with theoretical predictions over the range 0.64 _< n _< 0.93. It is also somewhat puzzling that their results of the surface averaged Nusselt number in carbopol solutions are virtually indistinguishable from the Newtonian correlations provided the cylinder diameter is used instead of cylinder radius! Ng and co-workers [214,217-219] have reported extensive results on the free thermal convection from horizontal (thin) wires to pseudoplastic and viscoelastic fluids for the constant heat flux condition on the wire surface. These experimental studies differ from the analysis of Acrivos [179] and the work of Gentry and Wollersheim [216] in an important way that the wire diameters were of the same order as the botmdary layer thickness. For pseudoplastic carbopol solutions, their final correlation is: i
Nu = (0.761 + 0.413n)
Ra :r
(45)
in the range 0.001 _
In a series of papers, Acrivos and co-workers [174,175,179,223] developed a theoretical fiamework for the laminar boundary layer heat and mass transfer from two dimensional objects including a horizontal cylinder submerged in power law fluids. Their expression for the surface-averaged Nusselt number is of the form: I
i
Nu = c(n) Re~ ?g5 Pr-
(46)
1469
where c(n) accounts for both varying degree of shear thinning behaviour, and different shapes. Subsequently, many other investigators [224,225] have employed different solution procedures to arrive at substantially similar conclusions [165].Mizushina and Usui [226,227]argued that the aforementioned analysis are flawed due to the so called zero defect at the separation and stagnlation points. They obviated this difficulty by combining the approaches of Kannan-Pohlhousen [228] for momentum boundary layer and of Dienemann [229] for thermal boundary layer to obtain approximate results for the power law and Powell-Eyring model fluids. Mixed convection from horizontal cylinders submerged in power law fluids has been investigated amongst others by Shenoy [230] and Wang and Kleinstreuer [231]. The first experimental study in this field is that of Shah et al. [223] who measured the local values of heat transfer coefficient and reported good agreement with theoretical predictions, especially for large Prandtl numbers. Mizushina et al. [232,233] measured both the rate of heat and mass transfer from horizontal cylinders oriented nonnal to flow. Their data show good correspondence with equation (44) and they correlated the constant c(n) as: c(n) = 0.72 n~
(47)
Similarly, Luikov et al. [234,235] reported tile values of mass transfer coefficients from cylinders oriented normal to the flow of polymer solutions in laminar region. For power law fluids, the point of separation moved downslream ca. 0 = 155 ~ Also, their data do not seem to conform to the generic foma of equation (44). Thus, the value of c(n) increased from 0.31 to 0.5 as the value of the flow behaviour index dropped from unity to 0.88. The conesponding change in the value of the exponent of Reynolds number ranged from 0.52 to 0.39. Surprisingly, similar trends are also present in the heat transfer data of Takahashi et al. [236] who reported the exponent of the Reynolds number to be 0.35 in the range I > n > 0.78 as opposed to the predicted values of 0.5 and 0.55 in this range. Kumar et al. [237] also reported extensively on mass transfer fiom cylinders placed in cross-flow configuration in aqueous CMC solutions. Indeed, this study elucidates the importance of the wall effects and the aspect ratio of the cylinder. Subsequently, Ghosh et al. [238]have reviewed the literature and put forward the following correlation for heat and mass transfer fi'om a cylinder in cross-flow to power law media: Nu~ (or Shl) = A Rep'" (Prp o1" Scp)1/3
(48)
1470
where A = 2.26 and m = 1/3 for Rep < 10 and A = 0.785 and m = 1/2 for Rep > 10. Equalion (48) was fotmd to be successfid not only in bringing together the heat and mass transfer data but also the resulls for Newtonian and power law liquids up to Re--- 25000, as can be seen in Figure 9. Considerable interest has also been shown in studying the forced convection from horizonlral cylinders immersed in drag reducing polymer solutions, though results of different investigators often do not agree even qualitatively. For inslance, James and Acosta [239] and James and Gupta [240] found that at low flow rates, the drag coefficient and heat transfer coefficient for cylinders placed nonnal to the flow of PEO solutions virlually coincided with the values for water but beyond a critical flowrate (presumably Deborah or Weissenberg number), both levelled off to constant values whereas the more recent data[241] show no such behaviour with a Separan AP-302 solution. Numerous plausible mechanisms including the viscoelasticity, boundary-layer thickening due Io the solid-like behaviour at high defonnation rates, etc. have been postulated; however, while none of these has proved to be completely satisfactory, Ruckenstein and Ramgopal [242] developed an empirical correlation for such anomalous data. Likewise, a few investigators have studied heathnass transfer from horizontal cylinders rotating in drag-reducing polymer solutions [243,244] and immersed in boiling non-Newtonian polymer solutions [245]. 7.3 Plates
7.3.1 Free Convection The theoretical treatment of Acrivos [179] results in a qualitatively similar expression for the mean Nusselt number as that for a cylinder, equation (44). Shenoy and Mashelkar [161] have treated in great detail free convection fi'om vertical and horizontal plates (with isolhennal as well as constant heat flux boundary conditions) in non-Newtonian fluids whence only the key findings are summarised here. For instance, the scant data [216,246,247] for isothermal surface are in line with the predictions[179,248].Shenoy and Ulbrecht [249] presented an inlegral solution for laminar thermal convection from a vertical surface to power law fluids under a variety of boundary conditions on the plate surface, i.e., constant heat flux, constant temperature and temperature varying with distance. For the constant temperature condition, their expression for the mean Nusselt number (averaged over the plate length) is given by: (2n + I)
Nul = fl(n)
3n + 1 ,-3. ,~ GrL2~"~'~ Pr,~.*, 2n +
(49)
1471
10 2 I
_ Equa,, 8, 785 J Ur~
I0 ~
A = 2.26 ,m
1,1
m
m = 1/3 1 0 -2
10"3
I.
I
10 "3
I
_
10 4
I
.....
I0 x
i
i
I0 3
Reynolds Number, Rep
Figure 9. Alternate correlation for heat and mass transfer from a sphere to power law fluids. where n
(50a)
2 A. (10/3) ~j"
1
with An =
(50b)
3n 6
f2(An) =
'~;-'cz,AJ
(50C)
J=i
cx~ = 1/15; or2 = -5/42;0t3=3/28 ot4 = - 1/18; (x5 = 1/63; 3/1540 13-6
--
"
(50d)
Shenoy and Ulbrecht [249] reported good agreement between their data and predictions. No analogous mass transfer results are available in the literature. This sub-section is concluded by noting that most theoretical studies in this field may be regarded as direct descendants of the pioneering work of Acrivos [179], except for the fact that a variety of teclmiques have been employed to seek approximate solutions.
1472
7.3.2 F o r c e d Convection
Little work has been reported on the forced-convective transport fonn a horizontal plate/plane surface to non-Newtonian media since the study of Acrivos et al. [174,175]. For the conditions when the thermal boundary layer is thinner than the momentum boundary layer (a condition which is fulfilled over a substantial part of the plate except near the leading edge). In this region (x --+ 0), Acrivos et al. derived the following expression for the local Nusselt number: ":
Nux =
x
Re"~n*,~ Pr"2
(51 )
More recently, a complete numerical solution for laminar heat transfer from a plane surface (aligned) immersed in Newtonian and power-law fluids has been carried out by Pittman et ai. [250]. The predictions and experiments show good match for detailed temperature profiles and the Nusselt number for the constant heat flux condition on the plate in the range (30 < Re < 2000; 50 < Pr < 400). The corresponding mass transfer problem has been studied by Mishra et al. [251], who employed the von Kannan integral method to solve the approximate boundary-layer equation for power law liquids. The average Sherwood number is given by: (52)
Sh = Co(n) Re~_''~n+'~ Sc,, ''~ where Co(n) =
:3 ( ( 3 (4.64).:,~. ~ ,~ 1.5 (n + 1) 2n + l
2 n
i
(53)
+
Both Ghosh et al. [252] and Luikov et al. [234,235] have re-cast the results of Acrivos et al. [174] into the equivalent mass transfer problem. While the expression of Acrivos et al. [174] is similar to equation (52), the one due to Luikov et al. is given below: ShL = (0.474 + 0.436n - 0.12112) Re~ ("~ '~ Sc 1/3
(54)
Luikov et al. [234,235] reported good agreement between equation (52) and limited data. Subsequently, Ghosh et al. [252] have reconciled the limited data on mass transfer from plane surfaces to a single correlation as:
1473
SIlL =
3.23
Re~ Sc~ L ,,3
0.2 < ReL < 100
10 3
--
i
~
I
(55)
100 < ReL < 5000
= 1.12 Rel '': Scl ''~
I"l
I
I
I
I
~
~
I
I
~ 'I
I
I
t
.I
l
~
t
t
! 4m
10 ~
~
10-1 I J 1 0 .2
~_ I
I
~
I
Slope = 1/3
J
~ ]
t
10 ~ Reynolds
10 2 Number,
!
10 4
Re*L
Figure 10. Correlation for mass transfer from a plate to power law fluids (plate length 10 - 90 ram). Figure l0 shows the overall conelation for different lengths of plates immersed in water as well as in a series of CMC solutions. The co~xesponding range of Schmidt number is 840 _< Sc~, _<2 x ] 0 6 while the power law index varies fiom 1 to 0.89 only. Analogous boundary layer-analyses from continuously moving surfaces and/or to the other type of generalised Newtonian fluids have been reported by Sharma and Adelman [253], Lin and Shih [254], Gorla [255,256]. On the other hand, Skelland [257] and Mishra and Mishra [258] have used the Chilton-Colburn analogy to obtain approximate expressions for heat and mass transfer in turbulent boundary layers. Forced convective heat transfer to power law fluids flow over a rotating non-isothermal sphere has been analyzed [259] and as expected the rotation yields higher values of the Nusselt number, as also noted previously by other investigators [231]. More recently, Ruckenstein [260] has re-examined
1474
transport in laminar boundary layers and the viscoelasticity is shown to moderate the role of the Reynolds numebr which is qualitatively consistent wilh the observations of James and Acosta [239]. In concluding this sub-section, it is perhaps appropriate to add that although this review has mainly focussed on heat and mass transfer from three simple configurations, namely, a sphere, a cylinder and a plate, considerable information is also available on mass transfer to non-Newtonian fluids from rotating spheres, cylinders, disks, cones, inclined plates and other shapes. All these studies, however, have been motivated largely with a view to measure molecular diffitsion coefficients [261 ]. 8. H E A T AND MASS TRANSFER IN P A C K E D AND FLUIDIZED BEDS
Packed and fluidized beds are widely used in chemical and polymer processing applications to affect a variety of operations including mixing, filtration and chemical reactions. Despite their overwhelming pragmatic significance in the handling of non-Newtonian materials, very little information is available on liquid-solid as well as bed-to-wall heat and mass transport with non-Newtonian systems. For instance, Kawase and Ulbrecht [201] developed a semi-theoretical expression for Sherwood number describing the mass transfer between power law liquids and a packed bed of spheres and they reported satisfactory correspondence between their predictions and limited data. On the other hand, Kumar and Upadhyay [262] and Wronski and Szembek-St6eger [263] carried out experimental works on liquid-solid mass transfer in packed beds and put forward the following empirical correlations. Kumar-Upadhyay ej =
0.765 0.365 + Rel '~: Re, ~"
(56)
Wronski-Szembek-St0eger: ~j=
1 0.097 Re ~ + 0.75 Re~ 6'
(57)
Figure I I shows the overall CO~Telation in terms of the j-factor, correspondence between the two independent sets of data as well as between equations (56) and (57) is seen to be satisfacto~7 over the ranges 10-4 _< Re~ _< 100; 800 _< Sc~ _<
1475
72000. Note that equations (56)and (57) are also valid for short cylindrical pellets (L/D <--- 1) provided an equal volume sphere diameter is used in lieu of sphere diameter. Hilal et al. [264], on the other hand, proposed the use of (6/a,.) where av is the specific surface area. Coppola and BNam [265] have studied mass transfer from a bed of screens to power law fluids.
500 Equation (56)
100
"~
10 Equation (57)
1.0
0.1 I
10"4
1
10 "2
I
....
l
I
,
10 ~
r
102
J
500
Reynolds Number, Re1
Figure 11. Power law liquid-solid mass transfer in packed beds. The effect of drag reducing additives on mass transfer in packed bed reactors has been studied by Sedahmed et al. [266]. Depending upon the type and dose of the additive, they reported up to 50% reduction in mass transfer. Similar deterioration in gas-liquid mass transfer has also been reported by others [267]. Likewise, convective heat and mass transport in fluidised beds have received only very scant attention. For instance, Kumar and Upadhyay [262] asserted equation (56) to be applicable to fluidised beds also. More recently, Hwang et al. [268] reported new experimental data on liquid-solid mass transfer in fluidised beds and found equation (56) to over-estimate the value of j factor at low Reynolds numbers (see Figure 12). They found it necessary to develop the following new correlation (0.01 __n >__0.63): log (~j) = 0.169- 0.455 log (Re~)- 0.0661 (log(Re~))2
(58)
1476
No such results are available on heat transfer and for viscoelastic media. 50 Equation (58)
10 O
Equation (56)
0.1 |
0.01 [0.001
I
,
! 0.10
t
~
.
10
500
Reynolds Number, Re I
Figure 12. Overall correlation for liquid-solid mass transfer in fluidized beds. 9. ]VIISCELLANEOUS GEOMETRIES Beside the simplified geometries dealt with in the preceding sections, numerous other configurations have been studied in the literature. While, it is not possible to include all such studies in this review; some key references are provided here for the benefit of the interested reader. The subject of heat transfer in mechanically agitated vessels used for the heating/cooling of process streams has been reviewed by several authors [74,269,270]. The CO~Tesponding literature on gas-liquid mass transfer with non-Newtonian liquids in bubble columns and packed towers has been summarized in many papers [271-274]. Heat transfer in boiling non-Newtonian fluids has only begun to receive systelnatic attention [275280] and thus far only preliminary results are available on different flow patterns and critical heat flux, etc. Similarly, the thennal perfonnance of compact heat exchangers and their fouling characteristics with non-Newtonian polymer solutions have been studied only in a cursory manner [145,281,282]. Thermal effects in the extrusion of food and polymer extrusion have been reviewed recently by Jaluria [283].
1477
10. CONCLUDING REMARKS AND FUTURE NEEDS
From the aforementioned discussion, it is clear that in the laminar region tlle shear thinning enhances heat transfer and reduces pressure drop as compared with a Newtonian fluid. Shear thickening produces the opposite effect. The temperature dependent viscosity facilitates the heating of fluids while it impedes the efficacy of cooling of fluids for shear thinning materials. Likewise, a significant degree of viscous dissipation aids in the cooling of fluids with the opposite effect in the heating. Furthermore, in situations with Br > 0, inespeclive of the heat transfer mechanism at the wall, the local Nusselt number attains an asymptotic value, about three times the value predicted for forced convection. In horizontal flow, the natural convection increases both heat transfer as well as pressure loss above those expected from the pure forced convection analyses. While from a heat transfer standpoint, the circular geometry is the most efficient one but when pressure loss is also of concern, short pipe lengths are to be preferred. Besides, under certain conditions, square pipes offer a good compromise between high heat transfer rates and low pressure drops, particularly for shear thickening fluids. Experimental work in this area of duct flow is trailing behind the great strides made in theoretical developments. Heat and mass transfer in laminar boundary layer flows have also witnessed remarkable years of progress. Consequently, sound theoretical frameworks are now available for the prediction of heat and mass transfer from simple geometries such as spheres, cylinders and plates, at least to purely viscous type nonNewtonian materials. Available experimental results, while essentially in agreelnent with the predictions, do not seem to cover anywhere near as wide ranges of kinematic conditions as encompassed by the corresponding expressions for Newtonian media. Considerable scope exists fi~rther developments in this rugged terrain of nonNewtonian fluid mechanics. For instance, the role of fluid elasticity is not at all clear both in internal flows in ducts as well as in boundary layer flows. For instance, secondary flows are ta~own to exist in non-circular ducts even in creeping regime but it is not yet fully known how these impact on heat/mass transfer. The question of stability has hardly evoked any interest. Furthermore, most heat transfer studies employ either the constant wall temperature or constant heat flux condition, whereas in practice the actual conditions may be somewhere in between these two idealizations. Also, the flow patterns and heat/mass transfer phenomena in systems of practical importance including packed and fluidized beds, heat exchangers, boiling, etc. merit much more attention than it has received in the past.
1478
11. NOMENCLATURE geometric shape factor, Table 7 B width of plate, m Br Brinkman number, = (D/2) x,,,("+l~"~/q,,,j.tl/",. b geometric shape factor, Table 7, - or coefficient in temperature dependent viscosity equation, I/K Cp heat capacity, J/kg K d particle diameter, m D tube diameter, m molecular diffusivity of A in B, m2/s DAB hydraulic diameter, m DH E activation energy for viscous flow, J/kg K, equation (5) f fanning friction factor, D (- Ap)/2L 9V 2 a
Gr Grh Gro Grl Grim Gr' Gz
g h
J k kc L 1"11 In lIl o
Nu Nul = 2Nu n q Pe Pem
Grashof number, = I3AT D 3p2g/~teff2 Grashof number, = (p2d3gl3AT/m2)(d2/002"2 modified Grashof number for sphere/cylinder, = p2Rn+2(13gAT)2-n/m2 modified Grashof number, p2d3g[ST/m2(d2/ot)2'~-2 Grashof number for mass transfer, = p2d3gAp/ln2(d2/ot)2n2 modified Grashof number, = p213gqw(Dh/2)2"+2V2"2n/k~t2 or p213g AT (Dh/2) 2n+l V22n/la2, ) =
Graetz number, = m Cp/kL acceleration due to gravity, m/s 2 heat transfer coefficient, Whn2K j-factor, = (kdV)Sc "2/3 thermal conductivity of fluid, W/InK mass transfer coefficient, m/s length in flow direction, m mass flowrate of fluid, kg/s power law consistency index, Pa.s" value of m at temperature To, Pa.s" Nusselt number, = lff)/k for pipes and hR/k for spheres and cylinders Nusselt number for spheres and cylinders power law index heat flux, W/in 2 Peclet number, p Cp VD/2k Peclet number for mass transfer (dV/DAB)
1479 Pr Pr'
Prandtl number, Cp Prandtl number = Cp m (V/Dh)"-l/k
Pra
Prandtlnumber'-Cpm l k[ 3n4,,+
Prh Pro Pr~
Prandtl number = (pC~r (1-'')/(l+,,) Prandtl number = (pCp/k)(ln/p)2/(n+l)R r 1-,,)/(l+n)(Rg[3AT)S(n-1)/2(,+I) Prandtl number = (mot/d2)n-l/pot Universal gas constant, Pa.mS/mole.K tube/sphere/cylinder radius, m Rayleigh number, = Gro Pro
R R Ra
8V]""D
R ep
Reynolds number, oV 2-n D"/8n-'m( a n + 1.]" 4n Reynolds number, pV2-n d"/m
Rel
modified Reynolds number, Rep (3,, 4nn +1)
r
radial coordinate, m Schimdt number, = (mDAB/d2)"-J/pDAB Schimdt number, = (m/pDAB)(d/V)I-" Stanton number, h/pV Cv temperature, K bulk temperature of fluid, K outlet fluid temperature, K inlet fluid temperature, K wall temperature, K mean velocity in pipe, m/s point velocity in z-direction, m/s angular velocity in 0-direction, m/s dimensionless axial distance, x/D dimensionless axial coordinate X/Pe axial coordinate, m
Rea
SClm
Scp St T Tb T~ To Zw
V Vz Vo X X* Z
Greek Symbols ot Ot.'k
13W 5
Ap
thennal diffilsivity, m2/s constant in equation (22) coefficient of thennal expansion, K l coefficient in viscosity equation (20), K -~ ((3n + 1)/4n) pressure drop, Pa
(12 (1 - e) / e")""
1480
porosity of packed or fluidised beds rl ~tcff ~-to
P 17rz B "Co
Subscripts b H1 H2 iso L S SIIS
T vp w X Z
"roB/Tw
viscosity, Pa.s effective viscosity, = re(V/d) nl, (Pa. s) viscosity at reference temperature, Pa.s dimensionless radial coordinate, r/R density, kg/m 3 shear stress, Pa Bingham yield stress, Pa wall shear stress, Pa volume fraction bulk pertains to thermal condition H l pertains to thermal condition H~_ pertains to isothennal case liquid solid suspension constant temperature variable property case wall condition local value local value average value
REFERENCES 1. Porter,J.E.,Trans.Inst.Chem.Engrs.,49(1971 ) 1. 2. Cho,Y.I.,Hartnett,J.P.,Adv.Heat Transf.,15(1982)59. 3. Paul,D.D.,Abdel-Khalik,S.I.,J.Rheol.,27( 1983)59. 4. Pmd'h~176176176 Interfacial Sci.,93(1983)274. 5. Irvine,Jr.,T.F.,Kim,I.,Cho,K.,Gori,F.,Exp.Heat Transl.,1 (1987) 155. 6. Hu,R.Y.Z.,Wang,A.T.A.,Hartnett,J.P.,Exp.Thennal Fluid Sci.,4(1991 )723. 7. Ishiguro,S.,Hartnett,J.P.,Int.Colmn.Heat Mass Transf.,l 9(1992)285.
8. Rodrigue,D.,DeKee,D.,ChanManFong,C.F.,J.Non-Newt.Fluid Mech.,66(1996)213. 9. Loulou,T.,Peerhossaini,H.,Bardon,j.p., Int.J.Heat Mass Transf., 35(1992)2557. 10. Shin,S., IntC~ 23(1996)665.
1481
11. Hubbard,D.W.,Encyclopedia of Fluid Mech.,6(1987)chapter 3. 12. Ghosh,U.K., Kumar,S.,Upadlayay,S.N.,J.Chem.Eng.Data,36(1991 )413. 13. Wickramasinghe,S.R.,Boger,D.V.,Pratt,H.R.C.,Stevens,G.W., Chem. Engng. Sci.,46(1991)641. 14. Wasan,D.T.,Lynch,M.A.,Claad,K.J.,Srinivasan,N.,AlChE J.,] 8(1972)928. 15. Clough,S.B.,Read,H.E.,Melzner,A.B.,Behn,V.C.,AIChE .1.,8(1962)346. 16. Perez,J.F.,Sandall,O.C.,AIChE J.,l 9(1973)1073. 17. Tadmor,Z.,Gogos,G.,Principles of Polymer Processing,Wiley,New York(1979). 18. Brandrup,J.,lmmergut,E.H. (eds.),Polymer Handbook,3rd ed., Wiley,NewYork(] 989). 19. Domininghaus,H.,Plastics for Engineers:Materials,Properties and Applications,Hanser,Munich(1993). 20. Kulkami,M.G.,Mashelkar,R.A.,Polymer, 22(1981 )867. 21. Duda,J.L. Vrentas,J.S.,Ju,S.T.,Liu,H.T.,A1ChE J.,28(1982)279. 22. Kulkami,M.G.,Mashelkar,R.A., Chem.Eng.Sci.,38(1983)925. 23. ]wai,Y.,Arai,Y.,J.Chem.Eng.Jpn.,22(1989)155. 24. On" Jr.,C.,Dallavalle,J.M.,Chem.Eng.Prog.Symp.Ser.,50,No.9(1954)29. 25. Yareef,B.M.,Colloid J.(USSR),6(1940)545. 26. Skelland,A.H.P., Non-Newtonian Flow and Heat Transfer,Wiley,New York(1967). 27. Bruggemann,D.A.G.,Ann.Phys.(keipzig),24(1935)636. 28. Raiaiah,J.,Andrews,G.,Ruckenstein,E.,Gupta,R.K., Chem.Eng. Sci.,47(1992)3863. 29. Dutta,A.,Mashelkar,R.A.,Adv.Heat Transf., 18(1987)16 I. 30. Feny,J.D.,Viscoelastic Properties of Polymers,Wiley,New York,3rd ed.(1980). 31. Astarita,G.,Mashelkar,R.A.,The Chem.Eng.,Feb(1977)100. 32. Kakac,S.,Shah,R.K.,Aung,W.(editors),Handbook of Single Phase Convective Heat Transfer,Wiley,New York (1987). 33. Lawal,A.,Mujumdar,A.S., Adv.Transp.Processes,5(1987)352. 34. Hartnett,J.P., Kostic,M., Adv.Heat Transf., 19(1989)247. 35. Kays,W.M.,Convective Heat and Mass Transfer,Mc-Graw Hill,New York (1966). 36. Bird'R'B"Annstr~ of Polymeric Liquids,Vol I,Wiley,New York(1977). Ilnd ed. (1987). 37. Grigull,V.,Chem.lng.Tech.,28(1956)553. 38. Skelland,A.H.P., lnd.Eng.Chem.Fundam.,6(1967)148. 39. Rohsenow,W.M.,Choi,H.,Heat,Mass and Momentum Transfer,Prentice Hall,Englewood Cliffs,NJ(1961 ).
1482
Inman,R.M.,NASA Tech. Note D-2674 (1965). Matsuhisa,S.,Bird,R.B.,AIChE J.,l 1(1965)588. Lyche ,B. C. ,Bird,R .B.,Chem.Eng.Sci.,6( 1956)35. Sestak,J.L.,Charles,M.E.,Chem.Eng.Prog.Symp.Ser.,64,No.82(1968)212. Joshi,S.D.,Bergles,A.E.,AIChE J.,27(1981 )872. Johnston,P.R., Mathl.Comput.Modelling,19(1994)1. Tonini,R.D.,Lemcoff,N.O., lnt.Comm.Heat MassTransf.,8(1981)425. Faghri,M.,Welty,J.R.,AIChE J.,23(1977)288. Pigford,R.L., Chem.Eng.Prog.Symp.Ser.,51(1955)79. Wissler,E.H.,Schechter,R.S.,Chem.Eng.Prog.Symp.Ser.,55 (1959)203. BlackwelI,B.F.,J.Heat Transfer (ASME),107(1985)466. Johnston,P.R.,Int.J.Heat Mass Transf.,34(1991 ) 1209. 52. Moudachirou,l.,Lebouche,M.,Devienne,R.,Wamle-und Stoffubertragung,29( 1994)243. 53. Metzner,A.B., Vauglm,R.D., Houghton,G.L.,AIChE J.,3(1957)92. 54. Christiansen,E.B.,Craig,S.E.,AIChE J.,8(1962)154. 55. Hanks,R.W.,Christiansen,E.B.,AIChE J.,7(1961 )519. 56. Chrisliansen,E.B., Jensen,G.E., AIChE J., 15(1969)504. 57. Christiansen,E.B., Jensen,G.E., Tao,F.S., AIChE J.,l 2(1966)1196. 58. Fonest,G.,Wilkinson,W.L.,Trans.lnst.Chem.Eng.,51 (1973)331. 59. Forrest,G.,Wilkinson,W.L.,Trans.lnst.Chmn.Eng.,52(1974)10. 60. Kwant,P.B., Zwaneveld,A.,Dijkstra,F.C.,Chem.Eng.Sci.,28(1973)1303. 61. Mahalingam,R.,Coulson,J.M.,Can.J.Chem.Eng.,53(1975)589. 62. Mahalingam,R., Tilton,L.O., Coulson,J.M., Chem.Eng.Sci.,30(1975)921. 63. Toor,H.L., AIChE J.,4(1958)319. 64. Gill,W.N.,Appl.Sci.Res.,l 1A(1962)10. Also see AIChE J.,8(1962)137. 65. Tyabin,N.V.,Shishlyannikov,V.V.,Dachin,O.H.,Torner, R.V.,Sth Int. Heal Transf. Conf.,2( 1974)213. 66. Richardson,S.M., Int.J.Heat Mass Transf.,22(1979) 1417. 67. Shih,Y.P.,Tsou,J.D., Chem.Engmg.Sci.,33(1978)55. 68. Glyglaszewsky,P.,Nowak,Z.,Stacharska-Targosz,J.,Wanne-und Stoffilbertragung, l 4(1980)81. 69. Dinh,S.M.,Annstrong,R.C.,AIChE J.,28(1982)294. 70. Manglik,R.M., Prusa,J., J.Thennophysics Heat Transf.,9(1995)733. 71. Prusa'J"Manglik'RM"NumHeal Transf.,26A(1994) 199. 72. Hammad,K.J.,Vradis,G.C.,Inl.J.Heal Mass Transf.,39(1996)1555. 73. Vradis,G.C.,Dougher,J.,Kumar,S., lnl.J.Heal MassTransf.,36(1993)543. 74. Caweau,P.J.,DeKee,D.,Chhabra,R.P.,Rheology of Polymeric Systems: Principles and Applications,Hanser,Munich(1997). 75. Melzner,A.B.,Adv.Heat Transf.,2(1965)357.
40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.
1483
76. Scheele,G.F.,Green,H.L.,AIChE J.,l 2(1966)737. 77. DeYoung,S.H.,Scheele,G.F.,AIChE J., 16(1970)712. 78. Gori,F.,J.Heat Transf.(ASME),100(1978)220. 79. Gori,F.,Int.J.Heat Mass Transf.,21(1978)247. 80. Mamer,W.J.,McMillen,H.K., Chem.Eng.Sci.,27(1972)473. 81. Gee,R. E. ,Lyon,J.B.,Ind.Eng Chem.,4 9( 1957)959. 82. Hirai,E.,AIChE J.,5(1959)130. 83. Schechter,R.S.,Wissler,E.H., Nucl.Sci.Engng.,6(1959)371. 84. Henning,C.D.,Yang,W.J.,Appl. Sci.Res., 18(1968)336. 85. Dakshinamurty, V.,Chem.Engng.Commun., 126( 1993)127. 86. Schenk, J.,van Laar, J.,Appl.Sci.Res., 7(1959)449. 87. McKillop,A.A., Int.J.Heat Mass Transf.,7(1964)853. 88. Atkinson,B.,Goldstein,S.,Moden Developments in Fluid Dynamics,Oxford Uni 9Press, Oxford(1938). 89. McKillop,A.A., Harper,J.C.,Bader,H.J.,Korayem,A.Y.,lnt.J.Heat Mass Transf., 13(1970)901. 90. Samant,A.B.,Mamer,W.J.,Nucl.Sci.Engng.,43(1971 )241. 91. Lin,T.,Shah,V.L.,6th Int.Heat Transf.Conf.,5(1978)317. 92. Victor,S.A.,Shah,V.L., Int.J.Heat MassTransf.,l 9(1976)777. 93. Oliver,D.R.,Jenson, V.G., Chem.Eng.Sci.,l 9(1964)115. 94.Mizushina,T.,lto,R.,Kuriwake,Y.,Yahikazawa,K.,KagakuKogaku,31 (1967) 250. 95. Sylvester,N.D.,Rosen,S.L., AIChE J.,l 6(1970)697. 96. Bassett,C.E.,Welty,J.R.,AIChE J.,21 (1975)699. 97. Scirocco, V. ,Devienne,R.,Lebouche,M.,Int. J.HeatMa ssTrans.,28( 1985)91. 98. Deshpande,S.D.,Bishop,A.A.,Chem.Engqag.Commun.,52(1987)339. 99. Filkova,l.,Lawal,A.,Koziskova,B.,Mujumdar,A.S., J.Food Engng., 6(1987)143. 100. N guyen, V.T.,Ouardighi,A.E.,Lebouche,M., C.R.Acad. Sci.Paris,t.319,SerieIl(1994)983. 101. Lin,C.-X.,Ko, S.-Y.,Tsou,F.K.,Int.J.Heat Mass Transf.,39(1996)503. 102. Chann,S.E.,lnd.Eng.Chem.Fundanl.,l (1962)79. 103. Matthys,E.F.,Sabersky,R.H., lnt.J.Heat Mass Transl.,31 (1988) 1913. 104. Ng,K.S.,Hartnett,J.P.,Tung,T.T.,17th Heat Transf.Conf. (1977)p.74. 105. Bird,R.B.,Chem.lng.Tech.,31 (1959)569. 106. Mehta,D.C., DOE Report # DOE/OR/03054-58(DE84013792),Int.Coal Refining Co.,Allentown,PA(1984). 107. Metzner,A.B., Friend,P.S.,Ind.Eng.Chem.,51 (1959)879. 108. Dodge,D.W.,Metzner,A.B.,AIChE J.,5(1959)189. 109. Yoo,S.S.,PhD Dissertation,Uni.illinois at Chicago Circle,IL(1974).
1484
11 0 .Kawase, Y., Shen oy,A. V., Wak abay sh i,K., Can .J. Chem. Eng., 72 ( ! 994 ) 798. II 1. Kawase,Y.,Moo-Young,M.,lnt. Comm.Heat Mass Transf., 19(1992)485. 112. Thomas,D.G., AIChE J.,6(1960)631. ll3. Quader,A.K.M.A.,Wilkinson,W.L.,lnt.J.Multiphase Flow,7(198! )545. 114. Dimant,Y.,Poreh,M.,Adv.Heat Transf., 12(1976)77. ll5. Kostic,M., Int.J.Heat Mass Transf.,37(suppl. 1) (1994)133. 116. Payvar,P.,Appl. Sci.Res.,27(1973)297. 117. Tien,C., Can.J.Chem.Engng.,40(1962) 130. 118. Vlachopoulous,J.,Keung,C.K.J., A1ChE J., 18(1972) 1272. 119. Suckow,W.H.,Hrycak,P.,Griskey,R.G.,Polym.Engng.Sci., 11 (1971 )40 I. 120. Ybarra,R.M.,Eckerl,R.E., AIChE J.,26(1980)751. 121. Dang, V.-D.,Can.J.Chem.Engng.,58( 1980)401. 122. Dang,V.-D., J.Heat Transf.(ASME),105(1983)542. 123. Kwant,P.B.,Van Ravenstein,T.N.M.,Chem.Eng. Sci.,28(1973) 1935. 124. Shulman,Z.P.,Zaltsgendler,E.A.,4th lnt.Heat Transf.Conf.,4(1970)Rh 7. 125. Yau,J.,Tien,C., Can.J.Chem.Engng.,41 (1963) 139. 126. Chandraputla,A.R.,Sastri,V.M.K.,lnt.j. Heat Mass Transf.,20(1977) 1315. 127. Chandraputla,A.R.,Sastri,V.M.K., 6th lnt.HeatTransf.Conf.,5(1978)323. 128. Chandraputla,A.R.,Sastri,V.M.K., Num.Heat Transl., 1(1978)243. 129. Lawal,A.,Mujumdar,A.S.,Num.Heat Transf.,8(1985)217. 130. Gao,S.X.,Hartnett,J.P.,Int. Commun .Heat Mass Tran sf., 19( 1992)673. 131. Gao,S.X.,Hartnett,J.P.,Int.Commun.Heat Mass Transf.,20(1993) 197. 132. Gao,S.X.,Hartnett,J.P.,Int.J.Heat Mass Transf.,39(1996) 1317. 133. Gingrich,W.K.,Cho,Y.l. Shyy,W.,Int.J.Heat Mass Transl.,35(1991 ) ] ] 95. 134. Shin,S.,Cho,Y.]., Gingrich,W.K.,Shyy,W., lnt.J.Heat Ma ssTran sf. ,36( 1993 )4365. 135. Hartnett,J.P., Kostic,M.,Int.J.Heat Mass Transf.,28(1985)1147. 136. Hartnett,J.P., Xie,C.,Zhong,T.,Int.Conf.Heat Transf. & Energy Conserv.,Shenyang,PRC,October (1988). 137. Xie,C.,Hartnett,J.P., Ind.Eng.Chem.Res.,31 (1992)727. 138. Shin,S.,Cho,Y.l., Int.J.Heat Mass Transf.,37(suppl. 1)(1994)19. 139. Kostic,M.,Hartnett,J.P.,ZAMM,66(1986)239. 140. Rao,B.K.,Exp.Heat Transf.,2(1989)201. 141. Cheng,J.A.,PhD Disserlation,SUNY,Stony Brook,NY(1984). 142. Lawal,A.,Ph D Dissertation,McGill Universily,Montreal,Canada(1985). 143. Lawal,A.,Mujumdar,A.S., Warme-und - Stoffubellragung,27( 1992)437. 144. Etemad,S.G.,Mujumdar,A. S.,Nassef, R.,Appl.Math.Modell.,20(1996)898. 145. Manglik,R.M.,Ding,J., lnt.J.Heat Mass Transf.,40(1997) 1379. 146. Jarzebski,A.B.,Wilkinson,W.L.,J.Non-Newt.Fluid.Mech.,8(1981 )239. 147. Christiansen,E.B., Kelsey,S.J.,Chem.Eng. Sci.,28(1973) 1099.
1485
148. Rao,B.K.,Exp.Heat Transf.,6(1993) 189.
149. Pak,B.,Cho,Y.I.,Choi,S.U.S., lnt.J.Heat Mass Transf.,34(1991 ) 1195. 150. Wronski,S.,Jastrzebski,M., Int.J.Heat MassTransf.,33(1990) 1. 151. Kozicki,W.,Tiu,C.,Can.J.Chem.Engng.,45(1967) 127. 152. Dunwoody,N.T.,Hamill,T.A.,ZAMP,30(1979)587. 153. Dunwoody,N.T.,Hamill,T.A., lnt.J.Heat Mass Transf.,23(1980)943. 154. Oliver,D.R.,Trans.Inst.Chem.Engns.,47(1969)T18. 155. Oliver,D.R.,Karim,R.B., Can.J.Chem.Engng.,49(1971 )236. 156. Oliver,D.R., Asghar, S.M., Trans.Inst.Chem.Engrs., 54(1976)218. 157. Mena,B.,Best,G.,Bautista,P.,Sanchez,T.,Rheol.Acta, 17(1978)454. 158. Lawal,A., J.Heat Transf.(ASME) 111(1989)399. 159. Naccache,M.,Souza Mendes,P.R.,lnt.J. Heat Fluid Flow, l 7(1996)613. 160. Naccache,M.,Souza Mendes,P.R., J.Thermophysics Heat Transf., 11(1997)98. 161. Shenoy,A.V.,Mashelkar,R.A.,Adv.Heat Transf.,15(1982)143. 162. Chhabra,R.P.,Bubbles,Drops and Particles in non-Newtonian Fluids,CRC Press, Boca Raton,FL(1993). 163. Chhabra, R.P.,Adv.Heat Transf.,23(1993) 187. 164. Chhabra,R.P.,Adv.Transport Processes,9(1993)501. 165. Ghosh,U.K., Chhabra,R.P.,Upadhyay,S.N.,Adv.HeatTrans.,25(1994)251. 166. Schowalter,W.R., AIChE J.,6(1960)24. 167. Bizzell,G.D.,Sl attery,J. C.,Chem.Eng. Sci., 17( 1962) 177. 168. Na,T.Y.,Hansen,A.G., lnt.J.Heat Mass Transf.,9(1966)261. 169. Lee,S.Y.,Ames,W.F.,AIChE J.,12(1966)700. 170. Chen,J.L.S.,Radulovic,P.T.,J.Heat Transf(ASME).,95( 1973)498. 171 Denn,M.M.,Chem.Eng. Sci.,22(1967)395. 172. Serth,R.W., AIChE J.,19(1973)1275. 173. Verlna,R .L.,Rheol.Acta, 16( 1977) 510. 174. Acrivos,A.,Shah,M.J.,Petersen,E.E.,AIChE J.,6(1960)312. 175. Acrivos,A., Shah,M. J. ,Pet ersen,E .E.,Chem .Eng. Sci. ,20( 1965) 101. 176. Nakayama,A.,Koyama,H., Warme-und-Stoffubertragung,20(1988)29. 177. Nakayama,A., Shenoy,A.V., Int.J.Heat Fluid Flow, 12(1991)336. 178. Nakayama,A., Shenoy,A. V., J.Heat Transf.(ASME), 114(1992) 127 179. Acrivos,A.,AIChE J.,6(1960)584. 180. Stewart,W.E., Int.J.HeatMassTransf., 14(1971 ) 1013. 181. Liew,K. S.,Adelman,M .,Can. J. Ch era. En gqlg.,53( 1975)494. 182. Amato,W.S.,Tien,C.,Int.J.Heat Mass Transl.,19(1976)1257. 183. Alhamdan,A.,Sastry,S.K.,J.Food Process Engng., 13(1990) 113. Process Engng., 184. Awuah'G'B"Ramaswamy'H'S"Simps~176176 16(1 993)39.
1486
185. Lee,T.-L., Donatelli,A.A.,lnd.Eng.Chem. Res.,28(1989) 1056.
186. Churchill,S.W.,Chem.Engng.Commun.,24(1983)339. 187. Westerberg,K.W.,Finlayson,B.A.,Num.Heat Transfer, l 7A(1990)329. 188. Shanna,O.P.,Bhatnagar,R.K.,ZAMM,55(1975)235. 189. Kawase,Y., Mashelkar,R.A., Uibrecht,J.J.,Int.J.Multiphase Flow, 8(1982)433. 190. Ogawa,K.,Kuroda,C.,lnoue,l.,J.Chem.Engng.Jpn., 17(1984)654. 191. Morris,S.,J.Fluid.Mech., 124(1982) 1. 192. Andersson,H.l.,lnt.J.Heat Fluid Flow.,9(1988)343. 193. Wang,T.-Y.,Kleinstreuer,C.,Int.J.Heat Fluid Flow,9(1988) 182. 194. Meissner,D.L.,Jeng,D.R.,DeWitt,K.J., Int.J.Heat Mass Transf., 37(1994)1475. 195. Kawase,Y., Ulbrecht,J.J., Chem.En~nlg.Commun.,8(1981 )213. 196. Lochiel,A.C.,Calderbank,P.H.,Chem.Eng.Sci., 19(1964)471. 197. Moo-Young,M., Hirose,T.,Ali,S.,Proc.5th hat.Con g.Rheol.,Kyoto, 1( 1968)233. 198. Hyde,M.A.,Donatelli,A.A.,Ind.Eng.Chem.Fundam.,22( 1983)502. 199. Carberry,J.J.,AIChE J.,6(1960)460. 200. Mixon,F.O.,Carberry,J.J., Chem.Eng.Sci., 13(1960)30. 201. Kawase,Y., Ulbrecht,J.J., Electrochimica Acta,28(1983)643. 202. Kumar, S.,Mall,B.K.,Upadhyay,S.N.,Lett.Heat Mass Transf.,7(1980)55. 203. Chandarana,D.l.,Gavin,A.,Wheaton,F.W.,J.Food Process En~nag., 13(1990)191. 204. Zuritz,C.A.,McCoy,S.C.,Sastry,S.K.,J.Food Eng.,11(1990)159. 205. Bhamidipati'S"Singh'R'K"Trans ASAE,38(1995)857. 206. Yamanaka,A.,Mitsuishi,N.,Heat Transf.-Jpn.Res.,6(1977)85. 207. Zitoun,K.B.,Sastry,S.K.,J.Food Process Engug.,l 7(1994)209 & 229. 208. Balasubramaniam,V.M.,Sastry,S.K.,J.Food Engng.,23(1994) 169. 209. Gadonna,J.P.,Pain,J.P.,Barigou,M.,Trans.l.Chem.E.,C74(1996)27.
210. Baptista,P.N.,Oliveira,F.A.R.,Oliveira,J.C.,Sastry,S.K.,J.Food Engng.,31 (1997)199. Also see ibid 125. 211 .Ghosh,U.K.,Kumar, S.,Upadhyay,S.N.,Polym.Plast.Tech.Eng.,31 (1992)271. 21 2. Keey,R.B.,Mandeno,P.,Tuoc,T.K.,l.Chem.E. Sym.Ser.,33(1971 )53. 21 3. Lal,P.,Upadhyay,S.N.,Chem.Eng. Sci.,36(1981) 1865. 21 4. Ng,M.L., Hartnett,J.P., lnt.Comm.Heat Mass Transf.,l 3(1986)115. 21 5. Gorla,R.S.R.,Polym.-Plast.Teclmol.Eng.,30(1991 )37. 21 6. Gentry'CC"W~ Transf.(ASME),96(1974)3. 21 7. Ng,M.L., Hartnett,J.P., lnt.J.Heat Mass Transf.,31 (1988)441. 21 8. Ng,M.L., Hartnett,J.P., J.Heat Transf.(ASME), 108(I 986)790. 21 9. Ng,M.L., Hartnett,J.P., Int.Comm.Heat Mass Transf.,l 5(1988)293.
1487
220. Lyons,D.W.,White,J.W.,Hatcher,J.D.,Ind.Eng.Chem.Fund.,l 1(1972)586. 221. Shenoy,A.V.,Mashelkar,R.A.,Chem.Engng.Sci.,33(1978)769. 222. Chhabra,R.P.,Proc. 14th Nat.Heat & Mass Transf. Conf.,Kanpur (1997)in press. 223. Shah,M.J.,Petersen,E.E.,Acrivos,A., AIChE J.,8(1962)542. 224. Kim,B.K.,Lee,H.S.,Korean J.Chem.Eng.,6(1989)227. 225. Wolf,C.J.,Szewcyzk,A.A.,Proc.3rd Int.Heat Tranf.Conf. ,Chicago,1( 1966)388. 226. Mizushina,T., Usui,H.,Heat Transf.-Jap.Res.,7(1978a)83. 227. Mizushina,T., Usui,H., Kagaku Kogaku Ronbunshu,4(1978b)166. 228. Schlichting,H.,BoundaryLayer Theory,McGraw Hill,New York(1968). 229. Dienemann,W.,ZAMM,33(1953)89. 230. Shenoy,A.V., AIChE J.,26(1980)505. 231. Wang,T.-Y.,Kleinstreuer,C., J.Heat Transf(ASME).,l 12(1990)939. 232. Mizushina,T., Usui,H.,Kato,T., Kagaku Kogaku Ron.,4(1978)173. 233. Mizushina,T., Usui,H.,Kato,T., Kagaku Kogaku Ron.,4(1978)214. 234. Luikov,A.V.,Shuhnan,Z.P.,Puris,B.l.,Heat Transf.Sov.Res., 1(1969) 121. 235. Luikov,A.V.,Shuhnan,Z.P.,Puris,B.I., Zhdanovich,N.V.,Prog.Heat Mass Transf.,2(1969)262. 236. Takahashi'K"Maeda'M"lkai'S"Preprint 14th Heat Transf.Sym.Jpn. (1977)A-306. 237. Kumar,S., Tripathi,P.K., Upadhyay,S.N.,Lett.Heat MassTransf., 7(1980)43. 238. Ghosh,U.K.,Gupta,S.N.,Kumar,S.,Upadhyay,S.N., Int.J.Heat Mass Transf.,29(1986b)955. 239. James,D.F.,Acosta,A.J.,J.Fluid Mech.,42(1970)269. 240. James,D.F.,Gupta,O.P.,Chem.Eng.Prog.Sym.Ser.,67,No.111 (1971 )62. 241. Hoyt,J.W.,Sellin,R.H.J.,Exp.Heat Transf.,2(1989)113. 242. Ruckenstein,E., Ramgopal,A., J.Non-Newt.Fluid Mech.,l 7(1985) 145. 243. A1Taweel,A.M.,Sedahmed,H.G.,Abdel-Khalik,A.,Farag,H.A., Chem.Eng.J., 15(1978)81. 244. Kawase,Y., Ulbrecht,J.J., J.Appl.Electrochem.,13(1983)289. 245. Garg,N. S.,Tripathi,G.,Ind.J.Technol., 19(1981) 131. 246. Fuj ii,T.,Bull.JSME.,2(1959)365. 247. Reilly,I.G.,Tien,C.,Adehnan,M.,Can. J. Chenl.Engqlg.,43(1965)157. 248. Huang,M._j.,Chen,C.K.,int.j.Heat Mass Transf.,33(1990)119. 249. Shenoy,A.V.,Ulbrecht,J.J.,Chem. Engng.Commun.,3( 1979)303. 250. Pittman'J'F'T"Richards~ Int.J.Heat Mass Tran sf.,37(suppl. 1)( 1994)333. 251. Mishra,l.M., Singh,B.,Mishra,P.,lndian J.Technol.,14(1976)322.
1489
A ONE-DIMENSIONAL M O D E L FOR VISCOELASTIC DIFFUSION IN POLYMERS C.J. Durning 1, P.H. Tang I and R.A. Cairncross z 1Department of Chemical Engineering and Applied Chemistry, Columbia University, New YorkNY 10027 2Department of Chemical Engineering, Drexel University, Philadelphia PA 19104
1. I N T R O D U C T I O N Very complex motions can result when a viscoelastic fluid is put under external forces. In general, for a given set of forces one has to solve continuity, momentum and constitutive equations simultaneously to determine the details. The majority of effort along these lines has focused on incompressible media, which execute isochoric motions; viscoelastic flows with a strong dilational component have not received as much attention, primarily because there are relatively few applications demanding their study. One situation involving viscoelastic media where the resulting flow is dominated by the dilational component is inter-diffusion. Here, flow occurs in a mixture with relative motion between components, driven by a combination of thermodynamic and mechanical forces. The simplest case involves just two components with no applied mechanical tractions; the relative motion is mutual diffusion, generally resulting in flow of each component dominated by dilation. We discuss this situation for one-dimensional motions in the case of a binary mixture with one viscoelatic component and the other an ordinary viscous fluid. A quantitative understanding of mutual diffusion in concentrated polymer/viscousfluid mixtures is of great practical importance, in order to understand and control a number of polymer processing operations and end-use applications. For example, it
1490
is a controlling physical process in devolatilization, fiber spinning, coating, drying, microlithography and several control release schemes; accurately modeling the process is essential to the design and control of these operations. Experimental studies indicate that unsteady, one-dimensional diffusion in concentrated polymer/viscous-fluid mixtures ranges from ordinary Fickian behavior to strongly non-Fickian depending on the conditions (see, for example, references [ 1] -[4] ). At temperatures far above the glass transition (greater than 50~ above Ta), diffusion follows Fick's law on all accessible length scales, reminiscent of the behavior in mixtures of simple viscous fluids. However, at lower temperatures and high polymer densities, the dynamics begin to deviate from Fickian on small length scales. Dramatic non-Fickian effects appear on relatively large scales throughout the concentrated regime, near and below the glass transition. It is known that the polymer's viscoelasticity, causes the deviations from Fickian response [5, 6], hence the non-Fickian diffusion in such cases is often called viscoelastic diffusion. In this work we demonstrate the capabilities of a relatively simple one-dimensional model to account for the most obvious viscoelastic, non-Fickian effects observed experimentally. This model allows a straitforward calculation of the diffusioninduced flow of the polymer component during mutual diffusion. Mutual diffusion in concentrated polymer/viscous-fluid mixtures is often studied by a vapor sorption experiment. Since this technique is the main focus of our modeling effort we describe it briefly, along with key experimental findings. The mixture is confined in a thin film, initially in equilibrium with a large reservoir of the vapor of the viscous fluid at activity a - . At time t = 0 the activity is suddenly increased to a +, driving the system to a new equilibrium, with kinetics controlled by diffusion into the mixture. This causes "swelling" or distention of the film in the lateral direction, corresponding to longitudinal flow of polymer in the mixture. The kinetics are usually tracked by measuring the evolution of concemration profiles or by following the mass of the mixture as a function of time. For a "differential" or "interval" sorption, a + - a - << 1 so that a very small disturbance from the initial state occurs. For "integral" sorption, a - -- 0 and a + ~ O(1) so that a relatively large disturbance from the initial state results. During an integral sorption, the transport properties can vary considerably, since the local composition usually changes significantly, so the response is typically non-linear. This may or may not be true for a differential sorption; it depends on how much the composition changes as
1491 a result of the incremem a + - a - . In principle, if a + - a - is small enough, linear response is observed. Of all the results from sorption documented, two stand out as unequivocal evidence for viscoelastic behavior during diffusion in polymer/viscous-fluid systems: Twostage sorption, observed in differential sorptions on thin films of concentrated solutions, and Case II transport, observed during imegral sorption of strong swelling agents and solvents in films of dry, glassy polymer. Odani et al. [7-9], Billovits and Durning [10, 11] and Tang et al. [12] carried out differential sorptions on thin films (2 - 10#m) of concentrated polystyrene solutions over a range of compositions around the glass transition (T9) for the mixture. Twostage sorption was observed in all three studies just below Tg 9 The fluid mass absorbed during sorption increased in two distinct steps. This effect seems to result from an initial elastic longitudinal swelling of polymer in the mixture to the applied thermodynamic driving force, followed by a protracted relaxation, controlled by the same microscopic mechanisms involved in the viscoelastic mechanical response of the mixture. The other benchmark observation is of Case II transport. Here, a relatively thick (~ O ( l m m ) ) , dry, glassy plate is suddenly exposed to an interactive fluid at near unit activity. The most widely studied systems are alchohols in poly(methyl methacrylate) and alkanes in polystyrene (see references [ 13] -[ 17] ). A sharp fluid concentration front develops, separating highly distented, uniformly swollen polymer from nearly dry, glassy polymer; the front propagates into the sample at constant speed. Hence, Case II differs radically from ordinary diffusion, where disturbances in concentration travel with t 1/2 and spread with time. It is clearly a non-linear effect, since it is not seen for the same systems in linear perturbation experiments, such as differential sorption. In principle, it should be described by the same theory adequate for linear response, i.e. for two-stage sorption, but including key nonlinearities. A number of years ago we derived from thermodynamic arguments a linearized, one-dimensional model for mutual diffusion in concentrated polymer/fluid mixtures, which includes the effect of viscoelasticity [18] . In this work, the model is extended to nonlinear cases by introducing composition dependencies in the dominant relaxation time of the mixture and in the mutual diffusion coefficient. There are four material parameters in the model, which are evaluated for two systems where careful sorption experiments have been carried out and sufficient auxiliary data exists: Polystyrene -ethylbenzene (PS-EB) and poly(methylmethacrylate)-methanol
1492
(PMMA-MeOH). The model's predictions are compared with observations in both systems, of two-stage sorption in the PS-EB system for differential sorption in thin films just below Tg, and of Case II sorption in the PMMA-MeOH system for integral sorption in relatively thick films at room temperature. Because of the practical importance of the Case II process, in microlithograpy (see reference [19] , for example) and in controlled release technologies (see reference [20], for example), we also present a systematic numerical study of the effect of the four material parameters in the model on the Case II process.
2. MATHEMATICAL MODEL 2.1 Diffusion Equation and Auxiliary Conditions for Sorption We consider one-dimensional sorption in a thin film surrounded by an infinite fluidvapor reservoir. The film occupies 0 _< x _< l in the dry state along a lab-fixed coordinate, x. During sorption, the fluid diffuses into the film along x, distending the polymer outward along the same direction. The one-dimensional kinematics and an assumption of no volume change on mixing [ 18] allow one to analyze the process in terms of the fluid concentration in the film, governed by a diffusion equation derived from the equation of continuity for the fluid and a constitutive law for the diffusion flux; momentum balances and the polymer continuity equation are not needed. Using "polymer material" coordinates [21 ] the fluid continuity equation is:
OC = ot
OJx ox
(1)
where X, C, and J x mean position, concentration and liquid flux in the material system. The expression for J z derived in reference [ 18] is
Jx -- - D ( C ) ~ x
0 - D'(C)-~-~g(C)
r
t'
OC' _, ;C')-~dt .
(2)
Here, r characterizes the linear viscoelastic response of the mixture in shear; the notation r t'; C') means that it is a functional of the concentration history between t and t'. 9(C) - G(C)/Go with G(C) being the instantaneous shear modulus of mixture and Go being G(C - 0), D(C) - D12(pzV2) 2 with D12 being the binary mutual diffusion coefficient and
D'(C) - D ( C ) V I ( 0 f / 0 w 1)-lp~GO/(RTw2).
(3)
1493
In equation (3), R T f is the e__quilibrium (i.e. thermostatic) fluid chemical potential, relative to the pure fluid, Vi (V/) means the partial molar (specific) volume of component i (i = i means fluid, i = 2 means polymer). Substituting equation (2) into equation (1) gives the diffusion equation for C. Note that D, D t, 9 and 05 all depend on C, which introduces nonlinearities, making solution of the diffusion equation challenging. One should note that the fluid concentration C is related to the local dilation of the polymer during sorption, relative to its initial state" The Jacobian of the polymer deformation with the dry state as reference is CV1 + 1. Hence, one can directly calculate the entire polymer flow history from the concentration field. It turns out that in order to use certain time-integrator software packages to solve the diffusion equation numerically, one needs to eliminate the time derivative inside the memory integral in equation (2), using an integration by parts, which permits replacement of the integral by C - q~(t, 0; C')Co
-
fo
t -~ 0r (t, t' ; C' )C'dt.
(4)
where Co is the initial fluid concentration in the polymer film; for an initially dry film Co equals zero. Boundary and initial conditions need to be specified in order to solve for C. For the boundary conditions, at the interfaces between the mixture and the reservoir, we assume continuity of the fluid chemical potential, #1, which can be written in terms of the fluid activity in the reservoir, tfl, as In ~ - In
Cgq + Q
c/)(t,t'; C') dCb dt' oo
(5)
where Cb means the boundary concentrations, C(X = 0, 1). The second term on the right in equation (5) corresponds to a relaxing (nonequilibrium) contribution to the fluid's chemical potential in the mixture. The first term on the right corresponds to the choice f - ln(C/C~q) for the thermostatic potential, implying the approximation of Henry's law for the equilibrium isotherm between the vapor reservior and the condensed mixture. C~q is a constant corresponding to the fluid concentration in the film at equilibrium with a reservoir at unit activity and Q - ( 1c~- ~ ) ( ~1 ) , where c~ is explained below. Equation (5) proves awkward with certain integration schemes. Integration by parts in equation (5) gives and alternative form
1494 Cb In r - In C Oq+ Q [r
t t';C;)Cs o - Q
r
,t'
' ' ' ; Cb)Cbdt
(6)
which requires an initial value for Cb. Taking the limit for t ~ 0 gives a ( cb In Cb + c or 1 - ~ Cgq
Co ) - - 0 co
(7)
defining the initial value. So far, the development can accommodate any viscoelastic model. Hereafter, we assume single exponential relaxation with a concentration dependem relaxation time t dr"
r
c') - - ~ s,, . - ~ .
(8)
In order to include the most important nonlinearities in the simplest way, we adopt the following assumptions, which approximate experimental results in the concentrated regime. (i) The relaxation time decreases exponentially with concentration: T(C) = TO exp - m C
(9)
where m is a constant, and TO is the relaxation time for the dry polymer. (ii) The diffusion coefficient increases exponentially with concentration: D ( C ) = Do exp k C
(10)
where k is a constant and Do is the diffusion coefficient of the dry polymer. (iii) The instantaneous shear modulus of mixture, G, is independent of concentration, which implies g(c) - 1. Equation (3) then gives: D ' ( C ) " D~o(C/C~
(ll)
expkC
with (12)
D~ -- DoGoV1 ~ p 2 V 2 C ~
2.2
Scaling
Introduce the dimensionless quantities u -
C
COeq ;
x--
X
T;
8-
t(Do + D~o) 12
;
K-
kC~q
;
M - mC ~
eq"
(13)
1495
where l is the dry film thickness. After substituting the expressions for D(C), "r(C), and r C'), and using equation (4), the diffusion equation becomes:
Ou 0 Os -OI - Ox (1 - a + au) exp K u OuO--x aOaO . x u exp K u Ox with
I -- Ouo exp +
(14)
(/0 s0 exp~- M u ~) -
Jo~ exp(Mu')
exp
( I- , ~~ 0 e~'' x: M~ u " / u_' d s '
(15)
where a --
D6 D~ + Do'
0 - T0(D0 + D6) 12
(16)
are dimensionless constants. Equation (6) becomes
~ ( Ub-o)
lnr where
~
(17)
C~/C;~
--
For numerical solutions, it is easier to work with differential equations. Equation (15) is the solution of
OI I 0---~= exp M u ( u - ~).
(18)
Here, 0I has an important physical meaning. It represents the dimensionless nonequilibrium contribution to the local fluid chemical potential. Differentiation of equation (17) gives
dub = _
d~
Ub exp Mub
In u~.
0[~ + (1---z%)~]
r
(19)
The initial conditions on u are
u--uo
@ t--O forO<x
< l.
(20)
The initial conditions needed for I are
I(x,s-O)-O
for O<x<
l.
(21)
1496
The initial conditions at the boundaries can be obtained from the scaled form of equation (7). The experimentally measurable relative fluid uptake, W(s), can be calculated from
W(s) -
Jo.1(u(z, s) -
uo)dz.
(22)
Caimcross and Duming [22] derived an equivalent dimensionless representation, based on equation (2), i.e. integration of the memory integral by parts (equation (4)) was not done. Somewhat different boundary conditions were employed in that study (see subsequent discussion).
3. N U M E R I C A L M E T H O D S We aimed to solve equations (14)-(21) for the broadest possible range of the dimensionless parameters c~, 0, K, M, uo and ~b. In the Case II limit, steep moving concentration fronts are expected. This demands a numerical scheme able to handle "stiff" problems effectively. Consequently, finite element spatial discretization, together with an ODE integrator designed to handle stiff initial value problems was the approach chosen. Two different finite-element schemes were used. The first employs orthogonal collocation. The details are more or less the same as described by Fu and Duming [23] and Tang et al. [ 12]. A solution is constructed by discretizing space into equal subintervals and approximating the spatial dependence with piece-wise continuous Hermite cubic polynomial basis functions. Orthogonal collocation is then applied to each subinterval to convert the PDE system into a coupled system of ODEs for the time dependent coefficients in the polynomial approximations to u(z, s) and I(z, s). We used LSODI [24] for the ODE solver, which requires that the system be in linearly implicit form; equations (14) and (18), employing equation (4), collapse to a linearly implicit ODE system when discretized. The second scheme is that discussed by Caimcross and Durning [22] . The dimensionless form of the diffusion equation retaining the memory integral in equation (2) is integrated directly by Galerkin's method, i.e. equation (4) is not used, using quadratic basis functions and a differential/algebraic equations systems solver, DASSL [25] . Actually, the boundary condition used with this scheme at the reservoir/mixture interface is somewhat different from equation (19): A "surface evaporation" condition is imposed where the external fluid flux is put proportional
1497
to the fluid activity drop between the film surface and the bulk phase and set equal to the flux in the film at the surface, given by equation (2). The proportionality constant appearing in the external flux at the surface is a mass transfer coefficient; for very large values of the mass transfer coefficient, the surface evaporation condition gives that the surface and bulk phase fluid activities nearly match, i.e. it reduces to our surface condition, equation (19). In fact, calculations based on the two schemes agree closely when the (dimensionless) external mass transfer coefficient in the second scheme (a Biot number) is set to large values ( ~ O(103)) and all other model parameters are set equal (see below). Values of the Biot number ~ O(103) were used for all of the calculations by the Galerkin scheme.
4. N U M E R I C A L RESULTS There are six dimensionless parameters in the model, a, 0, K, M, u0 and ~b. The first four characterize the polymer-fluid system, while the last two define the driving force for the sorption. For all the calculations, we fixed the number of subintervals at 30. As a basic benchmark we solved the case a - 0, K - M - 0 and recovered linear Fickian diffusion, as expected. By trial calculation, we found that the numerical solutions were most suseptible to instabilities for integral sorption (u0 -- 0.0 and = 1.0). In this case, when a is near 0, K and M must be less than 4 in order to avoid instability of the collocation scheme. When a is away from 0, M can be increased to 5, while K can go up to 12 before numerical instabilities occur with collocation. The code based on Galerkin's method enjoyed a wider range of stability in parameter space for integral sorptions than the one based on collocation, permitting values of both M and K up to 12 for a wide range of a and 0. A comparison of concentration profiles predicted by the collocation and Galerkin schemes for the demanding case of integral sorption near the Case II limit (a = 0.9, 0 = 0.01, K = 5.0, M = 5.0, u0 = 0.0, and~b = 1.0)showed excellent agreement between the two codes. Subsequent remarks refer to results from collocation, except where noted.
4.1 Predictions of Two Stage Sorption and Case II Transport Billovits and Durning [10, 11] reported a careful set of differential sorption experiments in the poly(styrene)-ethylbenzene (PS-EB) system. They clearly observed non-Fickian, viscoelastic effects, including two-stage sorption, just below Tg. The first four model parameters ~, 0, M and K can be calculated for the PS-EB
1498
system from published data; the details are given in Appendix D of Tang [26]. For a 5# thick PS film, at 40~ we find c~ -~ 0.97, 0 ~ 432, M ~ 3 5 , / ( ~ 17. To model the conditions where two-stage weight uptake was observed, we picked u0 - 0.03, and ~ - 0.1, which corresponds to run R4 in reference [10] . This choice corresponds to the solvent mass fraction starting at w I = 0.018 and ending at w~- - 0.059, somewhat too large of a change for purely linear response, i.e. the response is weakly non-linear. Figure (la) shows u(x, s) while figure (lb) shows the corresponding dimensionless weight uptake, W(s) vs. x/~, which is clearly a two -stage weight uptake curve. The concentration profiles show that the two-stage sorption process is controlled by the viscoelastic response in surface concentration, which which was first recognized by Long and Richman [27]. Consider next liquid methanol (MeOH) in poly(methylmethacrylate) (PMMA). It is known that Case II transport occurs when liquid methanol contacts a relatively thick ( ~ lmm) PMMA film at room temperature (see reference [ 14], for example). Under these circumstance, u0 -- 0, and ~b = 1, i.e. we are considering an integral sorption. From the physical properties provided by Thomas and Windle [14] we find c~ ~ 0.86, 0 -~ 0.0087. We adopt K = 6.2, as suggested by Thomas and Windle. According to Thomas and Windle, M = 13.0; however, M = 5.0 was used due to the limitation of the collocation scheme. Consider the concentration profiles predicted for this system, shown in figure (2a). Note two key features, characteristic of Case II [ 14-16]. First, the surface concentration relaxes with time. Second, a steep concentration front develops and propagates into the film at nearly constant speed after the surface equilibrates. Figure (2b) shows the corresponding (dimensionless) weight uptake, W(s) vs. s. The main, increasing part of the plot is nearly linear, and there is an induction time visible as illustrated by the construction on the figure; both features are additional signatures of the Case II process. Clearly, the predictions for the PMMA-liquid MeOH system capture all the characteristic features of Case II transport noted in previous work. The forgoing calculations demonstrate that the model considered here can predict the two most well documented manifestations of viscoelasticity in sorption experiments: Two-stage uptake in differential sorption and Case II transport in integral sorption. The remaining sections focus on the effect of the material parameters a, 0, M and K on the predictions of the model in the Case II limit, assuming integral sorption, i.e. u0 - 0 and ~b = 1.
1499
0.12
0.10
-~.
0.08 9.08
0.06
0.04
0.02 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.20
0.16
0.12
0.08
0.04 -, 0
,
,
_,,
l_.,
3
, ....
L
.... ,.
|
6
9
,_
,
,
I
9
- J - ~
-~
i
12
S~r2
Figure 1. (a). Concentration profiles (u(z, s)) from the non-linear model with parameters for the PS-EB system studied by Billovits and D u m i n g [ 10] 9a -- 0.97, 0 -- 432, K - 17, M - 35, uo = 0.03, ~b - 0.1; (b). Weight uptake kinetics ( W vs. s) from the non-linear model with the same set of parameters.
1500
.0
_
_
.
rM
0.8
g-
0.6
0.4
24
02
s=O.O 12
O0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0
0.8
0.6
0.4
0.2
0.0 0.00
si
0.01
0.02
0.03
0.04
0.05
0.06
Figure 2. (a). Concentration profiles (u(x, s)) from the non-linear model with parameters for the PMMA-MeOH system studied by Thomas and Windle [14] : c~ = 0.86, 0 = 0.0087, K = 6.2, M = 5, Uo = 0.0, ~ = 1.0.; (b). Weight uptake kinetics ( W vs. s) from the non-linear model with the same set of parameters.
1501 4.2 Surface Concentration in the Case II Limit It is a well-known feature of Case II transport that the surface concentration increases at a finite rate during the initial stages of sorption. This effect is predicted by the model. A dimensionless induction time, si, characterizing the time scale for surface equilibration, can be calculated from the numerical Ub vs. s plots using a construction analogous to that on the weight gain plot in figure (2b). Hui et al. [ 15], and later Duming et al. [28], developed asymptotic results for si for a model of Case II transport by Thomas and Windle [ 14] (TW model hereafter) corresponding to a small Deborah number limit of the model considered here (see reference [23] for a discussion). The asymptotic result for si is
1
s~ ~
M >> 1.
(23)
MlnM
Duming et al. [28] also show si ~ 0
(24)
according to the TW model. In the present model, three parameters a, 0, and M appear in the equation governing surface concentration, equation (19). One sees that the effect of 0 is only to rescale s, which implies that si ~ 0 as in the TW model (equation (24)). The parameter M describes the dependence of the relaxation time on concentration. Figure (3) shows its effect on surface swelling with a fixed at 0.99, and 0 set to 1.0; lnM ranges from 2 to 3.25. When in M < 1.5 ( M < 4.48), there is no discemable induction time; the Ub vs. s curves in this case do not show upward curvature. For In M >_ 1.5 ( M >_ 4.48), autoacceleration of Ub occurs, and an induction time can be calculated using a process analogous to that illustrated in figure (2b). Figure (4) plots si vs. ( M in M ) -1 for a - 0.99 and shows a linear relationship. Taking differences in 0 into account, the slope agrees in order of magnitude with the value obtained by Fu and Duming [23] from the TW model. In the current model the parameter a also affects the induction time. a shows up as a measure of the instantaneous elasticity of the polymer in the mixture; there is no analog for a in the TW model, since it presumes the polymer is purely viscous. a -- 0 means a perfectly compliant polymer and corresponds to ordinary diffusion; for finite a instantaneous elasticity and relaxation effects are switched on. Figure (5) shows the effect of a on surface concentration with M fixed at 10.0. For finite a one sees an initial jump, permitted by the instantaneous elasticity, which decreases with
1502
increasing c~. In the linearized version of the model, the surface concentration jumps initially to 1 - a (Tang et al. [ 12] ). In the non-linear version, the value given by the dimensionless version of equation (7) is near, but not exactly equal to 1 - c~. From figure (5) one sees that when a is less than 0.1, the relaxation of UD is minimal and there is no discernible induction time. As c~ increases from 0.1 to 0.9, the amplitude of the relaxation in Ub increases, induction times si appear and increase with a. The si increase monotonically with a, but the numerical data do not suggest an unambiguous analytical representation for si(a).
1.0
!
I
'
0.9 0.8 0.7 .Q
0.6 0.5 0.4 0.3 0.2
A
M = 12.18
v
M= 15.64
0.1 0.0
,
0
~
~
I
2
,
~
,
I
4
J
,
J
I
6
,
J
'
8
Figure 3. Effect of M on surface concentration (Ub vs. s) for a -- 0.99 and 0 -- 1.0.
1503
9
,
,
!
,
,
,
!
3
0
i
0.00
0.02
0.04
i
I
I
=
0.06
|
0.08
1/MIn(M)
Figure 4. Linear relation between induction time from surface concentration kinetics, si, and ( M In M) -1. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0"3 I
a
a=0.3
0.2 ~ [ 0.1 [
"
a=?.5
o
a=0.7 a=0.9
/
0.0 0.000
0.005
0.010
0.015
Figure 5. Effect of a on surface concentration kinetics O - 1.0.
0.020
(Ub VS. 8) for M - 10.0 and
1504
4.3 Profiles and Weight Uptake in the Case II Limit In what follows we systematically investigate the effects of a, 0, M and K on the concentration profiles (u(s, x)), nonequilibrium chemical potential profiles (OI/Os(x, s) ), weight uptake kinetics (W(s)) and several other characteristic features, including the concentration front's speed and the induction time based on weight gain kinetics. All of the calculations are on parameter ranges where Case II transport is predicted, u0 and ~ are fixed at 0 and 1, respectively, which corresponds to a dry polymer film suddenly exposed to unit fluid activity.
4. 3.1
Effect of a
Trial calculation showed that concentration profiles only begin to resemble Case-II when a is near 1. To determine its effect on the process, we varied it over the range 0.9 to 0.999. Provided 0, M and K also have appropriate values (see later sections), the weight uptake plots in this range are near linear with time, and show an induction as in figure (2b). From this, we conclude that in order to observe Case-II phenomena D~ >> Do (see equation (16)). Figure (6) shows the effect of a on weight uptake in detail for 0 = 0.01, K - M -- 5.0. From the plot one sees that, when a is small, there is practically no induction. The induction time shows up more clearly when a increases; at the same time the weight uptake curves become more nearly linear with time. Induction times for the process were calculated from figure (6) as illustrated in figure (2b); The results are qualitatively, consistent with those for induction times determined from Ub, i.e. the induction times from weight gain kinetics and surface concentration both increase monotonically with a, but they do not agree quantitatively. The quantitative discrepancy warns that experimental induction times determined from weight gain measurements not be interpreted in terms of surface concentration kinetics. Figures (7) show u and 0~ profiles for a - 0.9 with 0 - 0.01 and K and M fixed at 5. Recall ~oI is the (dimensionless) nonequilibrium contribution to the local chemical potential, and is part of the driving force for diffusion. Initially a large peak in 0I develops at the surface. As time increases, the peak decays, broadens, and then propagates into the film with fixed shape, at nearly constant speed. Eventually, the two peaks from both sides of the film combine into one at the film center and slowly relax while concentration reaches equilibrium. The peak in 0I is responsible for the dominant feature of Case II transport, i.e.
1505
1.0
0.8
0.6
0.4
/
/
/
/
=
o~= 0.92
.
-o.oo
i. i 0.0
,
0.0
I
,
,
0.2
Figure 6. Effect of a on W vs. s for 0 1.0.
,
,
,
0.4
0.01, K -
0.6
5.0, M -
5.0, Uo -
0.0,
the sharp concentration front which establishes near the surface after the external activity is switched on and propagates into the polymer film at a constant speed. We conducted systematic calculations to investigate the effect of a on the key features of OI the from: The front speed v and the values of u and 8-; at the front (uf and (o~):), oI Figure with the front's position being defined by the position of the maximum in N. (8) displays a typical trace of u: and (oi) i with time, for M - / f i - 5.0, a - 0.9 and 0 = 0.01. The plot shows that, after an initial induction at the surface both u / and ( ~ ) I achieve nearly steady values and move at nearly constant speed into the film. When the front reaches the center of the film, (oi N ) : stops and sinks to zero gradually. At the same time, uf relaxes to the equilibrium concentration. Two measures of the moving front speed were calculated. One, v:, is the slope of the linear part of front position versus time, and the other, Vw, is the slope of the linear portion of W(s) vs. s. Since the moving front invades the film from both sides, vw corresponds to about twice Vv (the values ofvf and vw/2 agree within • 8%).
1506
1.0
0.8
0.6
0.4
0.2
0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.20 0.18 0.16
s = 0.0004
0.14
o.12 o.lo
s=oos
s=o.o3
s= 0.011 /~/I
0.08 0.06 0.04 0.02 0.00 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 7. (a) Concentration profiles (u(z, s)) determined by two different numerical schemes. Hollow symbols correspond to collocation method while filled symbols correspond to Galerkin method. The model parameters are a = 0.9, 0 -- 0.01, K = 5.0, M = 5.0, Uo = 0.0, ~b = 1.0.; (b). c9I/c3s profiles from collocation for the same set of parameters.
1507 Results for effects of a on both v~ and "of with K = M = 5.0 and 0 = 0.01 are summarized in figure (9), which shows that the v decrease linearly with increasing a. That plot suggests v f ~ (1 - a) since the speed gets vanishly small near a - 1.
0.3 ~
Uf
0.2
? 0.1
0.0
0.0
0.1
Fig 8. u f , ( O I / O s ) s vs. x ; a
4.3.2
0.2
0.3
0.4
0.5
- 0.9, 0 -- 0.01, K - M - 5.0, Uo - 0.0, r -
1.0.
Effect of O
0 is a Deborah number for this model. It corresponds to the ratio of a characteristic relaxation time of the polymer to the characteristic diffusion time, evaluated for the dry film. For nonlinear diffusion, the behavior cannot be anticipated from 0 along, as in the linear limit, since the actual relaxation and diffusion times vary with concentration making the instantaneous, local value of the Deborah number vary significantly. However, for this model in the special case M = K the concentration dependence of the relaxation and diffusion times cancel, and 0 does correspond to the dimensionless relaxation time at all compositions. It happens that M ___ K is required for Case II (see later), so we may view 0 as a key parameter for Case II.
1508
14[:
....
I ....
t ....
I ....
= ....
d -i
12O ~ 10 8 [-
4_
~.
Vw
vf
-2 0.90
0.92
0.94
0.96
0.98
Gt
Figure 9. v f and v~ vs. a with 0 - 0.01, K - 5.0, M - 5.0, Uo - 0.0, r - 1.0.
We found that 0 must be _< O(0.01) in order to predict Case II, consistent with Fu and Durning's [23] analysis which derived the TW model of Case II as a small Deborah number limit of the model considered here. We picked the range 0.008 _< 0 _< 0.022 to study the effect of 0 on the process. From calculations of concentration profiles and weight uptake with a -- 0.9, / ( = M = 5.0, one finds that an increase of 0 causes a proportional increase in the induction time calculated from weight ~ain kinetics, i.e. si ~ 0, exactly as anticipated from analysis of the surface concentration. 0 also decreases front speed, and therefore the slope of the linear portion of W ( s ) vs s plots. The results for v f and vw are plotted in figure (10); we find v f ~ 0 -0.52 under these conditions. Fu and Durning [23] studied the TW model numerically and found v ~ 0 -~ identical with the dependence of in the current model in the Case II limit. One can anticipate this result from dimensional analysis.
1509
1.2
9
9
l--
w
1
w
i
i
w
1.1 1.0 0.9 c
0.8 b.O 9
z
0.7 0.6
Iogvf
0.5 L 0.4
-
3
1
I
I
-5.0
I
I
I
-4.5
-4.0
-3.5
logo
Figure 10. vf and v~ vs. 0 with a - 0.9, .K - 5.0, M - 5.0, Uo - 0.0, ~b - 1.0.
4.3.3
Effects of M and K
The parameter M controls how fast the local relaxation time decreases with concentration; it characterizes one of the nonlinearities in the model. The effect of M on weight uptake kinetics was determined for a -- 0.9, 0 = 0.01 and K -- 5.0 for 2 < M < 10. The data showed that increasing M decreases the induction time, in qualitative, but not quantitative agreement with the result from the analysis of surface concentration (figure (4)). When M is large, the average relaxation time in the surface layers is low, and it takes less time to establish equilibrium at the surface. Increasing M also increases the slope of the linear portion of weight uptake plots, corresponding to an increase in front speed. The effect of M on the front speed is summarized figure (11) for a = 0.9, 0 = 0.01 and K = 5.0; the data for, M = 6, 8, 10, were generated by Galerkin's method. The front speed increases monotonically with M, with the data for vf being nearly linear with M. Duming et al.'s [28] asymptotic prediction
1510
from the TW model, that v ~ ( K + M ) a/2 is consistent with figure (11).
22
. . . .
1
'
'
'"
I
. . . .
I
. . . .
I
. . . .
I
'
'
'
r
20
II
18
16 ~
vw
14 12
r-I
lO ~
_
o
-
8. 6~
El
-
9
[3
ooOO
. ,
0
-
0
0
4 2
9
n
=
j
,
I
2
,
,
,
I
0 O0
1
4
. . . .
9
-
v,
:
I
6
,
,
,
,
I
8
,
,
=
i
I
10
. . . .
12
M
Figure 11. Dependence of v / a n d vw on M with a - 0.9, 0 - 0.01, K - 5.0, Uo 0.0, ~b - 1.0. Filled symbols represent values calculated by Galerkin's method.
The numerical solutions give evidence that M has to be sufficiently large in order to predict Case II 9 We calculated u and 0I profiles with 3/1 -- 2 and other parameters as above. By comparing the results with figure (7), one finds that the decrease of M from 5 to 2 suppresses the Case II features. The decrease results in considerable broadening of the concentration front; at the same time, the peak in 0I continuously decays at M = 2 and spreads progressively more along the spatial coordinate during sorption. The parameter K govems how fast the diffusion coefficient increases with concentration and characterizes the other key nonlinearity in the model. Figure (12) shows the dependence of induction time from weight gain on K with ce = 0.9, 0 = 0.01 and M = 5.0. The four data points in the high K range were generated
1511
by the Galerkin code. One expects that the induction time from weight gain be independent of K, if it truly characterizes the surface relaxation process, i.e. if it corresponds closely to the induction calculated from the surface relaxation kinetics (e.g. figure (4)). However, we find that the induction time from weight gain increases linearly with K until near the value of M (5.0 in figure (12)) after which it becomes independent of K as expected. This indicates that diffusion in the surface layers can limit the induction time determined from weight gain kinetics. We note, however, that compared with the effect of M, which reduces the induction time even if the value exceeds K, the influence of K on the induction from weight gain is much weaker, as one expects intuitively. Calculations show a roughly linear relationship between front speed and K, as shown in figure (13) for M = 5, a - 0.9, 0 = 0.01. This finding is again consistent with that by a perturbation analysis ofthe TW model, which predicts v ~ ( K + M ) 1/2. Calculations showed that in order to have Case II, M needs to be at least 2 - 3 and K has to be at least equal to M. For example, it was found that if K is kept at 5.0, when M rises above 8, the predicted behavior deviates from Case II in that the linear weight uptake kinetics were not predicted. This is because when M exceeds K, the front speed becomes so fast that diffusion behind the front quickly becomes rate limiting. Consequently, the front begins to show diffusive dynamics (v ~ sl/2).
5. C O N C L U S I O N A one-dimensional nonlinear model for viscoelastic diffusion in concentrated polymer-fluid mixtures was constructed by an ad-hoc generalization of the linear response model by Durning and Tabor [18] . The nonlinearities were introduced by retaining concentration dependencies of physical properties. In order to get a numerical solution for sorption in films, the diffuson equation was cast as coupled partial differential equations by introducing a new dependent variable. Finite element methods were used to discretize the spatial domain, converting the PDEs into an ODE system, which was solved by a time integrator package. In one scheme, orthogonal collocation on Hermite cubic basis functions was used to discretize and LSODI was adopted to do the time integration. An alternative integration technique employed Galerkin's method on quadratic basis functions together with the DASSL integrator. The two schemes were shown to agree. There are six dimensionless parameters in the model, a, 0, M, K, u0 and ~b. The
1512
0.0092 0.0088
O
9
9
@
0.0084 0.0080 0.0076
0.0072
3
4
5
6
7
8
Figure 12. Dependence of the induction time from weight gain kinetics, si, on K with a - 0.9, 0 - 0.01, K - 5.0, Uo - 0.0, ~b - 1.0. Filled symbols represent values calculated by Galerkin's method.
first four characterize the mixture and the last two define the initial and final states for sorption. We first investigated the predictions for two well-studied situations, where all the parameters could be calculated apriori: Differential sorption in polystyreneethylbenzene (PS-EB) solutions and integral sorption in poly(methylmethacrylate)liquid methanol (PMMA-MeOH). For PS-EB, the two-stage sorption process was correctly predicted for differential sorptions in thin films at concentrations just below Tg. For PMMA-MeOH, the Case II diffusion was correctly predicted for immersion conditions in thick, dry plates at room temperature. The calculation shows that the model can predict the most well documented and striking non-Fickian effects observed in sorption w i t h o u t empiricism. A systematic investigation was conducted of the effects of the materials parameters
1513
22
'""
'
'
I
'
i
,
,
I
'
"'
'
I
'
;
;
'
I
'
"'
'
I ''w
'
'
"
i
20 18 16 Vw
14 12 10 8
0
9
D
9 0
6 0
0
0
9
9
Vf
4 4
5
6
7
8
9
Figure 13. Dependence of vf and vw on K with a - 0.9, 0 - 0.01, M - 5.0, 1.0. Filled symbols represent values calculated by Galerkin's method.
Uo - 0.0, ~b -
on Case II diffusion. The study should facilitate analytical asymptotic work, and provide guidelines for design and control of systems relying on Case II. Case II diffusion appears only when a, a measure of the instantaneous elasticity of the system, is close to 1. This implies D~ > > Do and physically means that the osmotic modulus is weak compared to the mixture's shear modulus. The situation occurs when the fluid is a poor solvent or swelling agent. It was found that the front's speed decreases nearly linearly with increasing a. 0, the diffusion Deborah number, has to be ~ O(0.01) for Case II to appear, indicating that Case II is a slow-motion limit of the model. 0 affects the front speed according to v ~ 0 -1/2. The numerical study shows that strong nonlinearities in the relaxation time and diffusion coefficient, represented by large M and /4, are both essential for the prediction of Case II transport. Importantly, the values of M and K should be about the same, and at least 2 - 3. I f M exceeds K by too much, the process rapidly becomes
1514
controlled by diffusion behind the moving front. Regarding the numerical methods used, collocation can handle parameters K an M only up to about 5, beyond which the calculation becomes unstable. For modelin more severe non-linearities, one should employ Galerkin's method, which seems t have a wider range of stability in parameter space. It was demonstrated that the model predictions qualitatively match the behavk observed experimentally. A forthcoming publication [29] reports in detail o practical procedures for evaluating the model parameters, and on the quantitatN capabilities of the model.
6. Acknowledgment 1LAC acknowledges support from Sandia National Laboratory.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Alfrey, T., E. E Gumee, and W. G. Lloyd, ~ Polym. Sr C, 12, 249 (1966). Crank, J., and G. S. Park, Diffusion in Polymers, Academic Press, Londc (1968). Frisch, H. L., Poly Eng. & Sr 20, 2 (1980). Vrentas, J.S., and J.L. Duda, Encyclopedia of Polymer Science ar, Engineering, 5, 36 (1986). Vrentas, J. S., C. M. Jarzebski, and J. L. Duda, AIChE J., 21, 94 (1975). Vrentas, J.S., and J.L. Duda, J. Polym. Sci., Polym. Phys. Ed., 15, 441 (1977~ Odani, H., S. Kida, M. Kurata, and M. Tamura, Bull. Chem. Soc. Japan, 3, 571 (1961). Odani, H., J. Hayashi and M. Tamura, Bull. Chem. Soc. Japan, 34, 817 (19611 Odani, H., S. Kida and M. Tamura, Bull. Chem. Soc. Japan, 39, 2378 (1966) Billovits, G.E, and C. J. Durning, Macromolecules, 26, 6927 (1993). Billovits, G.E, and C. J. Durning, Macromolecules, 27, 7630 (1994). Tang, PH, C.J. Durning, C.J. Guo, and D. DeKee, Polymer, 38, 1845 (1997). Hopfenberg, H.B., J. Memb. Sci., 3, 215 (1978). Thomas N. L., and A. H. Windle, Polymer, 23,529 (1982). Hui, C. Y, K.C. Wu, R.C. Lasky, and E. J. Kramer, J. Appl. Phys., 61, 51"~ (1987a). Hui, C. Y, K.C. Wu, R.C. Lasky, and E. J. Kramer, J. Appl. Phys., 61, 51"~
1515
17. 18. 19. 20. 21. 22. 23. 24.
25.
26.
27. 28. 29.
(1987b). Durning, C.J., M.M. Hassan, K.W. Lee and H.M. Tong, Macromolecules, 28, 4234 (1995). Durning, C. J., and M. Tabor, Macromolecules, 19, 2220 (1986). Rodriguez, E, P.D. Krasicky, and R.J. Groele, Solid State Tech., 28, 125 (1985). Roseman, T. J., and S. Z. Mansdorf, Eds., Controlled Release Delivery Systems, Marcel Dekker, New York (1983). Billovits, G.E, and C. J. Durning, Chem. Eng. Comm., 82, 21 (1989). Cairncross, R.A., and C.J. Durning, AIChE J., 42, 2415 (1996). Fu, T. Z., and C. J. Durning, AIChE J., 39, 6 (1993). Painter, J. E, and A. C. Hindmarsh, "Livermore Solver for Ordinary Differential Equations (Implicit Form)," Lawrence Livermore National Laboratory Report L-316 (1982). Brenan, K.E., S.L. Cambell, and L. Petzhold, Numerical Solution of InitialValue Problems in Differential-Algebraic Equations, Elsevier, New York (1989). Tang, P.H., " Analysis of Differential and Integral Sorption in Concentrated Polymer Solutions," PhD Thesis, Department of Chemical Engineering, Materials Science and Mining, Columbia University, New York (October, 1995). Long, EA., and D. Richman, J. Am. Chem. Sot., 82, 513 (1960). Durning, C.J., D.S. Cohen, and D.A. Edwards, AIChE J., 42, 2025 (1996). Huang, S.-J., and C.J. Durning, J. Polym. Sci., Part B: Polymer Physics in press.
1488
252. Ghosh,U.K.,Dey,K.N.,Gupta,S.N.,KtHnar,S.,Upadhyay,S.N.,Chem. Engng .Commun.,43( 1986a)335. 253. Shanna,K.K.,Adelman,M.,Can.J.Chem.Engng.,47(1969)553. 254. Lin,H.-T.,Sllih,Y.-P.,Chem.Engng.Commtm.,4(1980)557. Also 7(1980)327. 255. Gorla,R.S.R.,Chem.Eng.Commun.,49(1986)13. 256. Gorla,R.S.R.,Polym.-Plast.Technol.Eng.,30(1991)75. 257. Skelland,A.H.P., AIChE J.,l 2(1966)69. 258. Mishra,I.M.,Mishra,P.,Indian J.Technol., 14(1976)375. 259. Kim,H.W.,Esseniyi,A.J.,J.Thennophysics Heat Transf.,7(1993)581. 260. Ruckenstein,E., Ind.Eng.Chem.Res.,33(1994)2331. 261. Coppola,L.,Bohm,U., Int.Commtm.Heat Mass Transf.,13(1986)77. 262. Kumar,S., Upadhyay, S.N., lnd.Eng.Chem.Fundam.,19(1980)75. 263. Wronski,S.,Szembek-Stoeger,M.,Inzyni. Chem.Procesowa,4(1988)627. 264. Hiial,M.,Bnmjail,D.,Colniti,J.,J.Appl.Electrochem.,21(1991 ) 1087. 265. Coppola,L.,Bohm,U.,Chem.Eng.Sci.,40(1985)1594. 266. Sedahmed'G'H"Mans~ J.App.Electrochem., 17(1987)583. 267. Potucek,F.,Stesjkal,J., Chem.Eng.Sci.,44(1989)194. 268. Hwang,S.-J.,Liu,C.-B.,Lu,W.-J.,Chem. Eng.J.,52(1993)131. 269. Edwards'M'F"Wilkins~ Chem.Eng.,No.257( 1972)310. 270. Desplanchaes,H.,Llinar,J.R.,Chevalier,J.L.,Can..l.Cllem.Eng.,58(1980)160. 271. Schugrel,K.,Adv.Biochem.Engng.,l9(1981 )71. 272. Moo-Young,M.,Blanch,H.W.,Adv.Biochem.Engng.,19(1981 )1. 273. Deckwer,W.D.,NguyenTien,K.,Schumpe,A.,Serpemen,Y.,Biotech. Bioeng.,24(1982)461. 274. Chhabra,R.P.,Ghosla,U.K.,Kawase,Y.,Upadhyay,S.N.,in Multiphase Reactor and Polylnerisation System Hydrodynamics, ed.N.P.Cheremisinoff, Gulf, Houston (1996) p.539. 275. Hartnett,J.P. ,Hu,R.Y.Z., lnt.Comm.Heat Mass Transf.,l 3(1986)627. 276. Wang,A.T.,Hartnett,J.P., Wanne-und-Stoffubertragung,27(1992)245. 277. Floquet-Muhr,L.,Midoux,N.,Chera.Engllg.Process.,33( 1994)459. 278. Desplanchaes,H.,Gaston_Bonhomme,Y.,Chevalier,j.L., Int.Chem.Eng., 34(1 994)225. 279. Liu,J.,Ye,L.,Liu,H.,J.Chem.Engng.Jpn.,28(1995a)210. 280. Liu,J.,Ye,L.,Liu,H., Int.Colmn..Heat Mass Transf.,22(1995b)359. 281. Rene,F.,Leuliet,J.C.,Lalande,M.,Chem.Eng.Res.Des.,69C(1991 ) 115. 282. Fossa,M.,Yagliafico,L.A.,Exp.Thennal Fluid.Sci.,10(1995)221. 283. Jaluria,Y.,Adv.Heat Transl.,28(1996)145.
