Mechanics of mixtures

(2J2)

We define

= =

Dd>

d

+ [ [grad(/> g r a d < ( (x ] v /((xx ,t), ^ x ' t )*)]' ' ^'

~~~Dt=di ~Dt ==4>{*,t) > ,*) +

(2 13)

-

and thus or D/Dt is the variation of 4> with respect to time as noted by an observer at x moving with the mean velocity of the mixture. The acceleration a* of the i-th constituent is defined through

a

1

(2-14)

= -£>

a =

92r

at2'

and the velocity gradient L l through <9vz <9v* L' = W Vv* «= - .

(2.15)

&T'

Here, we designate V as the gradient from an Eulerian point of view, instead of the symbol grad. We observe that care should be exercised to keep the above notation in mind since in most texts in continuum mechanics as well as the papers cited in this book the symbol V is used to denote the gradient with respect to the reference configuration. It immediately follows from Eqs. 2.2, 2.6, 2.11 and 2.15 that 1 Fi (F 1 ) iy\ . U -u == Y'{F

(2.16) l

l

The symmetric part and the skew-symmetric part of L are denoted by D and W , respectively, and thus 1 l ; T D« = i [L D* i/ + + ((L l / ))T ]] ,, ~ 2

(2.17)

1 1 W l == - V W L - (L*) T . - « • "2

(2.18)

and

A much more detailed treatment of the kinematical quantities and the relation­ ships between the kinematical quantities for mixtures can be found in the articles by Bowen [56] and Atkin and Craine [64]. Also, mixture theory is general enough to deal with the situation when the constituents have a microstructure (cf. Allen and Kline [92], Twiss and Eringen [93,94], Dunwoody [95], Passman [96]). Here, we shall not consider the motion of such mixtures. We shall also not consider chemically reacting mixtures in our book; we refer the reader to the works of Nunziato and Walsh [97] and Bedford and Drumheller [98] for details regarding the same.

MECHANICS OF

10

2.2

MIXTURES

Partial Tractions and Partial Stresses

Let 5 be a surface in the body to which the point x belongs. Let ns denote the normal to the surface S at the point x. In the case of a single constituent we have the notion of the traction at x (that depends on n s ) and the notion of the stress that follows from Cauchy's theorem (cf. Truesdell and Toupin [37]). In the case of a mixture, we assume the existence of a partial traction vector t* and a partial stress tensor a' associated with each constituent of the mixture, where t' ={al)Tns.

(2.19)

We define the total traction t and the total stress a for the mixture through t = X>\

(2.20)

\

(2-21)

t =
(2.22)

and

so that

In the early literature on mixtures, there was considerable discussion on how to define the total stress in the mixture. We shall not get into a detailed discussion of it here but refer the reader to Bowen [52] and Green and Naghdi [47] for the same. In the following subsections, we will present the basic balance laws for mixtures, but it is not our purpose here to reproduce in great detail a discussion of these laws or the various quantities that appear in them as they can be found elsewhere (cf. Truesdell and Toupin [37], Bowen [56], Atkin and Craine [65]). On the contrary, after introducing the quintessential details of the theory with brevity, we would like to devote much of our efforts in putting the theory to practice. This is not to downplay the various subtle issues that raise their heads within the context of mixture theory. However, as these issues have been dealt with in the references mentioned above, we do not feel the necessity to repeat them here.

2.3

Balance of Mass

The theory of mixtures can take into account the possible interconversion of mass due to thermal interactions or chemical reactions between the constituents. We associate with each constituent the mass supply for that constituent rri1 that represents the production of the i-th constituent due to these actions. Then if V is any fixed volume in the mixture, the balance of mass requires that

PRELIMINARIES

11

d_

f ffdv + f p V • da. = f rrtdv, (2.23) dt J-p Jarp Jp J-p Jar J-p where dV denotes the boundary of V. If the integrands are continuous and smooth, then appealing to standard arguments based on continuity, we can obtain the appro­ priate local form of the balance of mass £>('>/>' — — - + p'divv* = m\

(2.24)

In addition to the balance of mass of each constituent, we also have the balance of mass for the mixture as a whole and thus d

— [ pdv+ [ pv ■ da = 0, at Jp Jdv

(2.25)

whose local form is p + pdivv = 0.

(2.26)

We note that the balance of mass for the mixture as a whole has the same form as that for a single constituent continuum. Also, it follows from summing Eq. 2.24 over all i = 1, ..., n and Eq. 2.26 that n

^m

l

= 0,

(2.27)

i=i

which states that there is no net production of mass. The above representations for the balance of mass are from an Eulerian point of view. However, in many applications involving the diffusion of fluids through a solid, it is convenient to express the balance of mass for the solid constituent in the Lagrangian form. In this situation, the global balance of mass for the z-th constituent takes the form

4- f pldV = [ mldV,

(2.28)

dt Jn Jn here, d/dt denotes the material (total) time derivative. The local form of the above balance equation is p 1 detF i = mMetF*

2.4

Balance of Linear Momentum

The balance of linear momentum for the z-th constituent states that

(2.29)

MECHANICS OF MIXTURES

12

- [ pWdv + I p V v 1 ■ da = f Uf dt Jv Jav Jar v '

da + [ ( p V + m' + m V ) ^.. Jv v '

(2.30)

In the above equation, b* is the specific external body force, and m ' is the momentum supply due to the interaction between the i-th constituent and other constituents (interactive body force). The term m V is the contribution to the momentum due to the mass production m1 of the i-th constituent. If the integrands of the above integrals are continuous and smooth, then it follows from standard arguments that — ( p V ) + div ( p V ® v l ) = div (<7<)r 4- p'W + m' + m V .

(2.31)

It immediately follows from Eqs. 2.24 and 2.31 that

(t^jf- = dw [jf

+ pV + m'.

(2.32)

We shaU find the above form for the balance of linear momentum particularly conve­ nient for later use. Summing Eq. 2.32 over the index i = 1, ..., n and using Eq. 2.4, we obtain pv = div

±

(pV

v*) 1 4- ±

9

(m' + m V ) + pb,

(2.33)

i=l

where

b = igpV.

(2.34)

We note that the above equations will have the same form as that for a single con­ stituent continuum if we define a stress tensor a through n


(/v' ®v') >

(2.35)

provided that = 0.

(2.36)

i=l

This prompted Truesdell [35] to define the total stress in the mixture through Eq. 2.35 rather than Eq. 2.21. Equation 2.36 is a statement to the effect that the net momentum supply to the mixture due to all the constituents is zero. A few brief remarks about the momentum supply m1 to the i-th constituent would be appropriate at this juncture. During the diffusion of one constituent through oth­ ers, various interaction effects come into play (see Johnson, Massoudi and Rajagopal

PRELIMINARIES

13

[99] for a detailed discussion of the interaction mechanisms), and these effects depend on the nature of the constituents and the manner in which the diffusion process takes place. For the sake of simplicity, we shall make the following remarks mainly related to a mixture of a linearly viscous fluid and solid particles or a mixture of two fluids. It would not be an exaggeration to say that the constitutive relation for the interaction plays a central and critical role in the historical development of this field. Early work on the movement of a solid body in an inviscid fluid is due to Thomson and Tait [100] and Kirchhoff [101]. This was followed by numerous papers by Lamb [102], Craig [103], Greenhill [104] and others. As we are not interested in the diffusion of an Euler fluid, we shall not get into a detailed discussion of the same here but refer the reader to the treatise by Lamb [105]. A common interaction force is that of drag which is usually a generalization of Stokes' seminal work in which he obtained the resistance offered to a slowly falling solid sphere in a fluid (cf. Stokes [106]) and that of Oberbeck [107] and Oseen [108] for ellipsoidal particles. The problem of a sphere moving unsteadily in a linearly viscous fluid was studied by Boussinesq [109], Basset [110] and Rayleigh [111]. The resistance due to the motion of a slightly deformed sphere moving slowly in a linearly viscous fluid has been given by Brenner [112]. Cunningham [113] provided a correction to the Stokes drag due to slip between the particle and the fluid. Brinkman [114] obtained a correction to Stokes' formula for the drag, due to the presence of a dispersion of particles in the fluid, based on the volume fraction of the dispersed particles (see also Tarn [115]). For concentrated suspensions, Richardson and Zaki [116] and Kuwabara [117] have determined the forces experi­ enced by a system of particles, which would be the basis for a more general form for the interaction force. The pioneering work of Hadamard [118,119], Rybczynski [120] and Boussinesq [121] that considered the drag due to a fluid flowing past a sphere of another fluid has spawned considerable work that would be particularly appropriate when dealing with the diffusion of one fluid through another and problems involving droplets and aerosols. Hsieh [122] has studied the arbitrary motion of a single spheri­ cal bubble in a linearly viscous fluid and Plesset and Mitchell [123] have investigated the stability of the spherical shape of the bubble. The dynamics of non-spherical bub­ bles in a linearly viscous fluid has been studied by Hsieh [124]. Taylor and Acrivos [125] studied the flow past a deformed fluid sphere when inertial effects are significant. An extensive and thorough discussion of the dynamics of bubbly liquids can be found in the review articles by Hsieh [126] and Plesset and Prosperetti [127]. Another interaction effect that often has to be taken into account is that due to a constituent accelerating through others (the virtual mass effect). A succinct account of its origins can be found in the book by Birkhoff [128]. As the relative acceleration between two particles is not frame indifferent, care has to be exercised in introducing the effect, and for instance in an early study of fluidization an incorrect choice for this effect was made (cf. Murray [129], Anderson and Jackson [130], Jackson [131]). A frame indifferent measure of the relative acceleration has been provided by Drew and Segel [132], El-Kaissy [133], Drew, Cheng and Lahey [134] and Drew and Lahey

14

MECHANICS

OF

MIXTURES

[135]. Another interaction term that is usually introduced in problems involving mixtures is the 'lift' due to the other constituents, on a particular constituent (see Rubinow and Keller [136], Saflman [137], Drew [138]). The accumulation of particles in an annular region in Poiseuille flow of a fluid in a pipe has been explained using the notion of lift (see Segre and Silberberg [139]). When one considers the flow of a fluid past a sphere, one recognizes that the motion of the fluid can cause the sphere to spin. This would lead us to include such an interaction mechanism when treating the flow of mixtures. The spinning which is caused is usually referred to as the Magnus effect and the interaction force that is introduced the Magnus force. This type of interaction between the constituents might be of particular significance when considering the flow of fluids infused with particles. A detailed discussion of the Magnus effect can be found in the article by Barkla and Auchterlonie [140]. For a particle undergoing unsteady motion through a fluid, Basset [110] computed an integral that depends on the entire flow history of the particle and that is a measure of the resistance to flow and can be considered as a correction to the Stokes drag due to the unsteadiness of motion. This interaction force can become important when the fluid flow is turbulent leading to considerable fluctuations in the flow field (cf. Ahmadi and Goldschmidt [141]). Of course, when we have a mixture the above interaction force has to be generalized to take into account a collection of particles (cf. Zuber [142]). In addition to the above interaction forces, we might have to consider interaction forces that come into play due to density gradients. All the above interactions have mechanical origins. For problems within a thermomechanical context we cannot rule out interaction forces that arise due to the thermal fields of the constituents. For instance, temperature gradients can be the source of interaction forces (cf. Phillips [143], Kempers [144], Barton and Subramanian [145]). A detailed discussion of many of the above effects of interaction with copious references can be found in Happel and Brenner [146] and Johnson, Massoudi and Rajagopal [99].

2.5

Balance of Angular M o m e n t u m

The balance of angular momentum for the i-th constituent takes the form

— / x x pW'dv + [ x x p ' v V • da = [ x x v( V ) T da ot Jv Jav J&P I + J [x x (p'b 1 + m1' + m V ) + M"] dv,

(2.37)

where M ' is the angular momentum supply to the i-th constituent. The term x x m*

15

PRELIMINARIES

denotes the angular momentum supply to the i-th constituent due to the momen­ tum supply m*, while x x m V is the angular momentum supply due to the mass conversion and x x p*b l the angular momentum supply due to the external body force. In the case of a single constituent continuum the balance of angular momentum, in the absence of body couples, leads to the condition that the stress tensor is symmetric. However, in the case of a mixture, the balance of angular momentum for the i-th constituent leads to M'=(
(2.38)

Thus, only in the absence of the angular momentum supply M ' does the partial stress tensor of the i-th constituent become symmetric. The balance of angular momentum for the mixture can be defined in an analogous manner to that of the linear momentum. If we require that the form be similar to that of a single continuum in that x x pv = div (x x &T) + x x pb,

(2.39)

we need n

£M l

(2.40)

= 0.

1=1

2.6

Conservation of Energy

Let e \ q \ r\ e\ denote the specific internal energy, the heat flux vector, the radiant heat supply and the energy supply, respectively, for the i-th constituent. Then, the conservation of energy for the i-th constituent of the mixture reads

1/,". ( , + i v ' . v > - M^-Vv*)v*. = j

(
da

( p V + p V • b«) dv

+ / J [ e : W * ^ v W ) W V ] < f e ,

(2.41)

The term m^e1 + ± v l V ) denotes the energy supply due to the mass that is produced, and m* ■ v accounts for rhe conttrbutton to the energy due to the momentum supply. Standard arguments lead to the local form of the conservation of energy | U Li + h V • v A l + div [> ( £ + \ ^ ■ V l ) V] = div ( o V - q1) dt +pV + p V • b ; + el + m l U + - v l • v ; ) + m1 • V .

(2.42)

MECHANICS

16

OF

MIXTURES

Combining Eqs. 2.24, 2.32 and 2.42 results in

d_

— ( p V ) + div ( p W ) = -divq* + tr ( V l / ) (2.43)

+ p V + e\ + m V .

This is the equation governing the internal energy density of the i-th constituent. Let us introduce

EA l ,

pe

q = £q\

^ q + EAV, i=i

pr = E P V -

(2.44)

i=l

K we further suppose that E ^ + m ^ e ' + iv'-v'j+m'.v1

0,

(2.45)

then, summing Eq. 2.43 over i = 1, ..., n yields pi = Y,

2.7

tr ( V l / ) - m i • v ; - - m V

v;

divh + pr.

(2.46)

Second Law of T h e r m o d y n a m i c s

Even within the purview of a single constituent continuum, there has been much debate as to what constitutes an appropriate statement of the second law of thermo­ dynamics. Whether the statement holds for a system or a body as a whole or to every subpart of the body is a matter of some disputation. Even the basic notion of en­ tropy, and whether it is a primitive or derived quantity and whether it can be defined uniquely, is the subject of controversy. Also, the meaning and status of temperature and the notion of hotness manifold has been the object of considerable discussion (see Fosdick and Rajagopal [149]). When mixtures of constituents are involved, we face additional dilemmas, for instance, whether the entropy inequality holds for each constituent or only for the mixture as a whole. These and related issues have been dis­ cussed in detail by Green and Naghdi [44,45,47], Bowen [52], Bowen and Wiese [147], Miiller [58], Dunwoody and Miiller [148]. Here, we shall assume the Clausius-Duhem inequality as the statement of the second law, and furthermore we shall assume that the inequality be required to hold locally and for the mixture as a whole. This is of course a stringent assumption, but as such an assumption has proved useful in vari­ ous other theories of continua, we shall also adopt it here. This does not necessarily

PRELIMINARIES

17

sanction its applicability to mixture theory nor prove its validity. However, in the absence of a more compelling alternative, we feel justified in adopting such a stance. As we shall see later, such an approach leads, for the problems considered in this book, to results that are in keeping with physical expectations. Let 77* and T* denote the specific entropy and the temperature of the i-th con­ stituent, respectively. Then the global form of the entropy inequality for the mixture as a whole is given by

ltLp^^Lp^-d^--LtXT)-d&+Lt^dv-

{2A7)

The local form of equation 2.47 is

I (f>v) + * (|>¥v) + a. (g <±£™) _ I £ , , (,48) Using Eq. 2.43 to eliminate r% from Eq. 2.48 results in

UhWr)-^'^-

- Pl (^

- v 1 • V [p{ (A1 - A)] - div [plAV

-- A ) : i)i/]

- p{A%w) + e* + m V

(q' + , W u ' ) . v r l , 0 i

(249)

where A1 = e* — Tlrf is the specific Helmholtz free energy of the i-th constituent and A = SiLi P% ^ I P i s * n e specific Helmholtz free energy of the mixture. For the mixture in which all the constituents have the same temperature field T, that is, T" = T, i = 1, ..., n, equation 2.49, by virtue of Eq. 2.45, reduces to

-p (i + ,f) + EJtr [(„' - p' (A' - A) I) L<] - imV . v' - [m- + V (,' (A- - A))] . V - <-'

+

' , T ; ' " ' ' - V r } > 0,

(2.50)

with =

1 -

y_ zz U=i=i Pp V

(251)

pP The entropy inequality 2.47 is quite general and allows for a different temperature field to be associated with each constituent. However, in all the problems considered

MECHANICS

18

OF

MIXTURES

herein, we shall a s s u m e a single c o m m o n t e m p e r a t u r e for all t h e c o n s t i t u e n t s . Read­ ers interested in t r e a t m e n t s w h e r e a different t e m p e r a t u r e field is associated w i t h each c o n s t i t u e n t a r e referred t o t h e works by I n g r a m a n d E r i n g e n [150], Steel [151], Bowen a n d G a r c i a [152,153], C r a i n e , G r e e n a n d N a g h d i [154], G u r t i n a n d P e n h a [155], Bed­ ford a n d S t e r n [156], Bowen a n d R a n k i n [157], Pecker a n d Deresiewicz [158], Bowen a n d C h e n [159], a n d Bowen a n d Reinicke [160].

2.8

Volume Additivity Constraint

Consider two fluids t h a t a r e incompressible in their p u r e s t a t e s a n d s u p p o s e t h a t we m i x t h e m t o g e t h e r . T h e n t h e volume of t h e m i x t u r e would e q u a l t h e volumes of t h e two c o n s t i t u e n t s . 1 T h i s v o l u m e a d d i t i v i t y in t h e m i x t u r e places restrictions on t h e m o t i o n of t h e m i x t u r e . Mills [161] was t h e first t o s t u d y this issue in d e t a i l . As we are p a r t i c u l a r l y interested in a m i x t u r e of a p o r o u s solid c o n s t i t u e n t a n d a fluid c o n s t i t u e n t , w i t h o u t m a s s conversion, we shall discuss t h e consequence of t h e restriction of v o l u m e a d d i t i v i t y in this case. Let us s u p p o s e t h a t t h e solid h a s a n a t ­ u r a l p o r o u s s t a t e w i t h uniform macroscopic porosity f3. T h e n t h e v o l u m e additivity constraint takes t h e form (cf. Shi [77]) P3

Pf

!L(l-0) + !L. = l, PO

(2.52)

PR

where p j = (1 — j3) psR is t h e effective m a s s density of t h e solid in its n a t u r a l p o r o u s s t a t e . Here, we i n t r o d u c e p j since we will consider t h e n a t u r a l p o r o u s s t a t e as t h e reference s t a t e of t h e a s s u m e d solid c o n t i n u u m in which t h e solid c o n t i n u u m is defor­ m a t i o n free. Notice t h a t in t h e case of (3 ^ 0 t h e a s s u m e d solid c o n t i n u u m does not u n d e r g o deformations (ps = pp) if t h e a s s u m e d fluid c o n t i n u u m h a s m a s s density of p? = f3pR, t h a t is, t h e fluid j u s t fills t h e macroscopic voids c o n t a i n e d in t h e p o r o u s solid. It follows from Eq. 2.52 t h a t 1 - /3 dp'

1

dp1

at ' p ^ = 0 '

(2 53)

- / v p * + i - V p / = 0. Po pR

(2.54)

PI

-

and

'Even in the case of a single constituent, incompressibility is an idealization. In the case of a mixture of two incompressible constituents, there is an inherent difficulty at the conceptual level; the two constituents have to occupy the whole volume of the mixture as they have to co-exist at each point in the mixture, which would violate the assumption that each constituent is incompressible. However, within the context of homogenized equivalent constituents such an idea can be given meaning.

PRELIMINARIES

19

Combining Eqs. 2.53, 2.54 and 2.24 and assuming mf = m" = 0 yields divv* + — " — ^ d i v v ' + V ( — ) • ( v s - v / ) = 0.

(2.55)

Po Similarly, for a mixture of two incompressible fluid constituents, we have (cf. Mills

[161]) P1 P2 £- + £- = 1, PR

(2.56)

PR

and ^ d i v v 1 + 4 d i w 2 + V (4] PR

PR

\PRJ

■ (v 1 - v 2 ) = 0. V

;

(2.57)

While we shall use the volume additivity constraint to study several problems involving a mixture of a solid infused with a fluid, a mixture of two fluids and a mixture of a fluid and solid particles, there are other constraints that could be used when considering mixtures. We notice that the volume additivity constraint leads to a Lagrange multiplier into the field equations, and the constraint does no work. Green, Naghdi and Trapp [162] have considered internal constraints which produce no entropy in thermodynamic processes. The interesting study of Liu [163] which shows how to exploit the entropy inequal­ ity with the Lagrange multiplier method warrants special mention. However, we shall not delve into this rather lengthy procedure because our main interest is to obtain the specific constitutive forms consistent with the entropy inequahty. The recent paper by Hutter and Svedsen [164] discusses several issues for constrained mixtures. So far we have summarized the basic field quantities and the basic balance equa­ tions in the theory of mixtures. The theory is not complete because it contains undetermined quantities such as m1,
Chapter 3 D I F F U S I O N OF A FLUID T H R O U G H A SOLID U N D E R G O I N G LARGE DEFORMATIONS: CONSTITUTIVE RESPONSE FUNCTIONS The studies on the flow of fluids through porous solids are too numerous to document, the books by Bear [165] and Freeze and Cherry [166] contain details pertinent to the problems of flows of fluids through solids undergoing small deformations, and the paper by Raats [167] presents applications pertinent to diffusion through soils and discusses many other papers devoted to this area. The many papers by Biot [168, 169,170], Hassanizadeh and Gray [171,172,173], Hassanizadeh [174,175], and Katsube and Carroll [176] consider the flow through a porous media, but are restricted to the solids undergoing small deformations. In this chapter, we shall be especially interested in constructing a constitutive model for diffusion wherein the solid undergoes large deformations, and we shall study this within the context of the equations documented in the previous chapter. There are several papers in the polymer science literature, both theoretical and experimental, dedicated to the diffusion of gases and liquids through polymers (see Hartley [177,178], Bagley and Long [179], Richman and Long [180], van Amerongen [181], Sourirajan [182], Henley, Li and Long [183], Rickles [184], Friedlander and Rickles [185], Merten [186], Kopecek and Sourirajan [187], Paul and Ebra-Lima [188], Astarita and Joshi [189], Astarita and Sarti [190], Neogi [191,192], Kim and Neogi [193], Neogi, Kim and Yang [194]). However, none of these studies are within the context of mixture theory. Also, here we shall be interested in viscous fluids diffusing 21

22

MECHANICS OF MIXTURES

through an elastic sohd undergoing large deformations, while many of the above pa­ pers concern the diffusion of gases or liquids through membranes with a view towards separation and purification. Consider the diffusion of a Newtonian fluid through a porous elastic solid. The sohd constituent has a natural reference state in which the solid contains macroscopic randomly-distributed voids, has uniform macroscopic porosity j3 and is undeformed. The diffusion process will be assumed to take place under isothermal conditions with­ out conversion of mass between the solid and the fluid. The balance equations for the sohd and the fluid are, following Eqs. 2.24, 2.29, 2.32, 2.36, 2.40 and 2.50, p ' d e t F 3 = po,

(3.1)

^ + div(PM)=o, dt

(3.2)

flWv

dW {asf - m + p3b,

(3.3)

n(/)v/ , .T Pf—Qf~ = div (
(3.4)

r

Dt

(a' + aff

=
(3.5)

and -pA + tr [{a' - p3 (A3 - A) I) L9] + tr [{af - pt (Af 3

3

A)

i) I / ]

3

+ [m - V (p (A - A))} ■ (v - v>) > 0.

(3.6)

Introducing a" = a3 - p3 (A3 - A)l,

af = af - p* [A* - A) I,

m=m-X/(p'(A3-A)),

(3.7)

we can rewrite Eqs. 3.3 through 3.6 in the form: P P1

3

-pr=div(^)

D

= div ( y j + af)

T

- m + p 3 b,

(3.8)

+m + pfb,

(3.9)

=a3 + a^

(3.10)

CONSTITUTIVE

MODEL OF A FLUID DIFFUSION THROUGH A SOLID

23

and pA + tr (cr3L3) + tr ( V l / ) + m ■ (v s - vf)

> 0.

(3.11)

We shall assume that the solid and the fluid are incompressible in their pure states. Then, the motion of the mixture has to satisfy the volume additivity constraint 2.52. Therefore, we have

-0) + ^ MlPo «l

(3.12)

= l,

and

-pA + tr (a3L3) + tr ( ^ / L / ) + m ■ (v s - v ' ) 1

+P —trl/ + Pi

p\.Tf PpRT

t,L' +, v niP K -(v-vO 'Wo

>o,

(3.13)

where P is the Lagrange multiplier associated with the constraint 3.12. We now need to postulate the constitutive structures for A3, A*, A, cf", a1 and m. From our experience with both an elastic solid single continuum and a linearly viscous fluid continuum, we might expect the constitutive response functions to depend on p3, F3, pf and I / . We also expect the various interactive mechanisms that were discussed in Chapter 2 to come into play and thus the constitutive relations may also depend on the relative velocity v* — v-^, a properly frame invariant form of the relative acceleration a and possibly other quantities like higher spatial gradients of F 3 and the velocity gradient of the solid I / . For the sake of simplicity, and motivated by earlier studies by Shi [77] and Shi, Rajagopal and Wineman [78], we shall assume that (A3,Af,A)

= (^,A',A)(/,F',v*-v'),

(
(3.14)

(3.15)

where a ==

£)(/) £J(/V V" Dt Dt

Z><3M X>Wv/

3.16 v Dt ' ' is frame indifferent (cf. Adkins [39]). Furthermore, we shall consider a special form for A, that is, A = Ao(pf,F°)

- — ,' £>£

+ ^ H ( p ' , F ' ) ( V - v>) ■ (v« - v ' ) .

(3.17)

24

MECHANICS OF MIXTURES

Here H is symmetric and positive semi-definite or it is identically zero. A simple justification for Eq. 3.17 is that it can be thought of as a Taylor expansion for A around v" — v ' = 0. Later, we shall see that H is related to the virtual mass effect, and thus we will refer to it as the virtual mass tensor. Substituting Eq. 3.17 into Eq. 3.13 yields

,fc n ^*Ur~ ^ PF*dW > an ~" 22pFP 'ikfc~dW a n ^^mm"" mml) \\v"n ~-vl) 3 ( 3 -p' v{) H v{) + P^S%J] v' jk (v'k --vQ+P^ jk(vi~P (v? [vi -~-vl)H "5*jA Po Po t 1 fdHmn , , f9A0 + - vL) (v'„ -vf)!** + j 6 - + 22ppPP ~-~ * mw1 y( n" - " v - ) 6 i j p^++ P ppP dp7dpf dp~rdpf ((vV™

*u

} -pf -p (v? (vf --v>)H - v() H]k (vtk-- VQ + P - l - ^ s J v*n jk(v' - v i ) + p l1-PPR - / p ^ \J f f dAodpf 1 ssdp dpfdH , L_ p3,dAodp dHmn ,s ,vL ,\ mn( , s, B p + + p p d fdx - vi)(< -vl) 2pdXk dPf yv~ P k

r- w ar* - ~2 a^rw ^ - > K - *) +pf P

^ a ^1

i a^a^

K -pfdF^dH P -(v^

dFfj dxk

^ dF^ dxk

(Dlfttf

f0

Defining cf , a

) (

/}

2 dxk a ^ v m zX»>«n 3 +P a

-"M-l**--D^J -pH-{ m 3 s0

(

vl) 2 Pdxk dF% v 1 " -vi) *W (r < » -*V

dAodF

0]

( < -- -^)>o. a^ (7j\ (« a^ # *°' (3-18)

and m° through

- ~vl)V *« = ^ % + ^ | f + \pF*^Ff (< ~vi)(v' " V-) (< " ") <j

=

n

+ps s(v' +p (i* t--vl)H w/) fl^ - v() + *f3, jk(v'(t£ k-vl)+rr°°

„ 1 4 =- T^7 **i = i-p 'Pp

/

(

S 8ii RP R

-

(3.19)

1 f9A0c sdHmn(s PP S Sii Pp V pp PP v r5p7
- vi) ((vt < "-<)

^

f

V (v;-vl)H ±P (vf - v() jkH (v° vQ al°, + all jk (v' k-vl) k- +

(3-20) (

and

m mfc =

P

p a (/p-\ dAodpf 1l sdpf8H dpfas^f . °\ + .OAodpf mn(s

d

««* \pfi/ ^

an

V a^fc

/

W

s

,

-<)(<--^) 2 P axfc a ^ ^ m

i sdF;0dHmnl ,

f W a

P -pp adFf^ «W^"" K ■«fl " <) n ai* ~ 2 ldxsxfck-dF?a n ^K "v"^

CONSTITUTIVE MODEL OF A FLUID DIFFUSION THROUGH A SOLID

25

( 3 - 21 )

+ H

' »[-Dr—Drl+ <

equation 3.18 becomes

a*°V»„ + 4 % / „ +m£ (wjJ - VQ > 0.

(3.22)

This equation will be called the reduced entropy inequality. Based on Eq. 3.22, we may propose (cf. Shi, Rajagopal and Wineman [78])

W3° = 7i (trD 3 ) I + 2 M l D ' +

72

(trD>) I + 2/z 2 D / - c ( w * - W ' ) ,

(3.23)

°f° = 73 (trD 3 ) I + 2 M 3 D ' +

74

( t r D ' ) I + 2 / J 4 D ' - c ( w > - W')

(3.24)

,

and P" PS I , f) mr=a—-j\v--V). (3.25) PoPR A lengthy but straightforward application of the reduced entropy inequality leads to

Pi > 0,

7J

2 + -Mi > 0,

2 74 + -M4 > 0,

/i 4 > 0, 2

4/Ul/i4 > (/u2 + /i 3 ) ,

' ! -n +T7M1J (73 + ~Pz; 3 7 V 3 c > 0, a > 0.

2 . 72 + 73 + - (Ma + Ms) (3.26)

In later applications, we will often choose cr3° = a^° = 0, that is, we will not incorporate the effects of viscosity in a" and a* This does not mean that the fluid diffusing through the solid is inviscid. The effect of the viscosity enters the theory through the interaction term m ° in Eq. 3.21 and Eq. 3.25 that accounts for the drag. The virtual mass tensor H will be assumed to be given by pH = p3pfU0.

(3.27)

In this equation, H 0 is constant, symmetric and positive semi-definite or zero. The assumption is motivated by the fact that in the hydrodynamics of a solid body moving in a Newtonian fluid, the virtual mass is proportional to the mass density of the fluid and the volume occupied by the solid body.

MECHANICS OF MIXTURES

26

The anisotropy of the solid can be captured by choosing the appropriate form for A0 in terms of the appropriate integrity basis (cf. Spencer [195]). If the solid is orthotropic with xr, x2 and a 3 -axes as its three principal directions, the appropriate form for ^o is (see Dai [196], Dai, Rajagopal and Wineman [84]) A0 = A0 (pf, I, II, EU,E22,

£33, {El2f

, (El3f

, (E23f)

,

(3.28)

where

/ = tr [F3 (F*)T] ,

II = i {[tr (Fs ( F ' ) T ) ] 2 - tr (Fs (F'fY} ,

E = - [(Fa)TF3-l] .

(3.29)

In the derivation of Eq. 3.28, we have appealed to the principle of material frame indifference, and the third principal invariant III = det \F3 (F3) = Jpo/p" does not appear in the expression for A0, as it can be replaced in terms of p? by virtue of Eq. 3.12. If the solid is transversely isotropic with a;3-axis being its principal direction, then A0 takes the form (cf. Dai and Rajagopal [83]), AQ = A0 [pf, I, II, E33, {El3f

+ (E23f)

.

(3.30)

In general, for the diffusion of a fluid through an anisotropic solid, the form for the specific Helmholtz free energy is too complicated and involves too many material constants for whose values we have very little, if any, experimental guidance. In fact, even in the simplest of problems of a transversely isotropic solid considered by Dai and Rajagopal [83], the expression for A0 is A0 = Kt (I - 3) + #2^33 + K3 {II -S) + KA(I+K6 (E213 + E\3) + a A

+ a 2 l?j)

.

3) 2 +

K ^ (3.31)

We see that there are as many as eight material constants. If the solid is isotropic, equation 3.30 reduces to (cf. Shi, Rajagopal and Wineman [78], Gandhi, Rajagopal and Wineman [79]), A0 = A0(p},I,Il)

.

(3.32)

Generally, we have to determine AQ from experiments. However, such experimen­ tal data is not available at the present moment. Shi, Rajagopal and Wineman [78], based on the theoretical work of Wall and Flory [197] on rubber elasticity, suggested the following constitutive expression for A0 for the mixture of an incompressible New­ tonian fluid and an incompressible isotropic rubber-like elastic solid with (3 = 0 and H = 0,

CONSTITUTIVE MODEL OF A FLUID DIFFUSION THROUGH A SOLID

A> K I+ lInn (1 Ao = K J - -33 + f l -- ^ - j

P'JI

,

J

27

(3.33)

where A' = 1ZT/2MC, 1Z is the gas constant, T the absolute temperature of the mixture, Mc the molecular weight between cross-links of the solid. Later, Gandhi, Rajagopal and Wineman [79] proposed another expression, A0 = K(I-3)

+

TIT ri —

-v

b ( l - v ) + X(l-i/)

(3.34)

Here, Vx is the molar volume of the fluid, x ls a mixing parameter which depends on the particular solid-fluid combination and v = p3 j p3R is the solid volume fraction. The models 3.33 and 3.34 are modifications to account for the infusion of a fluid in a neo-Hookean material. Generalizations to the Mooney model (see Mooney [198]) and the model due to Gumbrell, Mullins and Rivlin [199] have been used to study swelling. Of course, there is no dearth of models which could be used to study the problem of swelling or diffusion through nonlinearly elastic solids by appropriately modifying the numerous models available in the literature (cf. Valanis and Landel [200], Ogden [201], Knowles [202]). In order to provide closure to the theory, we also need to know A* or A3. In general, A* and A3 depend on p', F" and v" — v-' (see Eq. 3.14), and there is no reason to suppose A* and A3 have a similar structure. We shall assume special forms for A! and A' in later applications for the sake of simplicity and the purpose of illustration. The above formulation will be applied to study the problems of steady diffusion, unsteady diffusion with the existence of a moving singular surface and wave propa­ gations in the next three chapters. It would be appropriate for us to conclude this chapter by pointing out that the above formulation reduces to the constitutive formulation of an incompressible elastic solid undergoing large deformations in the absence of the fluid (cf. Green and Adkins [204]). In fact, much of the argument presented in this chapter is based on the pioneering papers of Rivlin [205,206,207] and the subsequent work by Smith and Rivlin [208], Smith [209], Pipkin and Wineman [210,211] and Truesdell [212] and generalizes them to account for the existence of a fluid.

Chapter 4 STEADY STATE P R O B L E M S As we have observed earlier, much of the developments in the theory of mixtures, since 1957, have been mainly restricted to the general formulations of the basic equations and constitutive models. Very few studies have been carried out which apply the theory to solve practical problems. One main reason for this lacunae of concrete applications is the difficulty in specifying boundary conditions for problems involving mixtures. In the case of a mixture composed of a solid constituent and a fluid constituent, while there are the balance of linear momentum equations for each constituent, usu­ ally only the total traction on the boundary of the mixture is known and this, in general, is not sufficient to determine the motion of the mixture. Since the individual partial tractions are not known, one needs a method for splitting the total traction on the boundary, and this should be derived on a physically sound basis. One way to achieve this is to assume that the boundary of the mixture under consideration is in a saturated state (cf. Treloar [81], Rajagopal, Wineman and Gandhi [82]). Such a state is characterized by the variation of the Gibbs free energy of dilution being zero, which leads to a condition that relates the stretch ratios of the solid and the applied surface tractions (the details are presented in Section 4.1). Following this treatment, several steady diffusion problems of a fluid through an isotropic (or anisotropic) nonlinearly elastic solid have been analyzed (cf. Shi, Rajagopal and Wineman [78], Rajagopal, Shi and Wineman [80], Gandhi, Rajagopal and Wineman [79], Gandhi, Usman, Wine­ man and Rajagopal [216], Rajagopal and Wineman [203], Dai and Rajagopal [83], Dai, Rajagopal and Wineman [84]). In this chapter, another way of furnishing boundary conditions to study the dif­ fusion of a fluid through an elastic solid is proposed, that is, the partial traction of the fluid constituent on the boundary is assumed to be equal to the product of the surface fraction occupied by the fluid on the boundary and the fluid pressure in the pure fluid region. Furthermore, the surface fraction of the fluid on the boundary is assumed to be equal to the volume fraction of the fluid on the boundary. This method of sphtting the traction may be expected to be appropriate for processes in which the 29

MECHANICS OF MIXTURES

30

diffusion is slow. A detailed treatment of the method of splitting the total traction is presented in Section 4.1. Based on this idea and the constitutive model presented in Chapter 3, the steady diffusion of a fluid through an isotropic rubber slab and the steady diffusion of a fluid through an isotropic rubber cylindrical annulus are studied in Section 4.2 and Section 4.3, respectively. In these applications, the diffusion processes will be assumed to be steady, isothermal and slow, the virtual mass effect and the external body forces will be neglected and the specific Helmholtz free energy function A is assumed to be described by Eq. 3.34. It is hoped that the results presented in this chapter will help to determine, albeit partially, the applicability of different constitutive models proposed and to design experiments to find material parameters that appear in these constitutive models.

4.1

Boundary Conditions

As we have remarked earlier, one of the thorny obstacles when it comes to putting mixture theory to use, is our inability to prescribe boundary conditions for traction boundary value problems. In the following, we shall discuss two different ways of augmenting the boundary conditions. The first method appeals to thermodynamics and derives a relation amongst the boundary tractions, the deformation of the solid constituent of the mixture, the mass densities of the constituents, etc., that holds when the mixture on the boundary is saturated (cf. Rajagopal et al. [82]). While such a boundary condition has been used with some degree of success in studying the large deformations of solids infused with fluids there are situations wherein it is inapplicable; in fact it leads to inconsistencies in the of boundary value problems A second method of augmenting boundary conditions has been nrnnnsed recently based on the sohtting of the traction adduced to above that hi LeW mechanical underpinning (cf Tao and Raiaeopal \S5]) It is also possihle iTJZZ boundary conditions by requiring continu tv in the'chemical n o W ^ l h ! ! S not investigate this possibility here. potential,

Boundary condition based on t h e notion of saturation Before embarking into the method for generating boundary conditions, a few words about the swelling of solids are warranted. As we shall be concerned with the diffusion of solvents through rubber and polymeric materials, our discussions will be pertinent to the phenomenon of swelling of such materials. In studying elastic solids from the perspective of continuum mechanics, we approach it from a phenomenological point of view. Rubber like elastic solids have also been modeled using a statistical approach (cf. Kuhn [217], Wall [218], James and Guth [219,220], Flory [221,222], Wall and Flory [197], Treloar [81,223], Yasuda et al. [224]), and in fact it is one of the few great success stories where phenomenological and statistical modelling speak with one voice. The study of the swelling of rubber due to a solvent, which is in

STEADY

STATE

PROBLEMS

31

equilibrium, was first studied by Flory [225] and Huggins [226]. They determined the configurational entropy due to mixing, using statistical methods, and used this to determine a formula for the Gibbs free energy of dilution. This equation bears the name the Flory-Huggins equation. The Gibbs free energy of dilution is assumed to be the sum of a part due to mixing and the other due to the strain, and setting the variation in the Gibbs free energy of dilution to be zero at equilibrium leads to a relation between the applied traction and the stretch in the solid (cf. Flory and Rehner [227], Gee [228]). The phenomenon of swelling from the above perspective is discussed in great detail in the book by Treloar [81]. Our discussion of boundary conditions here draws liberally from the previous work on swelling, but the problem is studied within the context of mixtures, and the intent is completely different; the aim here is to develop boundary conditions. Let us consider an incompressible isotropic solid in contact with an incompressible fluid in a fluid bath. The experimental work of Southern and Thomas [229] indicates that as the fluid diffuses, the boundary is saturated. By saturation, we mean a state in which a small element of the solid adjacent to the fluid is in a state in which it cannot absorb any more fluid, that is whatever fluid enters the elemental volume along the boundary has to exit through the elemental volume so that there is no accumulation of the fluid. Southern and Thomas [229] observed "When a sheet of rubber vulcanizate is immersed in a solvent, the surface layer takes up its equilibrium amount of the liquid, virtually instantaneously, the liquid subsequently diffuses into the bulk of the rubber.'' We interpret the notion of equilibrium to indicate that the mixture adjacent to the fluid, in its swollen state (extent of deformation) and under the applied stress, is in a saturated state. Thus, an elemental volume of the mixture in such a saturated state cannot be diluted further, i.e., it cannot absorb any additional fluid. This condition can be expressed in the form that the variation of the Gibbs free energy of dilution is zero. Assuming that the whole mixture is in a state of saturation, Treloar [81] derived an equation for the saturated state that is essentially the Flory-Huggins equation. And such an approach leads to the governing equation (field equation) for the mix­ ture as a whole. Our intent here is quite different. The appropriate field equations are the balance equations presented in Chapter 3. However, if the boundary alone can be assumed to be saturated, then a procedure similar to that adopted by Treloar [81] leads to a boundary condition, which can be used to augment the usual trac­ tion boundary condition. The fact that we use the notion of saturation to generate boundary conditions, and not field equations, cannot be overemphasized. Suppose we have a cube of material that is in the same state of deformation as that of a material element on the boundary of the mixture. The cube infused with the fluid in its swollen state is in equilibrium and the deformation of the cube is homogeneous. Let us now carry our thought experiment a little further. We shall suppose that the saturated cube which is in equilibrium is isolated in the sense that no fluid could enter or leave it (we could think of the cube infused with the fluid to

MECHANICS

32

OF

MIXTURES

be covered by an impermeable elastic sheet). Furthermore, suppose that we shghtly deform the cube and this deformation is carried out isothermally. Since the cube is isolated, by the first law of thermodynamics, we have 8W + 8Q = Ss%

(4.1)

where SW is the work done on the swollen cube, SQ is the heat transfered and Se* the change in the internal energy of the swollen cube. Since the Helmholtz free energy A* is defined through A' = e'-Tn',

(4.2)

SW + SQ = 8 A* + TSrf.

(4.3)

we have

In deriving the above equality, we have used the fact that the process is isothermal at absolute temperature T. If we now suppose that the deformation is reversible, and this is a critical assumption, then SQ = T6n\

(4.4)

SW = SA*.

(4.5)

and thus equation 4.3 reduces to

Let us take our thought experiment one step further. To the cube that is in equilibrium in a saturated state, of total mass m and specific Helmholtz free energy of the mixture A, let us suppose that we add a small amount of mass of fluid such that the new mass of the cube is m + 5m and the corresponding specific Helmholtz free energy is A + SA, so that the Helmholtz free energy of the cube is (m + Sm)(A + 8A). Ignoring higher order terms and equating the variation in the work done to the variation in the Helmholtz free energy of the cube, we obtain SW = m (SA) + (Sin) A.

(4.6)

The above relationship of course holds only under the special assumptions that axe supposed to hold in our thought experiment. In a real problem, an element on the boundary of the mixture adjacent to the fluid may or may not meet such restrictions. Yet, we shall assume that the relationship 4.6 holds, which leads to a relation on the boundary that can be used as a boundary condition. It is worthwhile to remark that equation 4.5 is derived on the assumption that we have a closed system, but equation 4.6 is obtained by treating the system as open, as we have added a mass 8m. Whether such an approximation is reasonable can only be borne out by the predictions of the assumption.

33

STEADY STATE PROBLEMS

The above motivation is slightly different from the derivation provided by Treloar [81] where in a saturated mixture, the variation in the Gibbs free energy of dilution being zero is employed. Under appropriate assumptions and approximations this also leads to Eq. 4.6. In the approach used by Treloar, however, we are faced with defining the pressure P of the swollen cube, which is taken to be the pressure of the bath. However, in mixtures of non-linear materials the pressure and the normal stresses are not the same and certain care has t o be exercised. We shall not get into a detailed discussion of these issues here as it can be found in standard treatises devoted to non-linear mechanics. Moreover, Treloar does not have a mixture in our sense and there is no cognizance of partial tractions and their distinction from the total traction to which the fluid pressure in the bath is related. Now, consider a unit cube, which after absorbing the fluid and attaining a state of saturation is of length At, A2 and A3 in the Xi, £ 2 a n d £3 directions, respectively. Suppose the tractions acting along the x% direction on the appropriate faces of the cube are given by t^. Let us suppose that the edge of the cube A; elongates by 5A; due to the addition of a small amount of the fluid. A trivial calculation yields SW —
(4.7)

Next, since A = A(l,II,pJ),

(4.8)

we obtain

dA

8A

dA „ ,

W> ++ — ->7 V =; -& > H ' +/ + &dpi ' ■. 81 dll

SA =

"

<">

Recall that I = A2 + A2 + A|,

(4.10)

II = (AtA,) 2 + (A2A3)2 + (A 3 Aj) 2 ,

(4.11)

7 / / = (A 1 A 2 A 3 ) 2 ,

(4.12)

and by virtue of the volume additivity constraint 3.12

"'^('-vbJ' Using Eqs. 4.10 through 4.13 in Eq. 4.9, we find that

< 4 - 13 '

MECHANICS OF MIXTURES

34

8A = 2 ^ (A^Ai + A 2 a 2 + A 3 a 3 ) + 2 — [(\l + \l) X18X1 + (A| + Aj) A25A2 4- (A? + A*) A35A3

dA

PR

(A 2 A 3 5Aj + A 3 Ai5A 2 + A X A 2 5A 3 )

(4.14)

m = pAiA 2 A 3 ,

(4.15)


which by v i r t u e of E q s . 4.12 a n d 4.13 is given by f .

m

PR ~ PR \ A X \ I Aj/\2A3

PR +

\A \A A\

(4.16)

1 2 3-

Thus 8m = p'R (XlX28X3 + X3X18X2 +

X2X38Xi).

(4.17)

Substituting Eqs. 4.7, 4.14, 4.16 and 4.17 into Eq. 4.6, we obtain 9A, 011A2A3 - 2:-jX1m f

dA - 2-QJJ (AJ + A3) Aim

dA

X2X3m

f

-^V(A7W~ f

2

]

TAl

<9A dA / 2 2\ o-22A3Ai - 2 — A 2 m - 2 — (A 3 + \ ) X2m f 9A ~pRR„J<\

+

Pfi

X3X1m

f

\ \ \2 ~

PRAX^

5A,

dp' (AiA 2 A 3 ) cL4, 9A / , 2\
XiX2m

~PR—,T.

\ \

N2 "

"'^(AiAaAa)

PRA^X2 8X3 = 0.

(4.18)

Now resorting to a standard argument of varying the 8Xi arbitrarily, we estabhsh that


PRA

(pR-pf)P

dA cM , 9A 0 dA s. + 2p dp~f m^ + 2pWi^-2pmx-"WiAi

,

„.

(4 19)

-

STEADY STATE PROBLEMS

35

where i = 1, 2 or 3 and no sum on i is intended. It follows from Eq. 4.19 that 9 ff

=

PR+{pR~

Pf) P~

A

dA \ „

1 + 2/0 f — + / B 81 dll

dA

an

, B2

(4.20)

At this juncture, it is important to pause and recognize several interesting features implied by Eq. 4.20. The stress in the saturated cube is not indeterminate, and is akin to that in a compressible material and the following remarks of Treloar [81] are relevant: "... whereas in an incompressible rubber (i.e., whether unswollen or swollen to a specified state of fixed extent) the volume and hence the state of strain is un­ affected by the superposition of an arbitrary hydrostatic pressure; in a compressible rubber or in a rubber considered to be in equilibrium with respect to the swelling liquid, this is no longer true, since any superimposed hydrostatic pressure will reduce the volume of liquid content. In this aspect, therefore, a swollen rubber in contin­ uous equilibrium with a surrounding liquid, may be regarded from a purely formal standpoint, as having mechanical properties equivalent to those of a compressible material." However, we note that if the swollen mixture is a closed isolated system, then it would not be akin to a compressible material, as the fluid cannot leave the system. The above thought experiment is assumed to hold only for elements belonging to the boundary that are adjacent to the fluid and assumed to be in a state of saturation, and not to material points in the interior which may or may not by saturated. This leads to a boundary condition, rather than a condition that holds throughout the mixture. It is also imperative to recognize that several simplifying assumptions have to be made in arriving at Eq. 4.20, assumptions that need to be reexamined carefully. Even so, when such an approach was conceived by Rajagopal et al. [82], it was felt that in the absence of any other means of generating boundary conditions, it was worthwhile determining where such an approach, inexact as it is, might lead us. Rajagopal and Wineman and their co-workers have used such an approach to solve a variety of bound­ ary value problems to study fluids diffusing through isotropic and anisotropic solids undergoing large deformations (cf. Shi, Rajagopal and Wineman [78], Rajagopal, Shi and Wineman [80], Gandhi, Rajagopal and Wineman [79], Dai and Rajagopal [83], Dai, Rajagopal and Wineman [84]). While the resolution of boundary value prob­ lems involving mixtures, with the boundary conditions augmented as above, leads to reasonable results in most cases, we shall show that it also leads to a difficulty which can only be reconciled by assumptions that may or may not obtain at the boundary. To illustrate the difficulty, let us consider the steady diffusion of an incompressible fluid through an isotropic rubber slab. The analysis to be presented in Section 4.2 indicates that the total normal stress component cr33 is constant throughout the slab. However, Gandhi et al. [79] assigned different values to <J33 at the top and the bottom of the slab, i.e.,

36

MECHANICS

a

^\z=H0

OF

= ~1h,

MIXTURES

(4-21)

and a

&\z=Q = -1o-

(4-22)

A way of resolving this c o n t r a d i c t i o n is b y n o t a p p l y i n g t h e b o u n d a r y condition 4.22 b e c a u s e of t h e presence of t h e rigid p o r o u s p l a t e . Therefore, we need t o seek new m e t h o d s t o a u g m e n t b o u n d a r y conditions. I n t h e following, w e shall present a purely mechanical m e t h o d t o achieve this goal. It is also i m p o r t a n t t o recognize t h a t t h e p r o b l e m u n d e r consideration m a y n o t a d m i t t h e simple s t r u c t u r e a s s u m e d for t h e deformation a n d flow field, a n d t h e b o u n d a r y conditions t h e r e b y lead t o n o inconsistencies.

Method of splitting the total traction on boundary Consider t h e m i x t u r e c o m p o s e d of a n incompressible elastic solid a n d a n incompress­ ible fluid. Let x b e a point on t h e b o u n d a r y of t h e m i x t u r e region a n d let t* a n d t ' d e n o t e t h e p a r t i a l surface t r a c t i o n s at x associated with t h e solid a n d t h e fluid, respectively, t h a t is, (
(CT/)Tn = t / ,

(4.23)

where n is t h e u n i t o u t w a r d n o r m a l vector a t x . W e usually know t h e t o t a l surface t r a c t i o n t 3 + t' a t x , in a traction b o u n d a r y value p r o b l e m . W e now p r o c e e d t o provide a m e t h o d for d e t e r m i n i n g t h e individual p a r t i a l t r a c t i o n s t * a n d t s Consider a small c u b e 8V of m i x t u r e t o which x belongs. L e t pf b e t h e m a s s density of t h e fluid a t x a n d 8V* d e n o t e t h e volume occupied b y t h e fluid c o n s t i t u e n t in 8V Trivially 8Vf f

sv ppif ~sv ~ P n S

(4.24;

here t h e incompressibility of t h e fluid c o n s t i t u e n t in its reference s t a t e h a s b e e n used. Now a s s u m e t h a t t h e b o u n d a r y surface fraction 8S*/6S occupied by t h e fluid constituent a t x is given b y 8Sf

_ 6V*

~8S ~ W

(4 25)

-

and t* a t x is given b y 8S*_. tf = 8S

(4.26)

STEADY STATE PROBLEMS

37

where t is the total applied surface traction at x. Then, equations 4.23 through 4.26 yield (
(4.27)

(v'fn

(4-28)

Similarly,

= ^-t. PR

Notice that equations 4.23, 4.27 and 4.28 are consistent with the volume additivity constraint 3.12. Thus, one could take Eq. 4.27 as an additional boundary condition to ensure closure to the problem. The condition 4.27 is clearly different from Eq. 4.20 proposed by Rajagopal et al. [82] as we have invoked no notion of saturation. In this sense, the condition 4.27 is more general in that it can be employed even when the boundary is not saturated. However, even when the boundary is saturated, the condition 4.27 is not equivalent to the condition 4.20. Moreover, it is not clear whether these two methods are the only ones that are available. For instance, we could require that the chemical potential be continuous across the boundary. The only means of determining which of these boundary conditions is more appropriate is to carefully conduct a series of experiments and to compare the predictions of the theory for the various boundary conditions in order to evaluate their efficacy. It is with this purpose in mind that we reinvestigate the boundary value problems studied earlier by using the saturation boundary condition, with the new boundary condition proposed above. We shall then compare the predictions of the theory with experimental results.

4.2

Diffusion of a Fluid Through a Rubber Slab

Consider an incompressible isotropic elastic solid slab of thickness H0 with /3 = 0 (Figure 4.1). The bottom of the slab is supported on a rigid porous plate. Let an incompressible fluid diffuse through the solid slab steadily and slowly under the pres­ sure difference qh — qo where <#, and q0 are the fluid pressures in the pure fluid regions above and below the slab, respectively. Let us suppose that the slab is deformed due to the infusion of the fluid and the external stretching, and let the velocity of the fluid be given by v* = vly = 0,

v* = v(z),

(4.29)

and the deformation of the solid be given by x = XX,

y = AY,

z = f(Z).

(4.30)

CO

00

Figure 4.1 Configuration of solid membrane and mixture layer.

I o

I

STEADY STATE PROBLEMS

39

Here [yl, v£, ti/J is the diffusion velocity of the fluid through the slab, (X, Y, Z) and (x, y, z) are the coordinates of a solid particle in the undeformed reference configuration and in the current deformed configuration of the mixture, respectively (Figure 4.1), and A is the stretch ratio in the x-y plane. The Cauchy-Green stretch tensor B = F 3 (F 3 ) associated with the deformation of the solid constituent is obtained from Eq. 4.30, A2 0 0 A2 0 0

B =

0 0 f2

(4.31)

where the prime denotes differentiation with respect to Z. Thus, the principal invari­ ants of B are / = 2A2 +

fa,

(4.32)

II = A2 (A2 + 2/' 2 ) ,

(4.33)

/ / / = A 4 /' 2 .

(4.34)

and

The balance of mass equations for the solid and the fluid constituents are PR P = A2/''

(4.35)

and 0. sW=' dz

(4.36)

Integrating Eq. 4.36 yields (4.37)

F,

p-'v

where F is the fluid mass flux in the z direction. It follows from Eqs. 4.29 and 4.30, 3.19 and 3.20 that the partial stress components for the solid and the fluid constituents are given by 5

5

° U = ^22 -

_fl-P 5

+ 2pK\2

(4.38)

PR Pn

^3 = -4^+ 2 ^/' 2 '

(4.39)

PR "12 "- a23 = °31 - °I

(4.40)

MECHANICS OF MLKTURES

40

4

-

"22 -

°33 -

Pf --1P-

- PPfA,

(4.41)

PR

r[2 = a{z = 4

= 0,

(4.42)

where A A= = -^-~[]n(l-u) V {PR) ^

2 ^}, - ! / )+ !u/ ++ XX^ ]:

- = 4 = (AY)"1 = 1-4PR

K

(4-43)

(4-44)

PR

Here, we have assumed that A* = A3 We shall comment on the shortcoming of this assumption later. The components of the diffusive body force, by virtue of Eqs. 3.21 and 3.25, are mi = ra2 = 0,

(4-45)

m3 = 4 (4) P + P'^f " ?iKTfn - a ^

( 4 - 46 )

dz \pRJ dz dz p'RpfR Substituting Eqs. 4.38 through 4.42 and 4.45 and 4.46 into the balance of linear momentum equations 3.3 and 3.4 results in 9P = 0,

„

$P -

(4.47)

,

= 0,

- 4 f + T ( 2 ^' 2 ) - ^"-f + ^ pj, dz

dz ^

'

dz

,

(4.48)

+ <44« = 0, dz

psR p'R

(4-49)

-P + 2pKfri - ppfA) = 0. (4.50) dz In the derivation of Eqs. 4.49 and 4.50, the inertia of the fluid has been neglected based on the assumption that the diffusion process is slow. That this assumption is reasonable is borne out by the experimental results of Paul and Ebra-Lima [188] where the rate of diffusion is a few cubic centimetres per day. Equations 4.47, 4.48 and 4.50 yield P = P(z),

(4.51)

STEADY STATE PROBLEMS

41

033 = ~P + 2pKf'2 - ppsA = constant,

(4.52)

that is, the total normal stress component 033 is constant throughout the slab. Eliminating P from Eqs. 4.49 and 4.50 and using Eq. 4.37 results in

4 ± (2PA7-) + 4 ^ WA) - p>Ad-f + pfK^ + a ^ ^ = 0, PRdz

K

PRdzK

'

dz

dz

(4.53)

pRP'R

or

P

- (2pKf2)'

Pfi

(ppfA)'

+4 '

PR

- p'Ap" + 2p'Kf'f''

K

+^

f

'

= 0.

(4.54)

PRPR

Introducing dimensionless variables through

7

Z

}

f H0

n0 P_

**

P pR

7_PRA

P = ^ &

-/ S

A =^ , = « '

P p'

<*FH0

P =f

^ . "(PR)'*

(4-55)

equation 4.54 can be rewritten as pf {2pfa)'

+ p" (ppfA)'

- ^-p-Af

+ 2pff'f"

+ £ = 0,

(4.56)

where the prime denotes differentiation with respect to Z. The expressions for p°, p' and p in terms of / ' can be obtained from Eqs. 4.35 and 3.12, P8 = - ^ , A 2 /' P =1

-, A 2 /'

(4.57) (4.53)

and ~ -

1

1 -u (& - i ) > + ( | - ' )/ A ^ 2-/'

W

(4 69)

'

To solve for / from Eq. 4.56, three boundary conditions for / are required because F is also undetermined. Clearly, at the top surface corresponding to z = f(H0) (or Z = H0), the total applied traction can be prescribed as (Figure 4.1)

MECHANICS

42


at z = f(H0).

OF

MIXTURES

(4.60)

We now require that the additional condition 4.27 hold, that is, «& = - T " ® »

at z = /{#<,)•

(4.61)

It follows from Eqs. 4.35, 4.39, 4.41, 4.60 and 4.61 that /

S

+ 2 ^ = °>

a t Z = l.

(4.62)

At the bottom of the slab 2 = 0 (or Z = 0), we have a rigid porous plate and the solid constituent is assumed to adhere to the plate, that is, /(0) = 0.

(4.63)

Furthermore, we shall assume that the permeability of the porous plate is such that it is as though the layer of the mixture were in contact with the pure fluid below the porous plate in the sense that 4z = - 4 ? o ,

at z = 0,

(4.64)

PR

which is derived from Eq. 4.27, that is, the porous plate has little effect on the flow of the fluid constituent. Now, equations 4.52 and 4.60 imply that - P + 2pKf-

- ppfA = -qh,

at Z = 0.

(4.65)

Z = 0.

(4.66)

Thus, it follows from Eqs. 4.41, 4.64 and 4.65 that 2pf'2 + p°pA = tZJh.,

at

PRK

Equations 4.62, 4.63 and 4.66 are the three boundary conditions for / given by Eq. 4.56. The ordinary differential equation solver D T P T B from the International Mathe­ matical and Statistical Library (IMSL) was used to solve Eqs. 4.56, 4.62, 4.63 and 4.66 numerically. This subroutine is based on a fifth and sixth order Runge-Kutta method and a multiple shooting method. The error tolerance of the numerical solu­ tion was of the order of 10~ 6 . The material constants used in the numerical computation were chosen based on the experiments of toluene diffusing through gum rubber by Paul and Ebra-Lima [188]: p'R = 0.862 S m/cc, p}R = 0.869gm/cc, V1 = 106cc/mol, Mc = 9l5Lgm/mol, X = 0.414, a = 1013gm/cc- s. The absolute temperature was T = 303.16/f, the universal gas constant TZ = 8.317 x 10 dyne ■ cm/Tnol ■ K and the original thickness of the slab H0 = 0.0625cm.

STEADY

STATE

PROBLEMS

Figure 4.2 Variation of flux with pressure difference (Layer).

43

44

MECHANICS

Figure 4.3 Variation of mixture layer thickness with pressure difference.

OF

MIXTURES

STEADY

STATE

PROBLEMS

Figure 4.4 Distribution of fluid density in layer.

45

46

MECHANICS OF MIXTURES

Figure 4.2 shows the experimental data for the fluid flux versus pressure difference by Paul and Ebra-Lima [188] which correspond to the case of A = 1.739 and the flux versus pressure difference curves of the numerical results for A = 1 and A = 1.739. The numerical results agree qualitatively with the experimental results. The quantitative difference may be attributed to the omission of the viscous effect of the fluid in the constitutive equations 3.19, 3.20 and 4.64, the structure of A in Eq. 3.34, the assumption that A' = A3, and the volume additivity assumption 3.12. Obviously, stretching (A > 1) increases the fluid flux through the slab. The relation between the thickness of the mixture layer and the pressure difference is presented in Figure 4.3. Stretching decreases the thickness of the mixture layer. Figure 4.4 shows the distribution of the dimensionless fluid mass density p* with the material coordinate Z for different pressure differences. Corresponding to a large pressure difference, p* varies significantly with Z. We like to comment that the assumption that A, = Af is of course hardly correct and we note that the assumption results in As = A. While each constituent depends on the kinematical and thermal variables associated with both constituents, we do not expect the form of the specific Helmholtz free energy to be the same for both constituents. This assumption is of course extreme, but we find that even in this case, the prediction of the theory is reasonable. We would be remiss if we did not point out that the theory does not indicate a ceiling flux as clearly as the results with the saturation boundary condition (cf. Shi et al. [78], Gandhi et al. [79]). It is also worth pointing out that in the analysis of Shi et al., Gandhi et al. and the subsequent work, because of the way in which the boundary condition is generated and the structure of the theory, the constitutive expressions for A, and Af are not needed and all that is necessary is knowledge of the form for the specific Helmholtz free energy for the mixture A. While this can be viewed as a strength, it is also ignores an important aspect of the theory, namely the issue of closure.

4.3

Diffusion of a Fluid T h r o u g h a R u b b e r Annulus

Consider an incompressible isotropic elastic solid cylindrical annulus with inner radius Ri and outer radius R0, which may be axially stretched (Figure 4.5). The fluid pressure <£ on the inner cylindrical surface is different from the fluid pressure q0 on the outer cylindrical surface, and the fluid diffuses radially through the annulus under this pressure difference. As the solid annulus absorbs the fluid and swells, the inner cylindrical surface R = Ri changes its position to ri = r(Ri) and the outer cylindrical surface R = R0 to r0 = r(R0). Assume that the diffusion process is steady, isothermal and slow and the inertia of the fluid motion can be neglected, and that the deformation of the solid constituent and the motion of the fluid constituent in

STEADY

STATE

PROBLEMS

a

'i S
3

E

.a 9

T3

0

*S c

3 M

M

'■3

« M

5 U

47

MECHANICS OF MIXTURES

48 the region of the mixture are given by r = r(R),

XZ,

6 = 0,

(4.67)

and vfr =

v{ = v{ = 0,

v(r),

(4.68)

where (R, 0, Z) are the coordinates of a solid particle in the undeformed reference state and ( r , 6, z) the coordinates of the same particle in the current deformed mixture, A being the constant stretch ratio in the axial direction. The Cauchy-Green stretch tensor B of the solid constituent is obtained from Eq. 4.67, X2r

0

0 0

A2 0 0 A2

B =

0

(4.69)

where dr dR' and the principal invariants of B are K

=

Xe=

r R'

(4.70)

/ = A2 + XI + A2,

(4.71)

/ / = (ArA9)2 + (A(,A)2 + (A r A) 2 ,

(4.72)

III = (A r A e A) 2 .

(4.73)

The balance of mass equations for the solid constituent and the fluid constituent yield 3

P

PR

z

\

r

(4.74)

\ Q \

and

d

(rpfv) dr Equation 4.75 can be integrated to give rp v

= 0.

F 2TT'

(4.75)

(4.76)

where F is the constant which denotes the mass flux of the fluid constituent through a unit length of the annulus. Combining Eqs. 4.74 and 3.12, we obtain

STEADY STATE PROBLEMS

49

pt=pf o =

1

(4(4.77) 77)

« i -db)'

1

p

1

PR(l-

-

)

ATAQA/

and f

,

p-

P'R-PR

(4.78)

ArA«A

From the constitutive equations 3.19 and 3.20, the form assumed for A in 3.34 and the assumption that A3 = A, the partial stress components for the solid constituent are given by
+ 2pKXl + 2pK\l

r,9,z, i = r,6, z,

(4.79)

PR

=< =< °i, = <e =

= 0, =

(4.80)

and the partial stress components for the fluid constituent are given by

4 = -VPR

pf vl = -^yP-ppfA, - PPfA,

i = r,0,z, =r,8,z,

(4.81)

PR

4 82 (4.82)

==o, 4B ==4,4,==4,4,= °>

(- )

4Bwhere A A

2

+ XxS], v] , = 7 ^ T - ±[ln(l-v)+v- u) + v + =

(4.83)

VX»21

(PR)

1 ^ 4 p"f = 3-^ =_ 1-4.P* A AgA

(4.84)

1

p'RR

ArrAflA

ppfR^

The total stress components of the mixture are f f an 2pK\\2 -- ppPP A, A Cii = = -P + 2pK\

i = r,9,z, r,6,z,

cr
(4.85) (4.86) (4.86)

The diffusive body force components are given by p ' pf d p fK {Xl+Xt)A A - a a TV /K± (4)p+P°A -£™* = -T ( 4 ) + "*¥ V^ dr \PR) dr ~ ? T ( * + «) " PRPR mT =

d

dr\pR)

dr dr \ me e = m = ro mzz = 0.

(4 87) (4.87)

-

p f i p£

(4.88)

The balance of linear momentum equations for the solid 3.3 and for the fluid 3.4 reduce to

50

MECHANICS OF MIXTURES

__£dp pftdr pR dr

+

d ^ dr v

A

2 )

/ y

+

-dpf 2p^ ^ . r vv - A'J ) - drdr

j

g

f

,2\ p s p^ sf , , i i / . , xf) + Pp ( TK— A — (Xl Af T++ Aj + a — ~vTV == 0, dr V r '/ p-^ fp / eJ

dr^ r

(4.89) (4.89)

pR p R

'

and (4.90) - § + ± (2pKXl - pp'A) + ** (XI -- %) XI) = o. 0. r yv dr dr vv 7 dr dr '' r in Eliminating Eqs. 4.74 Eqs. 4.89 4.90 and Eliminating Pfrom P from Eqs. 4.89 and and 4.90 and using using Eqs. 4.74 and and 4.76 4.76 results results in

0(^ )+ ^ + ,)-

-psA--f- dr

P' d ,

-~~r \pp

/J

9

pRdr V + /PJ2pA'(A 2 p t f , --A2)+ x ,« ^ == 00,, 9/ r v p£ r V"r •) 27rp^i?A 2npfRRKX 2X rA2A Ph 2

(4.91) (4.91)

or or p' d

^

A * l )) -- ^ ^ + (W * ++

(2«) +

(2pKX*r)+pfK± P1^

P3 d (P/A) Pfi dR

+g^

, p'2pA"A r ^ 2

+

+

+ ^^^-^ + p£ *A„
(^) (4.92) (4.92)

Introducing dimensionless quantities R R = —,

r =

Ro

(IT* r dr dR'

„^

Kr

=^

s

T r R' R

P3'-

Ae

° ^i'w

x =

Pf f p pP Pf = f Q = —— f PR f p PPR R 'P Pn R PR Cv /* aF ^ _ p(i^ A p -

3 p _ P

T

s '

A=

ir>

F=

(4yv (PR)2K

(4(4.93) 93)

.

we find find that that 22pf^(pXl)+p^ ^ K(fXl) +tf±:$(J* + A3) _ *rA& rJ H dR

dR

'

Pn dR

r (X:2r - \i) X2e+) — +— = ^L-- = - = 0. 0. R +^ V 2-KRXIX RX *>1 2KRXIX

+ 2P~~ e

sf

Also, p% p°, pp and p can be expressed as

+ ?J= (p~pfA) ^WA) (4.94) (4.94) V

'

STEADY STATE PROBLEMS

51

P* =

1

(4.95)

ArAsA

1

p' = l

(4.96)

ArA#A

and P= 1 +

PP-PR XAeX

(4.97)

Equation 4.94 is a differential equation for r(R). However, for the convenience of numerical computations, it can be taken as a differentia] equation for Ar and Xe, and another differential equation for Xr and Xe can be derived from Eq. 4.70: d\e

_K-\e

dR~

R

(4 QS^

•

Now, Equations 4.94 and 4.98 are two equations for XT and Xe . In order to solve the above equations, we need three boundary conditions since the non-dimensional flux F is also unknown. We once again invoke the traction splitting assumption 4.27 to conclude that at R = R,, PR

(4.99)

'

a'rr + a}„ = -q;, Pf

at R = Ri, at R = R0,

(4.100) (4.101)

PR "


-q0,

at R = R0. (4.102)

which lead to at R = — ,

2X1 + p»A = 0,

H0

(4.103)

and 2 A r 2 + p M = 0,

at/U1.

Next, integrating Eq. 4.90 subject to Eqs. 4.100 and 4.102 results in 2pAr f1 ?fh. (tfr _ \A dR = q°~qi

h

XeR ^

''

" fi&K

(4.104)

(4.105)

52

MECHANICS OF MIXTURES

Now, Equations 4.94 and 4.98 can be solved for Ar and \g with the boundary conditions 4.103 through 4.105. Essentially, the same computational method as that used in Section 4.2 was employed. The values of the material parameters are the same as those used in Section 4.2. Figure 4.6 shows the dimensionless fluid flux versus fluid pressure difference curves for the case of A = 1 and A = 1.739 with Ri/R0 = 0.9. Depending on q0 — qi > 0 or q0 — 5; < 0, the fluid diffuses radially inward or outward. Clearly the outward flux of the fluid is larger than the inward flux under the same absolute value for the pressure difference. In the case of the fluid diffusing inward, the fluid flux increases as the pressure difference increases, and eventually it reaches a ceiling flux. Stretching the annulus axially increases the fluid flux for the outward diffusion, but for the inward diffusion, this is true only for some range of pressure difference. This phenomenon may be explained as follows. Under a given pressure difference, the fluid flux is proportional to the stretching ratio A and to the diffusion area. According to Figure 4.7, the diffusion area corresponding to A = 1 is relatively larger than that corresponding to A = 1.739, as the pressure difference q0 — qi increases. Thus, if the pressure difference for inward diffusion exceeds some value, the flux corresponding to A = 1.739 is smaller than the flux for A = 1. The variation of fluid mass density p? with the material coordinate R for different pressure differences is presented in Figure 4.8. Obviously, p' does not depend strongly on R and p' decreases as qa — qi increases.

STEADY

STATE

PROBLEMS

Figure 4.6 Variation of flux with pressure difference (Annulus).

53

54

MECHANICS

OF

MIXTURES

Figure 4.7 Variation of annular mixture domain with pressure difference.

STEADY

STATE

PROBLEMS

Figure 4.8 Radial distribution of fluid density.

55

Chapter 5 DIFFUSING SINGULAR SURFACE There has been a great deal of work on the problem of phase transition which is concerned with the movement of a singular front. The treatise of Crank [231] and the article by Fasano and Primicerio [232] provide authoritative reviews of this topic. Much of the early mathematical works on the subject trace their origin to Stefan's seminal work [233] in this area. The popular mathematical approach to study such problems now is the phase-field approach, which appeals to an order parameter whose extreme values are taken to represent the pure liquid and solid phases. The heat con­ duction equation is modified, and an additional equation that describes the evolution of the order parameter provides closure to the problem. The original work on such an approach is due to Landau [235] (see also Hohenberg and Halperin [236], Lan­ dau and Lifschitz [237], Fix [238], and Caginalp [239] for a detailed discussion of the theory). The inadequacies of the Landau-Ginsburg theory were discussed with great clarity by Stanley [240]. Recently, Baldoni and Rajagopal [241] have developed a the­ ory to study the problem of phase transitions that account for the different material symmetries of the different phases. In this chapter, we discuss an unsteady diffusion process which involves a moving singular surface but without phase transition. Behind the singular surface, there is a mixture of a solid and a fluid, while ahead of the moving singular surface there might be a single solid or a mixture of a solid and a fluid with properties different from the mixture that is present behind the surface. In the treatment of this unsteady diffusion process, in addition to describing the governing equations for the mixtures ahead of and behind the singular surface, we also need to know the evolution equations for the singular surface. We will derive the evolution equations governing the motion of the singular sur­ face in Section 5.1 based on the results presented in Chapter 2. Then, these evolution equations are taken as free boundary conditions to solve two concrete problems in Section 5.2 and Section 5.3, that is, the unsteady diffusion of a fluid through a non57

MECHANICS OF

58

MIXTURES

linearly elastic slab and a nonlinearly elastic cylindrical annulus. The ideas developed here can be generalized to the case of a singular surface in a mixture of two fluids or the movement of a singular surface behind which we have a mixture of two fluids while ahead of the surface we have a single fluid. Such a problem models the spreading of a fluid contaminant in a pure fluid.

5.1

Analysis of a Time-dependent Diffusing Sin­ gular Surface

The physical nature of a system in which a fluid (denoted by subscript 2) is diffusing unsteadily through a porous solid medium (denoted by subscripts 1, 3, and s) contain­ ing another fluid (denoted by subscript 4) may be described in terms of three regions: two mixture regions W~ and W + , one of which (W~) consists of solid 1 and fluid 2, and another ( W + ) consists of solid 3 and fluid 4, and a transitional layer region VV between W~ and W + consisting of solid s and a fluid (denoted by subscript / ) which results from the contact of fluid 2 and fluid 4 (these two fluids are immiscible) (Figure 5.1) (cf. Defay and Prigogine [242]). The mechanical properties of the layer W* are different from those of the two regions W~, and W + because the molecular interactions among fluids 2, 4 and solid s in W" give rise to the surface stresses, such as surface tension. On either side of the transitional layer, the usual laws of mixture theory outlined in Chapter 2 can be applied. Physically, the surfaces W~ flW' and W ' n W + are not necessarily smooth (Figure 5.1). However, in order to simplify the mathematical treatment, we shall replace the physical transitional layer W with an idealized layer W ' having two continuous smooth surfaces W~ D W* and W f D VV+, where W~ and VV+ are the corresponding idealized regions of W~ and W + , respectively (Figure 5.2). Also, we shall assume that each region W~, W , or W + is occupied by a solid continuum and a fluid continuum simultaneously. Theoretically, the transitional layer VV' can be replaced by a surface called the dif­ fusing singular surface provided some surface properties are introduced appropriately. In the rest of this section, we will carry out this treatment. In the following we shall present a thermodynamical treatment of the problem; chemical, electromagnetic and other such effects being ignored.

Basic Integral Balance Equations in W' Let Vt be any part of the mixture containing VV* (Figure 5.2). The balance laws for mass, linear momentum, moment of momentum, and energy for the fluids and the solids, and the entropy inequality for the whole mixture in Vt are given as follows:

DIFFUSING SINGULAR

SURFACE

Figure 5.1 Physical configuration of transitional layer.

59

60

MECHANICS

OF

Figure 5.2 Idealized configuration of transitional layer and singular surface.

MIXTURES

DIFFUSING SINGULAR

SURFACE

61

Balance of m a s s For the fluids:

i Lp2dv+1Lp4dv+it

For the solids:

L hdv=°-

i Lpid"+it Lpadv+i LpAv=°-

(5 i}

-

^

f

Here V := VV" n Vt, V := VV' n Pt, and V+ := VV+ D Vt, Pl, p2 are the solid mass density and the fluid mass density in V~, respectively, p 3 , p 4 are the solid mass density and the fluid mass density in V + , respectively, and ps, pf the solid mass density and the fluid mass density in V , respectively. Balance of linear m o m e n t u m The balance of linear momentum for the fluids and the solids takes the form: d

It

Jv_ P2Vmidv + - j v +

P4V{4)idv

= I

Pzb{i)idv + /

+ /

m,(2)idv + j

+ I

o-(2)jinjda + I

+ - jvi

p4bWidv

+ /

m(4)idv + / a^j^rijda

pfVftdv

(

pfbfldv

rhfidv + / afjirijda,

(5.3)

and

|

Jv

Plv{1)idv

+ - Jv+ p3v{3)xdv

= / _ Pib(i)idv + / + / +

m^dv

+ /

_
+ ± Jv> p3v3%dv

p3b(3)idv + /

(

psb3idv

rri(3)idv + / ^ m3idv a

(3)jinjda

+ / ^ aeiinjda.

(5.4)

In the above equations, all the components have been expressed with respect to a rectangular Cartesian coordinate system. Here, f(i)i, f(2)« a r e the components of the velocities of the solid and the fluid in V~, respectively, V(3)i, f(4)i the components of the velocities of the solid and the fluid in V + , respectively, vai, Vfi the components of the velocities of the solid and the fluid in V', respectively. Also, fyxjj, b(2)i> ^(3)ii k(4);> fc^i and bfi are the components of the external body forces, m^jj, m^i, Ti(3)i, Ti(4)i, m s i, rn.fi a r e the components of appropriate momentum supplies, and C(i)ji, C(2)ji,

MECHANICS OF MIXTURES

62


(5.5)

Balance of m o m e n t of m o m e n t u m The balance of moment of momentum for the fluids takes the form:

— /

eijkXjp2V(2)kdv + -T7 =

ZijkXjPiV^kdv

£ijkXjP2b(2)kdv + /

eijhXjPtb^kdv

+ / _ Sijk^irn(2)kdv + /

£ijkXjin^kdv

_ £ijkXj<7(2)iknida -f /

+ J S_

+J

+ — I + /

eiikXjpfbfkdv

+ / ^ £ijkXjrn,fkdv

£ijkXja^iknida

+ /

^+

M{2)tdv + J

eijkXjpfVfkdv

£ijkXjafiknida

<J S'

M{4)idv + J Mjidv.

(5.6)

For the solids the balance of moment of momentum takes the form:

— J _ EijkXjP^^kdv =

+ — /

_eijkxjp1b(1)kdv+

+ / _ eijkXjm^kdv +

EifrXjPsV^dv

+ -Jl I

ZijkXjPsV3kdv

£ijkXjp3b(3)kdv + + I

Cijk®jm(S)kdv + /

_ £ijkXj
eiikXjp3bshdv (

e^Xjrn&dv

£ijk^j^(3)iknida + /

+ J _ M(1)tdv + y + M{3)ldv + / Msidu.

Eiikx0aAknida

(5.7)

Here Af^j, M(2)i are the components of the moment of momentum supplies to the solid and the fluid in V~, respectively, M^)i, M(4v the components of the moment of momentum supplies to the solid and the fluid in V + , respectively, and M,;, M/; the components of the moment of momentum supplies to the solid and the fluid in V', respectively. Also, £;_,-/, is the alternator tensor.

DIFFUSING SINGULAR SURFACE

63

C o n s e r v a t i o n of e n e r g y Let £j, £2 be the internal energy densities of the solid and the fluid in V - , respectively, £ 3 , £4 the internal energy densities of the solid and the fluid in V + , respectively, and £,, if the internal energy densities of the solid and the fluid in Vt, respectively. Also let 5(i)i, 9(2)i, "?(3)i> a n d 9/i denote the components of the surface heat fluxes, »*U r2, T3, T4, r„, and 77 the external heat supplies, and e3l, E , 2 , £,3, £„4, £S3, £»/ the energy supplies. Then the conservation of energy for the fluids is

jt X- P 2 ( £2+ 2 v(2) ^) dv+jt Lpi ( e 4 + \ v ^ w ) d v +

It J

PS

\

f +

2^ /iS/ V dv

=

J-

P2

hi)ivWdv

+ I

pJ>(4)iV(i)idv + I j pfhfiVjidv +

+ /

m(4)iW(4)j(fw + / ( rh^Vfidv +

+ / -

+

rn(2)iV(2)zdv a^2)ji'"(2)injda

cr(4)jt^(4)injrfa + / o-fjiVfiUjda — /

q^ymda

/ q(i\iTiida — j quriida + I p2r2dv + / p4r4dt; 7s+ Js' JvJV+

+ / ptrtdv + / £a2 + / £ 34 d" + / e.jdv, Jv JvJv+ Jv and the conservation of energy for the solids is

(5.8)

jt X- p i ( £ i + r ^ v ^ ) dv+jt Lp3 ( £ 3 + 2 v { * m ) d v +11 / ? * ( £ « + o ^ ^ ) r f u = / pi f c (i)^(i)i d u di J v 2 Jv+ I P3b(3)iV(3)idv + I ^ p3bsiv3idv + J m^v^ydv + / TO(3)iH(3)idv + / ( m3ivndv

+J _

+ /

cr(3)jiV(3)injda + / a3Jivnnjda

-

q^Uida

- / ^ qsiriida +

-

a^^v^ynjda _

piTXdv + I

q^n.da p3r3dv

+ I Psr3dv + [ e3ldv + e33dv + / eS3dv. (5.9) Jv JvJV+ JV We shall assume that there is no net accumulation of energy in V', that is, / ( (rhfiVfi + m3iv3Z + e,} + sS3) dv = 0.

(5.10)

MECHANICS

64

OF

MIXTURES

Entropy inequality The entropy inequality places restrictions on the allowable forms of the constitutive expressions. In our analysis, we prefer to postulate the entropy inequality for the mixture as a whole. Thus

P\V\dv-\ — [ P\V\dv-\

&

p2-q2dv-\ ptfi2
/

p-i-qsdv / p-i-qsdv

+±f "l+±[dt Jv „ * + dt- *Jv/ « * . dt Jv+ --

dt Jv+ f

f Jsz Jsz

qil)iUi

T Tx

+

t

f

g(4)i"j ■

Js* T4 Jst~TT f ^^d JV- ~lT Jv+ I14 JV+ 4

dt Jv q{2)%ni

da

f Jsz Jsz

qfjUj

T T2

2

da

f

U ff Js'~Ti

da +4-

f —d Jv- ~TT JV

If

dt Jv q(3)ini

da da

f da 7s+ T T3 is* 3

qsjUj

Js> Js> T.f. ++

f Jv+

—d T3 dV

Jv

1l3

(5.11)

where ifr, r,2 are the entropy densities of the solid and the fluid in V~, respectively, 773, r}4 the entropy densities of the solid and the fluid in V+, respectively, and 77,, rjf the entropy densities of the solid and the fluid in V', respectively, and Tu T2, T3, T4, f„ ff are the appropriate absolute temperatures. It can be shown that (cf. Eringen [243])

4 f $m«fe *mifo = /f ^P-dv ^±dv dt JvJv~ at dt JvJv~ at

-l '

f

* ( » ) & =J V/

a

+ f $MMvv(1)i nida (1)inida Jsz Jsz

! < 2 ) ^ +J/S

OL

--- f

$(1)$v(1} vfin7da, fin7da,

(5.12)

mda ${2)vftn-da, *${2) MvwW * * * - JI/S *(2)Zf.n-da,

(5.13)

Js-

d v 9 dJ dv+ (5.14) H^~-L ft L ^ = L >+L lr Ut*^*{3)V{3*■-/„ ^da *»««»?*». ~ L *******

wdv dv 4 {4)dv = L ^rdv + Js- *wv^da J t l \+ *

it L * =L ^&r +L *( >w*^ --LL*ww**> *wW**> (5-15) (5-15) where $(1), $(3) are quantities associated with solid 1 and sohd 3, respectively, and $ ( 2 ) , $(4) are quantities associated with fluid 2 and fluid 4, respectively. The surfaces 5", 5+ and their normals n " and n+ are indicated in Figure 5.2.

DIFFUSING SINGULAR SURFACE

65

Now, taking $ ( i ) = ph pivWi, SijkXjPiv^, pi(et + \v{VjiV(t)i), and ptqh respectively, / = 1, 2, 3, 4, and substituting the corresponding equalities 5.12 through 5.15 into the corresponding equations 5.1 through 5.11, and letting S~ approach S~ and S+ approach S + , we obtain

I _P2 (v(2)i ~ vfi^j nida+

j

= 0,

(5.16)

Pi (vti)i ~ Vfi) njda + I +P3 («(3)i - Vfi) njda + — / _ p 3 di; = 0,

(5.17)

/

I + p 4 (f(4); - Vfi) nfda + — j _ pfdv

P2V(2), (v(2)j - Vfj) njda + J^ p4v{4)l

(v{i)j - vf])

njda

d r PfVf;:dv dt Jv = f

(pfbfi + m / ; ) dv + I _ cr^injda

+J

+

o-(4)3,njda

+ / o-funjda, Js1

J_ PiV(i)i (vW

- vfj) nj da + j

(5.18)

s +

p3v{3)i

(w(3y - vh)

njda

+17 / PsVndv dt Jv< = j

IV

+

f

(p,K

+ ™*») dv + Js_ cril)j%n-da + J^

a(3)jinjda

/ a3jiUjda,

J t{mfi+msl)dv

(5.19)

= 0,

(5.20)

/ _ £ijkXjP2V(2)k [V(2)l - «//) njda + /

£ijkXjP4V(4)k [V(4)l - £/() V

= /

£

+ /

y't a ; j \P~fbfk + ra/fc) dv +

d a

+ 17 /

(

^jkXjPfVfkdv

eijkXj
£zjkXj
+ / Mj%dv,

(5.21)

66

MECHANICS OF MIXTURES

/ _ £ijk%jPiV(i)k (w(i)i - vn\ Js

nl da

+ /

£,jkXjP3V(3)k («(3)1 - W/i) n ; + ^ o + J l / ,

= /

ey/c^' ( p A * + "*, fc J dv +

+ /

EijkXj
J~\?

+ /

£

ijkXjP3v3kdv

eijkXja^)iknYda

J s

eijkXj<7siknida

+ /

Msidv,

(5.22)

L

Vf}) "n3J'da P2 ([e 2V(2).«(2)lj («(a)i (v(2)j -"" W/j) j ^ _ P2 E22 + + 2^(2)^(2)^ <^ + y s + P4 (^4 + -V(4)iV(4)iJ

(v(4)j - vfj)

J* ('> + fH"fi) = I

Js

t

[pfbfi

dv

+ rrifij Vfidv + I

o-(2)j,V(2)in]da

+ /

&{4)jiV{4)irijda + I a-fjiVfiUjda

- /

1(4)in^da - J qfinida

Pi [EI + -V(iyv{iyi)

+ I

- Vfj)

njda

+ j s + P3 [£3 + ~v{3)iv{3)i)

(y{3)j

- v{j)

r ~

+ ~VsiVai)

+

^{3)jiv(3)infda

~ J

+

q{3)infda

J

t

- j

+ j (

q(2)in~da

+ esf) dv,

(5.23)

n+da

dv

+ ^ / p> \' = at J Jv* vsidv + I t [Psbsi + mst) + J

— /

(p}r}

(v(1)j

d

n+da

i

v(i)jiV(i)iTiJda

aai%vsin0da

qsin,da

-

I

q(l)in^da

+ / ^ (p3r3 + £,„} dv,

[rUfiVfi + rn3iv,i + esJ + i33) dv = 0,

and

j s _ [P1V1 (v(i)i ~ Vfi) + P2V2 ("(s)i - «/*)]

riida

(5.24)

(5.25)

DIFFUSING SINGULAR SURFACE

67

+ / s + [fWa (v(3), -Vfi)

4 P4I74 \V(4)i --

+

PsVsdv

jtL^fdv+

>-ut

+

dtJv

Vfi)\ ri^da

t+thda

■>da-L{-1 (&+y\ riida 4 [ (h:s + MAdv. Js-\Tf

T2 )

Jv \ Tf

TJ

(5.26)

Ts J

Governing Integral Equations on t h e Diffusing Singular Sur­ face Because the transitional layer W ' has a very small thickness, say 6, the layer can be approximated by a singular surface <S which has appropriate surface properties. In this way, the integral equations 5.16 to 5.26 on V will be transformed into the cor­ responding integral equations on <S n 7?t and this transformation will greatly simplify the mathematical analysis. The surface <S will be called the diffusing singular surface. Let us construct S in W ' such that S is the mid-surface (Figure 5.2). In W f , we establish a three dimensional coordinate system (Z1, I2, I3) such that the axes I1 and Z2 are embedded in S or surfaces parallel to S and the third axis I3 is normal to both I1 and I . The corresponding contravariant metric tensor g is ' a11

M

21

a 0

a12 a22 0

0 0 1

(5.27)

Furthermore, (Z1, I2, 0) can be considered as a surface coordinate system on 5 with contravariant metric tensor a =

a11 a21

a12 '

a22

(5.28) l3=0

Next, we introduce the surface properties for the fluid / and the solid s on S in the following manner: (i) surface mass densities

-=l~Pidl*

Pf =

Ps

= psdl3;

(5.29)

(ii) surface velocities Vf% = — PfVfidl3, Of Jj . . .

vsi = — / p3vsidl3; P, Ji

(5.30)

68

MECHANICS OF MIXTURES

(iii) surface body forces bfi = — pfbfidl3, Pf Js

bsi = — / p3b3idl3\ pa Js

(5.31)

rh„ = / m^dl3;

(5.32)

»laNa = f ^njdl3;

(5.33)

(iv) surface momentum supplies mfi

= / mfidl3,

(v) surface stress tensors (&y,ff™) crfNa = / ofrijdl3,

where N is the unit vector on S tangent to S and normal to T = S' n S, a = 1, 2 (Figure 5.2). (vi) surface moment of momentum supplies

Mft = J Mfidl3,

Msi = J Msidl3;

(5.34)

£„ = — I p3e3dl3; P, J6

(5.35)

%Na = J q3ln%dl3;

(5.36)

r3 = — /p 3 ? 3 dl 3 ; pa Js

(5.37)

e33 = J i3Sdl3;

(5.38)

VB = — f PsVsdl3\ p3 Js

(5.39)

fs = i - /p a f 3 dZ 3 . Pa •/*

(5.40)

(vii) surface internal energy densities ef = — / Pf£fdl3, Pf J* (viii) surface heat fluxes (
q]Na = J qfin4l3, (ix) surface external heat supplies *f = — /Pfrfdl 3 , P/ Js (x) surface energy supplies

e,f = J esfdl3, (xi) specific surface entropies Vf = — I PfVfdl3, Pf Js (xii) surface temperatures ff == -i fpfffdl3, P/ >/«

DIFFUSING SINGULAR SURFACE

69

With the above definitions and the fact that the thickness 8 of the transitional layer Wl is very small, we have / pgdv = / pqdl3 da = p„da, JV Jsnv, Us J Jsnv,

L ^ > = L , il~p^dl3}da=L,**>•

(5.41)

^5-42)

Jvi~pgbgdv = lnv^gda,

(5.43)

/ rrfdv = I m'ada, JV y Jsnv, a J

afrijda = J \Jafnjdl3]

(5.44)

dr = j agaNadr,

(5.45)

/ £ijkXjp~gvgkdv = / EijkXjPgVghda, JVl Jsnv,

(5.46)

/ eijkx:pgbgkdv = j eijkx3pgbgkda, Jv* Jsnv,

(5-47)

/ EijkXjmgkdv Jv

= / eijkXjrhgkda, Jsnv,

J ^ £ijkXj
(5.48)

Sij^a^NcdF,

(5.49)

/ M~dv = / Mgida, Jv JSnV,

L~p°( * » + f ^ ) d v = L v ,fi°( * « + r A ) da>

J

i

(5.50)

(5 5i)

-

/ PgbgiVgidV = / pgbgiVgidd, JV Jsnv,

(5.52)

I rnJvgidv = I m™iL' Jsnv,

(5.53)

dfvgiUida = fr{J6

Jt q^nja = jjj

/

Kwdl3^

dr = jr vg,»gaNadr,

g^dl3} dr = jf £JV«dr,

Pg^gdv = /

pgrgda,

(5.54)

(5.55)

(5.56)

70

MECHANICS

OF MIXTURES

/ esgdv = / e3gda, Jv JsrfPt I PgVgdv = / JV<

(5.57)

Pgrjgda,

(5.58)

JSCiV,

I PX^dv = [ ?&da, (5.59) Jv Tg Jsnr, Tg where g corresponds to either / or s. Substituting the above equations into Eqs. 5.16 through 5.26, approximating S~ and 5 + with S n Vt because of the smallness of 6, and using n + = —n~ = n results in

Jsnv, t P2 W ~ VW) +

Jsnv, ^

^

~" "W)

Pi

(WW " */)]

+ P3

^'3) " ^

M

° + A XnP, ^

" ^

+

dt Xn*. ^

Lnv, W ) (^ " ^2>) + P4V'4) W« " ^)1 "jda =

LOP,

+

=

°'

(5 60)

=

°'

(5-61)

'

* jL>, ^^

W + */) ^ + L P , (""PI + °w) n >

+ / a)aNadr,

(5.62)

JL>, W ) $ ~"»)) +p3 ^ 3) w»)" ^)] V a + « JL>. ^•do JSnVt

(P&

+ m\) da+ [ '

JSnV,

-°ih +*&)>n,jda

("

+ / 3-?Nadr,

(5.63) y

jT

(m> + »fcj) da = 0,

EyfcX3 [p2u(fc2) ( ^ - w|2)) + Pitfi) (v(4) - 4 ) ]

(5.64)

n rfa

'

+ "JI / eijkx3pjVkfda at JsnV, = Jsnv

{eiih^

+ jrei3kxia)aNadr,

[ ( - o § j + ojjj) nj + P/S/ + f n j ] + M / t } da (5.65)

DIFFUSING SINGULAR

Jsnv, £lJkXJ d + J! at

SURFACE

71

\.PlVW O5' ~ ^f1') t t £ / ijkXJp3v3da Jsnv, {etjkxj

= J

[eijkxj

+ PaV

W ("W ~ **')]

(-o\^

nida

+ crl{k4)) n, + p3bk + rhk] + Mai} da

+ j e,3kxj
(5.66)

Lnvf2 ( £2 + 2^2^(2)*) (^ " V^

PA. (£4 + 2^(4)V(4).J («(*) - #/) " j d a

+ J

+

Ujda

dt JsnV, Pi (*

= / (pfblvfi Jsnv, v

+ 2 W Jda +

m'.VfAda '

[(~cr(2),;(2)i + °il)vw)

+ /

n

o + (9(2) - 9(4)) "i] da

+ f (ZfTf + e3}) da+ f (o-fv,, - <%) Nadr, Jsnv, Jr x '

Xnp/1 (£l + 2^ 1 ) V ( 1 ) 7 ( ^ " V^ + !snVt + 17 /

(5.67)

nida

^ ( £ 3 + 2«(3)"(3).) («(») - «/) » > ^ (e 3 + - ^ v 3 ; ) d a

= / (p,6*ft,;+ ml3v3i) da JsnV, v ' + j

[(-^i)^(i)» + ofa)vw)

" j + (9(i) " 9(3)) ",-] da

+ / (&*. + if..) da + / (a\avsi - C ) A W Jsnv, Jr x ' ( (rh)vl% + rh%i + e3j + X„) da = 0, Jsnv, v ' and

J

[piVl (#/ - VU) + P2V2 (v) - v\2))] n3da

(5-68) (5.69)

MECHANICS OF MIXTURES

72

LJ" 3 " 3 ^"

J - Vf) + P4V1 (v(4) -~ v f)\ rijda

+ 17 / P / ^ / d a + 37 / P.V>da at Jsrypi at Jsnv, 9(1)

>-L

9(2)

9(3) _ 9(4) \ +PJH_ + P£± da T4)n>+ ff + f.

J\ + n

JsnVt

/(f

%

f *§-) NadR TJ

(5.70)

Governing Differential Equations on <5 We now derive a set of differential equations from Eqs. 5.60 through 5.70. To achieve this goal, we need to apply Green's theorem, i.e., a

a

(5.71) f$> Nadr=[ t> ,ada, Jr Jsnv, where $ a denotes the physical quantities of the fluid / or the solid s on S, and we need to know the representations for d/dtfSnVt <&fda and d/dtfSnVt $3
j $fda = f fe + ( V X " ) * */ " 2Xmt)/„$/ It JSnVt JSnVt I at '

da,

(5.72)

2Kn9m*. da,

(5.73)

v

and

~ [ f>sda = f -~ + (vneaaa0) ,0 $ . v dt Jsnv, JsnPt at '

where Vfn and ii m are the velocity components of the fluid / and the solid s on 5 in the direction of the normal n, respectively. The definitions of t'a, aa>3 and Km are given in Appendix A. Inserting Eqs. 5.71 through 5.73 into Eqs. 5.60 through 5.70 with appropriate choices of <£, $y and $„ (for example, we can set $^ = p} in Eq. 5.72 and insert the result into Eq. 5.60, set $ , = p„ in Eq. 5.73 and insert the result into Eq. 5.61, and set $ / = jiffi in Eq. 5.72 and <£a = a)" in Eq. 5.71 and insert the results into Eq. 5.62, etc.), and letting S n Vt approach zero, we find that

p2 (y) - i>(2)) ni + p4 (y'w - v}) m +

dp, ^dT dt

+

^ f ^ )

+tt -

2K

™Pf*f» = °-

(5.74)

73

DIFFUSING SINGULAR SURFACE

Pi (y) - vl{1)) m + p3 (yli3) - vlfJ m dp.

+^T dt + (<***•>€} * P . - 2Kmfi.*m = 0,

(5.75)

P2V\2) (y3f - V3(2)) Uj + p4v\4) (vfq - vf) Tlj +~ (m)

+ («"%**«) ,ppfVf -

iKmp}v)vfn

= psl) + m) + (-afa) + of;,) nj + (»?) ,a ,

(5.76)

Piv\x) {ff - v3(1)) rij + p3v\3) (v»3) - Vf) Uj

+^M) + ( o< *v£)*^ -

2Kmp,v\vm

= P £ +K + (-*& + 4 ) "i + (*?) -,

(5.77)

m^ + m* = 0,

(5.78)

ziikVJ&f + M /t = 0,

(5.79)

+M Mnn = ijkt3a
(5.80)

e

P2 f 62 + -«(2)V(2)>J (*>/ - f(2)) n J +P4 (e 4 + 2 V (4) v w) (^(4) - *j) no + Jt PS (£"/ + 2 ^ * ) + {a^Vf^)

,apf \if + -«}%) "- 2JBTm^/ f Uf + -Vfifi)

= Pftyvfi + nCfVfi + {~^2)v(2)i + rflfwi) rij + (<J(2) - 9(4)) «i + P/*7 + e3f + (Sftifi - # ) , « ,

Pi (ei + 2 U (i)^(i)i) («/ ~ VU)

n

i

+P3 (^3 + ~ W(3)V{3)ij (V(3) - * j ) Wj + ^ p 3 ( e s + -w*v3-

(5.81)

MECHANICS OF MIXTURES

74

+ {aa^ajti),ap3(s3 + h%?J - 2Kmp3 {e3 + ^v%i) #«n = & $ # « + m * ^ + (-ofj)^(i)i + ^3)"(3)i) ni + (gl(l) - 9 * s) ) n, + p,r s + £88 + ( T ^ - q«) „ ,

(5-82)

Ayi5/j + m*v^ + e,/ + e „ = 0,

(5.83)

and

PiVi {ft -- Vp^) Uj + P2V2 {ff ~ V(2)) UJ +PsV3 (vfS) ~ v]) nj + p4V4 (w(4) - Vf) rij +

di ^ ^

+

(a°^^t") ' " ^

+-jr (PsVs) + (a^VsjPc) (j%) ~\Ti >

?m T2

9(3) T3

- ( | + f)„.

~~

2K

™PtffVfn

tfpA

-

2Kmp3rj,vsn

ih\ T4 )

'

pftf ff

tJ. f,

(,M,

In the derivation of Eqs. 5.79 and 5.80, we have used Eqs. 5.76 and 5.77. Equations 5.74 through 5.84 govern the evolution of the diffusing singular surface in the mixture. Notice that there are many surface properties contained in Eqs. 5.74 to 5.84 like fhlf, rhls, &f, a'3a, Mfi, Msi, ef, e3, qj, q™, esf, e33, rjf and rj3 which are assumed to be given by constitutive functions. Obviously, it is not an easy task to determine these constitutive functions because they depend on the nature of the fluids and the solids on both sides of the singular surface and the interactions amongst them. Furthermore, we need to apply the axiom of material frame-indifference to cover the case of general two-dimensional surfaces imbedded in three-dimensional spaces. Due to these reasons, we will consider only two specific unsteady diffusion problems which involve diffusing singular surfaces based on the constitutive functions 3.19 through 3.21 and the evolution equations 5.74 through 5.84 on S in the next two sections. The first problem is the unsteady diffusion of an incompressible fluid through an incompressible nonlinearly elastic solid slab with a diffusing singular front which moves as time goes on. Ahead of the front there exists a pure solid region and behind the front there is a mixture of a fluid and a solid. The second problem is the unsteady diffusion of an incompressible fluid through an incompressible nonlinearly elastic solid annulus which also involves a moving diffusing singular front, a pure solid region ahead of the front and a mixture region behind the front. We will apply Eqs. 3.1 through 3.4 and 3.19 through 3.21 and 3.33 to describe the motions of the fluid

DIFFUSING SINGULAR SURFACE

75

and the solid in the mixture region and use Eqs. 5.74 through 5.84 to describe the evolution of the singular front. Moreover, we will assume that the diffusion processes are slow and isothermal, thus the nonlinear inertial terms in the governing equations for the mixture and the motion of the pure solid region may be neglected and the virtual mass effect and thermal effects will not be included.

5.2

Unsteady Diffusion of a Fluid Through a Nonlinearly Elastic Solid Slab

Consider a thick, incompressible, nonlinearly elastic solid slab with one end, say z = 0, fixed (Figure 5.3), with an incompressible fluid diffusing through it unsteadily. Assume that in the region of the fluid and solid mixture the motion of the solid is given by X = x,

Y = y,

Z = f(z,t),

(5.85)

where (X, Y, Z) and (x, y, z) denote the initial and current positions of the same solid material particle, respectively. Notice that it is customary to represent the motion of a solid body by defining the current coordinates of the solid body in terms of the reference coordinates of the body and time. We shall, however, find it convenient to define the inverse mapping. Since the motion \ in Eq. 5.85 is invertible, we can indeed do this. Thus, care should be exercised in relating the terms which appear in the subsequent equations in this section and the corresponding equations in Chapter 3. The velocity components of the solid are vt = 0,

Vi

= 0,

«5 = | , ( Z

>

t )

S S

- ( g )

1

g .

(5.86)

Also, we shall assume that the motion of the fluid in the region occupied by the mixture is given by v{ = 0,

v( = 0,

v£=g(z,t),

(5.87)

where (v{, v{, v() are the velocity components of the fluid in the x, y and z directions, respectively. Substituting Eq. 5.85 into the balance of mass equation for the solid, Eq. 3.1, gives P° = Psn?f, (5-88) az where p3R, p' are the densities of the solid in the reference state and in the mixed state, respectively. It follows from Eqs. 5.85, 5.88 and 3.3 that the balance of linear momentum equation for the solid takes the form

~J

a>

Figure 5.3 Flow domain (Slab).

Ii 1 1o o

I

77

DIFFUSING SINGULAR SURFACE

„d2f dp'dz dp' (dz\2 2 - -W- ^tm-^{m) p

d , =^33—3,

, (5.89)

where
d2f

d

Pfl-^F = - ^ 3 3 + ^ 3 -

(5.90)

Substituting Eq. 5.87 into the balance of mass equation for the fluid 3.2 yields

^

+! K ) = 0 ,

(5.91)

where pf is the density of the fluid in the mixture. Also the combination of Eq. 5.87 with the balance of linear momentum equation for the fluid 3.4 results in

pf9

i = hL+Tn^

(5 92)

-

where 0-33 is the partial stress component of the fluid. Here, we have once again neglected the nonlinear convective term since the fluid velocity is assumed to be small. For the problem under consideration, it is assumed that the individual volumes of the two constituents are additive, so the constraint 3.12 holds (with (3 = 0) P-+-j = l. (5.93) PR PR Before developing the specific differential equations, we first discuss the boundary and initial conditions for the motion. (i) At z = 0 We will assume that the solid constituent is fixed in this plane and the fluid input flux is prescribed as a function of time t. Thus, f(0,t)

= 0,

(5.94)

pf(0,t)g(0,t)=pfRVo(t).

(5.95)

(ii) At z = s(t) (singular front)

MECHANICS OF MIXTURES

78

The motion of the singular front S is described by Eqs. 5.74 through 5.84. The unit normal vector on <S is n = (0, 0, 1). Across this plane the densities of the fluid and the solid have jumps. Also p 4 = 8 (this assumption cannot hold in the initial time period of the diffusion), we have t ~ O(lfe). Based on the above results, we obtain dpf/dt. ~ 0(e 2 ), dpa/dt ~ 0(e 2 ). We also assume that 4 , ,
4

+ OU ~CT(3)33= 0,

-■33

p' (A' - A3) - ( l - £ - \

m / 3 = 0,

^ 3 3 + mf31 (yi - v'3) < 0,

(5.96)

(5.97)

(5.98)

(5.99)

where Us is the speed of the singular plane <S, (7(3)33 is the stress component in the pure solid region ahead of <S, m/ 3 is the surface momentum supply component to the fluid, A3 = e' — Tr]3, A3 = e3 — T-q3 are the Helmholtz free energy densities of the solid in the mixture region and in the pure solid region, respectively, and T is the absolute temperature of the mixture. Next, we introduce a function H as follows: H := mf3 + p' (A 3 - A3) - (1 - fL j c7(3)33-

(5.100)

Then equation 5.99 becomes H(v(-v33)<0.

(5.101)

Expanding H in aTaylor series in Eq. 5.101 and retaining only the linear terms, we can approximate H by H = A, + 7o (v{ - v33) ,

(5.102)

DIFFUSING SINGULAR SURFACE

79

where /30 and 70 are non-positive functions generally (since v[ — v% > 0 on the plane S). But we will treat them as constants. We shall assume that the deformation in the pure solid region may be neglected. Thus, A 3 = 0, and equations 5.93, 5.97, 5.98, 5.100 and 5.102 give 3

'A' -po-lo

f

(v( - v°) + 4 - 0 & - Zjo'a = 0. PR

(5.103)

PR

The position of the singular plane <S is determined by ds dt

(5.104)

Us.

The combination of Eqs. 5.96 and 5.104 yields ds , , . jt=g{s(t),t)

,

(5.105)

with (5.106)

s(0) = 0.

Equations 5.103, 5.105 and 5.106 can be used as a set of conditions on the evolving boundary S in our problem and thus in effect play the role of boundary conditions. It follows from Eqs. 5.88, 5.91, 5.93 through 5.95 that z

9i ,t)

V0(t) +

21 dt

(l.

dfY1 dz) '

(5.107)

Now we restrict the solid to be an incompressible isotropic nonlinearly elastic material such that the specific Helmholtz free energy for the mixture is given by Eq. 3.33. Thus, from Eqs. 3.19 through 3.21 with the neglect of the virtual mass effect and Eqs. 5.85 to 5.88 and 5.107, we can get the expressions for £T|3, a^3, and m3,

«lz = P' (AB ~ A)

J ^33

d ^-P + 2K PR + (PR - pRt\ f - )Tz_ dz

_ -p s (A' ~ A) - (l +K Pi + [PR ~

fdfY2 \-8-z)

(5.108)

dz)

df\ (a/y1

f\df

(-

dz) \dz)

(5.109)

MECHANICS OF MIXTURES

80

m3

97

+K

= 9-^(A°~A»-^P

,d2f

^

'^-y-m'u

+2Kp}f

(5.110)

%+%™ A= A=KK

2 d (?p\~- l-^4 + ( |i) {) '(¥) \dz) ln

\dz)

(5-ln)

■

Substituting Eqs. 5.108 through 5.110 into Eqs. 5.90 and 5.92 and eliminating the pressure P by cross differentiation results in

*M('-jrfi^-* (-2)" [^-er; + *(£)is

a2/

PR V0{t) + ^

= #

^ 2

+

^(^r+(«-^(sr-(«-*(sr + PR [4A

82f

(5.112)

~ (PR ~ PR) dz2'

This equation has to be solved with the following initial and boundary conditions resulting from Eqs. 5.94, 5.103, 5.105 through 5.109 and 5.111,

mt)--

/ ( 0 , i ) == 0,

(5.113) (5.113)

df 2j( Y\( ° + vR ' 4p 2P \~d~ 10 *(|{) ("« " «) ( 1 0 +PR-2PR + 2PR%

+(,.-,() (*)•-*&.(§£)♦£ + :KV'lt){l-%) ' 7o df

(I) +( 1 -I) ds ds

= Tt = Jt

= 0,

11 [a/ _,, .1 / 8/Y a/r (>- ■ a J ' [dI + Vo{t)\[1~Tz) '

'fH

2 = s(f) ,

s(£) 22== sW

'

(5.114)

(5.115)

(5-115)

DIFFUSING SINGULAR SURFACE

81

s(0) = 0.

(5.116)

We shall solve Eqs. 5.112 through 5.116 numerically. In our numerical work, the derivatives with respect to z are approximated by the central or backward finite difference formula, and the derivatives with respect to t are approximated by the backward finite difference formula. Thus, we can get a set of difference equations for f(z,t) at discrete spatial and temporal points. Because this problem has a moving boundary z = s(t), we shall fix only the spatial difference step length Az which is not necessarily constant and allow the temporal difference step length At to be determined by Az and the speed Us of the singular front S. The main computational idea is as follows: assume that the solution for f(z,t) until time r has been obtained and the singular front position z = s ( r ) is known, then the front speed Us = VI(S(T),T) = g(s(r) , r ) can be determined by Eq. 5.107. Increasing z = s(r) to s ( r ) + Az, we get a new spatial region from z = 0 to z = S(T) + Az. Corresponding to this increase in the spatial region, the temporal difference step length At at t = T is approximately equal to Az/Us. With the above results for f(z,t) (0 < z < s(r)), Az and At, we can iteratively solve the difference equations for / to determine f(z,t) (0 < z < s ( r ) + Az, t = T + At). In this computation we choose Az = 0.01/3 to 0.02. The error tolerance of the numerical solution is of the order of 10~ 6 Following Shi, Rajagopal and Wineman [78], we assume that the values for the ma­ terial parameters are: psR = 0.862(gm/cc), pR = 0.869(gm/cc), M c = 9151(gm/mole), K = 8.317 x 107(dyne - cm/mole - K), T = 303.16(ii'), a = K, /?0 = 7 o = - 0 . 1 xK. We use three different flux inputs Q0 = pRV0 for the fluid: V0 — 2e~3t, e~3t and e - 3 £ /4(cm/fer). These inputs imply that after some time, the supply of fluid is negligible. Notice that equations 5.112 through 5.116 have a static solution / = Cz where the constant C is determined by pR,p R and /30/K. This solution corresponds to the steady state solution as t —» oo. In this state the densities p' and pf are constant. Figures 5.4 through 5.8 are concerned with the propagation of the singular front in the solid and the motion of the mixture. Figure 5.4 depicts the position of the singular front as a function of time, for a variety of values of the input fluid fluxes. In all the three cases considered, the front comes to rest asymptotically with time. Physically this is to be expected as the influx at the other end z = 0 tends to zero exponentially in time. Notice that the speed with which the singular front moves, namely the slope of the curves, is quite steep initially, and in fact monotonically decreases with time. A very interesting phenomenon is documented in Figure 5.5. Initially, as the fluid is being pumped into the solid material, there is an accumulation of the fluid near the boundary at z = 0 before the fluid gets dispersed further into the solid. This in turn causes the density of the solid near z = 0 in the mixture to initially decrease before it increases to an asymptotic value. However, at the singular front z = S(T), there is no such 'choking' of the fluid and thus the solid density increases monotonically to an asymptotic value. Also, if the input fluid flux is small such as V0

82

MECHANICS

Figure 5.4 Propagation of singular front with time (Slab).

OF

MIXTURES

DIFFUSING SINGULAR

SURFACE

Figure 5.5 Variation of solid density with time at z = 0 and z = s(i).

83

84

MECHANICS

Figure 5.6 Variation of solid density with z at various instants.

OF

MIXTURES

DIFFUSING SINGULAR

SURFACE

Figure 5.7 Variation of stretch A- with time at various positions.

85

86

MECHANICS

Figure 5.8 Variation of stretch with position Z at various instants.

OF

MIXTURES

87

DIFFUSING SINGULAR SURFACE

= e~3t/4(cm/hr), this phenomenon of local fluid accumulation does not exist. Figure 5.6 shows the variation of the density in the solid at various instants of time. While there are density gradients as the fluid is diffusing through the solid, we see, as is to be expected, that at large times, after the fluid input is shut off, the density of the solid reaches a uniform value asymptotically. Notice that the asymptotic value of p"/p3R is less than unity, which reflects the fact that there is fluid infused in the solid. Figure 5.7 shows the variation of the stretch ratio \ z as a function of t at various positions. We see that the stretch ratio initially increases but then decreases monotonically to an asymptotic value. Once again, the small increase in the stretch ratio initially is due to the 'choking' of the fluid. In Figure 5.8 we see how the stretch ratio varies with Z at various instants. We find that there are steep gradients associated with the transient flow while a uniform asymptotic value is reached at large times.

5.3

Unsteady Diffusion of a Fluid Through a Nonlinearly Elastic Solid Cylindrical Annulus

In this section, we shall consider the motion in which an incompressible fluid is diffus­ ing unsteadily and radially through a thick cylindrical annulus of an incompressible isotropic non-linear elastic solid with an inner radius r = a (Figure 5.9). Further­ more, we shall assume that the motions of the solid and the fluid constituents in the mixture region are given, respectively, by R = f(r,t),

0 = 6,

v* = 9(r,t),

vf9=0,

Z = \z,

(5.117)

and v( = 0.

(5.118)

Here (r, 9, z) represents, in a cylindrical coordinate system, the deformed current position of a solid material particle which in the undeformed reference state was at (R, 0 , Z), and (i;/, Vg, v{) are the velocity components of the fluid in r, 8 and z directions, respectively. We shall assume that the stretch ratio A in the Z direction is a constant. The velocity of the solid is determined from Eq. 5.117 as

df (d/y1 < ~= ~ % ( % y \ dt\dr)

«* = 0,

< = 0.

(5,19)

Let pR, pR be the mass densities of the solid and the fluid before forming the mixture, respectively. The densities of the solid and the fluid in the mixed state are p" and pf, respectively. The partial stresses of the solid and the fluid in the mixture are a\ and o[-, respectively, and m^ is the momentum supply in the mixture. On substituting Eq. 5.117 into Eqs. 3.1 and 3.3 and Eq. 5.118 into Eqs. 3.2 and 3.4, we obtain the following equations

88

MECHANICS OF

Figure 5.9 Flow domain (Cylindrical annulus).

MIXTURES

DIFFUSING SINGULAR SURFACE

89

pP'3 = pR*fr-l% or or _ do*„ a3rr - a'ee -mmr, ~d¥ = ^rdr + —r r -

-*Mr-& =

-p*Xfr

dp* ~dt

1 d (rpfg) == 0, r dr

(5-120)

(5 121)

-

(5.122)

f

dal da* ]
£PR+ 4PR= ° -

(5-124)

f(a,t) f{": *) === a,a,

(5.125)

p\a,t)g(a,t)= p\a ,t)g(a, PR = PRVo(t) ■ •*)V0={t).

(5.126)

EL + EL = 0.

PR PR The above equations can be solved once we assume specific constitutive equations for the constituents and proper initial and boundary conditions. We first discuss the issue of boundary conditions: (i) At r = a We shall assume that the solid constituent is fixed at this surface and the fluid input flux is given as a function of time t. Then

and

(ii) At r = s(t) (singular front) In this problem, the surface r = s(t) is moving radially at a speed of Us = ds/dt. Across the surface S, the densities of the fluid and the solid have jumps. Assume that the thickness b of the physical transitional layer is small and that the process of diffusion is slow and that there is no surface tension due to the presence of a single fluid. Then, an argument similar to that in Section 5.2 results in Us=g(r,t), = g(r,t), Us

(5.127)


(5.128)

a{rr-m cr/ - fr=0, rhfT = 0,

(5.129)

90

MECHANICS OF MIXTURES

and B(v*-v>)<0,

(5.130)

H := p> {A° - A3) - (l - P- J
(5.131)

where

Here, A' is the specific Helmholtz free energy of the solid in the mixture region, <7(3)rr, A3 and p3 are the the stress component, specific Helmholtz free energy, and mass density in the pure solid region, respectively. Also, rhfr is the component of the surface momentum supply. Since the solid constituent is incompressible and the deformation of the solid in the pure solid region is to be neglected for the sake of simplicity, p3 = psR, A3 = 0. Similar to Eq. 5.102, the function H denned in Eq. 5.131 is assumed to have the form H = f30 + j0(v!-v°r),

(5.132)

where /3o and 7 0 are non-positive constants. The combination of Eqs. 5.128, 5.129, 5.131 and 5.132 yields

(5-133)

p'A' -ft-7o U ~ v°) + 4*4 - V " = °' PR PR The motion of the singular surface S is governed by ~ = Us = g(s(t) ,t),

(5.134)

with s(0)=a. 8(0) =-- a.

(5.135) .135) (5

Equations 5.133 through 5.135 furnish boundary conditions at r = s(t). It follows from Eqs 5.120, 5.122, 5.124 through 5.126 that

-K)~'

1 9(r,t)== A\ff£ / ^ ++aV aV g(r,t) 0(t) r- (l - £\ 0(t)

(5. (5.136) 136)

.

In the following, we study the unsteady diffusion with the constitutive model 3.19 through 3.21 and 3.33. Substituting Eqs. 5.117 through 5.119 into Eqs. 3.19 through 3.21 gives

+ 2K{p° + pf)(?A -fr+**M{%Y>

*l=p*{A°-A)-^P Kr = p°(A°--A)-

,

.137) (5(5.137)

DIFFUSING SINGULAR

SURFACE

91

<4e = p3 {A' -A)~f-P

+ 2K U + +f) r2f~\ V

PR

o± = -p' (A' -A)~4P

+K l - ) ^ PR

4, = -P°(A*-A)-^-P

(5.138)

'

\PR)

+ K(^) PR

(P° + pf) ,

PR

V

(5.139)

'

E^4V( + pf)/

\PR/

(5.(50)

PR

~^-«.-£(£)'«S -Kp>

Or

9t

PRPR

df

\dr

„2 / /--22 +

a7

2 x-2 ♦'/

df fdf

P3 Pf *——f

A= K

|l

+ g(r,t)

A-2^3+

(5.141)

(5.142)

b

*

&

where p', ps a n d g(r,t) a r e given by E q s . 5.120, 5.124 a n d 5.136, respectively. Here, we have neglected t h e v i r t u a l m a s s effect. S u b s t i t u t i n g E q s . 5.137, 5.138 a n d 5.141 i n t o E q . 5.121 a n d E q s . 5.139 t h r o u g h 5.141 i n t o E q . 5.123 a n d e l i m i n a t i n g t h e p r e s s u r e P by cross differentiation results in

2

xd

f w

- ^ - ppf

(p3-Pfi)A/r

K

f_dg_ PRdt

P^P^ a pRPR

«7

£1. pRdr

d + 2K— or

+ 2Kr-x [ps + pf)

~

PR

+ Kpf4Or

fl

■■ '-

T h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n s are:

'dl(diY\ dt \dr J \PR)

r2f~ r j

PR

K

(rt)

^

')

dp3 ~dr~ (5.143)

MECHANICS OF MIXTURES

92

(5.144)

f(a,*) = a,

Kp'

! r + ^ 2 + A"2-3+ln(AHr

-^m"

+ K(p' +

7o 9(r,t) +

Tt = r~ [ f m

df fdf

dt \dr

+

0,

o(i)J y1 ~ yR) '

v=

pf)^j-p0

PR

(5.145)

s(t),

r=s(t)

'

(5.146)

and the initial condition is s(0) = a.

(5.147)

Equation 5.145 is derived from Eqs. 5.133, 5.137, 5.139 and 5.142. We shall solve Eqs. 5.143 through 5.147 numerically. The computational scheme is similar to that used in Section 5.2. For the computation, we choose pR = 0.862(pm/cc), pfR = 0.869(5m/cc), Mc = 9l5l(gm/mole), 11 = 8.317 x 107(dyne - sm/mole - K), T = 303.16(AT), a = K, Po = 7o = - 0 . 1 xK, A = 1 or 0.85, a = l(cm) or 10(cm), and V0 = e-3t(cm/hr). Figure 5.10 shows how the non-dimensional solid density varies with time at the boundary r = a, for various stretch ratios and cylinder radii. We find that initiaUy, the density drops and then it increases to an asymptotic steady state value. This is the same phenomenon of local accumulation of fluid that we have observed in the case of slab presented in Figure 5.5. Continued influx of the fluid and the pressure gradients associated with it tend to disperse the fluid more evenly, causing the solid density in the mixture to increase towards its asymptotic value. Figure 5.11 depicts the motion of the singular front with time for various values of the stretch ratio. We find that the singular front s(t) tends toward an asymptotic value, and this is to be expected because the fluid influx is given by V0 = e"3t which implies that after reasonable time the volumetric influx is close to zero. Figure 5.12 shows how the non-dimensionalized solid density varies with the spatial coordinate for various instants of time. It is interesting to note that the final steady solid mass density in the mixture region does not vary monotonically with the spatial coordinate but reaches a minimum value in the interior. Also, this final steady solid mass density distribution is in contrast with the one shown in Figure 5.6 in the case of slab where the final steady solid mass density is of uniform value. This difference reflects the effect of the curvature of the cylindrical annulus. Figure 5.13 portrays the variation of the stretch

DIFFUSING SINGULAR

SURFACE

93

ratio \R in the R direction, for various values of R, with time, and once again we see that initially the stretch ratio increases and reaches a maximum and then decreases to an asymptotic value. This result is consistent with the results presented in Figure 5.10 which implies that the fluid content increases initially in the region near the boundary R = a, which results in a decrease of the solid density, causing the material to swell and thereby increasing its stretch ratio in the R direction.

94

MECHANICS OF

Figure 5.10 Variation of solid density with time at r = a.

MIXTURES

DIFFUSING SINGULAR

SURFACE

Figure 5.11 Propagation of singular front with time (Cylindrical annulus).

95

96

MECHANICS OF

Figure 5.12 Variation of solid density with position r at various instants.

MIXTURES

DIFFUSING SINGULAR

SURFACE

Figure 5.13 Variation of stretch \R with time at various positions.

97

Chapter 6 WAVE P R O P A G A T I O N IN SOLIDS I N F U S E D W I T H FLUIDS In this chapter, we present three results related to wave propagation in elastic solids infused with fluids based on the formulation in Chapter 3. The first result is the derivation of Biot's wave equations in a compressible porous elastic solid infused with a compressible fluid. The second is a counterpart of Biot's wave equation, that is, wave propagation in an incompressible porous elastic solid containing an incompressible fluid with the volume additivity constraint holding. The final result is the wave propagation in the mixture of an incompressible nonlinear elastic solid and an incompressible fluid where the solid undergoes small deformations on top of finite homogeneous deformations which result from simple tri-axial loading. In the 1950s, Biot [87,88] proposed a theory of wave propagation in fluid-saturated porous elastic solids based on linearized elasticity. One feature of the theory is the inclusion of the virtual mass effect. Moreover, the theory predicts the existence of two dilatational wave speeds, which was confirmed by recent experiments (cf. Plona [246], Berryman [247]). We shall proceed to obtain Biot's wave equations within the context of Chapter 3 under some special assumptions. Biot's theory applies to a mixture of a compressible solid and a compressible fluid. We expect a different structure to the theory when we impose the restriction that the mixture obeys the volume additivity constraint 3.12. We will see that this constraint allows only a single dilatational wave speed. The result on the linear wave motions in a finitely homogeneously deformed elastic solid infused with a fluid is a, straightforward extension of the standard for­ mulation in finite elasticity (cf. Adkins [39], Green, Rivlin and Shield [248], Eringen and Suhubi [249], Ie§an [250]).

99

MECHANICS OF

100

6.1

MIXTURES

Biot's Equations

Consider the mixture to be composed of a homogeneous, compressible and isotropic elastic porous solid constituent with randomly distributed constant macroscopic poros­ ity (3 in its reference state Q,R and a homogeneous compressible fluid constituent (Figure 6.1). For such a mixture, (1) in general the volume additivity constraint 3.12 does not apply : because of the compressibility of both the solid and the fluid constituents in their reference states Q3R and fifR, and thus we can set P = 0 in Eqs. 3.13, 3.18 through 3.21; and (2) it follows from material frame indifference, the isotropy of the solid constituent and the random distribution of porosity in its reference state that A0 = A0(pf,I,II,III,0,P),

(6-1)

where

B = FS(F3)T,

7/=i[/2-tr(B2)],

/ = tr(B),

III = d e t ( B ) .

(6.2)

Clearly, equation 6.1 results in „ , 8A0

F and

r

„9A)

2III

fdAa

r

8 +2

-iFfk = 8m « {-w

dAo 8F^_ ^fdAo 8F>0 dxk~\dl

+

8A» 811

+

8A0\ 81II)

+

+ ~TjjBxiBrdBni,k

T9A0\

B

8A0 „

„

,

2 B B

' m ) - ~ m * «> ,k

_ (dAo \8II+

jMo_\ 8111 j

■

i3

,

^

%3

* (6.4)

We now introduce the concept of displacement to be consistent with Biot's for­ mulation. Let the displacement of a typical solid particle X 3 be given by U =

s X

(Xs,i)-X%

(6.5)

and the displacement of a typical fluid particle X-^ be given by V =

1

/ X

(X',£)-X'.

(6.6)

However, it is of course possible for a mixture of two compressible constituents to undergo an isochoric motion.

WAVE PROPAGATION

IN SOLIDS INFUSED WITH FLUIDS

Figure 6.1 Mixture of a porous solid and a fluid without initial solid deformations.

101

102

MECHANICS OF MIXTURES

In order to show that Biot's wave equations are imbedded in the preceding for­ mulation in Chapter 3, we make the following assumptions: (i) the displacements U, V and their gradients d\J/dX% 8V/dXf, and the velocities Vs = dxs/dt, vf = f f dx /dt and their gradients dv* /dx. and dv j&x. are small in terms of the norm ||U|| = y/UkUk, ||9U/i9X s || = JUi,j Ui,j, etc.; (ii) the process is isothermal; (iii) the virtual mass tensor H is given by H = HVI, where H„ is a non-negative constant; and (iv) as0 = af0 = 0 and m° = 6(v* — v-M, where b is non-negative as required by the reduced entropy inequality 3.22. Now, equations 6.5 and 6.6 result in

Ff, :=

dx!

dxl

vl -

dX{ Bij^Sij

ij + Ui,j ,

(6.7)

Sij + viy

(6.8)

:s

8X?

+ U^+U^,

(6.9)

whence I = 3 + 2Uk,k,

11 = 3 + 4Uk,k,

111 = 1 + 2Uk,k.

(6.10)

We have neglected the nonlinear terms in Eqs. 6.9 and 6.10 based on assumption (i). Let p£ and p(, be, respectively, the solid mass density and the fluid mass density in their reference state J?0 (Figure 6.1), then the balance of mass for the solid and the fluid yields p* =

Po det(F')

Pof 1

(6.11)

~Uk,k),

and p' =

Pi det(F/)

p'o (i - Vk,k k

(6.12)

.

On expanding the terms dA^/dl, dA0/dII, dAo/dlll and dA^/dpf contained in Eqs. 6.3, 6.4, 3.19 through 3.21 in a Taylor series around B0 := (p1, / , / / , / / / ) j ^ - = (po, 3, 3, 1), and using Eqs. 6.10 and 6.12, we get dAp dl

dAp dl

+ 2 Bo

92

^ + 2 ^ + 8P

SPA0 dpfdl

dldll

PoVkik,

d2Ao dldlll

Bo

u,k,k (6.13)

WAVE PROPAGATION IN SOLIDS INFUSED WITH FLUIDS

dA0 dAo dll ~~ " dll

Bo

„ / d2A0 \dldll d 2A 0 dpidll

dAo dill

^Ao "dIP

din

Bo

d*Ao \ dlldlll)

u,fc,fc Bo

(6.14) Bo

„ / d2A0

dA0

103

[didiii d 2A 0 dpi dill

d2A0

d2Ao

"diwiii

anp

o

u,k,k Bo

(6.15)

PO Vlt >fc ! Bo

and dAp dpi

dAp dpi

( d2A0 \dpidl

Bo

d2Ao dpidll

d2Ao

d(Piy

2

Ap \ dpidlll)

Uk,k Bo

(6.16)

PoVk,kBo

Here, once again, the nonlinear terms have been neglected due to assumption (i). Substituting Eqs. 6.9, 6.10, 6.13 through 6.15 into Eqs. 6.3 and 6.4, and neglecting the nonlinear terms results in dAo ' dF jk

dAp dl

k

4

A dip

f Po

+2

dAo dll

^ din2

t a2AQ [dpidl dAp

di

Bo

A didii a2Ao dpidll

dAo\ +

(dAp

9A0 dill

an)

,

dA^

'' an j % didin

d2Ap \ dpidlll)

5Mo

din dp 2 d Ao \ Uk,k an an i Bo vk,k k

Sii

Bo

(Uiri+Uhi),

(6.17)

Bo

and dAodI% dF?- dxk

/dAo \dl

dAo dll

dAo_ dill

U

j:jk

(6.18)

Bo

Next, substituting Eqs. 6.11, 6.12, 6.16 to 6.18 into Eqs. 3.19 to 3.21 and using assumptions (i), (iii), (iv) and P = 0, we have

MECHANICS OF MIXTURES 104

^

/3 ^

(a + - + ^ ^ ] ^ / 3+ 2 ( ^ + +^ ) ' }

1 + 1 a°litj == W+Q + *%* + («> + - + *> ^ +

*^

(6.19)

(6 19)

^ '

'

(6.20) (6-20) Vfe fc 5 dy at = ((70 ' )> ''« ' 7 o + TiPfe'fe +T2

and 3Uk a V rB ,3fc++«aV) + 6 ^ dt # mfcfc = = aaiit/j £W 2 j 'w* ;;*+*>(

dVk\ dt Qt )

2

d'vkM \ A a /fa £4 ^ # f Po")#» + (P0 (po + +P o ) ^ li,v -dt^2 T ' 0t* j '

(6.21) (6-21)

where / s\(dA°+?dAo ft::=2(p°0 + pi){-Qj-+2dII

(6.22) (6 22) -

L ) \ . + dIIl)\Bo

+ +

^

ft := 2 (PO + PO) i ^ f + 2 a/7 a m ; |Bo

+ ,^{^^^^ ^h ft + ^^)|-(^^)l

, (6-23)

/aA) a A ) \

(6.24)

p 5\di ) ^ ++9 7dii) 7 J BBo0 ' = 2 ( p00 + + Po) ft:=2(p

ft

y

A . A 70 == := -pi -Po (po+poj (Po + PoJ 7o

dAo0 dA -g^J ^ 7

(6.25)

)

Bo ' Bo

2 fly, ,, A/ / S A o0 _ 9 M o \_ \ W Aoo ^,„„_ ^d*A d*A„ A 8A + 22 + f 0 22p{ (p + Po) \ + + dpWII dpWIIl) Qp!QI ^ B - "o (PS0 + Po) [-^JQI dptdlll) 77i 1 := dpWII BoBo = P^Po POPO -Q^f

1

(6-26) (6.26)

Bo

72 :

f(o>±2AdM

+(Po)2(Po + ^ ) ^ 4 ? | '

Vro i dip1) Bo ,.(dAo, / a A9dAodAo\\ o ^ dA0 , j ^ M l + + -2Po ^ \^r ++ 22dll dIIl)\ dlll)\j a :=-2 ai i : = Po dII Bo'

-y2-=Po{Po + lPo)

(6.27) (6 2?)

-

r IBo^\H0J Q f\

(6.28)

and ; , ,a_Ao| S M poPo '-- w\Bo-dpl\B-

d

aa22 :■= ="PoPo

(6.29)

WAVE PROPAGATION IN SOLIDS INFUSED WITH FLUIDS

105

The third term b (dUk/dt - dVk/dt) in Eq. 6.21 is usually referred to as the drag term and the constant b is called the drag coefficient. The last term (p'0 + pi) Hv (d2Uk/dt2 - d2Vk/dt2) in Eq. 6.21 describes the virtual mass effect. Now substituting Eqs. 6.11, 6.12, 6.19 through 6.21 into the bal ance of linear momentum equations 3.8 and 3.9 for the solid and the fluid, and neglecting the nonlinear terms yields

[* + <* +4*] £?-<* +*)*£+»(£-£) = /32Ui,kk + ( A + ft - a i ) Uk,ki + (on + 7i) Vh,H ,

(6.30)

and r / is f\ rr^92Vi [pi + (pfi + Pfo) H.] —

, d2U, fs - (pg + Pi) H v ^

, (dUi dV{\ - b [ - £ - - £ )

= (a 1 +7i)Cfc,JK + («a + 7 2 ) ^ » « -

(6-31)

Equations 6.30 and 6.31 have the same structure as Biot's wave equations [87] and so allow for the possibility of two distinct dilatational wave speeds. Thus, Biot's wave equations are imbedded as special approximations in our formulation. However, there are some basic differences between our formulation and Biot's. First, the momentum supply m which describes the momentum transfer between the solid and the fluid is introduced in the formulation here. Actually, it is through the momentum supply ex­ pression 6.21 that the drag effect and the virtual mass effect enter the wave equations 6.30 and 6.31. But in Biot's theory, no such quantity is explicitly presented. Second, the expressions for the stress tensors a" and a' in our formulation are different from those of Biot's. We next discuss the wave propagation in the mixture where both the solid con­ stituent and the fluid constituent are incompressible in their reference states H3R and flfR, respectively. The analysis will show that the possibility of two distinct dilata­ tional wave speeds is due to the compressibility of the solid and the fluid constituents in their reference states.

6.2

Counterpart of Biot's Equations

Let the mixture be composed of an incompressible, isotropic elastic porous solid constituent with randomly distributed constant macroscopic porosity (3 in its reference state HSR and an incompressible linearly viscous fluid constituent (Figure 6.1). Due to the incompressibility of both the solid and the fluid constituents in their respective reference states QSR and QJR (Figure 6.1), the volume additivity constraint 3.12 has to be applied here,

106

MECHANICS OF MEiTURES

(1-/3)4 + 4 = 1>

(6-32)

P° PR where pR, the fluid mass density in the reference state J2/.R, is related to pi by (Figure 6.1) p{ = PpR-

(6.33)

Thus, instead of Eq. 6.1, AQ takes the form Ao = Ao(pf,I,II,9,0).

(6.34)

Here, A0 does not depend on III in virtue of Eqs. 3.1 and 6.32. It follows from Eq. 6.34 that F.dAo 'kdF3'k

JAo _=

«dAo *

&Ao\ fMo

\ a/

a///

8Ap 8F* _ (dAp

dAo\

2

dAo

(6.35)

2

^iBtkBk»

an) '* an

ij ij,k

y "

and

dFfj dxk \ di

dAp

'

(6.36)

As in Section 6.1, we introduce the displacement field U for the solid U = xa(Xs,t)-Xa,

(6.37)

and the displacement field V for the fluid V =

/ x

(x/,t) -X'.

(6.38)

Now, we make the same assumptions here as (i) to (iv) in Section 6.1. Thus, we have F% = Stj + Uuj ,

(6.39)

F* = % + VUi ,

(6.40)

BiJ=Sii 1 = 3 + 2Uk,k,

+ Ui,i+Ujri, 11 = 3 + 4Uk,k,

p' = p'0(l-Uk,k), and

(6.41) (6.42) (6.43)

WAVE PROPAGATION IN SOLIDS INFUSED WITH FLUIDS

pf = pS0{\-

107

(6.44)

Vk,k ) .

Next, combining Eqs. 6.32, 6.33, 6.43 and 6.44 yields (l-0)Uk,k+0Vk,k

(6.45)

= O.

We follow a procedure similar to that adopted earlier and we expand dA^/dl, dAo/dll, and dA0/dpf in Taylor series at B0 := (pf, I, 77)| ref = (pf0,3, 3) with the help of Eqs. 6.44 and 6.45, dAp _ 8A0

d2Ao dl2

d2Ao dldll

d2A0

d2Ao 2+

1T~ ~dT

+

dAp _ dAo dll ~ dll

+ dwn mT 2

Bo

+4

(1-/3) pi d2A0 0

dpfdJ pf0

(1-/3)

d2Ao

0

asm

(1-0)

pi d2Ao

0

d{plf

Uk,k ,

(6.46)

Uk,k,

(6.47)

So

and

dA0 _ dAQ ~dp~i~ dpi

+

d2Ao 2

d2A0 + 4

dpm

dp7dn

+

V,k-k

■

(6.48)

Bo

Substituting Eqs. 6.41, 6.42, 6.46 and 6.47 into Eqs. 6.35 and 6.36 and neglecting the nonlinear terms gives dAo =2

UdAo

dAo\

+2

^ {{-aT {l~0)d

p . !dA0 1

+

m)

[ 3A0 2

( d Ao \dpldl dAo

~W + a/7

d2A0 +2

82A0 +8 +8

d2A0

[ m ^P ^P 8laTi

Bo 2

2

8 A0 \ dpfdll)

\ Uk,k } Sij (6.99)

[U%n +Uiri),

and 8A0dF>3 8F& dxk

(dAo \ dl

dA0 dll

Uj,jk ■

(6.50)

Next, substituting Eqs. 6.43 through 6.45, 6.48 through 6.50 into Eqs. 3.19 through 3.21 and using assumptions (i), (iii) and (iv) and Eq. 6.33 results in 3£ = (-P + 0o + 0iUhth ) Sy + 02 (Ui,d +Ujn ) ,

(6.51)

MECHANICS

108

(-^TQP 4at == (i V

OF MIXTURES

(6.52) (6-52)

7+ fou + % ) 6ij, k,k + liU TiDi,*J

and

d2vk\ ) , - W ) '

dV \ - » ) -

TJ

d2 U k , d k + Po)pl)H m = ctJJ^ (j£ - ~ k) + (p' (p00 + Hv v(( ^ d 2 - ^ rnhk = alUJ,+b 3k+b[—^-

(6.53) (6.53)

where

ft_lfr!+^)(^+^)

P\ := 2

(6.54)

f-?*-* {^m 2

Sy*o dll

+2 ( ^ )

9 A dP

(l-p-)PUd*A0 /? \dptdi P [dpfdl

Bo

d2A, dIP

.nd*A,\ dpwii)\ "dpfdii)

' B Boo

ft (p0 + ^ p^)) f 5^2 + | ^ ) P2 :== 2 U

1,9/

a//J

7o :—d{ := -Po/ fio (p 0 + + Po')d) ^jW 7o

/

[

0

,

d'Ao "dldll (6.55) ;

v

'

(6.56)

So

,

5

(6.57) (6-57)

Bo

((i-/3)(ps 1-/3)(PS + + 2PS) 2pi)l

aA, 5A, - -aoP /T Bo ^ LL p \Bpt 2 2 5o r d Ao d A0 Po (Po + Po) dpfdl ' dpfdii

771i :== Po Po PS Po

-a a

(I-P)PI

0

a2A0l 55(( Pp/)/ )22JJ 8 o '1

l(6.58)

• >

5Ao dAo] dAo] 2dAo (6.59) | 1 L ^ i r T - 2 ^ - 44 - ^ . (6.59) 1 p/9 dpf V dl dll\ BBo By appealing to Eqs. 6.43 through 6.45 and 6.51 through 6.53, we find that equations 3.8 and 3.9 can be written as ff \(l-B)p" r ( l - / 3 )0dAa p;34>

= Po <*i--=Po "i

~ -w~ - a7/J

WAVE PROPAGATION IN SOLIDS INFUSED WITH FLUIDS

[P; + (PS+POO ff.j ^

(PS+POO a .

2

9 t

109

; +&(

OVA

a

; ~ dt )

= - P „ +/32t/„fcfc + (A +(32- at) Ujtji,

(6.60)

and

-(^-')^-(§-^:

[poN (PS + P O O ^ ] ^ 13

P„ +

(a1+7i)[/JUl.

(6.61)

Comparing equations 6.60 and 6.61, which govern the propagation of waves in a solid infused with a fluid that is subject to the volume additivity constraint, with equations 6.30 and 6.31, which govern the propagation of waves in a swollen solid that is not constrained by the volume additivity constraint, we see that the former contain an additional pressure term V P due to the volume additivity constraint and that V U and V V are dependent through Eq. 6.45. Furthermore, equations 6.45, 6.60 and 6.61 only allow single dilatational wave speed, which can be shown as follows. Let (6.62)

£ : = Uk,k, then, it follows from Eqs. 6.45, 6.60 and 6.61 that

U + p£)Hj0ZE

Po +

V

bdE

T — H^r f | ^ r = -p*« + (ft + 2& - «i) EM

(6.63)

and

t

Pi,

Pi

+ (Pi+Pi)Hv 02E bdE -W'p-dt fi

0 P,kk + (a1+ll)EM, 1^s

(6-64)

which in t u m give, on eliminating P,kk,

ftS 1 -

, (1-/3U

(p°o + Pfo)Hv d2E

■/ 3 ( f t +P2 f t + + i )P(l-P) --^-7i 1-/3

w

F

,kk

'

+

b

8E

j[rT)dt

(6.65)

MECHANICS OF

110

MIXTURES

If b = 0, the above equation can be rewritten in the form ^-=c2E,kk,

(6.66)

which implies that there can only be one dilatational wave speed in the mixture that obeys the volume additivity constraint, in marked contrast to Biot's theory that allows for two dilatational wave speeds.

6.3

Wave P r o p a g a t i o n in Anisotropic Swollen Mix­ tures

In this section, we shall generalize the result established in Section 6.2 to a mixture of an incompressible, transversely isotropic and nonlinearly elastic porous solid con­ stituent with randomly distributed constant macroscopic porosity (3 in its reference state Q,R and an incompressible linearly viscous fluid where the solid has undergone a finitely homogeneous deformation due to the absorption and the subsequent swelling and external stretching (Figure 6.2). The superposed wave motion on the swollen mixture will be assumed small in both speed and amplitude which will allow us to linearize. In fact, the formulation to be presented is a direct extension of the linearized theory of small deformation superposed on finite deformations in the theory of finite elasticity (cf. Adkins [39], Green, Rivlin and Shield [248], Eringen and Suhubi [249], Ie§an [250]). Suppose that the swollen mixture has a static basic state O^ (Figure 6.2) in which the velocity of the fluid is v/(b) =

= o,

(6.67)

and the deformation of the solid is z f = AjXf,

46)=A2X2*,

x f = A 3 Xl,

(6.68)

where X s denotes the position of a sohd particle in a rectangular coordinate system associated with f2s0 and x' 6 ' the position of the same sohd particle in the rectangular coordinate system associated with J?t, (Figure 6.2), Ai, A2, and A3 being positive constants characterizing the stretch of the sohd in the static basic state. The stretch ratios Ai, A2 and A3 are not completely independent and the relationship will be given later. It follows from Eq. 6.68 that p»(6) . S x «

dX.*

Ai 0 0 0 A2 0 0 0 A3 .

(6.69)

WAVE PROPAGATION

IN SOLIDS INFUSED WITH FLUIDS

Figure 6.2 Mixture of a porous solid and a fluid with initial solid deformations.

111

112

MECHANICS OF MIXTURES

B (6)

gW = -

(F

3(6)

)

_

F

F.(b)

s(t,)

fF.(b)\T

A? 0 0 \\ 0 0

=

0 0 A| j

' A2 - 1 0 0 A| - 1 0 0

1 - I ~ 2

0 0 Aj - 1

(6.70)

(6.71)

and the principal invariants of B ' 6 ' are given by Jib)

=

trB(k) =

//("> = -[(/("))2--tr(BC')) " 2

A2 +

A2 +

A

2;

(6.72)

(A 2 A 3 ) 2 + (A 3 A 1 ) 2 + (A I A 2 ) 2

(6.73)

and IIIW

= detB (!>) = (A!A 2 A 3 ) 2 .

(6.74)

3

Let p ^ and p*^ denote, respectively, the sohd mass density and the fluid mass density in the basic state, then with the balance of mass equation for the solid 3.1 and the volume additivity constraint 3.12, we have Po AiA2A3

'(b)

(6.75)

and P>«» = ^ | I

1-/3 Ai A2A3y

(6.76)

Since both p1^ and pR are positive in Eq. 6.76, we have A1A2A3>l-/3.

(6.77)

Now, we shall superimpose a small perturbance to the basic state of the mixture of the following form for the solid

Xl(x«,t) - AiX; U = X *(x< b >,t)-x< h > =

b) - A X| xlfc ,t) xS(xW,t) 2

(6.78)

~ -^3^1

where U is the perturbed displacement of the solid from the basic state, and a per­ turbance V =

x

>(x«t)-xM

(6.79)

WAVE PROPAGATION

IN SOLIDS INFUSED WITH FLUIDS

113

to the fluid particles in f!\>. Here, by a small perturbance, we mean that

dUi

>

max

dxf

ij = 1,2,3} < 1 ,

(6.80)

a n d t h e n o n l i n e a r t e r m s involved in forming t h e p r o d u c t s of <9U / dt, &V / dt, d\J j dxf®, dV I <9x, <92U I dt&xW, d2V / dtdx®, a n d <92U / dx<6>2 can b e neglected. B a s e d o n E q s . 6.78 a n d 6.80, we h a v e

F" =

_ p«(b) I <9X»

Ai t/i,i X\U2,I Al"3,l

A2Ui,2 A 2 [/ 2 , 2 A2L/3,2

X3Ui,3 X3U2,3 A 3 t73, 3

(6.81)

B = F 3 ( F ' ) T = B(b) 2A2C/1,1

^ f / a , ! +A2f7j.,2

AjUa.i+A3C/i,3

-^1^2)1+^2^1,2 -^1^3,1 +A 3 C/i,3

2A 2 C2,2 A2?/3,2 +A 3 ?/2,3

A2f/ 3 , 2 +A 3 L r 2 ,3 2A 3 [/ 3 , 3

E = - ((Fs)TF'-l)

+

1 2

(6.82)

= E 00

2A?t/x,i AiA 2 (?7i, 2 +t/ 2 , AiA3(C/i,3+£/3,i)

AiA2(f7ll2+t/2jl) 2XW2^2,2

A 2 A 3 (f/2,3 +t^3,2 ,

A2A3(?72,3+^3)2:

2A3Z73,3

/ = t r B = / « + 2 ( A ^ i , i +\22U2,2

AiA 3 ((7i, 3 +[/3,i:

+\23U3,3),

, (6.83)

(6.84)

II = I (7 2 - t r B 2 ) = 7 J « + 2A2 (A2 +A 2 ) Di„ +2A 2 (A2 +A 2 ) U2,2 +2A 2 (A2 + A2) U3,3 ,

(6.85)

and / / / = detB = III(b) + 2 (AiA2A3)2 Uk,k ,

(6.86)

where Ui,j := dUJdxj. The solid mass density in the perturbed state is P =

Po

fin

/*> _

Ph .

,

x

Aj A2A3

U

„ k,k

,

(6.87)

114

MECHANICS OF MIXTURES

and the fluid mass density in the perturbed state is obtained from the volume addi­ tivity constraint 3.12 as Pf = Pfm+{\~f)/RUk*.

(6.88)

The above analysis holds for any mixture of a solid and a fluid that obeys the volume additivity constraint. In the following treatment, we will restrict ourselves to a mixture in which the nonlinearly elastic porous solid constituent is transversely isotropic with the principal direction along the i 3 -axis and the fluid constituent is incompressible and hnearly viscous. We assume that the constitutive equations 3.19 through 3.21 apply. Thus A, = A) (pf, I, II, E33, Ex\ + E?3, p)

(6.89)

by Eq. 3.30. Now let B0:=(pW,lW,IlW,E&\Eir

(6.90)

+ EiT)

denote the static basic state of the swollen mixture. By expanding dAojdp*, dAo/dl, dAo/dll, dAo/dE^, and dA0/d(E23 + E^) in a Taylor series around B0, using Eqs. 6.83 through 6.85 and 6.88, and neglecting the nonlinear terms, we have

dAo a,4o=8A« 5A dpi dpf
r

2

f

( i ~ /P?R ) ^ d*Ao a A, , \(l~P) Bo Bo

A

2 ' [ A XA22A A!A A33 d{pf) a( P /) 2

Mo0 3 2^a^ "1'dpfdl dpfai

2 d2A [/ 4-[(1-£)/>& \Q-&A 5 A, °

v 9 2 Ao '

2 (A22 +4-A A2'2;1 -dpfdil ^*£-1 + 2A 2A2(A

B0

AiA2A3

+ <,t,+ 2 1 °(^)^s]

d(pf)~

u2,2

ISn

+

+

r [(i-/?)/>£ ( i - / ? ) P £ aa22A> A, 22

(p/) [ AiA A:A22A A33 d9(p/)

;

2 a ^ 2 a A, 3 3

"'V<97 dpfdl

^W+*)^+*^]L>

2 2 2 5 + 2A32(A > ' 11 + A 2 ; ) / / ° + As Sp/a/J dpfdE33,

^dAo 4 o = ^dAo 4o dl ~ dl dl

+

+ Bo Bo

Bo

f 2 Ul-p)pid*A d*A \{l-P)p d*A0 0d A0 R + r [ AiA ap/97 dP XlX2A3 X dpfdl "' ' dP 2 3

( 1 - / 3 ) ^ 92A 22 + 2A?12(A +A A3j2)2 d) Ao ^ 1 Uur + l 1 1 ' ^ ^ ^ 00 ^1,1 1(A 2 + l r t T+ + 2A dldll\ 8o [ AAiA ^A dpfdl 2As 3
(691)

-

WAVE PROPAGATION

IN SOLIDS INFUSED WITH FLUIDS

115


+ + 2xf-^+2Xl(Xl+Xl)^\

U22,2 2

-I Bo

} Ul-f3) ■ ( i - / 3P)Rpd*A ^ a 20A 0 [ AA11AA22AA33 39pp/ a^/ /

+

n2 2X3

+

d2

( \ a . X2\ Ao ^ + xv aiaii

dA dA0 a4o 0 = OAo

dii dll

"'

gMp a2 A , dP a/2

33

.2 3

'

^A, dWE33

22 A, + [" ( i1--/ 3 )) pp ji a Ap

^(A

2

'a/a// "'xdldll

A 2 )^£

+

Bo

(l-(3)pR AiA2A3

d2A0 dpfdii

AiAsAs

Sp/S//

+ 2A, (A2 + A3j — j ^ tflrt + [

+

(6.92)

A o 2 A 2^^Ao

dii [ x,x x33 dpfdii dll Bo A!A22A dpfdii Bo +

u3,3, -I Bo Oo

< ^ + »5 «+*?)££] U ,2 2

Bo

r

+

(i-f3)PfR

a2A,

2 2 9 A

"'2dIdII 2 2 d A,

AjAsAg v a n [ ( l - / 3 ) p j gMp

&4> ■ ^ ( A 2 A2)g£- . v2dIWE a/a//

+ [ AjAjAg + a P ^a// + 2A2(A2 + A

2

) ^

3

+

2

33

2

A

^ - ] | -I

2

A,0 d^ dA0 dA^^dA^ , [r(i-/3)p£ (I-/?)P& a A -TAiAsAs dpfdE dE dE3333 33 a£ a/533 33Bo Bo ^[ AjAsAs ap^a^s

u3,3,

(6.93)

t/3,3,

(6.93)

Bn Do

2 aa2A, ^

2

"' x 8/9^3 d!dE33 l

2 2 i S9^Mo° 4-AA3j2^9 /-7 a*i ? 3*3 jU^ UUltl ++[(f1x-(^/Xs3J )-pM +. 2A 2XX (A [X2 + dpfdE^

+ «.'(*+ *)«/£.]

AjA2A3

Bo

+ + + +

ov2 ^ ^ 0 2X2 dWE33

,

2

M

,2\ C^O ^yX3]Xi>dndE33\

AiAjAs

U2,2

]

r[ ( 1i - / 3 )) pPj^ a22A A 0,

dpfdExs

Bo

a22Ap Ao

2

2

dPfdE33

"' didE33

2A [X, + A j

2 a / / a £+ 3 3 [ 3AJAJAS dpfdE 33

1

2

u3,3,

^ 2\

dE 3 dIdE 33

(6 .94)

Bo

and

and

dA0 d(E 3 + E!3) 2

dAo ~ d{El3 + El3)

+

■(1-/3) p i a 2 Ao A!A2A3 dpfd (E22A3 + E\3) \(1-P)PR 9 V

aylp _ a^to Bo 2 d(E?a + E\3) ~ d{Ei3 + EId 3A) -0Bo [ AXA2A3 ap/a(£ 2 3 + Eis) + 2K El a2A) + Ei3)\ dId{E d22A ,J 3 ^^) 1 3 0 + -E23y+ ~\{\ 2 \dIId{ + 2Al

a/a ( £ 2 3 + £ 2 2 3 ) +

2Al

^

^

a//0(i3?, + i&)]

Bo

B/

l!l

MECHANICS OF MIXTURES

116

+

d2 Ao

(l-p)pR

2

A, A2A3 dptd (E

2

+ E 3)

3

d2A0 ■ did (E23 + Eh)

+ 2A22

2

d Ao

+ 2X1 AI+A? diw(E2 3

u,2)2

+ E 23

d2Ao a2A0 + 2A + dpfdiE^ + E^) ' " " » 3 I 0 ( £ » j + £&) 2 d Ao + 2A2(A2+A2) dIW(E23 + E23) d2Ag U2,2. + A;3 dE33d{E23 + E23)\ (l-f3)PfR AlAjAs

(6.95)

Substituting Eqs. 6.78, 6.79, 6.81 through 6.84, 6.87, 6.88, and 6.91 through 6.95 into 3.19 through 3.21 and using Eq. 6.80, we get ^11 = -PpP + Ao + A l P i . l +A 2 ^2,2 +f3l3U3,3 ,

(6.96)

°22 = -PpP + &0 + @2lUul +/322U2,2 +/?23^3,3 ,

(6.97)


(6.98)

a

=

Pl2\2U\,2 +P\22\U2,i ,

(6.99)

Cl3 = /3l313f^l53+A33lf^3!2 i

(6.100)

\2

^M

'' "

1-/3

'<92£/i

d2VA

ft p P + 10 + 71^1)1 +72^2,2 +73^3,3 -

' ' ~W ~ ~W J +

, . d2V2 ™2 = # | — dt2

m3 = H

(6.101)

P2323^2i3 +P2332^3i2 ,

1

^

T7

=

d2u3 dt

2

,(dVi

dV1\

I ~8T ~ ~dt I +

d2V2\ , (dU2 dV2\ 5 ^2 "jI + M\ "ST dt dt " "ST dt +

d2v3\ dt2 j

(du3 dv3 \ dt

dt

ai

l m +a2

ai?Jl 12

'

£y>

(6.102)

^ 2 ' 2 1 +Q3[/3,31 , (6.103)

+ a 2 C / 2' 2 2 +"3^3,32 ,

+ "1^1)13+a2^2,23+"3^3,33,

(6.104)

(6.105)

WAVE PROPAGATION

IN SOLIDS INFUSED WITH FLUIDS

117

where Pp '—

1

(6.106)

AiA2A3'

"a^ (*»-A?) 22 m (b) (p*> fto := - Po ^ T++2A 2A (^>++ p pM) ^ Aot= ) [ ^ f+ AiA2A3 *^ dl (/ -A-)^7 +

io (6.107)

2 a£ * ^]L« 33

(6- 7)

So

2

A, ^

2A Po 2 A 2 [[(l-/?)p£-p ( l - / ? ) p £ - p 00]] ^'a A Po /r(6) AiAjAs + AiA2A3 AjA2A3 [ dl + V

''

,. fMo 2 N ^

M a//

I a^ ' + 2 A 2 ( p ^ + p ^ ' ) { 2 A 2 ^ ( l --M i3 + idEa + ± | ^ a ] I + 2A2 (p'W + p^>) | 2 A ^ (1 - StJ)

+ +

+

2

a/

+217

X2 N J/

5i4o , 1 9 4 all A3 a E M

/

% j

~dT V i)~dll YdE~ (l-/?)p^a2A0 PA, f(M Aj A,A 2 A 3 a P / 9 / ' " a/ 2 + i A / 7

A2) '1 " ( 1 - ^ ) 4 d2A0 A 3 a 7 a ^ ^ + (/(6)-A') A!A2A3 ap/au

92A

° didll

,<>^>-<>^t/«

2 r(i-/J)p£ „.2a PAo a 2 A, (l-/3)pJR 8a2A> A0 + - [ A,A2A3 ap/as* ' "x>didE33

1i

^^m>¥}.:

+ 2A2 ( / « - ;

(6.108)

+ 2 [ AaA2A3 a p / a ^ + >dIdETs ^:=2A

2

(p^

+

p^)

'a^ a/

+C"->s->Oi£*] ■

_ := 2 ( / » A * :=7o2A Pm) [ ^ 1-/3+ V

12 2 §^ £ +i F ^ % + (KH _ A - A )+ "n f)Po+^)^

Bo

5

] [ ,

7;

7i '

-

Po A!A2A3 AiA2A3

-

(i-/?)(^' + 2 p ^ ) ) p ^ p ^ ) p 0dpi aA AjAjAa AiA2A3

dp!

(6.110)

(6.109)

(6 no)

^ I ^ -PoJ O( i -T )*^^^!*' /?)(p^ + 2p^))p^-p^)p0aA :

(6.109)

BQ

MECHANICS

118

_ pfW (p-W + / W ) \(l-[3)pR

d*A0

.2 d Ao a A, AjJ^S-J 2A^(i«-^J^L ( / « » - \ 2if\ + +3 o + + 3 dp'dE dpfdii t

A

cV3/

2

2

«v

l3

dPfdE3333

i dpfdii

,

(6.111) (6.111)

7

J BoBo

m

= pMpMHn, H H := P^p H0,

(6.112) (6-112)

a f (l - 1 " ^ ) , A1A2A3/ AiA \ 2A3 y A1A2A3 AiA2A3y

(6.113)

b6 := ■ — —

f

<^i <*♦ :=

(l-(3)P^P R Po > x% H r~r^ ^ 1 ^ 2 A3 —; A1A2A3 AiA2A3 AXA2A3

- 2A2p/) -2A^[^

MIXTURES

, _25M0

[ A^AS a( P /) 2 '

2

OF

8Ao ~^~f B„ Op'! Bo dp

+ ( / < ^XA ? ) ^ ++ i ^ ]3 [

[dl + ( 7

>)dII

2dEj >\

;

(6.114)

Bo

where P0 is the pressure of the mixture in the basic state and p is the pressure perturbance of the mixture in the perturbed state, no summation is intended in the above expressions. With the above expressions for p", pf,a>, af and m , it follows from the balance of mass equation for the fluid 3.2 and the balance of linear momentum equations for the solid and the fluid 3.8 and 3.9, that (l-/3)/3pU -/3)/3 {l-/3)f3 + [l-(l [l--(l-/3)fi = 0, 0, k,k,k p]V pU p]V k,kk,k

(ps(b) + H) ^ at2

d2^

d€-

(dUY

9VA

[ dt "dt)

= ) c / l#1,11 , n + A+A 2 12212#1,22 # l , 2 2 + A+0131 3 1 3 #SlU , 3U33 3 = -/3pPl -fo* l ++ (/?„ (Ai --a 1<*,) + (Aa + A a a - «.) #2,21 #2,21 ++ (As ++ Ass: A331 --« «3) #3,31, 3 ) #3,31 (Aa+Aa», M) r^92^ (P°ib) + H) d^2 -

iP^ + H)-^-

2

v2 ,rdd2V2

(6.115)

(du , [9U2 2

(6.116) (6.116)

9V22\ \ dV

+b -H— (— ~ - _ ] ~H dt- +b {dt

= -AP>2+A221#2,U -ppP,2+0llnU2iu U ,33 + ((3 (A222 --a2a)t/2,22+A323#2 2) U2,22+/323 = !33a 3 + (Al {fin +

f31212 + 3A332 Os) #3,32, ++ A212 -- a-1 a,)U ) # 1 , 1ul2 + +( A(As 3+A 3 2 -" -03) #3,3

(P°w + H) ^

-

d2v3 dP

(du3 \ dt

(6.117) (6.117)

8V3\

"dTJ

== --PpP,3 A P , 3 +A331 +P1331 #3,11 t/3,11 +A332#3,22 +^2332^3,22 + + (As IP33 " "- «,) a 3 J #3,33 L/3,33 +f ((Ai A l ++AAsia 3 1 3 - a ai )a )# l#,l1! l33++((AA22++A^2323 ^ 2 3 --a 2a)2[) / C/ 3, 2 , 22,23,

(6.118)

WAVE PROPAGATION

IN SOLIDS INFUSED WITH FLUIDS

(*«+<£/1 1 I ,

ff^

at2

r, ~ Pp

du

b( b

119

avA

>

\at

dt )

|p,i + (ai + 7 i ) ^ i , n (6.119)

+ ("2 + 72) U2,2i + (a 3 + 73) l ^ ,

{Pm + H) ^

-

= - h

Hd

2

U2 dt2

V dt

dV2 dt

5 - A> ) P,2+ (<*! + 7l)^l,12

1-0

+ (a2 + 72) t/2,22 + (as + 73) f/3,32 ,

/ f(b) , ^d2y3

2[/

(6.120)

\ (p^ + ^ ^„d- 3^ ^, /0tf - t ^<9K -^ j 1

3

3

fl, P , 3 + ( « l + 7l)^l,13

^1-/3

^

+ (a2 + 72) (72,23 + (a3 + 7s) C/3,33 •

(6.121)

In the case of the solid constituent being isotropic and Aj = A2 = A3 = A, dAo dE33

dAp d(E23 + E23) a,=a2

0,

= a3)

(6.122)

(6.123)

0io = 020 = 030,

(6.124)

01212 = 01221 = 01313 = 01331 = 02323 = 02332,

(6.125)

012 = A s = A l = 023 = 031 = 032,

(6.126)

011 =022 = 0 3 3 = 012+20l2 1 2 ,

(6.127)

7i = 72 = 7s-

(6.128)

120

MECHANICS OF MIXTURES

Thus equations 6.115 through 6.121 reduce to (1 -/3)(3pUk,k + [l - (1 -(3)/3p}Vk,k=0,

(p3{b) + H)

d2 i U dt2

H92Vi2 + b (9U' dt \ dt

(6.129)

dVi \ dt J

= ~PPp,i +Pi2i2Ui,kk + (Aa + A212 - « i ) Ukrki,

(6.130)

-
l f(b)

-

= -(j~-p,)p*

+ {<*1 + T1)UhM.

(6.131)

Clearly, equations 6.129 through 6.131 reduce to Eqs. 6.45, 6.60 and 6.61 if A = 1. In fact, with 0:=l-(l-f3)f3p, -0)fr

(6.132)

and p:=f3pP,

(6.133)

equations 6.129 to 6.131 can be rewritten as (l-p)uk,k+pVk,k=0,

{ ,(

p b) + H)dm2 -

2 ud Vt

L

/»Ki

(6.134)

dVA

= ~Pn +/3i2i2Ui:kk + (/?12 + / ? „ „ - «j) Uhlhi,

t^^rr.^Vi {p 3(b) +

(P

+

H

(6.135)

d2Ui (dUi 8VA d2V 9VA H H) h ^ ' ~~d^~ dt2H~dF~ " \~dt~^) ~7»

P=p,i + (a1+ll)Uk)ki, 1-/3

(6.136)

which have the same structure as Eqs. 6.45, 6.60 and 6.61. This implies that the uniform extension of the solid can be treated with an equivalent porosity. Moreover,

WAVE PROPAGATION IN SOLIDS INFUSED WITH FLUIDS

A. = y3 < i <=> 0 > P-

121

( 6 - 137 )

Also, a similar treatment can be carried out for Eqs. 6.115 through 6.121. The above formulation is restricted to the case in which the solid has been subject to simple tri-axial loading. A similar procedure could be carried out for other and more general basic deformations. It would be appropriate to end this chapter with a few remarks on the study of propagation of waves in general mixtures. In fact, most early studies of mixtures were confined to the propagation of waves, under the assumption that the mixture occupies all of R 3 , as we do not have to deal with the problem of specifying boundary conditions or the difficulties associated with the reflection of waves. There have been several studies on the propagation of waves through a mixture of two fluid constituents (cf. Knops and Steel [251], Craine and Johnson [252], Bowen and Rankin [157], Bowen and Doria [253], Bowen and Chen [159,254,255], Kuznetsov et al. [256], Drumheller and Bedford [257]), through a mixture of two solids (cf. Aron and Sava [258], Aron [259]), and through a solid infused with a fluid (cf. Steel [260], Bowen and Chen [261], Bowen and Wright [262]) as well as that through a fluid infused with solid particles (cf. Jackson [263], Murray [129], Garg and Pritchett [264], Hill and Bedford [265], Homsy, El-Kaissy and Didwania [266], Batra, Bedford and McCarthy [267]). We shall not discuss these studies in detail here but refer the reader to the aforementioned papers for the same.

Chapter 7 M I X T U R E OF T W O N E W T O N I A N FLUIDS There have been a number of studies on the mechanics of mixtures comprising two Newtonian fluids within the context of mixture theory (cf. Adkins [39], Green and Naghdi [44], Mills [161,268], Bowen [52], Miiller [58,269], Doria [270], Faizullaev [271], Bartlet and Home [272], Craine [273], Atkin and Craine [65], Beevers and Craine [274], Liu and Miiller [275]). These studies have resolved issues pertaining to which variables need to be included in the constitutive formulation for a certain class of fluid mixtures (cf. Bowen [52], Miiller [58]) and some constitutive models have been proposed (cf. Mills [161], Craine [273]), and Williams and Sampaio [276] have discussed the nature of the viscosity of a mixture of two fluids. Extensive literature abounds that pertains to bubbly fluids, which within the context of mixture theory would be considered as a mixture of a gas and a liquid. We have cited some of this literature when discussing the various interaction terms that come into play in the mechanics of mixtures in Chapter 2. As there have been many comprehensive review articles and books devoted to this subject (cf. Soo [90], Wallis [277], van Wijngaarden [278], Batchelor [279], Miskis and Ting [280]), we shall not present the results treated therein; rather we would like to present a different aspect of the problem here. Our purpose in this chapter is to extend previous formulations on the mechanics of a mixture of two immiscible incompressible Newtonian fluids by appealing to a splitting method due to Soo (cf. Soo [90]). We also take into account the effect of quantities such as the virtual mass and the interfacial density, which are not usually incorporated in previous theories. Furthermore, certain symmetry relations among the coefficients that appear in the constitutive equations are proposed for a class of mixtures. Using this framework, we study two special motions within the context of our theory, which, together with experimental data, will give some guidance regarding the determination of the coefficients contained in the proposed constitutive functions. 123

MECHANICS OF MIXTURES

124

7.1

Basic Balance Equations

Consider the mixture of two immiscible incompressible Newtonian fluids in the ab­ sence of chemical reactions. We shall denote the motions related to the two fluids by superscripts 1 and 2, respectively. Then, isothermal processes of the mixture are governed by the balance equations 2.24, 2.32, 2.36, 2.40, 2.50, and 2.56, that is, .(i)

at p)t

+ div (V1 V 1 ') = 0,

(7.1)

+ div(p<2V2>)=0,

(7.2)

i) (^L

+ [VvW] v*1') = div ( ^ ) ) T - m + pWb,

(7.3)

») ( ^

+ [ w W ] v<2)] = div (b,

(7.4)

(^)+^)) T = *«+*«, "

LL

(1) T"

PR

(7.5) (7.6)

—1

(2) -

*>

PR

and -pA + tr ( s W V v W ) + tr (<7<2>Vv<2>) + m • (V 1 * - v<2>)

+P ^ d i v v ^

^divv(2)

+

+

V

^

.(v^-V 2 ))

> 0.

(7.7)

Here, jyM

=

^(1) _ p ( D ( A ( D _

A

)

I;

jyd)

= CT(2)

m = m-V[p(1)(^1)-A)] .

_ p (2) ( A (2) _ ^

j

(7.8)

Equation 7.7 combines Eqs. 2.50 and 2.57 by virtue of the volume additivity con­ straint 7.6, P being the Lagrange multiplier. An additional quantity, the interfacial density s (cf. Drew and Segel [132]), which denotes the averaged area of the interface between the two fluid constituents per unit volume will be used in our analysis. We shall describe the evolution of s by ds — +div(sv) = h

(7.9)

MIXTURE OF TWO NEWTONIAN

FLUIDS

125

where a constitutive relationship has to be provided for h. The relevance of h can be traced to geometric configurational changes such as breaking or coalescence or flattening of either fluid constituent as the mixture deforms. The importance of s lies in the fact that the two fluid constituents interact through the interface and s represents the averaged area of the interface per unit volume. Two additional issues related to Eqs. 7.7 and 7.9 deserve to be addressed here. Thus far, in the formulation of a theory for the mixture, we have neglected the interfacial properties such as surface tension and the work and energy associated with changing the interfacial density s, so no terms directly related to 5 appear in the energy equation 2.46 and consequently no such terms appear in the entropy inequality 7.7. Also, there is another way to describe the evolution of s which is more general than Eq. 7.9, that is, we can express the right-hand side of Eq. 7.9 in the form of a flux term plus a source term. The inclusion of such a flux term will generally increase the order of the evolution equation of s, and also complicate the constitutive formulation. Therefore, in order to simplify the treatment, we shall not consider these issues. In the next section, we shall discuss the constitutive variables that are necessary for the determination of A ( 1 ) , A^2\ A, a ( 1 ) , o=(2), m and h.

7.2

Constitutive Variables

To construct a constitutive model for the mixture of two immiscible incompressible Newtonian fluid constituents, we first need to know which variables should be included in the formulation. Clearly, the constitutive theory for a Newtonian fluid suggests that the velocity gradient of each constituent has to figure in the constitutive rela­ tions. Also, the velocity difference between the two constituents is important from a hydrodynamic point of view, as the drag on a particle due to the flow of a fluid past it depends on the relative velocity. Furthermore, the mass density gradient of each fluid has to be included, and the rationale for this is provided by the analysis of Miiller [58]. Interestingly, a unified argument can be made for the possible inclusion of the above mentioned variables in the formulation of the constitutive modelling based on Soo's analysis [90]. This argument is indirect in that instead of considering a general mixture of two immiscible Newtonian fluids, we restrict our argument to a special and simple mixture of two Newtonian fluids, namely a mixture which acts like a Newtonian fluid. For this mixture, we know the constitutive expression for its Cauchy stress tensor. Then, based on this expression, we infer which variables should be included when determining the constitutive functions for its constituents. Of course, we may not need to formulate the constitutive equations for the individual motion of each constituent of this special mixture. The following discussion will give us some insight into choosing the constitutive variables for the motion of a mixture composed of two immiscible Newtonian fluid constituents. We should however bear in mind that various types of constituents, not individually Newtonian, could be added together to form a Newtonian mixture, but this is not germane to the discussion here.

126

MECHANICS OF MIXTURES

For the sake of simplicity, let us imagine that a Newtonian fluid such as water or air consists of two mixed constituents, one 'dyed' 'red' in color (denoted by superscript 1) and another 'dyed' 'white' in color (denoted by superscript 2) without the possibility for the conversion of mass between them. According to the formulation in Chapter 2, we can introduce the fields of mass densities p*1' and p^2K velocities v' 1 ) and v<2>, Cauchy stress tensors a^ and a^2\ momentum supply rn associated with the 'red' and 'white' constituents such that ^

dt

(7.10)

+ div ( ^ ' v W ) = 0,

( p ^ v ' 1 ) ) + div ( p ^ v ™ ® v' 1 ') = div (a^f

-m

+ 0«b,

(7.11)

and ^

dt

+ div ( p « v « ) = 0,

(p<2>v<2>) + div (p<2>v<2> ® v<2)) = div (?<2>)T + m + p<2>b.

(7.12) (7.13)

To treat the 'mixture' as a continuum, it is physically reasonable to assume that the mass density of the 'mixture' is given by p := pW + p<2>.

(7.14)

Therefore, equations 7.10 through 7.13 yield dp dt

dt

div (p^v* 1 ) + P<2)v<2>) = 0,

(7.15)

feO>vv<2>) + div i ( p ^ v ' 1 ' + p<2'v<2)) ® (p^v* 1 ) + P<2)v<2>) 2

'(1^2) / 2 \ J (v< ) - vW) ® (v<2) - v'1'

= div (5 (1) (i) + ?< )f - 2 ^ 2

7.16)

The velocity v of the 'mixture' is defined through

v := - (p^v' 1 ' + p<2M2>) , p\

(7.17)

'

based on Eq. 7.15. Thus, combining Eqs. 7.15 to 7.17 gives — + d i v ( p v ) = 0,

(7.18)

MIXTURE

OF TWO NEWTONIAN

FLUIDS

127

— (pv) + div (pv v) = div (ffM + 5 « ) 3 p^ 1 )^ 2 )

(v<2> - v* 1 ') ® (v<2> - v*1))

pb.

(7.19)

Because the 'mixture' here is actually a Newtonian fluid, we have — (pv) + div (pv ® v) = div ( - p i + 2£D) + pb,

(7.20)

where p is the pressure, jl the viscosity, and D is the rate of deformation tensor which can be calculated through Eq. 7.17, as p< 2 >-

D := 1 [Vv fVv + f( V v )f 1] = ^ D * 1 ' + ^ D < 2 '

+2p-L

| ^ i ) [(v<2) - vW) ® Vp<2> + Vp<2> ® (v<2> - v* 1 ')] -p<2' [(v<2> - v*1)) ® Vp^1) + V p ^ ® (v<2> - v ' 1 ' ) ] } .

(7.21)

Here D*1) and D<2) are the rate of deformation tensors associated with constituents 1 and 2, respectively. Comparing Eqs. 7.19 and 7.20 results in

(ffM + ? ( 2 ) ) T _ IJEI

(v<2> - V*1)) ® (v<2> - V<1>)

-pi + 2pD.

(7.22)

Therefore, we conclude that

- p i + - i — ^ D ( 1 ) + -*—--D(2) p" P

flrW + ?<2>

_ i. ■ - ( # - v W )

P

+L

8

2 1 ( v >>(- ) _ vvW !

( p « [(v<2> - v « ) ® Vp<2> + Vp<2> ® (v<2> - vW)] _ p<2) [(v( 2 ) - v*1)) ® V p ^ + V p « 0 (v<2> - v' 1 *)]} .

(7.23)

It is clear from Eq. 7.23 that the mass density gradient and velocity gradient of each constituent and the velocity difference between the constituents have to be included in the constitutive formulation if the mixture is to act like a Newtonian fluid as a whole. Naturally, we may reason that these variables are also to be present in

MECHANICS OF MIXTURES

128

the constitutive model for a mixture composed of two immiscible Newtonian fluids. In addition, the virtual mass effect and the interfacial density need to be included in general but these terms are absent in the above analysis. We now turn to seek the structures of A, a^K c^2, in and h.

7.3

Constitutive Modelling

We shall assume that A=A[P,S,V^-V^),

(7.24)

c-'-^-|p,Vp,,,Vs,v(1)-v(2),D(1),D(2),W,^^-^^V

/

J

n(2) v ( 1 )

I'""

nWv(2)\

tf» = ^ A V M , V S , v W - v ( ) , D W 1 D P ) , W , ^ L - f i ^ - J , n(2)v(i)

r)(1)v(2)N

nr = o i ( A V M , V s , v W - v ( 2 > , D W , D W 1 W , ^ - ^ - | )

(7.26)

,7-,,

and

A = ^,Vp,5,Vs,v(1>-v(2),D(1),D(2),W,^^-ii^j,

(7.28)

where

-v< 2 ))f},

1 W =I { v (v' (vM - vv(2(2>) - [V [V (v' (v'11)) -- v < 2 > ) f } , W ::= \ {V ) >) -

(7.29)

and D(1> and D(2> are the rate of deformation tensors associated with fluids 1 and 2, respectively. D^v^/Dt - D^v^/Dt accounts for the virtual mass effect. The dependence of A, c1'1', o^2', m and h on p^', pfc and the viscosities fi^ and /x'2) of the fluid constituents in their reference states is implied in Eqs. 7.24 through 7.28. While the relative acceleration a^1' — a' 2 ' is not frame invariant, the quantity D ^ M 1 ' / D t — Z) ( 1 M 2 VDt is frame invariant. We also note that W is frame invariant though W(1> and W ( 2 ) are not. We know that A depends on v ( 1 ) — v ( 2 ) in terms of |v ( 1 ) v(2) . Therefore, we follow the procedure adopted in Chapter 3 and expand A in a Taylor series, i.e.,

MIXTURE OF TWO NEWTONIAN FLUIDS

129

2) (1) (2) A= -A0(p,s,p£\p%\^),pW) = A 3 (p,.,P^, /0 k ,M ,M )

+i/fo(p,S)P^,pk2),M(1),M(2))

( v « - v<2>) • (v<» - v<2>) ,

(7.30)

> 0.

(7.31)

with Ao > 0,

ff0

We will see later that # 0 is related to the virtual mass effect. For the mixture in which both the constituents have similar geometrical configurations such as drop-like or net-like distributions, we are motivated to assume that

A>(p,,«,M W « ) = ^(p^pMV'V25) ,

(7-32)

H0(p,s,p{*\&\^\^)

(7.33)

and = H0(p,s,p%\p%\^\^) .

These two equalities describe the fact that both the constituents are incompressible Newtonian fluids with similar structure so that they are of equal status in the constitutive formulation, that is, we can assign either superscript 1 or superscript 2 to any one of them without changing the structure of the constitutive model. Substituting Eq. 7.30 into Eq. 7.7 yields

H

rr dA a^o0

a^o i / s # 0 aF 0 \ i (1) + + S^7 ++ 2 (,pp^p~ ++s~dT) {«ff+,» + [pp^~ ^ 7 " ^ 2 [ ~ap~ s~dT) Il v i

p

ri1

dAo 1 / dH0

dH0\ |

|2 ( /22)>,2]

m

Sji I] *

-«f)+C^j

- , W i f 0 ( ^ _ , (2)\ f ) )/((1) , f ) - - « f ) + £ | p %nfrf l^ -pMHc(vPL(2) f (2)

«),[r (2

;

dA

o <9^o

9A 11// dH <9^o d#o 0 0

*-^P

^*-* " \

VJ '' / ' U

' a^0 p P)P dp dP

+ m, + p

yni

D<

s

G>Ao

dA <9A>\ 0\ I1 (m 1)_

Op

OS J 11

1 / 5ff0 p

ds 22 v \ dP dp ds

_ P„ /^A) / ^ +l Ii ^ £l |vv ^( 1-) v- v' (2 2' '!r2N l /Ui >>00. .

U,

%

p™ PH J i '° S

MTiA , s

dds )) 'l

-rr- rvra-}H'>-

" V da ^ 22 ds a , \I

, 2,2J 1|2 (2) - V1, '

!I /; "

v

(1)

(2)|2_

v

/P«\

'' J. V P / "

(2)\

(7.34)

130

MECHANICS

OF

MIXTURES

Based on this inequality and Eqs. 7.23 and 7.25 through 7.28, we may choose one specific constitutive model as follows: P{1) n

-(11 f !?W =

9

m [ -P(1)

.1^

^0

&40

+5^7

+i*+x,+4«[(vl-.-v,»)8v(^) 8H0 8H0\ (1) i1 /f d# dff I a) f2v,2l 0 0\ (2) I2 + + — 2lIpp—— "a7 +h5S——— ^7jrI V^" ' — V^ *

+ 2a<1'D(1) + 2ai1)D<2> -

+V

f4

1)

(|) 8(v™ -.„)]

(i) (2)) +««£«+««}] + ^ 1 l^kk T 2 ^Jcfc | 1 (1)n(1)

n

a

T

'(v ( 1 ) -v ( 2 ) )® V f—)

+(^

+ 4-)(Xv.,)

® (VW - v(2>) + a'1' [(v' 1 ' - v(2>) ® V , + V , ® (v' 1 ' - v<2))] (7.35)

- 2a 8 W,

I

PR

+

L

d

P

4S

8Ao

^7 v(1,

+ K^ ^)iv(2)-

i1+ a M +4 2 ) D -} 1

+ 24 2) D< 2 ) + 2 a f D « + 4 2 ) [(v<2> - vW)

e

V

^

+v|),(^-^] + ( ? % 0 + ^)(Xv«) 8 (v« - v « ) + 4

2 )

[(v<2> - v « ) » Vs + V , ® ( v « - v « ) ]

+ 2a 8 W,

m =:

V

(7.36)

p(D\

and

sdAv

- D "' v l "j + i,(v») vW)

*ra

ifdHo dHo\ i (1) _ ( 2 ) | 2 2 \ dp ds ,(D 2 ( a « D « + 4 2) D< 2 >) V IE_ ( a W D W + 42>D<2>) ( v « - v<2>) Ph - 2 (4 1) D< 1 ) - 4 2 ) D W ) V , + a,W ( v « - v<2>) (7.37) P

^4o <9p

/z}(») v 0)

9s

MIXTURE OF TWO NEWTONIAN FLUIDS

(dAQ h=-=-w^-awP ft +I^r_vp>r 1 ).-g—

131

ldH0 v(1) 2 ds

-v
(7,8;

The coefficients in the above equations, by virtue of the entropy inequality, meet J2) 4 2 ) >0,

« -> n ai^+fa^

> 0,

441M2)>(aW + al2')2,

a « + \a?

> 0,

4 ^ + | - i - j ^ + § * ' j > [«?> + 4 2 ! + | («f> + 42))J , a 8 > 0,

a 10 > 0,

6 > 0.

(7.39)

Here the coefficients of 3 , af\ a8, ag, aw and b are functions of p, s, $ , pg>, pW and p<2), and a 10 also depends on Vp, Vs, | v « - V(2>|, D*1), and D<2), and b also depends on I v ^ - V ( 2 )|. As in Eqs. 7.32 and 7.33, we may assume for the mixtures specified there that aV=a%(P,s,P$,P%\^\^) a

(2)

-^(p^p^PflV2',^1')'

i = l,...,7,

(7.40)

( 1 ) ( ) ( 2) a i ,p R ,M(1).A'(2)) , 8(p,*,pM B(p,sJ*\p%\^\^) p gW \ / x « V W) === aa8(p,5,p

LL

a*(p,*,P$A\nm, 22

J = -

aJP:s,P%\P%\^\^),

11

2) ) ( 2 )(2) (1 a1100 (p, (p,SS,Vp,Vs,|v( ,Vp,Vs,|v( )-v( )-v( )|,D( )|,D(2 )2)),D( ,P^,M ^ ( 1 )) D( 1 )1)),pk pk 2 ) ,p^ ,M )M 1 2 1 2 1) 2) (1) == aa1 0 (p, ,M ( 2 ) ), 5 ,Vp,V 5 ,|v( )-v( )|,D( ),D( ),^ ,pk ,M 10^,5,Vp,V5,|v^-v^|,D^,D^,p^\p^,^,M^)

b{p, S^-v%PfJ$,^\^) b(p, ,\^-^\,p^,p^,^, S

l

a«)

= b( = hip.s, P,s,\^-^\,ti\ v W - v < 3 ,P^%\^\^). f i ^ , ^ ■

(7.41)

The above formulation is restricted essentially to the laminar flow regime of the mixture. For a theory broad enough to include non-laminar flows, we have to intro­ duce the concept of fluctuations, and essentiaUy extend the above ideas (see Chapter 8 and Ishii [281]).

7.4

Two Special Motions

Having introduced a specific constitutive model from Eqs. 7.35 through 7.38, we now study two elementary problems within the context of this model. We first consider

MECHANICS OF MIXTURES

132

the possibility of the mixture being in a static state under the action of a gravitational field of b = -ge3 in the unbounded three dimensional space R3. Let P (1) = W%\

P{2) = (1 - «*) P(R\ * = -o. v*1' = v<2> = 0,

(7.42)

where u0 g (0,1) and s 0 > 0 are constants. Substituting Eq. 7.42 into Eqs. 7.35 through 7.37, we get
a^

-y0P

- P(1)

( dAo

(2)

=

8A0\

i dA0

( 1 - ^ P - ^ ^

(7.43)

I,

dA0

t

(7.44)

+ s-^

and (7.45)

fn = 0. Then equations 7.1 through 7.4 reduce to P,i = 0,

P,2=0,

P,3-

-$9

"PR

9,

(7.46)

that (1)

- PR o{2) —

o PR

(7.47)

This implies that the mixture cannot be in the static state given by Eq. 7.42 in which both the fluids have uniform mass density distributions and the interfacial density is constant unless pR — pR . This is completely in keeping with the physical expectation based on the effect of buoyancy. Next let us consider the steady fully developed flow of the mixture in a straight circular pipe of radius R, with the neglect of b. We shall assume that the velocities and densities of the two constituents in a circular cylindrical coordinate system attached to the axis of the pipe are given by

vW = w W ( r ) e 2 I

P

(2)

= P

(2)

v< 2 >=u,< 2 >(r)e,, W,

p^ = pW(r) ,

5 = s(r).

(7.48)

We shall assume that both the fluids adhere to the boundary so that v (1 '(Fc) = 0,

v ( 2 ) (fl) = 0.

(7.49)

Furthermore, we shall assign some values for the densities of the constituents on the boundary such as

MIXTURE OF TWO NEWTONIAN FLUIDS

pw{R)

= uopl[',

133

(2) pW{R) = (1 - vo) p%\

s{R) = SO, s0

(7.50)

where ez is t h e base vector along the 2-axis, and u0 £ (0,1) and s0 > 0 are constants. Of course, this implies that for different values assigned for the densities on the boundary, we can have different solutions. Substituting Eq. 7.48 into Eqs. 7.35 through 7.38 results in {1) <9A < M 0 1I /(ddH -d) (i) F'p dA °0 + s dA° H°0
(7 51) (7.51)

2

5&> oty = =

0, o,

(7.52)

-'^(f)

^)=a(!)^)+a(!)^2)+4!)(u;(1)_

1 w ( 1 ) , « ) ,, +4 * * («« (w« -- u/2>) ~ +h aa8s±+4 | : ( (w^ "-- «w^) 1

(7.53)

4J sfc} == o,0,

-(11 di

<4-

(1) P " ( 1 ) r, p

(1) (i)

— P — 0^ ' PR

8A0 r' dA° p

-di

+s

(7.54) (7-54)

8A0 dA0

^

+

1 (/ dH0 8H0 1

2{p-dJ

(7.55) (7-55)

+ ^ ) ( - ( 1 ) - - (- 2- ") )) 2* ]' ' J 41 = = o, 0, *£>

_ fml ,

(l)^W ti\dw(1) ( 1 )

i)(fti) (2) ((i)(fui

(i) (!) /

(7.56) (7-56)

((11))

(2)\

(2A

d

(p^\ K ^

(2 +41J)' (u,W1 («,« +4 (u,« - «»(2') >) -- -- aa8^(«,«-- u,«) u>(2)) , ,

g=W = = 0, o,

^y

(7.57) (7.57)

(7.58)

134

MECHANICS OF MIXTURES

9A 0

^ = - ^ p - p " PR +

.

9

5-

dA0

, 1 / <9tf0 +

a7 2V

P

2 ^ ) ( ^ > - « » ) " + (/>M#O + 4 I ) ) K ) - ^ ) ) ,

" 8A0

^} = -P4)P-P(2) P^pPR

+s

8A0

^7

+ ^)K>-W«)2"

+

1 /

affc

■

2{p^p-

(7.60)

}

«S> = o,

(7.61)

<# = 42) *;w + 42) *f' + 42) (-(2) -(1)) .? f p

(2)

\

v ar ar ' dr \ P ^ V 2 2 +a< (xW< ) - w^) $ + a 8 ^ (W( ) - w^) , ' dr dr ^ ' 2)

4? = o, -(2)

P(2)

v

Pa

(7.62)

(7.63)

' 8A0 dp

(2)

(7.59)

8A0 ds

+ ^)K'-^)2

1 / 2 \

8H0 dp

)

(7.64)

4? = o,

(7.65)

^ = 4 2 ) c f f ) + a f d f ) + 4 2) M 2) - ( 1 ) ) ' f ppk2\V (2)

ar

v

ar

' dr \

^ > (v w « _ w « ) £ _ a * v(„,<»>_ tuW), ' ar

4

2) z

= o,

dr

(7.66)

'

(7.67)

MIXTURE OF TWO NEWTONIAN FLUIDS

H? = — m p ~ f>(2)P PR

135

dAc dAo 1 / 9 i / 0 hs 1— o dp da 2 V dp

:%)(---)•] -mT

=

_

d

(f>(i)\

P

*Us>r

+ (pWH0 + a?)(wW~WW)\ wW)\

r' ddA* A°

-p p

^

+ s

(7.68)

dA0 i1 /( 8H 8H00 9Ao ~d7 + 2 \p~dp~

^)(" B --")'] s ( f )

_

K

4

. ,

r

+

4

,

r

)

K

, _ . ,

)

(7.69)

- - 0 9 ( ^ ( 1 ) - « ( 2 ) ) — ( ^ ( 1 ) " W^) ,

m« = 0,

(7.70)

m^-^^+4^)±(£l\ dr \ W) p

P

dr ~'a7

i3)dwW\

ds dr ) dr'

(7.71)

and h = —aiop

dA0 ds

ldH0, 2 ds \W

{i)_w(2)Y

y

(7.72)

Furthermore, on substituting the above expressions into the balance equations 7.3, 7.4 and 7.9 and carrying out some simple calculations, we find that P = Cz + * ( r ) ,

d§

* ~

d r(p [,** + ^ 5

(

dp

i fp^2 + . « (xw « _ W(2))211, ds 2 \ dp ds J 'JJ +

T*l {'(2) [ ^ + slt + W-W + slt) V* ~ w(1))2]}

(7.73)

(7.74)

MECHANICS OF MIXTURES

136

_Pm±

r (1) [Mo + s Mo + 1 / ^ P + T

- pw& dr }[ +P

[dp dp

04, r" 8AQ

2\

b

ds ds

a84, AC 1I (/ dH <9#00 b b

dr dr

dH aff„\ 0\

dr dr JJ vy

2 W)

(i)1

+ s*M , ds ds}\ j

22 l\ dp dp

w _w ^ l } 2

1)N M

/ n) _ ( 2(2n /V \ ) \ ' l d<*(P d~r\7)

( m

w

' 22

+-) A — (w(u/ ' .-- ™< ™<>)>) = = 0, 0, + - a 99 («;« (w(1) --_ W ^Pwi ) 1 d d ff r

2 ((2)dlU 2 ) ^( ( )2 )

[4

33

M ( 2A«W )d™(1)

dr + a i dr

1 \

' .J J

2

(7.75) (7.75)

(2) , (2) (2) ,/ (2)

+4,H)-

(1)v

d /p (2 >\

_ (i)\w I

w +4 ( 2)^- - ^ J +f *£(«>»^C^ — )]} +4 2) («/ 2 2 p( d f r«^\ v\ \ pV J>i 1d ,,,\ dd (/ Pp^> {1)dv,w (1)<w > (1U , (2A ^>+a?(^z

a

)) ^ ( ^ "-^O^+^K') S + as|: K ' -- ^w(2) )J}1

+a +41' (u,'1' - w(2)

/ »«_ r.A / ( D ^ ( 1 ) +6, («W -w<2))5

^

-02;-°-

_ ^1)d 3^„a?)^j=0)

(4 X

(1)

( 2 ) ^ ( 2 ) \ dd / > ( 1 ) \

dr ' % dr Jdr

(7.76) (7.76)

2) 1 !,2) ( (41* -+ 42)') a^ ± £ f(«i" )(^ | (§) (4" " -- "«<»>) + (4 ' ++ 44 )(«' dr r + (4 + *>) > * ( $ ) v

+ ( -^F->^-)^( )»( W *, » M- -««««)) J£ === i}<>,

(7.77) ^(7.77)

and W) i o■ [ ^^++~ I( ^ t «K< ,I _ ) -„« « ) ' ] = =0 0. . %o a

(7.78)

o , ™ gradient „ „ j ; » „ t along „i™,„ n,^, , v ; , i j;-=,.+; Here C is the constant n~<>° pressure the axial direction. Equation 7.77 has been derived based on the boundedness of cr« and a£> in the flow domain. The solutions to the above equations depend strongly on the coefficients. In the case of H0 = constant and a61] = a^ = a8 = a9 = a10 = 0, equations 7.75 through 7.77 allow the solution: W = **W PW = W##,,

2) pV2) = (l-v p« ( l -0,)p%\ 0)pk ,

ss = sQ0,

(7.79)

MECHANICS OF MIXTURES

136

_Pm±

1 \

r (1) [Mo +s Mo + 1 / ^ + s * M , w _ w ^ l } -%i{p" w dr [ [dp dp 8s ds 22 Y \ dp dp ds ds}\ j ' .J J P

p

+P

dAc r' 8AQ

2 \ 2\

+

T

dH00 adAo AC I1 /( <9#

b b

dr dr

dH aff 00\\

y

dr dr J y

2

1

2

/( n) 2 w _- u / (2n ' ) 2 l <* /V

1)N

'' 22

(1) +-)>) = = 0, + - a 99 (u; -- ™<>) — (u/ ' -- ™< Li ' dr *

2(2) wi 1 d d ff r ((2)dlU 2 ) ^( ) [4 dr

PW P

A«W M (2)rf^ +
(1)

, (2) (2) ,/ (2) (2)

(7.75) (7.75)

(1)v

+4,H)-'"

d /p (2 >\ }

dr-{$) (1) w W)£J +f ^^( C I ^ >^ - -U)—) )]} +a?(wM+433 ( ^_-W^ dr

> , d ( v\ k-r-^r-^K'-^^s)

2 p( d ff r {1)dv,w p(2>>i1 d

+a +4 1 '

(1) <w

2

(2A

(1U

P

w(2) )) ^ ((u,'^' -"-^O^+^K') S + as|: K ' -- ^w(2) )J}} 1

/ »«_ r.A / ( D ^ ( 1 ) +■ b, («W - w < 2 ) ) - 05

I

-02;-°-

_a^d 3 ^ „ a ? ) ^ j = 0 )

-{ " l

2) (■?' + 42))' ^ dr dr + (4 + *>)' ±£

(«»" -

v

(£)

( 2 ) ^ ( 2 ) \ dd / > ( 1 ) \ dr J dr

dr ' %

(7.76) (7.76)

2 (*> + 42)) ( ^ -- ™< « >) ) || ( (§) | ) +f (<,<»+<.<»)(„<»

+K(^-^)(»W-««)£ ( ^ - ^ ) ( » W - « « ) J ==^O, i*,

(7.77) ^(7.77)

and and + 1 2 W i o^[ ^ + ~ ^( t( «- <( I ) )- "«-™<2>) ) ' ] '= =0 0. . %o a

(7.78)

Here C is the constant pressure gradient along the axial direction. Equation 7.77 has been derived based on the boundedness of cr« and
2) pV2) = = (l-v p« ( l -0,)p%\ , 0)pk

ss = sQ0 ,

(7.79)

138

MECHANICS OF MIXTURES

[286]), which can possibly be described by our model. Recently, there have been several studies on the mechanics of oil and water emulsions, specifically with respect to applications to problems in lubrications (cf. Chamniprasart et al. [287], Wang et al. [288], Al-arabi et al. [289]), that appeal to mixture theory. Literature related to gas-liquid mixtures can be found in Gouse [290].

Chapter 8 M I X T U R E OF A FLUID A N D SOLID PARTICLES Because of their importance in industrial applications such as fluidized beds, pneu­ matic transportation, reactors, combustors, particle sedimentation and so on, the two-constituent flows of a mixture composed of a Newtonian fluid and solid particles have been the subject of numerous theoretical studies (cf. Zenz and Othmer [291], Murray [129,292], Soo [90], Anderson and Jackson [130,293], Buyevich [294,295,296], Drew and Segel [132,297], Passman and Nunziato [298], Bedford and Drumheller [69], Elghobashi and Abou-Arab [299], Passman, Nunziato and Walsh [300], Ahmadi and Ma [301], Massoudi [302,303], Rajagopal, Massoudi and Ekmann [304], Johnson, Massoudi and Rajagopal [99,305,306,307]). These studies use either mixture theory (cf. Bedford and Drumheller [69], Passman et al. [300], Ahmadi and Ma [301]) or averag­ ing methods (cf. Anderson and Jackson [130], Buyevich [294], Drew and Segel [132]). Some formulations have included the virtual mass effect (cf. Anderson and Jackson [130]), and some have tried to incorporate turbulence (cf. Elghobashi and Abou-Arab [299], Ahmadi and Ma [301]). Many of the studies appeal to the principle of phase separation, an assumption first introduced by Adkins [39] and used extensively by others (cf. Drew and Segel [132], Bedford and Drumheller [69], Passman et al. [300]). Meanwhile, pertaining to experiments, there is a considerable amount of literature on the migration of particles within a mixture that is flowing. An excellent review of the early literature can be found in the articles by Brenner [308] and Goldsmith and Mason [309]. Aspects pertinent to the formation of boundary layers in the flow of suspensions can be found in the review article by Lee [310] (see also Marble [311], Singleton [312], Liu [313]). The behavior of suspensions undergoing shear flows has been investigated by various authors with a view towards explaining the interesting experimental findings of Segre and Silberberg [314,315] (in this connection see Halow and Willis [316], Tachibana [317], Ho and Leal [318], Leighton and Acrivos [319]). In a parallel development, the seminal ideas of Einstein [320,321,322] and sub­ sequent work by Jeffrey [323], Taylor [282], Guth and Simha [324], Burgers [325], 139

140

MECHANICS OF MIXTURES

Mooney [326], Brenner [327], Brodnyan [328], and Batchelor [329,330] and others have given rise to a method for modelling suspensions and the interested reader can find numerous papers based on this perspective in the fluid mechanics literature. Also, the flow of gases with small spheres has been studied using a kinetic theory approach by Knudsen and Weber [331], Epstein [332] and others. The theory presented here, though based on the theory of mixtures, is markedly different from that due to Massoudi [302,303] and Johnson et al. [99] who also base their modelling on the theory of mixtures. They, however, model the partial stresses in the mixture to reflect the fact that the mixture is composed of a Newtonian fluid and a granular solid; the model reducing to that for a Newtonian fluid in one limit and the constitutive form of a granular material studied by Goodman and Cowin (see Goodman and Cowin [333], Savage [334], Hutter and Rajagopal [335]) at the ether limit; the model is not intended to account for the turbulence that is usually observed in the flows involving such mixtures. It is not surprising that so far no formulation of equations for the averaged motion of the mixture has yet emerged which can describe general flows of the mixture rather satisfactorily. Direct observation of the motions of a realistic mixture without chemical reactions reveals that (1) a lot of finite size solid particles are dispersed in the mixture; (2) the solid constituent is discrete and the fluid constituent is not continuous in the flow domain, and the position of the interface between these two constituents changes constantly; (3) the physical motion of the mixture is turbulent, in at least a part of the flow domain, due to collisions among particles and between particles and the solid parts of the boundary of the flow domain, the irregular shapes and distributions of particles in space and in time, and the sudden contractions and expansions of fluid flow 'tunnels' formed by particles; and (4) particles spin, coalesce or break. Clearly, to construct a set of rather simple and reliable averaged equations of motion for the mixture is not an easy task, and given the fact that turbulent flows of a single constituent Newtonian fluid is not yet well understood at this point in time, the modelling of turbulent mixtures is a much more daunting task. Because our concern is to apply the theory of mixtures to model the behavior of the suspension and because the averaging approach (space, time, space-time, sta­ tistical) has been a primary and fruitful tool in studying two phase flows and since the resultant equations for the motion of the mixture have been used successfully to analyze real flows, it is appropriate to make some comments here on the differences between these two approaches and on the philosophy of averaging. As pointed out by Johnson et al. [99] , the primary conceptual difference "is the location of the averaging step in the development of the balance equations. Mixture theory implicitly assumes a locally averaged field before any equations are written; averaging makes it an ex­ plicit step in the development of the conservation equations." Both approaches need constitutive assumptions to determine the unknown terms contained in the averaged balance equations. Conceptually, the undetermined terms presented in the averag­ ing approach have seemingly clearer physical origins and meanings. However, in the

MIXTURE

OF A FLUID AND SOLID

PARTICLES

141

averaged modelling, this advantage is compromised by the fact that (1) we have to approximate two discontinuously distributed constituents with two interacting continua since we associate each constituent with a set of partial differential equations which hold in the entire flow domain of the mixture; (2) this approximation has to be accounted for in the structures of those undetermined terms; and (3) the effect of the approximation on those structures is not as clear as initially promised by the averag­ ing approach. Among those averaging schemes, the space-time averaging developed by Drew [336] is especially interesting in that it uses two integrations in space and two integrations in time to get differentiability of the averaged fields in both space and time. However, we know that the only practical way of guaranteeing smooth averaged fields is by assigning appropriate structures to the constitutive quantities contained in the averaged balance equations, that is, an averaging scheme itself may have lit­ tle, if any, effect on the smoothness of averaged fields. This is why other averaging schemes of single space integration or single time integration also yield smooth aver­ aged fields governing the motion. Compared with space and time averaging methods, the statistical ensemble averaging seems rather abstract. Also, there are some subtle issues with regard to time averaging which we shall not get into detail here. For example, as we require the constitutive quantities to meet the principle of material frame indifference, we need to know the relation between the averaged velocities v j and v*f where vy is the velocity of the fluid constituent associated with a frame of reference x and v j is the velocity of the fluid constituent associated with another frame of reference x* = Q(i) x + c(i) with Q Q T = Q T Q = I. Based on the usual time averaging, we cannot ensure that the commonly used relation v) = Qvy + Q x + c holds for arbitrary orthogonal Q(i) and c(t). Similar difficulties also arise with the space averaging in two-phase flows. In this chapter, we shall discuss some general issues concerning modelling of a mixture composed of a Newtonian fluid (to be called fluid constituent) and dispersed solid particles (to be called solid constituent) without thermal, chemical and other such effects coming into play, within the context of mixture theory. Before embarking on this, we first present an overview of the basic formulation of turbulence modelling in a single Newtonian fluid in Section 8.1. As adduced to before, the motions of a real mixture of a fluid constituent and a solid constituent are inherently turbulent. Thus, it is imperative that any complete formulation of the modelling has to come to grips with the phenomenon of turbulence, and the brief overview will help in appreciating the modelling provided later. We will introduce the basic field quantities and formulate the averaged balance equations for the mixture in Section 8.2, which generalizes the formulation presented in Chapter 2 to include the interfacial density, the solid spin, the averaged fluctuation correlations, the averaged fluctuation energies and the equations governing the evolution of these quantities. Section 8.3 is devoted to a discussion of some issues related to the constitutive modelling such as the principle of phase separation, immiscibility, the volume fraction constraint, the entropy inequality along with viscous dissipation and the constitutive variables. Finally, in Section 8.4,

MECHANICS OF MIXTURES

142

we propose a specific model which takes into account the effects of turbulence, drag and virtual mass.

8.1

Turbulence Modelling in Newtonian Fluids

Turbulence modelling of an incompressible Newtonian fluid has long been a topic of extensive research ever since Reynolds' pioneering work in this area (cf. Reynolds [337,338], Rivlin [339], Jones and Launder [340], Mellor and Herring [341], Hinze [342], Launder, Reece and Rodi [343], Lumley [344,345,346], Marshall and Naghdi [347,348,349], Speziale [350,351], Yakhot et at. [352], and Huang and Rajagopal [353]). Though some specific flows can be modelled quite successfully, a model that has a broad range of applicability is yet to be established. For this reason, we shall review only some basic ideas and procedures on this topic. The arguments presented here are based on an unpublished paper by Tao, Chen and Rajagopal [354] and the aim is to construct a unified model for the averaged motions of a Newtonian fluid. For a historical development of the subject and a comprehensive treatment of this topic, see the detailed works by Mellor and Herring [341], Hinze [342], Lumley [345] and Speziale [351]. Readers interested in the treatment of turbulence through statistical mechanics or renormalization group methods are referred to the works by Kolmogorov [355], Rotta [356], Kraichnan [357], Davidov [358,359], Wyld [360], Monin and Yaglom [361], Yakhot and Orszag [362], Yoshizawa [363], Yakhot and Smith [364], and McComb [365]. Assume that the equation of conservation of mass and the Navier-Stokes equations can describe all the isothermal flows of an incompressible Newtonian fluid under appropriate initial and boundary conditions, i.e., divw = 0, dw at

+ div (w ® w)

(8.1)

-Vq + ^Vw,

(8.2)

where w and q are the velocity and pressure of the fluid, respectively, p is the mass density and p is the viscosity. Following Reynolds, we introduce the average and fluctuating fields w = V + v,

q = P + p,

(8.3)

subject to v = 0,

p = 0,

(8.4)

where V and P are the averaged velocity and pressure, v and p the fluctuations of velocity and pressure, respectively, and $ denotes the average of $. It follows from Eqs. 8.1 and 8.2 that (cf. Hinze [342])

MIXTURE

OF A FLUID AND SOLID

PARTICLES

143

Vi,5 = 0,

p [dtVt + {ViVj)^}

(8.5)

= -P,i +pVujj

-p(vfv-h

,

p [dtWvj + (W>jVk),k ] = P (Wvj),hh -p (vjVk~Vi,k

(8.6)

+WVk~Vj,k)

- 2/tu i , fc u J -, fc + Aijkfr+anj,

P \8tViVj,k

+ (ViVj,kVt),l

] =p (viVj,k),U

-p (viVj,kVi,l

+ vivJ7iVi,k +TWiVj,ki)

P [dtVi,kVj,i

+ (vi,kVj,iVm),m

+Vj,ivrn,kVi,rn

+Vi,kVj,mVm,i

+vmVi,kVj,lrn)

+ r

(8.7)

+ViVl,kVj,l

+ Bijkl,i

] - p (uj,fcUj,i), mm -p +Vj,ivi,rnVrn,k

+Pijk,

(8.8)

{vi,kvm,iVj,m

+vmVj,[Vi,km

(8-9)

-hwi where A y * ■■= ~ (vipSjk

+ VjpSik)

- pViVjVk,

(8.10)

aij:=(vi,j+vj,i)p,

(8.11)

BjjM : = -vJp—Ski - pViVj,kvi,

(8.12)

Ajfe : = vjP,ki - viP'jk

~ pvtVj,i vhk - 2pv{,i Vj,ki,

^ikjlm •'= - ViP,jiSkm

- Vjp,ik5im

- p (vi:k Vj,i vm

+Vi,k vm,i VJ + Vj,i vm,k Vi),

Jihfl

:

= ViP,jkl

+ VjPrikl + P {Vi,km Vm,l Vj + Vj,lm

-2A«>i,fcm«j,lm, a n d dt : = d/dt.

(8.13)

(8-14)

» m , t «»)

(8.15)

MECHANICS

144

OF MIXTURES

We may assume that the above flux and source terms Ayj,, Bijki, ^ikjlm, OHJ, /?#&, and -yn,ji are to be given by the constitutive equations which depend on the quantities, \i, u[Vj, ViVj,/,, Vi,i,Vj,i, as well as their derivatives in space and in time, for example,

J-(A,d, Bj,

T, 7, VA, V d , VB, V/3, Vf, V 7 , A, a, B,(3, T, 7,

VA, V a , VB, V/3, V r , V 7 , V V A , V V a , V V B , VV/3, V V I \ V V 7 , V, — v ® v, — v ® Vv, —Vv ® Vv, V V , V — v ® v, V — v ® Vv, at at at at at ,d-, V — V v ® Vv, V , v ® v, v ® Vv, V v ® Vv, V V , V v ® v, V v Vv, at V V v ® Vv, V V V , V V v (g v, V V v ® Vv, V V V v ® V v ) = 0, (8.16) where 4> = d^/dt = 5 ^ / 5 i + V • V. Clearly, this assumption implies that we close the averaged model by including the second order correlations for v and V v as field variables and accumulating all the other correlation terms into the flux and source terms which involve the pressure fluctuation p, second order or higher spatial derivatives of velocity fluctuation v, and third and higher order correlations (see definitions 8.10 through 8.15). Meanwhile, the derivatives of the averaged quantities with respect to space in Eq. 8.16 are up to second order. Notice that v ® v and v ® V v are not completely independent of each other due to the following relation (W5J),A; = VtVj,k + VjVi,k.

(8-17)

Moreover, some restrictions on those flux and source terms can be derived by virtue of their definitions. First, A, a, B, /?, T, and 7 have the following symmetry properties: Ajik

= Kjk,

(8.18)

a-ji = ai:j,

(8.19)

Fjlikm — ^ikjlm,

(8.20)

Ijiik = likji

(8.21)

and -*- iijjmim

'Jiijj

— "•

yo.ZZJ 3

Second, they have to satisfy the following constraint (*): I.;' "" C R be a flow domain. If there exists an open subset U C V such th v~"v = 0 in U, • en v = 0,

MIXTURE OF A FLUID AND SOLID PARTICLES

145

v ® v = 0, v ® V v = 0 and V v ® V v = 0 in U, which in turn require that A = 0, a = 0, B = 0, /3 = 0, T = 0, and 7 = 0 in U. At this juncture, some remarks about this constraint are called for: (1) the NavierStokes equations will be embedded in the model to be formulated satisfying this constraint, thus this makes it possible to have a model which could describe a flow consisting of laminar, intermittent and turbulent regimes, and furthermore, the model may have the property of bifurcation, that is, under some conditions, such as at low Reynolds numbers, the model may yield stable solutions with v v = 0 in T> (laminar flow) which are the same as those for the Navier-Stokes equations; and if subject to some other conditions, like high Reynolds numbers, the model may result in stable solutions with v ® v ^ 0 in T> and unstable solutions with v ® v = 0 in V, the latter corresponding to those for the Navier-Stokes equations. (2) Due to this embedding, only up to second order spatial derivatives (of V especially) would be included in the modelling of the averaged motions to avoid singularities. (3) If a flow in T> has no initial disturbances, then it will remain fluctuation free unless a disturbance occurs on the boundary of the flow domain T>. Other restrictions on A, a, B, j3, T, and 7 can also be motivated such as (1) They have to make

DJIJI > 0, ViVj,j = 0,

v2v2 >0,

V3W3 >0,

vk,kVj,[ = 0,

v%,kVi,h > 0, (8.23)

realizable (cf. Speziale [351]). However, it may not be surprising that some of the conditions in Eq. 8.23 might be violated locally in an averaged model because the averaged model provides but an approximate description of real turbulent flows. (2) They can be required to obey the principle of material frame indifference if we consider v = w — V as a velocity difference, which is the case in statistical ensemble averaging. We are aware that there are many subtle issues involving the requirement of frameinvariance within the context of turbulence. We shall not get into a detailed discussion of these issues here. We refer the reader to Speziale [351] and also the comments in Marshall and Naghdi [347] for a discussion of this issue. Although this second order closure scheme seems quite reasonable, there are a number of difficulties associated with it. Firstly, we have to deal with a formidably large number of equations; secondly, we have to determine constitutive equations involving many variables; finally, we have to prescribe physically sound initial and boundary conditions for v ® v, v V v , V v ® V v and so on. Due to these and other reasons, we prefer to consider a much simpler formulation following the methodology used in formulating k - e type models (cf. Speziale [351]). We may take this special formulation as the one derived from the above second order closure scheme by choosing proper constitutive equations for the flux and source terms in Eqs. 8.7 through 8.9. Let us introduce k and e through

MECHANICS OF MIXTURES

146

k:~-ViVi, 2 Equations 8.7 and 8.9 reduce to

e := -Vi,kVi,k. p

(8-24)

p (dtk + Vjk,j ) = p,k,jj ~pv]v^Vj,k -pe + Ak,k,

p (dte + VJE^ ) = fie,jj -2fi (vj^v^Vj^

(8.25)

+vk,jVk,iVi,j )

( 8 - 26 )

+ v?o^Vhki +fk,k +7, where Ak :=

1

2 ""'

l

-AJ3k,

f* := -Tjm, P

(8.27) (8.28)

7 := -1»». (8-29) P In this k - e type model, (V, P, k,e) are taken as the primitive variables. Thus, we may assume that {vivj, ViVj,k,Vi,k VjH, Ai,f i , 7 / =

{iyjGiiksjHifcjij/ijffij^^fcjejVVjVfe,

Ve, V,fe,i, VVV, W i t , W E , VV, Vfe, Vs).

(8.30)

Actually, equation 8.30 is a special case of Eq. 8.16. We shall also assume that v ® v, v ® Vv, Vv ® Vv, A, T, and 7 depend on the same variables as there is no reason to assume otherwise a priori. Furthermore, V, fc, s, their gradients, their material derivatives, and the gradients of these gradients and material derivatives appear in the constitutive expressions for v ® v, v ® Vv, Vv ® Vv, A, F, and 7. This may be considered as an extension of the standard idea of equipresence that is customary in modelling. As discussed earlier, a unified model would be preferable in which the spatial derivatives of V, k, and e are of at most second order. Therefore, equation 8.30 reduces to t p j = Fij (V, ft, e, VV, Vfe, Ve, V,fc,e) ,

W^k

(8.31)

= Gijk (V,fc,e, VV, Vfe, Ve, V,fc,e, VVV, VVfc, W e , VV, Vfc, Ve) ,

(8.32)

MIXTURE OF A FLUID AND SOLID PARTICLES

147

Vhk Vj,i = Hikji ( V ,fc,e, V V , Vfc, Ve, V , k, e, V V V , W A i , W e , V V , Vk, Ve) ,

(8.33)

hi ~ U ( V , k, e, V V , Vk, Ve, V , k, e) ,

(8.34)

f i = gt ( V , k, e, V V , Vk, Ve, V , k, e) ,

(8.35)

and

7 =

fe(v,/c,e,VV,V/c,Ve,V,fc,e, V V V , VVfc, W e , V V , Vfc, Ve) .

(8.36)

Now, if we require that the principle of material frame indifference holds, then it follows from Eqs. 8.31 through 8.36: ,QapeiP ,k,ej

Fij^k^e^^QpsD^TQapk,/} =

£ 5 U<xf3i ™>a ) £>a ) ^? £ ) 7

Wim^jn^mni^}

Gi-jh [k, c, Qa-yQpsD^S, Qapk,0

Qa-/Q/36k,-,6 ,Q =

^imWjn^chp^m-np

, Qa0£,0

, k, £, Qa^Qpa-Q-i-r^^T

{ ™) £ > Dafii

■ffiM (k> e , QaiQpsD^S,

"-)a > £ ? a 5 ™> ^> 'a)/3-y 7 " ^ a / 3 > c )at/3 ?

(8.38)

Qapk,0

Qa-yQ{36kn6 ,
k,a/3 ,6,^,^/3,

,

QccyQpsA-yS, Qaf}k,p , Qapi,f3 J

A j ^ j ^ a , £,a ) ,

=

(O.o ( J

, Qa0E,f3 ,fc,e,

QasQpaQ

QpsA-yS, Qapktf , Qaf)£,0 j £

( "-; ; ^(X0i ™,a > £5Q > "'I £ ? *« 3/3-7 )

k,0,e,a}

,

/ i (&, e ) Qa-tQpsD-tS, Qapkrf

(8.39)

, Qaf}£,0 , k, £j

= Qijf^(k,e,Da0,k,a,£,a,k,e)

,

(8.40)

148

MECHANICS OF MIXTURES

gi(k, e, QcnQpsDys, Qa0k,0 , Qap£,0 , k, ej — Qij9j yK->ei *Ja0> K,a , £,„ ,K,ej

(8.41)

,

and

h\k,£,QaJQ0sDjS,Qafik,0 Qa-tQpsk^S

, Qa/3£,0 ,k,£,

QasQ/3aQ-rr^6-,a-r ,

I Qar,Q(36£,i6 , QcryQ'f36A-yS, Qa0^,0 , Qa0£,0 j

= h[k,£,Dap,k,a

,e,a,k,e,

V^,^ ,k,a/3 ,e,ap ,Aa0,k,a

f

= 1,

D := i [VV + ( V V ) r ] ,

■ \T

+ VV + 2 ( V V ) T V V .

,i,a ) ,

(8.42)

where

A:=(VV)

(8.43)

These equations demonstrate that F , G, H , f, g, and h are isotropic functions, which can be represented as follows (cf. Spencer [195]), (8.44)

ViVj = -k&ij + a i Dtj + ...,

ViVj,k = - (k,k Sij - -k,i SjhJ + - (k,j 8M ~ -k,i + ■z
Vi,kVj,k

rjtik i r. \*j iota ®ki

\l pp

[k,ij ---k

O LL

+ C3 [e,ij

\p vk,i vk,j =--e8xj

Sjk)

—-E,ae,Oij\

+ c4

6-)l+.

a1k,k+a2e,k+...,

(8.45)

(8.46)

6i3

+ d3 ( em ~3 e ">" 6ij) + d* lAii - 3 K a S n ) + -.., Ak =

■

S-)

(A--~A

/ 1 + diDij + d2 \krij ~-k,aa

*i iota ®jk j +

(8.47)

(8.48)

MIXTURE OF A FLUID AND SOLID

PARTICLES

149

Tk=f31k,k+f32£,k+...,

(8.49)

7 = h\^k,e, fc,e, Aaa, k,aa , e, a a , k,aa , e,aa ,...j .

(8.50)

A special case of the above formulation would be like a k - type model in which equations 8.5, 8.6, and 8.25 are included and w^uj, Afc and e are assumed to be given by vTv]= Ftj(k,k,D,Vk)

,

(8.51)

Afc = / f c (fe,*,D,Vfc),

(8.52)

e = e(k, k, D , Vfe, V V V , VVA;, Vfc) > 0.

(8.53)

and

The above discussions are based on the assumption that the Navier-Stokes equa­ tions can describe all the isothermal flows of an incompressible Newtonian fluid. Another possible formulation for the modelling of a Newtonian fluid is to get rid of that assumption and to write down directly the balance equations for the basic av­ eraged fields such as velocity, velocity fluctuation correlations and so on. Then, an interplay between the constitutive theory and experiments will provide a means for determining the flux and the source terms contained in those balance equations. We might require that the averaged model reduces to the Navier-Stokes equations under the condition of vanishing fluctuations, but we will not require that the Navier-Stokes equations hold for all the isothermal flows of the fluid. Actually, this idea is inherent in the above formulation: though we start from the Navier-Stokes equations, that we seek the structures of the undetermined quantities A, a, B, j3, T and 7 in Eqs. 8.7 through 8.9 by additional constitutive theory may compromise the very idea of taking the Navier-Stokes equations as the foundation. So far we have outlined some of the basic ideas and procedures in the modelling of the averaged equations for a single constituent incompressible Newtonian fluid. These ideas will provide some guidance in the modelling of the averaged equations for a mixture of a Newtonian fluid and solid particles, but they will also reveal to us the need to revise these ideas to adapt to the physics of the mixture, as will become clear in the following sections.

8.2

Balance Equations for the Mixture

Consider a mixture composed of a Newtonian fluid (denoted by subscript / ) and uniform rigid solid spheres of radius RQ (denoted by subscript s) that flows in the domain T>. Let TZ denote the collection of all reahzations of the motion of the mixture

MECHANICS OF MIXTURES

150

in 23 subject to certain initial and boundary conditions, and let m be a proper measure on 1Z and tp be the probability distribution function on 1Z. such that

L


(8.54)

JH

Clearly, corresponding to each realization r £ R , there exist (1) a solid particle constituent distribution function x{xi^\r) m ^ x ["> °°) with , . X(x, ' r '

_

f 1, if a: is occupied by particles at t, \ o , otherwise;

(2) an interface distribution function xs(x,t;r)

Xs{x,t;r) = | o '

, ■. ^ ' '

in 23 x [0, oo) with

if x G S(t;r), otherwise,

(8.56)

where S(t; r) := cl {x £ T> : x(x,t;r) = 1} n cl {x £ 2? : x( a; )*; r ') = 0} is the inter­ face between the fluid constituent and the solid constituent; (3) a mass density field p(x, t; r) in 23 x [0, oo); and (4) a velocity field v(x, t; r) in 23 x [0, oo). The interface S(t; r) is singular, across which p has a jump and on which x and p may not be de­ fined. From x and p, we can get the mass density distributions pj and ps associated with r 6 TZ, Pt = ?{l-X),

(8-57)

Ps = PX-

(8.58)

The velocity field V/ of the fluid constituent and the velocity field v s of the solid constituent are given by v

/ =

v,

v s = v,

on23/

{ a ; e 2 3 : X ( a ; , £ ; r ) = 0},

(8.59)

on23 3 = { a ; e 2 3 : x ( : c , t ; r ) = l } ,

(8.60)

=

and the spin field il„ of the solid constituent is given by ft, = - c u r l v ,

on 23,.

(8.61)

We may now approximate both the fluid constituent and the solid constituent with two coexisting interacting continua by introducing the averaged fields for the solid volume fraction 7, interfacial density s (see Drew and Segel [132]), mass densities Pf, pa, velocities v^, v,, and solid spin ft, (see Eringen [366], Ariman, Turk and Sylvester [368]) in 23 x [0,00) through

MIXTURE

OF A FLUID AND SOLID

PARTICLES

151

7 0 M ) := / x{x,t;r)
(8.62)

s(x, t) := y

(x, t; r)