Cavitation in Non-Newtonian Fluids: With Biomedical and Bioengineering Applications

Cavitation in Non-Newtonian Fluids Emil-Alexandru Brujan Cavitation in Non-Newtonian Fluids With Biomedical and Bioe...

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Cavitation in Non-Newtonian Fluids

Emil-Alexandru Brujan

Cavitation in Non-Newtonian Fluids With Biomedical and Bioengineering Applications

123

Emil-Alexandru Brujan University Politechnica of Bucharest Department of Hydraulics Spl. Independentei 313, sector 6 060042 Bucharest Romania [email protected]

ISBN 978-3-642-15342-6 e-ISBN 978-3-642-15343-3 DOI 10.1007/978-3-642-15343-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010935497 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Cavitation is the formation of voids or bubbles containing vapour and gas in an otherwise homogeneous fluid in regions where the pressure falls locally to that of the vapour pressure corresponding to the ambient temperature. The regions of low pressure may be associated with either a high fluid velocity or vibrations. Cavitation is an important factor in many areas of science and engineering, including acoustics, biomedicine, botany, chemistry and hydraulics. It occurs in many industrial processes such as cleaning, lubrication, printing and coating. While much of the research effort into cavitation has been stimulated by its occurrence in pumps and other fluid mechanical devices involving high speed flows, cavitation is also an important factor in the life of plants and animals, including humans. Several books and review articles have addressed general aspects of bubble dynamics and cavitation in Newtonian fluids but there is, at present, no book devoted to the elucidation of these phenomena in non-Newtonian fluids. The proposed book is intended to provide such a resource, its significance being that non-Newtonian fluids are far more prevalent in the rapidly emerging fields of biomedicine and bioengineering, in addition to being widely encountered in the process industries. The objective of this book is to present a comprehensive perspective of cavitation and bubble dynamics from the stand point of non-Newtonian fluid mechanics, physics, chemical engineering and biomedical engineering. In the last three decades this field has expanded tremendously and new advances have been made in all fronts. Those that affect the basic understanding of cavitation and bubble dynamics in non-Newtonian fluids are described in this book. It is essential to understand that the effects of non-Newtonian properties on bubble dynamics and cavitation are fundamentally different from those of Newtonian fluids. Arguably the most significant effect arises from the dramatic increase in viscosity of polymer solutions in an extensional flow, such as that generated about a spherical bubble during its growth or collapse phase. Specifically, polymers, which are randomly-oriented coils in the absence of an imposed flow-field, are pulled apart and may increase their length by three orders of magnitude in the direction of extension. As a result, the solution can sustain much greater stresses, and pinching is stopped in regions where polymers are stretched. Furthermore, many biological fluids, such as blood, synovial fluid, and saliva, have non-Newtonian properties and

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Preface

can display significant viscoelastic behaviour. Therefore, this is an important topic because cavitation is playing an increasingly important role in the development of modern ultrasound and laser-assisted surgical procedures. Despite their increasing bioengineering applications, a comprehensive presentation of the fundamental processes involved in bubble dynamics and cavitation in non-Newtonian fluids has not appeared in the scientific literature. This is not surprising, as the elements required for an understanding of the relevant processes are wide-ranging. Consequently, researchers who investigate cavitation phenomenon in non-Newtonian fluids originate from several disciplines. Moreover, the resulting scientific reports are often narrow in scope and scattered in journals whose foci range from the physical sciences and engineering to medical sciences. The purpose of this book is to provide, for the first time, an improved mechanistic understanding of bubble dynamics and cavitation in non-Newtonian fluids. The book starts with a concise but readable introduction into non-Newtonian fluids with a special emphasis on biological fluids (blood, synovial liquid, saliva, and cell constituents). A distinct chapter is devoted to nucleation and its role on cavitation inception. The dynamics of spherical and non-spherical bubbles oscillating in non-Newtonian fluids are examined using various mathematical models. One main message here is that the introduction of ideas from theoretical studies of nonlinear acoustics and modern optical techniques has led to some major revisions in our understanding of this topic. Two chapters are devoted to hydrodynamic cavitation and cavitation erosion, with special emphasis on the mechanisms of cavitation erosion in non-Newtonian fluids. The second part of the book describes the role of cavitation and bubbles in the therapeutic applications of ultrasound and laser surgery. Whenever laser pulses are used to ablate or disrupt tissue in a liquid environment, cavitation bubbles are produced which interact with the tissue. The interaction between cavitation bubbles and tissue may cause collateral damage to sensitive tissue structures in the vicinity of the laser focus, and it may also contribute in several ways to ablation and cutting. These situations are encountered in laser angioplasty and transmyocardial laser revascularization. Cavitation is also one of the most exploited bioeffects of ultrasound for therapeutic advantage. In both cases, the violent implosion of cavitation bubbles can lead to the generation of shock waves, high-velocity liquid jets, free radical species, and strong shear forces that can damage the nearby tissue. Knowledge of these physical mechanisms is therefore of vital importance and would provide a framework wherein novel and improved surgical techniques can be developed. This field is as interdisciplinary as any, and the numerous disciplines involved will continue to overlook and reinvent each others’ work. My hope in this book is to attempt to bridge the various communities involved, and to convey the interest, elegance, and variety of physical phenomena that manifest themselves on the micrometer and microsecond scales. This book is offered to mechanical engineers, chemical engineers and biomedical engineers; it can be used for self study, as well as in conjunction with a lecture course. I would like to gratefully acknowledge the advice and help I received from Professor Alfred Vogel (Institute of Biomedical Optics, University of Lübeck),

Preface

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Professor Yoichiro Matsumoto (University of Tokyo), Professor Gary A. Williams (University California Los Angeles), and Professor J.R. Blake (University of Birmingham). I also appreciate fruitful conversations with and kind help I received from Professor Werner Lauterborn (Göttingen University), Dr. Teiichiro Ikeda (Hitachi Ltd), Dr. Kester Nahen (Heidelberg Engineering GmbH), and Peter Schmidt. Bucharest, Romania June 2010

Emil-Alexandru Brujan

Contents

1 Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Newtonian Fluids . . . . . . . . . . . . . . . 1.1.2 Non-Newtonian Fluids . . . . . . . . . . . . 1.2 Non-Newtonian Fluid Behaviour . . . . . . . . . . . 1.2.1 Simple Flows . . . . . . . . . . . . . . . . . 1.2.2 Intrinsic Viscosity and Solution Classification 1.2.3 Dimensionless Numbers . . . . . . . . . . . 1.2.4 Constitutive Equations . . . . . . . . . . . . 1.3 Rheometry . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Shear Rheometry . . . . . . . . . . . . . . . 1.3.2 Extensional Rheometry . . . . . . . . . . . . 1.3.3 Microrheology Measurement Techniques . . . 1.4 Particular Non-Newtonian Fluids . . . . . . . . . . . 1.4.1 Blood . . . . . . . . . . . . . . . . . . . . . 1.4.2 Synovial Fluid . . . . . . . . . . . . . . . . . 1.4.3 Saliva . . . . . . . . . . . . . . . . . . . . . 1.4.4 Cell Constituents . . . . . . . . . . . . . . . 1.4.5 Other Viscoelastic Biological Fluids . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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1 1 1 4 7 7 12 13 15 25 25 29 32 34 34 37 40 41 43 43

2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Nucleation Models . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nuclei Distribution . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Distribution of Cavitation Nuclei in Water . . . . . . . 2.2.2 Distribution of Cavitation Nuclei in Polymer Solutions 2.2.3 Cavitation Nuclei in Blood . . . . . . . . . . . . . . . 2.3 Tensile Strength . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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49 49 53 53 54 55 57 59

3 Bubble Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Spherical Bubble Dynamics . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Equations of Bubble Dynamics . . . . . . . . . .

63 63 63

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65 81 82 86 91 92 98 98 99 101 107 110 112

4 Hydrodynamic Cavitation . . . . . . . . . . . . . . . . . . . . . . 4.1 Non-cavitating Flows . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Drag Reduction . . . . . . . . . . . . . . . . . . . . . 4.1.2 Reduction of Pressure Drop in Flows Through Orifices 4.1.3 Vortex Inhibition . . . . . . . . . . . . . . . . . . . . 4.2 Cavitating Flows . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Cavitation Number . . . . . . . . . . . . . . . . . . . 4.2.2 Jet Cavitation . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Cavitation Around Blunt Bodies . . . . . . . . . . . . 4.2.4 Vortex Cavitation . . . . . . . . . . . . . . . . . . . . 4.2.5 Cavitation in Confined Spaces . . . . . . . . . . . . . 4.2.6 Mechanisms of Cavitation Suppression by Polymer Additives . . . . . . . . . . . . . . . . . . 4.3 Estimation of Extensional Viscosity . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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117 118 118 121 123 123 124 126 129 134 143

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148 149 150

5 Cavitation Erosion . . . . . . . . . . . . . . . . . . . . . . . 5.1 Cavitation Erosion in Non-Newtonian Fluids . . . . . . . 5.2 Mechanisms of Cavitation Damage in Newtonian Fluids 5.3 Reduction of Cavitation Erosion in Polymer Solutions . . References . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

3.3 3.4 3.5

3.1.2 The Equations of Motion for the Bubble Radius . 3.1.3 Heat and Mass Transfer Through the Bubble Wall 3.1.4 Experimental Results . . . . . . . . . . . . . . . 3.1.5 Bubbles in a Sound-Irradiated Liquid . . . . . . . Aspherical Bubble Dynamics . . . . . . . . . . . . . . . 3.2.1 Bubbles Near a Rigid Wall . . . . . . . . . . . . 3.2.2 Bubbles Between Two Rigid Walls . . . . . . . . 3.2.3 Bubbles in a Shear Flow . . . . . . . . . . . . . 3.2.4 Shock-Wave Bubble Interaction . . . . . . . . . Bubbles Near an Elastic Boundary . . . . . . . . . . . . Bubbles in Tissue Phantoms . . . . . . . . . . . . . . . Estimation of Extensional Viscosity . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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155 156 163 171 172

6 Cardiovascular Cavitation . . . . . . . . . . . . . . . . . . . 6.1 Cavitation for Ultrasonic Surgery . . . . . . . . . . . . . . 6.1.1 Sonothrombolysis . . . . . . . . . . . . . . . . . . 6.1.2 Ultrasound Contrast Agents . . . . . . . . . . . . . 6.2 Cavitation in Laser Surgery . . . . . . . . . . . . . . . . . 6.2.1 Transmyocardial Laser Revascularization . . . . . 6.2.2 Laser Angioplasty . . . . . . . . . . . . . . . . . . 6.3 Cavitation in Mechanical Heart Valves . . . . . . . . . . . 6.3.1 Detection of Cavitation in Mechanical Heart Valves

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175 175 175 177 199 199 202 206 206

Contents

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6.3.2 Mechanisms of Cavitation Inception in Mechanical Heart Valves . . . . . . . . . . . . . . . . . . . . . 6.3.3 Collateral Effects Induced by Cavitation . . . . . . Gas Embolism . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Treatment Strategies for Gas Embolism . . . . . . 6.4.2 Gas Embolotherapy . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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208 209 210 211 211 214

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7 Nanocavitation for Cell Surgery . . . . . . . . . . . . . . 7.1 Cavitation Induced by Femtosecond Laser Pulses . . . 7.1.1 Numerical Simulations . . . . . . . . . . . . . 7.1.2 Experimental Results . . . . . . . . . . . . . . 7.2 Cavitation During Plasmonic Photothermal Therapy . . 7.2.1 Nanoparticles and Surface Plasmon Resonance 7.2.2 Bubble Dynamics . . . . . . . . . . . . . . . . 7.2.3 Biological Effects of Cavitation . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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225 226 226 228 229 231 233 241 245

8 Cavitation in Other Non-Newtonian Biological Fluids . . 8.1 Cavitation in Saliva . . . . . . . . . . . . . . . . . . . 8.1.1 Cavitation During Ultrasonic Plaque Removal . 8.1.2 Cavitation During Passive Ultrasonic Irrigation of the Root Canal . . . . . . . . . . . . . . . . 8.1.3 Cavitation During Laser Activated Irrigation of the Root Canal . . . . . . . . . . . . . . . . 8.1.4 Cavitation During Orthognathic Surgery of the Mandible . . . . . . . . . . . . . . . . . 8.2 Cavitation in Synovial Liquid . . . . . . . . . . . . . . 8.3 Cavitation in Aqueous Humor . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Non-Newtonian Fluids

A fluid can be defined as a material that deforms continually under the application of an external force. In other words, a fluid can flow and has no rigid three-dimensional structure. An ideal fluid may be defined as one in which there is no friction. Thus the forces acting on any internal section of the fluid are purely pressure forces, even during motion. In a real fluid, shearing (tangential) and extensional forces always come into play whenever motion takes place, thus given rise to fluid friction, because these forces oppose the movement of one particle relative to another. These friction forces are due to a property of the fluid called viscosity. The friction forces in fluid flow result from the cohesion and momentum interchange between the molecules in the fluid. The viscosity of most of the fluids we encounter in every day life is independent of the applied external force. There is, however, a large class of fluids with a fundamental different behaviour. This happens, for example, whenever the fluid contains polymer macromolecules, even if they are present in minute concentrations. Two properties are responsible for this behaviour. On one hand, polymers change the viscosity of the suspension by changing their shape depending on the type of flow. On the other hand, polymer have long relaxation times associated with them, which are on same order as the time scale of the flow, and allow the polymers to respond to the flow with a corresponding time delay. Other complex systems consisting of several phases, such as suspensions or emulsions and most of the biological fluids, behave in a similar manner. In the following, we will focus on some of the most important aspects of the flow of this class of fluids.

1.1 Definitions 1.1.1 Newtonian Fluids An important parameter that characterize the behaviour of fluids is viscosity because it relates the local stresses in a moving fluid to the rate of deformation of the fluid element. When a fluid is sheared, it begins to move at a rate of deformation inversely proportional to viscosity. To better understand the concept of shear viscosity we assume the model illustrated in Fig. 1.1. Two solid parallel plates are set on the top E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_1,

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1 Non-Newtonian Fluids

Fig. 1.1 Illustrative example of shear viscosity

of each other with a liquid film of thickness Y between them. The lower plate is at rest, and the upper plate can be set in motion by a force F resulting in velocity U. The movement of the upper plane first sets the immediately adjacent layer of liquid molecules into motion; this layer transmits the action to the subsequent layers underneath it because of the intermolecular forces between the liquid molecules. In a steady state, the velocities of these layers range from U (the layer closest to the moving plate) to 0 (the layer closest to the stationary plate). The applied force acts on an area, A, of the liquid surface, inducing a shear stress (F/A). The displacement of liquid at the top plate, x, relative to the thickness of the film is called shear strain (x/L), and the shear strain per unit time is called the shear rate (U/Y). If the distance Y is not too large or the velocity U too high, the velocity gradient will be a straight line. It was shown that for a large class of fluids F ∼

AU . Y

(1.1)

It may be seen from similar triangles in Fig. 1.1 that U/Y can be replaced by the velocity gradient du/dy. If a constant of proportionality η is now introduced, the shearing stress between any two thin sheets of fluid may be expressed by τ=

U du F =η =η . A Y dy

(1.2)

In transposed form it serves to define the proportionality constant η=

τ , du/dy

(1.3)

which is called the dynamic coefficient of viscosity. The term du/dy = γ˙ is called the shear rate. The dimensions of dynamic viscosity are force per unit area divided by velocity gradient or shear rate. In the metric system the dimensions of dynamic viscosity is Pa·s. A widely used unit for viscosity in the metric system is the poise (P). The poise = 0.1 Ns/m2 . The centipoise (cP) (= 0.01 P = mNs/m2 ) is frequently a more convenient unit. It has a further advantage that the dynamic viscosity of water at 20◦ C is 1 cP. Thus the value of the viscosity in centipoises is an indication of the

1.1

Definitions

3

viscosity of the fluid relative to that of water at 20◦ C. In many problems involving viscosity there frequently appears the value of viscosity divided by density. This is defined as kinematic viscosity, ν, so called because force is not involved, the only dimensions being length and time, as in kinematics. Thus v=

η . ρ

(1.4)

In SI units, kinematic viscosity is measured in m2 /s while in the metric system the common units are cm2 /s, also called the stoke (St). The centistoke (cSt) (0.01 St) is often a more convenient unit because the viscosity of water at 20◦ C is 1 cSt. A fluid for which the constant of proportionality (i.e., the viscosity) does not change with rate of deformation is said to be a Newtonian fluid and can be represented by a straight line in Fig. 1.2. The slope of this line is determined by the viscosity. The ideal fluid, with no viscosity, is represented by the horizontal axis, while the true elastic solid is represented by the vertical axis. A plastic body which sustains a certain amount of stress before suffering a plastic flow can be shown by a straight line intersecting the vertical axis at the yield stress. The relationship between stress and deformation rate given in Eq. (1.3) represents a constitutive equation of the fluid in a simple shear flow. We can generalize this result by saying that, in simple fluids, the stress on a material is determined by the history of the deformation involving only gradients of the first order or more exactly by the relative deformation tensor as every fluid is isotropic. A general constitutive equation which describes the mechanics of materials in classical fluid mechanics can be written as: tij = −pδij + τij + λv ekk δij ,

Fig. 1.2 Rheological behaviour of materials

(1.5)

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1 Non-Newtonian Fluids

or, using the unit tensor I, T = −pI + τ + λv (trE)I,

(1.6)

where T(x, t) denotes the symmetric Cauchy–Green stress tensor at position x and time t, p(x, t) is the pressure in the fluid, τ is the extra stress tensor, λv is the volume viscosity, and E is the rate of deformation tensor of the velocity field u(x, t): 1 eij = 2

∂uj ∂ui + ∂xj ∂xi

(1.7)

or E(x, t) =

1 (∇u) + (∇u)T , 2

(1.8)

where trE = eii = ∂ui /∂xi . The extra stress tensor can be written as τ = η(I1 , I2 , I3 )E.

(1.9)

The apparent viscosity η in the above equation is a function of the first, second and third invariants of the rate of deformation tensor: I1 = eii , I2 =

1 (eii ejj − eij eij ), I3 = det(eij ). 2

(1.10)

For incompressible fluids, the first invariant I1 becomes identically equal to zero. The third invariant I3 vanishes for simple shear flows. The apparent viscosity then is a function of the second invariant I2 alone, and Eq. (1.9) can be written in a simplified form as τ = η(I2 )E.

(1.11)

If the fluid does not undergo a volume change, i.e. it is incompressible, then the last term on the right-hand side of Eq. (1.6) drops out and the volume viscosity has no role to play.

1.1.2 Non-Newtonian Fluids There is a certain class of fluids, called non-Newtonian fluids, in which the viscosity η varies with the shear rate. A particular feature of many non-Newtonian fluids is the retention of a fading “memory” of their flow history which is termed elasticity. Typical representatives of non-Newtonian fluids are liquids which are formed either partly or wholly of macromolecules (polymers), or two phase

1.1

Definitions

5

materials, like, for example, high concentration suspensions of solid particles in a liquid carrier solution. There are various types of non-Newtonian fluids. Pseudoplastic fluids are those fluids for which viscosity decreases with increasing shear rate and hence are often referred to as shear-thinning fluids. These fluids are found in many real fluids, such as polymer melts and solutions or glass melt. When the viscosity increases with shear rate the fluids are referred to as dilatant or shear-thickening fluids. These fluids are less common than with pseudoplastic fluids. Dilatant fluids have been found to closely approximate the behaviour of some real fluids, such as starch in water and an appropriate mixture of sand and water. For pseudoplastic and dilatant fluids, the shear rate at any given point is solely dependent upon the instantaneous shear stress, and the duration of shear does not play any role so far as the viscosity is concerned. Many of these fluids exhibits a constant viscosity at very small shear rates (referred to as zero-shear viscosity, η0 ) and at very large shear rates (referred to as infinite-shear viscosity, η∞ ). Some fluids do not flow unless the stress applied exceeds a certain value referred to as the yield stress. These fluids are termed fluids with yield stress or viscoplastic fluids. The variation of the shear stress with shear rate for pseudoplastic and dilatant fluids with and without yield stress is shown in Fig. 1.3. Viscoelastic fluids are those fluids that possess the added feature of elasticity apart from viscosity. These fluids have a certain amount of energy stored inside them as strain energy thereby showing a partial elastic recovery upon the removal of a deforming stress. In the case of thixotropic fluids, the shear stress decreases with time at a constant shear rate. An example of a thixotropic material is non-drip paint, which becomes thin after being stirred for a time, but does not run on the wall when it is brushed on. By contrast, when the shear stress increases with time at a constant shear rate the fluids are referred to as rheopectic fluids. Some clay suspensions exhibit rheopectic behaviour. Figure 1.4 shows a schematic of the thixotropic

Fig. 1.3 Rheological behaviour of non-Newtonian fluids

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1 Non-Newtonian Fluids

Fig. 1.4 Rheopectic and thixotropic fluids

and rheopectic fluid behaviour. In the case of thixotropic and rheopectic fluids, the shear rate is a function of the magnitude and duration of shear, and the time lapse between consecutive applications of shear stress. Viscoelastic fluids have some additional features. When a viscoelastic fluid is suddenly strained and then the strain is maintained constant afterward, the corresponding stresses induced in the fluid decrease with time. This phenomenon is called stress relaxation. If the fluid is suddenly stressed and then the stress is maintained constant afterward, the fluid continues to deform, and the phenomenon is called creep. If the fluid is subjected to a cycling loading, the stress–strain relationship in the loading process is usually somewhat different from that in the unloading process, and the phenomenon is called hysteresis. There is a distinctive difference in flow behaviour between Newtonian and nonNewtonian fluids to an extent that, at time, certain aspects of non-Newtonian flow behaviour may seem abnormal or even paradoxical. For example, when a rod is rotated in an elastic non-Newtonian fluid, the fluid climbs up the rod against the force of gravity. This is because the rotational force acting in a horizontal plane produces a normal force at right angles to that plane. The tendency of a fluid to flow in a direction normal to the direction of shear stress is known as the Weissenberg effect. Another effect caused by viscoelasticity is the die swell effect of the fluid as it leaves a die exit. This expansion is an elastic response of the fluid to energy stored when its shape changes while entering the die. This energy is released as the fluid leaves the die and causes a swelling effect normal to the direction of flow in the die. It has been also observed that, when a viscoelastic fluid flows in a tube with a sudden contraction, bubbles with a certain diameter come to a sudden stop right at the entrance of the contraction along the centerline before finally passing through after a hold time. This behaviour has been termed the Uebler effect.

1.2

Non-Newtonian Fluid Behaviour

7

1.2 Non-Newtonian Fluid Behaviour A Newtonian fluid requires only a single material parameter to relate the internal stress to the applied strain. For non-Newtonian fluids, more elaborate constitutive equations, containing several material parameters, are needed to describe the response of these fluids to complex, time-dependent flows. There exists no general model, i.e., no universal constitutive equation that describes all non-Newtonian fluid behaviour. Currently successful theories are either restricted to very specific, simple flows, especially generalizations of simple shear flow and extensional flow, for which rheological data can be used to develop empirical models, or to very dilute solutions for which the microscale dynamics is dominated by the motion of simple, isolated macromolecules. This section deals with the description of the nature and diversity of material response to simple shearing and extensional flows. The analysis of experimental methods for measuring these quantities is presented in the next section.

1.2.1 Simple Flows We shall now examine some simple flow fields of fluids. Simple flow fields are required to determine the material properties of the fluids and these are separated in three groups: steady simple shear, small-amplitude oscillatory, and extensional flow. 1.2.1.1 Steady Simple Shear Flow The most common flow is steady simple shear flow, represented in rectangular Cartesian coordinates by: ux = γ˙ y,

uy = uz = 0,

(1.12)

where (ux , uy , uz ) are the velocity components in the x, y, and z directions, and γ˙ = dux /dy. For steady shear flow (sometimes called a viscometric flow) the shear rate is independent of time; it is presumed that the shear rate has been constant for such a long time that all the stresses in the fluid are time-independent. The extra stress tensor in such a flow is thus defined by ⎞ τxx τxy 0 τ = ⎝ τyx τyy 0 ⎠ , 0 0 τzz ⎛

(1.13)

where τxy = τyx are called the shear stress components, and τxx , τyy , and τ zz are called the normal stress components. The corresponding stress distribution for a non-Newtonian fluid can be written in the form τxy = τ (γ˙ ) = η(γ˙ )γ˙ ,

(1.14)

8

1 Non-Newtonian Fluids

τxx − τyy = N1 (γ˙ ),

(1.15)

τyy − τzz = N2 (γ˙ ),

(1.16)

where N1 and N2 are the first and second normal stress differences. For a Newtonian fluid, η is a constant and N1 and N2 are zero. The variation of η with shear rate and non-zero values of N1 and N2 are manifestations of non-Newtonian viscoelastic behaviour. The second normal stress difference N2 , however, receives less attention due to difficulties in its measurement and for the smallness of its value. For many non-Newtonian fluids, the value of N2 would be usually an order of magnitude smaller than that of N2 . The viscosity function η, the primary and secondary normal stress coefficients ψ1 , and ψ2 , respectively, are the three parameters which completely determine the state of stress in any steady simple shear flow. They are often referred to as the viscometric functions. The normal stress coefficients are defined as follows: τxx − τyy = ψ1 (γ˙ )γ˙ 2 ,

(1.17)

τyy − τzz = ψ2 (γ˙ )γ˙ 2 ,

(1.18)

and

and are also functions of the magnitude of the strain rate. The first and second normal stress coefficients do not change in sign when the direction of the strain rate changes. The primary normal stress coefficient is used to characterize the elasticity of a non-Newtonian fluid. A constant primary normal stress coefficient is obtained when the primary normal stress varies quadratically with shear rate. 1.2.1.2 Small-Amplitude Oscillatory Shear Flow Small-amplitude oscillatory shear flow provides another mean to characterize a viscoelastic fluid. The oscillatory tests belong to the general framework of dynamic characterization of viscoelastic fluids in which both stress and strain vary harmonically with time. The dynamic properties of viscoleastic fluids are of considerable importance because they can be directly related to the viscous and elastic parameters derived from such measurements. Oscillatory tests involve the measurement of the response of the fluid to a small amplitude sinusoidal oscillation. The applied strain and strain rates are given by γ (t) = γ0 sin(ωt),

(1.19)

γ˙ (t) = γ0 ω cos(ωt) = γ˙0 cos(ωt),

(1.20)

and

1.2

Non-Newtonian Fluid Behaviour

9

where γ 0 is the amplitude of the applied strain, γ˙0 is the shear rate amplitude, and ω is the frequency. The resulting shear stress may be given in terms of amplitude, τ0 , and phase shift, δ = π2 − φ, as follows: τxy (t) = τ0 sin(ωt + δ),

(1.21)

τxy (t) = τ0 ω cos (ωt − φ).

(1.22)

and

These equations may be expanded and rewritten in terms of the in-phase and outof-phase parts of the shear stress and placed in terms of four viscoelastic material functions as

τxy (t) = γ0 G sin(ωt) + G cos(ωt) ,

(1.23)

τxy (t) = γ0 ω η cos(ωt) + η sin(ωt) .

(1.24)

The storage modulus, G , is defined as the stress in-phase with the strain in a sinusoidal shear deformation divided by the strain and is a measure of the elastic energy stored in the system at a particular frequency. G represents the solid like response of a material and, for a perfectly elastic solid, is equal to the constant shear modulus, G, for a perfectly elastic solid with the loss modulus equal to zero. Similarly, the loss modulus, G , is defined as the stress 90◦ out-of-phase with the strain divided by the strain, and is a measure of the energy dissipated as a function of frequency. G represents the viscous component or liquid-like response of a material to a deformation. The dynamic viscosity, η , and dynamic rigidity, η , are related to G and G by G , ω G η = . ω

η =

(1.25) (1.26)

The material functions G , G , η , and η are referred to as the linear viscoelastic properties because they are determined from the shear stress which is linear in strain for small deformations. It should be noted that as the frequency approaches zero, η approaches η0 and 2G /ω2 approaches ψ1,0 (the zero-shear-rate value of ψ1 ). Correspondingly, the loss modulus is asymptotic to η0 ω as ω → 0. A method of comparing the storage and loss modulus is made by the calculation of the loss tangent defined as tan δ =

G , G

and represents the phase angle between stress and strain.

(1.27)

10

1 Non-Newtonian Fluids

For more detail on these and other linear viscoelastic properties, standard references should be consulted (for example, Bird et al. 1987). 1.2.1.3 Extensional Flow Shear measurements are not sufficient to characterize the behaviour of nonNewtonian liquids and must be supplemented by measurements obtained in extension or extension-like deformations. An extensional flow is one in which fluid elements are stretched or extended without being rotated or sheared. Extensional flow can be visualized as that occurring when a material is longitudinally stretched as in fiber spinning. In this case, the extension occurs in a single direction and the related flow is termed uniaxial extension. Extension of material takes place in processing operation as well, such as film blowing and flat-film extrusion. Here, the extension occurs in two directions and the flow is referred to as biaxial extension in one case and planar extension in the other. In biaxial extension, the material is stretched in two directions and compressed in the other. In planar extension, the material is stretched in one direction, held to the same dimension in a second, and compressed in the third. A schematic representation of the three types of extensional flow fields is shown in Fig. 1.5. Uniaxial Extensional Flow In a uniaxial extensional flow, the dimension of the fluid elements changes in only one direction. The velocity components are: ux = ε˙ x,

ε˙ uy = − y, 2

where ε˙ = dux /dx is a constant strain tensor is ⎛ τxx τ=⎝ 0 0

ε˙ uz = − z, 2

(1.28)

rate, and the corresponding extra stress ⎞ 0 0 τyy 0 ⎠ . 0 τzz

Fig. 1.5 Extensional flow fields: (a) uniaxial, (b) biaxial, (c) planar

(1.29)

1.2

Non-Newtonian Fluid Behaviour

11

The corresponding stress distribution can be written in the form τxx − τyy = τxx − τzz = ηE (˙ε)˙ε ,

(1.30)

τij = 0, i = j,

(1.31)

where ηE is the uniaxial extensional viscosity. Fluids are considered extensionalthinning if ηE decreases with increasing ε˙ . They are considered extensionalthickening if ηE increases with ε˙ . These terms are analogous to shear-thinning and shear-thickening used to describe changes in viscosity with shear rate. The uniaxial extensional viscosity is frequently qualitatively different from shear viscosity. For example, highly elastic polymer solutions that posses a shear viscosity that decreases in shear often exhibit uniaxial extensional viscosity that increases with strain rate. In most applications, the extensional viscosity is presented in terms of a Trouton ratio which is defined conveniently to be the ratio of extensional viscosity to the shear viscosity, Tr = ηE /η. For calculating Trouton ratio in uniaxial extensional flow, √ the shear viscosity should be evaluated at a shear rate numerically equal to 3˙ε . This result is obtained by comparing extensional and shear viscosities at equal values of the second invariant of the rate of deformation tensor. The Trouton ratio, which takes the constant value 3 for Newtonian liquids and shear-thinning inelastic liquids, is found to be a strong function of strain rate ε˙ in many viscoelastic liquids, with very high values, of about 104 , possible in extreme cases.

Biaxial Extensional Flow In biaxial extensional flow, the dimensions of the fluid elements change drastically but they change only in two directions. The velocity field in simple biaxial extensional flow is given by ux = ε˙ B x,

uy = ε˙ B y,

uz = −2˙εB z.

(1.32)

The corresponding stress distribution is τxx − τzz = τyy − τzz = ηEB (˙εB )˙εB ,

(1.33)

where ηEB is the biaxial extensional viscosity. The Trouton number for the case of biaxial extensional flow can be calculated as TrEB = ηEB /η. For calculating Trouton ratio in a biaxial extensional √ flow, the shear viscosity should be evaluated at a shear rate numerically equal to 12˙ε. For a Newtonian fluid, TrEB = 6.

12

1 Non-Newtonian Fluids

Planar Extensional Flow Planar extensional flow is the type of flow where there is no deformation in one direction. The velocity field is represented by ux = ε˙ P x,

uy = −˙εP y,

uz = 0.

(1.34)

In this case, the stress distribution is given as τxx − τyy = ηEP (˙εP ) ε˙ P ,

(1.35)

where ηEP is the planar extensional viscosity. The Trouton number for the case of planar extensional flow can be calculated as TrP = ηP /η. For calculating Trouton ratio in a panar extensional flow, the shear viscosity should be evaluated at a shear rate numerically equal to 2˙ε . For a Newtonian fluid, TrP = 4. It is difficult to generate planar extensional flow and experimental tests of this type are less common than those involving uniaxial or biaxial extensional flows.

1.2.2 Intrinsic Viscosity and Solution Classification The intrinsic viscosity is another parameter that characterize the behaviour of nonNewtonian fluids. The intrinsic viscosity, [η], of a polymer solution is defined as the zero concentration limit of the reduced viscosity, ηred = ηsp /c, where c is the polymer concentration and ηsp is the specific viscosity. The specific viscosity is defined as the relative polymer contribution to viscosity ηsp = (η0 − ηs )/μs , where η0 is the zero-shear viscosity and ηs is the solvent viscosity. The intrinsic viscosity can thus be expressed as: [η] = lim ηred = lim c→0

c→0

η0 − ηs . cηs

(1.36)

Note that the intrinsic viscosity has dimensions of reciprocal concentration. The intrinsic viscosity is determined graphically by plotting ηred versus c and extrapolating to zero concentration. It is also found that extrapolation to zero concentration of the inherent viscosity, ηinh = 1c ln(ηsp + 1), can also be used to determine the intrinsic viscosity and the same result for [η] must be achieved. The most common relation between specific viscosity and polymer concentration is that of Huggins (1942), ηsp = [η] + k [η]2 c, c

(1.37)

where k is the Huggins slope constant. The alternative expression of Kraemer (1938)

1.2

Non-Newtonian Fluid Behaviour

13

1 ηsp ln = [η] − k [η]2 c, c c

(1.38)

where k is the Kraemer constant, may also be used. Huggins slope constant and Kraemer constant are related by k + k = 0.5. The intrinsic viscosity can be used to determine the viscosity molecular weight, Mη , using the Mark-Houwink equation as follows (Bird et al. 1987) [η] = kMηα ,

(1.39)

where k and α are determined from a double logarithmic plot of intrinsic viscosity and molecular weight. These parameters have been published for many systems by Bandrup and Immergut (1975). The polymer solutions are regarded as dilute when there is no interaction between molecules. A standard method to evaluate whether a polymer solution is dilute is to determine a dimensionless concentration of polymer solution which can be given by either [η]c (Flory 1953) or cNA V/Mw (Doi and Edwards 1986), where c is the polymer concentration, NA is the Avogadro’s number, V is the volume occupied by a polymer molecule, and Mw is the average molecular weight. Flexible polymers tend to occupy a spherical region in solution such that V = 4π R3h /3. In the case of rigid molecules, the spherical region required such that the large aspect ratio molecule can freely rotate without interaction with its neighbours is calculated from the molecule length such that V = π L3 /6, where L is the length of the molecule. The length L can be determined using relations and

given by Broersma (1960) Young et al. (1978) for rigid molecules, L = Rh 2δ − 0.19 − (8.24/δ) + 12/δ 2 , where δ = ln (L/r) is the aspect ratio of a rod and r is the radius of the rigid rod. The polymer solution is regarded as dilute when both dimensionless concentrations are less than unity. When one of the dimensionless concentrations is larger than 1, the polymer solution is considered semi-dilute.

1.2.3 Dimensionless Numbers Fluid dynamics is parametrized by a series of dimensionless numbers expressing the relative importance of various physical phenomena. These include, for example, the Reynolds number, addressing inertial effects, the Froude number, describing gravity-driven flows, the Weber number, addressing the importance of surface tension forces, the Grashof number, addressing buoyancy effects, or the Mach number, describing the importance of liquid compressibility. In the specific case of non-Newtonian fluids, three additional sets of non-dimensional parameters are generated, namely the Weissenberg number, the Deborah number, and the elasticity number, describing elastic effects. The dimensionless numbers are particularly useful for scaling arguments, for consolidating experimental, analytical, and numerical results into a compact form, and also for cataloging various flow regimes.

14

1 Non-Newtonian Fluids

1.2.3.1 Reynolds Number Of all dimensionless numbers encountered in fluid dynamics, the Reynolds number is the one most often mentioned in connection with non-Newtonian fluids. The Reynolds number represents the ratio of inertia forces to viscous forces and has the expression: Re =

LU ρLU = , ν η

(1.40)

where L is a linear dimension that may be any length that is significant in the flow pattern and U is the flow velocity. For example, for a pipe completely filled, L might be either the diameter or the radius, and the numerical value of Re will vary accordingly. 1.2.3.2 Weissenberg Number The Weissenberg number is defined as Wi = τfluid e˙ or

τfluid γ˙ ,

(1.41)

which relates the relaxation time of the viscoelastic liquid to the flow deformation time, either inverse extension rate 1/˙ε or shear rate 1/γ˙ . When Wi is small, the liquid relaxes before the flow deforms it significantly, and perturbations to equilibrium are small. As Wi approaches 1, the liquid does not have time to relax and is deformed significantly. 1.2.3.3 Deborah Number Another relevant time scale, τflow , characteristic of the flow geometry may also exist. For example, a channel that contracts over a length L0 introduces a geometric time scale τflow = L0 /U0 required for a liquid to transverse it with velocity U0 . The flow time scale τflow can be long or short compared with the liquid relaxation time, τfluid , resulting in a dimensionless ratio known as the Deborah number De =

τfluid . τ flow

(1.42)

For small De values, the material responses like a fluid, while for large De values, we have a solid-like response. In the limit, when De = 0 one has a Newtonian fluid, and when De = ∞, an elastic solid. The usage of De and Wi can vary. Some references use Wi exclusively to describe shear flows and use De for the general case, whereas others use Wi for local flow time scales due to a local shear and De for global flow time scales due to residence time in flow.

1.2

Non-Newtonian Fluid Behaviour

15

1.2.3.4 Elasticity Number As the flow velocity increases, elastic effects become stronger and De and We increase. However, the Reynolds number Re increases in the same way, so that inertial effects become more important as well. The elasticity number El = De/Re =

τfluid η , ρL2

(1.43)

where L is a dimension setting the shear rate, expresses the relative importance of elastic to inertial effects. Significantly, El depends only on the geometry and material properties of the fluid, and is independent of flow rate. For example, extrusion of polymer melts corresponds to El >> 1, whereas processing flows for dilute polymer solutions (such as spin-casting) typically correspond to El << 1.

1.2.4 Constitutive Equations A constitutive equation is required to describe the extra stress tensor τ that governs the motion of a non-Newtonian fluid. Numerous constitutive equations have been proposed to describe various classes of non-Newtonian fluids and a few of the simplest are described in this section. The books by Bird et al. (1987) and Larson (1988) are recommended for more in depth discussion on constitutive models. 1.2.4.1 Purely Viscous Fluids When the fluid is relatively inelastic, the generalized Newtonian model (1.11) can be used to describe the change in viscosity with shear rate of non-Newtonian fluids. The Power Law Model The simplest generalized Newtonian model is the power law model which describes the non-Newtonian viscosity as η = K γ˙ n ,

(1.44)

where K is referred to as the consistency index and n is the power law exponent. For the special case of a Newtonian fluid (n = 1), the consistency index K is identically equal to the viscosity of the fluid. When the magnitude of n < 1 the fluid is shearthinning, and when n > 1 the fluid is shear-thickening. The power-law model is the most well-known and widely-used empiricism in engineering work, because a wide variety of flow problems have been solved analytically for it. One can often get a rough estimate of the effect of the non-Newtonian viscosity by making a calculation based on the power-law model. One shortcoming of the power law model is that it does not describe the low shear and high shear rate constant viscosity data of

16

1 Non-Newtonian Fluids

shear-thinning or shear-thickening fluids. For n < 1, this model presents a problem when the shear rate tends to zero because the fluid viscosity becomes infinite. The Carreau Model A more sophisticated model is the Carreau model given as η0 − η∞ η = η∞ +

N , 1 + (λc γ˙ )

(1.45)

where λc is a time constant and N is a dimensionless exponent. At low shear rates, the model predicts Newtonian properties with a constant zero-shear viscosity, η0 , while at high shear rates, it predicts a limiting and constant infinite-shear viscosity, η∞ . The Carreau model can be modified to include a term due to yield stress. For example, the Carreau model with a yield term given by η=

ηp τ0 +

N , γ˙ 1 + (λc γ˙ )

(1.46)

where τ0 is the yield stress and ηp is the plateau viscosity, was employed in the study of the rheological behaviour of glass-filled polymers (Poslinski et al. 1988). The Casson Model The Casson model given by √ γ˙ =

0

√ τ − τ0 2 √ η

, for τ ≥ τ0 , , for τ < τ0

(1.47)

where τ0 is the yield stress, captures both the yield stress and shear dependent viscosity of a fluid. This model reduces to a Newtonian fluid when τ0 = 0. Equation (1.47) indicates that a finite yield stress is required before flow can start. This yield stress results in a plug flow and the velocity distribution shaped like a blunted parabola that is so typical of blood flow in small diameter vessels. The Casson model was originally developed to describe the flow of printing ink through capillaries and was later applied to other fluids containing chain like particles. The Casson equation has also proven useful for the description of the flow of blood on both glass and fibrin surfaces. 1.2.4.2 Viscoelastic Fluids A large number of constitutive equations have been proposed to describe the viscoelastic behaviour of non-Newtonian fluids. The Maxwell and Oldroyd-B models have had a popularity far beyond expectation and anticipation. Their relative simplicity has obviously been an attraction, especially in the case of numerical

1.2

Non-Newtonian Fluid Behaviour

17

simulation of viscoelastic flows, where simple models have been essential in the development of numerical strategies. Other important viscoelastic models that have been used extensively are the dumbbell models and the KBKZ model. The Maxwell Model The simplest constitutive model to account for fluid elasticity is the Maxwell model which considers the fluid as being both viscous and elastic. The Maxwell equation is given by: τ+λ

∂τ = 2ηE, ∂t

(1.48)

where λ is the relaxation time and η is the constant shear viscosity. For steady-state motions this equation simplifies to the Newtonian fluid with viscosity η. By replacing the time derivative with the convected time derivative, the upper convected Maxwell model is obtained which is given as ∇

τ + λ τ = 2ηE,

(1.49)

∇

where the upper convected derivative τ is defined by ∇

τ=

∂τ + (u · ∇)τ − (∇u)T τ − τ (∇u) . ∂t

(1.50)

For steady simple shear flow, the Maxwell relaxation time is λ=

ψ1 N1 , = 2 2η 2ηγ˙

(1.51)

while in small-amplitude oscillatory flow, the viscoelastic properties for this model are given by G =

ληω2 , 1 + λ2 ω 2

(1.52)

η =

η . 1 + λ2 ω 2

(1.53)

and

At low frequency, G is predicted to vary quadratically with frequency while it approaches a constant value at high frequencies. The uniaxial extensional viscosity for the upper convected Maxwell model is ηE = 3η

1 . (1 + λ˙ε) (1 − 2λ˙ε )

(1.54)

18

1 Non-Newtonian Fluids

This model predicts strain rate thickening behaviour, but the predicted extensional viscosity asymptotes to infinity when ε˙ = 1 (2λ) . The upper convective Maxwell model exhibits many of the qualitative behaviours of viscoelastic fluids, including normal stresses in shear, extension thickening, and elastic recovery. However, it does not exhibit shear thinning. To get a reasonable match to viscoelastic behaviour, one must introduce some additional nonliniarities by altering the model in the form ∇

Y · τ + λ τ = 2ηE.

(1.55)

Two models that are widely used are the Giesekus model, which has Y=I+

αλ τ, η

and the Phan-Thien-Tanner model, for which ελ Y = exp tr(τ) I. η

(1.56)

(1.57)

Each of these models adds another dimensionless parameter, α or ε, that control the nonlinearity. A multi-mode Maxwell model may also be used to allow the material functions to be predicted more accurately by adjusting the parameters in each mode. The extra stress tensor is expressed, in this case, as a combination of several relaxation times as τ=

n

τi ,

(1.58)

i=1

where each τi is described by ∇

τι + λi τi = 2ηi E.

(1.59)

The Oldroyd-B Model The Maxwell model may be extended to obtain a more useful constitutive equation by including the convected time derivative of the rate of deformation tensor. This way the Oldroyd-B constitutive model is obtained which is described by ∇ ∇ τ + λ1 τ = 2η E + λ2 E ,

(1.60)

where λ1 and λ2 are the time constants (relaxation and retardation) and the viscosity has also a constant value. We observe that, by setting λ2 = 0, the above equation

1.2

Non-Newtonian Fluid Behaviour

19

reduces to the upper convected Maxwell model. The Oldroyd-B model qualitatively describes many features of the so-called Boger fluids (elastic fluids with almost constant viscosity). The material functions of this model are defined as ψ1 = 2η (λ1 − λ2 ) ,

ψ2 = 0,

(1.61)

while the linear viscoelastic properties are given by G =

(λ1 − λ2 )ηω2 , 1 + λ21 ω2

(1.62)

and

η =

1 + λ1 λ2 ω 2 η 1 + λ21 ω2

.

(1.63)

As in the case of Maxwell model, the Oldryod-B model predicts that at low frequencies the storage modulus varies quadratically with frequency while at high frequencies a constant value is obtained. The equation for the uniaxial extensional viscosity is given by ηE = 3η

1 − λ2 ε˙ − 2λ1 λ2 ε˙ 2 , 1 − λ1 ε˙ − 2λ21 ε˙ 2

(1.64)

and, therefore, the extensional viscosity asymptotes to infinity when ε˙ = 1 (2λ1 ) . In non-convected form, the Oldroyd-B model is referred to as the Jeffreys model which is given by ∂τ ∂E = 2η E + λ2 . τ + λ1 ∂t ∂t

(1.65)

It is interesting to note that this equation was originally proposed for the study of wave propagation in the earth’s mantle (Jeffreys 1929). The Dumbbell Model In elastic dumbbell models a polymer is described as two beads connected by a Hookean spring. The beads represent the ends of the molecule and their separation is a measure of the extension. The beads experience a hydrodynamic drag force, a Brownian force due to thermal fluctuations of the fluid, and an elastic force due to the spring connecting one bead to the other. It can be further assumed that the polymer solution is sufficiently dilute that the polymer molecules do not interact with one another. The polymer contribution to the stress tensor is ∇

τp + λH τp = nkB TλH E,

(1.66)

20

1 Non-Newtonian Fluids

where n is the number density of molecules, kB is the Boltzman constant, T is the temperature, and λH is the relaxation time for a Hookean dumbbell. The Hookean relaxation time is defined in terms of a friction coefficient of the beads, ς , and a Hookean spring constant, H, as λH = ς/(4H). The Oldroyd-B constitutive equation may be derived from the elastic dumbbell model with the following relations used to determine the material functions for steady shear flow: η = ηs + nkTλH

,

ψ1 = 2nkTλ2H

,

ψ2 = 0,

(1.67)

while the relaxation time is given as a function of the intrinsic viscosity as λ1 =

[η] ηs Mw , Rg T

(1.68)

The retardation time in the where Rg is the universal gas constant (8.314 J/(kg·mol)). Oldroyd-B model is given by λ2 = λ1 ηs η . The relaxation time in the Maxwell constitutive model, λ, is related to the Oldroyd characteristic times by λ = λ1 − λ2 . The elastic dumbbell model is only suitable to use for flexible polymers, such as polyacrylamide. The rigid dumbbell model may be used to describe rigid or semi-rigid molecules (such as, DNA in a helix configuration, xanthan gum, or carboxymethylcellulose) in solution. It accounts for the orientability of the rod-like macromolecules in the flow field while ignoring molecular stretching and bending motions which are not considered significant for this class of macromolecules. The macromolecule is represented by two beads joined by a massless rod, with the solvent presumed to only interact at the beads. The rigid dumbbell relaxation time is given by λD =

1 m [η] ηs Mw , m= , Rg T m1 + m2

(1.69)

where the values of m1 and m2 depend on the details of the model, as listed by Ferry (1980). The extensional viscosity approaches a constant at relatively low extension rates such that as ε˙ → ∞ the limiting extensional viscosity is ηe = 3ηs + 6ckB TλD ,

(1.70)

where c is the polymer concentration. The BKBZ Model The KBKZ model is an integral type constitutive equation proposed by Kaye (1962) and Bernstein et al. (1963). The time-integral constitutive equation of the KBKZ model is

1.2

Non-Newtonian Fluid Behaviour

t σp =

21

μ t − t H (I1 , I2 ) B t, t dt ,

(1.71)

−∞

where σp is the polymer contribution to the extra stress tensor, μ t, t is the linear memory function, H(I1 , I2 ) is a non-linear damping function, and viscoelastic B t, t is the Finger strain tensor given by ⎛ 2 ⎞ ς t, t 0 0 B t, t = ⎝ 0 ς 2m t, t 0 ⎠, −2(m+1) 0 0 ς t, t

ς t, t = exp ε˙ 0 t − t ,

(1.72)

(1.73)

where m = –0.5 for uniaxial extension, m = 0 for planar extension, m = 1 for biaxial extension, and ς is the extension ratio. The memory function is expressed as an exponentially fading term while the strain function can be written as (Papanastasiou et al. 1983) a t − t μ t, t = exp − , λ λ H(I1 , I2 ) =

α , (α − 3) + βI1 + (1 − β) I2

(1.74) (1.75)

where α and β are adjustable parameters determined from the shear and extensional results, respectively. The set of parameters {λ, a} are the conventional relaxation time and weight, which can be evaluated from simple rheological tests such as stress relaxation or sinusoidal oscillations. Several other forms of the damping term are known in literature such as those provided by Wagner (1976), Wagner and Demarmels (1990): 1 , √ 1 + α (I1 − 3)(I2 − 3)

H(I1 , I2 ) = exp −β αI1 + (1 − α)I2 − 3 . H(I1 , I2 ) =

(1.76) (1.77)

This constitutive equation has been found to accurately predict transient and shear modes of simple shear, uniaxial extension and biaxial extension at low, moderate, and high strains and rates of strain (Papanastasiou et al. 1983). Example: Material Functions for the Oldroyd-B Model In a steady simple shear flow ux = γ˙ y,

uy = 0,

uz = 0,

with γ˙ = dux dy .

(1)

22

1 Non-Newtonian Fluids

The rate of deformation and extra stress tensors have the following expressions: ⎛ 01 1 1 ∇u + ∇uT = ⎝ 0 0 E= 2 2 00

⎞ ⎛ ⎞ ⎛ ⎞ 0 000 010 1 1 0 ⎠ γ˙ + ⎝ 1 0 0 ⎠ γ˙ = ⎝ 1 0 0 ⎠ γ˙ , 2 000 2 000 0

⎛ ⎞⎛ ⎞ 010 010 1 du E= − ∇u · E + E · ∇uT = − ⎝ 0 0 0 ⎠ ⎝ 1 0 0 ⎠ γ˙ 2 dt 2 000 000 ⎛ ⎞ ⎛ ⎞⎛ ⎞ 100 010 000 1 − ⎝ 1 0 0 ⎠ ⎝ 1 0 0 ⎠ γ˙ 2 = − ⎝ 0 0 0 ⎠ γ˙ 2 , 2 000 000 000

(2)

∇

(3)

⎞ τxx τxy 0 τ = ⎝ τyx τyy 0 ⎠ , 0 0 τzz

(4)

⎛ ⎞⎛ ⎞ 010 τxx τxy 0 dτ − ∇u · τ + τ · ∇uT = − ⎝ 0 0 0 ⎠ ⎝ τyx τyy 0 ⎠ γ˙ τ= dt 000 0 0 τzz ⎛ ⎞⎛ ⎞ ⎛ ⎞ τxx τxy 0 000 2τxy τyy 0 − ⎝ τyx τyy 0 ⎠ ⎝ 1 0 0 ⎠ γ˙ = ⎝ τyy 0 0 ⎠ γ˙ . 000 0 0 τzz 0 0 0

(5)

⎛

and ∇

Replacing these results into the equation of the rheological model, we can write ⎛ ⎛ ⎞⎤ ⎞ ⎞ ⎞ ⎡ ⎛ τxy τyy 0 100 τxx τxy 0 010 1 ⎝ τyx τyy 0 ⎠ − λ1 γ˙ ⎝ τyy 0 0 ⎠ = 2ηγ˙ ⎣ ⎝ 1 0 0 ⎠ − λ2 γ˙ ⎝ 0 0 0 ⎠⎦ , (6) 2 000 000 0 0 τzz 0 0 0 ⎛

from which it follows that τxx − 2λ1 γ˙ τxy = −2ηλ2 γ˙ 2 τxy − λ1 γ˙ τyy = ηλ2 γ˙

,

(7)

so that τxy = ηγ˙ , τxx − τyy = N1 = 2η(λ1 − λ2 ) γ˙ 2 , τyy − τzz = N2 = 0,

(8)

1.2

Non-Newtonian Fluid Behaviour

23

and ψ1 = 2η (λ1 − λ2 ) , ψ2 = 0.

(9)

For an upper convective Maxwell fluid, λ2 = 0, and λ = ψ1 2η . For a small amplitude oscillatory shearing flow

γ˙ (t) = γ0 ω cos (ωt) , τxy = γ0 G sin (ωt) + G cos (ωt) .

(10)

The Oldroyd-B equation becomes

G sin (ωt) + G cos (ωt) + λ1 ω G cos (ωt) − G sin (ωt) = ηω [cos (ωt) − λ2 ω cos (ωt)] .

(11)

We obtain

G − λ1 ωG = −λ2 ω2 η G + λ1 ωG = ωη

,

(12)

and 1 + λ1 λ2 ω2 ηω (λ1 − λ2 ) ηω2 = , G = , 1 + λ21 ω2 1 + λ21 ω2 1 + λ1 λ2 ω 2 η (λ1 − λ2 ) ηω η = , η = . 2 2 1 + λ1 ω 1 + λ21 ω2

G

(13)

For the Upper Convective Maxwell fluid we have G =

ληω2 ηω η ληω , G = , η = , η = . 1 + λω2 1 + λω2 1 + λω2 1 + λω2

(14)

In a steady uniaxial extensional flow 1 1 ux = ε˙ x, uy = − ε˙ y, uz = − ε˙ y. 2 2

(15)

The rate of deformation tensor and the extra stress tensor become ⎞ ⎞ ⎛ ⎛ ⎛ ⎞ 1 0 0 1 0 0 2 0 0 1⎝ 1 1 0 −1 2 0 ⎠ ε˙ + ⎝ 0 −1 2 0 ⎠ ε˙ = ⎝ 0 −1 0 ⎠ ε˙ , E= 2 0 0 2 0 0 2 0 0 −1 −1 2 −1 2 (16)

24

1 Non-Newtonian Fluids

⎞⎛ ⎞ 1 0 0 2 0 0 1 0 ⎠ ⎝ 0 −1 0 ⎠ ε˙ 2 E = − ⎝ 0 −1 2 2 0 0 0 0 −1 −1 2 ⎞ ⎛ ⎞⎛ ⎛ 1 0 0 2 0 0 40 1⎝ 1 0 −1 0 ⎠ ⎝ 0 −1 2 0 ⎠ ε˙ 2 = − ⎝ 0 1 − 2 0 0 −1 2 00 0 0 −1 2 ⎛

∇

⎞ 0 0 ⎠ ε˙ 2 , 1

(17)

⎛

⎞ τxx 0 0 τ = ⎝ 0 τyy 0 ⎠ , 0 0 τzz

(18)

and ⎞⎛ ⎞ 1 0 0 τxx 0 0 τ = − ⎝ 0 −1 2 0 ⎠ ⎝ 0 τyy 0 ⎠ ε˙ 0 0 τzz 0 0 −1 2 ⎞ ⎛ ⎛ ⎞⎛ ⎞ τxx 0 0 1 0 0 τxx 0 0 1 − ⎝ 0 τyy 0 ⎠ ⎝ 0 −1 2 0 ⎠ ε˙ = −2 ⎝ 0 − 2 τyy 0 ⎠ ε˙ . 0 0 τzz 0 0 −1 2 0 0 − 12 τzz ⎛

∇

(19)

Replacing these results into the equation of the rheological model, we can write in the case of a steady uniaxial extensional flow ⎞ ⎛ ⎞ 0 τxx 0 τxx 0 0 1 ⎟ ⎝ 0 τyy 0 ⎠ − 2λ1 ε˙ ⎜ ⎝ 0 − τyy 0 ⎠ 2 0 0 τzz 0 0 − 12 τzz ⎡ ⎛ ⎛ ⎞⎤ ⎞ 400 2 0 0 1 1 = 2η˙ε ⎣ ⎝ 0 −1 0 ⎠ − λ2 ε˙ ⎝ 0 1 0 ⎠⎦ , 2 0 0 −1 2 001 ⎛

(20)

and ⎧ 2 ⎪ ⎨τxx − 2λ1 ε˙ τxx = 2η˙ε − 4ηλ2 ε˙ τyy + λ1 ε˙ τyy = −η˙ε − ηλ2 ε˙ 2 ⎪ ⎩ τzz + λ1 ε˙ τzz = −η˙ε − ηλ2 ε˙ 2

.

(21)

Thus, the uniaxial extensional viscosity is given by ηE =

τxx − τyy 1 − λ2 ε˙ (1 + 2λ1 ε˙ ) = 3η . ε˙ (1 − 2λ1 ε˙ ) (1 + λ1 ε˙ )

(22)

1.3

Rheometry

25

For an upper convective Maxwell model, λ2 = 0, and ηE = 3η

1 . (1 − 2λ˙ε ) (1 + λ˙ε)

(23)

1.3 Rheometry While the rheological behaviour of Newtonian fluids is completely determined by the constant viscosity, η, the situation for non-Newtonian fluids is much more complicated. The rheological characterization of non-Newtonian fluids is widely acknowledged to be far from straightforward. Even the apparently simple determination of a shear rate versus shear stress relationship is difficult as the shear rate can only be determined directly if it is constant throughout the measuring device employed. Rheological measurements may be further complicated by nonlinear and thixotropic properties. The most commonly techniques used for measuring the viscoelastic properties of non-Newtonian fluids are considered below.

1.3.1 Shear Rheometry In the case of shear rheometry, the shear flow is generated between a moving and a fixed rigid surface or by a pressure difference over a tube (Fig. 1.6). Classic examples of shear flow geometries belonging to the first group include cone and plate and concentric cylinder. An example of shear flow geometry belonging to the second group is capillary or Poiseuille flow.

Fig. 1.6 Shear rheometers. (a) Cone and plate, (b) Concentric cylinder, (c) Capillary rheometer

26

1 Non-Newtonian Fluids

1.3.1.1 Cone and Plate Rheometer The principal features of the cone and plate rheometer are shown schematically in Fig. 1.6a. The fluid sample, whose rheological properties are to be measured, is trapped between the circular conical disc at the top and the circular horizontal plate at the bottom. Two types of cone and plate rheometers are used for steady shear measurements, either a constant rate rheometer or a constant stress rheometer. In the constant rate instrument, the plate is rotated at a constant rate and the resulting shear stress is determined from the measurement of torque, M, on the cone. The shear rate, shear stress, τxy , and viscosity, η are given by γ˙ ∼ = , θ τxy =

μ (γ˙ ) =

3M , 2π R3

τxy 3M θ = , γ˙ 2π R3

(1.78)

(1.79)

(1.80)

where is the angular rotation rate of the plate, θ is the cone angle and R is the radius of the cone and plate. The θ angle between the cone and the plate is assumed to be small. Typically, θ is less than 4◦ . The small cone angle ensures that the shear rate is constant throughout the shearing gap, this being of particular interest when investigating time-dependent systems because all elements of the fluid sample experience the same shear history, but the small angle can lead to serious errors arising from eccentricities and misalignment. In the characterization of viscoelastic fluids, a force may result from the rotation of the plate which acts to separate the plates. The total thrust, F, on the bottom plate may then be used to determine the primary normal stress difference and typically placed in terms of a primary normal stress coefficient as N1 =

2F = ψ1 (γ˙ ) γ˙ 2 , π R2

(1.81)

and ψ1 (γ˙ ) =

2F θ 2 . π R2 2

(1.82)

All the above relationships are obtained under the assumption of negligible fluid inertial and edge effects, including surface tension. For accurate measurements, corrections for these possible errors are recommended in Carreau et al. (1997). The cone and plate viscometer can be used for oscillatory shear measurements as well. In this case, the fluid sample is deformed by an oscillatory driver and the amplitude of the sinusoidal deformation is measured by a strain transducer. The force

1.3

Rheometry

27

deforming the fluid sample is measured by the small deformation of a relatively rigid spring to which is attached a stress transducer. On account of the energy dissipated by the fluid, a phase difference develops between the stress and the strain. The material functions are determined from the amplitudes of stress and strain and the phase angle between them. The major advantage of this type of viscometer is the constant shear rate throughout all the liquid. This is because, at a fixed radial position, the circumferential viscosity varies linearly across the gap between the cone and the plate. A consequence of this fact is that the cone and plate viscometer is well suited for time-dependent measurements, such as dynamic measurements or transient measurements that imply a step change in the rate of shear strain. On the other hand, it should be noted that measurements can usually be made at relatively low shear rates. With increasing shear rate, there is a tendency for the development of secondary flows in the fluid and the fluid will crawl out of the instrument under the influence of centrifugal forces and elastic instabilities. 1.3.1.2 Concentric Cylinders Rheometer Another rheometer commonly used to determine the apparent viscosity of nonNewtonian fluids is the concentric cylinder or Couette flow rheometer, schematically depicted in Fig. 1.6b. Typically the outer cylinder rotates with an angular speed and the torgue M on the inner cylinder, which is usually suspended from a torsion wire or bar, is measured. The apparent viscosity of the fluid is given by η=

M R2o − R2i 4π LR2o R2i

,

(1.83)

where Ro and Ri are the radii of the outer and inner cylinder, respectively. A narrow gap approximation is typically involved to avoid a priori selection of a rheologicalmodel. It is recommended that the narrow gap approximation only be used for Ri Ro ≥ 0.99 . The main sources of error in the concentric cylinder rheometer arise from end effects, wall slip, inertia and secondary flows, viscous heating effects and eccentricities due to misalignment of the geometry. To minimize end effects the lower end of the inner cylinder is a truncated cone. Secondary flows are of particular interest in the controlled stress instruments which usually employ a rotating inner cylinder. In this case, inertial forces cause an axisymmetric cellular secondary motion (Taylor vortices). The dissipation of energy by these vortices leads to overestimation of the torque. For a Newtonian fluid in a narrow gap, the stability criterion is (Macosko 1994) ρ 2 2 (Ro − Ri )3 Ri < 3400, whereas, for non-Newtonian fluids, the stability limit increases.

(1.84)

28

1 Non-Newtonian Fluids

The maximum value of the shear rate achievable with concentric cylinders viscometers is, in most of the cases, not an instrument limitation but depends on the viscosity of the fluid sample. For high viscosity fluids, viscous heating may become a problem. For low viscosity fluids, the upper limit may be set by the occurrence of secondary flows. Usually, the maximum value of the shear rate is about 102 s–1 . At the other end of scale, it is possible to go down to shear rates as low as 10–2 s–1 , especially with high viscosity fluids. 1.3.1.3 Capillary Rheometer This method involves the laminar flow of a fluid through a small tube (Fig. 1.6c). In this case, the shear rate γ˙ has a maximum at the wall and zero in the centre of the flow. The flow is therefore non-homogeneous and capillary rheometers are restricted to measuring steady shear functions, i.e. steady shear stress – shear rate behaviour for time independent fluids. For an ideal viscometer, the flow rate is given by π R3 Q= 3 τw

τw τ 2 γ˙ (τ ) dτ ,

(1.85)

0

where τw = (R/2) (−p/L) is the shear stress at the wall of the tube, R is the tube radius, L is the tube length, and p is the pressure drop over the length L. For a Newtonian fluid, γ˙ (τ ) = τ/η, this equation yields π R4 Q= 8η

p − , L

(1.86)

from which η can be calculated using a value of Q obtained for a single value of (−p/L). For flow of unknown form, Eq. (1.85) yields (see, for example, White 1995) γ˙ (τw ) =

3n + 1 4Q , 4n π R3

(1.87)

with n =

d log τw

, d log π4Q 3 R

(1.88)

which is known as the Weissenberg–Rabinowitsch equation. For shear-thinning fluids, the apparent shear rate at the wall is less than the true shear rate. Thus at some radius, ς R, the true shear rate of a non-Newtonian fluid of apparent viscosity equals that of a Newtonian fluid of the same viscosity. The stress at this radius, ς τw , is independent of fluid properties and thus the true viscosity at this radius equals the apparent viscosity at the wall and the viscosity calculated from Eq. (1.86) is the true viscosity at a stress ς τw . Laun (1983) indicated that this method for correcting viscosity is as accurate as the Weissenberg–Rabinowitsch

1.3

Rheometry

29

method. Errors may occur due to wall slip, e.g. in the case of concentrated dispersions where the layer of particles may be more dilute near the wall than in the bulk flow. The layer near the wall has a much lower viscosity, resulting in an apparent slippage of the bulk fluid along the wall. In addition to these effects, viscous dissipation heating, fluid compressibility, change of viscosity with pressure, and flow instabilities can introduce errors into capillary viscometer measurements. The temperature rise associated with viscous dissipation can be reduced by using a smaller diameter capillary, since for shear-thinning fluids the rate of heat conduction to the wall increases more rapidly than viscous heat generation with decreasing tube radius. The intrinsic viscosity of polymer solutions is typically determined by measuring the viscosity using capillary viscometers due to their ability to precisely detect small differences in viscosity at low polymer concentration. Incorrect determination of the viscosity and, therefore, intrinsic viscosity can arise if the polymer solution is shear thinning and measurements must be made at low shear rates such that the viscosity equates to the zero-shear viscosity.

1.3.2 Extensional Rheometry As discussed in Sect. 1.2.1, extensional flow is fundamentally different from shear flow and extensional viscosity is a different material function from shear viscosity. The major difficulty in this type of rheometry is to generate a purely extensional flow, especially for low-viscosity fluids. In most of the cases, different measuring techniques give different results. There are, however, several types of flow geometries that can approximate a purely extensional flow, such as squeezing flow, stagnation point, entrance flow, or sheet stretching (Macosko 1994). Here we limit our attention to the opposed-jet and filament stretching techniques which are the most popular and promising methods for studying extensional properties of non-Newtonian fluids. 1.3.2.1 Opposed-Jet Rheometer The opposed jet rheometry was first introduced by Fuller et al. (1987) and a schematic diagram of the device is shown in Fig. 1.7. Fluid is drawn into opposed jets with the right nozzle arm fixed while the left nozzle arm is free to rotate about a pivot. The fluid exerts a hydrodynamic force onto the nozzles during flow, which is balanced by applying a torque, TM , to the pivot arm to prevent movement of the left arm. The force, FR used to balance the hydrodynamic force is related to the fluid extensional viscosity and can be used to define an apparent extensional stress difference (τc = τzz − τxx ) as: τc =

TM FR = , A AL

(1.89)

where L is the length of the lever arm between the nozzle and transducer, R is the nozzle radius, and A = π R2 is the area of the nozzle opening. Assuming a uniform

30

1 Non-Newtonian Fluids

Fig. 1.7 Opposed-jet rheometer

jet entrance velocity, the apparent extensional rate, ε˙ , in the flow field defined in terms of the volumetric flow rate through a nozzle, Q, and the gap between the nozzles, dn , is ε˙ =

Q . Adn

(1.90)

The apparent extensional viscosity can then be calculated according to ηEa =

τc . ε˙

(1.91)

One problem with the opposed-jet apparatus is that an upturn in the extensional viscosity is measured at high rates of strain for Newtonian fluids of low viscosity which Hermansky and Boger (1995) associated with fluid inertia. By introducing a correction coefficient, they indicated the following relationship between the measured and corrected Trouton ratio ηa a(R) ρdn2 ηc = E − ε˙ , η η 4π LR2 η

(1.92)

where ηc is the corrected extensional viscosity and η is the shear viscosity. The parameter a(R) was found to be constant for a particular jet. It should be noted here that before the correction is applied to a viscoelastic fluid, it is recommended that it be applied to a Newtonian fluid with a similar shear viscosity to the viscoelastic fluid, in order to ensure accuracy. Corrections are not required for fluids with a shear viscosity larger than about 50 mPa·s.

1.3

Rheometry

31

1.3.2.2 Filament Stretching Rheometer The filament stretching device for determining the steady uniaxial extensional viscosity was first developed by Tirtaatmadja and Sridhar (1995). A depiction of the instrument is shown in Fig. 1.8. The fluid sample is held between two disks which move apart at an increasing rate so that the extension rate along the filament midpoint is held constant. The instrument has the advantage over the opposed-et apparatus by producing a flow field which is a very close approximation to pure uniaxial extensional flow. The extensional stress is derived from the surface tension of the fluid, σ , reducing the calculation of extensional viscosity to ηEa =

σ σ/Rm =− , ε˙ 2dRm /dt

(1.93)

where Rm is the midpoint radius of the filament. The instantaneous deformation rate experienced by the fluid element at the axial midplane can be determined in a filament rheometer in real-time using the high resolution laser micrometer which measures Rm (t). Numerous experimental variants have been developed, and by precisely controlling the endplate displacement profile it is now possible to reliably attain the desired kinematics. The results can be best represented on a “master curve” showing the evolution of the imposed axial strain on the endplate versus the resulting radial strain at the midplane. The excellent review by McKinley and Sridhar (2002) surveys some of the recent developments in filament stretching extensional rheometry.

Fig. 1.8 Filament stretching rheometer

32

1 Non-Newtonian Fluids

1.3.3 Microrheology Measurement Techniques The traditional rheometers described so far measure the rheological properties of fluids using milliliter-scale material samples. In contrast, microrheology is tipically concerned with flows around microscale and nanoscale particles that are embedded within a very small (microliter or even nanoliter) volume of the test fluid. Moreover, conventional rheometers provide an average measurement of the bulk response, and do not allow for local measurements in inhomogeneous systems. To address these issues, a new type of microrheology measurement techniques has emerged. Two classes of microrheology tests can be distinguished. The first class of microrheology tests exploits the Brownian motion of the tracer particles and is termed passive microrheology. Because no external forces are applied, passive experiments always operate in the linear viscoelastic regime and are suitable for soft media. The second class uses active manipulation of the particles by applying an external driving force. These techniques include, for example, the use of magnetic or optical tweezers. Active methods are useful if large stresses have to be applied to stiff media as well as for investigations of nonlinear response and non-equilibrium phenomena. 1.3.3.1 Passive Measurement Methods Passive microrheology is based on an extension of the concepts of Brownian motion of particles in simple liquids. The motion of particles within a liquid can be quantified with the diffusion coefficient, D, which is a measure of how rapidly particles execute a thermally driven random walk. Given the particle size, temperature, and viscosity, η, the diffusion coefficient in a viscous liquid can be determined by the Stokes–Einstein relation D=

kB T , 6π aη

(1.94)

where kB is Boltzmann’s constant, a is the particle radius, and T is the absolute temperature. In the above equation, it is assumed that particles are spherical and rigid and no heterogeneities exist. The dynamics of particle motion are usually described by the time-dependent mean-square displacement, r2 (t). When particles diffuse through a test fluid or are transported in a non-diffusive manner the mean-square displacement becomes nonlinear with time and can be described with a time-dependent power law, r2 (t) ∝ tα . The slope of the log–log plot of the r2 (t), denoted by α (also referred to as the diffusive exponent), describes the mode of motion a particle is undergoing and is defined for physical processes between 0 ≤ α ≤ 2. The time-dependent mean-square displacement can be used to obtain rheological properties of a complex fluid microenvironment. The Stokes–Einstein relation correlates the particle radius, a, and the term r2 (t) provide the creep compliance D=

π a r2 (t) . kB T

(1.95)

1.3

Rheometry

33

The time-dependent creep compliance, or material deformation under a stepincrease in stress, can be directly obtained from the mean-square displacement. This provides a measure of the viscosity or the elastic modulus in viscous or elastic samples, respectively. In addition, a method was developed to estimate the elastic, storage modulus and the loss modulus based on the logarithmic slope of the mean-square displacement (Mason 2000). Another passive microrheology test is the fluorescence correlation spectroscopy which is based on the principles of dynamic light scattering. Fluorescence correlation spectroscopy uses a laser beam focused on a small volume within the test fluid and photon detectors to record fluctuations in fluorescence resulting from the movement of fluorescent molecules into and out of the volume (Hess et al. 2002). The method is well suited for the study of viscoelasticity within a cell. 1.3.3.2 Active Measurement Methods In addition to the passive techniques described above, fluids may be externally manipulated using active microrheology techniques. Externally applied forces, acting on particles in a test fluid, result in local stress and movement of the particles through more elastic regions. Active forces can be applied to particles in a test fluid through magnetic and laser tweezers. For example, paramagnetic and ferromagnetic microbeads can be manipulated by magnetic-field gradients and used to apply large forces in a viscoelastic fluid. Magnetic-field gradients applied to paramagnetic beads can generate pulling forces (Bausch et al. 1998; Karcher et al. 2003), whereas their application to ferromagnetic particles can generate torsional forces (Fabry et al. 2001, 2003). Forces up to 10 nN (Karcher et al. 2003) can be generated using paramagnetic beads, and forces of several pN (Trepat et al. 2007) can be generated using ferromagnetic beads. Similarly, laser tweezers have been used to manipulate particles, cells, and bacteria (Ashkin et al. 1987; Ashkin and Dziedzic 1987) by applying small forces to them and then measuring their displacements with high accuracy. Trapped particles can be restricted to a specific region and passively monitored (Tolic-Norrelykke et al. 2004), or an active force can be locally applied and its effects on internal structure measured. Laser tweezers have been used to trap spherical, polymeric particles or naturally occurring granules within cells (Tolic-Norrelykke et al. 2004). To measure rheological properties, optical tweezers are used to apply a stress locally by moving the laser beam and dragging the trapped particle through the surrounding material; the resultant bead displacement is interpreted in terms of viscoelastic response. Elasticity measurements are possible by applying a constant force with the optical tweezers and measuring the resultant displacement of the particle. Alternatively, local frequencydependent rheological properties can be measured by oscillating the laser position with an external steerable mirror and measuring the amplitude of the bead motion and the phase shift with respect to the driving force (Ou-Yang 1999). This method produces forces lower than 100 pN and can be used to measure the cell’s linear response (Peterman et al. 2003). Magnetic tweezers have the advantage over their optical counterparts that they generate no heat in the sample examined, can have a

34

1 Non-Newtonian Fluids

uniform force field over the entire field of view, and can orient objects regardless of their geometry. They do have the disadvantage that it is difficult to make multiple independent traps.

1.4 Particular Non-Newtonian Fluids We turn now to a particular class of non-Newtonian fluids, namely the biological fluids. They are rheologically complex due to their multi-component nature. The complexity of the biological fluids relies on the fact that such fluids are active, and can rearrange their microstructure to produce different properties, in order to achieve a precise function. As a result they are both elastic and viscous. The biological fluids whose rheology has been most studied are human blood, synovial fluid, saliva, and the cell constituents, cytoplasm and cytoskeleton. Blood is undoubtedly the most important biological fluid and its rheology is interesting from both theoretical and applied points of view.

1.4.1 Blood Blood is a suspension of cells in plasma. Plasma represents about 55% of the total blood volume. It is composed of mostly water (92% by volume), and contains dissolved proteins (6–8%), glucose, lipids, mineral ions, hormones, and carbon dioxide. Blood plasma has a density of approximately 1,025 kg/m3 (Lentner 1979). The cells are red blood cells (erythrocytes), white blood cells (leukocytes) of several types, and plateles. The red blood cells are biconcave disks, some 8.5 μm in diameter and of maximum thickness 2.5 μm. The cells consist of a highly flexible membrane, filled with a concentrated haemoglobin solution. The membrane, consisting of a lipid bilayer and a cytoskeleton (a network of protein molecules), exhibits viscoelastic properties. The elastic shear modulus (about 6×10–6 N/m) is several orders of magnitude lower that the modulus of isotropic dilatation (about 0.5 N/m) and so the membrane shears rapidly but resists area changes. Also, bending resistance is small unless very small radii of curvature are involved; the bending modulus is about 1.8×10–19 Nm (Evans 1983). Normal human blood has a hematocrit (volume fraction of red cells) of about 45%, and so red cells strongly influence the flow properties of blood. White blood cells are comparable in size to red blood cells but much less numerous. They are much stiffer than red blood cells and may contribute significantly to microvascular flow resistance (Schmid-Schönbein et al. 1981). There are several classes of white blood cells, e.g., granulocytes, which include neutrophils, basophils and eosinophils, monocytes, lymphocytes, macrophages, and phagocytes. They vary in size and properties, e.g., a typical inactivated neutrophil is approximately spherical in shape with a diameter of about 8 μm. The mechanical properties of white blood cells have been discussed by Schmid-Schönbein (1990). Platelets are discoid particles with a diameter of about 2 μm. Platelets, which are essential to the blood clotting process, are much

1.4

Particular Non-Newtonian Fluids

35

smaller than the red cells and do not contribute significantly to flow resistance. They are preferentially distributed near microvessel walls, probably as a result of hydrodynamic interaction with red cells (Tangelder et al. 1985). Under physiological conditions white blood cells occupy 1/600 of total cell volume while platelets occupy approximately 1/800 of total cell volume. The viscosity of plasma has been shown to be invariant with γ˙ (Newtonian fluid) and is dependent mainly on protein content and temperature. In normal conditions, the viscosity of plasma is about 1.1 cP. Whole blood is a non-Newtonian fluid. At small values of shear rate (γ˙ < 0.5 s–1 ), the apparent viscosity of whole blood shows a zero-shear plateau followed, at higher shear rates, by a decrease of viscosity with the shear rate. Blood is therefore a shear-thinning fluid. At very high values of shear rate (γ˙ > 102 s–1 ), the apparent viscosity of blood is almost constant indicating an infinite-shear plateau. At normal hematocrit content and at a temperature of 37◦ C, the zero-shear viscosity of blood is as high as 120 cP (Chmiel and Walitza 1980) while the infinite-shear viscosity is about 4 cP (MacKintosh and Walker 1973; Lowe and Barbenel 1988). The rheology of blood is primarily determined by the behaviour of the red blood cells at different shear rates. At sufficiently low shear rates the red blood cells agglomerate into column-like structures called rouleaux and the concentration of fibrinogen and immunoglobulins in the plasma is known to have an important role in this process (Baskurt and Meiselman 2003). At yet lower shear rates these rouleaux may develop branches and at even lower shear rates complex networks of red blood cells may be observed (Samsel and Perelson 1982; Baskurt and Meiselman 2003). At rest, human blood cells form a gel all over the sample and some researchers claim experimental evidence for the existence of a yield stress (Thurston 1993; Picart et al. 1998). Red cell aggregation or rouleaux formation is induced by many macromolecules, and particularly, fibrinogen and imunoglobulins contained in plasma (Brooks et al. 1970). On the other hand, increase in shear breaks down the bridging lattice and reduces the rouleaux length, with minimal aggregation for shear rate above 102 s–1 . As aggregates are broken down with increasing shear, the number of individual red cells increases. These align with the flow direction, causing further reduction in viscosity for larger values of shear rates. Chien (1970) has shown that for shear rate values up to 1 s–1 , aggregation dominates the viscous behaviour, whereas in the range 1–100 s–1 , deformation of red blood cells is the dominant factor. A comparison of this shear thinning characteristic of blood viscosity in the presence and absence of aggregating agents suggests that about 75% of the viscosity decrease is a result of the disruption of red cell aggregates, and 25% is due to red cell deformation in response to increased shear stresses (Lipowski 2005). At a given shear rate, blood viscosity rises exponentially with increasing red blood cell concentration (hematocrit) to a degree dependent on prevailing γ˙ . Blood viscosity is higher in men than in women because of the men’s higher hematocrit level. Furthermore, blood viscosity decreases with increasing temperature (Eckmann et al. 2000). The blood viscosity decreases with decreasing the diameter of the vessel. Fahreus and Lindqvist (1931) were the first to indicate that the flow resistance of blood in a cylindrical tube cannot be predicted on the basis of the viscosity of the blood as

36

1 Non-Newtonian Fluids

measured in large scale rheometers. A large number of publications has addressed the dependence of apparent blood viscosity on tube diameter and hematocrit. The results of 18 studies were combined to a parametric description of apparent blood viscosity relative to the viscosity of plasma (relative apparent blood viscosity) as a function of tube diameter, D, and hematocrit, H, according to the equation (Pries et al. 1992), η = 1 + (0.45 − 1)

(1 − H)c − 1 . (1 − 0.45)c − 1

(1.96)

Here, η0.45 the relative apparent blood viscosity for a fixed hematocrit of 45%, is given by η0.45 = 220e−1.3D + 3.2 − 2.44e−0.06D

0.645

,

(1.97)

and c describes the shape of the viscosity dependence on hematocrit

−0.075D

c = 0.8 + e

1 −1 + 1 + 10−11 D12

+

1 . 1 + 10−11 D12

(1.98)

The apparent blood viscosity exhibits a strong decrease with decreasing tube diameter reaching a minimum at about 7 μm. At this value, the apparent viscosity of blood with a hematocrit of 45% is only 25% higher than that of plasma. Only at diameters below about 3.5 μm does the apparent viscosity increase above the level seen in large vessels. In the absence of shear, red blood cells only collide rarely so that aggregation tends to be a slow process (Samsel and Perelson 1982). Aggregation and disaggregation take place over differing non-zero time scales. Blood is therefore thixotropic (Huang et al. 1975), in the sense that when a step increase in shear rate is applied to blood the viscosity is a decreasing function of time. Blood thixotropy is exhibited mostly at low shear rates (up to 10 s–1 ) (Huang et al. 1987) and has a fairly long time scale. For example, initial resistance to flow start-up returns only slowly to normal blood, with well over 1 min of standing required (Mewis 1979). This suggests that thixotropy is of secondary importance in pulsatile blood flow, which has a time scale of approximately 1 s. Whole blood viscosity is an important physiological parameter (see, for a detailed discussion, Baskurt 2003 and the references therein). For example, the viscosity of whole blood was associated with coronary arterial diseases. Whole blood viscosity is significantly higher in patients with peripheral arterial disease than that in healthy controls. Other researchers investigated correlation between the hemorheological parameters and stroke. They reported that stroke patients showed two or more elevated rheological parameters, which included whole blood viscosity, plasma viscosity, red blood cell and plate aggregation, red blood cell rigidity, and hematocrit. It was also reported that both whole blood viscosity and plasma viscosity are significantly higher in patients with essential hypertension than in

1.4

Particular Non-Newtonian Fluids

37

healthy people. In diabetics, whole blood viscosity, plasma viscosity, and hematocrit are elevated, whereas red blood cell deformability is decreased. There is also a direct connection between whole blood viscosity and smoking, age, and gender. It was found that smoking and aging might cause the elevated blood viscosity. In addition, it was reported that male blood possessed higher blood viscosity, red blood cell aggregability, and red blood cell rigidity than premenopausal female blood, which may be attributed to monthly blood-loss. Blood is also a viscoelastic fluid. The deformations of the erythrocytes in flow and the storage and release of elastic energy that this implies, as well as the dissipation in blood due primarily to evolution of the erythrocyte networks (at low shear rate) and internal friction (at higher shear rates) (Anand and Rajagopal 2004), give rise to the viscoelastic character of blood (Chien et al. 1975). At normal hematocrit values, the viscous component, η , of the complex viscosity predominates over the elastic component, η (Chmiel and Walitza 1980). This suggests that blood viscoelasticity also has a secondary impact on blood flow at physiological hematocrit values. Thurston (1979) indicated that both components of the complex viscosity have relatively constant values for shear rates below 1.5 s–1 . In this range, the elastic and viscous components are approximately 3.9 mPa·s and 11.5 mPa·s, respectively. As the shear rate is increased beyond this level, blood displays a nonlinear viscoelastic behaviour, i.e. η and η are dependent on shear rate. The value of η starts dropping rapidly as the shear rate is increased beyond this level, diminishing to 0.1 mPa·s by 16 s–1 . This sharp decrease is connected to the breakdown of the blood microstructure formed by red blood cell aggregates. The speed of sound in blood was investigated by Bakke et al. (1975), where the following equation is given for the dependence of whole blood on hematocrit at a temperature of 37◦ C c = 1541.82 + 0.98 × H,

(1.99)

where c is sound speed in m/s and H is the hematocrit. At normal hematocrit content, this equation gives a value of c of 1,586 m/s. This is in close agreement with values reported by other authors; e.g., 1,590 m/s (Hughes et al. 1979) and 1,584.2 m/s (Collings and Bajenov1987). The sound speed in whole blood increases with temperature at a rate of c T = 1.3 m/s/◦ C. The surface tension of whole blood at normal hematocrit content is 5.6×10–2 N/m (Lentner 1979). This value was, however, measured at 24◦ C and no data at normal body temperature are available in literature. The density of whole blood is approximately 1,060 kg/m3 (Lentner 1979).

1.4.2 Synovial Fluid Synovial fluid is found in the diarthrodial joints where it forms a thin viscous film over the surface of the synovium and articular cartilage in the joint space. The composition of the synovial fluid is almost identical to that of plasma with

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the exception of the large polymers like fibrinogen or larger globulins, which are reduced or not found at all in the synovial fluid due to the sieving action of the synovial capillary walls. Synovial fluid can be distinguished from plasma by the presence of hyaluronic acid and lubricin. These two molecules are the major determinants of synovial fluid viscosity and are of key importance for one of its main functions, which is to act as lubricant of the joint surfaces. Hyaluronic acid, or hyaluronan, is the most abundant glycosaminoglycan in mammalian tissue. It is present in high concentrations in connective tissue, such as skin, vitreous humor, cartilage, and umbilical cord, but the largest single reservoir is the synovial fluid of the diarthrodial joints, where concentrations of 0.5–4 mg/ml are achieved (Laurent and Fraser 1992; Fraser et al. 1997). The high concentration of hyaluronic acid in synovial fluid is essential for normal joint function, because hyaluronan confers exceptional viscoelasticity and lubricating properties to synovial fluid, particularly during high shear conditions. Under dynamic loading of diarthrodial joints, shear thinning and a reduction in viscosity occur because of decreased physical entanglements of hyaluronan molecules and their realignment to directions more parallel with the axis of articulation. These unique non-Newtonian rheological properties of hyaluronan not only reduce wear and attrition of articular cartilage during joint motion (Balazs et al. 1967; Balazs and Denlinger 1985) but also stabilize joints at low shear rates (Cullis-Hill and Ghosh 1987). At high loads, however, hyaluronan is not an effective lubricant. Here lubricin, which interacts with surfaceactive phospholipids, seems to have an important role (Simkin 1985). Furthermore, hyaluronan in the synovial fluid also bonds the opposing surfaces of the joints to each other. This creates tensile strength with little or no shear strength and enables opposing surfaces to slide freely across each other but limits their distraction (Wooley et al. 2005). The volume of the synovial fluid in normal human joints is quite small with approximately 0.5–2.0 ml (Dewire and Einhorn 2001; Mason et al. 1999). The synovial fluid undergoes continuous turnover by trans-synovial flow into synovial lymph vessels. As a result, water and protein in the synovial fluid are replaced within a period of 2 h or less. The turnover of hyaluronan is considerably slower with complete replacement of hyaluronan within about 38 h (Mason et al. 1999). The density of the synovial fluid is ρ = 1008–1015 kg/m3 (Duck 1990). Rheological studies have documented three types of non-Newtonian properties for the synovial fluid: shear-thinning, elasticity, and rheopexy. King (1966) seems to be the first who tested synovial fluids of bullocks using a cone-and-plate rheometer. He found that, at very small shear rates (<10–1 s–1 ), the apparent viscosity of the synovial fluid from the knee joint is constant at a value of about 10 Ns/m2 and then decreases with increasing shear rate. King also observed that the knee joint fluid has a larger apparent viscosity than that of ankle fluid. For example, the apparent viscosity of the ankle joint fluid shows a zero shear viscosity of about 10–1 Ns/m2 which manifests for values of the shear rate of up to 1 s–1 . At very large values of the shear rate the apparent viscosity of both knee and ankle joint fluids tends to become equal to that of water. Similar rheological

1.4

Particular Non-Newtonian Fluids

39

studies were later conducted by Davies and Palfrey (1968) using synovial fluids obtained from patients with joint effusions. They confirmed the shear-thinning character of synovial fluid viscosity and presented some evidence that the lowest values of the apparent viscosity were obtained in the case of synovial fluids taken from patients with rheumatoid arthritis. Synovial fluid rheology has long been utilized in the study of rheumatic diseases (see, for example, ScottBlair et al. 1954) and significantly lower viscosities are often noted in these cases (Gomez and Thurston 1993; Schurz 1996). Under inflammatory conditions of arthritic diseases, such as osteoarthritis or rheumatoid arthritis, high molar mass hyaluronic acid is degraded by reactive oxygen species leading to a reduction of the synovial fluid viscosity. The smaller viscosity of the synovial fluid impairs its lubricant and shock absorbing properties leading finally to deteriorated joint movement (Soltés et al. 2006). Dynamic tests on the rheological properties of synovial fluid were conducted by Balazs and his co-workers using both healthy and arthritic human synovial fluids (Balazs 1968; Gibbs et al. 1968). They found that the synovial fluids of both healthy young and old subjects when exposed to shear stress at low frequencies behave as viscous fluids, but when the frequency increases, the fluids become more and more elastic. At a given frequency, the values of the storage modulus G and the loss modulus G become equal and the transition from the viscous fluid to elastic body occurs. For healthy young persons (27–34 years) the cross-over value is about 0.2 Hz, but increases to about 1 Hz for old persons (52–78 years). Similar results were reported by Rwei et al. (2008). The importance of this transition from viscous to an elastic state is that it occurs at a frequency rate that is present in joints under various types of movement. When a person loads the knee in a standing up position, the input frequency is low and the viscous component of the synovial fluid predominates over the elastic one. When the person is walking, running, or jumping, the input frequency becomes faster and faster and the elastic component becomes dominant. It absorbs the mechanical energy and thereby protects the cartilage and the synovial cells from mechanical damage. Balazs and co-workers also noted that the synovial fluid from healthy young donors shows more elasticity than that of old persons. On the other hand, the arthritic human synovial fluid loses almost all of its elastic properties. They explained this result by the lower concentration, lower average molecular weight, and changes in the conformation of the hyaluronan molecules in the arthritic joint fluid. Oates et al. (2006) indicated that, at small shear rates (γ˙ ≤ 10s−1 ), the synovial fluid is rheopectic, i.e. the stress grows in time during steady shear. At high shear rates, rheopexy was still observed but it is significantly less pronounced than at low shear rates. This flow characteristic is caused by protein aggregation, and the total stress is enhanced by entanglement of this tenuous protein network with the long-chain polysaccharide sodium hyaluronate under physiological conditions. This structure builds faster under quiescent conditions and applications of steady shear retards structure growth, with slower rates of structure growth at higher shear rates.

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1.4.3 Saliva Saliva is mainly composed of water (99.5%), proteins (0.3%) and inorganic and trace substances (0.2%) (Humphrey and Williamson 2001; van Nieuw Amerongen et al. 2005). The proteins in saliva are mainly constituted by glycoproteins, enzymes, immunoglobulins, and a wide range of peptides, such as cystatins, statherin, histatins, proline-rich proteins, with antimicrobial activities (van Nieuw Amerongen et al. 2005). The inorganic fraction of saliva contains the usual electrolytes (sodium, potassium, chloride and bicarbonate) of the body fluids but at different concentrations making saliva a hypotonic fluid. Saliva is produced by the contra-lateral major glands, i.e. parotidglands (Par), submandibular glands (SM), and sublingual glands (SL), and minor salivary glands present in the mucosa of the tongue (von Ebner glands), cheeks, lips and palate (Pal) (Silvers and Som 1998; Young and van Lennep 1978). Whole saliva is formed primarily from salivary gland secretions, but also blood, oral tissues, microorganisms, and food remnants can be contributors to the salivary fluid (Schipper et al. 2007). Whole saliva is also a pseudoplastic fluid, i.e. the viscosity of saliva decreases upon increasing shear rate (Schwartz 1987; Vissink et al. 1984; van der Reijden et al. 1993; Levine et al. 1987). For example, the apparent viscosity of whole saliva at a shear rate of γ˙ = 0.02s−1 is η = 100 mPa·s and decreases at η = 2.5 mPa·s at γ˙ = 95s−1 (van der Reijden et al. 1993). The main reason for the shear thinning character of whole saliva is the presence of large glycoproteins, like mucins, causing a weak gel character of saliva (Veerman et al. 1989). This is supported by different studies showing that the viscosity of Par saliva, which does not contain high molecular weight mucins, is shear rate independent with a viscosity slightly higher than that of water (η = 1.3 mPa·s at γ˙ = 230s−1 ) (Veerman et al. 1989; Levine et al. 1987). In addition, treatment of saliva by homogenisation destroys the weak gel and resulted in a 3 to 4 fold decrease in viscosity (Veerman et al. 1989). Another parameter affecting saliva viscosity is the pH. Lowering the pH resulted in a decrease in viscosity of whole unstimulated saliva and a small viscosity increase of stimulated saliva (Nordbo et al. 1984). The viscous component η of viscoelasticity of whole saliva dominates the elastic component η. Reported values in literature are η = 1.5 mPa·s and η = 0.6 mPa·s at shear rates between 1 and 300 s–1 and at a temperature of 37◦ C (Levine et al. 1987). The viscosity of SM and Pal saliva was shown to be hardly dependent on the shear rate opposite to SL saliva showing a clear shear-thinning behaviour (Levine et al. 1987). Moreover, at similar viscosity, SM saliva has a lower elasticity than SL saliva. Typical values are η = 0.4–0.6 mPa·s for SM saliva and η = 1.8–4.9 mPa·s for SL saliva (Levine et al. 1987). The high viscosity at low shear rates of SL saliva prevents dehydration of the mucosa of the floor of the mouth. On the other hand, the high elasticity of SL saliva, in combination with appropriate adhesion to the oral mucosa, may provide a high retention of SL saliva. Rantonen and Meurman (1998) investigated the effect of collection time and within-subject variations of viscosity and flow rate of whole saliva. Unstimulated saliva viscosity showed the minimum viscosity at the end of the afternoon

1.4

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41

(measuring times from 8 to 20 h) and significant within-subject variation, whereas no effect of day time was seen for stimulated saliva viscosity. Flow rate and viscosity were positively correlated for both stimulated and unstimulated saliva. It is also interesting to note that saliva viscosity decreases upon storage within a few hours (Kusy and Schafer 1995). Bacterial glycosidases and proteases breakdown the macromolecular organisation of mucins leading to the formation of a protein precipitate accompanied by a decrease in viscosity and agglutinating activity (Sato et al. 1983). However, stimulated saliva viscosity was found to be enzymatically stable upon storage at room temperature for at least 30 min (Rantonen and Meurman 1998). Zussman et al. (2007) measured the relaxation time λ of saliva secreted from the different glands, at rest or under stimulation and at different ages, in an uniaxial extensional flow. They found that submandibular/sublingual salivary elasticity was significantly higher than that of parotid saliva, especially under stimulation. For example, the values of the relaxation time at rest are λ = 1 ms for whole saliva, λ = 3.58 ms for submandibular/sublingual saliva, and λ = 1.08 ms for parotid saliva. The corresponding values obtained under stimulation are λ = 3.46 ms for whole saliva, λ = 18.7 ms for submandibular/sublingual saliva, and λ = 1.31 ms for parotid saliva. They noted that the significant difference in the elasticity of the parotid and submandibular/sublingual saliva may have resulted from the difference in their protein profiles, such as mucins and glycoproteins, which are much more prevalent in submandibular/sublingual saliva. In addition, an age-related increase in relaxation time was demonstrated. This increased viscoelasticity of whole saliva in the elderly may result from a reduction in salivary watery content, which results in increased salivary protein concentration.

1.4.4 Cell Constituents Cells are the fundamental structural and functional unit of tissues and organs. There are about 200 different types of cells in the human body. A typical cell consists of a membrane, a cytoplasm (i.e. fluid-like cytosol, structural cytoskeleton and dispersed organelles), and a nucleus which contains the chromosomal DNA. The cell membrane consists primarily of a phospholipid bilayer with many embedded proteins that serve a host of functions: channels, gates, anchoring sites, and receptors for target molecules. In many cells, the structural integrity of the cell membrane is augmented by a sub-membranous cortical network or layer of actin filaments. The cytosol makes up about 70% of the cell volume and is composed of water, salts and organic molecules. The cytoskeleton is a dynamic structure that maintains cell shape, protects the cell, enables cellular motion, and plays important roles in both intracellular transport and cellular division. Actin filaments, intermediate filaments and microtubules are the three primary structural proteins of the cytoskeleton. Specifically, actin filaments are about 7–9 nm in diameter and thought to be extensible and flexible. They form by the polymerization of globular, monomeric actin (G-actin) into a twisted strand of filamentous

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actin (F-actin) having a barbed end and a pointed end. The filament grows at the barbed end, whereas polymerization occurs preferentially at the pointed ends. Intermediate filaments are often described as rope-like structures about 10 nm in diameter that appear to play an important structural role throughout the cytoplasm. Microtubules exist as long cylinders about 25 nm in diameter, and they appear to have a higher bending stiffness than the other two primary filaments. They are polymerized filaments constructed from monomers of α- and β-tubulin in a helical arrangement. Microtubules are highly dynamic, undergoing constant polymerization and depolymerization, so that their half-lives are typically only a few minutes (Mitchison and Kirschner 1984). Organelles are membrane-bound compartments within the cell that have specific functions. For example, mitochondria provide the cell with usable energy to perform its many functions. The smooth and rough endoplasmic reticulum are sites for the synthesis of proteins, lipids and steroids. Finally, the lysosomes and perioxisomes are responsible for the hydrolytic degradation of various substances within the cell. The ability of a cell to produce and remove various substances within the confines of its cell membrane as well as within the extracellular matrix in which it resides is fundamental to much of its activity. The rheological characteristics of the cell cytoplasm have been studied by several methods, and some of them are discussed in Sect. 1.3.3. One popular approach to measure cytoplasmic viscosity was the direct observation of the displacement of microinjected submicronic magnetic particles and macromolecules in the cell (see, for example, Valberg and Albertini 1985; Dembo and Harlow 1986). Valberg and Feldman (1987) reported a comprehensive study of the cytoplasmic viscosity of pulmonary macrophages. They found that the apparent viscosity at higher shear rates (5×10–2 s–1 ) is 254 Pa·s, while at very low shear rates (10–3 s–1 ) the corresponding value is 2,745 Pa·s. The marked shear dependence of η indicates that the cytoplasm is distinctively non-Newtonian with a pseudoplastic characteristic of apparent viscosity. Similar results have been later reported by Bausch et al. (1998) for 3T3 murine fibroblasts (η = 2×103 Pa·s), Bausch et al. (1999) for J774 macrophages (η = 2×102 Pa·s), and Sato et al. (1984) for the cytoplasmic viscosity of the axoplasm of squid axon (η = 104 –105 Pa·s). The cytoplasm of other cell types, such as, leukocytes and neutrophils, has similar properties (Evans and Yeung 1989; Heidemann et al. 1999). However, large regional variations in viscosity have been found within a cell (Yanai et al. 1999; Laurent et al. 2005). These high values are a consequence of the mechanical barriers imposed by the mesh-like structure of the cytoskeletal network. Much smaller values were reported when only the aqueous domain of cell cytoplasm was investigated. Typical values vary in the range 2×10–3 to 5×10–2 Pa·s (Lepock et al. 1983; Mastro and Keith 1984). The rheology of cytoplasmic extracts prepared from Xenopus laevis eggs was investigated by Valentine et al. (2005). At macroscopic length scales, cytoplasm is a soft viscoelastic solid with an elastic modulus in the range of 2–10 Pa, and a considerable viscous modulus of 0.5–5 Pa. Actin and microtubules cooperate to withstand shear deformation. Disruption of the microtubule network significantly weakens the elastic response, and the disassembly of actin filaments completely

References

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prevents gelation. To measure the microscopic properties of the extract, they used a multiple particle tracking technique to observe the thermal motions of embedded colloidal particles. At microscopic length scales, the elastic filaments do not contribute and the sample is predominantly viscous with an apparent viscosity of about 20 mPa·s. Fabry et al. (2001, 2003) performed dynamic tests using magnetic twisting cytometry to measure the frequency dependence of the storage modulus, G (ω), and the loss modulus, G (ω), of the cytoskeleton. The storage modulus was observed to increase with increasing frequency, ω, according to a power law, G ∝ ω0.2 . The loss modulus also increases with increasing frequency and follows the same power law in the range of 0.01–10 Hz. Above 10 Hz, however, the same power-law behaviour was not observed. For ω < 103 Hz, the storage modulus is larger than the loss modulus indicating that the elastic component dominates the viscoelastic properties of the cytoplasm. For ω > 103 Hz, the loss modulus becomes dominant. Experiments were also performed by introducing various drugs into the cell in order to create contraction or relaxation in the cytoskeleton. Similar qualitative properties were observed. The storage modulus increased with increasing frequency as a power law and the loss modulus also increased with increasing frequency with the same power law and same exponent up to frequencies of 10 Hz.

1.4.5 Other Viscoelastic Biological Fluids Some other biological fluids have non-Newtonian properties as well. For example, mucus from the respiratory tract is a viscoelastic fluid. Its viscoelasticity is influenced by bacteria and bacterial DNA. Cervical mucus and semen are other examples of viscoelastic biological fluids. A rheological description of these fluids is, however, beyond the scope of this book because there is no biomedical or bioengineering application where cavitation might occur in these fluids. The reader interested in the viscoelastic properties of these fluids are referred to the book of Fung (1993).

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Macosko,C.W. 1994 Rheology: Principles, Measurements and Applications. VCH Publishers, New York. Mason, T. G. 2000 Estimating the viscoelastic moduli of complex fluids using the generalized Stokes-Einstein equation. Rheol. Acta. 39, 371–378. Mason, R.M., Levick, J.R., Coleman, P.J., Scott, D. 1999 Biochemistry of synovium and synovial fluid. In Biology of the Synovial Joint (Eds. C.W. Archer, M. Benjamin, B. Caterson, and J.R. Ralphs). Harwood Academic, Amsterdam, pp. 253–264. Mastro, A.M., Keith, A.D. 1984 Diffusion in the aqueous compartment. J. Cell Biol. 99, 180–187. McKinley, G.H., Sridhar, T. 2002 Filament-stretching rheometry of complex fluids. Annu. Rev. Flui Mech. 34, 375–415. Mewis, J. 1979 Thixotropy – a general review. J. Non Newt. Fluid Mech. 6, 1–20. Mitchison, T., Kirschner, M. 1984 Dynamic instability of microtubule growth. Nature 312, 237–242. Nordbo, H., Darwish, S., Bhatnagar, R.S. 1984 Salivary viscosity and lubrication: influence of pH and calcium. Scand. J. Dent. Res. 92, 306–314. Oates, K.M.N., Krause, W.E., Jones, R.L., Colby, R.H. 2006 Rheopexy of synovial fluid and protein aggregation. J. R. Soc. Interface 22, 167–174. Ou-Yang, H.D. 1999 Design and applications of oscillating optical tweezers for direct measurements of colloidal forces. In Colloid-Polymer Interactions: From Fundamentals to Practice (Eds. R.S. Farinato and P.L. Dubin). Wiley, New York, pp. 385–405. Papanastasiou, A.C., Scriven L.E., Macosko, C.W. 1983 An integral constitutive equation for mixed flows: viscoelastic characterization. J. Rheol. 27, 387–410. Peterman, E.J.G., van Dijk, M.A., Kapitein, L.C., Schmidt, C.F. 2003 Extending the bandwidth of optical-tweezers interferometry. Rev. Sci. Instrum. 74, 3246–3249. Picart, C., Piau, J.-M., Galliard, H., Carpentier, P. 1998 Human blood shear yield stress and its hematocrit dependence. J. Rheol. 42, 1–12. Poslinski, A.J., Ryan, M.E., Gupta, R.K., seshadri, S.G., Frechette, F.J. 1988 Rheological behaviour of filled polymeric systems. 1. Yield stress and shear-thinning effects. J. Rheol. 32, 703–735. Pries, A.R., Fritzsche, A., Ley, K., Gaehtgens, P. 1992 Redistribution of red blood cell flow in microcirculatory networks by hemodilution. Circ. Res. 70, 1113–1121. Rantonen, P.J.F., Meurman, J.H. 1998 Viscosity of whole saliva. Acta Odontol. Scand. 56, 210–214. Rwei, S.P., Chen, S.W., Mao, C.F., Fang, H.W. 2008 Viscoelasticity and wearability of hyaluronate solutions. Biochem. Eng. J. 40, 211–217. Samsel, R.W., Perelson, A.S.1982 Kinetics of rouleau formation. Biophys. J. 37, 493–514. Sato, M., Wong, T.Z., Brown, D.T., Allen, R.D. 1984 Rheological properties of living cytoplasm: a preliminary investigation of squid axoplasm (Loligo pealei). Cell Motil. 4, 7–23. Sato, S., Koga, T., Inoue, M. 1983 Degradation of the microbial and salivary components participating in human dental plaque formation by proteases elaborated by plaque bacteria. Arch. Oral Biol. 28, 211–216. Schipper, R.G., Silletti, E., Vingerhoeds, M.H. 2007 Saliva as research material: biochemical, physicochemical and practical aspects. Arch. Oral Biol. 52, 1114–1135. Schurz, J. 1996 Rheology of synovial fluids and substitute polymers. J. Mat. Sci. Pure Appl. Chem. A 33, 1249–1262. ScottBlair, G.W., Williams, P.O., Fletcher, E.T.D., Markham, R.L. 1954 On the flow of certain pathological human synovial effusions through narrow tubes. Biochem. J. 56, 504–508. Schmid-Schönbein, G.W. 1990 Leukocyte biophysics. Cell Biophys. 12, 107–135. Schmid-Schönbein, G.W., Sung, K.P., Tözeren, H., Skalak, R., Chien, S. 1981 Passive mechanical properties of human leukocytes. Biophys. J. 36, 243–256. Schwartz, W.H. 1987 The rheology of saliva. J. Dent. Res. 66, 660–664. Silvers, A.R., Som, P.M. 1998 Salivary glands. Radiol. Clin. North Am. 36, 941.

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Simkin, P.A. 1985 Synovial physiology. In Arthritis and Allied Conditions. A Textbook of Rheumatology (Ed. D.J. McCarty). Lea and Febiger, Philadelphia, pp. 196–209. Soltés, L., Mendichi, R., Kogan, G., Schiller, J., Stankovská, M., Arnhold, J. 2006 Degradative action of reactive oxygen species on hyaluroran. Biomacromolecules 7, 659–668. Tangelder, G.J., Teirlinck, H.C., Slaaf, D.W., Reneman, R.S. 1985 Distribution of blood-platelets flowin in arterioles. Am. J. Physiol. 248, H318–H323. Tirtaatmadja, V., Sridhar, T. 1995 A filament stretching device for measurement of extensional viscosity. J. Rheol. 37, 1081–1102. Thurston, G.B. 1979 Erythrocity rapidity as a factor in blood rheology. Viscoelastic dilatancy. J. Rheol. 23, 703–719. Thurston, G.B. 1993 The elastic yield stress of human blood. Biomed. Sci. Instrum. 29, 87–93. Tolic-Norrelykke, I. M., E. L. Munteanu, E.L., Thon, G., Oddershede, L., Berg-Sorensen, K. 2004 Anomalous diffusion in living yeast cells. Phys. Rev. Lett. 93, 078102. Trepat, X., Deng, L., An, S.S., Navajas, D., Tschumperlin, D.J., et al. 2007 Universal physical responses to stretch in the living cell. Nature 447, 592–595. Valberg, P.A., Albertini, D.F. 1985 Cytoplasmic motions, rheology, and structure probed by a novel magnetic particle method. J. Cell Biol. 101, 130–140. Valberg, P.A., Feldman, H.A. 1987 Magnetic particle motions within living cell – Measurement of cytoplasmic viscosity and motile activity. Biophys. J. 52, 551–561. Valentine, M.T., Perlman, Z.E., Mitchison, T.J., Weitz, D.A. 2005 Mechanical properties of Xenopus egg cytoplasmic extracts. Biophys. J. 88, 680–689. van der Reijden, W.A., Veerman, E.C.I., van Nieuw Amerongen, A. 1993 Shear rate dependent viscoelastic behaviour of human glandular salivas. Biorheology 30, 141–152. van Nieuw Amerongen, A., Bolscher, J.G.M., Veerman, E.C.I. 2005 Salivary proteins: protective and diagnostic value in cariology? Caries Res. 38, 247–253. Veerman, E.C.I., Valentijn-Benz, M., van Nieuw Amerongen, A. 1989 Viscosity of human salivary mucins: effect of pH and ionic strength and role of sialic acid. J. Biol. Buccale 17, 297–306. Vissink, A., Waterman, H.A., Gravenmade, E.J., Panders, A.K., Vermey, A. 1984 Rheological properties of saliva substitutes containing mucin, carboxymethylcellulose or polyethylenoxide. J. Oral Pathol. 13, 22–28. Wagner, M.H. 1976 Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt. Rheol. Acta 15, 136–142. Wagner, M.H., Demarmels, A. 1990 A constitutive analysis of extensional flows of polyisobutylene. J. Rheol. 34, 943–958. White, J.L. 1995 Rubber Processing: Technology, Materials, Principles. Hanser, Cincinnati. Wooley, P.H., Grimm, M.J., Radin, E.L. 2005 The structure and function of joints. In Arthritis and Allied Conditions. A Textbook of Rheumatology (Eds. W.J. Koopman and L.W. Moreland). Lippincott Williams and Wilkins, Philadelphia, pp. 149–173. Yanai, M., Butler, J.P., Suzuki, T., Kanda, A., Kurachi, M., Tashiro, H., Sasaki, H. 1999 Intracellular elasticity and viscosity in the body, leading, and trailing regions of locomoting neutrophils. Am. J. Physiol. 277, C432–C440. Young, C.Y., Missel, P.J., Mazer, N.A., Benedek, G.B. 1978 deduction of micellar shape from angular dissymmetry measurements of light scattered from aqueous sodium didecyl sulfate solutions at high sodium chloride concentrations. J. Phys. Chem. 82, 1375–1378. Young, J.A., van Lennep, E.W. 1978 The morphology of the salivary glands. Academic Press, London. Zussman, E., Yarin, A.L., Nagler, R.M. 2007 Age- and flow-dependency of salivary viscoelasticity. J. Dent. Res. 86, 281–285.

Chapter 2

Nucleation

Cavitation is critically dependent on the existence of nucleation sites. Cavitation starts when these nuclei enter a low-pressure region where the equilibrium between the various forces acting on the nuclei surface cannot be established. As a result, bubbles appear at discrete spots in low-pressure regions, grow quickly to relatively large size, and suddenly implode as they are swept into regions of higher pressure. In most conventional engineering contexts, the prediction and control of nucleation sites is very uncertain even when dealing with a simple liquid like water. Here we present data on the nuclei distribution in more complex fluids, such as polymer aqueous solutions and blood.

2.1 Nucleation Models Nucleation is the onset of a phase transition in a small region of a medium. The phase transition can be the formation of a tiny bubble in a liquid or of a droplet in saturated vapour. There are two main types of nucleation models: homogeneous nucleation and heterogeneous nucleation. Homogeneous nucleation takes place in a liquid phase without the prior presence of additional phases (Fuerth 1941; Church 2002). It is a consequence of the distribution of thermal energy among the molecules comprising a volume of liquid. Because some molecules will be more energetic than others, random processes will occasionally produce groupings of higher energy molecules. If the average energy is high enough, such a grouping of molecules represents an inclusion consisting of gas and vapour in the bulk of the liquid. Because statistical fluctuations in the distribution of thermal energy occur continuously, the small gas or vapour inclusions are constantly forming and disappearing. Such cavitation nuclei are, however, unstable. A gas bubble will dissolve in an undersaturated solution and the effect of surface tension will cause it to dissolve in a saturated solution. In supersaturated solutions, a bubble can be in equilibrium because the tendency for the bubble to dissolve due to surface tension is opposed by the tendency for the bubble to grow by diffusion of gas into it. This equilibrium is unstable; the bubble will grow or dissolve depending on whether the perturbation increases or decreases the bubble’s radius relative E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_2,

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to its equilibrium radius (Epstein and Plesset 1950). Therefore, a liquid would be free of bubbles after a short period of time. This does not imply that gas bubbles could not serve as cavitation nuclei. It does imply, however, that in order for gas bubbles to serve as cavitation nuclei, they must be stabilized at a size small enough to prevent their rising to the surface of the liquid, yet large enough so that they will grow when exposed to negative pressure as low as a few bars. In other words, a stabilization mechanism must exist for a gas bubble before it can act as a cavitation nucleus. Various types of stabilizing skins have been proposed. These skins usually consist of contaminants which somehow deposit themselves on the bubble’s surface and counteract the surface tension. Fox and Herzfeld (1954) proposed that surface active organic molecules could form a rigid skin around a gas bubble. This skin would be impermeable to gas diffusion and would be mechanically strong enough to withstand moderate hydrostatic pressures. Rather than the rigid skin of organic molecules proposed by Fox and Herzfeld, Yount (1979, 1982) has developed a stabilization theory in which the dissolution of gas bubbles is halted by a non-rigid organic skin. This so-called varying-permeability model, which employs a skin of surface-active molecules to stabilize the nucleus, has been mainly applied to bubble formation in supersaturated liquids. Although this model was originally used to explain bubble formation in gelatin upon rapid decompression, Yount noted that it can be applied to bubbles in water as well. It can also be applied to polymer solutions because polymers have surfactant properties. Surfactants present a barrier to mass transport and reduce the surface tension at a liquid-gas interface (Borwankar and Wassan 1983). Both mechanisms increase the stability of nuclei against dissolution (Porter et al. 2004). Furthermore, since surfactants are prelevant in biological systems, this model may be particularly important for applications involving decompression sickness and medical ultrasonics. In heterogeneous nucleation small pockets of gas are stabilized at the bottom of the cracks or crevices found on hydrophobic solid impurities in the liquid (Strasberg 1959; Apfel 1970; Atchley and Prosperetti 1989). Liquids normally contain a large number of solid impurities with a very irregular surface consisting of grooves or pits (Crum 1979). As is schematically shown in Fig. 2.1, a crevice stabilized gas nucleus can have an interface that is concave towards the liquid. Due to surface tension, the pressure of the gas in the nucleus can therefore be less than the pressure in

Fig. 2.1 The crevice model of nucleation: (a) Stabilization mechanism of nuclei. (b) Nucleus starts to grow into a bubble when the pressure in the surrounding liquid is reduced

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Nucleation Models

51

the liquid, and if gas diffuses from the nucleus, so long as the contact line is pinned, the concavity will increase, reducing the pressure of gas. Hence such a nucleus can persist without dissolving completely into the liquid. The origin of such nuclei has been explained by considering the flow of a liquid onto a hydrophobic surface with crevices (Atchley and Prosperetti 1989). The crevice model is useful for explaining the hysteresis effect of pressurization on cavitation threshold (Crum 1980). The cavitation threshold increases because pressurization causes the crevice to shrink and gas diffuses into the surrounding liquid. After the pressure is released, a smaller pocket of gas exists in the crevice requiring a larger negative pressure to produce nucleation. Another explanation of the origin and persistence of nuclei is that ordering of liquid molecules adjacent to solid surfaces leads to local hydrophobicity in regions of concavity of an otherwise non-hydrophobic surface (Mørch 2000). This explanation suggests that the resulting voids have interfaces which are convex toward the liquids, and that their persistence is due to a resonant behaviour forced by ambient vibrations. Cavitation nuclei are not always permanently stabilized. Short-lived nuclei can also formed by radiation. Although many theories have been proposed to explain this phenomenon, the one that seems to have the most experimental support is the thermal spike model (Seitz 1958). In this model, a positive ion is created by the radiation-matter interaction. This ion quickly liberates its energy, generating neighbouring atoms that are thermally excited. If tension exists within the liquid, this region can produce a vapour bubble that expands and eventually results in a cavitation event. Example: Classical Theory of Homogeneous Nucleation According to classical theory of homogeneous nucleation (see, for example, Frenkel 1955), a nucleus is spontaneously generated as a result of density fluctuations in the metastable liquid phase in the form of a small vapour bubble of radius r. A minimum reversible work required to form a nucleus of new phase depends on the radius of the bubble and arrives a maximum at the critical radius rc . The nucleation rate J, which determines the average number of nuclei formed in a unit volume of the metastable phase per unit time, is proportional to the probability of having a critical nucleus J = J0 exp (−Wmin /kB T),

(1)

where Wmin =

4 2 πr σ 3 c

(2)

determines the nucleation barrier, which is equal to the minimum reversible work required to form a critical size nucleus, σ is the surface tension, kB is the Boltzmann constant, and J0 is a factor which does not depend on the critical radius rc and changes only slightly with the depth of penetration into the metastable state. All

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modifications introduced in the theory later do not change the general result of the classical theory given by Eqs. (1) and (2) and concern only details of calculations of the kinetic prefactor J0 and the nucleation barrier Wmin in different cases of metastable states (see, for example, Debenedetti 1996). In the classical theory of homogeneous nucleation the critical radius and the nucleation barrier can be calculated with the Gibbs equations (Landau and Lifshitz 1980) rc = Wmin =

2σ ν , μ(P) − μ (P)

σ 3ν2 16π , 3 [μ(P) − μ (P)]2

(3) (4)

where P is the bulk phase pressure, v is a specific volume of the nucleus, and μ (P) and μ(P) are the chemical potentials of the nucleus and of the metastable bulk fluid phase, respectively. A nucleus with radius less than a critical size rc requires energy for further growth and usually disappears without reaching the critical size. A nucleus with radius larger than rc grows freely with decrease of free energy, and a phase transition into a thermodynamically stable vapour phase takes place. Equations (3) and (4) can be applied for the formation of liquid droplets in supercooled vapour as well as for the formation of vapour bubbles in superheated liquid at moderate positive pressures and in the critical region. In metastable liquids at low temperatures considerable negative pressures are observed. In this case, Eqs. (3) and (4) are not more applicable. Particularly for large negative pressures the equations for rc and Wmin developed by Fisher (1948) can be used rc = − Wmin =

2σ , P

16π σ 3 . 3 P2

(5) (6)

However, these equations in turn fail at zero and small positive pressures where they give an unphysical divergence rc → ∞ and Wmin → ∞. A better result in this region can be obtained with the theory developed by Blander and Katz (1975). They obtained that the nucleation barrier Wmin is defined by Eq. (2), and found for the critical radius rc =

2σ v ∼ 2σ , = ∗ PV − P (P − P) δ

(7)

where P∗ is the saturation pressure at given temperature T. The correction factor δ takes into account the effect of the pressure P in the metastable liquid on the vapour pressure pv in the nucleus and is given by the equation

2.2

Nuclei Distribution

53

ρV 1 δ∼ + =1− ρL 2

ρV ρL

2 ,

(8)

where ρ L is the density of the liquid and ρ V the density of the vapour. Equations (7) and (8) are accurate for values of P at least up to 0.1 MPa, but are not valid in the critical region where the ratio ρV /ρL ∼ = 1 is not small and analytical expansion (8) is not more applicable. In the theory of homogeneous nucleation the mean time of formation of a critical nucleus in a volume V (9) tM = (JV)−1 determines the lifetime of the metastable state. The homogeneous nucleation limit of the metastable state is determined as a locus of the constant lifetime tM = const.

2.2 Nuclei Distribution The basic questions we want to answer in this section are how big are the nuclei and how many are these of each size. Data are presented for water, various polymer solutions, and blood. No information is available in the literature for the case of synovial liquid and saliva.

2.2.1 Distribution of Cavitation Nuclei in Water Several methods have been used to investigate the distribution of cavitation nuclei in water. Yilmaz et al. (1976) and Ben-Yosef et al. (1975) used the light scattering method, Gates and Bacon (1978) used a holographic technique, while Gavrilov (1969) used acoustic methods. Measurements of nuclei distribution using a Coulter counter were performed by Ahmed and Hammitt (1972), Pynn et al. (1976) and Oba et al. (1980). The Coulter counter detects change in electrical conductance of a small aperture as fluid containing cavitation nuclei is drawn through. A typical apparatus has one or more microchannels that separate two chambers containing electrolyte solutions. When a nucleus flows through one of the microchannels, it results in the electrical resistance change of the liquid filled microchannel. This resistance change can be recorded as voltage pulses, which can be correlated to the size of cavitation nuclei. Another direct measurement of the presence of cavitation nuclei is achieved when a liquid sample is passed through a region of known low pressure. Nuclei with radii that exceed a certain value radius will cavitate. The event rate of these cavitating bubbles can then be counted by visual observations. Moreover, when a cavitating bubble is convected to a region of higher pressure downstream, it will collapse producing an acoustic emission. The noise pulses can be detected and counted, giving another independent measurement of the nuclei. Devices that measure nuclei through inducing cavitation events are called cavitation susceptibility meters (Chambers et al. 1999).

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Some attempts were made to obtain a relationship that describes the distribution of cavitation nuclei in water. Gavrilov (1969) reported that the number of bubble nuclei is inversely proportional to the nuclei radius. Ahmed and Hammitt (1972) indicated that the distribution of cavitation nuclei can be described by pV , 4.838 σ v2/3 + pv

N(v) =

(2.1)

where N(v) is the number of nuclei of volume v, V the total gas content, σ the surface tension, and p is the ambient pressure. If 4.838 σ v2/3 << pv then N(v) ∝ d−3 , and if 4.838 σ v2/3 >> pv then N(v) ∝ d−2 , where d is the gas nuclei diameter. Shima et al. (1985) indicated that, in the range 2 μm < d < 20 μm, the gas nuclei distribution in water can be described by N(d) =

M , dn

(2.2)

where M is a constant. They found that the values of the exponent n lies between 2 and 4, in agreement with the results of Gavrilov (1969) who indicated n = 3.5. In a later study, Shima and Sakai (1987) obtained a more general equation for the size distribution of bubble nuclei in the form: N (r) =

M − nK (ln r−ln α)2 e 2 , rn

(2.3)

where r is the nuclei radius and M, n, K, and α are constants. They found good agreement with the experimental results reported by Ahmed and Hammitt (1972), Ben-Yosef et al. (1975), and Klaestrup-Kristensen et al. (1978).

2.2.2 Distribution of Cavitation Nuclei in Polymer Solutions Oba et al. (1980) and Shima et al. (1985) have measured the distribution of cavitation nuclei in water and various polymer solutions using a Coulter counter. They indicated that the size range of the nuclei is from 2 to 50 μm in radius, and the number of small nuclei below 7 μm represents more than 50% from the total number of nuclei. Oba et al. (1980) investigated the influence of polyethylene oxide concentration on the nuclei size distribution (Fig. 2.2). They found that the number of nuclei increases with the polymer concentration for nuclei diameters smaller 14 μm. For a diameter of about 12 μm, the number of bubble nuclei is one order of magnitude larger than in the case of water. However, for nuclei diameters larger than 14 μm, a significant reduction of the number of cavitation nuclei was observed. For a diameter of 35 μm, the number of nuclei in the 100 ppm polyethylene solution is one order of magnitude smaller than in the case of water.

2.2

Nuclei Distribution

55

Fig. 2.2 Nuclei distribution in a polyethylene oxide (PEO) aqueous solution. Adapted from Oba et al. (1980)

Shima et al. (1985) measured the cavitation nuclei distribution, in the range 2–20 μm, in three polymer aqueous solutions, namely a 100 ppm polyethylene oxide (Polyox) aqueous solution, a 2,000 ppm hydroxyethylcelullose aqueous solution, and a 50 ppm polyacrylamide aqueous solution (Fig. 2.3). For nuclei diameters larger than 3 μm they also found a decrease of the number of bubble nuclei in comparison to the case of water. The largest reduction was observed in the polyacrylamide and polyethylene solutions, while the results obtained in the hydroxyethylcelullose solution are almost similar to the case of water. They also indicated that the scaling law between the number of bubble nuclei and the nuclei diameter is not affected by the polymer additives.

2.2.3 Cavitation Nuclei in Blood The first attempts to detect cavitation in blood within the abdominal aorta of dogs exposed in vivo to lithotripsy have not proved successful although cavitation was observed in blood under in vitro conditions (Williams et al. 1988). Similar observations have been made by Deng et al. (1996). Lee et al. (1993) investigated bubble formation in the inferior vena cavae of dead rats after 6–15 h exposures to air at 12.3 MPa and decompression to 0.1 MPa at 1.36 MPa/min. Bubbles were detected by light microscopy, buoyancy, and underwater dissection. No bubbles were formed in 42 blood-filled vena cavae that were

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Nucleation

Fig. 2.3 Nuclei distribution in various polymer aqueous solutions. Adapted from Shima et al. (1985)

isolated from the minor circulation by ligatures, but bubbles were always observed in unisolated vena cavae. Their results indicate that nuclei are not present in blood, even at supersaturations that are significantly higher than those experienced in vivo. One explanation for this result is that the continuous filtration of impurities by the body allows the presence of cavitation nuclei in only minute amounts, and only in particular sites. This observation concurs with the finding that the cavitation threshold for water doubles upon filtration to 2 μm (Greenspan and Tschiegg 1967). More recently, Chambers et al. (1999) investigated the nuclei characteristics of blood using a cavitation susceptibility meter in an ex vivo sheep model. This hydrodynamic method measures the nuclei threshold pressure by subjecting the fluid to a certain characterized flow. All nuclei with a critical pressure higher than the minimum pressure within the device will cavitate, and the number of activated nuclei was determined by counting the cavitation events. The nuclei concentration of blood was measured to be at most 2.7 nuclei per litre and the authors estimated that the radius of the nuclei is on the order of 0.3 μm. However, they noted that these values may be even lower in an in vivo situation. Chappel and Payne (2006) suggested that cavitation nuclei could originate from tissues or microcapillaries and migrate into blood circulation. The contact between adjoining endothelial cells on the capillary walls could be a site for crevice nuclei. The effect of muscular contraction on crevices might be expected to squeeze the gas pocket and potentially cause the release of bubbles. While the concept of in

2.3

Tensile Strength

57

vivo hydrophobic crevices remains a theoretical possibility, none have yet been identified. No bubble formation was observed when isolated endothelium in contact with blood was decompressed (Lee et al. 1993). The extravascular space could be an alternative location: as extravascular gas nuclei expand, they might rupture capillaries, thereby seeding the blood with gas (Vann 2004). It has been also suggested that musculoskeletal activity could generate surfaceactive molecules that stabilize the nuclei and increase their lifetime (Hills 1992). On the other hand, there have been studies that demonstrate the beneficial effect of surfactants on bubble elimination. The addition of surfactants to blood makes it feasible to manipulate interfacial stresses and prevent or reduce formation of the adhesion responsible for trapping intravascular gas bubbles. In vivo studies have shown that the addition of surfactants favorably alters the patterns of deposition and accelerate the rates of clearance of bubbles (Suzuki et al. 2004). While surfactants could play a role first in nuclei stabilization, they could also be involved at last in vascular bubble elimination.

2.3 Tensile Strength It is important to realise that cavitation is not necessarily a consequence of the reduction of pressure to the liquid’s vapour pressure, the latter being the equilibrium pressure, at a specified temperature, of the liquid’s vapour in contact with an existing free surface. Cavity formation in a homogeneous liquid requires a stress sufficiently large to rupture the liquid. This stress represents the tensile strength of the liquid at that temperature (Brennen 1995; Trevena 1987; Young 1989). Several methods have been employed to obtain the tensile strength of water. The first to be used was the Berthelot tube technique: a vessel is filled with liquid water at high temperature and positive pressure, then sealed and cooled down at constant volume. The liquid sample follows an isochore and is brought to negative pressure. Berthelot claimed that he had reached –5 MPa in a glass ampoule completely filled with pure water (Berthelot 1850). Another method was designed by Briggs (1950): by spinning a glass capillary filled with water, he obtained a minimum value of the tensile stress of –27.7 MPa at 10◦ C; the tension falls to a much smaller value at lower temperature (down to –2 MPa at 0◦ C). Shock tube and bullet piston experiments generate negative pressure by reflection of a compression wave travelling in water at an appropriate boundary. This type of experiments has been reconsidered several times and the presently accepted results are around –10 MPa (Williams and Williams 2000). It is worth noting here that even larger values of the tensile strength of water were obtained. Zheng et al. (1991) used an improved version of the static Berthelot method by using synthetic water inclusions in quartz. A quartz crystal with cracks is autoclaved in the presence of liquid water. Water fills the cracks which then heal at high temperature, thus providing low density water in a small Berthelot tube. They reported a maximum tension of –140 MPa at 43◦ C. This result is similar to that obtained by Roedder (1967) who reached –100 MPa with water inclusions in natural rocks.

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Nucleation

Despite numerous studies, the precise role of non-Newtonian properties in determining cavitation threshold remains unclear. Most previous work in this area has considered polymer solutions – fluids made non-Newtonian by polymeric additives (Trevena 1987). Under conditions of dynamic stressing by pulses of tension there is evidence that polymer additives can lower cavitation threshold. An example has been reported by Sedgewick and Trevena (1978) who studied the cavitation properties of water containing polyacrylamide additives by the bullet-piston reflection method. Williams and Williams (2000) have shown that the latter method, which involves the conversion of a compressional pulse to a rarefaction at the free surface of a column of liquid, provides realistic estimates of tensile strength for water and other Newtonian fluids (Williams and Williams 2002). The experimental arrangement used by Williams and Williams (2002) consists of a cylindrical, stainless steel tube closed at its lower end by a piston (Fig. 2.4a). The piston’s lower surface is coupled to a stun-gun which generates a pressure pulse in a column of liquid within the tube. The upper flange connects the tube to a regulated oxygen-free nitrogen supply and a pressure gauge. Pressure changes within the liquid are monitored using three dynamic pressure transducers mounted

Fig. 2.4 The bullet-piston method for estimating the tensile strength of liquids. (a) Schematic of the cavitation threshold apparatus. (b) Pressure record obtained from a pressure transducer in a experiment on a sample of distilled water. (c) Cavitation threshold of distilled water. Reproduced with permission from Williams and Williams (2002). © IOP Publishing Ltd

References

59

in mechanically isolated ports in the wall of the tube. The main features of a typical pressure record obtained from a pressure transducer in an experiment on a sample of distilled water are shown in Fig. 2.4b, in which the data are presented in terms of transducer output in unscaled ADC units (positive values correspond to positive pressure and vice versa). A pressure pulse (feature “1” in Fig. 2.4b) is followed immediately by a tension pulse (“2”) and thereafter the record comprises “secondary” pressure-tension cycles (“3–4”, “5–6”, etc.) associated with cavitational activity. The method involves regulating a static pressure, Ps , in the space above the liquid, Ps being increased gradually in a series of dynamic stressing experiments. From the dynamic pressure records a measurement is made of the time delay, τ i , between the peak incident pressure (“1” in Fig. 2.4b) and the first pressure pulse arising from cavity collapse (“3” in Fig. 2.4b). Under tension, cavities grow from pre-existing nuclei within the liquid and eventually collapse and rebound, emitting a pressure wave into the liquid as they do so. Hence the interval τ i , which encompasses the attainment of maximum cavity radius and its subsequent decrease to a minimum value, is reduced by increasing Ps (τ i therefore provides a convenient measure of cavitational activity). The experiment involves the transmission of tension by the liquid to the face of the piston and it follows that in the case of experiments in which cavitation is detected, the magnitude of the tension transmitted by the liquid is sufficient to develop a transient, net negative pressure in the presence of a background static pressure Ps . Thus an estimate of the magnitude of tension capable of being transmitted by the liquid can be obtained from a knowledge of Ps . The time delay, τ o , between pulses corresponding to “1” and “2” in Fig. 2.4b represents the time required for the upward travelling pressure wave to return, as tension, to the lower transducer location. It also represents the smallest time interval for which a cavity growth-collapse cycle could occur (given that a bubble would have to grow and collapse infinitely quickly in order that τ i = τ o ). Thus the tensile strength can be estimated by extrapolation of the data in Fig. 2.4c to that value of the pressure Ps at which τ i = τ o , this condition representing the complete suppression of cavitation. Bullet-piston work has demonstrated a reduction of liquid effective tensile strength in non-Newtonian polymer solutions, the reduction increasing with increasing polymer concentration (Williams and Williams 2002). However, when this system was investigated using an ab initio technique, the cavitation threshold was found to be increased by the same polymer additive (Overton et al. 1984; Brown and Williams 2000). When subjected to quasi-static stressing (in a modified Berthelot tube) the presence of polymer made no discernible difference to the effective tensile strength of the liquid (Trevena 1987).

References Ahmed, O., Hammitt, F.G. 1972 Cavitation nuclei size distribution in high speed water tunnel under cavitating and non-cavitating conditions. Univ. Michigan ORA Rep. UMICH 013570-23-T. Apfel, R.E. 1970 The Role of impurities in cavitation-threshold determination. J. Acoust. Soc. Am. 48, 1179–1186.

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Atchley, A.A., Prosperetti, A. 1989 The crevice model of bubble nucleation. J. Acoust. Soc. Am. 86, 1065–1084. Berthelot, M. 1850 Sur quelques phenomenes de dilatation force des liquids. Ann. Chim. Phys. 30, 232–237. Ben-Yosef, N., Ginis, O., Mahlab, P., Weity, A. 1975 Bubble size distribution measurement by Doppler viscometer. J. Appl. Phys. 46, 738–740. Blander, M., Katz, J.L. 1975 Bubble nucleation in liquids. AIChE J. 21, 833–848. Borwankar, R.P., Wassan, D.T. 1983 The kinetics of adsorption of surface active agents at gasliquid interface. Chem. Engng Sci. 25, 1637–1649. Brennen, C.E. 1995 Cavitation and Bubble Dynamics. Oxford University Press, Oxford. Briggs, L.J. 1950 Limiting negative pressure of water. J. Appl. Phys. 21, 721–722. Brown, S.W.J., Williams, P.R. 2000 The tensile behaviour of elastic liquids under dynamic stressing. J. Non Newt. Fluid Mech. 90, 1–11. Chambers, S.D., Bartlett, R.H., Ceccio, S.L. 1999 Determination of the in vivo cavitation nuclei characteristics of blood. ASAIO J. 45, 541–549. Chappel, M.A., Payne, S.J. 2006 A physiological model of gas pockets in crevices and their behaviour under compression. Respir. Physiol. Neurobiol. 152, 100–114. Church, C.C. 2002 Spontaneous homogeneous nucleation, inertial cavitation and the safety of diagnostic ultrasound. Ultrasound Med. Biol. 10, 1349–1364. Crum, L.A. 1979 Tensile strength of water. Nature 278, 148–149. Crum, L.A. 1980 Acoustic cavitation threshold in water. In Cavitation Inhomogeneities in Underwater Acoustics (Ed. W. Lauterborn), Springer, New York, pp. 84–89. Debenedetti, P.G. 1996 Metastable Liquids. Concepts and Principles. Princeton University Press, Princeton. Deng, C.X., Xu, Q., Apfel, R.E., Holland, C.K. 1996 In vitro measurements of inertial cavitation thresholds in human blood. Ultrasound Med. Biol. 22, 939–948. Epstein, P., Plesset, M. 1950 On the stability of gas bubbles in liquid-gas solutions. J. Chem. Phys. 18, 1505–1509. Fisher, J.C. 1948 The fracture of liquids. J. Appl. Phys. 19, 1062–1067. Frenkel, J. 1955 Kinetic Theory of Liquids. Dover, New York. Fox, F., Herzfeld, K. 1954 Gas bubbles with organic skin as cavitation nuclei. J. Acoust. Soc. Am. 26, 984–989. Fuerth, R. 1941 On the theory of the liquid state. Proc. Camb. Philosph. Soc. 37, 252–290. Gates, E.M., Bacon, J. 1978 A note on the determination of cavitation nuclei distributions by holography. J. Ship Res. 22, 29–31. Gavrilov, L.R. 1969 On the size distribution of gas bubbles in water. Sov. Phys. Acoust. 15, 22–24. Greenspan M., Tschiegg, C.E. 1967 Radiation-induced acoustic cavitation; apparatus and some results. J. Res. NBS 71C, 299–312. Hills, B.A. 1992 A hydrophobic oligolamellar lining to the vascular lumen in some organs. Undersea Biomed. Res. 19, 107–120. Klaestrup-Kristensen, J., Hansson, I., Mørch, K.A. 1978 A simple-model for cavitation erosion of metals. J. Phys. D Appl. Phys. 11, 899–912. Landau, L.D., Lifshitz, E.M. 1980 Statistical Physics. Pergamon, New York. Lee, Y.C., Wu, Y.C., Gerth, W.A., et al. 1993 Absence of intravascular bubble nucleation in dead rats. Undersea Hyperb. Med. 20, 289–296. Mørch, K.A. 2000 Cavitation nuclei and bubble formation – a dynamic liquid-solid interface problem. J. Fluids Eng. 122, 494–498. Oba, R., Kim, K.T., Niitsuma, H., Ikohagi, T., Sato, R. 1980 Cavitation-nuclei measurements by a newly made Coulter-counter without adding salt in water. Rep. Inst. High Speed Mech. Tohoku Univ. 43, 163–176. Overton, G.D.N., Williams, P.R., Trevena, D.H. 1984 The influence of cavitation history and entrained gas on liquid tensile strength. J. Phys. D Appl. Phys. 17, 979–987. Porter, T.M., Crum, L.A., Stayton, P.S., Hoffman, A.S. 2004 Effect of polymer surface activity on cavitation nuclei stability against dissolution. J. Acoust. Soc. Am. 116, 721–728.

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Pynn, J.J., Hammitt, F.G., Keller, A. 1976 Microbubble spectra and superheat in water and sodium, including effect of fast neutron irradiation. J. Fluids Eng. 98, 87–97. Roedder, E. 1967 Metastable superheated ice in liquid-water inclusions under high negative pressure. Science 155, 1413–1417. Sedgewick, S.A., Trevena, D.H. 1978 Breaking tensions of dilute polyacrylamide solutions. J. Phys. D Appl. Phys. 11, 2517–2526. Seitz, F. 1958 On the theory of bubble chambers. Phys. Fluids 1, 2–13. Shima, A., Tsujino, T., Tanaka, J. 1985 On the equation for the size distribution of bubble nuclei in liquids. Rep. Inst. High Speed Mech. Tohoku Univ. 50, 59–66. Shima, A., Sakai, I. 1987 On the equation for the size distribution of bubble nuclei in liquids (Report 2). Rep. Inst. High Speed Mech. Tohoku Univ. 54, 51–59. Strasberg, M. 1959 Onset of ultrasonic cavitation in tap water. J. Acoust. Soc. Am. 31, 163–169. Suzuki, A„ Armstead, S.C., Eckmann, D.M. 2004 Surfactant reduction in embolism bubble adhesion and endothelial damage. Anesthesiology 101, 97–103. Trevena, D.H. 1987 Cavitation and Tension in Liquids. Adam Hilger, Bristol. Vann, R.D. 2004 Mechanisms and risks of decompression. Diving Medicine, Saunders, Philadelphia, pp. 127–164. Williams, A.R., Delius, M., Miller, D.L., Schwarze, W. 1988 Investigation of cavitation in flowing media by lithotripter shock-waves both in vitro and in vivo. Ultrasound Med. Biol. 15, 53–60. Williams, P.R., Williams, R.L. 2000 On the anomalously low values of the tensile strength of water. Proc. R. Soc. A 456, 1321–1332. Williams, P.R., Williams, R.L. 2002 Cavitation of liquids under dynamic stressing by pulses of tension. J. Phys. D Appl. Phys. 35, 2222–2230. Yilmaz, E., Hammitt, F.G., Keller, A. 1976 Cavitation inception thresholds in water and nuclei spectra by light-scattering technique. J. Acoust. Soc. Am. 59, 329–338. Young, F.R. 1989 Cavitation. McGraw-Hill, New York. Yount, D. 1979 Skins of varying permeability: a stabilization mechanism for gas cavitation nuclei. J. Acoust. Soc. Am. 65, 1429–1439. Yount, D. 1982 On the evolution, generation, and regeneration of gas cavitation nuclei. J. Acoust. Soc. Am. 71, 1473–1481. Zheng, Q., Durben D.J., Wolf G.H., Angell, C.A. 1991 Liquids at large negative pressures: water at the homogeneous limit. Science 254, 829–832.

Chapter 3

Bubble Dynamics

The main goal of the investigations on bubble dynamics is to describe the velocity field and the pressure distribution in the liquid surrounding the bubble. In this section we describe the effect of the viscoelastic properties of the liquid on the behaviour of cavitation bubbles situated in a liquid of infinite extent or near a rigid boundary. As a special case, we will consider the interaction of individual cavitation bubbles situated in water with boundary materials with elastic/plastic properties. Due to the difficulty of the problem most of the theoretical work on bubble dynamics in non-Newtonian fluids was restricted to the case of spherical bubbles. The experimental studies, on the other hand, made use of high speed cameras to observe the growth and collapse of both spherical and non-spherical bubbles in nonNewtonian fluids. Two types of experiments have been conducted: in the first type, a cavitation bubble is collapsed after its expansion by the ambient pressure in the surrounding fluid. In this case, the bubble is generated by laser or electrical discharge. Alternatively, a stable gas bubble, usually of a size large enough to be visible, is compressed by a positive pressure pulse.

3.1 Spherical Bubble Dynamics The investigation of the dynamics of spherical cavitation bubbles is of no direct interest for the explanation of cavitation erosion, because bubbles close enough to a boundary to cause damage will always collapse aspherically. Nevertheless, it provides the basis for the interpretation of data obtained for the asymmetrical collapse of bubbles in non-Newtonian fluids and is to date the only means of comparing experimental results with theory.

3.1.1 General Equations of Bubble Dynamics Consider a spherical bubble of initial radius R0 situated in a compressible viscoelastic liquid. Until the reference time, t = 0, the pressure is uniform at p∞ and the liquid is at rest. At t = 0, the pressure inside the bubble is decreased instantaneously E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_3,

63

64

3

Bubble Dynamics

to p0 and the bubble begins to collapse due to the pressure difference between the inside and outside of the bubble. The bubble keeps its spherical shape throughout the motion and the centre of the bubble remains fixed and is the centre of a spherically symmetric coordinate system. In principle, the quantities associated with the bubble collapse, such as velocity and pressure, can be determined from the solution of the conservation equations of continuum mechanics inside and outside of the bubble joined together by suitable boundary conditions at the bubble interface. Neglecting the effects of gravity, gas diffusion and heat conduction through the bubble wall, the governing equations may be expressed as follows: (i) Continuity: ∂p ∂(ρvr ) ρvr + +2 = 0, ∂t ∂r r

(3.1)

1 ∂p 1 ∂vr ∂vr + vr =− − (∇ · τ)r , ∂t ∂r ρ ∂r ρ

(3.2)

(ii) Momentum:

where ν r is the radial component of the velocity field, ρ, the liquid density, p(r, t) is the pressure in the liquid, and τ is the extra stress tensor. (iii) Equation of state for the liquid: A widely used equation of state for liquids is the Tait form:

p+B = p∞ + B

ρ ρ∞

n ,

(3.3)

where the subscript ∞ refers to the values at infinity, and B and n are constants having, for water, the values n = 7.15 and B = 3,049.13 atm. (iv) Equation of state for the gas inside the bubble: pi = p0

R0 R

3κ ,

(3.4)

where κ is the polytropic index. (v) Boundary conditions at the bubble wall (r = R(t)): Kinematic boundary condition: vr (t) =

dR ˙ = R, dt

(3.5)

Dynamic boundary condition: pB (t) = pi (t) −

2σ − (τrr )r=R , R

(3.6)

where pB is the pressure on the liquid at the bubble wall and σ is the surface tension.

3.1

Spherical Bubble Dynamics

65

Several comments relevant to bubble dynamics in non-Newtonian liquids are appropriate here. In a compressible liquid the extra stress tensor consists of two parts. The first part is the shear stress tensor τs that depends on the rate-of-strain tensor. For a purely viscous liquid, this tensor has the form tr(γ˙ )I , τs = 2η γ˙ − 3

(3.7)

where η is the shear viscosity of the liquid, I the unit tensor, and γ˙ is the shear rate. The second part is the isotropic tensor τi = f0 I with f0 being a function of invariants of the rate-of-strain tensor, i.e., f0 = f0 (I1 , I2 , I3 ), where I1 = tr(γ˙ ),

2 I2 = tr(γ˙ ) − tr γ˙ 2 , and I3 = Det(γ˙ ). For Newtonian and linear viscoelastic liquids τi has the form τi = λv tr(γ˙ )I,

(3.8)

where λv is the second coefficient of viscosity. For non-linear viscoelastic liquids, where the shear stress tensor has a finite trace, tr(τ) = 0, there is an additional contribution to the mean pressure p¯ = −tr[−pI + τ] that results in its variation from the pressure p in the liquid surrounding the bubble. We further note that Eq. (3.3) applies only to isentropic changes, but can be applied with reasonable accuracy in general since n is independent of entropy and B and ρ∞ are only slowly varying functions of entropy. Finally, Eq. (3.6) assumes that the gas-liquid interface is “clean” i.e., the only molecules present are those of the gas and the surrounding liquid. However where surfactants are adsorbed onto the bubble surface, a surface stress term needs to be added to Eq. (3.6) which includes the effects of surface viscosity and surface tension gradients. The latter occurs when the concentration of surfactant molecules on bubble surface is not constant resulting in an additional radial force that arise from the variation in the concentration of surface active molecules. A further approximation that was introduced in (3.6) is the neglect of the surface viscous term which, in the case of a spherical symmetric motion, is defined as ˙ 2 , where αs is the surface dilatational viscosity (Aris 1989). While τrr,s = 4αs R/R this procedure is justified for dilute surfactant solutions, it may be noted here that the predictions of a pure interface model are of interest in themselves in view of the frequent use of such a model in the study of bubble dynamics in non-Newtonian liquids.

3.1.2 The Equations of Motion for the Bubble Radius Here we shall restrict ourselves only to the case of linear viscoelastic liquids for which the extra stress tensor is traceless i.e., the sum of the normal stress components is zero. It should be emphasized here that these models are not entirely satisfactory for the description of viscoelastic flow behaviour. However, studies of idealized models may provide a qualitative insight for more realistic systems, and also quantitative results about their intermediate asymptotic behaviour. Moreover,

66

3

Bubble Dynamics

these models have the main advantage of being tractable and, thus, they allow us to obtain an elegant solution by reducing the problem to a non-linear differential equation. The “near field” is a region surrounding the bubble with typical dimension R, the bubble radius, the “far field” scales with a typical length c∞ T, where c∞ is the speed of sound in the liquid and T a characteristic time, such as the collapse time. If ˙ with R˙ a typical radial velocity of the bubble one assumes that R is of the order RT, wall, the ratio of length scales is just the Mach number of the bubble wall motion. Once cast in these terms it is clear that, to lowest order, the near-field dynamics are essentially incompressible while the far field is governed by linear acoustics. The picture becomes considerably more intricate for a non-linear viscoelastic liquid, however (Khismatulin and Nadim 2002). The analysis leads unambiguously to the following equation for the radius of a spherical bubble situated in a linear viscoelastic liquid (Brujan 1998, 1999, 2001, 2009a): ∞

∂τrr 3τrr 1 1 2 ... 3 2 3 ˙ ¨ ˙ ˙ ¨ + dr, (3.9) R R + 6RRR + 2R = H − RR + R − 2 c∞ ρ∞ ∂r r R

where H is the liquid enthalpy at the bubble wall n(p∞ + B) H= (n − 1)ρ∞

!

pB + B p∞ + B

(n−1)/n

" −1 ,

(3.10)

˙ 2. with τrr evaluated in the near-field where vr = R2 R/r The striking feature of Eq. (3.9) is the appearance of the third-order derivative of the bubble radius with respect to time. This is just a consequence of using ¨ − R/c∞ ) ≈ Taylor series expansions to express retarded-time quantities, e.g. R(t ... ¨R(t) − (R/c∞ ) R . A similar term arises in Lorentz’s theory of electrons. Lorentz was ... considering periodic displacements x at frequency ω and thus set x ≈ −ω2 x˙ and identified this term with radiation damping. Later researchers, however, were deeply puzzled by this third derivative although there is nothing mysterious about it (Brujan 2001). For c∞ → ∞ the incompressible formulation is recovered, namely: 3 1 RR¨ + R˙ 2 = H − 2 ρ∞

∞ R

∂τrr 3τrr + dr, ∂r r

(3.11)

which, in the case of a Newtonian fluid, is known as the Rayleigh–Plesset formula˙ = α(R2 R) ˙ + (1 − α)(R2 R) ˙ and uses the tion. Furthermore, if one writes (R2 R) incompressible formulation in the form ˙ ˙ 2 1 (R2 R) 1 (R2 R) − =H− R 2 R4 ρ∞

∞ R

∂τrr 3τrr + dr ∂r r

(3.12)

3.1

Spherical Bubble Dynamics

67

to evaluate the first term and (3.11) to express the third derivative of the radius which appears on expanding the second term, one finds 3α + 1 α+1 3 2 1−α R ¨ ˙ ˙ ˙ ˙ ˙ RR 1 − R + R 1− R =H 1+ R + H c∞ 2 3c∞ c∞ c∞ ∞ ∞ ∂τrr ∂τrr 3τrr 3τrr 1−α 1 1 R d + + R˙ 1+ − dr − dr, ρ∞ c∞ ∂r r ρ∞ c∞ dt ∂r r R

R

(3.13) which represents an extension of the general Keller–Herring equation to the case of a bubble in a linear viscoelastic liquid. For a Newtonian liquid, by taking α = 0, Eq. (3.13) becomes identical to the equation proposed by Keller and Kolodner (1956), while the value α = 1 brings it into the form suggested by Herring (see, for example, Trilling 1952). It will be noted that, by dropping terms in c−1 ∞ , Eq. (3.13) reduces to Eq. (3.11), which is therefore seen to have an error of the order c−1 ∞ . The arbitrary parameter α (which does not seem to have any physical meaning) must, of course, be of order 1 so as not to destroy the order of accuracy of the approximate Eq. (3.13). Because of the presence of the third time derivative of the radius, the form (3.9) of the radial equation is hardly more attractive than (3.13), if for nothing else than for ¨ Actually, this is a minor difficulty the need to prescribe an initial condition for R. since, to the same order of accuracy in the bubble wall Mach number, an initial condition for R¨ can be obtained by substituting the given initial conditions for R and R˙ in the incompressible formulation (6). However, in view of its uniqueness (Brujan 1999), it is proper to consider Eq. (3.9) the fundamental form of the motion equation of a spherical bubble in a compressible linear viscoelastic liquid. With reference to Eq. (3.13) it should be noted that a related equation is that due to Gilmore (see, for example, Prosperetti and Lezzi 1986): R˙ 3 R˙ R R˙ R˙ ˙ + R˙ 2 1 − =H 1+ + 1− H RR¨ 1 − C 2 C C C C ∞ (3.14) ∞ R˙ ∂τrr ∂τrr 3τrr 1 R d 3τrr 1 1+ + dr − + dr, − ρ∞ C ∂r r ρ∞ C dt ∂r r R

R

whereby C is the speed of sound at the bubble wall C = [c2∞ + (n − 1)H]1/2 ,

(3.15)

and whose derivation relies on the Kirkwood–Bethe approximation (Kirkwood and Bethe 1942; Knapp et al. 1970). In this approach, the speed of sound C is not constant, but depends on H. This allows one to model the increase of the speed of sound with increasing pressure around the bubble, which leads to significantly reduced Mach numbers at bubble collapse.

68

3

Bubble Dynamics

To close the mathematical formulation an equation for the shear stress in terms of the rate-of-strain is necessary. Several examples on obtaining the equation of motion for the bubble radius for some constitutive models are given below. Example 3.1: Equation of Motion for Bubble Radius in Terms of Pressure #p Using the Taylor series expansion and the definition of enthalpy, h = p∞ dp/ρ, we may write H=

pB − p∞ ρ∞

1 p − p∞ 1− . 2 ρ∞ c2∞

(1)

With this result and using the dynamic boundary condition (3.6), the equation of motion for the bubble radius in terms of pressure is found to be ∞

3 1 3 2 τrr 1 2 ... 2σ 3 ˙ ¨ ˙ ¨ ˙ pi (t) − − p∞ − dr, R R + 6RRR + 2R = RR+ R − 2 c∞ ρ∞ R ρ∞ r R

(2) where, in the case of an adiabatic evolution of the gas inside the bubble, pi (t) is given by Eq. (3.4). Example 3.2: Equation of Motion for Bubble Radius for a Newtonian Fluid In the case of a Newtonian fluid τrr = −2η

R2 R˙ ∂vr = 4η 3 , ∂r r

(1)

and ∞ 3

R˙ τrr dr = 4η . r R

(2)

R

Thus, the equation of motion for the bubble radius in a Newtonian fluid written in terms of pressure becomes

3 2 1 1 2 ... 2σ R˙ 3 ˙ ¨ ˙ ¨ ˙ RR + R − pi (t) − − p∞ − 4η . (3) R R + 6RRR + 2R = 2 c∞ ρ∞ R R After some time of oscillation, due to acoustic and viscous dissipation, the trajectory R = R(t) move towards an equilibrium position characterized by the equi... librium radius of the bubble Re . This value may be obtained imposing R˙ = R¨ = R = 0 in Eq. (3) to find

3.1

Spherical Bubble Dynamics

R3γ e +

69

2σ 3γ −1 p0 3γ R + R = 0. p∞ e p∞ 0

(4)

Suppose now that the bubble oscillates with small amplitude. Then a solution R of this equation may be given as R = Re (1 + δ), |δ| << 1,

(5)

where Re is the equilibrium radius of the bubble. The linearized solution of the motion equation valid to order c−1 ∞ is obtained by substituting Eq. (5) into Eq. (3) ˙ If this procedure is carried out and neglecting the...higher-order terms of δ and δ. a term containing δ appears in the resulting equation. To avoid this we note that, since the error in Eq. (3) is of the order c−2 ∞ , it is sufficient to approximate this term correctly to order 1. For this purpose, after introduction of (5) we differentiate the incompressible equation in the form (3) with c∞ → ∞ and use the result to ... eliminate δ to find 1+

4η ρ∞ c∞ Re

% 4η 1 2σ 3γ p∞ + (3γ − 1) δ˙ + Re ρ∞ R2e ρ∞ c∞ R2e 1 2σ + 3γ p∞ + (3γ − 1) δ = 0. Re ρ∞ R2e

δ¨ +

$

(6)

The natural frequency of the damped oscillator defined by Eq. (5) may be derived as

1 2η 2 2σ 3γ p∞ + (3γ − 1) − ρ∞ Re ρ∞ Re &&1/2 3 2η 2 1 2η 2σ 3γ p∞ + (3γ − 1) −4 − , c∞ ρ∞ Re ρ∞ Re ρ∞ Re

1 f0 = 2π Re

(7)

assuming that the condition 4η/(c∞ ρ∞ Re ) << 1 is satisfied. The equation describing the pressure distribution in the liquid surrounding the bubble can be obtained as (for a detailed derivation see Lezzi and Prosperetti 1987): p(r) = p∞ + ρ∞

R2 R¨ + 2RR˙ 2 (ξ ) ρ∞ 2 ... ρ∞ R4 R˙ 2 − R R + 6RR˙ R¨ + 2R˙ 3 (ξ ) − , r c∞ 2 r4 (8)

where ξ = t − (r − R)/c∞ and, unless indicated, all the R have the argument t.

70

3

Bubble Dynamics

Example 3.3: Equation of Motion for Bubble Radius in an Inelastic Shear-Thinning Fluid Consider now an inelastic shear-thinning fluid characterized by the Williamson rheological model given as η = η∞ +

η0 − η∞ √ n , 1 + I2 / k

(1)

where η∞ is the infinite-shear viscosity, η0 is the zero-shear viscosity, k and n are the Williamson model parameters and I2 is the second invariant of the rate of deformation tensor, ! " v 2 ∂vr 2 r I2 = 2 +2 . (2) ∂r r In this case ∞ 3 R

τrr r−4 R˙ dr = 4η∞ + 12R2 R˙ (η0 − η∞ ) ,a = r R 1 + a/r3n ∞

' √ )2 2 3 2 (( (( R R˙ . (3) k

R

With y=

a/r3n 1 + a/r3n

(4)

we have ∞ R

r−4 1 = − a−1/n 3n 3n 1 + a/r

0

(1 − y)−1/n y(1/n)−1 dy,

(5)

a/R3n 1+a/R3n

and writing (1 − y)−1/n = 1 +

∞

s+1

Cs+1/n ys+1

(6)

s=0

we find ∞ R

! ' √ ( ( )n "−1/n 2 3 (R˙ ( r−4 1 = 1 + 3R k R 1 + a/r3n ⎧ ! ' √ ( ( )n "−(s+1) ⎫ ∞ ⎨ ⎬ R˙ 2 3 (R˙ ( 1 s+1 . × 1+ 1+ Cs+1/n ⎩ ⎭R ns + n + 1 k R s=0

(7)

3.1

Spherical Bubble Dynamics

71

Thus, the final form of the equation of motion for bubble radius in an inelastic shear-thinning fluid characterized by the Williamson rheological model is

1 2 ... 3 2 ¨ R R + 6RR˙ R¨ + 2R˙ 3 RR+ R˙ − 2 c ⎡∞ ! ' √ ( ( )n "−1/n 2 3 (R˙ ( R˙ 1 ⎣ 2σ − p∞ − 4η∞ − 4 (η0 − η∞ ) 1 + = pi (t) − ρ∞ R R k R ⎧ ! ' √ ( ( )n "−(s+1) ⎫ ⎤ ∞ ⎨ ⎬ R˙ 1 2 3 (R˙ ( s+1 ⎦. × 1+ 1+ Cs+1/n ⎩ ⎭R ns + n + 1 k R s=0

(8) For η0 = η∞ = η (η being the Newtonian viscosity) this equation is identical with that obtained for a Newtonian fluid. The equation describing the equilibrium radius of the bubble is identical to that in a Newtonian fluid while the equation describing the natural frequency of the bubble is similar to that in a Newtonian fluid but with η∞ instead of η. The equation describing the pressure distribution in the liquid surrounding the bubble is in this case (see Brujan 1998):

R2 R¨ + 2RR˙ 2 (ξ ) ρ∞ R4 R˙ 2 ρ∞ 2 ... R R + 6RR˙ R¨ + 2R˙ 3 (ξ ) − − r c∞ 2 r4 ! ' √ ! ' √ )n "−1 )n "−1/n R2 R˙ 2 3 (( 2 (( 2 3 (( 2 (( ˙ ˙ − (η0 − η∞ ) 1 + + 4 − η R R 1 + ) (η (R ( (R ( 0 ∞ kr3 r3 kr3 ⎫ ⎧ ! ' √ ( ( )n "−(s+1) ∞ ⎬ R2 R˙ ⎨ 2 3 (R˙ ( 1 s+1 1 + × 1+ . Cs+1/n ⎭ r3 ⎩ ns + n + 1 k R

p(r) = p∞ + ρ∞

s=0

(9)

For η0 = η∞ = η this equation reduces to the case of a Newtonian fluid. Example 3.4: Equation of Motion for Bubble Radius in Linear Elastic Fluid We first consider the three-parameter, linear Oldroyd model given by τrr + λ1

Dτrr Derr = 2η err + λ2 , Dt Dt

(1)

where D/Dt = ∂/∂t + vr ∂/∂r, λ1 is a characteristic relaxation time (for the stress), η, the viscosity coefficient, λ2 , a characteristic retardation time (i.e., relaxation time for strain) and err = ∂vr /∂r is the strain rate. It should be noted that the assumption of a single relaxation time λ1 is over simplistic, even if the polymers are monodispersed. Rather, one would expect a long chain to have a distribution of time scales, corresponding to various subchains that compose the polymer. In principle, there is no problem in incorporating such a distribution of time scales in the model, but

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it would violate the fundamental desideratum of simplicity. Usually, one chooses λ1 to be some average of those time scales, but perhaps it is more reasonable to assume that strong flows will be dominated by the longest relaxation time scale of the system. After transformation to a Lagrangian coordinates system, by using y=

1 3 r − R3 , 3

we find D ∂ ∂ ∂ ∂ = + vr = − R2 R˙ + Dt ∂t ∂r ∂t ∂y and err = −

(2) R2 R˙ ∂ ∂ r2 = , 2 ∂y ∂t r

2R2 R˙ . 3y + R3

(3)

(4)

The Oldroyd equation becomes dτrr 2RR˙ 2 + R2 R¨ R2 R˙ , + λ2 τrr + λ1 = 4η dt 3y + R3 3y + R3

(5)

with the solution (for λ1 = 0) 4η τrr = λ1

t 0

ξ −t exp λ1

& ˙ ) + λ2 R2 (ξ )R(ξ ¨ ) + 2R(ξ )R˙ 2 (ξ ) R2 (ξ )R(ξ dξ . (6) 3y + R3 (ξ )

Using this last result we obtain the equation of motion for bubble radius in a threeparameter, linear Oldroyd model as

1 1 2 ... 2σ 12η 3 pi (t) − − p∞ − R R + 6RR˙ R¨ + 2R˙ 3 = RR¨ + R˙ 2 − 2 c∞ ρ∞ R λ1 & " (7)

2 2 ˙ t ¨ ) + 2R(ξ )R˙ 2 (ξ ) R (ξ )R(ξ ) + λ2 R (ξ )R(ξ R ξ −t ln exp dξ . λ1 R(ξ ) R3 − R3 (ξ ) 0

The pressure equation in the liquid surrounding the bubble reads (Brujan 1999): 2

R R¨ + 2RR˙ 2 (ξ ) ρ∞ 2 ... ρ∞ R4 R˙ 2 − R R + 6RR˙ R¨ + 2R˙ 3 (ξ ) − p(r) = p∞ + ρ∞ r c∞ 2 r4 t

. 4η ξ −t - 2 ˙ ) + λ2 R2 (ξ )R(ξ ¨ ) + 2R(ξ )R˙ 2 (ξ ) R (ξ )R(ξ − exp λ1 λ1 0 1 1 1 − ln dξ . × 3 (8) r − R3 − R3 (ξ ) R3 − R3 (ξ ) r3 − R3 − R3 (ξ )

3.1

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73

As a second example consider the Jeffreys rheological model given as τrr + λ1

∂τrr ∂err = 2η err + λ2 . ∂t ∂t

(9)

Using the Laplace transform, the rheological equation becomes (1 + λ1 s)Trr = −2η(1 + λ2 s)εrr , Trr (s) = L[τrr (t)], εrr (s) = L[err (t)].

(10)

Using the Laplace inverse transform, we have τrr = −2ηL

−1

1 + λ2 s εrr , 1 + λ1 s

(11)

and then from the convolution theorem for Laplace tranform t τrr = −2

G(t − ¯t)err (¯r, ¯t)d¯t,

(12)

0

where G(t) = ηL

−1

1 + λ2 s λ2 −t/λ1 λ2 1 1− e , =η δ(t) + 1 + λ1 s λ1 λ1 λ1

(13)

with G(t − ¯t) the relaxation modulus and δ(t) the Dirac function. Using the last two equations, the normal stress component in the r direction may be obtained as t τrr = −2η 0

λ2 1 δ t − ¯t + λ1 λ1

λ2 −(t−¯t)/λ 1 1− e err (¯r, ¯t)d¯t. λ1

(14)

Considering Lagrangian coordinates, the history of the strain component can be obtained as err (r, t) = −

˙ ¯t) 2R2 (¯t)R( , r3 − R3 − R3 (¯t)

(15)

thus t τrr = 4η 0

λ2 1 δ(t − ¯t) + λ1 λ1

˙ ¯t) λ2 −(t−¯t)/λ1 2R2 (¯t)R( 1− e d¯t, λ1 r3 − R3 − R3 (¯t)

(16)

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and, therefore, ∞ 3

τrr dr = 12η r

R

t 0

×

λ2 1 λ2 δ(t − ¯t) + (1 − )e−(t−¯t)/λ 1 λ1 λ1 λ1

˙ ¯t) 2R2 (¯t)R( R3

− R3 (¯t)

ln

(17)

R d¯t. R(¯t)

Using the last result, the equation of motion for the bubble radius in a Jeffreys fluid is $ 1 2 ... 1 2σ 3 2 3 ˙ ¨ ˙ ¨ ˙ pi (t) − (R R + 6RRR + 2R ) = RR+ R − − p∞ 2 c∞ ρ∞ R ⎫ t ˙ ¯t) λ2 λ2 −(t−¯t)/λ1 2R2 (¯t)R( R ⎬ 1 1− e d¯t . ln − 12η δ(t − ¯t) + λ1 λ1 λ1 R3 − R3 (¯t) R(¯t) ⎭ 0

(18) The pressure equation in the liquid surrounding the bubble reads (Shima et al. 1988): p(r) = p∞ + ρ∞

R2 R¨ + 2RR˙ 2 (ξ ) ρ∞ 2 ... − R R + 6RR˙ R¨ + 2R˙ 3 (ξ ) r c∞

ρ∞ R4 R˙ 2 − − 4η 2 r4

t 0

˙ ¯t) 2R2 (¯t)R( r3 − R3 − R3 (¯t)

−

λ2 1 δ(t − ¯t) + λ1 λ1

λ2 −(t−¯t)/λ1 1− e λ1

(19)

˙ ¯t) R2 (¯t)R( r3 − R3 − R3 (¯t) d¯t. ln R3 − R3 (¯t) r3

Example 3.5: Non-Dimensional Form of the Equations of Motion for the Bubble Radius We make use of the following dimensionless variables, indicated by an asterisk, defined as R = R0 R∗ , t = Tt∗ , H = U 2 H∗ , C = c∞ C∗ , τrr = ρ∞ U 2 τrr∗ ,

(1)

where U = (p∞ / ρ∞ )1/2 is of the order of the bubble wall velocity and T = R0 /U. With these definitions the Gilmore formulation becomes

3.1

Spherical Bubble Dynamics

75

R∗ R∗ R∗ 3 2 1−ε = H∗ 1 + ε + R∗ 1 − ε C 2 C∗ C∗ ∞ R ∂τrr∗ R∗ R˙ 3τrr∗ 1 − ∗ H∗ − 1 + ε dr∗ +ε + C∗ C∗ C ∂r∗ r∗

R∗ R∗

R∗ d −ε C∗ dt∗

∞ R∗

∂τrr∗ 3τrr∗ + ∂r∗ r∗

R∗

dr∗ ,

.1/2 C∗ = 1 + ε2 (n − 1)H∗ , ε2 H∗ = (n − 1)

!

(2)

pB∗ + B p∞ + B

(3) "

(n−1)/n

−1 ,

(4)

where ε is of the order of bubble wall Mach number. The nondimensional form of Eq. (3.9) reads R∗ R∗ +

3 2 3 R − ε(R2∗ R ∗ + 6R∗ R∗ R∗ + 2R∗ ) = H∗ − 2 ∗

∞ R∗

∂τrr∗ 3τrr∗ dr∗ , (5) + ∂r∗ r∗

while that of Eq. (3.13) is

3 ε R∗ R∗ 1 − ε(α + 1)R∗ + R2 ∗ 1 − (3α + 1)R∗ = H∗ 1 + ε(1 − α)R∗ 2 3 ∞ ∂τrr∗ 3τrr∗ dr + εR∗ H∗ − 1 + ε(1 − α)R + ∂r∗ r∗ d − εR∗ dt∗

∞ R

R∗

∂τrr∗ 3τrr∗ + ∂r∗ r∗

(6)

dr∗ .

For example, for the three-parameter, linear Oldroyd model, the stress component becomes t∗ 4η ξ ∗ − t∗ exp τrr∗ = Re De De 0 (7)

& ˙ ∗ ) + χ De R2∗ (ξ∗ )R∗ (ξ∗ ) + 2R∗ (ξ∗ )R2 R2∗ (ξ∗ )R(ξ (ξ ) ∗ ∗ × dξ∗ , r∗3 − R3∗ + R3∗ (ξ∗ ) where χ = λ2 /λ1 . The introduction of viscoelastic liquids into the bubble dynamics analysis creates two independent sets of parameters: the Reynolds number, defined as

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Re = Rmax ρ ∞ U/η, and the Deborah number which is defined as the ratio of the characteristic time of the fluid and the characteristic time of the bubble collapse, De = λ1 U/Rmax . 3.1.2.1 Bubble Behaviour in Non-Newtonian Purely Viscous Fluids A large number of theoretical studies on the behaviour of spherical bubbles in purely viscous liquids have been published. The power-law model was adopted by Yang and Yeh (1966) and Shima and Tsujino (1976) in their investigations. In addition, the Casson model (Shima and Tsujino 1978), the Ellis model (Shima and Tsujino 1980a), Sisko model (Shima and Tsujino 1977), the Carreau model (Shima and Tsujino 1980b), the Powell-Eyring model (Shima and Tsujino 1981), the Shima model (Shima and Tsujino 1982), the Sutterby model (Shima et al. 1984a), the Williamson model (Brujan 1993, 2000), and the Bueche model (Brujan 1994a) have been applied. Brujan (1998) derived the equation of motion for a spherical bubble and the pressure equation in a compressible purely viscous liquid by using the Williamson model which well represents the rheological properties of carboxymethylcelullose (CMC) and hydroxyethylcelullose (HEC) polymer aqueous solutions. The apparent viscosity of the fluids is modelled by the Williamson rheological equation (see Example 3.3). The physical properties of polymer solutions and water are listed in Table 3.1. Typical results of the calculations are shown in Fig. 3.1 which illustrates the maximum velocity reached by the bubble wall during first collapse (left) and maximum pressure at the bubble wall (right) as a function of the minimum bubble radius at the end of the first collapse. It was demonstrated that, for values of the maximum bubble radius smaller than 10–1 mm, the shear-thinning characteristic of liquid viscosity strongly influences the behaviour of the bubble and the rheological parameter with the strongest influence is the infinite-shear viscosity, η∞ . For larger bubbles, in spite of the considerable differences of the apparent viscosity of the liquid, η, the behaviour of the bubble remains the same as that of an equivalent Newtonian fluid with a viscosity η∞ . Similar results were reported by Shima and Tsujino (1977, 1980a, b, 1982), and Shima et al. (1984a) using incompressible formulations of bubble dynamics. The effect of polymer additives leads to a significant decrease of the maximum values of the bubble wall velocity and pressure at the bubble wall and to a prolongation of the first collapse time of the bubble. On the other Table 3.1 Rheological data of hydroxyethylcellulose (HEC) and carboxymethylellulose (CMC) aqueous solutions modelled with the Williamson relationship

Water 0.5% HEC 0.5% CMC

η0 (Pa·s)

η∞ (Pa·s)

k (s–1 )

n

ρ∞ (kg/m3 )

σ (N/m)

c∞ (m/s)

9.8×10–1

1.01×10–3 4.02×10–3 6.2×10–3

0.035 0.0395

0.772 0.746

999.64 1,002 1,002

0.0725 0.07 0.07

1,496 1,496 1,496

4.4

3.1

Spherical Bubble Dynamics

77

Fig. 3.1 Maximum dimensionless velocity of the bubble wall during first bubble collapse, Vmax /U, where U = (p∞ /ρ ∞ )1/2 , (top) and maximum pressure at the bubble wall, pmax /p∞ , (bottom) versus minimum bubble radius at the end of first bubble collapse, Rmin . The far-right points correspond to an initial bubble radius R0 = 1 mm and the far-left ones to R0 =10–2 mm. The full symbols are the results of incompressible formulation and the open ones of compressible formulation. Circles: water, triangles: 0.5% hydroxyethylcelullose solution and squares: 0.5% carboxymethylcelullose solution. The calculations were conducted for p0 /p∞ = 10–4 . Reproduced with permission from Brujan (1998). © IOP Publishing Ltd

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Fig. 3.2 Pressure distribution in the liquid after the collapse of a bubble with initial radius R0 = 10–2 mm situated in water (solid line) and 0.5% carboxymethylcelullose (CMC) solution (dashed line with one point) for p0 /p∞ = 10–4 . The dimensionless time is defined as t∗ = t(p∞ /ρ ∞ )1/2 /R0 . The dotted line indicates the position of the bubble wall. Reproduced with permission from Brujan (1998). © IOP Publishing Ltd

hand, it was found that, for values of the initial bubble radius R0 > 10–1 mm, sound emission is the main damping mechanism in spherical bubble collapse. Liquid compressibility plays an important role in the formation of shock waves during the rebounding phase of the bubble. Hickling and Plesset (1964) where the first to suggest, from numerical calculations in Newtonian liquids, that the amplitude of the spherical acoustic wave is inversely proportional to the distance r from the collapse centre. As can be seen in Fig. 3.2, in the range 10–2 mm < Rmax < 1 mm, the 1/r law of pressure attenuation through the liquid is not affected by the shearthinning characteristic of liquid viscosity. 3.1.2.2 Bubble Behaviour in Linear Viscoelastic Fluids The earliest theoretical treatment of bubble collapse in incompressible linear viscoelastic liquids is that of Fogler and Goddard (1970) who considered the collapse of a spherical bubble in a liquid model including stress accumulation with fading memory. Later, Tanasawa and Yang (1970), Yang and Lawson (1974), Ting (1975, 1977), McComb and Ayyash (1980), Tsujino et al. (1988b), and Agarwal (2002) used an Oldroyd model, and Shima et al. (1988) a Jeffreys model. More recently, Ichihara et al. (2004), and Ichihara (2008) studied the bubble oscillation in the context of magma fragmentation using a linear Maxwell model. A theoretical treatment of spherical bubble dynamics in a compressible linear viscoelastic fluid was formulated by Brujan (1999). In this study, as in the incompressible formulations of Tanasawa and Yang (1970), Yang and Lawson (1974),

3.1

Spherical Bubble Dynamics

79

Fig. 3.3 The effect of Deborah number on the dimensionless maximum velocity attained during the first collapse, |R∗ |min = |R˙ min |/R0 , and dimensionless minimum radius at the end of the collapse, R∗ min = Rmin /R0 , for χ = λ2 /λ1 = 10–1 . The filled symbols indicate the results obtained using the incompressible formulation, the open ones using the compressible formulation. Circles: Newtonian liquid, diamonds: De = 10–2 , squares: De = 10–1 , triangles (): De = 1, triangles (∇): De = 10 and hexagons: inviscid liquid. Reproduced with permission from Brujan (1999). © Elsevier B.V.

Tsujino et al. (1988b) and Shima et al. (1988), the three-parameter, linear Oldroyd model was employed to represent the rheological behaviour of a viscoelastic liquid (see Example 3.4). The effect of Deborah number on the behaviour of a spherical bubble is illustrated in Fig. 3.3, which shows the maximum dimensionless velocity of the bubble wall plotted as a function of the minimum bubble radius, for three values of the Reynolds number Re = 10, 102 , 103 and λ2 /λ1 = 10−1 . Figure 3.4 shows the influence of the ratio λ2 / λ1 on the maximum dimensionless velocity of the bubble wall and minimum radius of the bubble, for three values of the Reynolds number Re = 10, 102 , 103 and De = 10. It can be seen that the liquid elasticity accelerates the bubble collapse, in agreement with the predictions of Ting (1975), Tsujino et al. (1988b), and Agarwal (2002) while the effect of liquid viscosity and retardation time is to decelerate the bubble collapse. These results further indicate that, under conditions comparable to those existing during cavitation, the effect of liquid rheology on spherical bubble dynamics is negligible for values of the Reynolds number larger than 102 and the only significant influence is that of liquid compressibility. The noticeable effect of liquid rheology was found only for Reynolds-values smaller than 102 . In both situations, as in the case of a shear-thinning fluid, the 1/r law of pressure attenuation through the liquid is not affected by the viscoelastic properties of the liquid. The results presented by Fogler and Goddard (1970) show that fluid elasticity can have an important effect on bubble collapse. However, for conditions similar to

80

3

Bubble Dynamics

Fig. 3.4 The effect of ratio χ = λ2 /λ1 on the dimensionless maximum velocity attained during the first collapse, |R∗ |min = |R˙ min |/R0 , and dimensionless minimum radius at the end of the collapse, R∗ min = Rmin /R0 , for De = 10. The filled symbols indicate the results obtained using the incompressible formulation, the open ones using the compressible formulation. Circles: Newtonian liquid, diamonds: χ = 10–1 , squares: χ = 10–2 , triangles χ = 0, and hexagons: inviscid liquid. Reproduced with permission from Brujan (1999). © Elsevier B.V.

cavitation, one would not expect to be in a parameter range where differences from Newtonian response are appreciable. In fact, when the characteristic time for bubble collapse is in the microsecond range, as it is for cavitation, Rayleigh–Plesset inertial solution appears to be entirely satisfactory. An analysis of surface-tension driven oscillations of a bubble was performed by Inge and Bark (1982), who also restricted the rheology to linear viscoelasticity. They found that the effects of elasticity are small, and comparable to viscous effects. We close our discussion of spherical bubble dynamics in quiescent viscoelastic liquids with the important theoretical contribution of Ryskin (1990). By incorporating the polymer-induced stress calculated using a “yo–yo” model which accounts for the unravelling of the polymer molecules, Ryskin computed the growth and collapse phase of a vapour bubble. He concluded that the growth of the bubble is not affected by the polymer, but the final stage of collapse is. He showed that there is a total arrest of the collapse, with the bubble wall velocity reduced to nearly zero, when the bubble radius becomes about 10% of the radius at the initiation of collapse.

3.1.2.3 Bubble Behaviour in Non-linear Viscoelastic Liquids The numerical simulation of spherical bubble collapse in non-linear viscoelastic liquid is complicated by the fact that the extra stress tensor has a finite trace. In contrast to the case of a linear viscoelastic liquid the extra stress tensor has two

3.1

Spherical Bubble Dynamics

81

Fig. 3.5 Influence of Deborah number on bubble collapse in an upper convective Maxwell liquid for Re = 10. Reproduced with permission from Kim (1994). © Elsevier B.V.

components instead of one and, therefore, the problem cannot be reduced to a single differential equation. Kim (1994) solved the continuity and momentum equations in a Lagrangian frame for the study of the free oscillations of a spherical bubble in an upper convective Maxwell liquid (see Eq. (1.49)). He implemented the Galerkin-finite element method for solving these equations and compared some of his results with those obtained by Fogler and Goddard (1970). The significant parameters of his study are the Reynolds and Deborah numbers. He found that, for values of the Reynolds number smaller than 10, the fluid elasticity accelerates the collapse in the early stage of the collapse while in the later stages it retards the collapse. Figure 3.5 shows an example of bubble oscillation for Re = 10. He also noted that, with increasing the values of the Deborah number, the bubble behaviour follows closely that in an ideal liquid. The differences between a viscoelastic and an ideal liquid becomes smaller and smaller as the Reynolds number or the Deborah number increases. Figure 3.6 illustrates this trend for the case De = 0.02 and several values of Re between 1 and 10. Similar trends were reported by Brutyan and Krapivsky (1991) who used an Oldroyd model, and Shulman and Levitsky (1987) and Jimenez-Fernandez and Crespo (2006) who investigated the behaviour of spherical bubbles in an Oldroyd-B liquid and upper convective Maxwell liquid, respectively.

3.1.3 Heat and Mass Transfer Through the Bubble Wall Ting (1977) employed an Oldroyd three-constant model with characteristic relaxation and retardation times multiplying the covariant convected time derivatives (Schowalter 1978) of the stress and strain rate, respectively. He allowed for thermal effects due to the phase change of water being evaporated or condensed. The resulting integro-differential equation was solved numerically for the case of a 500 ppm

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Bubble Dynamics

Fig. 3.6 Bubble collapse in an upper convective Maxwell liquid for De = 0.2. Reproduced with permission from Kim (1994). © Elsevier B.V.

solution of polyethylene oxide. He concluded that viscoelasticity has a very limited retardation effect on bubble growth and collapse, provided the material constants are compatible with dilute polymer solutions properties. It also appears from the work of Ting that the effects of heat and mass transfer are not important under cavitation conditions. Zana and Leal (1975) numerically solved the conservation equations of mass and momentum along with a gas diffusion equation for a single bubble collapse. A complicated constitutive equation incorporating several material parameters was employed and the results were compared with the corresponding Newtonian case. They found that viscoelastic effects coupled with gas diffusion had profoundly impacted only the dissolution of gas bubbles. For a study of some other situations where diffusive effects are important, the reader is referred to the work of Burman and Jameson (1978), Yoo and Han (1982), Shulman and Levitsky (1992), and Venerus et al. (1998).

3.1.4 Experimental Results A convenient method to produce a single bubble in a liquid is to focus a short pulse of laser light into the liquid. Depending on the focal spot size, the transverse mode structure of the laser, the pulse duration, and the light intensity a small or several small volumes of liquid are rapidly heated, in nanoseconds, picoseconds or femtoseconds, according to the laser and the pulse width employed. Figure 3.7 shows a high-speed photographic record of the dynamics of a laserinduced bubble in water and the corresponding hydrophone signals, at a distance r = 10 mm from the laser focus, for a value of the laser pulse energy EL = 10 mJ. When the laser-induced stress transients possess a sufficiently short rise time, their propagation results in the formation of a shock wave. The large pressure in the laser-induced vapour bubble leads to a very rapid expansion that overshoots the

3.1

Spherical Bubble Dynamics

83

Fig. 3.7 Bubble dynamics in water and the corresponding pressure signal measured at a distance of 10 mm from the laser focus for a laser pulse energy EL = 10 mJ. The first frame was taken 15 μs after the moment of optical breakdown, and the frame interval is 20 μs. Frame width 4 mm

equilibrium state, in which the internal bubble pressure equals the hydrostatic pressure. The increasing difference between the hydrostatic pressure and the falling internal bubble pressure then decelerates the expansion and brings it to a halt. At this point, the kinetic energy of the liquid during bubble expansion has been transformed into the potential energy of the expanded bubble. The bubble energy is related to the radius of the bubble at its maximum expansion, Rmax , and the difference between the hydrostatic pressure, p∞ , and the vapour pressure, pv , inside the bubble by: EB =

4π (p∞ − pv )R3max . 3

(3.16)

The expanded bubble collapses again due to the static background fluid pressure. The collapse compresses the bubble content into a very small volume, thus generating a very high pressure that can exceed 1 GPa for an approximately spherical bubble collapse (Vogel et al. 1989; Brujan et al. 2008). The rebound of the compressed bubble interior leads to the emission of a strong pressure transient into the surrounding liquid that can evolve into a shock wave. Even a third pressure transient generated during second bubble collapse can be observed in this figure. The time from optical breakdown to the first collapse is denoted by 2Tc , where Tc is the collapse time that is proportional to the maximum bubble radius Rmax . The Rayleigh collapse time is given by (Rayleigh 1917) / Tc = 0.915Rmax

ρ∞ , p∞ − pv

(3.17)

where ρ ∞ is the liquid density, and was derived by Lord Rayleigh for the case of an empty bubble without surface tension and viscosity, collapsing under a constant pressure. It has been found that for spherical laser-produced bubbles expanding and contracting under the action of the static ambient pressure in water under normal conditions, the expansion phase and the contraction phase are to a high degree symmetrical so that the time from generation to first collapse is twice the Rayleigh collapse time.

84

3

Bubble Dynamics

Fig. 3.8 Oscillation time of a spherical bubble in a 0.5% polyacrylamide (PAM) solution and 0.5% carboxymethylcelullose (CMC) solution. The solid line represents twice Rayleigh’s collapse time. Reproduced with permission from Brujan and Williams (2006). © Elsevier B.V.

The results of numerous experiments conducted to investigate the behaviour of laser-generated bubbles in viscoelastic liquids (specifically, carboxymethycellulose and polyacrylamide aqueous solutions in concentration of 0.5%) have been described by Brujan et al. (1996). It was observed that, for bubbles whose maximum radius is larger than 0.5 mm, the polymer additives, even in the case of polyacrylamide for which the aqueous solution display marked viscoelastic effects, did not affect the behaviour of bubbles in any significant way, and the duration of the oscillation time (2Tc ) is equal to the Rayleigh time. However, for bubbles whose maximum radius is smaller than 0.5 mm, a slight prolongation of the oscillation time was observed, which increases with decreasing maximum bubble radius (Fig. 3.8). In this figure, the maximal bubble radius is shown as a function of the laser pulse energy (Brujan and Williams 2006). No difference between the case of water and both polymer solutions is seen indicating that the growth phase of the bubble is not affected by the polymer additive. The scaling law for the size of the bubble oscillating in both polymer solutions is the same as that in water, namely, the maximum bubble radius is proportional to the cube root of the laser pulse energy (Fig. 3.9). This scaling law applies, however, only to laser pulse energies larger than 2 mJ, well above the breakdown threshold. For lower values, the energy dependence of the bubble size is stronger. In all previous experimental studies, no significant influence of the polymer additives on spherical bubbles was observed. Ting and Ellis (1974) used polyethylene oxide (PEO) and Guar Gum aqueous solutions in concentration as high as 1,000 ppm, Chahine and Fruman (1979) used distilled water and a 250 ppm solution of PEO (Polyox WSR 301) with a viscosity two times larger than that of water, and Kezios and Schowalter (1986) used different polymer solutions whose viscosity was up to 10–2 Pa·s. They indicated that the time and amplitude of the first and second rebounds were unaffected by the polymer additive. It should be noted here

3.1

Spherical Bubble Dynamics

85

Fig. 3.9 Maximal cavitation bubble radius, Rmax , as a function of the laser pulse energy, EL . The slope of the straight line gives the scaling law for the bubble radius at energy values well above the breakdown threshold. The same scaling law applies in water and polymer solutions: 1/3 Rmax ∝ EL . Reproduced with permission from Brujan and Williams (2006). © Elsevier B.V.

that the bubbles generated in their experiments were extremely large, with a maximum radius Rmax > 1 mm. The negligible effect of polymer additives on growth and collapse of spherical bubbles has also been noted by Hara (1983). More recently, Bazilevskii et al. (2003) have investigated the growth and collapse of spherical bubbles with maximum radii of about 0.1 mm generated in polyacryamide aqueous solutions in concentrations of up to 0.6%. They noted that the growth phase of the bubble is not affected by the polymer additive and, at high polymer concentration, they also observed a slight increase of the collapse time of the bubble in comparison to the case of water. It is worth noting here that a direct comparison between experiments and numerical results is difficult owing to the limitations in the constitutive equations used and/or in the rheological data presented in all of the above-mentioned studies. It is clear, however, from the experimental work, that even a strong shear-thinning component of fluid viscosity and a high degree of elasticity of the fluid surrounding the bubble cannot influence the collapse of spherical bubbles dramatically. The maximum radius of the bubbles generated in these experiments is larger than 10–1 mm and the viscosity of the polymer solutions used as testing liquids is smaller than 10–2 Pa·s, so that the Reynolds number associated with the bubble motion is larger than 102 . Obviously, the collapse of such large bubbles is dominated by inertia, irrespective of any details of fluid rheology. It should be also noted that a significant reduction of the maximum bubble size can be obtained by using laser pulses of picosecond or femtosecond duration. Such a short pulse offers the possibility to produce bubbles with a maximum radius of the order of 10–2 mm. Using such small bubbles, it is possible to achieve small enough values of Reynolds number to detect the influence of liquid rheology even in the case of dilute polymer solutions. Numerical predictions in spherical bubble dynamics is possible, but there is a need for experimental results using well-characterised liquids which can be described by more sophisticated constitutive models than those that have been used previously.

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3.1.5 Bubbles in a Sound-Irradiated Liquid A spherical bubble in a liquid can be viewed as an oscillator that can be set into radial oscillations by a sound field. For very small sound pressure amplitudes these oscillations can be considered as being linear about the equilibrium radius of the bubble. The response then is that of a linear oscillator. Going up in the driving amplitude will bring out the effects of non-linearity manifesting themselves in the occurrence of several resonances (Lauterborn 1976). The behaviour of a bubble in a sound field can be described by the theoretical models outlined in section. The theoretical description starts with a bubble nuclei with radius R0 . At time t = 0, the pressure inside the bubble nuclei, p0 , is balanced by the static pressure in the surrounding liquid, P0 , and the surface tension, σ: p0 = P0 +

2σ . R

(3.18)

Provided that the bubble in the viscoelastic liquid is subjected to a periodically varying pressure, the pressure p∞ far from the bubble can be expressed by p∞ = P0 (1 + A sin 2π ft),

(3.19)

where A is the ratio of the pulsating pressure amplitude to the static pressure and f is the frequency of the pulsating pressure. This section presents some general characteristics of the motion of a single gas bubble driven sinusoidally in non-Newtonian liquids. 3.1.5.1 Frequency Response Curves The behaviour of a single spherical bubble situated in a sound field and in a purely viscous liquid was investigated by Shima et al. (1985), Tsujino et al. (1988a), and Brujan (1994b). On the other hand, Shima et al. (1986) studied the bubble oscillations using a linear viscoelastic relationship to describe the liquid rheology. Figure 3.10 shows an example of frequency response curves of a bubble situated in a Williamson liquid (see Example 3.3), as predicted by the incompressible formulation of Brujan (1994b) at A = 0.4, for water and polyethylene oxide (PEO) solutions in concentration of up to 1.5%. Here, the normalized maximum radius, (Rmax − R0 )/R0 , during one period of the driving frequency after the solution has reached steady state, is plotted as a function of the ratio between the frequency of the sound field f and the resonance frequency of the bubble f0 . The maximum response occurs when f /f0 is nearly equal to 1. Other peaks are seen at or near f /f0 = 1/2, 1/3, 1/4. They are known as the harmonics of the resonance response. These peaks have been labeled with an expression m/n, known as the order of the resonance, according to the notation introduced by Lauterborn (1976). The case m = 1, n = 2, 3, . . . , denotes the well known harmonics, the resonances n = 1 and m = 2, 3, . . ., are called subharmonics of order 1/2, 1/3, . . . The resonances n = 2, 3, . . . , and

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Fig. 3.10 Frequency response curve of a spherical bubble oscillating in water (dashed line), 0.5% carboxymethylcelullose (CMC) solution (solid line), 1% CMC solution (dotted line with one point), and 1.5% CMC solution (dotted line with two points). The initial bubble radius is R0 = 0.1 mm and the amplitude of the oscillating pressure field is A = 0.4. Reproduced with permission from Brujan (1994b). © Elsevier B.V.

m = 2, 3, ..., are called ultraharmonics. It is clear from this figure that the resonances are strongly damped or even suppressed with increasing polymer concentration. Whereas the harmonic resonances of order 2/1 and 3/1, respectively, are found in water and in all polymer solutions, the subharmonic resonance of order 4/1 is not found in the 1% PEO solution while the subharmonic resonance of order 1/2 is found only in water and in the 0.5% PEO solution. For f /f0 = 0.641, the ultraharmonic resonance of order 3/2 was found only in water. The numerical calculations indicated that the rheological parameter which is influential in this respect is the infinite-shear viscosity, η∞ . The larger the value of η∞ , the smaller are the values of the normalized bubble radius during one period of bubble oscillation leading finally to the observed damping of the resonances. We also note that the non-linearity of the bubble oscillation has a softening effect. The values of f /f0 at the point of primary resonance move to the low frequency side and this value increases with the polymer concentration. For example, the value of f /f0 at the primary resonance is 0.8 for a 0.5% PEO solution, 0.855 for a 1% PEO solution and 0.865 for a 1.5% PEO solution, respectively. It was also found that the increase of polymer concentration leads to a reduction of the maximum pressure inside the bubble. Similar observations have been made by Shima et al. (1985) and Tsujino et al. (1988a) who considered the bubble oscillations in a Powell–Eyring liquid and in a Carreau-like liquid, respectively.

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Fig. 3.11 Influence of the relaxation time of the liquid λ1 on the frequency response curve of a spherical bubble oscillating in a viscoelastic liquid. The liquid rheology is described by the three-parameter linear Oldroyd model. Newtonian liquid (dashed line), λ1 = 10−5 s (solid line), λ1 = 10−4 s (dotted line with one point), and λ1 = 10−3 s (dotted line with two points). Other values adopted in the numerical calculations are: initial bubble radius R0 = 0.1 mm, amplitude of the oscillating pressure field A = 0.2, viscosity η = 10−1 Pa·s, and retardation time λ2 = 10−5 s. Reproduced with permission from Shima et al. (1986). © Elsevier B.V.

Shima et al. (1986) obtained the frequency response curves of spherical bubbles using a three-parameter linear Oldroyd model (see Example 3.4). They found that the harmonic and subharmonic resonances are more easily generated in elastic liquids and the normalized maximum radius, (Rmax − R0 )/R0 , increases with the relaxation time of the liquid λ1 (Fig. 3.11). For example, for A = 0.2, η = 0.1 Pa·s and λ2 = 10−5 s, the values of (Rmax − R0 )/R0 at the primary resonance is with a factor of 6 larger than in the case of a Newtonian liquid. On the other hand, the increase of the retardation time λ1 leads to a decrease of the normalized bubble radius and to a strong damping of the resonances (Fig. 3.12). In particular, the subharmonic resonance of order 1/2 and the harmonic resonances of order 3/1 and 4/1 are the most affected ones. More generally, the authors noted that for λ1 /.λ2 > 10 the values of (Rmax − R0 )/R0 are larger than the corresponding values in a Newtonian liquid, while for λ1 /.λ2 < 1, (Rmax − R0 )/R0 is smaller. Similar trends have been observed for the pressure at the bubble wall. Recently, numerical investigations on the non-linear bubble oscillations in viscoelastic liquids have been carried out by Allen and Roy (2000a, b), using the linear Jeffreys model, as well as the upper convective Maxwell model, and

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Fig. 3.12 Influence of the retardation time of the liquid λ2 on the frequency response curve of a spherical bubble oscillating in a viscoelastic liquid. The liquid rheology is described by the three-parameter linear Oldroyd model. Newtonian liquid (dashed line), λ2 = 10−4 s (solid line), and λ2 = 10−5 s (dotted line with one point). Other values adopted in the numerical calculations are: initial bubble radius R0 = 0.1 mm, amplitude of pressure field A = 0.2, viscosity η = 10−1 Pa·s, relaxation time λ1 = 10−4 s. Reproduced with permission from Shima et al. (1986). © Elsevier B.V.

Jimenez-Fernandez and Crespo (2005) who used a differential constitutive equation with an interpolated time derivative which includes the Oldroyd-B model and the upper convective Maxwell model as particular cases. Their results confirm the previous trend quoted above on subharmonics enhancement in elastic liquids. It was also shown that the fluid elasticity produces a significant growth of the amplitude of bubble oscillations. Figure 3.13 illustrates an example of the influence of liquid elasticity on the oscillation of a spherical bubble situated in an upper convective Maxwell fluid (Jimenez-Fernandez and Crespo 2005). Here the Deborah number is defined as De = 2π f λ, where λ is the relaxation time of the fluid, while the Reynolds number is Re = 2π f ρR20 /η, with ρ and η the density and viscosity of the liquid, respectively. 3.1.5.2 Chaotic Oscillations Up to this point, all quantities have been given a single value for each solution if, after reaching steady state, the solutions have the same period as the driving pressure. But this not always the case, especially at high pressure amplitudes where the non-linear effects are more prominent. One of the most significant developments in bubble dynamics is the realization that the bubble response to a time-periodic pressure field can be chaotic, even when the bubble is assumed to remain spherical. Parlitz et al. (1990) have shown that the notions of strange attractors and perioddoubling bifurcations can all be illustrated in the Rayleigh–Plesset framework.

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Fig. 3.13 Normalized bubble radius, R/R0 , versus time for a bubble oscillating in an upper convective Maxwell fluid for Re = 0.63, and two values of the Deborah number, De =1 (lower values) and De = 2. Time is denoted in acoustic periods 1/f. The initial bubble radius is R0 = 1 μm, and the amplitude and frequency of the pressure field are A = 0.4 and f = 3 MHz, respectively. Reproduced with permission from Jimenez-Fernandez and Crespo (2005). © Elsevier B.V.

Due to the complexity of the problem the numerical investigations on the chaotic oscillations of spherical bubbles in non-Newtonian liquids were restricted only to the case of purely viscous liquids (Brujan 2009b). It was shown that the chaotic oscillations of the bubble are suppressed by the polymer additives and the infiniteshear viscosity of the liquid is the rheological parameter with the strongest influence in this respect. This trend is illustrated in Fig. 3.14 which shows the bifurcation diagrams of a bubble oscillating in water and carboxymethylcellulose (CMC) solutions for a value of the pressure ratio A = 0.95. The rheological data of the polymer solutions modelled with the Williamson relationship are listed in Table 3.2. Saddle-node bifurcations, period-doubling cascades and strange attractors occur when the bubble oscillates in water (Fig. 3.14a). When the bubble oscillates in the 0.5% CMC solution (Fig. 3.14c) no period-doubling cascades and strange attractors are visible in the range of f /.f0 -values investigated. Only saddle-node bifurcations are observed in connection with the resonances R2,1 and R1,1 and the ultraharmonic resonances R3,2 , R5,2 while the subharmonic resonance R1,2 is suppressed. As the polymer concentration is further increased to 1% (Fig. 3.14d), even the saddle-node bifurcations vanish and the bubble oscillates in a stationary state with a period equal to that of the driving field. Examples of chaotic oscillations of a single spherical bubble situated in a viscoelastic liquid are given by Jimenez-Fernandez and Crespo (2005) and Naude and Mendez (2008). They concluded that liquid elasticity may enhance the chaotic oscillations of bubbles even at moderate values of the driving pressure field. No influence of liquid elasticity on the number of collapses in a fixed amount of time was observed.

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Fig. 3.14 Bifurcation diagrams of a spherical bubble situated in water and carboxymethylcellulose (CMC) aqueous solutions. (a) water, (b) 0.2% CMC solution, (c) 0.5% CMC solution, and (d) 1% CMC solution. The initial bubble radius is R0 = 10–2 mm and the amplitude of the oscillating pressure field is A = 0.95. Reproduced with permission from Brujan (2009b). © S. Hirzel Verlag Table 3.2 Rheological data of carboxymethylcellulose (CMC) aqueous solutions modelled with the Williamson relationship

Water 0.2% CMC 0.5% CMC 1.0% CMC

η0 (Pa·s)

η∞ (Pa·s)

k (s–1 )

n

ρ∞ (kg/m3 )

σ (N/m)

2.026×10–2 3.183×10–2 4.902

10–3 4.72×10–3 1.28×10–2 3.28×10–2

51.4 10.12 1.051

0.715 0.625 0.619

999.6 1, 000.2 1, 002 1, 005

0.072 0.075 0.074 0.073

3.2 Aspherical Bubble Dynamics While the events during bubble generation are not influenced by the viscoelastic properties of the fluid, the subsequent bubble dynamics is primarily influenced by the boundary conditions in the neighbourhood of the bubble and the properties of the fluid. A spherical bubble produced in an unconfined liquid retains its spherical shape while oscillating and the bubble collapse takes place at the site of bubble formation. When the bubble oscillates under asymmetric boundary conditions, it is usually exposed to pressure gradients. This leads to a faster collapse of the bubble section exposed to a higher pressure and to the formation of a liquid jet even for an initially spherical bubble. When the bubble collapses in the vicinity of a rigid boundary, the jet is directed toward the boundary (Plesset and Chapman 1971; Blake et al. 1986; Brujan et al. 2002). The pressure gradient causing the jet formation is

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due to the low-pressure region between bubble and rigid wall developing during bubble collapse. During the initial collapse phase, the bubble acquires the form of a prolate spheroid. This shape also contributes to the formation of the liquid jet. A bubble oscillating between two parallel rigid walls is subjected to two opposite pressure-gradient forces and the collapse is characterised by the formation of two liquid jets that are directed toward each wall (Chahine 1982).

3.2.1 Bubbles Near a Rigid Wall Of utmost interest is the case of a bubble near a rigid boundary because bubbles are the source of cavitation erosion. The use of a normalised distance γ = s / Rmax where s is the distance of the bubble inception from the boundary has proven advantageous to classify bubble dynamics near a plane rigid boundary. Bubbles with different Rmax but the same γ -value exhibit similar dynamics, thus giving the chance to specify the degree of asymmetry of bubble collapse: cavitation bubbles with a small value of γ are more influenced by the boundary, thus collapsing with a more pronounced shape variation, than those with a large value for which collapse is more sphere-like. This statement, however, does not apply to bubbles too close to the boundary, where γ ≈ 0 and the bubble adopts a hemispherical shape, i.e. approaches a spherical symmetry again.

3.2.1.1 Behaviour of the Jet Figure 3.15 shows a series of high-speed photographic records of bubble motion in water, a 0.5% carboxymethylcellulose (CMC) solution with a weak elastic component, and a 0.5% polyacrylamide (PAM) solution with a strong elastic component for the case where γ = 3.17 (Brujan et al. 1996). The liquid jet, which is developed on the upper side of the bubble leading to the protrusion of the lower bubble wall, can be seen in the case of bubbles situated in water (top sequence). A similar bubble shape is found in the CMC solution, but, in this case, the jet is not as strong as in the case of water. The most interesting behaviour for a bubble situated in the vicinity of a rigid boundary was found for the case of the PAM solution. The liquid jet is not observed and a flat form of the bubble shape is the dominant aspect of bubble motion after the first collapse. In the case of the PAM solution, the maximum jet velocity was found to be 88 m/s, a value which represents about 78% of the corresponding velocity in water (113 m/s). For the CMC solution the jet velocity, 102 m/s, is almost the same as that for the case of water. Similar observations have been made when γ was reduced to 1.67 (Fig. 3.16). At first sight the addition of polymers into water has a less significant effect on the bubble collapse because the liquid jet inside the bubble was observed for water and both polymer solutions. Even the protrusion sticking upwards out of the bubble, formerly called counterjet by Vogel et al. (1989), can be seen in water and in the 0.5% CMC solution. However, a significant influence of the polymer additives was noted for the velocity of the re-entrant jet.

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Fig. 3.15 Picture sequences of the behaviour of a laser-induced bubble near a rigid wall in water and polymer solutions for γ = 3.17. Top: water; Middle: 0.5% carboxymethylcelullose solution; Bottom: 0.5% polyacrylamide solution. Frame interval 4.8 μs, frame width 1.7 mm. Reproduced with permission from Brujan et al. (1996). © S. Hirzel Verlag

Whereas in the case of water the maximum jet velocity is 104 m/s, only 63 m/s was measured for the PAM solution. In a previous experimental study, Chahine and Fruman (1979) indicated that although bubble growth is not sensitive to addition of 250 ppm of polyethylene oxide to water, the collapse sequence and the shape near a rigid boundary are appreciable affected. In particular, they also observed that the polymer additive introduces a retardation effect over the initiation of the re-entering jet developed during bubble collapse. 3.2.1.2 Acoustic Emission Upon Bubble Collapse Even bubble collapse in contact with a rigid boundary is accompanied by the emission of shock waves. Figure 3.17 shows a high-speed photographic sequence of the final stage of bubble collapse in water taken with 200 million frames/s (Brujan et al. 2008). Two shock waves are emitted upon the first collapse of the bubble. From the

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Fig. 3.16 Picture sequences of the behaviour of a laser-induced bubble near a rigid wall in water and polymer solutions for γ = 1.67. Top: water; Middle: 0.5% carboxymethylcelullose solution; Bottom: 0.5% polyacrylamide solution. Frame interval 4.8 μs, frame width 1.7mm. Reproduced with permission from Brujan et al. (1996). © S. Hirzel Verlag

radius of the shock waves, the time interval between the emissions was calculated as t ≈ 100 ns. The first wave (the shock wave with the larger diameter indicated by the black arrowhead) is created by the impact of the high-speed liquid microjet onto the rigid wall, and the second one (indicated by the white arrowheads) as a consequence of the strong compression of the bubble content at its minimum volume. The microjet-induced shock wave is, however, so weak that it is barely visible on the photographic frames. Since it was substantiated that the viscoelastic properties of the surrounding liquid might affect the collapse of a cavitation bubble situated near a rigid boundary, further studies have investigated the dependence of the pressure amplitude of the acoustic transients emitted during bubble collapse with γ (Brujan et al. 2004; Brujan 2008). In these studies, two polymer solutions were investigated, namely a polyacrylamide (PAM) aqueous solution and a carboxymethylcellulose (CMC) aqueous solution, both in a concentration of 0.5%. The extensional properties, in the form of an apparent Trouton ratio (Tr = ηe /η), for both polymer solutions were measured in uniaxial extension using a Rheometric RFX opposed-jet apparatus with 1 mm diameter nozzles. The general behaviour of the PAM solution is that it is extension

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Fig. 3.17 A high-speed photographic sequence in side view showing the propagation of the shock waves emitted upon the collapse of a cavitation bubble attached to rigid boundary. The shock wave indicated by the black arrowhead is generated at the impact of the liquid jet developed in an earlier stage of bubble collapse onto the rigid wall. The shock wave indicated by the white arrowheads is generated at the minimum bubble volume. The rigid boundary is located in the right-hand side of each frame. Sequence taken with 200 million frames/s and an exposure time of 5 ns. Reproduced with permission from Brujan et al. (2008). © Elsevier B.V.

rate thickening, which is a general characteristic for flexible polymers. The apparent Trouton ratio for the PAM solution was initially at a value of Tr ≈ 4.5 at low extension rates and then it increased to attain a maximum of Tr ≈ 70 at extension rates of ε˙ ≈ 4, 000 s–1 , indicating a strong elastic component. The apparent Trouton ratio for the CMC solution was relatively constant at a value of about 5 for all the extension rates investigated, indicating a relatively less elastic behaviour of the polymer solution. Figure 3.18 shows the amplitude of the acoustic transients emitted during first bubble collapse, pmax , as a function of γ in water and both polymer solutions. It can be seen that the largest values of the maximum amplitude of the acoustic transients

Fig. 3.18 Pressure amplitude of the acoustic transients emitted during first bubble collapse as a function of γ . The pressure values are measured at a distance of 10 mm from the ultrasound focus. Reproduced with permission from Brujan et al. (2004). © American Institute of Physics

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Fig. 3.19 Maximum jet velocity as a function of the stand-off parameter γ for bubbles situated in water, 0.5% carboxymethylcellulose (CMC) aqueous solution, and 0.5% polyacrylamide (PAM) aqueous solution. The jet velocity is averaged over 3 μs. Reproduced with permission from Brujan et al. (2004). © American Institute of Physics

are obtained in water. For the relatively less elastic 0.5% CMC solution, the bubble dynamics do not differ substantially from that in water and the maximum amplitude of the acoustic transients emitted during bubble collapse is almost similar to that in water. For the elastic 0.5% PAM solution, however, a significant reduction of pmax was observed. We further note that the most pronounced reduction of the shock pressure in the PAM solution was observed for γ < 0.6 and γ > 1.5. Figure 3.19 shows that the velocity of the liquid jet developed during the final stage of bubble collapse range from about 10 m/s up to 50 m/s. Furthermore, the jet velocity shows a dependence on γ similar to the pressure amplitude of the acoustic transients emitted during bubble collapse: There is a minimum for values γ ≈ 1 and the jet velocity decreases with increasing the extensional viscosity of the liquid. The effect of the viscoelastic properties of the liquid on the sound emission during first bubble collapse can be understood in a heuristic manner. A spherical bubble generated in a liquid of infinite extent retains its spherical shape while oscillating. When the bubble is formed near a rigid boundary, the collapse is associated with the formation of a high-speed liquid jet directed towards the boundary. However, examination of the high-speed photographic sequences shows that the bubble remains near spherical for much of its collapse period (between 90 and 95% depending on γ ), only developing significant non-sphericity at the end of the pulsation. The flow is thus predominantly uniaxial in extension during most of the collapse and the viscosity of both polymer solutions is significantly larger than that of water. Therefore, a large part of the maximum potential energy of the bubble is dissipated during the collapse phase due to an increased resistance to extensional flow which is conferred upon the surrounding liquid by the polymer additive. Consequently, less energy is available for bubble collapse, the bubble content becomes less compressed than in the case of water, and the pressure amplitude of the shock wave is diminished. For large γ -values, the retarding effect of the rigid boundary on the fluid during collapse is small. Therefore, the bubble remains nearly spherical and the liquid jet develops only in a very late stage of the collapse. For γ < 0.6, the bubble is nearly hemispherical and the flow is directed towards the bubble center for most parts of the bubble

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surface, as in the case of a spherical collapse. In both cases, the bubble assumes spherical symmetry for most of the collapse, thus the fluid elements experience a strong uniaxial extensional flow and therefore the energy dissipation during bubble collapse is the largest. The explanation for the significant reduction of the jet velocity is similar as for the acoustic transients emitted during bubble collapse. The presence of the polymer additive confers on the solution an ability to sustain higher extensional stresses than its Newtonian counterpart. This enhanced resistance to extensional deformation reduces the intensity of the re-entrant liquid jet developed during bubble collapse. For γ < 0.6 and γ > 1.5, where the spherical symmetry is preserved during most of bubble collapse, the extensional flow becomes dominant and the reduction of the jet velocity is the largest. Cheny and Walters (1999) have also reported studies of the role of fluid viscoelasticity in the development of liquid jets. In their work, the addition of small amounts of polyacrylamide to a Newtonian solvent was found to lead to an orderof-magnitude reduction in the length of the ascending vertical jet formed when a sphere is dropped into a reservoir of liquid. By analysing the evolution of the shape of these jets, Cheny and Walters concluded that the deformation of the fluid involved substantial elongation. While it is clearly important to bear in mind the different circumstances involved in the work described by Cheny and Walters and that described herein, the role played by elongation during the evolution of the jets in these different experiments is noteworthy, particularly in the case of non-Newtonian fluids. 3.2.1.3 Theoretical Description Using a perturbation approach, Hara and Schowalter (1984) investigated the effect of viscoelasticity on the dynamics of single nonspherical bubbles situated in a quiescent viscoelastic liquid. The constitutive equation they used is of the Maxwell type, similar to that used by Fogler and Goddard (1970). They showed that the effects of fluid rheology on nonspherical bubble dynamics are larger than on spherical bubbles. Nevertheless, growth and collapse of bubbles in an initially unstressed liquid remain dominated by inertia. Their method is, however, limited only to small oscillations of the bubble and cannot describe the motion of the re-entrant liquid jet. It is well known that boundary integral methods are particularly well suited to this class of problems as they involve discretization of the boundaries only. However, application of this method is possible only in the creeping and potential flow limits. The restriction of the boundary integral methods to potential flow problems precludes an exact accounting of the role viscoelastic effects play in the dynamics of cavitation bubbles near boundaries. It is, however, possible to include weak viscoelastic effects in the boundary integral formulation if it is assumed that these effects are limited to a thin region near the interface so that the bulk of the fluid remains irrotational. Lundgren and Mansour (1988) performed an analysis to include weak viscous effects in a boundary integral simulation for an oscillating drop while Boulton-Stone (1995) applied the boundary integral method to study the effect of surfactants on the behaviour of bursting gas bubbles.

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The development of computer codes that would permit the calculation of bubble collapse in a viscoelastic fluid and near a rigid boundary has been slow. Owing to the difficulties involved in implementing both moving boundaries and viscoelasticity, resolution has not been possible anywhere near the experimentally attainable limit, even with present-day computers. Numerical simulations could contribute to a better understanding of the dynamics by providing pressure contours and velocity vectors in the liquid surrounding the bubble which are not easily accessible through experiments.

3.2.2 Bubbles Between Two Rigid Walls When a bubble is initiated between two parallel rigid walls an annular flow is developed during bubble collapse. For a sufficiently small distance between the walls, the annular flow leads to bubble splitting and the formation of two opposing liquid jets directed towards each wall (Chahine 1982). On the other hand, for a sufficiently large distance between the walls the bubble achieves a prolate shape during collapse leading to the formation of two high-speed liquid jets of equal velocity directed towards the bubble centre (Brujan et al. 2005). Chahine and Morine (1980) conducted several tests using bubbles generated in water, and 125 and 250 ppm of polyethylene oxide, respectively. They found that, although the growth phase of the bubble is unaffected by the polymer additive, the lengthening effect on the oscillation period of the bubble is significantly reduced in the case of polymer solutions and the departure from sphericity of the bubbles is considerably delayed. No results were presented by these authors with respect to the influence of polymer additive on the velocity of the liquid jets formed after bubble splitting.

3.2.3 Bubbles in a Shear Flow Virtually all of the previous observations and analyses have focussed on bubble collapse in a quiescent liquid, despite the fact that a number of experimenters have commented on the deformation of cavitation bubbles by the flow (see, for example, Blake et al. 1977). Some of the early observations of individual travelling cavitation bubbles by Knapp and Hollander (1948) make mention of the deformation of the bubbles by the flow. A detailed investigation of the effect of a controlled shear flow on the deformation of laser-generated bubbles was conducted by Kezios and Schowalter (1986) using polyacrylamide (PAM) and polyethylene oxide (PEO) solutions in concentrations of up to 2,000 ppm. The main purpose of their work was to understand the role played by a pre-existing stress field at the moment when cavitation bubbles are generated. They demonstrated that the departure from sphericity is significantly reduced in polymer solutions, in particular in the highly elastic PAM solutions. They also noted that increasing the concentration beyond a critical value reverses the results and they

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speculated that this can be caused by the relative increase of the solution viscosity as compared to its elasticity. Ligneul (1987) also performed experiments with spark-generated bubbles in the shear layer developed by a rotating cylinder. By comparing the behaviour in water and solutions of polyethylene oxide with 50 and 250 ppm concentration, he concluded that the influence of the polymer additive is to maintain sphericity during bubble collapse. The effect of viscoelasticity on cavitation characteristics in flow between eccentric cylinders in relative rotation has been reported by Ashrafi et al. (2001) who found that for low speeds of rotation, the liquid’s free surface departed progressively from the initial horizontal (rest) configuration. With further increase in rotational speed, a provocative fingering mechanism appeared, generating a series of cavities, the number of which increased with rotational speed and eccentricity. The elastic liquids were found to generate more cells than their Newtonian equivalents, the shape of the cavities exhibiting distinctive cusp-like extremities. In this study, fluid elasticity was found to promote cavitation.

3.2.4 Shock-Wave Bubble Interaction The interaction of a shock wave with a bubble in a liquid is of special interest because of the shock-induced formation of a high-speed liquid jet. When a shock wave reaches a resting bubble, it will be almost completely reflected due to the sharp increase in acoustic impedance at the bubble wall. The resulting momentum transfer accelerates the bubble wall and starts the collapse from this side. Together with focusing effects during the collapse stage, this situation finally leads to the formation of a fast liquid jet in the direction of wave propagation. Shima et al. (1984b) examined the shock-induced collapse of bubbles situated in water and polyacrylamide aqueous solutions. They used the streak technique to visualise the collapse phase of the bubble and restricted their investigations only to collapse time. The bubble radius in this experiment was varied between 0.01 and 1 mm. They observed that, for bubbles smaller than 0.05 mm, the collapse time in polyacrylamide solutions with concentration of 0.05 and 0.1% is shorter than that in water (Fig. 3.20). They explained this result as a consequence of the relaxation effect of the polymer solutions. However, no indication was given about the evolution of the liquid jet. Experimental studies of the evolution of shock-induced jets in viscoelastic liquids were conducted by Williams et al. (1998) and Barrow et al. (2004a, b). They found that the jet developed during shock-induced bubble collapse in viscoelastic liquids is either markedly reduced or even suppressed at high values of liquid elasticity. Examples are shown in Figs. 3.21 and 3.22 which illustrates the jet evolution in an elastic liquid and its Newtonian counterpart. Their results support the conclusion that the reduced velocity and final length of such jets in elastic liquids, relative to their Newtonian counterparts, is due to an increased resistance to extensional flow.

100 Fig. 3.20 Collapse time of shock-induced bubbles in water and polyacrylamide solutions in concentration 0.05 and 0.1%. The solid line indicates the theoretical result in water. Reproduced with permission from Shima et al. (1984b). © American Institute of Physics

Fig. 3.21 Shock-induced jet formation in a Newtonian liquid with a shear viscosity 0.3 Pa · s. Frame interval 500 μs. Reproduced with permission from Williams et al. (1998). © Elsevier B.V.

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Bubbles Near an Elastic Boundary

101

Fig. 3.22 Shock-induced jet formation in an elastic liquid with a shear viscosity 0.3 Pa·s. Frame interval 500 μs. Reproduced with permission from Williams et al. (1998). © Elsevier B.V.

3.3 Bubbles Near an Elastic Boundary Not only properties of the liquid surrounding the bubble and distance to the boundary, but also the elastic properties of the nearby boundary material strongly influence bubble dynamics. Gibson (1968) observed 40 years ago that under certain conditions the liquid jet formed during bubble collapse near an elastic boundary as well as the bubble migration are both directed away from the boundary. Since jet impact and the high pressures developed during bubble collapse near a rigid boundary were known as majors factors causing cavitation erosion, the use of compliant boundaries was considered as a means of preventing erosion. This conclusion was also supported by Gibson and Blake (1982) who examined the behaviour of spark-generated bubbles in the vicinity of rigid boundaries with rubber coatings. Blake and Gibson (1987) observed that, for some range of coating properties, no re-entrant jet towards or away from the boundary is developed during bubble collapse. In this case, the bubble collapses from the sides forming an hour-glass shape which can eventually lead to bubble splitting. Similar observations were made by Shima et al. (1989), Duncan and Zhang (1991), Duncan et al. (1996), Shaw et al. (1999), and Kodama and Tomita (2000).

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Fig. 3.23 Interaction of a laser-produced bubble with an elastic boundary with elastic modulus E = 0.25 MPa for γ = 1.14. The boundary is located at the top of each frame. A liquid jet directed away from the boundary develops during bubble collapse. Frame interval 20 μs, frame width 3.5 mm. Reproduced with permission from Brujan et al. (2001a). © Cambridge University Press

Recently, Brujan et al. (2001a, b) have conducted experiments on the interaction of laser-produced bubbles with elastic boundaries. Polyacrylamide (PAA) gel samples, whose elastic properties were controlled by modifying the water content of the sample, were used as elastic boundary materials. The main features of the interaction of bubbles with elastic boundaries are illustrated in Figs. 3.23, 3.24 and 3.25 for a value of the elastic modulus E = 0.25 MPa (Brujan et al. 2001a). The motion of a bubble situated relatively far away from the boundary (γ = 1.14, Fig. 3.23), is characterized by the generation of a high-speed liquid jet and a migration of the bubble away from the boundary. A bubble produced closer to the boundary (γ = 0.62, Fig. 3.24) generates an annular flow in an early stage of bubble collapse leading to bubble splitting during the final stage of collapse and the formation of two axial jets flowing towards and away from the boundary. When the bubble is generated almost at the surface of the boundary (γ = 0.04, Fig. 3.25), a hemispherical cavity is formed in the PAA sample. The sample rebounds early during the growth phase of the bubble and a very strong PAA jet develops which, finally, penetrates the bubble wall opposite to the boundary. No bubble splitting occurs, and the bubble migrates away from the boundary during rebound. The complex behaviour of bubbles near elastic boundaries can be understood by considering the forces acting on the bubble surface. As in the case of a rigid wall, the bubble oscillation is associated with a pressure gradient towards the wall due to the low pressure region between bubble and wall developing during collapse (Bjerknes force). Unlike the rigid wall, however, the material is deformed during bubble expansion, it rebounds, and thus creates a flow and pressure gradient directed away from the wall. The counteracting forces lead to a flattening of the bubble, and

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Fig. 3.24 Interaction of a laser-produced bubble with an elastic boundary with elastic modulus E = 0.25 MPa for γ = 0.62. The boundary is located at the top of each frame. The elastic boundary is compressed during bubble expansion and elevated during bubble collapse. The collapse results in bubble splitting with the formation of two liquid jets in opposite directions. The liquid jet directed towards the boundary penetrates the elastic boundary. Frame interval 20 μs, frame width 3.5 mm. Reproduced with permission from Brujan et al. (2001a). © Cambridge University Press

Fig. 3.25 Interaction of a laser-produced bubble with an elastic boundary with elastic modulus E = 0.25 MPa for γ = 0.04. The boundary is located at the top of each frame. Strong jet-like ejection of boundary material and suppression of liquid jet penetration into the boundary are the main features of the interaction. Frame interval 20 μs, frame width 3.5 mm. Reproduced with permission from Brujan et al. (2001a). © Cambridge University Press

the collapse of the oblate bubble then results in an annular jet, bubble splitting, and the formation of two axial jets in opposite direction. When the opposing forces are not equally strong, the bubble splits into unequal parts, and the jet originating from the larger bubble part is stronger than the other jet in opposite direction. In the limit, however, where one force is much stronger than the other, only one axial jet is formed. For large γ -values, only an axial jet flow directed away from the boundary

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Fig. 3.26 Jetting behaviour of bubbles near polyacrylamide samples with different water content and elastic modulus, respectively: 95% ↔ 0.017 MPa, 85% ↔ 0.124 MPa, 80% ↔ 0.25 MPa, 70% ↔ 0.4 MPa, 60% ↔ 1.04 MPa, 50% ↔ 2.03 MPa. Symbols: , no jet formation; , liquid jet away from the boundary; ∇, liquid jet towards the boundary; ♦, liquid jets away from and towards the boundary; , liquid jets away from and towards the boundary, with jet penetration into the PAA sample; , liquid jets away from and towards the boundary, and jet-like ejection of PAA material; , liquid jets away and towards the boundary, with liquid jet penetration into the boundary and jetlike ejection of PAA material; , liquid jet away from the boundary and jet-like ejection of PAA into the liquid. The dashed line surrounds the bubble splitting region, and the dotted line denotes the state where the “centre of gravity” of the two-bubble system does not migrate. Reproduced with permission from Brujan et al. (2001b). © Cambridge University Press

was observed (Fig. 3.23). This is because the flow induced by the rebounding PAA sample is stronger than the Bjerknes attractive force caused by the low pressure between bubble and boundary. With decreasing γ -value, the strength of the Bjerknes force increases faster than the flow from the boundary. The highest jet velocity is achieved for γ -values around γ = 0.6, where an asymmetric annular jet leads to the formation of a strong axial jet towards the elastic boundary. Photographic series taken with 5 million frames/s yielded a peak velocity for the jet directed towards the boundary as high as 960 m/s. This value is ten times higher than the jet velocity reached near a rigid boundary at a similar γ -value (Brujan et al. 2001b). Figure 3.26 gives an overview of the jetting behaviour as a function of the elastic modulus of the boundary E and the normalized bubble-boundary distance γ (Brujan et al. 2001b). In the region between 85 and 70% water content, corresponding to an elastic modulus of 0.12 MPa < E < 0.4 MPa, the elastic response of the deformed boundary is particularly strong. The rebound of the boundary after its deformation during bubble expansion leads to the formation of a jet flow directed away from the boundary – either as a unidirectional jet or as one component of a pair of jets flowing in opposite directions. An unidirectional liquid jet directed away

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105

Fig. 3.27 Jet velocity for bubbles collapsing near polyacrylamide samples with different water content and elastic modulus (95% ↔ 0.017 MPa, 85% ↔ 0.124 MPa, 80% ↔ 0.25 MPa, 70% ↔ 0.4 MPa, 60% ↔ 1.04 MPa, 50% ↔ 2.03 MPa) and dimensionless bubble-boundary distance γ for jets flowing towards the elastic boundary. The jet velocity is mediated over 1 μs. Reproduced with permission from Brujan et al. (2001b). © Cambridge University Press

from the boundary is only observed for very large and very small γ -values. In the intermediate γ -range, bubble splitting occurs and, besides the jet away from the boundary, a fast jet directed towards the boundary is formed. A characteristic feature of the bubble-splitting-region is the extremely high velocity of the jet directed towards the boundary (Fig. 3.27). The high jet velocity results in a penetration of the PAA samples with water content between 70 and 85%. With larger elastic modulus (E = 2.03 MPa, at 50% water content), the bubble dynamics starts to resemble the behaviour near a rigid wall: the PAA is so stiff that the jet is always directed towards the boundary. With a smaller elastic modulus (E = 0.017 MPa, at 95% water content), the behaviour becomes more similar to the dynamics in an infinite liquid. The elastic response of the boundary is, however, still strong enough to cause formation of a liquid jet directed away from the boundary. Using sophisticated numerical calculations Klaseboer and Khoo (2004), Pei et al. (2005), Klaseboer et al. (2006), and Turangan et al. (2006) were able to simulate many of the dynamical features of bubble oscillations near elastic boundaries. This includes formation of the annular flow, bubble splitting, and the generation of the liquid jets flowing in opposite direction. An example is shown in Fig. 3.28 for the case of a bubble oscillating near an elastic material with an elastic modulus E = 0.405 MPa when the relative distance between bubble and boundary is γ = 0.88. Very recently, Miao and Gracewski (2008) investigated numerically the interaction of cavitation bubbles with elastic boundaries by combining finite element and boundary element codes. The interesting result of their work is the characterization of the bubble behaviour situated in an elastic tube and in a sound-irradiated

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Fig. 3.28 Numerical simulation of the behaviour of a cavitation bubble situated near an elastic interface for γ = 0.88 and E = 0.405 MPa. The dimensionless times are 0.00, 0.773 (largest deformation of the interface), 0.973 (near maximum bubble volume), 1.594, 1.809, 1.889, 1.916 (bubble splits up), and 1.920 (downward jet developing in lowest bubble). Reproduced with permission from Klaseboer and Khoo (2004). © American Institute of Physics

liquid. Figure 3.29 illustrates an example of the bubble and tube shapes when they are exposed to an ultrasound frequency of 1 MHz and an ultrasound pressure of 0.2 MPa. This situation is very similar to that encountered in the medical application of ultrasound contrast agents (see Chap. 6). The authors found that the presence of the tube inhibits bubble expansion and the maximum equivalent bubble radius

Fig. 3.29 The behaviour of a bubble situated in an elastic tube and in an ultrasonically irradiated liquid. Bubble and tube shapes and hoop stress distribution within the tube wall are illustrated at a series of time points. The inner tube radius is 4 μm, the elastic modulus of the tube is 10 MPa, and the initial bubble radius is 1 μm. Reproduced with permission from Miao and Gracewski (2008). © Springer Science + Business Media

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107

decreases with decreasing the tube radius. The maximum hoop stress in the tube occurs at the inner tube surface, well before the bubble reaches the maximum radius. During the initial collapse stage the bubble takes an elongated shape in a direction perpendicular to the tube axis that leads to the formation of a radial jet in the final collapse stage.

3.4 Bubbles in Tissue Phantoms An important issue that has only recently received attention is the influence of the mechanical properties of biological tissue on the stress wave and cavitation dynamics following pulsed laser ablation and optical breakdown. Jansen et al. (1996), Asshauer et al. (1997), and Delacretaz and Walsh (1997) presented first studies on holmium-laser-induced bubble formation in more realistic tissue phantoms. Vogel et al. (1999) compared the optical breakdown dynamics in water and real tissue. They observed that the optical breakdown in corneal tissue is accompanied by a strong tensile stress wave which is absent in water. Moreover, the bubble expansion phase is shortened and the stress wave originating from the bubble collapse in water is missing. In a very recent study, Brujan and Vogel (2006) investigated the dynamics of laser-induced cavitation bubbles in tissue phantoms consisting of transparent gels of polyacryalmide (PAA) with a water content of 95, 85, 80 and 70%, respectively. The measured values of the elastic modulus, E, density, ρ, sound velocity, c0 , and yield strength, Y0 , of the polyacrylamide samples are shown in Table 3.3. Figure 3.30 shows a collection of high-speed photographic records of the bubble dynamics and the corresponding hydrophone signals for a value of the laser pulse energy EL ≈ 1 mJ. The hydrophone signals were recorded simultaneously with the documented image series. With increasing elastic modulus of the samples, both maximum size and oscillation period of the bubble decrease. For the PAA sample with 70% water content (Fig. 3.30d), the bubble dynamics is characterized by a strongly damped behaviour where no pronounced collapse and rebound are observed after bubble expansion. The largest pressure transient is produced during optical breakdown. This transient is followed by a weaker pressure transient associated with the first bubble collapse. In the case of water, the second transient is almost as high as the first one. In contrast, for the PAA sample with 70% water content, the pressure signal is characterized by the absence of a transient emitted during bubble collapse, which is a consequence of strong damping of the bubble oscillation. Table 3.3 Values of the elastic modulus E, plastic yield strength at large strain rates Y0 , density ρ, and sound velocity c0 of the PAA samples at 20◦ C and ambient pressure Sample

Water

PAA-95% water

PAA-85% water

PAA-80% water

PAA-70% water

E (MPa) Y0 (MPa) ρ (kg/m3 ) c0 (m/s)

– –

0.017 ± 0.001 – 1,012 ± 10 1,518 ± 15

0.124 ± 0.004 20 1,032 ± 10 1,560 ± 15

0.3 ± 0.01 60 1,050 ± 10 1,575 ± 15

0.4 ± 0.01 80 1,073 ± 10 1,605 ± 15

998 1,483

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Fig. 3.30 Cavitation bubble dynamics in water and polyacrylamide samples with different water content c, and the corresponding pressure signal measured at a distance of 10 mm from the laser focus. The laser pulse energy was about 1 mJ. (a) Water, EL = 0.93 mJ; (b) c = 95%, EL = 1.36 mJ; (c) c = 80%, EL = 1.24 mJ; (d) c = 70%, EL = 1.05 mJ. The first frame was taken 15 μs after the moment of optical breakdown, and the frame interval is 20 μs. Frame width 4 mm. Reproduced with permission from Brujan and Vogel (2006). © Cambridge University Press

At equal laser pulse energy, the maximum radius of the cavitation bubble decreases with decreasing water content of the sample, i.e. with increasing elastic modulus (Fig. 3.31). The scaling law for the bubble size in the PAA samples is the same as that for water, namely, the maximum bubble radius is proportional to the cube root of the laser pulse energy. This scaling law applies, however, only to laser pulse energies larger than 2 mJ, well above the breakdown threshold. The damping of the bubble oscillation in PAA samples results in a reduction of the amplitude of the stress transient emitted upon bubble collapse. The amplitude reduction is shown in Fig. 3.32 where the amplitude of the transient emitted during the first bubble collapse is plotted as a function of the laser pulse energy. In water and PAA sample with 95% water content, the measured values are fitted by a curve

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Fig. 3.31 Maximal cavitation bubble radius, Rmax , as a function of the laser pulse energy, EL . The slope of the straight lines gives the scaling law for the bubble radius. Reproduced with permission from Brujan and Vogel (2006). © Cambridge University Press

Fig. 3.32 Maximum amplitude of the shock wave emitted during first bubble collapse as a function of the laser pulse energy. Reproduced with permission from Brujan and Vogel (2006). © Cambridge University Press

(pc − p0 ) = aELb , with a = 1 and b = 0.38 for water, and a = 0.73 and b = 0.4 for the PAA sample with 95% water content. For the PAA sample with 80% water content, the energy dependence is quite different. Here, the measured values are fitted by a curve (pc − p0 ) = a(EL − EL,c )b , with a = 0.61, b = 0.34, and EL,c = 1.01 mJ. The fit parameter EL,c can be interpreted as a critical value of the laser pulse energy which determines the behaviour of the bubble: strongly damped behaviour occurs if EL ≤ EL,c , and damped oscillatory behaviour if EL > EL,c .

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3.5 Estimation of Extensional Viscosity The flow around an expanding or collapsing spherical bubble is an extensional flow. Uniaxial extension obtains in collapse, while biaxial extension obtains in growth. Not surprisingly, therefore, there have been some attempts to estimate the extensional viscosity from measurements on bubble growth or collapse. Ting (1975) was the first who pointed out that the extensional nature of the flow field during growth and collapse of a spherical bubble holds out the possibility of observing the anomalously large viscous effects associated with the stretching of high polymer solutions. Here we describe several strategies for estimating the extensional viscosity of dilute polymer solutions by exploiting the extensional nature of the flow field around a spherical bubble. Observation of the bubble radius as a function of time provides a means for determining the rate-of-strain tensor at the bubble wall, and one attempts to estimate the stress history from measurements of the pressure inside the bubble as a function of time. Neglecting inertial effects, the momentum equation, in the incompressible limit, gives (Pearson and Middleman 1977a, b) 2σ = −2 pi − p∞ − R

∞

τrr − −τθθ dr. r

(3.20)

R

One can now chose a constitutive equation to characterize the material, substitute it into the integral in Eq. (3.20), measure pi and R and determine rheological parameters appearing on the right-hand side of (3.20). Because the stress cannot be measured at the bubble wall, one must accept an integrated result, and hence a constitutive assumption must be invoked over the range of integration of the right-hand side of Eq. (3.20). In comparing experiment and theory, Pearson and Middleman (1977a, b) found it expedient to define an apparent extensional viscosity

ηE,app =

3R pi − p∞ − 4R˙

2σ R

,

(3.21)

and they found from experiments on bubble collapse in dilute polymer solutions that good agreement was obtained between Eq. (3.21) and the results predicted for extensional viscosity by the Tanner rupture model and a corrotational Maxwell model (Schowalter 1978). However, stretch rates were below 10 s–1 . In a subsequent paper, Papanastasiou et al. (1984) developed a robust numerical scheme that is compatible with a KBKZ constitutive equation. The equation of motion was solved by a Galerkin weighted-residual method using a finite element solution. Inertia was also neglected in their calculations. They checked the numerical results by evaluating rheological parameters from the measurements of Pearson and Middleman of shear and normal stress differences for a hydroxypropylcellulose solution. With these data they predicted the extensional behaviour shown in ˙ Fig. 3.33. Although the deformation rates, ε˙ = 2R/R, are rather small, the good

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Estimation of Extensional Viscosity

111

Fig. 3.33 The radius of a spherical bubble collapsing in a hydroxypropylcellulose solution as measured by Pearson and Middleman (1977b) and as predicted by the finite element analysis of Papanastasiou et al. (1984) (solid lines). Reproduced with permission from Papanastasiou et al. (1984). © Elsevier B.V.

agreement between predictions and the measurements of Pearson and Middleman is to be noted. They concluded that bubble collapse offers a suitable technique for measuring extensional viscosity at low elasticity levels. In a recent experiment, Barrow et al. (2004a) explored the extensional flow induced by liquid jets formed by the collapse of bubbles under cavitation-generated pressure waves. By determining the temporal evolution of such jets using high-speed photography they defined an uniaxial extensional strain as ε˙ =

dL 1 , dt L(t)

(3.22)

where dL/dt is the velocity of extenson with reference to the tip of the jet and L(t) is the instantaneous length of the jet, and an uniaxial extensional viscosity as ηE =

σE , ε˙

(3.23)

where σ E is the tensile stress in the liquid. To determine the average value of the tensile stress, σ E , developed in the jet, they estimating the tensile force from a knowledge of the liquid mass which replaces the annihilated volume of the gas bubble (based on its maximum diameter prior to collapse) and the subsequent deceleration of this mass of liquid (which forms the jet) during the jet’s extension. From a knowledge of the jet’s diameter, this force provides an estimate of the average tensile stress, σE , sustained by the liquid during its residence within the extensional flow, the latter being characterized by an average value of the rate of

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Fig. 3.34 Calculated values of the extensional viscosity, ηE , (normalized by the shear viscosity, η) for a Newtonian glycerol/water mixture and solutions of xantham gum. The broken line indicates the case of a Newtonian liquid (for which the Trouton number has the value 3). Reproduced with permission from Barrow et al. (2004a). © American Society of Mechanical Engineers

extension, ε˙ . The dominant feature in the jet dynamics is the high rate of extension, of the order of 103 s–1 , that characterize the jet flow. They investigated the extensional viscosity for a Newtonian glycerol/water mixture and solutions of xanthan gum in a maximum concentration of 50 ppm. The polymer solutions showed enhanced levels of resistance to extension, with values of ηE two orders of magnitude larger that the Newtonian counterpart (Fig. 3.34). A notable result of their investigations is the finding that the technique is sensitive to the influence of extremely small concentrations of high molecular weight polymeric additive (as low as 5 ppm for xanthan gum).

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Klaseboer, E., Turangan, C.K., Khoo B.C. 2006 Dynamic behaviour of a bubble near an elastic infinite interface. Int. J. Multiphase Flow 32, 1110–1122. Knapp, R.T., Daily, J.W., Hammitt, F.G. 1970 Cavitation. McGraw-Hill, New York. Knapp, R.T., Hollander, A. 1948 Laboratory investigations of the mechanisms of cavitation. Trans. ASME 70, 419. Kodama, T., Tomita, Y. 2000 Cavitation bubble behaviour and bubble-shock wave interaction near a gelatin surface as a study of in vivo bubble dynamics. Appl. Phys. B 70, 139–149. Lauterborn, W. 1976 Numerical investigation of nonlinear oscillations of gas bubbles. J. Acoust. Soc. Am. 59, 283–293. Lezzi, A., Prosperetti, A. 1987 Bubble dynamics in a compressible liquid. Part 2. Second-order theory. J. Fluid Mech. 185, 289–321. Ligneul, P. 1987 Study of the influence of the proximity of a moving solid wall on bubble dynamics. Phys. Fluids 30, 2280–2282. Lundgren, T.W., Mansour, N.N. 1988 Oscillations of drops in zero gravity with weak viscous effects. J. Fluid Mech. 194, 479–510. McComb, W., Ayyash, S. 1980 The production, pulsation and damping of small air bubbles in dilute polymer solutions. J. Phys. D Appl. Phys. 13, 773–787. Miao, H.Y., Gracewski, S.M. 2008 Coupled FEM and BEM code for simulating acoustically excited bubbles near deformable structures. Comput. Mech. 42, 95–106. Naude, J., Mendez, F. 2008 Periodic and chaotic oscillations of a bubble gas immersed in an Upper Convective Maxwell fluid. J. Non-Newt. Fluid Mech. 155, 30–38. Papanastasiou, A.C., Scriven, L.E., Macosko, C.W. 1984 Bubble growth and collapse in viscoelastic liquids analyzed. J. Non-Newt. Fluid Mech. 16, 53–75. Parlitz, U., English, V., Scheffczyk, C., Lauterborn, W. 1990 Bifurcation structure of bubble oscillators. J. Acoust. Soc. Am. 88, 1061–1077. Pearson, G., Middleman, S. 1977a Elongational flow behavior of viscoelastic liquids: part I. Modeling of bubble collapse. AIChE J. 23, 714–722. Pearson, G., Middleman, S. 1977b Elongational flow behavior of viscoelastic liquids: part II. Definition and measurement of apparent elongational viscosity. AIChE J. 23, 722–725. Pei, O.G., Cheong, K.B., Turangan, C., Klaseboer, E., Wan, F.S. 2005 Behaviour of oscillating bubbles near elastic membranes. An experimental and numerical study. Mod. Phys. Lett. B 19, 1579–1582. Plesset, M.S., Chapman, R.B. 1971 Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary. J. Fluid Mech. 47, 283–290. Prosperetti, A., Lezzi, A. 1986 Bubble dynamics in a compressible liquid. Part 1: first-order theory. J. Fluid Mech. 168, 457–478. Rayleigh, L. 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94–98. Ryskin, G. 1990 Dynamics and sound emission of a spherical cavitation bubble in a dilute polymer solution. J. Fluid Mech. 218, 239–263. Schowalter, W.R. 1978 Mechanics of Non-Newtonian Fluids. Pergamon, Oxford. Shaw, S.J., Jin, Y.H., Gentry, T.P., Emmony, D.C. 1999 Experimental observations of the interaction of a laser-generated cavitation bubble with a flexible membrane. Phys. Fluids 11, 2437–2439. Shima, A., Tsujino, T. 1976 The behavior of bubbles in polymer solutions. Chem. Eng. Sci. 31, 863–869. Shima, A., Tsujino, T. 1977 Behavior of bubbles in non-Newtonian lubricants. Trans. ASME J. Lub. Tech. 99, 455–461. Shima, A., Tsujino, T. 1978 The behaviour of gas bubbles in the Casson Fluid. J. Appl. Mech. 100, 37–42. Shima, A., Tsujino, T. 1980a The behaviour of bubbles in an Ellis fluid. Rep. Inst. High Speed Mech., Tohoku Univ. 42, 25–41. Shima, A., Tsujino, T. 1980b The behaviour of bubbles in a Carreau model fluid. Rep. Inst. High Speed Mech., Tohoku Univ. 42, 43–58.

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Shima, A., Tsujino, T. 1981 The effect of polymer concentration on the bubble behaviour and impulse pressure. Chem. Eng. Sci. 36, 931–935. Shima, A., Tsujino, T. 1982 On the dynamics of bubbles in polymer aqueous solutions. Appl. Sci. Res. 38, 255–263. Shima, A., Tsujino, T., Yokoyama, K. 1984a The behaviour of bubbles in Sutterby model fluids. Rep. Inst. High Speed Mech. Tohoku Univ. 49, 39–52. Shima, A., Tomita, Y., Ohno, T. 1984b The collapse of minute gas bubbles in a dilute polymer solution. Phys. Fluids 27, 539–540. Shima, A., Rajvanshi, S., Tsujino, T. 1985 Study of nonlinear oscillations of bubbles in PowellEyring Fluids. J. Acoust. Soc. Am. 77, 1702–1709. Shima, A., Tsujino, T., Nanjo, H. 1986 Non-linear oscillations of gas bubbles in viscoelastic fluids, Ultrasonics 24, 142–147. Shima, A., Tsujino, T., Oikawa, Y. 1988 The collapse of bubbles in viscoelastic fluids (The case of Jeffreys model fluid). Rep. Inst. High Speed Mech. Tohoku Univ. 55, 17–34. Shima, A., Tomita, Y., Gibson, D.C., Blake, J.R. 1989 The growth and collapse of cavitation bubbles near composite surfaces. J. Fluid Mech. 203, 199–214. Shulman, Z.P., Levitsky, S.P. 1987 Closure of a cavity in polymeric liquids. J. Eng. Phys. 53, 893–897. Shulman, Z.P., Levitsky, S.P. 1992 Heat/mass transfer and dynamics of bubbles in high-polymer solutions. I: free oscillations. Int. J. Heat Mass Transfer 35, 1077–1084. Tanasawa, I., Yang, W.J. 1970 Dynamic behaviours of a gas bubble in viscoelastic liquid. J. Appl. Phys. 41, 4526–4531. Ting, R.Y. 1975 Viscoelastic effect of polymers on single bubble dynamics, AIChE J. 21, 810–813. Ting, R.Y. 1977 Effect of polymer viscoelasticity on the initial growth of a vapour bubble from gas nuclei. Phys. Fluids 20, 1427–1431. Ting, R.Y., Ellis, A.T. 1974 Bubble growth in dilute polymer solutions. Phys. Fluids 17, 1461– 1462. Trilling, L. 1952 The collapse and rebound of a gas bubble. J. Appl. Phys. 23, 14–17. Tsujino, T., Shima, A., Oikawa, Y. 1988a Effect of polymer additives on the generation of subharmonic and harmonic bubble oscillations in an ultrasonically irradiated liquid. J. Sound Vib. 123, 171–184. Tsujino, T., Shima, A., Tanaka, J. 1988b Effect of viscoelasticity on bubble collapse and induced impulsive pressure. Rep. Inst. High Speed Mech. Tohoku Univ. 55, 1–15. Turangan, C. K., Ong, G.P., Klaseboer, E., Khoo B.C. 2006 Experimental and numerical study of transient bubble-elastic membrane interaction. J. App. Phys. 100, 054910. Venerus, D.C., Yala, N., Berstein, B. 1998 Analysis of diffusion-induced bubble growth in viscoelastic liquids. J. Non-Newt. Fluid Mech. 75, 55–75. Vogel, A., Lauterborn, W., Timm, R. 1989 Optical and acoustic investigations of the dynamics of laser-produced cavitation bubbles near a solid boundary. J. Fluid Mech. 206, 299–338. Vogel, A., Scammon, R.J., Godwin, R.P. 1999 Tensile stress generation by optical breakdown in tissue: experimental investigations and numerical simulations. Proc. SPIE 3601, 191–206. Williams, P.R., Williams, P.M., Brown, S.W.J. 1998 A study of liquid jets formed by bubble collapse under shock waves in elastic and Newtonian liquids. J. Non-Newt. Fluid Mech. 76, 307–325. Yang, W.-J., Lawson, M.L. 1974 Bubble pulsation and cavitation in viscoelastic liquids. J. Appl. Phys. 45, 754–758. Yang, W.-J., Yeh, H.-C. 1966 Theoretical study of bubble dynamics in purely viscous fluids. AIChE J. 12, 927–931. Yoo, H.J., Han, C.D. 1982 Oscillatory benavior of a gas bubble growing (or collapsing) in viscoelastic liquids. AIChE J. 28, 1002–1009. Zana, E., Leal, L.G. 1975 Dissolution of a stationary gas bubble in a quiescent, viscoelastic liquid. Ind. Eng. Chem. Fundam. 14, 175–182.

Chapter 4

Hydrodynamic Cavitation

Hydrodynamic cavitation is observed when large pressure differentials are generated within a moving liquid and is accompanied by a number of physical effects, erosion being most notable from a technological viewpoint. Cavitation bubbles can form in the low pressure region and be carried into the higher pressure region where they collapse, so that the surface of any body is acted on by pulsating pressure loads, eventually leading to the destruction of the surface. The collapse of the bubbles goes hand in hand with a cracking noise, giving the first indication of cavitation occurrence. The term incipient cavitation describes cavitation that is just barely detectable. The discernible bubbles of incipient cavitation are small, and the zone over which cavitation occurs is limited. Three cases of hydrodynamic cavitation arise (Young 1989). Traveling cavitation is a type composed of individual transient bubbles, which form in the liquid, as they expand, shrink, and then collapse. Such traveling transient bubbles may appear at the low-pressure points along a solid boundary or in the liquid interior either at the cores of moving vortices or in the high-turbulence region in a turbulent shear field. The term fixed cavitation refers to the situation that sometimes develops after inception, in which the liquid flow detaches from the rigid boundary of an immersed body or a flow passage to form a pocket or cavity attached to the boundary. The attached or fixed cavity is stable in a quasi-steady sense. Fixed cavities sometimes have the appearance of a highly turbulent boiling surface. In vortex cavitation, the bubbles are found in the core of vortices that form in zones of high shear. The cavitation may appear as traveling cavities or as a fixed cavity. Vortex cavitation is one of the earliest observed types, as it often occurs on the blade tips of ship propellers. In fact, this type of cavitation is often referred to as tip vortex cavitation. Tip vortex cavitation occurs not only in open propellers but also in ducted propellers such as those found in propeller pumps at hydrofoil tips. The consequences of cavitation in a flowing system are the reduction of hydrodynamic performance of liquid machinery – pumps, turbines and propellers – and hydraulic circuits, the emission of noise, the generation of vibration and the erosion of materials. A detailed characterization of hydrodynamic cavitation in water can be found in the books by Knapp et al. (1970), Young (1989), Brennen (1996), and Franc and Michel (2004). In this chapter, we focus on hydrodynamic cavitation in E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_4,

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non-Newtonian fluids, especially in the case of cavitation of submerged jets, on hemispherical bodies and hydrofoils, in vortices, or in very confined spaces. The latter situation can be used to estimate the extensional viscosity of the non-Newtonian fluids.

4.1 Non-cavitating Flows Before embarking on a discussion of the cavitation phenomenon in non-Newtonian fluids it is important that the reader has some minimal acquaintance with some non-Newtonian effects in non-cavitating flows. The purpose of this section is to demonstrate the striking qualitative difference between the flow behaviour of Newtonian and non-Newtonian liquids, such as polymer solutions. An appreciation of the qualitative behaviour of polymeric liquids is important as background information for the quantitative treatments to follow in the remainder of this chapter. Therefore, we consider in this section three interesting effects encountered in the flow of polymer solutions, namely drag reduction, reduction of pressure drop across an orifice, and vortex inhibition. These effects are manifestations of viscoelastic properties of non-Newtonian fluids and illustrate the change of the flow field induced by the addition of small amounts of polymers.

4.1.1 Drag Reduction Toms (1949) discovered that if he added a very small amount of polymethylmethacrylate, approximately 10 ppm by weight, to monochlorobenzene undergoing turbulent flow in a cylindrical tube, a substantial reduction in pressure drop at the given flow rate resulted. This reduction in pressure drop in the polymer solution relative to the pure solvent alone at the same flow rate is defined as drag reduction. Since then any number of polymer-solvent pairs have been found to show drag reduction. Three additional examples give an idea of the variety of possible systems: polyisobutylene in decalin, carboxymethylcellulose in water, and polyethylene oxide in water. Drag reduction results may be presented conveniently in terms of the Fanning friction factor f: f =

D 4L

2p , ρv2

(4.1)

where p is the modified pressure drop over a length L of the tube, D is the tube diameter, and v is the average velocity over the cross section of the tube. The friction factor is essentially a dimensionless pressure gradient, and it is a function only of the Reynolds number Re = ρvD/η for fully developed flow of Newtonian fluids. At the very tiny polymer concentrations of interest in drag reduction, the viscosity and density of the polymer solution differ only slightly from those of the pure

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solvent. Nonetheless, the effect of the polymer additive is to lower the value of the friction factor at a given Reynolds number, especially in the turbulent region. The amount by which the friction factor is lowered is a measure of the amount of drag reduction. Figure 4.1 shows the friction factor for water with and without a small amount of polyethylene oxide. Whereas the addition of only 5 ppm of polyethylene oxide to water gives a 40% reduction in f at Re = 105 , the viscosity of the solution is only 1% greater than the viscosity of the water alone. The mechanisms of polymer drag reduction in turbulent flows have been under investigation for several decades. In spite of the large amount of observational data available, the physical mechanism of drag reduction by polymers still remains unclear. Den Toonder (1995) and den Toonder et al. (1995) discussed the role of extensional viscosity in the mechanism of drag reduction by polymer additives. The aim of this paper was to test a hypothesis introduced by Lumley (1969), who was the first to suggest that the molecular extension of polymers is responsible for drag reduction. Lumley argued that this extension will take place in the flow outside the viscous sublayer, causing an increase in effective viscosity there. Using general scaling arguments, Lumley showed that then a reduction in overall drag will occur. Den Toonder et al. (1995) presented the results of a direct numerical simulation with a simplified polymer extension criterion to increase the viscosity locally. It was found that a mere increase in effective viscosity outside the viscous sublayer is in itself not enough to

Fig. 4.1 Friction factor for dilute aqueous solutions of polyethylene oxide. In the turbulent regime, the curves for the polymer solutions lie below that of the solvent and illustrate drag reduction. Reproduced with permission from Wang (1972). © American Chemical Society

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produce significant drag reduction, so that Lumley’s hypothesis should perhaps be made more specific. In particular, it should be noted that neither Lumley’s hypothesis, although based on the notion of polymer extension, nor the rather simple model used in den Toonder et al. (1995) contains any anisotropic stress effects caused by specific polymer orientations. It also interesting to mention here that the possibility of drag reduction, as being an anisotropic response of the flow to an anisotropic viscosity induced by elongated polymers, has been also suggested by Hinch (1977). De Gennes (1990) and Joseph (1990) suggested an interesting mechanism to explain drag reduction by polymers. A polymer solution, even a very dilute one, can be regarded as a viscoelastic fluid. In these fluids the viscosity takes care of diffusion and of the smoothing of shear discontinuities (“shear waves”), while on the other hand the elasticity is able to propagate these shear discontinuities. Moreover, in purely viscous fluids, the stress is always in phase with the rate of strain in the flow while, in viscoelastic fluids, this is generally not the case. This is related to the fact that polymers are in principle capable of storing elastic energy. In the view of Joseph (1990), the characteristic speed of shear waves in polymer solutions (see, also, Joseph et al. 1986) provides a natural cut-off for velocities which fluctuate at high frequencies. In fact the fluctuating velocities which are observed in turbulent flow of aqueous drag-reducing solutions are of the right order, namely a few centimeters per second, for such a cut-off to be important. This cut-off would then suppress the small eddies and presumably lead to drag reduction. De Gennes (1990) also states that the effects of polymers at high frequencies are described by an elastic modulus, resulting in a truncation of the turbulent velocity fluctuations at these frequencies. Using a simple scaling analysis and an elastic model for the polymer solution, he indeed finds that drag reduction might occur through such a mechanism. His analysis, however, suffers from the shortcomings that it is rather crude and it is not able to explain the detailed dynamics of wall turbulence. Experiments by Sasaki (1991, 1992), who measured the effectiveness for drag reduction of various polymers in combination with several kinds of solvents, indicated that the drag-reducing ability of polymer solutions tends to decrease when the polymers become more flexible. In a more recent study, Dubief et al. (2004) discussed in detail how the stretching and recoiling of polymer molecules in the near-wall flow can modify the usual self-sustained near-wall turbulence regeneration cycle involving buffer-layer vortices and viscous sub-layer streaks (Jimenez and Pinelli 1999). The polymer extracts energy from the buffer-layer vortices and releases it in the near-wall streaks leading to enhanced streamwise velocity fluctuations, reduced wall-normal velocity fluctuations and increased streamwise vortex spacing in the near-wall region. The near-wall shear stress balance is then a combination of viscous, Reynolds and polymer stresses. The state of maximum drag reduction is achieved when there is sufficient polymer in the near-wall region to produce a new self-sustained turbulence regeneration cycle. Once this state is achieved, further increases in polymer concentration do not reduce friction drag. Instead, higher polymer concentrations may increase friction drag via increased shear viscosity. The detailed interaction

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of the polymer with turbulent flow is still the object of study, but this phenomenological description is consistent with experimental observations of drag reduction, including the recent work of Warholic et al. (1999), and White et al. (2004).

4.1.2 Reduction of Pressure Drop in Flows Through Orifices Consider the flow through an orifice as illustrated in Fig. 4.2. Where the streamlines converge in approaching the orifice, they continue to converge beyond the upstream section of the orifice until they reach the section ab where they become parallel. Commonly this section is about 0.5d from the upstream edge of the opening, where d is the diameter of the orifice. The section ab is then a section of minimum area and is called the vena contracta. Jet velocity is defined as the average velocity at the vena contracta. The velocity at this section is practically constant across the section except for a small annular region around the outside. The pressure is practically constant across the diameter of the jet wherever the streamlines are parallel, and the pressure must be equal to that in the medium surrounding the jet at that section. The increased velocity in the vena contracta is accompanied by a reduction in pressure. Beyond the vena contracta the streamlines commonly diverge because of frictional effects. In this region the velocity is transformed back into pressure with slight friction loss. An essential characteristic of flows through an orifice is the pressure drop resulting from a sudden change in diameter. A significant reduction of the pressure drop was observed when small amounts of polymers have been added to water. Figure 4.3 illustrates the dimensionless pressure drop Kt =

Fig. 4.2 Flow through an orifice

2pt , ρv2

(4.2)

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Fig. 4.3 Dimensionless pressure drop Kt against Reynolds number Re for orifices. The data contain both the forward (front → back) and backward (back → front) flow results. The orifice was pasted over a hole of 1 or 3 mm in diameter, D, drilled in a base plate of length L and, thus, the forward and backward flows are not identical. Solid line shows the prediction of the Navier– Stokes equation, and a broken line is the line of Poiseuille flow. Reproduced with permission from Hasegawa et al. (2009). © American Institute of Physics

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against Reynolds number, Re = ρvD/η, where pt is the pressure drop across the orifice, ρ is the liquid density, η is the liquid viscosity, and v is the liquid mean velocity, for the case of water, a 50/50 mixture of glycerol and water with a viscosity of 10–2 Pa·s, and a viscoelastic polyethylene oxide (PEO) solution with a viscosity similar to that of water. The solution of PEO produced a lower pressure drop than water and the glycerol/water mixture. Hasegawa et al. (2009) considered the effect of several factors, including orifice shape, deformation of orifice foil, wall slip, transition, and liquid elasticity but they results suggest that the significant reduction in pressure drop may be caused by wall slip or the elasticity induced in a flow of high extensional rate.

4.1.3 Vortex Inhibition Consider a simple experiment in which water is discharged from a tank. The experiment is done by first filling a large tank with water, stirring the water to induce a circulation in the tank, and finally removing a plug from the center of the bottom to allow the water to drain. As the water empties from the tank, a very stable air core reaching all the way to the outlet forms, accompanied by a pronounced slurping sound. If a very small amount of polyethyleneoxide or polyacrylamide is added to the draining water, the air core suddenly disappears and the noise that goes with it ceases. Moreover, the volume flow rate out of the tank nearly doubles after the air core is eliminated, provided that the level in the tank is kept constant (see, for a detailed discussion, Bird et al. 1987). The mechanism for vortex inhibition is much easier to understand than that for drag reduction because the former involves laminar flow and the latter involves turbulent flow. The bulk of the flow out of the vortex tank occurs through a thin boundary layer along the bottom surface of the container. As polymer molecules flow from this thin layer into the exit tube they experience very large rates of strain which are sufficient to stretch them nearly to their fully extended configuration (see, for example, Bird et al. 1987). In the fully extended configuration, the polymer molecules cause sufficient increase in the tension along streamlines near the exit hole that a larger fraction of the total flow passes through the boundary layer. This results in a slight increase in the size of the viscous core around the axis of rotation and the disappearance of the air core. This understanding of the mechanism for vortex inhibition and the close connection between effectiveness of polymers in vortex inhibition and drag reduction lends credence to the currently accepted belief that molecular extension is also responsible for drag reduction.

4.2 Cavitating Flows The above-mentioned studies indicate a significant change of the local pressure in a viscoelastic liquid as a result of the modifications of the flow structure. We turn now to the main focus of this book and examine the cavitation phenomenon in

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flowing liquids. Most of experiments conducted on hydrodynamic cavitation in nonNewtonian fluids have used polymer solutions as test fluids. An excellent review on this topic was provided by Fruman (1999).

4.2.1 Cavitation Number A relative flow between an immersed object and the surrounding liquid results in a variation in pressure at a point on the object and the pressure p0 in the undisturbed liquid at some distance from the object, which is proportional to the square of the relative velocity, v0 . This can be written as the negative of the usual pressure coefficient Cp , namely, −Cp =

p0 − p , ρv20 /2

(4.3)

where ρ is the density of liquid, and p the pressure at a point on object. At some location on the object, p will have a minimum, pmin , so that (−Cp )min =

p0 − pmin . ρv20 /2

(4.4)

In the absence of cavitation (and if Reynolds-number effects are neglected), this value will depend only on the geometry of the object. It is easy to create a set of conditions such that pmin drops to some value at which cavitation exists. This can be accomplished by increasing the velocity v0 for a fixed value of the pressure p0 or by continuously lowering p0 with v0 held constant. Either procedure will result in lowering of the absolute values of the local pressure on the surface of the object. If surface tension is ignored, the pressure pmin will be the pressure of the contents of the cavitation bubble. If we now assume that cavitation will occur when the normal stresses at a point in the liquid are reduced to zero, pmin will equal the vapour pressure pv . We can define a cavitation number as σ =

p0 − pmin . ρv20 /2

(4.5)

The value at which cavitation inception occurs is designated as σi . The cavitation number represents an index of dynamic similarity of flow conditions under which cavitation occurs. The numerator of (4.5) is related to the net pressure or head of the flow. The denominator is the velocity pressure or head of the flow. The variations in pressure, which take place on the surface of the body, are due to changes in the velocity of the flow. Thus, the velocity pressure may be considered to be a measure of the pressure reductions that may occur to cause a cavity to form or expand.

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This relationship has been universally adopted as the parameter for comparison of cavitation events. However, the presence of gas nuclei, boundary layers, and turbulence will modify and often mask a departure of the critical pressure for cavitation occurrence from pv . The cavitation number assumes a definite value at each stage of development of cavitation on a particular body. For inception, σ = σi , while for advanced stages of cavitation, σ < σi . The cavitation inception number, σi , and the values of σ at subsequent stages of cavitation depend primarily on the geometry of the immersed object past which the liquid flows. It should be noted here that, depending on the way the experiments are performed – either by decreasing the pressure from a non-cavitating situation until cavitation is reached or by increasing the pressure from a cavitating situation until the non-cavitating condition is achieved – the inception, σi , or desinent, σd , cavitation numbers are obtained. The conditions that mark the boundary between no cavitation and detectable cavitation are not always identical. For example, the pressure of disappearance of cavitation has been generally found to be greater, and less variable, than the pressure of appearance (Knapp et al. 1970). The most complex factor contributing to the cavitation inception and development is the bubble dynamics, which is closely associated with the size, distribution and content of nuclei in a fluid. Based upon the assumption that the total gas content is composed of its dissolved and undissolved (free) components (Holl 1970; Rood 1991), the latter can act as cavitation nuclei, while the dissolved gas content affects the number, size and growth of the nuclei. Therefore, both forms of gas content play important roles in the development of cavitation bubbles in terms of the number, size and distribution of these bubbles. These parameters influence the pressure field of the flow and hence the inception of cavitation. The effect of the gas content on cavitation inception has been investigated by many investigators (see, for example, Holl 1960; Kuiper 1981; Arndt and Keller 1991). Tanibayashi et al. (1998) have indicated that the inception number of cavitation of a cylinder decreases with decreasing dissolved gas content. In a recent paper, Keller (2000) has demonstrated the effect of water quality on two-dimensional hydrofoils and head forms. The effect of the free-stream turbulence on the inception of cavitation has been realized by several investigators during tests with various bodies and its importance has been reported in a limited number of papers. A high turbulence level is known to cause early transition of the boundary layer, which, in turn, can lead to the complete elimination of the laminar separation. Arndt and George (1979) showed that the viscous effects associated with the laminar separation and transition of the boundary layer had a major effect on the inception of cavitation. The level of turbulence is expected to influence the inception of cavitation when the turbulence level can cause significant change in transition and laminar separation (Huang 1986). Keller (1979) indicated that the level of free-stream turbulence could be responsible for variations in the inception results of identical bodies tested in different facilities. Gates and Acosta (1979) analysed the results of experiments on an ellipsoidal head form and drew attention to the differences in the measured inception values, which could be as high as 300% for the same body tested in different tunnels. Gates and Acosta (1979) also related the differences to the variations in the levels of turbulence and

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performed experiments with three axisymmetric bodies in varying levels of freestream turbulence. These experiments indicated that the increase in the level of turbulence shifted the laminar separation point forward and, hence, enforced the boundary layer to become almost fully turbulent. Pauchet et al. (1996) have also shown that increasing the level of free-stream turbulence affects the inception of cavitation of hydrofoils. Cavitation inception does not necessarily occur in the vicinity of the lowest mean pressure on the surface of a cavitating body but rather, occurs in the region of the natural boundary layer transition (Pan et al. 1981). When the level of turbulence is too low, a change in the boundary layer structure achieved by artificial roughness elements can affect the inception of cavitation. In contrast to the effect of the freestream turbulence, the effect of roughness on cavitation inception has been studied extensively in the open literature by a number of investigators. For example, Kuiper (1981), Shen (1985), Huang (1986), Katz and Galdo (1989), and Pichon et al. (1997) have all demonstrated that roughness influences the conditions for the inception of cavitation.

4.2.2 Jet Cavitation Hydrodynamic cavitation in flow through constriction elements like orifices or nozzles has enticed considerable attention from the scientific community because of its inherent importance and applicability in multitudinous engineering situations. The primary requirement for hydrodynamic cavitation to occur in any system demands the reduction of the static pressure to a critical value for triggering the available nuclei. Any change (either abrupt or smooth) in the flow area introduces fluctuations in the velocity that is accompanied by concomitant vacillations in both the static and dynamic head. In fact, such an arrangement is the perfect breeding ground for cavitation nuclei, provided the apt hydrodynamic conditions are supported. The reduction in static pressure facilitates the rupture of the liquid, and expedites the formation of vapour- and gas filled cavities. The abrupt reduction in the flow area due to the presence of an orifice or a nozzle in the flow field manufactures a sharp drop in the static pressure and produces a concomitant acceleration of the liquid. Thus, the flow of liquids through an orifice provides an ideal situation wherein the sub-micrometer nuclei can grow in the low-pressure region and collapse once the pressure recovers downstream of the constriction element. Hoyt (1976) conducted several experiments with a nozzle situated 60 cm below the free surface of an open tank using polyethylene oxide aqueous solutions in concentration of up to 100 ppm. The cavitation number in their experiment was defined as σ =

2 (pr − pv ) , ρv2j

(4.6)

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where pr is the pressure in the tank where the jet is discharging and vj is the mean velocity of the jet. At the largest concentration, the polymer solution shows an increase of the infinite shear viscosity of about 25% as compared to that of water and a reduction of the surface tension with about 10%. His results indicate a sharp decrease of the incipient cavitation number when the concentration of the polymer solutions was increased up to 10 ppm; afterwards the reduction was not as rapid. Baker et al. (1976) obtained the desinent cavitation number, σd , instead of the inception cavitation number, σi , as in the case of Hoyt. Data were obtained for confined water jets generated by a 50.8 mm diameter sharp-edged orifice and a contoured nozzle of the same diameter. Data were acquired over a range of total air content from 2.0 to 12.0 ppm in water and a polyethylene oxide (WSR-301) aqueous solution in concentration of 100 ppm. The jet velocity was kept constant at 9.1 m/s resulting in a Reynolds number of 6×105 , larger than in the case of Hoyt experiments. Both in water and polymer solution the desinent cavitation number increases with the total gas content. The polymer caused a reduction in the desinent cavitation number and the reduction increased with air content. For example, for a total gas content of 10 ppm, the desinent cavitation number is about 1.1 in the case of water and about 0.8 in the case of a 100 ppm polyethylene oxide aqueous solution. However, below an air content of 4.0 ppm, the desinent cavitation number was apparently independent of air content. The differences in cavitation behaviour between Baker et al. (1976) and Hoyt (1976) results may be due to the larger size of the orifice, by near a factor of 10, the reduction of the jet velocity, by a factor of three, and the choice of desinent instead of the incipient cavitation number as a reference. In the case of pure water, an increase of the gas content resulted in a strong decrease of σd , especially for supersaturated conditions. Arndt et al. (1981) suggested an alternative explanation of the differences between Baker et al. (1976) and Hoyt (1976) results based on the difference in the hydrodynamic behaviour of jets depending on the values of the Reynolds number. His analysis is based on the fact that, for equal Reynolds number, the order of magnitude of the strain rate in the contraction of small nozzles will be larger than for the large nozzles. This effect is expected to lead to a significant viscoelastic influence on the pressure field which can explain the inhibition of cavitation. Oba et al. (1978) investigated the effect of polymer additives on cavitation in water flow through an orifice. Polyethylene oxide with a molecular weight 3×106 in concentration of up to 100 pmm was used as a test fluid. The rheological properties of the polymer solution are, however, not provided but it is well known that dilute solutions of flexible polymers, such as polyacrylamide or polyethylene oxide, exhibit very high values of the extensional viscosity. The orifice, with a diameter, d, of 8 mm, was mounted in a pipe with an inner diameter, D, of 20 mm so that the area ratio, d2 /D2 , is 0.16. The cavitation number in their experiment was defined as σ =

2 (pds − pv ) , ρv2t

(4.7)

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where pds is the pressure 25.4 mm downstream from the orifice, and vt is the mean throat velocity. The ranges in test conditions were 137–142 kPa for the upstream pressure, 2 × 104 – 2.6 × 104 for the Reynolds number upstream the orifice, and 7–12◦ C for the fluid temperature. The relative air content α/αs , where α is the total gas content and α s the saturated gas content, was measured before each test and varied over a range of 1.1–1.16. During each test the polymer degradation was negligible since the testing time was less than 10 min. They results indicate that very small amounts of polymer additives effectively suppress both the inception and the development of cavitation (Fig. 4.4). Near inception, cavitation bubbles were observed along the jet boundary 1D–2D downstream the orifice. With a decrease in the occurrence zone of cavitation, bubbles gradually approach the orifice, and the bubbles increase in size as well as in number. In any polymer solution tested, cavitation bubbles consist of non-spherical bubbles, such as irregular surface-, string-, as well as massive bubbles, and fine spherical bubbles. The occurrence of the fine spherical-, the irregular spherical as well as the string bubbles are strongly suppressed by the polymer additives. Oba et al. (1978) also investigated the influence of polymer additives on cavitation noise. They used a PZT probe fixed 40 mm downstream of the orifice where the maximum shock pressure was observed. Their results are illustrated in Fig. 4.5 for σ = 0.38. A relatively small amount of polymer (10 ppm) results in a considerable reduction of the time averaged value of shock pressures, P˜ sm , at 2–3 kHz. They also found a noticeable increase at 800 and 5 kHz, and a significant downward shift in the frequency where P˜ sm is a maximum. The reduction of P˜ sm seems to be related to

Fig. 4.4 Desinent cavitation number as a function of the polymer concentration. Also included in this figure are the results obtained by Hoyt (1976) and Baker et al. (1976). Reproduced with permission from Oba et al. (1978). © American Society of Mechanical Engineers

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Fig. 4.5 Shock spectra for various polymer concentrations. Reproduced with permission from Oba et al. (1978). © American Society of Mechanical Engineers

the fact that the cloud of fine bubbles are suppressed by the polymer. On the other hand, the increase of P˜ sm at 800 kHz is qualitatively related to an increase in irregularity, size and number of massive bubbles, while the increase of P˜ sm at 5 kHz to an increase in the number of string bubbles. In the 50 ppm solution, P˜ sm increases at 400 kHz and then decreases for larger frequencies. They explained this result by observing that the number of fine bubbles as well as the number of string bubbles is considerable reduced in the polymer solution. The effect of polymer additives (both drag-reducing and non drag-reducing) on jet cavitation was also studied by Hoyt and Taylor (1981) using a nozzle of only 2.92 mm diameter. Solutions of the drag-reducing additives, polyacrylamide and polyethylene oxide, at concentrations of 25 ppm, decreased the cavitation inception number and greatly changed the appearance of the cavitation bubbles. Solutions of the non drag-reducing polymer, Carbopol, produced cavitation bubbles having the same appearance on pure water and did not change the inception number. In pure water, the cavitation appearance resembles ragged groups of small bubbles with the overall impression of sharpness and roughness, but in drag-reducing polymer solutions the bubbles are larger, rounded, and of completely different appearance.

4.2.3 Cavitation Around Blunt Bodies The effect of polymer additives on cavitation inception on blunt bodies was first investigated by Hoyt (1966). Subsequent studies have been conducted by Ellis (1967), Ellis and Hoyt (1968), Ellis et al. (1970) and Baker et al. (1973) in homogeneous solutions, and van der Meulen (1973, 1976) and Gates and Acosta (1979) for injected solutions. Ellis and co-workers (1967, 1968, 1970) conducted a series of experiments to investigate the effect of polymer additives on both cavitation inception and its appearance on a hemisphere-nosed cylindrical body in a blowdown water tunnel.

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Using a stainless steel test body with a diameter of 6.35 mm, Ellis and co-workers detected the inception of cavitation in two ways. A laser beam was adjusted to just graze the surface of hemisphere nose in the region where cavitation first appears. Light scattered by the cavitation bubbles was detected by a photocell sensing light at about 90◦ from the laser beam direction. This method of cavitation detection was checked by acoustic observation, and very good agreement was obtained. The cavitation number was defined as 2 (p∞ − pv ) , ρv2∞

(4.8)

where p∞ and v∞ are the pressure and liquid velocity upstream the body, respectively. Tests were made with water (passed through a 0.4 μm filter), 50 and 100-ppm polyethylene oxide and a suspension of alga, porphyridium aerugineum. All tests were made with water containing the same amount of dissolved air, 17 ppm, and Table 4.1 gives the inception data obtained (averages of 4 runs). It can be seen that the polymer content of the water has a large effect on the cavitation inception point. The appearance of the cavitation bubble is also changed by the presence of polymers. There also seems to be a noticeable difference in the appearance of steady-state cavities in flows containing polymer solutions compared with observations at the same cavitation number in water (Hoyt 1966). The cavitation bubbles generated in polymer solutions are more striated, and appear to collapse less violently than the bubbles generated water. Walters (1972) has shown a similar lowering of cavitation inception index on a disk and he noted that the higher-frequency noise content of the bubble collapse spectra was diminished. Under very intense cavitation, the polymer solution produced a higher level of radiated noise at higher frequencies than water. Van der Meulen (1973) has shown that cavitation inception on a hemisphericalnosed stainless steel body in a water tunnel is greatly reduced by the presence of polyethylene oxide, while a Teflon-coated body shows a much smaller effect. In other work, Huang (1971) noted that the cavitation inception reduction with polyethylene oxide was much smaller when a large (100 mm in diameter) model was used in a water tunnel. Table 4.1 Cavitation inception number for different polymer (polyethylene oxide) concentration for a hemispherical-nosed cylindrical body Fluid

Tunnel velocity (m/s)

Cavitation inception number

Water 20 ppm Polyox 50 ppm Polyox 100 ppm Polyox Algae

12.55 13.40 14.18 13.70 12.88

0.73 0.50 0.39 0.41 0.66

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In an attempt to interpret these results, van der Meulen (1976) and Gates and Acosta (1979) visualized the boundary layer along the surface of the body in the region of separation. They noted that the polymer additives remove the laminar separation by stimulation of transition causing transition to turbulence at much lower Reynolds numbers than the pure solvent. This conclusion is further substantiated by the observation that there is essentially no effect on the cavitation characteristics of a Schiebe body on which there is no laminar separation. Moreover, as indicated by Gates and Acosta (1979), tripping the boundary layer transition on the hemispherical headform has been demonstrated to be more effective than the polymer additives in delaying cavitation occurrence. It seems, therefore, that the polymer additives act by modifying the behaviour of the boundary layer and have little effect on the behaviour of individual bubbles at inception. In a different type of experiments, Brennen (1970) investigated the influence of dilute polymer solutions (polyethylene oxide Polyox WSR 301, polyacrylamide Separan AP30, and guar gum) on the characteristics and appearance of the interface of well developed cavities generated behind a cylinder and spheres. He observed that the polymers caused a wavy instability of the wetted surface flow around the headforms at the initiation of the cavity. This instability can be related to the effect of the polymers on the transition and separation mentioned above. Bazin et al. (1976) presented result on the effect of the injection of very concentrated (5,000 and 10,000 ppm) solutions of polyethylene oxide on the surface of a cylinder downstream of the stagnation point. They found that the injection of the polymers inhibits the development of cavitation and that the noise level at inception stage is significantly reduced by the polymer additive. Ting (1978) investigated the influence of two drag-reducing polymers (polyethylene oxide PEO-FRA, and hydrolyzed polyacrylamide PAM-273) on cavitation inception around cylinders mounted on a rotating circular disk. One cylinder was placed 11.43 cm from the center of the disk and the other one at 13.02 cm, the angle between them being 90◦ . The cavitation number in this experiment was defined as σ =

2 (p − pv ) , ρu2

(4.9)

where the velocity u is the product of the disk rotating speed ω and the radial distance r where the cavitation inducer is located from the center of the disk. The cavitation inception number was plotted against a Reynolds number defined by Re = ρrωd/η, where d is the diameter of the cavitation inducer and η is the viscosity of water. The inception cavitation number as a function of the Reynolds number for the polyethylene oxide solution is plotted in Fig. 4.6. The filled symbols represent data corresponding to the inside cylinder (r = 11.43 cm) and the open symbols are those corresponding to the outside cylinder (r = 13.02 cm). It is clear that the polymer additive has a strong effect in reducing the values of the cavitation inception number. The suppression effect generally increases with increasing polymer concentration. For example, for the 500 ppm Polyox solution, the cavitation inception number is

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Fig. 4.6 Cavitation inception data for water and polyethylene oxide aqueous solution at (a) the inside and (b) the outside cavitation inducer. Reproduced with permission from Ting (1978). © American Institute of Physics

reduced to about 65% of that in water for Re = 105 . Similar results were obtained in the case of the polyacrylamide solution (Fig. 4.7). However, for the largest concentration, polyacrylamide seems to be more effective over a larger range of Reynolds numbers as a result, probably, of its capacity to sustain high shear rates with moderate degradation. Ting also noted that the appearance of cavitation bubbles in polymer solutions was more transparent than in water and showed a regular and smooth wavy pattern, as demonstrated by the experiments conducted by Brennen (1970). The suppression of cavitation inception in polymer solutions was explained by Ting (1978) by an increased extensional viscosity of polymer solutions due to the hydrodynamically interacting stretched molecules. He noted that when a fluid element is approaching the stagnation point of the cylinder, the flow field is one with a high deceleration rate. The axisymmetric compression that develops in the stagnation region generates high extensional stresses that lead to the suppression of cavitation inception. High flow gradients also develop as the fluid flows around the cylinder introducing high viscoelastic stresses which change the overall flow field. Reitzer et al. (1985) investigated the flow around a cylinder in an open loop tunnel. A 1,000 ppm polyethylene oxide solution was ejected upstream of the test

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Fig. 4.7 Cavitation inception data for water and polyacrylamide aqueous solution at (a) the inside and (b) the outside cavitation inducer. Reproduced with permission from Ting (1978). © American Institute of Physics

section with a flow rate such that the concentration of the polymer solution in the test section reached a concentration of 3 ppm. The main results of this study are summarized in Fig. 4.8, where the acoustic pressure is plotted as a function of the Reynolds and cavitation numbers. The Reynolds number was calculated from the mean velocity of the liquid upstream the cylinder, the viscosity of water and the diameter of the cylinder. At the very low polymer concentration used in their experiment (3 ppm) the viscosity of the solution was approximated to the viscosity of water. Before cavitation inception (Zone I; σ > 8.7), a low level acoustic pressure, slightly increasing with the Reynolds number, was observed at high cavitation coefficients. This acoustic pressure was attributed to the background noise of the tunnel. At a Reynolds number of Re = 85,000 (corresponding to σ = 8.7), inception of the cavitation occurs in water and the collapse of isolated bubbles has been visually observed. Beyond this value of the Reynolds number, cavitation develops and the acoustic pressure increases up to a maximum (Zone II; 8.7 > σ > 4.3), after which it decreases to a minimum corresponding to a Reynolds number of 1.24 × 105 (Zone III; 4.3 > σ > 3.2). The noise then increases very sharply, reaches another maximum much larger than the previous one (Zone IV; 3.2 > σ > 2.7), and decreases

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Fig. 4.8 Acoustic pressure as a function of cavitation and Reynolds numbers for water and a 3 ppm polyethylene oxide aqueous solution. Reproduced with permission from Reitzer et al. (1985). © Elsevier B.V.

as sharply as it had increased (Zone V; σ > 2.7). The most important observation in the polymer solution concerns the inception of the cavitation, which is delayed from Re = 8.5 × 105 to Re = 1.05 × 106 (corresponding to σ = 5.1). The polymer solution completely inhibits zone II, which is replaced by the continuation of zone I, while zone III, where the noise was decreasing in the case of water, is replaced by an increasing region that fits perfectly zone IV. From these data the authors conclude that the suppression of cavitation associated with the presence of the polymer molecules is more likely due to the modification of the flow field around the cylinder than an effect of the macromolecules on the dynamics of individual bubbles. However, no information on the size of the cavitation bubbles is provided. As we already discussed in the previous chapter only bubbles with a maximum radius smaller than 10–1 mm will be affected by the viscoelastic properties of the surrounding liquid.

4.2.4 Vortex Cavitation Vortical structures occur in a wide range of flows. These vary from the eddies in turbulent flows that occur randomly in time and space to more developed vortices that occur at the tips of hydrofoils and lifting surfaces (also called tip vortex) and at the hubs of propellers (also called hub vortex). Tip vortex cavitation is typically

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observed as the first form of cavitation in propeller flows. The prediction of the onset of this type of cavitation is particularly important in the design of “silent” propellers, as cavitating vortex represents a significant source of noise. 4.2.4.1 Tip Vortex Cavitation The prediction of tip vortex cavitation inception is a complex problem which has been approached by many investigators and which is of practical interest for marine propellers. As we have previously discussed, the inception cavitation number σi is supposed to be given by the minimum value of the pressure coefficient (−Cp )min . In the case of the trailing vortex which develops at the tip of a finite span hydrofoil, it is still a challenge to predict where the minimum pressure, pmin , will occur along the vortex path. Considerable insight can be gained from simple models of vortex of strength and viscous core radius a. In these cases, a minimum pressure coefficient can be defined by (Cp0 )min =

pmin − p0 2 , 1 2 ρ 2π a

(4.10)

where (Cp0 )min = −2 if a Rankine model is used and (Cp0 )min = −1.74 if a Lamb vortex is used. Classical theory for an elliptically loaded lifting surface shows that the strength of the vortex is related to the lift coefficient CL and the mid-span chord length c0 , U∞ c0

=

CL , 2

(4.11)

where U∞ is the undisturbed velocity. A generalization of these ideas leads to (Arndt 2002) (Cp )min = Cp0

CL 4π

2

c0 2 , a

(4.12)

where Cp0 depends on the circulation distribution. Thus, cavitation inception scales with lift (and hence circulation) and the inverse of the vortex core radius. The determination of the core radius remains, however, a difficult problem. Despite the uncertainty in the relationship for core radius, numerous studies indicate that σi scales with Rem , where m is generally accepted to be approximately 0.4. It should be noted here that the conditions of inception in the vortex core are sensitive to other parameters such as turbulence, water quality, and confinement. The influence of water quality, i.e. nuclei and dissolved gas content, on tip vortex cavitation was proved by several studies (Arndt and Keller 1991; Gowing et al. 1995). As for the effect of turbulence, Arndt et al. (1991) reached the conclusion that no complete correlation is possible without knowledge of the fluctuating levels of pressure in the vortex flow. Figure 4.9 gives a typical example of the effect of confinement on tip

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Fig. 4.9 Visualizations of tip vortex cavitation in water for various values of tip clearance, e. The position of the wall is indicated in right-hand side of the photographs. (a) e = 50 mm, (b) 20 mm, (c) 13 mm, (d) 4 mm, (e) 2 mm, (f) 0.5 mm. Reproduced with permission from Boulon et al. (1999). © Cambridge University Press

vortex cavitation in water. For a value of the cavitation parameter of 2.6, in the case of Fig. 4.9a, no cavitation occurs in the tip vortex as long as tip clearance remains above 20 mm. For a clearance of about 20 mm, a continuous vapour core suddenly appears. It is stable, but not attached to the tip, as shown in Fig. 4.9b. As the tip clearance is further reduced, the vapour core extends towards the foil tip and attaches

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to the tip (Fig. 4.9c). For even smaller values of the clearance (Fig. 4.9d), in addition to the tip vortex cavitation, a leading-edge cavity originates in the laminar separation bubble. The cavitating vortex is highly disturbed by the development of turbulence and it is barely visible for a clearance of only 2 mm (Fig. 4.9e). For highly confined flows (Fig. 4.9f), leading-edge cavitation, as well as tip vortex cavitation, practically vanish. In the general case of a non-Newtonian fluid the radial pressure gradient for a linear vortex is given by V2 ∂ (p − τrr ) τθθ − τrr =ρ θ + , ∂r r r

(4.13)

where Vθ is the tangential velocity and τrr and τθθ are the extra stresses in the radial and azimuthal direction, respectively. The pressure on the axis of the vortex can be obtained by integrated this equation. In a Newtonian fluid the extra stress terms cancel. In non-Newtonian viscoelastic fluids, these terms contribute to reduce the centrifugal effect with subsequent effects on cavitation inception and development. Inge and Bark (1983) conducted experiments with an elliptical wing having a maximum chord of 160 mm and half span of 238 mm. They reported results obtained by ejecting concentrated polymer solution into the water stream through an injector situated one meter upstream of the foil and for homogeneous polymer solutions with concentrations between 0.01 and 12 ppm of polyethylene oxide Polyox WSR 301. They results show very clearly that cavitation occurrence is significantly delayed for concentrations larger than 1 ppm. The experiments with the polymer solution ejection gave qualitatively similar results. The cavitation inhibition was theoretically ascribed to the normal stresses developed by the polymer solutions (Inge 1983). Fruman (1984) investigated the behaviour of the vortex of a NACA 16020 cross section elliptical foil in water and in polymer aqueous solutions ejected from a 0.5 mm diameter tube at a distance of 20 mm upstream of the tip. Ejecting a 500 ppm polyethylene solution when a large vapour tube occupies the vortex core considerably modifies the appearance of the cavitation. At an ejection velocity of about half of the free stream the continuous long cavity is reduced to a very short cavity of about half of a maximum chord length and only scattered isolated bubbles are carried downstream. By doubling the ejection velocity these entrained bubbles are eliminated and only the shortened cavity remains. The values of the inception cavitation number obtained with an ejection of 250 ppm polymer solution are smaller with about 30% than those corresponding to pure water. Fruman and Aflalo (1989) conducted experiments on the effect of drag-reducing polymer solutions on tip vortex cavitation of a finite span hydrofoil. They also conducted systematic measurements of the hydrodynamic forces on the hydrofoil and tangential velocities in the tip vortex. The cavitation number was defined as σ =

p∞ − pv , 2 0.5ρU∞

(4.14)

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where p∞ and U∞ are the pressure and velocity of the unperturbed flow, respectively. Cavitation desinence was determined visually and checked against noise measurements. Two hydrofoils of elliptical shape and symmetrical cross section were employed for generating the tip vortex. The first hydrofoil has a maximum chord of 40 mm and a half span of 60 mm and was provided with a 1 mm diameter injection orifice at the tip of the wing. The second hydrofoil has a maximum chord of 30 mm and a half span of 40 mm. Tests were conducted without and with a 1,000 ppm polyethylene oxide (Polyox WSR 301) solution ejection at the tip of the large foil, with pure water, and with an homogeneous solution of the same polymer with a concentration of about 10 ppm. They found that at equal incidence angle the desinent cavitation number with polymer ejection and in the homogeneous polymer solution is smaller than in pure water (Figs. 4.10 and 4.11). The measurements of the lift of the foil indicated that it was significantly reduced in the homogeneous polymer solution. The tangential velocities were also considerably reduced in both the core and the potential region. However, lift coefficients were not affected by the ejection process. In this case, the maximum tangential velocity of the vortex decreases, the size of the viscous core increases and the intensity of the vortex remains constant during polymer solution ejection. These results indicate that the mechanism of cavitation inhibition with polymer ejection from the tip of a hydrofoil and in homogeneous polymer solution is completely different. In the case of polymer ejection, it is due to a local modification of the tangential velocities of the vortex core without changing the lift (and hence circulation) of the foil. In the case of an homogeneous polymer solution, it is due to the modification of the lift (and hence circulation) of the foil and the associated change of the tangential velocities. Equal mass ejection rates of water, and water/glycerine solutions did not alter the tip vortex conditions. Cavitation inhibition was thus associated solely with the viscoelastic properties of the polymer solutions. It was further speculated that

Fig. 4.10 Desinent cavitation number versus incidence angle for water and semi-dilute polymer (Polyox WSR 301) ejection. Reproduced with permission from Fruman and Aflalo (1989). © American Society of Mechanical Engineers

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Fig. 4.11 Desinent cavitation number vs the incidence angle for water and homogeneous polymer (Polyox WSR 301) ejection. Reproduced with permission from Fruman and Aflalo (1989). © American Society of Mechanical Engineers

the jet from the ejection orifice swells in such a way that the roll-up of the potential flow occurs over an apparently rounded tip. The results obtained by Fruman and Aflalo (1989) have been confirmed by Chahine et al. (1993) in the case of polymer (Polyox WSR 301) ejected through orifices at the tip of the blades of a 29-cm-diameter propeller. With a polymer concentration of 3,000 ppm they were able to achieve critical cavitation number reduction of about 35%. As an example, Fig. 4.12 shows the cavitation number at inception as a function of the polymer concentration for the pure water conditions, for water/glycerin solution injection, and for Polyox solution injection. The viscoelastic properties of the polymer solution injected in the vortex core play a significant role in thickening the viscous core of the tip vortex and, thus, reduce the pressure drop at the vortex center without affecting circulation or lift. To obtain more information on the effect of polymer additives on the suppression of tip vortex cavitation, Fruman et al. (1995) conducted experimental investigations with an elliptical hydrofoil having a NACA 16020 cross section, a 3.8 aspect ratio, and a maximum chord length of 80 mm. They also conducted axial and tangential velocities measurements very close to the foil tip. The velocity profiles were measured along a direction, y, parallel to the span, positive outboard passing through the vortex axis. The downstream distance, x, was measured relative to the tip of the foil which was taken as the origin. Measurements were conducted for x /cmax of 0.125, 0.25, 0.5 and 1, where cmax is the maximum chord (Fig. 4.13). In this experiment, the polymer (polyethylene oxide Polyox WSR 301) solution was ejected through a port with a diameter of 1 mm situated at the foil tip. Figure 4.14 shows the nondimensional axial, Va V∞ , and tangential, Vt V∞ , velocity for an angle of attack of 10◦ and a free stream velocity V∞ = 12.5 m/s. The tangential velocities indicate a solid-body rotation region, where velocities increase

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Fig. 4.12 Influence of the concentration of injected Polyox solutions on tip vortex cavitation inception and comparisom with water and water/glycerin solution injection. Reproduced with permission from Chahine et al. (1993). © American Society of Mechanical Engineers

Fig. 4.13 Schematic of the test arrangement in the experiment of Fruman et al. (1995)

linearly with distance from the vortex axis, an intermediate transition region, and a potential region, where velocities are inversely proportional to the distance to the vortex axis. Only minor modifications of the tangential velocity profiles occur during water ejection. In contrast, the ejection of the polymer solution causes a significant reduction of the maximum tangential velocity and an appreciable increase in the size of the viscous core while the potential region remains unchanged. The radius of the viscous core increases significantly by about 70% near the tip of the foil. On the other hand, ejection of water or polymer solution causes a reduction of the axial velocities in the viscous core region. Very close to the tip, the effects of polymer solution are significantly larger than those of water. However, the effect of polymer additive becomes smaller and smaller when the distance from the foil tip increases.

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Fig. 4.14 Tangential and axial velocities as a function of distance to the vortex axis for different axial stations. Polymer (polyethylene oxide) concentration 1,000 ppm. Reproduced with permission from Fruman et al. (1995). © Japan Society of Naval Architects and Ocean Engineers

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The authors concluded that jet swelling caused by the relaxation of normal stresses associated with the viscoelastic properties of the polymer solution is responsible for the thickening of the viscous core which, in turn, causes the inhibition of the tip vortex cavitation. Latorre et al. (2004) carried out a theoretical analysis of tip vortex cavitation inception based the Rankine vortex model. In their analysis, the bubbles were assumed to be spherical and the non-Newtonian features of the polymer solution were assumed to only affect the vortex core radius. Their analysis shows that while polymer injection causes instability in small bubbles, its main effect is an increase in tip vortex core radius, resulting in the delay of tip vortex cavitation inception. Very recently, Zhang et al. (2009) investigated numerically the dynamics of a propeller tip vortex in water and in polymer solutions. Polymer injection, simulated with inclusion of polymer effects only in the tip vortex centerline region, results in higher pressures at the vortex center than in pure water. The pressure along the vortex centerline was found to first decrease then increase for both water and the polymer solutions. Starting from the propeller tip, polymer stresses along the vortex centerline increase dramatically and reach a maximum in the region close to the minimum pressure point. This pressure rise can explain tip vortex cavitation suppression with polymer injection, in agreement with previous experimental observations. 4.2.4.2 Cavitation in Vortex Chambers Hoyt (1978) investigated the effect of two polymer solution (polyethylene oxide and Carbopol) on the onset of cavitation in a vortex chamber where the liquid was injected tangentially from a single port and evacuated through an axial pipe. The results are summarized in Table 4.2. The incipient cavitation number was defined as the difference between the discharge pressure and the vapour pressure divided by the pressure difference across the device. There is a large inhibition effect for the polyethylene aqueous solution, even at very low concentration. The non dragreducing polymer, Carbopol, does not display an inhibition effect. Bismuth (1987) and Fruman et al. (1988) conducted several tests in a long vortex chamber where the fluid was introduced tangentially at one end through eight rectangular slits and evacuated axially at the other end. The reference pressure was Table 4.2 Incipient cavitation number and air content for tests in a vortex chamber Test liquid Deaerated water Tap water Polyetylene oxide Polyetylene oxide Polyetylene oxide Polyetylene oxide Carbopol

Concentration (ppm)

Incipient cavitation number

Air content/air content at saturation

8.2 12 16 17.5 20

0.253 1.48 0.547 0.513 0.1 0.107 1.3

0.284 0.701 0.755 0.709 0.709 0.785

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measured in a large tank situated downstream of the vortex chamber and used to determine the incipient cavitation number using, instead of pressure drop across the chamber as in Hoyt (1978), the kinematic head at the exit section. In contrast to the results of Hoyt (1978), for a polyethylene oxide aqueous solution in concentration of 10 ppm, the cavitation onset was advanced as compared to the case of water. It should be noted, however, that because of the different definition of the cavitation number, direct comparison of these results is difficult. If it is accepted that the drag reducing properties of the polymer solution should increases, at equal pressure drop, the flow rate, the results by Hoyt (1978) should correspond to a much larger inhibition effect. On the other hand, the velocity measurements indicated that the tangential velocity increased, as compared with pure water, when moving from the wall towards the center of the vortex. This increase was large enough to justify the enhanced cavitation characteristics of the flow. Bismuth (1987) also performed experiments by ejecting, through an orifice of 1 mm diameter situated at the end opposite to the evacuation, semi-dilute solutions of polyethylene oxide with a concentration of 10 ppm. The main result of his study is that the polymer ejection significantly inhibits cavitation onset. In well developed cavitation stages, the vapour tube that occur on the chamber axis in the case of pure water is made to disappear over increased distances when the rates of ejection of the polymer solution increase. A vortex chamber with tangential injection was also used by Barbier and Chahine (2009) in order to generate a central line vortex and observe its structure in water and various solutions of polymer and corn syrup. Measurements of the velocities, pressures, and thus the cavitation number were conducted using a particle image velocimetry system, pressure gauges, and Pitot tubes. Experiments were performed using water, different dilute concentrations of polymer (Polyethylene oxide Polyox WSR 301) solutions, and solutions with different concentrations of corn syrup for a large range of Reynolds numbers. The measurements and observations showed that cavitation inception at the vortex center was delayed when polymer and corn syrup solutions are used as compared to the experiments in water. However, contrary to reported observations with tip vortices, here the large scale vortex was found to rotate faster in the polymer and corn syrup solutions. This did not match with the previous observations of cavitation inception delay in the case of polymers and the conventional thinking about the relationship between pressures and velocities in a vortex line. This may be due to the observations that the velocity fluctuations and the turbulent kinetic energy in the viscous core region increased significantly in the polymer and corn syrup solutions.

4.2.5 Cavitation in Confined Spaces In a lubricated conjunction, cavitation occurs in the diverging part of the contact and it is responsible for a partial or complete collapse of the lubricated film. Consequently, it reduces the load-caring capacity of the interface and affects the lubricant film thickness, friction force and lubricant flow rate. Pockets (or cavities)

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of gas may interrupt the film, producing film rupture or cavitation. Dowson et al. (1980) propose a simple classification of the cavitation phenomena based on the main mechanism that governs it. They consider that there are two types of cavitation: “vaporous cavitation” and “gaseous cavitation”. The first type occurs when the lubricant pressure is reduced to its vapour pressure. The second is encountered when the lubricant pressure falls below the saturation pressure and dissolved gases are emitted from the solution. A pressure reduction below ambient conditions may either encourage suspended bubbles of gas to grow or draw gas into the lubricating film from an external source such as atmosphere. This form of gaseous cavitation is called ventilation (Dowson et al. 1980). The flow between a moving and a fixed wall separated by a micron size gap is characterized by very large pressure gradients in the flow direction. It is, therefore, expected that the extensional flow prevailing in the vicinity of the gap will promote the occurrence of an elastic contribution and modify the conditions for cavitation onset and development. Narumi and Hasegawa (1986) were probably the first who investigated the influence of the viscoelastic properties of non-Newtonian fluids on cavitation in very confined space. They considered the flow between a flat glass and a convex lens with a radius of curvature of about 2 m. The Newtonian fluids in their experiment consisted in a 10% glycerol aqueous solution (with a viscosity η = 1.08 mPa·s) and a 30% (η = 3.51 mPa·s) and 50% (η = 11.3 mPa·s) starch syrup aqueous solution, respectively. The non-Newtonian fluids are polyethylene oxide aqueous solutions in concentration of 100 and 200 ppm. The polymer solutions have a constant viscosity and elastic properties. The viscosity of both polymer solutions is similar to that of water. Their results indicate that the viscoelastic effects in the thin film flow leads to a significant displacement of the point of cavitation from the centre of contact (where film thickness is a minimum) and enhanced film thicknesses. Ouibrahim et al. (1996) have investigated the influence of polymer additives on cavitation generated in very confined spaces comprised between a rotating cylinder of radius R and a stationary flat plate, with a minimum gap, e, down to 5 μm. The test fluids are water and a 600 ppm polyethylene oxide (Polyox WSR 301) aqueous solution, known to be a very effective drag reduction agent in dilute solution. The polymer solution displays a slightly shear-thinning behaviour and the shear viscosity, for large shear rates (>103 s–1 ), is 2.05 mPa·s. The cavitation number is defined as σ =

Pref − pv 0.5ρ (ωR)2

,

(4.15)

where ω is the cylinder tangential velocity and Pref is a reference pressure. The gap, e, was varied between 5 and 20 μm while the tangential velocity, U = ωR, of the rotating cylinder was varied from zero to 22 m/s. The absolute pressure was decreased from 105 Pa to 8 × 103 Pa. It should be noted here that, for all experimental conditions, the product, (e/R)Re, where Re is the Reynolds number calculated with U and e, is much smaller than unity as lubrication theory requires.

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Cavitating Flows

145

The cavitation inception stage in water is characterized by the occurrence of intermittent spots, scattered along a thin band parallel to the cylinder axis, downstream of the minimum gap. When the cavitation number was decreased, the cavitation spots become less scattered, decrease in number, and take the shape of an arrowhead with the tip facing the upstream of the rotating cylinder. At even smaller cavitation numbers, a string of nearly regular cavities is formed, with a characteristic dimension of a few millimeters, separated by thin lateral liquid films. In the case of the Polyox solution and of incipient conditions, intermittent tiny cavitation spots in the shape of rod-like cells scattered along a very thin band parallel to the cylinder axis, appear downstream of the position of the minimum gap. When the cavitation number is decreased, a nearly continuous cavity, characterized by a practically straight leading edge and an undulating trailing edge, develops parallel to the axis of the rotating cylinder. This continuous cavity results from the lateral coalescence of the arrowhead cavities, present only for a very small range of the reference pressure. If the cavitation number is further reduced, the trailing edge waviness increases, and oscillations occur in the flow and axial directions. For even smaller values of the cavitation number, the movement of the cavity trailing edge becomes chaotic. The authors noted that the difference in morphology is not a consequence of the modification of the shear viscosity since tests conducted with a water/glycerin solution, with a viscosity of 11 mPa·s, show arrowhead cavities analogous to the ones observed in water. It seems to be related to the strong extensional flow prevailing in the confined space and the viscoelastic characteristics of the polymer solution. Figure 4.15 shows the inception cavitation number, σi , obtained for a gap e = 10 μm, in the case of water and Polyox solution. For equal Reynolds numbers, the polymer solution decreases the inception cavitation numbers as compared to the solvent results.

Fig. 4.15 Cavitation inception number as a function of the Reynolds number in a lubricating type flow for water and a 600 ppm polyethylene oxide (Polyox) aqueous solution. Reproduced with permission from Ouibrahim et al. (1996). © American Institute of Physics

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More recently, Seddon and Mullin (2008) studied the cavitation phenomenon in the lubrication layer between a rotating heavy sphere and a nearby wall of a rotating drum in a Newtonian fluid (silicone oil with a nominal kinematic viscosity of 103 mm2 /s), a shear-thinning fluid (high molecular weight silicone oil with a zero-shear viscosity of 1.3 × 104 mm2 /s), and a viscoelastic fluid (0.025% polystyrene Boger fluid with a zero-shear viscosity of 2.4 × 104 mm2 /s, and a relaxation time of about 3 s). Images of the vapour cavities in all three fluids are shown in Fig. 4.16. For the Newtonian fluid, an almost circular cavity was found between the sphere and wall in the downstream region of the flow. In the shear-thinning fluid, a pair of stable symmetric cavities was formed adjacent to each other separated by a tongue of fluid that was able to stretch across the cavity site and form a stable liquid bridge. However, the most intriguing cavitation occurred in the viscoelastic fluid because its increased elasticity allowed several tongues of fluid to stretch across the cavitation site in order to create many long thin cavities. In this case, a hierarchical structure of cavities was formed that was unstable and constantly changed shape.

Fig. 4.16 Examples of vapour cavitation in the downstream region of the lubricating layer between a sphere and wall. (a) and (d) are in the Newtonian fluid, (b) and (e) are in a shear-thinning fluid, and (c) and (f) are in a viscoelastic fluid. The scale-bar on image (a) represents 2 mm. The direction of flow is shown by the white arrow in (d). Reproduced with permission from Seddon and Mullin (2008). © American Institute of Physics

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147

Fig. 4.17 Time sequence of the vapour cavity ensemble created between sphere and wall in a Boger fluid. Important features include the unstable cavity patch ahead of the main cavitation site (at the bottom of each image), the division of the central large cavity, the subsequent interfacial motion between adjacent cavities, and the irregular trailing tail (at the top of each image). The scale bar on the first image represents 2 mm. Reproduced with permission from Seddon and Mullin (2008). © American Institute of Physics

A time sequence of images outlining the dynamical motion of the viscoelastic fluid cavity ensemble is shown in Fig. 4.17. Ahead of the cavitation site, at the bottom of each image in Fig. 4.17, patches of unstable cavitation were constantly created and annihilated. The pressure distribution in this part of the flow is not enough to cause persistent cavitation, indicating that the elasticity of the fluid was occasionally adding a tensile stress to the fluid to cause cavitation. At the main cavitation site, several cavities were formed adjacent to each other that underwent a creation-division-collapse process: a vapour bubble was formed in the center of the ensemble and split into two bubbles at its leading edge. These two new cavities then moved outwards and shrank. When the cavities reached the extremities of the ensemble, they disappeared completely. This is a continuous process where bubbles were created, split, and moved to the edges over a time scale of about 1 s. It is interesting to note here that the relaxation time of the Boger fluid is about 3 s. At very fast drum rotation speeds, the vapour cavities in the Boger fluid ceased to exist and are replaced by a single air bubble, shown in Fig. 4.16f. This implies a reduced negative pressure in the lubrication region. A similar observation was made by Ouibrahim et al. (1996), who showed that the contribution from elasticity to the normal stresses in a shear flow was such as to add a positive pressure.

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4.2.6 Mechanisms of Cavitation Suppression by Polymer Additives Most of the experimental investigations on hydrodynamic cavitation indicates that the polymer additives have a major influence on cavitation inception and development. Cavitation in orifice-discharge flows has been shown to be delayed when the polymers are present in the flow. Propeller cavitation is affected by polymers. Onset of cavitation in the wake of a cylinder was also inhibited, as is cavitation in a vortex. The mechanisms that cause the inhibition of the cavitation by polymer additives are not completely understood. There is, however, an almost general consensus on the suppression mechanisms in the case of cavitation around blunt bodies and in tip-vortex cavitation. In the case of flow around blunt bodies, the polymer acts to inhibit cavitation by an earlier transition to turbulence, thus suppressing the laminar separation region where cavitation is most likely. In the case of tip-vortex cavitation, ejection of polymer from the wing tip leads to a modification of the tangential velocity component large enough to justify a pressure increase in the vortex core and thus a retardation of the cavitation. In the case of a homogeneous polymer solution, the cavitation suppression is due to a modification of the lift (and hence the circulation) of the foil and the associated change of the tangential velocities. Comparison between different experiments and fluids is difficult because, in all experimental studies, the rheology of the fluids was not provided. Most likely, all previous experimental investigations have been performed using shear-thinning elastic fluids. If the fluid is shear-thinning, then it is difficult to distinguish between the effects of shear thinning and those of elasticity, especially when the Reynolds number is high. Consequently, it is difficult to ascertain the role of elasticity in all of the above-mentioned observations. Extensional effects often play a role in phenomena associated with viscoelastic fluids, such as in drag reduction where extensional viscosity is associated as the cause in the reduction of turbulence through the suppression of eddy formation (Roy et al. 2006). It is, therefore, expected that extensional viscosity will play an important role in the suppression of cavitation inception in viscoelastic fluids. The apparent Trouton ratio for polymer aqueous solutions may be significantly greater than the Newtonian Trouton ratio of Tr = 3 (Brujan et al. 2004). Therefore, the resistance to extension from the polymer solutions is higher than for water and this will have an effect on the flow kinematics. The greater resistance to extension for the polymer aqueous solutions when compared to water will result in a lowering of the velocity (and hence an increase of static pressure) in areas of high extension rate. The net result is the suppression of cavitation inception. The extensional viscosity is expected to play a major role in the flow through orifices and in lubricating type flows where the extensional component of the flow is dominant. We conclude this section by noting that there is a need for new experimental results and numerical predictions on the behaviour of cavitation in non-Newtonian fluids using well-characterized fluids which can be described by sophisticated constitutive models. Constant-viscosity elastic fluids, commonly referred to as Boger

4.3

Estimation of Extensional Viscosity

149

fluids, must be used in future experiments in order to ensure that changes in the flow kinematics are associated purely with fluid elasticity and cannot be confused with effects due to shear-thinning viscosity.

4.3 Estimation of Extensional Viscosity There have been many recent attempts to estimate the extensional viscosity of mobile liquids. It must be noted, however, that generating a purely extensional flow in the case of mobile fluids is virtually impossible. The most that one can hope to do is to generate flows with a high extensional component and to interpret the data in a way which captures that extensional component in a consistent way through a suitable defined extensional viscosity and strain rate. An unambiguous determination of the extensional viscosity of dilute polymer solutions is thus very difficult, perhaps impossible. However, since the extensional viscosity of dilute solutions of polymers exhibit very high values, the experimental methods with only semi-quantitative capabilities may be sufficient in some practical applications. In the case of an extensional flow, the elastic component of a non-Newtonian fluid has the net effect of adding a positive pressure to the Newtonian contribution which leads to a reduction of the incipient cavitation number. The added pressure can be estimated from the difference of incipient cavitation numbers at equal Reynolds number. Ouibrahim et al. (1996) used the flow in a very confined space comprised between a rotating cylinder of radius R and a stationary plate (see Sect. 4.2.5) to determine the extensional viscosity of a 600 ppm aqueous solution of polyethylene oxide. For this particular case, the added pressure, Pe , is given by Pe =

( 1 (( σip − σiw ( ρ(ωR)2 , 2

(4.16)

where σip and σiw is the inception cavitation number in the case of polymer solution and water, respectively, and ω is the angular velocity of the cylinder. The values of Pe are plotted in Fig. 4.18, together with an estimate of the extensional viscosity, defined as μe =

Pe , ∂u/∂x

(4.17)

where ∂u/∂x(≈ 70ω) is the maximum extensional strain rate. The results obtained by Ouibrahim et al. (1996) indicate that the elastic pressure contribution increases with the strain rate increasing while the extensional viscosity decreases. The latter is up to four orders of magnitude larger than the shear viscosity of water (a Trouton ratio as high as 104 ). A general characteristic of flexible polymers (such as polyethylene oxide) is that they are extension rate thickening (Barnes et al. 1989). However, Stokes (1998) has

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Fig. 4.18 Elastic contribution to the pressure and extensional viscosity of a 600 ppm aqueous solution of polyethylene oxide as a function of the extensional strain rate. Reproduced with permission from Ouibrahim et al. (1996). © American Institute of Physics

indicated that the extensional viscosity of flexible polymer solutions is initially at a value close to that of a Newtonian fluid (a value of the Trouton ratio close to 3) at low extension rates and then it increases to a maximum, at a certain value of the extensional strain rate, after which the extensional viscosity decreases. Thus, the results obtained by Ouibrahim et al. (1996) are, in the high extensional strain rate region, in qualitative agreement with the previously reported results. The mechanisms for the apparent decrease at high rates have not been established, but it may be associated with the polymer not having enough time to extend in the flow field. In contrast, a constant extensional viscosity is characteristic of rigid or semirigid macromolecules and with perfectly aligned rigid rods (see, for example, Ng et al. 1996). Large-aspect-ratio macromolecules and rigid rods align instantaneously with the flow field, even at relatively low extension rates, such that the extensional viscosity is relatively independent of extension rate.

References Arndt, R.E.A. 2002 Cavitation in vortical flows. Annu. Rev. Fluid Mech. 34, 143–175. Arndt, R.E.A., Keller, A.P. 1991 Water quality effects on cavitation inception in a trailing vortex. In Proceedings of the ASME, Cavitation ’91, Portland, OR, USA, pp. 1–9. ASME. Arndt, R.E.A., George, W.K. 1979 Pressure fields and cavitation in turbulent shear flows. In Proceedings of the Twelfth Symposium on Naval Hydrodynamics. Washington, DC, USA, pp. 327–339. Arndt, E.A., Hoyt, J.W., Baker, C.B. 1981 A brief survey of polymer effects on cavitation noise. ASME Cavitation and Polyphase Flow Forum. Boulder. pp. 70–73. Arndt, R.E.A., Arakeri, V.H., Higuchi, H. 1991 Some observations of tip-vortex cavitation. J. Fluid Mech. 229, 269–289. Baker, C.B., Arndt, R.E.A., Holl, J.W. 1973 Effect of various concentrations of WSR-301 polyethylene oxide in water upon the cavitation performance of 1/4-in and 2-in hemispherical nosed bodies. Applied Research Laboratory Technical Memo. The Pennsylvania State University, University Park, pp. 73–257.

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Baker, C.B., Holl, J.W., Arndt, R.E.A. 1976 Influence of gas content and polyethylene oxide additive upon confined jet cavitation in water. ASME Cavitation and Polyphase Flow Forum. New York. pp. 6–8. Barbier, C., Chahine, G. 2009 Experimental studies on the effects of viscosity and viscoelasticity on a line vortex cavitation. In Proceedings of the Seventh International Symposium On Cavitation CAV 2009. Ann Arbor, Michigan, USA. Barnes, H.A., Hutton, J.F., Walters, K. 1989 An Introduction to Rheology. Elsevier, New York. Bazin, V.A., Barabanova, Y.N., Pokhil’ko, A.F. 1976 Effect of dilute aqueous polymeric solutions on the onset of cavitation on a cylinder. Fluid Mech. Soviet Res. 5, 79–82. Bird, R.B., Armstrong, R.C., Hassager, O. 1987 Dynamics of Polymeric Liquids. Wiley, New York. Bismuth, D. 1987 Inhibition de la cavitation de tourbillon marginal par injection de solutions de polymères. PhD thesis. Univ. Paris VI. Boulon, O., Callenaere, M., Franc, J.P., Michel, J.M. 1999 An experimental study into the effect of confinement on tip vortex cavitation of an elliptical hydrofoil. J. Fluid Mech. 390, 1–23. Brennen, C.E. 1970 Some cavitation experiments with dilute polymer solutions. J. Fluid Mech. 44, 51–63. Brennen, C.E. 1996 Cavitation and Bubble Dynamics. Oxford University Press, Oxford. Brujan, E.A., Ikeda, T., Matsumoto, Y. 2004 Dynamics of ultrasound-induced cavitation bubbles in non-Newtonian liquids and near a rigid boundary. Phys. Fluids 16, 2402–2410. Chahine, G.L., Frederick, G.F., Bateman, R.D. 1993 Propeller tip vortex suppression using selective polymer ejection. J. Fluid Eng. 115, 497–503. Dowson, D., Smith, E.H., Taylor, C.M. 1980 An experimental study of hydrodynamic film rupture in a steadily-loaded, non-conformal contact. J. Mech. Eng. Sci. 33, 71–78. Dubief, Y., White, C.M., Terrapon, V.E., Shaqfeh, E.S.G., Moin, P., Lele, S.K. 2004 On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. Fluid Mech. 514, 271–280. Ellis, A.T. 1967 Some effects of macromolecules on cavitation inception and noise. California Institute of Technology Report 071585. Ellis, A.T., Hoyt, J.W. 1968 Some effects of macromolecules on cavitation inception. ASME Cavitation Forum. New York, pp. 1–5. Ellis, A.T., Waugh, J.G., Ting, R.Y. 1970 Cavitation suppression and stress effects in high-speed flows of water with dilute macromolecular additives. J. Basic Eng. 92, 459–466. Franc, J.P., Michel, J.M. 2004 Fundamentals of Cavitation. Kluwer, Dordrecht. Fruman, D.H. 1984 Tip vortex cavitation inhibition by polymer additives. Cavitation and Multiphase Flow Forum. FED vol. 9, ASME, New York, pp. 73–76. Fruman, D.H. 1999 Effects on non-Newtonian fluids on cavitation. Rheol. Ser. 8, 209–254. Fruman, D.H., Aflalo, S.S. 1989 Tip vortex cavitation inhibition by drag-reducing polymer solution. J. Fluid Eng. 111, 211–216. Fruman, D.H., Bismuth, D., Aflalo, S. 1988 Cavitation in a confined vortex. In AIAA, ASME, SIAM, and APS, National Fluid Dynamics Congress. Cincinnati, OH, pp. 1639–1645. Fruman, D.H., Pichon, T., Cerrutti, P. 1995 Effect of drag-reducing polymer solution ejection on tip vortex cavitation. J. Mar. Sci. Technol. 1, 13–23. Gates, E.M., Acosta, A.J. 1979 Some effects of several free-stream factors on cavitation inception of axisymmetric bodies. In Proceedings of the Twelfth Symposium on Naval Hydrodynamics. Washington, DC, USA, pp. 86–112. Gennes, P.G. de 1990 Introduction to Polymer Dynamics. Cambridge University Press, Cambridge. Gowing, S., Briançon-Marjollet, L., Fréchou, D., Godeffroy, V. 1995 Dissolved gas and nuclei effects on tip vortex cavitation inception and cavitation core size. International Symposium on Cavitation. Deauville, France. DCN Bassin d’Essais des Carènes. Hasegawa, T., Ushida, A., Narumi, T. 2009 Huge reduction in pressure drop of water, glycerol/water mixture, and aqueous solution of polyethylene oxide in high speed flows through micro-orifices. Phys. Fluids 21, 052002.

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Hinch, E.J. 1977 Mechanical models of dilute polymer solutions in strong flows. Phys. Fluids 20, S22–S30. Holl, J.W. 1960 An effect of air content on the occurrence of cavitation. J. Basic Eng. 82, 941–946. Holl, J.W. 1970 Nuclei and cavitation. J. Basic Eng. 92, 681–688. Hoyt, J.W. 1966 Effects of high-polymer solutions on a cavitating body. In Proceedings of the Eleventh International Towing Tank Conference. Tokyo. Hoyt, J.W. 1976 Effect of polymer additives on jet cavitation. J. Fluids Eng. 98, 106–112. Hoyt, J.W. 1978 Vortex cavitation in polymer solutions. Cavitation and Polyphase Flow Forum. ASME, pp. 17–18. Hoyt, J.W., Taylor, J.J. 1981 A photographic study of cavitation in jet flow. J. Fluids Eng. 103, 14–18. Huang, T.T. 1971 Comments on “Cavitation inception: the influence of roughness turbulence, and polymer additives.” Sixteenth American Towing Tank Conference. Sao Paulo, Brazil, Vol. 1, p. 6.10. Huang, T.T. 1986 The effects of turbulence stimulators on cavitation inception of axisymmetric headforms. J. Fluid Eng. 108, 261–268. Inge, C. 1983 Effect of polymer additives on tip vortex cavitation. Tech. Rep. TRITA-MEK 83–05, Roy. Inst. Tech. Sweden. Inge, C., Bark, G. 1983 Tip vortex cavitation in water and in dilute polymer solutions. Tech. Rep. TRITA-MEK 83–12, Roy. Inst. Tech. Sweden. Jimenez, J., Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359. Joseph, D.D. 1990 Fluid Dynamics of Viscoelastic Liquids. Springer, New York. Joseph, D.D., Narain, A., Riccius, O., Arney, M. 1986 Shear-wave speeds and elastic moduli for different liquids. Theory and experiments. J. Fluid Mech. 171, 289–338. Katz, J., Galdo, J. 1989 Effect of roughness on rollup of tip vortices on a rectangular hydrofoil. J. Aircraft 26, 247–253. Keller, A.P. 1979 Cavitation inception measurement and flow visualisation on axisymmetric bodies at two different free-stream turbulence levels and test procedure. In Proceedings of the ASME International Symposium on Cavitation Inception. New York, USA, pp. 63–74. Keller, A.P. 2000 Cavitation scale effects a representation of its visual appearance and empirically found relations. In Proceedings of the International Conference on Propeller Cavitation NCT’50. Newcastle upon Tyne, UK, pp. 357–380. Knapp, R.T., Daily J.W., Hammitt, F.G. 1970 Cavitation. MacGraw-Hill, New York. Kuiper, G. 1981 Cavitation inception on ship propeller models. PhD thesis, University of Delft, The Netherlands. Latorre, R., Muller, A., Billard, J.Y., Houlier, A. 2004 Investigation of the role of polymer on the delay of tip vortex cavitation. J. Fluid Eng. 126, 724–729. Lumley, J.L. 1969 Drag reduction by additives. Ann. Rev. Fluid Mech. 1, 367–384. Narumi, T., Hasegawa, T. 1986 Experimental study on the squeezing flow of viscoelastic fluids (1st Report, The effect of liquid properties on the flow between a spherical surface and a flat plate). Bull JSME 29, 3731–3736. Ng, S.L, Mun, R.P., Boger, D.V. 1996 Extensional viscosity measurements of dilute solutions of various polymers. J. Non-Newt. Fluid Mech. 65, 291–298. Oba, R., Ito, Y., Uranishi, K. 1978 Effect of polymer additives on cavitation development and noise in water flow through an orifice. J. Fluids Eng. 100, 493–499. Ouibrahim, A., Fruman, D.H., Gaudemer, R. 1996 Vapour cavitation in very confined spaces for Newtonian and non Newtonian fluids. Phys. Fluids 8, 1964–1971. Pauchet, A., Viot, X., Fruman, D. H. 1996 Effect of upstream turbulence on tip vortex roll-up and cavitation. In Proceedings of the ASME Fluids Engineering Division Conference. Atlanta. vol. 1, pp. 463–469. Pan, S.S., Yang, Z.M., Hsu, P.S. 1981 Cavitation inception tests on axisymmetric headforms. J. Fluid Eng. 103, 268–272.

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Pichon, T., Pauchet, A., Astolfi, A., Fruman, D.H., Billard, J.Y. 1997 Effect of tripping laminar-toturbulent boundary layer transition on tip vortex cavitation. J. Ship Res. 41, 1–9. Reitzer, H., Gebel, C., Scrivener, O. 1985 Effect of polymeric additives on cavitation and radiated noise in water flowing past a circular cylinder. J. Non Newt. Fluid Mech. 18, 71–79. Rood, E.P. 1991 Mechanisms of cavitation inception. J. Fluids Eng. 113, 163–174. Roy, A., Morozov, A., van Saarloos, W., Larson, R.G. 2006 Mechanism of polymer drag reduction using a low-dimensional model. Phys. Rev. Lett. 97, 23501. Sasaki, S. 1991 Drag reduction effect of rod-like polymer solutions. I. Influences of polymer concentration and rigidity of skeltal back bone. J. Phys. Soc. Japan 60, 868–878. Sasaki, S. 1992 Drag reduction effect of rod-like polymer solutions. III. Molecular weight dependence. J. Phys. Soc. Japan 61, 1960–1963. Seddon, J.R.T, Mullin, T. 2008 Cavitation in anisotropic fluids. Phys. Fluids 20, 023102. Shen, Y.T. 1985 Wing sections on hydrofoils. Part 3. Experimental verifications. J. Ship Res. 29, 39–50. Stokes, J.R. 1998 Swirling flow of viscoelastic fluids. PhD dissertation. University of Melbourne. Tanibayashi, H., Ogura, K., Matsuura, Y. 1998 On the cavitation occurring at the bottom of an accelerated circular cylinder. In Proceedings of the Third International Symposium on Cavitation, Cavitation’98. Grenoble, France, pp. 161–166. Ting, R.Y. 1978 Characteristics of flow cavitation in dilute solutions of polyethylene oxide and polyacrylamide. Phys. Fluids 21, 898–901. Toms, B.A. 1949 Observation on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. In Proceedings of the International Congress on Rheology (Holland, 1948). North-Holland, Amsterdam, 1949, pp. II.135–II.141. Toonder, J.M.J. den 1995 Drag reduction by polymer additives in a turbulent pipe flow: laboratory and numerical experiments. PhD thesis, Delft University of Technology. Toonder, J.M.J. den, Nieuwstadt, F.T.M., and Kuiken, G.D.C. 1995 The role of elongational viscosity in the mechanism of drag reduction by polymer additives. Appl. Sci. Res. 54, 95–123. van der Meulen, J.H.J. 1973 Cavitation suppression by polymer injection. ASME Cavitation and Polyphase Flow Forum. New York, pp. 48–51. van der Meulen, J.H.J. 1976 A holographic study of cavitation on axisymmetric bodies and the influence of polymer additives. Netherlans Ship Model Basin Publ. No. 509. Walters, R.R. 1972 Effect of high-molecular weight polymer additives on the characteristics of cavitation. Advanced Technology Center Inc., Dallas, Report No. B-94300/sTR-32. Warholic, M.D., Massah, H., Hanratty, T.J. 1999 Influence of drag-reducing polymers on turbulence, effects of Reynolds number, concentration, and mixing. Exp. Fluids 27, 461–472. Wang, C.B. 1972 Correlation of the friction factor for turbulent pipe flow of dilute polymer solutions. Ind. Eng. Chem. Fundam. 11, 546–551. White, C.M., Somandepalli, V.S.R., Mungal, M.G. 2004 The turbulence structure of drag reduced boundary layer flow. Exp. Fluids 36, 62–69. Young, F.R. 1989 Cavitation. McGraw-Hill, New York. Zhang, Q., Hsiao, C.T., Chahine, G. 2009 Numerical study of vortex cavitation suppression with polymer injection. In Proceedings of the Seventh International Symposium On Cavitation CAV 2009, Ann Arbor, Michigan, USA.

Chapter 5

Cavitation Erosion

Cavitational activity in close proximity to solid boundaries is known to lead to material damage and erosion. Such damage occurs, for example, at marine propellers, turbine blades, or in pumps, but it is also deemed to be involved in ultrasonic cleaning, in the wear of knee joints, and in a variety of cardiovascular applications of lasers and ultrasound. The evidence that most directly links surface damage to cavitation has come from experiments with hydrofoils in cavitation tunnels, which show that the maximum erosion along a hydrofoil surface correlates with the location of collapsing cavitation bubbles (Knapp et al. 1970). Various techniques have been used to investigate the cavitation erosion of solid surfaces. These include vibratory magnetostrictive devices, cavitating jets, spark and laser-induced bubbles, and fire-resistant fluids. The emphasis in much of this research has been on the classification of materials for resistance to cavitation and the determination of environmental effects such as temperature, physical properties of liquids, and pressure. Evolution of cavitation erosion depends on many parameters such as material properties, surface shape, liquid properties, and cavitation intensity. There is not a universal law describing the evolution of the erosion rate with the period of exposure to cavitation. But in most situations, a little mass loss is observed in an early stage of cavitation (incubation stage) (Fig. 5.1). This stage is followed by a period of significant increase of erosion rate (accumulation stage) or of a constant erosion rate (steady stage). After that, a decrease of erosion rate is often observed (attenuation stage). Incubation stage corresponds to a not detectable weight loss period. The energy developed during cavitation bubble collapse is dissipated by surface elastic or plastic deformation or by cracking for most metals with sometimes work hardening effect at the surface. The surface exhibits some modification like plastic flow, indentation traces, undulation, coarse slip band, and cracking. The determination of duration of this period depends on accuracy of weight measurements. Accumulation is, in most cases, a steady stage. When the work hardening limit is reached, continuous plastic deformation leads to detachment of material and propagation of cracks near the surface. This results in an acceleration of material removal rate. The worn surface becomes rougher with a large number of small pits and deep craters. The erosion rate can increase or remain constant depending on material properties and cavitation conditions. Neither the craters nor the pits are associated in any way with material nature, grain boundaries, slip lines, or any other structure feature. E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_5,

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Fig. 5.1 Mass loss rate as a function of the exposure time to cavitation

The duration of this stage can vary with cavitation resistance of material. There is no significant modification of surface morphology during this period. Attenuation stage is the final stage in which the decrease of erosion rate depends on many factors like material properties, and interaction between liquid flow and worn surface. This stage of cavitation occurs only under certain conditions. During cavitation tests with magnetostrictive devices, no significant attenuation stage has been observed for aluminium, copper based alloy, carbon steel, stainless steel, and titanium based alloy. A comprehensive review of cavitation erosion in Newtonian liquids is given by Young (1989).

5.1 Cavitation Erosion in Non-Newtonian Fluids Most of the experiments on cavitation damage in non-Newtonian fluids have used polymer aqueous solutions as test liquids and have been conducted using vibratory devices due to their high erosion rates. The tested specimen was fixed either to the vibratory amplifying horn (moving specimen), or at a small distance from it (stationary specimen). Damage is carried to the point of measurable weight loss and damage intensity is usually defined in terms of weight loss per unit time. A typical experimental arrangement used for investigating cavitation erosion is shown in Fig. 5.2. A complete description of the method is given in ASTM G3209. The basic principle is that the horn is vibrating vertically with high frequency in liquid to induce cavitation bubbles near or onto the specimen surface. When non-Newtonian liquids are tested, the specimen is, in most of the cases, made of aluminium. The degradation rates of the polymer solutions used in this type of experiments are quite large, so it may be assumed that the rheological properties of their solutions are significantly changed during test. Ultrasonic degradation of polymer solutions has been the subject of considerable research and an excellent review article has

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Cavitation Erosion in Non-Newtonian Fluids

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Fig. 5.2 The vibratory horn cavitation device

been published by Basedow and Ebert (1978). When solutions of polystyrene, polyacrylates, and nitrocellulose were exposed to ultrasound in organic solvents, an irreversible reduction of the viscosity was observed (Schmid and Rommel 1939; Schmid 1940). The initial decrease of viscosity slowed down as a function of sonication time, and a limiting value of the viscosity was reached. Similar observations have been made by Henglein and Gutierrez (1988) on the degradation of polyacrylamide in aqueous solutions. The ultrasonic degradation of a polymer solution is a non-random mechanical process resulting from hydrodynamic processes arising from cavitation. Polymer chains cleave preferentially near their center, higher molecular weight fractions degrade faster than ones of lower molecular weight, and a lower molecular weight limit exists below which no further degradation occurs (Basedow and Ebert 1978). The role of cavitation in the degradation of polystyrene in toluene and of hydroxyethylcellulose in water was first demonstrated by Weissler (1950, 1951), who observed that no depolymerization occurred at intensities below the threshold for cavitation and in solutions that had been degassed to prevent cavitation. In some cases frictional forces between polymer and solvent molecules may be sufficient for chain scission, and both mechanisms may occur simultaneously. The important conclusion of these studies is that ultrasonic cavitation degrades the polymer. Therefore, in this type of experiments, the test liquid must be discharged from the test vessel and, at the same time, replaced by fresh test liquid in order to avoid the alteration of the physical properties of polymer solution. Ashworth and Procter (1975) seem to be the first who conducted experiments on cavitation damage in polymer solutions. They used copper test specimens placed at 1.3 mm below the tip of an ultrasonic probe. Both in 100 and 1,000 ppm polyacrylamide solution they observed an increase of the cavitation erosion rate. For example, after exposure for 60 min to cavitation, the weight loss of the copper specimen in a 1,000 ppm polyacrylamide aqueous solution is almost two times larger than that in water.

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In a later paper, Shapoval and Shal’nev (1977) reported the results on cavitation damage generated after a cylinder placed in a cavitating pipe. They found that the weight loss decreased by an addition of 300 ppm polyacrylamide into water. Shima et al. (1985) conducted similar experiments to those performed by Ashworth and Procter, but used polyethylene oxide instead of polyacrylamide and placed the aluminium test specimen attached to the vibrating rod. They also recirculated the polymer solutions in the test vessel in order to avoid the polymer degradation. The polymer concentrations they used are 100, 500, and 1,000 ppm, respectively. During the first 15 min of the test, the weight loss in all polymer solutions is slightly larger than that in water. After 60 min of exposure to cavitation, they observed that the addition of small amounts of polymer (100 ppm) shows almost a similar behaviour to that in water and the weight loss decreases significantly with increasing the polymer concentration (Fig. 5.3). For example, the weight loss in the 1,000 ppm polyethylene oxide solution is three times smaller than that in water; for a fully degraded solution the results are very close to those for pure water. A stringy erosion pattern was found at the largest concentration of the polymer solution (1,000 ppm). In a subsequent study, Tsujino (1987) reported similar results from cavitation damage experiments using a 1,000 ppm polyethylene oxide aqueous solution. He further noted that when the amplitude of the vibrating horn is decreased from

Fig. 5.3 Effect of polyethylene oxide (Polyox) concentration on the weight loss. Reproduced with permission from Shima et al. (1985). © American Society of Mechanical Engineers

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38 to 25 μm, both in water and in the polymer solution the mass loss from the test specimen decreases (Fig. 5.4). Tsujino et al. (1986) performed experiments on cavitation erosion using several polymer aqueous solutions of varying elasticity levels (Fig. 5.5). For the polymer solutions with the highest levels of elasticity (guar gum, polyacrylamide, and polyethylene oxide (Polyox)) they found a clear reduction of the weight loss in comparison to the case of water after 60 min of exposure to cavitation. For example, in a 1,000 ppm polyethylene oxide solution, they noted a reduction of the weight loss with about 70% in comparison to the case of water. As in the previous experiments, the authors observed that, during the initial period of exposure to cavitation, the weight loss in all these solutions is larger as compared to water. This feature is very pronounced in the case of polyacrylamide and polyethylene oxide aqueous solutions. For the inelastic polymer solutions (carboxymethylcellulose and hydroxyethylcellulose), however, they found similar results to the case of water. Similar observations have been made by Urata (1998) who concluded that water dissolved polymers can suppress the cavitation erosion when the molecular weight of the polymer exceeds a certain level. Tsujino et al. (1986) also investigated the influence of polymer additives on the damaged area, Ad , of the specimen surface (Fig. 5.6). In water, the test specimen was uniformly damaged so that the damaged area reaches a nearly constant value

Fig. 5.4 Mass loss in water and in a 1,000 wppm Polyox solution. Test frequency f = 19.5 kHz, test temperature T = 295 K, and peak-to-peak amplitude 25 μm (circles) and 38 μm (triangles). Open symbols indicate mass loss in water while filled symbols indicate mass loss in a 1,000 Polyox aqueous solution. Reproduced with permission from Tsujino (1987). © Elsevier B.V.

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Fig. 5.5 Weight loss as a function of time in water and various polymer aqueous solutions in concentration of 1,000 ppm. The polymers tested are polyacrylamide (PAM), polyethylene oxide (Polyox), carboxymethylcellulose (CMC), hydroxyethylcellulose (HEC), and guar gum (GGM). Reproduced with permission from Tsujino et al. (1986). © Professional Engineering Publishing

at t = 5 min. In the carboxymethylcellulose aqueous solution, the damaged area increases rapidly for about 15 min and reaches a steady state at about 30 min; then the extent is larger than in the case of water. On the other hand, in the cases of polyacrylamide and Polyox aqueous solutions, the increasing rates of Ad are much smaller, and the values at t = 60 min are only one-third of the case of water. It is clear from Figs. 5.5 and 5.6 that the additions of polyacrylamide and Polyox are effective for suppression of cavitation damage after a long enough time of exposure to cavitation. Nanjo et al. (1986) conducted erosion tests on specimens with conical or cylindrical holes and used water and a 1,000 ppm polyethylene oxide aqueous solution as test liquids. After 5 min of exposure to cavitation, the walls of the conical holes were damaged in water, but not in the polymer solution. On the other hand, in water, cavitation erosion was observed over the entire inside walls of the cylindrical holes. In the polymer solution, the side walls and bottom surface of the cylindrical holes were undamaged and only the edges of both upper and lower surfaces of the cylinder have been heavily eroded. They concluded that, in polymer solutions, the regions with small radius of curvature are most susceptible to cavitation erosion and, therefore, cavitation erosion is more localized in polymer solutions than in water.

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Fig. 5.6 Damaged area as a function of time in water and various polymer aqueous solutions in concentration of 1,000 ppm. The polymers tested are polyacrylamide (PAM), polyethylene oxide (Polyox), carboxymethylcellulose (CMC), hydroxyethylcellulose (HEC), and guar gum (GGM). Reproduced with permission from Tsujino et al. (1986). © Professional Engineering Publishing

Tsujino et al. (2003) investigated cavitation erosion in polymer aqueous solutions (polyethylene oxide – Polyox) and water using a stationary aluminum plate in close proximity to the free end of a vibratory horn. They investigated the mass loss up to a time of 30 min of exposure to cavitation. They found that the mass loss in a 100 ppm polymer solution is smaller than in the case of water. However, for the largest polymer concentration used in their experiment (1,000 ppm), the mass loss is larger than in water. This result is a consequence of the position of bubbles in the space between vibratory tip and test specimen. Whereas in the 100 ppm polymer solution the cavitation bubbles were generated mostly on the vibratory tip, in the 1,000 ppm polymer solution, the bubbles were formed on the stationary specimens. Ashassi-Sorkhabi and Ghalebsaz-Jeddi (2006) and Ashassi-Sorkhabi et al. (2006) investigated the effect of polyethylene glycol additives, with different molecular weight, on the cavitation erosion of carbon steel in sulphuric acid and hydrochloric acid. They reported that the inhibition efficiency of cavitation erosion increases with the mean molecular weight of polymer and its concentration. In a more recent study, Brujan et al. (2008) investigated the cavitation erosion in polyacrylamide (PAA) aqueous solutions in concentration of up to 1%. Figure 5.7a–c give an overview of the damage to the specimen in water, 0.1% PAA

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Fig. 5.7 Erosion pattern on an aluminium specimen after exposure to cavitation in water [(a) and (d)], 0.1% PAA solution [(b) and (e)], and 1% PAA solution [(c) and (f)]. Frame width is 2.5 mm in (a), 16 mm in (b) and (c), 0.85 mm in (d), and 2.5 mm in (e) and (f). Reproduced with permission from Brujan et al. (2008). © Elsevier B.V.

solution, and 1% PAA solution, respectively. Figure 5.7d–f show specific regions of the specimen surface at increased magnification using a scanning electron microscope which provides visible contrast even with small depth differences. Damage patterns after 80 min exposure to cavitation in polymer solutions differ significantly from those in water. In water, the test specimens developed heavily-eroded areas with cavernous damage structures comprising holes over 0.8 mm in depth, in addition to which were smaller crater-like damage structures which occurred extensively over the specimen surface, with diameters of 10–40 μm and depths of 2–20 μm. The structure of the damage pattern changes distinctly as the polymer concentration is increased. At the highest polymer concentration, the damage consists of isolated, individual craters with large regions of the surface apparently undamaged. Whereas in water the pits cover the specimen homogeneously, nearly all pits accumulate along a few isolated lines at large polymer concentrations, the damage pattern appearing string-like. An increase of the polymer concentration leads to a decrease of the diameter of individual craters and whereas in water the maximal diameter of individual craters was 40 μm, it was 35 μm in the 0.1% PAA solution, and only 20 μm in the 1% PAA solution. The scanning electron micrographs shown in Fig. 5.8 reveal the morphology of the smallest craters found on the specimen. Both in water and in all the polymer solutions, several holes can be seen within the main indentation, e.g. the crater illustrated in Fig. 5.8a consists of two almost cylindrical holes, the first with a diameter of 25 μm and the second with a diameter of 16 μm. Even a third hole, with a diameter of 5 μm, can be seen at the bottom of this indentation. This observation suggests that even the smallest craters on a boundary

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Fig. 5.8 Morphology of the craters developed on an aluminium specimen surface after exposure to cavitation for 80 min in water [(a) and (b)] and 1% polyacrylamide aqueous solution [(c) and (d)]. Frame width is 60 μm in (a) and (c) and 30 μm in (b) and (d). Reproduced with permission from Brujan et al. (2008). © Elsevier B.V.

surface act as nucleation sites. Microscopic gas pockets, which are entrapped at the bottom of the craters, can grow to relatively large sizes during the expansion phase of the sound field. When the localized conditions change back to positive pressure, the potential energy gained in cavity growth is converted into kinetic energy as the interface accelerates to smaller radius. Although the total energy of the cavitation event may be small, the concentration of this energy into a very small volume results in an enormous energy density, with a large potential for damage. The examination of the damage pattern in polymer solutions showed the same gross result although not so prominent as in the case of water. It was also found that the weight loss decreases with increasing the polymer concentration and is one order of magnitude smaller in the 1% polyacrylamide solution than in the case of water.

5.2 Mechanisms of Cavitation Damage in Newtonian Fluids Nucleation is an important factor for any cavitation-mediated physical or chemical process, because the population of cavitation bubbles depends significantly on the population and the size distribution of suitable nuclei. Cavitation results from the

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expansion of these nuclei subjected to a negative pressure. At sufficiently large negative pressure, the growth and collapse of cavitation bubbles can lead to an enormous concentration of energy. The temperature and pressure experienced by the material contained within the imploding cavities can achieve values in excess of thousands of degrees and tens of kilobars, respectively. The first attempt to explain cavitation damage was Lord Rayleigh’s (1917) seminal analysis of the behaviour of an isolated spherical void collapsing in an incompressible liquid. An important conclusion of this study is that as the collapse nears completion, the pressure inside the liquid becomes indefinitely large. It is this mechanism, albeit extensively modified, which has lead to the association of bubble collapse with cavitation damage. In the case of bubble activity near a solid surface, the extent of cavitation damage depends not only on the population and the size distribution of the nuclei, but also on their relative distance from the surface, γ , and the properties of the nearby boundary. Depending on the stand-off factor γ the bubble undergoes different motions (Lauterborn and Bolle 1975; Shima et al. 1981; Tomita and Shima 1986; Blake and Gibson 1987; Vogel et al. 1989). For large γ values (γ > 5) the bubble remains almost spherical during first oscillation cycle. Bubble rebound leads to the emission of a strong pressure transient into the liquid that can evolve into a shock wave. Near a boundary material the collapse is asymmetric and associated with the formation of one or two high-speed liquid jets, directed towards or away from the boundary. Kornfeld and Suvorov (1944) were amongst the first to suggest that cavitation bubbles deform during collapse, and that damage is also caused by the impact of high-speed liquid microjets that strike the nearby surface during the collapse phase. Naudé and Ellis (1961) observed the formation of a liquid microjet in their classic photographic study and Benjamin and Ellis (1966) provided a theoretical discussion of the asymmetric collapse. They concluded that microjet formation and impact is important and probably the main factor for cavitation damage. The water-hammer or shock pressure, PWH , for impact on a perfectly rigid target, is given by

PWH

ρS cS = ρL cL v ρL cL + ρS cS

,

(5.1)

where ρ, c and v are the density, sound speed, and jet impact velocity, respectively, and subscript S refers to solid and subscript L refers to liquid. Such pressures are a consequence of the conservation of linear momentum at the impact site. Usually, ρS cS >> ρL cL , so PWH = ρL cL v. If the target is of the same acoustic impedance, the pressure is half this value. This loading at the impact site, in the case of a cylindrical jet, lasts only for a short period, τ , that it takes a release wave, generated at the contact edge of the jet, to reach the impact axis. This time can be expressed as τ = r/c where r is the radius of the jet. For example, a water-hammer pressure of about 150 MPa lasting 20 ns is generated by an impact velocity of 100 m/s with a jet radius of 30 μm. However, a number of experiments on bubble dynamics in water have cast doubt on this explanation. For example, Shutler and Mesler (1965) suggested that the damage is caused by the pressure pulse occurring at the minimum

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volume of a bubble collapsing near a rigid boundary. Fujikawa and Akamatsu (1980) have reported experiments in which a photoelastic material was used to observe cavitation induced stresses, while the associated acoustic pulses were also recorded. They confirmed that impulsive stresses in the material were initiated at the same moment as the acoustic pulse and concluded that the stress waves were not due to a microjet. In more recent studies, Tomita and Shima (1986) and Philipp and Lauterborn (1998) also suggested that the major causes of cavitation erosion are the high pressures and temperatures reached inside a bubble collapsing very close to a rigid wall. Detailed studies of cavitation erosion of rigid materials, generated by individual cavitation bubbles collapsing in a quiescent liquid, was conducted by Tomita and Shima (1986) and Philipp and Lauterborn (1998). Tomita and Shima (1986) found a circular damage pattern with many indentations around a circumference. Philipp and Lauterborn (1998) observed two distinct damage patterns – a shallow pit damage and a circular damage pattern – and concluded that damage generated during first bubble collapse will occur for γ ≤ 0.7, whereas for 0.9 ≤ γ < 2 cavitation erosion is due to the second collapse when the bubble is directly attached to the material surface. They concluded that the largest erosive force is caused by bubble collapse in direct contact with the rigid boundary, where pressures of up to several GPa and temperatures of about 8,000 K are reached inside the bubble (see, also, Brujan and Williams 2005). Bubbles in the range γ ≤ 0.3 and γ = 1.2 to 1.4 caused the greatest damage. Interestingly, a significant reduction of the damage was observed for 0.5 ≤ γ ≤ 1.1. This is mainly provoked by the “splash” effect which was found to occur after the liquid jet impact onto the rigid boundary (Tong et al. 1999; Brujan et al. 2002). When the liquid jet threads the bubble, the closeness of the boundary results in a radial flow away from the jet axis. This flow collides with the flow induced by the still contracting bubble and a “splash” is projected away from the boundary, in a direction opposite to the liquid jet motion. During the final stages of collapse, a large part of the kinetic energy of the radial flow into the bubble is transformed into kinetic energy of a rotational flow around the bubble. The content of the bubble becomes, therefore, less compressed and the sound emission is diminished. Experiments by Vogel et al. (1989) and Tomita and Shima (1986) showed that the collapse pressure is minimal around these γ -values. This way, the damage potential of the bubble is diminished even when the bubble is in direct contact with the boundary at the moment of the first collapse. Although it is clear now that the liquid jet developed during bubble collapse does not have a significant potential to produce erosion of metals, it can play an important role in fragmentation of brittle objects, such as renal calculi, dental tartar or intraocular lens. On one hand, the yield strength of brittle materials is much lower than that of metals. For example, Murata et al. (1977) reported compressive strengths of renal calculi to vary from 2 to 17 MPa and Burns et al. (1985) obtained similar values (2–8 MPa). Since impact velocities of the jet as small as 10 m/s develops localized pressures of about 15 MPa, this mechanism can be considered as a likely contributor to renal calculi disintegration. On the other hand, the action of pressure transients on metal surfaces and on brittle materials is very different. Intracorporeal stones,

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as well as dental tartar, are usually a conglomerate of crystalline and organic components and small voids. They are acoustically inhomogeneous, with many zones of different acoustic impedance. Whenever the pressure pulse coming from a zone with high acoustic impedance propagates into a zone with smaller impedance, it is partially reflected as a tensile wave. These waves have a high damage potential because stones are about 5 times more susceptible to tensile stress than to pressure (Rink et al. 1995). The very localized action of the pressure transient into the material and the inhomogeneity of the pressure wave propagation may, additionally, lead to shearing forces. Tensile stress and shearing forces will create cracks, enlarge pre-existing cracks and voids and, finally, lead to fracture. The distribution of the cavitation nuclei over the surface of the rigid surface is also important because of interaction with adjacent bubbles. In a cloud of bubbles, a greater probability for the occurrence of ultra-high-velocity jets is possible. Bubblesplitting leading to the formation of high-speed liquid jets due to the presence of other bubbles has been demonstrated by Blake et al. (1993). Another accelerating effect on jet velocity may be the interaction of an acoustic transient emitted by bubbles when collapsing in the neighbourhood of the jetting bubble (Dear et al. 1988; Bourne and Field 1992; Philipp et al. 1993) (Fig. 5.9). It is well known that a strong acoustic transient hitting a bubble will induce collapse, forming a jet that completely penetrates the bubble at its minimum volume. This process is independent of any boundary in the bubble’s vicinity and the jet direction is the same as the propagation direction of the acoustic wave. Dear et al. (1988) made cylindrical cavities in a gel to observe the collapse of these cavities when they are impinged by a shock wave. A striker was projected to impact the gel, and high-speed photography was used to record the behaviour of the cavities and jet formation under such impact. For an impact pressure of 260 MPa, a 3 mm bubble generated a jet with a velocity of about 400 m/s. Bourne and Field (1992) reported the results of a high-speed photographic study of cavities collapsed asymmetrically by shock waves of strengths in the range of 0.26–3.5 GPa. The collapse of a 3 mm cavity in gelatine under a shock of strength 0.26 GPa induces the formation of a jet with a velocity of 300 m/s. Under a shock strength of 1.88 GPa, the jet velocity is up to 5 km/s for a 6 mm cavity. Philipp et al. (1993) also used high-speed photography and observed jet formation in a gas cavity induced by lithotripter-generated shock waves. They used peak shock pressures of 65 and 102 MPa and reported a maximum jet velocity of up to 800 m/s. Bourne

Fig. 5.9 A high-speed jet travels across a 6 mm cavity under a 1.88 GPa shock from a plane-wave generator. The shock wave is visible at the bottom of each photographic frame as a dark band. The jet travels at approximately 5 km/s. Reproduced with permission from Bourne and Field (1999). © The Royal Society Publishing

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and Field (1995) carried out experiments on cavitation damage generated by shock wave-bubble interaction. The craters observed on the specimen surface exposed to cavitation in water are attributed to the impact of the shock-induced liquid jets onto the material surface. The authors concluded that, in this case, the high pressure and temperature of the gases inside the bubble, and the impact of the liquid jet onto the boundary material are responsible for the destructive action of cavitation bubbles. Theoretical and experimental studies confirmed that the pressure generated by a multiple interaction can be much higher than that caused by a single bubble. Hansson and Mørch (1980) performed numerical calculations along the collapse of a hemispherical cluster of cavities, related to the experimental observations by Ellis (1966). They showed that the collapse of each shell of cavities exposes the next inner shell to the hydrostatic pressure field which in turn initiates its collapse. At each stage, the energy of collapse is transferred to the inner shell resulting in a steady build-up of pressure. They demonstrated that this increased the collapse energy of the cavities at the centre of the cloud by an order of magnitude. An example of shock wave emission during the collapse of a hemispherical cloud of bubbles is shown in Fig. 5.10.

Fig. 5.10 Shock wave emission from a hemispherical cloud of bubbles attached to a rigid wall. The shock waves are visible in frames 9–11. Sequence taken with 2 million frames/s. Frame width 2.56 mm. Courtesy of E.A. Brujan, T. Ikeda, K. Yoshinaka, and Y. Matsumoto

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The situation is quite different for cavitation bubbles collapsing near elastic materials (Brujan et al. 2001a, b). This case is representative for the interaction between cavitation bubbles and biological tissues during medical applications of lasers and ultrasound. Three different mechanisms may contribute to cavitation erosion in this case, namely liquid jet penetration into the elastic material, jet-like ejection of boundary material into the surrounding liquid, and elevation and tearing of the material surface by the low pressure between bubble and boundary developing during bubble collapse. Liquid jet penetration into the material requires that the pressure generated by the impact of the liquid jet onto the boundary is sufficiently high to overcome the yield strength of the material. Jet-like ejection of the elastic material has three prerequisites. First, the material must be sufficiently deformed to allow geometric focusing effects during its rebound. Secondly, the elastic modulus of the material must be sufficiently large so that the restoring force caused by the elastic deformation is large enough to cause this jet formation. Finally, the plastic flow stress of the material and the ultimate tensile strength of the material must be exceeded. For bubbles close to the boundary, the late stage of the collapse is associated with a volcano-like uplifting of the boundary caused by the low pressure region developing between the collapsing bubble and the boundary. Elevation of the material surface has been also reported by Grimbergen et al. (1998), and Godwin et al. (1998) pointed out the role of bubble dynamics for an enhancement of pulsed laser ablation. While the suction force enhances the material removal only for very soft materials, the elastic rebound plays a role also for materials with moderate strength, and the jet impact can erode even hard materials with high mechanical strength. Example: Impact of a Liquid Jet on a Rigid Boundary We consider in this example the impact of a plane-ended liquid mass on a plane rigid surface (Lush 1983). The direction of motion of the liquid mass is at right angle to the surface, the plane end being parallel to the surface. When the liquid strikes this surface a normal shock wave is propagated against the liquid stream, and behind the shock the liquid velocity is reduced and the pressure increased. Assuming that the liquid mass is infinitely wide, the problem can be analyzed in one-dimensional terms and is most simply done in a reference frame moving with the shock (Fig. 1). If the velocity of the liquid is initially v, the velocity behind the shock is u and the velocity of the shock wave is –c, which is not necessarily equal to the sound speed, then, in the steady reference frame, the equation of continuity is ρ∞ (v + c) = ρ(u + c),

(1)

where ρ∞ is the ambient density of the liquid and ρ that behind the shock. If the pressure behind the shock is p, then by conservation of momentum it can be shown that p − p∞ = ρ∞ (v + c)(v − u),

(2)

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Fig. 1 Normal-shock configuration for (a) unsteady and (b) steady reference frame

where p∞ is the ambient pressure. Equations (1) and (2) can be solved provided that the relation between pressure and density is known. We consider the Tait equation of state p+B = p∞ + B

ρ ρ∞

n ,

(3)

where, for water, n ≈ 7 and B ≈ 300 MPa. Since the pressure behind the shock, p, is much larger than the ambient pressure we get

p v−u= ρ∞

p −1/n 1/2 1− 1+ . B

(4)

If the impact is with a rigid surface, then u is zero, and (4) gives the impact pressure in terms of the impact velocity. On the other hand, if we assume that the surface responds as a rigid solid until a certain compressive stress is reached and then behaves as a perfectly plastic solid, for which the stress will remain constant at a value pY , Eq. (4) can be rewritten to give the velocity deformation u in terms of the impact velocity v and the stress to produce plastic flow as u = v − v0 ,

(5a)

where

pY v0 = ρ∞

pY −1/n 1/2 1− 1+ . B

(5b)

It is clear that the impact velocity must exceed a certain critical value v0 before any plastic deformation can occur.

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Up to this point we have assumed that the liquid mass is infinite in width. In reality this will not be the case, and, as soon as the liquid strikes the surface, a release wave will propagate from the edge of the liquid mass towards the impact centre at the ambient sound speed c∞ (Fig. 2). Assuming the liquid mass to be a cylinder of radius a, the pressure at the centre will decreases after a time a/c∞ from the value given by (4) to the stagnation pressure ρ∞ v2 / 2, which will be an order of magnitude smaller unless exceptionally high impact velocities are encountered. To obtain the amount of deformation induced by jet impact we consider only the centre of impact, where the only motion will be normal to the surface and the depth of penetration has the maximum value. If it is assumed that the plastic flow is established immediately, and that it ceases as soon as the release wave reaches the centre, the time available for deformation will be simply equal to the time taken for the wave to travel across the radius of the liquid cylinder; i.e. a/c∞ . Since the deformation velocity given by (5) is constant, the depth d of penetration at the center is given simply by the product of the velocity u from (5) and the time available, i.e. by v − v0 d = . a c∞

(6)

For aluminium with 99.5% purity, the static value of the plastic flow stress is about 400 MPa and from (5b) it results that v0 is about 200 m/s. The corresponding value of pY in dynamic tests is about 1,300 MPa and, in this case, v0 is about 1,100 m/s. These values are much higher than the impact velocity of the liquid jet developed during bubble collapse near a rigid wall (around 80 m/s) and thus the jet

Fig. 2 Impact of plane-ended cylindrical liquid mass on a rigid surface

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impact cannot produce a plastic deformation of the wall material. For a conical jet, Lush (1983) has indicated that the impact pressure is p ≈ 2.9ρ∞ c∞ v. The pressure developed by the impact of a conical jet with a velocity of 80 m/s is about 350 MPa, which is again smaller than the plastic flow stress of aluminium. These considerations suggest that only shock-induced jets can produce material damage because their velocity (up to 5,000 m/s) is much higher than the critical value for plastic deformation v0 .

5.3 Reduction of Cavitation Erosion in Polymer Solutions With one exception (Ashworth and Procter 1975), all other experiments have indicated an inhibition of cavitation erosion by polymer additives. This is particularly obvious when the test specimen is exposed to cavitation for a long time (30 min or longer). The largest inhibition was observed in the case of aqueous solutions of flexible polymers such as polyacrylamide and polyethylelene. For short exposures to cavitation (less than 15 min), the cavitation erosion in aqueous solutions of flexible polymers is slightly larger than in the case of water. Similar results to the case of water have been observed in aqueous solutions of semi-rigid or rigid polymers such as carboxymethylcellulose, hydroxyethylcellulose, and guar gum. Three major factors can contribute to the reduction of cavitation erosion in polymer solutions: • Reduction of cavitation nuclei by polymer additives. In all polymer solutions investigated so far, a significant decrease of nuclei population with increasing polymer concentration was found (see Chap. 2). The decrease of nuclei population can increase the threshold for cavitation and, as a consequence, leads to a reduction of the erosion potential of cavitation. • Increase of the extensional viscosity of the solution by the polymer additives. Both jet evolution and bubble collapse involve a substantial component of extensional viscosity (see Chap. 3). The regions of highest extension are developed very close to the bubble wall during the final stage of collapse where the fluid is subjected to an extensional strain rate of about 105 s–1 (Brujan et al. 2004). The greater resistance to extension of the polymer solutions when compared to its Newtonian counterpart will result in a lowering of the velocity in areas of high extension rate. In particular, the velocity of the liquid jet developed during bubble collapse and the velocity of the bubble wall are markedly diminished. The jet formed during bubble collapse in polymer solutions is strongly decelerated on its way to the boundary and the impact velocity of the jet onto the specimen surface is smaller than in the case of a Newtonian fluid. In addition, a bubble collapsing in a polymer solution has a smaller compression, because more energy is dissipated during the collapse phase due to the increased resistance to extensional flow, and thus the pressure amplitude of the shock wave emitted during

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bubble collapse is diminished. The slowing of the collapse in this manner is also likely to reduce the ambient temperature in the gases inside the bubble. The reduction of cavitation erosion is higher in solutions of flexible polymers, such as polyacrylamide and polyethylene oxide, which exhibit extension rate thickenning behaviour. Cavitation erosion in aqueous solutions of semi-rigid polymers, such as carboxymethylcellulose, is almost similar to that in water. These polymer aqueous solutions are considered relatively inelastic with low values of the Trouton ratio (Barnes et al. 1989). Therefore, the resistance to extension from these fluids will be almost similar to that of water with only minor effects on the behaviour of cavitation bubbles near the conclusion of collapse. These features have been clearly observed in the case of laser-induced bubbles in polyacrylamide and carboxymethylcellulose aqueous solutions (Brujan 2008). Whereas in the case of the polyacrylamide solution (with a strong elastic component) the amplitude of the shock wave emmited during bubble collapse was diminished, no significant differences were observed in the carboxymethylcellulcose solution (with a weak elastic component) as compared to the case of water. • Polymer additives change the overall flow field. As a classic example we note that the structure of the boundary layer around bodies is significantly altered by high molecular weight polymers additives. This leads to an earlier transition to turbulence and elimination of laminar separation. This effect diminishes cavitation erosion since the region of re-attachement of the separated boundary layer is the critical zone for cavitation to occur due to intense pressure fluctuations in this region (Arakeri and Acosta 1981). Other examples of the suppression of cavitation inception are presented in Chap. 4. It was also noted that the extensional effects may also play an important role in the suppression of cavitation inception by changing the flow kinematics. It was speculated that the resistance to extension from the polymer solutions will result in a lowering of the velocity (and hence increase of static pressure) in areas of high extension rates.

References Arakeri, V.H., Acosta, A. 1981 Viscous effects in the inception of cavitation. J. Fluids Eng. 103, 280–287. Ashassi-Sorkhabi, H., Ghalebsaz-Jeddi, N. 2006 Effect of ultrasonically induced cavitation on inhibition behavior of polyethylene glycol on carbon steel corrosion. Ultrason. Sonochem. 13, 180–188. Ashassi-Sorkhabi, H., Ghalebsaz-Jeddi, N., Hashemadeh, F., Jahani, H. 2006 Corrosion inhibition of carbon steel in hydrochloric acid by some polyethylene glycols. Electrochim. Acta 51, 3848–3854. Ashworth, Y., Procter, R.P.M. 1975 Cavitation damage in dilute polymer solutions. Nature 258, 64–66. Barnes, H.A., Hutton, J.F., Walters, K. 1989 An Introduction to Rheology. Elsevier, New York. Basedow, A.M., Ebert, K.H. 1978 Ultrasonic degradation of polymers in solution. In Advances in Polymer Science, vol. 22 (Eds. H.J. Cantow et al.). Springer, New York, pp. 83–148. Benjamin, T.B., Ellis, A.T. 1966 The collapse of cavitation bubbles and the pressure thereby produced against solid boundaries. Phil. Trans. R. Soc. Lond. A 260, 221–240.

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Blake, J.R., Gibson, D.C. 1987 Cavitation bubbles near boundaries. Ann. Rev. Fluid Mech. 19, 99–123. Blake, J.R., Robinson, P.B., Shima, A., Tomita, Y. 1993 Interaction of two cavitation bubbles with a rigid boundary. J. Fluid Mech. 255, 707–721. Bourne, N.K., Field, J.E. 1992 Shock-induced collapse of single cavities in liquids. J. Fluid Mech. 244, 225–240. Bourne, N.K., Field, J.E. 1995 A high-speed photographic study of cavitation damage. J. Appl. Phys. 78, 4423–4427. Bourne, N.K., Field, J.E. 1999 Shock-induced collapse and luminescence by cavities. Phil. Trans. R. Soc. Lond. A 357, 295–311. Burns, J.R., Shoemaker, B.E., Gauthier, J.F., Finlayson, B. 1985 Hardness testing of urinary calculi. In Urolithiasis and Related Clinical Research (Eds. P.O. Schwille, L.H. Smith, W.G. Robertson, and W. Vahlensieck). Plenum, New York, pp. 181–185. Brujan, E.A. 2008 Shock wave emission from laser-induced cavitation bubbles in polymer solutions. Ultrasonics 48, 423–426. Brujan, E.A., Williams, G.A. 2005 Luminescence spectra of laser-induced cavitation bubbles near rigid boundaries. Phys. Rev. E 72, 016304. Brujan, E.A., Nahen, K., Schmidt, P., Vogel, A. 2001a Dynamics of laser-induced cavitation bubbles near an elastic boundary. J. Fluid Mech. 433, 251–281. Brujan, E.A., Nahen, K., Schmidt, P., Vogel, A. 2001b Dynamics of laser-induced cavitation bubbles near elastic boundaries: influence of the elastic modulus. J. Fluid Mech. 433, 283–314. Brujan, E.A., Keen, G.S., Vogel, A., Blake, J.R. 2002 The final stage of the collapse of a cavitation bubble close to rigid boundary. Phys. Fluids 14, 85–92. Brujan, E.A., Ikeda, T., Matsumoto, Y. 2004 Dynamics of ultrasound-induced cavitation bubbles in non-Newtonian liquids and near a rigid boundary. Phys. Fluids 16, 2402–2410. Brujan, E.A., Al-Hussany, A.F.H., Williams, R.L., Williams, P.R. 2008 Cavitation erosion in polymer aqueous solutions. Wear 264, 1035–1042. Dear, J.P., Field, J.E., Walton, A.J. 1988 Gas compression and jet formation in cavities collapsed by a shock wave. Nature 332, 505–508. Ellis, A.T. 1966 On jets and shock waves in cavitation. Proceedings of the Sixth Symposium On Naval Hydrodynamics. Washington, pp. 137–161. Fujikawa, S., Akamatsu, T. J. 1980 Effects of the non-equilibrium condensation of vapour on the pressure wave produced by the collapse of a bubble in a liquid. J. Fluid Mech. 97, 481–512. Godwin, R.P., Chapyak, E.J., Prahl, S.A., Shangguan, H. 1998 Laser mass ablation efficiency measurements indicate bubble-driven dynamics dominates laser thrombolysis. Proc. SPIE 3245, 2–11. Grimbergen, M.C.M., Verdaasdonk, R.M., van Swol, C.F.P. 1998 Correlation of thermal and mechanical effects of the holmium laser for various clinical applications. Proc. SPIE 3254, 69–79. Hansson, I., Mørch, K.A. 1980 The dynamics of cavity clusters in ultrasonic (vibratory) cavitation erosion. J. Appl. Phys. 51, 4651–4658. Henglein, A., Gutierrez, M. 1988 Sonolysis of polymers in aqueous solution. New observations on pyrolysis and mechanical degradation. J. Phys. Chem. 92, 3705–3707. Kornfeld, M., Suvorov, L. 1944 On the destructive action of cavities. J. Appl. Phys. 15, 495–506. Knapp, R.T., Daily, J.W., Hammitt, F.G. 1970 Cavitation. McGraw-Hill, New York. Lauterborn, W., Bolle, H. 1975 Experimental investigations of cavitation bubble collapse in the neighbourhood of a solid boundary. J. Fluid Mech. 72, 391–399. Lush, P.A. 1983 Impact of a liquid mass on a perfectly plastic solid. J. Fluid Mech. 135, 373–387. Murata, S., Watanabe, H., Takahashi, T., Watanabe, K., Oinuma, S. 1977 Studies on the application of microexplosion to medicine and biology. II. Construction and strength of urinary calculi. Jpn. J. Urol. 68, 249–256. Nanjo, H., Shima, A., Tsujino, T. 1986 Formation of damage pits by cavitation in a polymer solution. Nature 320, 516–517.

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Naudé, C. F., Ellis, A. T. 1961 On the mechanism of cavitation damage by nonhemispherical cavities collapsing in contact with a solid boundary. Trans. ASME D J. Basic Eng. 83, 648–656. Philipp, A., Delius, M., Scheffczyk, C., Vogel, A., Lauterborn, W. 1993 Interaction of lithotripter generated shock waves with air bubbles. J. Acoust. Soc. Am. 93, 2496–2509. Philipp, A., Lauterborn, W. 1998 Cavitation erosion by single laser-produced bubbles. J. Fluid Mech. 361, 75–116. Rayleigh, L. 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94–98. Rink, K., Delacrétaz, G., Salathé, R.P. 1995 Fragmentation process of curent laser lithotripters. Lasers Surg. Med. 16, 134–142. Schmid, H.G. 1940 Zerreiben von Makromolekalen, Versuch einer Erklarung der depolymerisierenden Wirkung der UItraschallwellen. Phys. Z. 41, 326–337. Schmid, G., Rommel, O. 1939 Zerreissen von Makromolekulen mit Ultraschall. Z Elektrochem. 45, 659–661. Shapoval, I.F., Shal’nev, K.K. 1977 Cavitation and erosion in polymer aqueous solutions of polyacrylamide. Sov. Phys. Dokl. 22, 635–637. Shima, A., Takayama, K., Tomita, Y., Miura, N. 1981 An experimental study on effects of a solid wall on the motion of bubbles and shock waves in bubble collapse. Acustica 48, 293–301. Shima, A., Tsujino, T., Nanjo, H., Miura, N. 1985 Cavitation damage in polymer aqueous solutions. J. Fluids Eng. 107, 134–138. Shutler, N.D., Mesler, R.B. 1965 A photographic study of the dynamics and damage capabilities of bubbles collapsing near solid boundaries. Trans. ASME D J. Basic Eng. 87, 511–517. Tomita, Y., Shima, A. 1986 Mechanisms of impulsive pressure generation and damage pit formation by bubble collapse. J. Fluid Mech. 169, 535–564. Tong, R.P., Schiffers, W.P., Shaw, S.J., Blake, J.R., Emmony, D.C. 1999 The role of ‘splashing’ in the collapse of a laser-generated cavity near a rigid boundary. J. Fluid Mech. 380, 339–361. Tsujino, T. 1987 Cavitation damage and noise spectra in a polymer solution. Ultrasonics 25, 67–72. Tsujino, T., Shima, A., Nanjo, H. 1986 Effects of various polymer additives on cavitation damage. Proc. Instn. Mech. Engrs 200, 231–235. Tsujino, T., Kenjiro, I., Takayuki, O. 2003 Study of polymer effect on cavitation damage in narrow clearance. Memoir. Fac. Educ. Nat. Sci. Kumamoto Univ. 52, 1–6. Urata, E. 1998 Cavitation erosion in various fluids. In Bath Workshop on Power Transmission and Motion Control (Eds. C.R. Burrows and K.A. Edge). Professional Engineering Publishing, Suffolk, pp. 269–284. Vogel, A., Lauterborn, W., Timm, R. 1989 Optical and acoustic measurements of the dynamics of laser-produced cavitation bubbles near a solid boundary. J. Fluid Mech. 206, 299–338. Weissler, A. 1950 Depolymerization by ultrasonic irradiation: the role of cavitation. J. Appl. Phys. 21, 171–173. Weissler, A. 1951 Cavitation in ultrasonic depolymerization. J. Acoust. Soc. Am. 23, 370. Young, F.R. 1989 Cavitation. McGraw-Hill, London.

Chapter 6

Cardiovascular Cavitation

Cavitation has been shown to play a key role in a wide array of novel therapeutic applications of ultrasound and lasers. Sometimes the mechanical effects associated with cavitation contribute to the intented surgical effect. More often, however, they are the source of unwanted collateral effects limiting the local confinement of ultrasound and laser surgery. Whether the mechanical effects are wanted or unwanted, a characterization of the cavitation effects is of interest for the optimization of the surgical procedures. In this section we review the modeling studies and experiments on cavitation effects that are most relevant in the context of diagnostic and therapeutic applications of ultrasound and lasers in the cardiovascular system. These include sonothrombolysis, diagnostic ultrasound with microbubble contrast agents, ultrasound-mediated gene transfer and drug delivery, transmyocardial laser revascularization, laser angioplasty, and gas embolotherapy.

6.1 Cavitation for Ultrasonic Surgery Ultrasound has been in use for the last three decades as a modality for diagnostic imaging in medicine. Recently, there have been numerous reports on the application of ultrasound energy for targeting or controlling gene or drug release in the cardiovascular system. This new concept of therapeutic ultrasound combined with genes and drugs has induced excitement in various medical fields. Ultrasound energy can also enhance the effects of thrombolytic agents and anticancer drugs. Although therapeutic ultrasound was considered a tool for hyperthermia or thermal ablation of tumors, the major mechanism for the recent ultrasound-assisted gene transfer and drug delivery is mainly mechanical and is induced by cavitation.

6.1.1 Sonothrombolysis Sonothrombolysis is the concept of augmentation of clot lysis by application of external ultrasound. The transducers employed for sonothrombolysis have varied from 20 to 5,000 kHz employing both pulse and sweep modes with intensities E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_6,

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ranging from 0.5 to 90 W/cm2 . Ultrasound energy, used in combination with thrombolitic drugs, such as tissue plasminogen activator (t-PA), has a beneficial effect in the dissolution of coronary thrombus. Recently, Polak (2004) has suggested several mechanisms for the effect of t-PA and cavitation on the dissolution of thrombus as a function of ultrasound energy (Fig. 6.1). At very low energies, ultrasound increases the total amount and depth of penetration of t-PA in the clot. At slightly higher energies, ultrasound can increase the number of exposed enzyme binding sites on the fibrin complex and causes reversible disaggregation of fibrin fibers thus increasing fluid permeation and improving fibrinolytic efficacy. Temperature elevation generated by ultrasound may also be responsible for accelerating thrombolysis. At high energies, ultrasound enhances clot thrombolysis through cavitation-induced direct damage and induction of blood flow microstreaming. Cavitation nuclei exist within clots and contribute partially or fully to sonothrombolysis (Everbach and Francis 2000). When ultrasound is applied, these nuclei vibrate and induce the occurrence of acoustic cavitation which, in turn, can produce the dissolution of thrombus by the high temperature and pressure generated during cavitation bubble collapse. Invasive catheters and some transcutaneous devices generate enough ultrasound energy to cause cavitation. For example, catheter-based ultrasound delivery systems use an external transducer which delivers the ultrasound in a continuous or pulsed

Fig. 6.1 Mechanism of action of tissue plasminogen activator (t-PA) and possible mechanisms of action of ultrasound energy in the dissolution of thrombus. Reproduced with permission from Polak (2004). © Massachusetts Medical Society

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mode. The rapid probe motion causes acoustic cavitation which has direct mechanical effect on the clot. This system was tried clinically (Hamm et al. 1994), and demonstrated that it could rapidly and effectively be used for treatment of patients with acute thrombotic coronary artery occlusion. It has also been shown to help in recanalization of chronic total occlusions of saphenous venous grafts (Rosenschein et al. 1999) as well as native coronary arteries (Rees and Michalis 1995).

6.1.2 Ultrasound Contrast Agents The concept of ultrasound contrast agent for diagnostic imaging consists of using contrast particles that are delivered in the area of interest. Contrast agents are administrated intravenously, circulates in the bloodstream, accumulates in the targeted area, and the agent signal is then used to demarcate the site and condition of the target tissue. Thus, a successful ultrasound contrast agent must be able to provide efficient backscatter of sound waves that are transmitted by an ultrasound system. Several types of contrast particles have been suggested for ultrasound imaging, including liquid-core micro-emulsions and nano-emulsions (Lanza et al. 1996), liposomes (Alkan-Onyuksel et al. 1996), and gas-filled microbubbles with the average size of several micrometers (Meza et al. 1996). Liquid contrast emulsion particles offer lower acoustic impedance mismatch but enhanced stability and prolonged circulation time (Lanza et al. 1996). Liposome-based agents were initially hypothesized to provide acoustic response due to their multi-lamellar lipid structure (Alkan-Onyuksel et al. 1996), but later it was shown that their acoustic response is comparable to that of liquid emulsions (Huang et al. 2002). Most effective design of ultrasound agent is the gas-filled microbubble. The rationale for using microbubbles as ultrasound contrast agents is based on their compressibility. As the gas inside the microbubble compresses during the passage of the pressure wave front and expands during rarefaction phase of the pressure wave, a strong acoustic signal is created by the radial oscillations of microbubbles. This ultrasound echo is detected by the imaging system and permits the visualization of the microbubbles only, without any artifact from the surrounding anatomical structures. 6.1.2.1 The Basic Principles of Microbubble Ultrasound Contrast Agents for Diagnostic Imaging Gramiak and Shah (1968) demonstrated the first ultrasound contrast enhancing agent in 1968 when they injected agitated saline during echocardiographic recordings of the aortic root and observed a cloud of echoes. These echoes were probably generated by small air-bubbles present in the contrast medium. These early contrast agents were too large and insufficiently stable to pass the pulmonary capillaries. First-generation contrast agents are air microbubbles (Nanda et al. 1997). Their main disadvantage is that air bubbles disappear in a few seconds after intravenous injection as the solubility of air in blood is high and the lungs filter the microbubbles with larger diameter (Boukaz et al. 1998; Mayer and Grayburn 2001). Sonification

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of human serum albumin produces albumin-stabilized air bubbles that are sufficiently stable and small to pass the pulmonary capillaries (Keller et al. 1987). However, they cannot resist arterial pressure gradients. Inert high-molecular-mass gases with low diffusion coefficients and low solubility have been used, in the second-generation contrast agents, to increase the stability of microbubbles under higher pressure (Raisinghani and DeMaria 2002). The low diffusivity keeps the gas in the bubble, and the low concentration of saturation results in rapid saturation of the blood with the gas and an alteration of the equilibrium condition such that the gas tends to remain in the microbubble. The gases found to most favourably exhibit these properties are the sulphur hexafluoride and perfluorocarbon. Encapsulation of high-molecular-mass gases with surfactants or proteins is now used with these gases to control their size distribution and to further improve stability (Raisinghani and DeMaria 2002). The imposition of a shell or another substance capable of altering the surface tension of a microbubble can both inhibit the diffusion of gas into the blood and enhance the pressure gradient that a microbubble can tolerate before dissolving. Several new agents in development use polymer shells whose thickness and flexibility can be more precisely controlled. For cellular and molecular noninvasive imaging, microbubbles can be targeted to regions of disease either by intrinsic properties of the shell constituents which interact with upregulated cell receptors, or by surface conjugation of specific ligands or antibodies that bind to disease-related markers (Klibanov 2006). A number of microbubble ultrasound contrast agents are commercially available or are in development. The initially marketed ultrasound contrast agent, AlbunexTM (Molecular Biosystems) is a room air bubble with an albumin shell and a mean diameter of about 4 μm. OptisonTM (Mallinckrodt) represents the refinement of AlbunexTM , and uses perfluoropropane with an albumin shell. EchovistTM (Schering AG) is a first-generation contrast agent and consists of a suspension of galactose microparticles in a solution of galactose at 20%. LevovistTM (Schering AG) consists of an air microbubble galactose within a fatty acid. DefinityTM (Bristol-Myers Squibb Medical Imaging) represents a perfluoropropane gas in a liposome shell with a mean diameter between 1 and 3 μm, while ImagentTM (Imcor) uses a surfactant to stabilize microbubbles filled with a combination of perflurohexane and air. The mean diameter of ImagentTM microbubbles is 5 μm and the concentration is 5×108 microbubbles/ml. SonazoidTM (Nycomed Amersham) consists of an aqueous dispersion of lipid-stabilized perfluorobutanefilled gas microbubbles with median volume diameter of approximately 3 μm. SonoVueTM (Bracco Diagnostics), or BR1, is a suspension of stabilized sulphur hexafluoride microbubbles coated by a highly elastic phospholipid monolayer shell with high bubble concentration (up to 5×108 bubbles/ml) and a mean diameter of 2.5 μm. BR14 (Bracco Diagnostics) is a third-generation ultrasound contrast agent that has undergone pre-clinical studies and is currently undergoing phase II studies in humans. It consists of perfluorocarbon-containing microbubbles stabilized by a phospholipid monolayer. The mean diameter of the microbubbles is 2.5–3.0 μm, and their mean concentration is 2 to 5×108 bubbles/ml. Targestar-P agents (Targeson Inc.) are microbubbles composed of a perfluorocarbon gas core

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encapsulated by a lipid shell. The agents are further stabilized and shielded by a layer of polyethylene glycol. Agents are suspended in aqueous saline at a concentration of approximately 1.5×109 particles/ml, and are packaged in glass vials with a perfluorocarbon gas headspace. These microbubbles have a median diameter of approximately 2.5 μm. Visistar-Integrin agents (Targeson Inc.) are also microbubbles composed of a perfluorocarbon gas core encapsulated by a lipid shell. The outer shell is derivatized with a peptide that selectively binds endothelial avβ3 integrin. The concentration of microbubbles is 1.5×109 particles/ml with median diameter of approximately 2.5 μm. AI-700TM (Acusphere) is a perfluorocarbon gas containing microbubble coated by a polymer shell. The mean diameter of the microbubbles is 2 μm and the concentration is 2×109 microbubbles/ml. BiSphereTM (POINT Biomedical) consists of a double polymer/albumin wall shell encapsulating air using bi-SphereTM technology. The inner shell, consisting of bio-degradable biopolymers, provides physical structure and controls acoustic response, while the outer albumin layer is designed to operate as a biological interface. The mean diameter of the bubbles is 4 μm. The double layer of the bubbles provides them additional resistance to ultrasound destruction. A new generation of ultrasound contrast agents with a similar technology platform is PB127. This product is composed of biSpheres containing nitrogen and has been engineered to break under specific ultrasound conditions. An important physical characteristic of the encapsulated microbubbles is that they oscillate during sonification. These oscillations are used for specific imaging techniques processing the resulting nonlinear signals with harmonic frequencies at medium acoustic pressure (around 0.1 MPa), and microbubble disintegration at high acoustic pressure (larger than 0.2 MPa) (von Bibra et al. 1999; Firschke et al. 1997). At medium acoustic pressure, microbubble oscillation produces stable nonlinear scattering with harmonic echoes that can be used to augment signal intensity of ultrasound contrast agents using various modalities, such as harmonic B-mode, pulse inversion imaging modalities, and harmonic power Doppler imaging (Firschke et al. 1997; Becher et al. 1997; Burns et al. 2000; Strobel et al. 2005). Subharmonic (Shi et al. 1999) and ultra-harmonic (Basude and Wheatley 2001) imaging methods have also been investigated. Current clinical applications of microbubbles, at medium acoustic pressure, include visualization of the cardiac chambers during echocardiography, especially in assessing left ventricular size and systolic performance at rest or during stress (Hundley et al. 1998; Rainbird et al. 2001), thrombus imaging (Unger et al. 1998), enhancing tumor detection with ultrasound (Ferrara et al. 2000), and obtaining information on microvascular blood volume and velocity (Wei et al. 1998). At high acoustic pressure, oscillations lead to disruption of microbubbles, causing high energy broadband nonlinear scattering. This phenomenon has been successful used for quantifying myocardial microperfusion using refill kinetics (Leong-Poi et al. 2001; Hansen et al. 2005). The success and versatility of microbubble contrast agents have caused them to be the subject of several review articles on the topic of their use in diagnostic procedures (Mayer and Grayburn 2001; Kaul 2001; Blomley et al. 2001; Raisinghani and DeMaria 2002; Bull 2007; Dijkmans et al. 2004; Lindner 2004).

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6.1.2.2 Dynamics of Encapsulated Microbubble Contrast Agent Extensions of the Rayleigh-Plesset Equation The dynamics of encapsulated microbubbles depend on the microbubble size, the mechanical properties of the shell, the compressibility and density of the gas inside the microbubble, the viscoelasticity and density of the surrounding medium, and the frequency and power of ultrasound applied. Existing theoretical models are based upon various forms of the Rayleigh-Plesset equation for spherical bubble oscillations, and attempt to take into account most of these parameters. The first model for the radial motion of a microbubble contrast agent was developed by de Jong (1993) (see also de Jong and Hoff 1993). In this model, which is based on semiempirical observations, a damping coefficient and a shell elasticity term is added in the Rayleigh-Plesset equation. However, neither a normal stress balance at the microbubble surface nor a rheological model for the shell is considered. A more rigorous theoretical treatment that considers the thickness of the encapsulating shell is due to Church (1995). This model is derived from conservation of radial momentum assuming the existence of two interfaces, one between the gas and the shell and another between the shell and the surrounding liquid (Fig. 6.2). The shell is modeled as an incompressible viscoelastic solid and the liquid is considered incompressible and Newtonian. Church model reads as: ! ) " ' 3 4R2 − R31 R1 3 ρL − ρS R1 ρL − ρS 2 + R˙ 1 + R1 R¨ 1 1 + ρS R2 2 ρS R2 2R32 ! ' ) 3κ R01 2σ1 2σ2 R˙ 1 VS μS + R31 1 − (p0 + pa ) − − −4 = pg,eq (6.1) ρS R1 R1 R2 R1 R32 " Re1 VS GS 1− −4 3 , R1 R2 where VS = R302 − R301 , and ! Re1 = R01

1 1+ 4GS

2σ1 2σ2 + R01 R02

'

R302 VS

)" .

(6.2)

In Eqs. (6.1) and (6.2) ρ L and ρ S are the densities of the liquid and shell, respectively, R01 and R02 are the initial radii of interfaces 1 and 2, respectively, κ the polytropic exponent, pg,eq the equilibrium pressure in the cavity, Re1 the unstrained equilibrium position of interface 1, σ 1 and σ 2 are the interfacial tension at interfaces 1 and 2, respectively, GS is the modulus of rigidity of the shell, ρ L and ρ S are the densities of the liquid and shell, respectively, μL and μS are the viscosities of the liquid and shell, respectively, p0 the ambient liquid pressure, and pa is the acoustic pressure. Church (1995) indicated that the values of the shell parameters for the

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Fig. 6.2 Schematic sketch of an encapsulated microbubble

AlbunexTM microbubbles are R01 – R02 = 15 nm (shell thickness), σ 2 = 5×10–3 N/m, GS = 88.8 MPa, μS = 1.77 Pa·s, and ρ s = 1,100 kg/m3 . A similar model was introduced by Allen et al. (2002) for the case of microbubbles coated with thick fluid shells, such as the MRX-552 agent developed by ImaRx Corporation which is a triacetin-shelled microbubble. The Church model was refined by Khismatullin and Nadim (2002), Stride and Saffari (2004), and Doinikov and Dayton (2007). Khismatullin and Nadim (2002) have considered the compressibility and viscoelastic properties of the liquid. As in the model derived by Church, the shell is modeled as an incompressible viscoleastic solid. In contrast to the Church model, the Oldroyd-B model was used to describe the viscoelastic properties of the liquid surrounding the bubble. Stride and Saffari (2004) consider the adhesion of the blood cells to the shell and a second viscoelastic layer that surrounds the shell was introduced in the mathematical formulation. The results of both numerical calculations and experiments indicated that the presence of the blood cells has only a small effect on the microbubble dynamics. The authors concluded that, for the purposes of microbubble design, it is justifiable to model the surrounding liquid as homogeneous and Newtonian. Doinikov and Dayton (2007) also considered the liquid compressibility but approximated the behaviour of the shell by a linear Maxwell constitutive equation and included the translation motion of the microbubble. In all these cases, however, very complicated equations for the radial oscillation of a microbubble were obtained. A much simpler and tractable model was introduced by Morgan (2001) (see also Wu et al. 2003; and Zheng et al. 2007) to study the dynamics of microbubbles encapsulated with a thin shell. This model is basically a modified Herring-Trilling equation for bubble dynamics in a slightly compressible Newtonian liquid:

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3κ R0 3κ 3 2σ 2χ 1− ρ RR¨ + R˙ 2 = p0 + + R˙ 2 R0 R0 R c 2 3R˙ 2σ R˙ 2χ R0 R˙ 1− 1− − − 4μ − R R c R R c R˙ − (p0 + pa ). − 12μS ε R(R − ε)

(6.3)

In this equation, ρ is the density of liquid, R is the radius of the encapsulated bubble, R0 the equilibrium radius, μ the liquid viscosity, μS the shell viscosity, χ the shell elasticity, σ the interfacial tension, ε the shell thickness, and c the speed of sound in the liquid. The estimated values of the shell parameters for the MP1950 microbubbles are χ = 0.5 N/m, ε = 1 nm, and εμs = 1 nm·Pa·s (Wu et al. 2003). Chatterjee and Sarkar (2003) (see also Tu et al. 2008) also developed a model of encapsulated microbubble dynamics that treats the thickness of the shell as infinite small, but considers only the interfacial tension at the bubble interface. Sarkar et al. (2005) refined this model to include a viscoelastic model of the interface that includes the surface dilatational elasticity. In the incompressible limit their model reads as (Sarkar et al. 2005): $ % 3κ R0 2σ R˙ 3 2 2σ (R0 ) ¨ ˙ − 4μ − ρ RR + R = p0 + 2 R0 R R R " ! 2 s ˙ 2E R R − − 1 − 4κs 2 − (p0 + pa ), R RE R

(6.4)

with σ = σ (R0 ) + ES [(R/RE )2 − 1] and RE = R0 [1 − (σ/ES )]−1/2 , where κs is the surface dilatational viscosity and Es is the surface dilatational elasticity. According to Sarkar et al. (2005), the values of the interfacial rheological parameters for the SonazoidTM microbubbles are κs = 10–8 Ns/m, σ (R0 ) = 0.019 N/m, and ES = 0.51 N/m. The dynamics of phospholipid-shelled microbubbles, such as SonoVueTM or BR14, was modelled by Marmottant et al. (2005). According to their model, if in the course of expansion the radius of the microbubble exceeds a threshold value, the shell breaks up, the surface tension becomes equal to that for unencapsulated bubble, and the shell elasticity becomes zero. A second threshold value is set for compression and if the radius of the microbubble goes below it, both the surface tension term and the elastic term vanish. The model has three parameters to describe the surface tension of the lipidic monolayer: the buckling radius of the bubble below which the surface buckles (zero surface tension), Rbuckling , an elastic compression modulus of the shell, λ, and a critical break-up tension, σbreak-up : 3κ R0 3κ 3 2 2σ (R0 ) ¨ ˙ ˙ 1− ρ RR + R = p0 + R 2 R0 R c R˙ 2σ (R) R˙ − 4κs 2 − (p0 + pa ), − 4μ − R R R

(6.5)

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where ⎧ 0 if R ≤ Rbuckling ⎪ ⎪ ⎨ 2 σ (R) = λ 2 R − 1 if Rbuckling < R < Rbreak-up . Rbuckling ⎪ ⎪ ⎩ σfluid if R ≥ Rruptured

(6.6)

The lower limit of the elastic regime is Rbuckling , while the upper limit radius is fixed by the maximum surface tension before rupture of the shell Rbreak-up = Rbuckling (1 + σbreak-up /λ)1/2 . When σbreak-up is reached the shell is ruptured and the interfacial tension becomes equal to the surface tension of the fluid surrounding the microbubble. The values of the interfacial parameters are χ = 1 N/m, κs = 15×10–9 Ns/m, and σbreak-up > 1 N/m for SonovueTM microbubbles, while for the BR14 microbubbles the corresponding values are χ = 1 N/m, κ s = 7.2×10–9 Ns/m, and σ break-up = 0.13 N/m (Marmottant et al. 2005). The translational motion of a microbubble in a fluid during insonation can be studied by solving a particle trajectory equation (Dayton et al. 2002): du 2 ˙ r − πρL R3 r − 2πρL |ur |ur R2 cd , ρb V X¨ = −V∇pa − 2πρL R2 Ru 3 dr

(6.7)

where ur = X˙ + ∇pa /ρL .

(6.8)

In the above equations, X˙ is the bubble translation velocity, ρb , ur , and V are the density of gas inside the bubble, the relative velocity between the bubble and liquid and the volume of the bubble, respectively. Cd is the drag coefficient determined by Reynolds number of liquid around the oscillating bubble, as defined in Dayton et al. (2002) and Watanabe and Kukita (1993): cd =

24 2R|uL −ub | ν

!

2R|uL − ub | 1 + 0.197 ν

0.63

−4

+ 2.6 × 10

2R|uL − ub | ν

1.38 " , (6.9)

where ν is the kinematic viscosity of the liquid surrounding the microbubble. The term on the left in Eq. (6.7) is the product of the mass of the bubble and its acceleration. The four terms on the right side of the equation describe the radiation force on a highly compressible bubble as a result of the acoustic pressure wave, the added mass as a result of the oscillating bubble wall, the added mass required to accelerate a rigid sphere in the surrounding fluid, and the quasistatic drag force, respectively. Example 6.1: Equation of Motion for a Microbubble Encapsulated with a Thin Membrane For a contaminated gas/liquid interface with a surface active substance, such as a surfactant, the interfacial stress is a function of two intrinsic properties of the

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interface, the surface shear viscosity, μs , and the surface dilatational viscosity, κ s . Consider, for simplicity, the case of a spherical bubble and a Newtonian interface, i.e. an interface for which the relationship between the viscous part of the surface stress and the surface rate of deformation is linear. The surface shear viscosity does not come into play in the present situation, because of the radial motion of the bubble. If the bubble surface is expanded at a constant dilatational rate λ˙ =

1 dA , A dt

(1)

where A is the area of the bubble, the constitutive law for the isotropic part of the surface stress is given by ˙ τrrs = σ + κ s λ.

(2)

We also note that in compression or expansion deformation of an insoluble monolayer, an elastic modulus is defined as the increase in surface tension for a small increase in area of a surface element at constant shape and curvature χ=

dσ . d ln A

(3)

The variation of surface tension with the bubble radius R is thus expressed as ' σ (R) = σ (R0 ) + χ

) R2 −1 , R20

(4)

which, for |R − R0 | << R0 , reads σ (R) ∼ = σ (R0 ) + 2χ

R −1 , R0

(5)

where R0 is the initial bubble radius. The balance of normal stresses at the interface can now be written R˙ R˙ 2σ (R) − 4μ − 4κs 2 R R R 1 R˙ ) R˙ 2σ (R 1 0 ∼ − 4χ − 4η − 4κs 2 , − = pg − R R0 R R R

pl = pg −

(6)

with pl the liquid pressure, pg the gas pressure in the bubble, and μ the surrounding liquid viscosity. Combining the Rayleigh-Plesset equation and a polytropic gas law with the boundary condition (5) we obtain the model for the bubble dynamics as

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3 1 2 ... RR¨ + R˙ 2 − (R R + 6RR˙ R¨ + 2R˙ 3 ) 2 c∞ R˙ 1 R˙ 2σ (R0 ) 1 1 pg − − ρ∞ − 4χ − 4η − 4κs 2 , = − ρ∞ R R0 R R R

(7)

which, in the incompressible limit, is similar to the model proposed by de Jong (1993) for thin elastic shells. Example 6.2: Equation of Motion for a Microbubble Encapsulated with a Thick Membrane Consider now a spherical gas bubble encapsulated by a thick shell. The geometry of the system is shown in Fig. 6.2. Assuming the surrounding liquid and the encapsulating layer to be incompressible, from the continuity equation it follows that both the velocity of the surrounding liquid and the velocity inside the bubble shell are subject to the equation ∇ · ν = 0,

(1)

where v stands for both of the above velocities. From this it follows further that v(r, t) =

R21 (t)R˙ 1 (t) , r2

(2)

where v(r, t) is the radial component of v, R1 (t) is the inner radius of the bubble shell, and the over-dot denotes the time derivative. If R1 ≤ r ≤ R2 , where R2 denotes the outer radius of the bubble, v is the velocity inside the encapsulating layer; if r > R2 , v is the velocity of the surrounding liquid. The assumption of incompressible shell gives the following equations: R32 − R31 = R320 − R310 , R21 R˙ 1 = R22 R˙ 2 ,

(3)

where R10 and R20 are, respectively, the inner and the outer radii of the bubble shell at rest. Conservation of radial momentum yields ρ

∂v ∂v +v ∂t ∂r

=−

∂p ∂τrr 3τrr + + , ∂r ∂r r

(4)

where ρ is equal to ρS or ρL , ρS and ρL , are respectively, the equilibrium densities of the shell and the liquid, p is the pressure, and τrr is the stress deviator in the shell or the liquid. The boundary conditions at the two interfaces are given by pg (R1 , t) = pS (R1 , t) − τrrS (R1 , t) +

2σ1 , R1

(5)

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6

pS (R2 , t) − τrrS (R2 , t) = ρL (R2 , t) − τrrL (R2 , t) +

Cardiovascular Cavitation

2σ2 + p0 + pa (t), R2

(6)

where pg (R1 , t) is the pressure of the gas inside the bubble, σ1 and σ2 are the surface tension coefficients for the corresponding interfaces, and pa (t) is the driving acoustic pressure at the location of the bubble. Integrating Eq. (4) over r from R1 to R2 using the parameters appropriate for the encapsulating layer and from R2 to ∞ using those appropriate or the surrounding liquid, assuming that the liquid pressure at infinity is equal to the hydrostatic pressure, p0 , and combining the resulting equation with Eq. (2), one obtains ! ) " ' 3 4R2 − R31 R1 3 ρL − ρS R1 ρL − ρS 2 + R˙ 1 + R1 R¨ 1 1 + ρS R2 2 ρS R2 2R32 ⎤ ⎡ R2 S ∞ L 1 ⎢ τrr (r, t) τrr (r, t) ⎥ 2σ1 2σ2 dr + 3 dr⎦ = − +3 ⎣pg (R1 , t) − (p0 + pa ) − ρS R1 R2 r r R1

R2

(7) Assuming that the surrounding liquid is a viscous Newtonian fluid, τrrL (r, t) is written as τrrL = 2ηL

∂v , ∂r

(8)

where ηL is the shear viscosity of the liquid. By using Eqs. (8) and (2), the second integral term in Eq. (7) is found to be ∞ 3 R2

R2 R˙ 1 τrrL (r, t) dr = −4ηL 1 3 . r R2

(9)

Consider now a viscoelastic shell whose rheology is described by the linear Maxwell constitutive equation τrrS + λ

∂v ∂τrrS = 2ηS , ∂t ∂t

(10)

where λ is the relaxation time and ηS is the shear viscosity of the shell. Substituting Eq. (2) into Eq. (10), one has τrrS + λ

R2 R˙ 1 ∂τrrS = −4ηS 1 3 . ∂t r

(11)

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This equation suggests that τrrS (r, t) can be written as τrrS = −4ηS

D(t) , r3

(12)

and, therefore, the function D(t) obeys the equation ˙ = R21 R˙ 1 . D(t) + λD(t)

(13)

Using Eqs. (12) and (2), the first integral term in Eq. (7) is calculated as R2 3 R1

D(t)(R320 − R310 ) τrrS (r, t) dr = −4ηS . r R31 R32

(14)

Substitution of Eqs. (9) and (14) into Eq. (7) yields ! ) " ' 3 4R2 − R31 R1 3 ρL − ρS ρL − ρ S R 1 2 ˙ ¨ + + R1 R1 R1 1 + ρS R2 2 ρS R2 2R32 ! " (15) D(t)(R320 − R310 ) R21 R˙ 1 2σ1 2σ2 1 − − 4ηL 3 − 4ηL pg (R1 , t) − (p0 + pa ) − , = ρS R1 R2 R R3 R3 2

1 2

where the function D(t) is calculated from Eq. (13). Resonance Frequency The linear resonance frequency of microbubble oscillation f0 is the frequency at which the bubble first harmonic response (linear amplitude-frequency response) has a local maximum. The linear resonance frequency of encapsulated microbubbles in the Church model is (Church 1995): % 2σ2 R301 ρL − ρS R01 −1/2 2σ1 1+ − 3κp0 − ρS R02 R01 R402 ! ) "&1/2 ' 3R301 R302 2σ1 VS GS 2σ2 1 +4 3 + . 1+ 1+ 3 4GS R01 R02 VS R02 R02

1 f0 = 2π

$

ρS R201

(6.10)

Equation (6.10) refers to a breathing mode of oscillation where the bubble simply pulsates. This is the frequency of oscillation of a zero-order spherical harmonic perturbation upon a spherical bubble. Putting ρL = ρS = ρ, R01 = R02 = R0 , σ2 = 0, and GS = 0 in this equation yields the natural frequency of the bubble in a Newtonian liquid (Lauterborn 1976). The resonance frequency of the encapsulated microbubbles increases approximately as the square root of the modulus of

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rigidity GS . Encapsulated microbubbles will resonate at higher frequencies than free bubbles of the same size, and therefore will tend to appear acoustically smaller than they actually are. QuantisonTM , which has the thickest and most rigid shell shows increased resonance frequency, while SonoVueTM , which has an encapsulation more flexible has a lower resonance frequency (Boukaz and de Jong 2007). Effects due to the density and surface tension of the shell are relatively minor by comparison to that produced by its elasticity (Church 1995). Khismatullin and Nadim (2002) found a decrease in the maximal resonance frequency with decreasing the speed of sound in the liquid (Fig. 6.3a). They also noted that the maximal resonance frequency is larger in a viscoelastic liquid than in a Newtonian liquid (Fig. 6.3b). Both effects are, however, small compared to the shell effect (Fig. 6.4). The resonance frequency of the encapsulated microbubbles in the Morgan model may be approximately expressed as (Wu et al. 2003): 1 f0 = 2π

&1/2 2(σ + χ ) 4μ + 12εμS /R0 3κ 2σ + 6χ − . (6.11) p0 + − R0 ρR20 ρR20 ρR30

It can be seen that the resonance frequency of the microbubble increases with increasing the shell elasticity and decreasing the thickness and viscosity of the shell. The most influential parameter is, however, the shell elasticity (Wu et al. 2003). Scattering Cross Section As scatter and reflection are exploited by ultrasound imaging, a contrast agent material has to possess a high scattering cross section in order to provide a significant scatter enhancement compared to the surrounding tissue. The scattering cross section may be defined as the ratio of the total acoustic power scattered by a microbubble at a particular frequency to the incoming acoustic intensity WS , Iinc

(6.12)

4πr2 |PS |2 , 2ρc

(6.13)

|Pa |2 , 2ρc

(6.14)

σS = with WS = and WS =

where PS is the amplitude of the scattered wave at a distance r from the emission center.

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Fig. 6.3 Effects of liquid compressibility and viscoelasticity on the resonance frequency of microbubble oscillation. Plots (a) and (b) show the resonance frequency as a function of bubble radius for different values of sound velocity c for a Newtonian liquid and of relaxation time λ1 at c = 1,500 m/s when the retardation time λ2 = 0, respectively. Other parameters are Gs = 88.8 MPa, μs = 1.77 kg/(ms), and shell thickness 15 nm. Reproduced with permission from Khismatullin and Nadim (2002). © American Institute of Physics

190 Fig. 6.4 Resonance frequency versus bubble radius for an encapsulated microbubble in a compressible Newtonian liquid (c = 1,500 m/s) for different values of (a) shell elasticity, Gs , and (b) shell viscosity, μs . Shell thickness 15 nm. Reproduced with permission from Khismatullin and Nadim (2002). © American Institute of Physics

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Fig. 6.5 Effect of liquid viscoelasticity on the second-harmonic scattering cross section. The amplitude of the acoustic pressure is pa = 0.3. Other parameters are Gs = 88.8 MPa, μs = 1.77 kg/(ms), and shell thickness 15 nm. Reproduced with permission from Khismatullin and Nadim (2002). © American Institute of Physics

The final form of the expressions for the scattering cross section depends on the model used to describe the radial oscillations of the microbubble. The resulting expressions for a thick-shelled microbubble can be found in Khismatullin and Nadim (2002) and for a thin-shelled microbubble in Wu et al. (2003). The numerical results obtained by Church (1995), Khismatullin and Nadim (2002), and Wu et al. (2003) indicate that the scattering cross section increases with increasing shell elasticity and decreasing shell viscosity and thickness. Scattering is higher in a viscoelastic liquid than in a Newtonian liquid but this effect is minor as compared to the shell effect (Fig. 6.5). 6.1.2.3 Potential Therapeutic Applications of Microbubble Ultrasound Contrast Agents As microbubble contrast agents developed, interest grew in understanding their interaction with propagating ultrasound waves and nearby biological tissue. Hypotheses of potential benefits from these interactions suggested that microbubble contrast agents loaded with therapeutic substances could be targeted for destruction with ultrasound and thus enhance diffusion-mediated delivery by increasing localized concentration of the substances. The ability to increase tissue permeability and concomitantly augment localized drug concentrations through targeted microbubble destruction has fuelled interest in developing efficient methods for delivering drugs and genetic material. Damage of cell membrane is a well-known biological effect of cavitation (Miller et al. 2002). The mechanical action of the cavitation bubbles typically causes cell

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lysis and disintegration. However, sub-lethal membrane damage also occurs, in which large molecules in the surrounding medium are able to pass in or out of the cell, followed by membrane sealing and cell survival. This allows foreign macromolecules to be trapped inside the cell. This ultrasound-mediated increase in cell membrane permeability has been termed sonoporation. It should be noted that sonoporation represents transient permeabilization, which can be indicated by trapping large fluorescent molecules inside the viable cells, and is different from the commonly noted permeabilization indicated by trypan blue or propidium iodide stains, which stain lysed, nonviable cells. Most investigators who have used ultrasound contrast agents for therapeutic applications worked with perfluorocarbon bubbles stabilized by an albumin or lipid shell. The main advantage of this type of contrast agents is their fragility when exposed to ultrasound. Microbubbles can be produced together with the bioactive substance, thus potentially incorporating it into the microbubble shell or lumen (Shohet et al. 2000; Frenkel et al. 2002; Erikson et al. 2003; Unger et al. 2002) (Fig. 6.6a, b), or microbubbles can be incubated with the bioactive substance, thus attaching the substance to the microbubble shell, presumably by electrostatic or weak non-covalent interactions (Lawrie et al. 2000; Pislaru et al. 2003; Mukherjee et al. 2000) (Fig. 6.6c). In several other studies microbubbles and the bioactive

Fig. 6.6 Illustrating the transfer modalities of active substances (drugs or genes) to tissue using microbubble ultrasound contrast agents. (a) Active substances are included in the gas-core region of the microbubble, (b) Active substances are incorporated in the shell of the microbubble, (c) Active substances are attached to the microbubble shell, (d) Microbubbles and the active substances are co-administrated in the targeted region

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substance were co-administered (Price et al. 1998; Song et al. 2002; Kondo et al. 2004) (Fig. 6.6d). The most widely investigated application is for gene transfer/gene therapy. A second application is for drug and protein delivery. Finally, ultrasound targeted microbubble destruction alone has been studied for therapeutic effects without any transported substance. Blood vessels are obvious targets for microbubbles and ultrasound, because they are the first tissue exposed to the microbubbles. Several in vitro and in vivo studies have been performed to evaluate transfection and the physiologic response to ultrasound-target microbubble destruction in vessels. Cultured vascular smooth muscle cells and endothelial cells were transfected with plasmids and microbubbles, showing 3,000-fold higher expression than obtained with naked DNA alone (Lawrie et al. 2000). Rat carotid arteries were transfected with anti-oncogene plasmids and microbubbles, resulting in a significant reduction of intimal proliferation (Taniyama et al. 2002a). Similarly, oligodeoxynucleotides were used with microbubbles to reduce intimal proliferation in balloon-injured rat carotids (Hashiya et al. 2004). Hynynen et al. (2001) has shown that transcranial application of ultrasound combined with intravenous administration of microbubbles in rabbits reversibly open the blood-brain barrier. They indicate that the mechanism responsible for opening the blood-brain barrier is most likely due to cavitation of microbubbles with ultrasound. Many potent drugs with severe adverse effects may be used more beneficially if local concentrations could be increased while keeping systemic concentrations low. Several studies demonstrated the potential for using ultrasound–microbubble interactions to deliver therapeutically functional substances to treat various cardiac pathologies through myocardial microcirculation. Figure 6.7 shows a schematic diagram of drug delivery or gene therapy to the heart. A diagnostic ultrasound transducer is placed on the patient’s chest. An ultrasound contrast agent bearing drug or genetic material has been administered intravenously. As the microbubbles enter the region of insonation, they distribute within the myocardial tissue via the vascular bed. The microbubbles cavitate within the capillaries of the myocardial tissue releasing the drug or genetic material (Unger et al. 2001). Vascular endothelial growth factor bound to albumin microbubbles was delivered to the heart using ultrasound. A 13-fold augmentation of cardiac vascular endothelial growth factor uptake was seen compared with systemic vascular endothelial growth factor administration (Mukherjee et al. 2000). A study using lipid microbubbles with luciferase protein demonstrated up to seven-fold augmented cardiac uptake of luciferase compared with systemic administration (Bekeredjian et al. 2005a). In a rat model of acute myocardial infarction, Kondo et al. (2004) utilized ultrasonic microbubble destruction to transfer systemically injected hepatocyte growth factor plasmid into myocardial cells to enhance capillary density and limit or negate left ventricular remodeling. Erikson et al. (2003) used low-frequency ultrasound (1 MHz) to release antisense oligonucleotides from albumin-shelled microbubbles, thereby facilitating oligonucleotide delivery to the myocardium. Recently, the vascular endothelial growth factor protein and its encoding gene have been administered in a canine (Zhou et al. 2002) and rat (Zhigang et al. 2004) model of myocardial infarction, respectively. In vitro studies have shown that microbubbles contrast agents can also

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Fig. 6.7 Schematic diagram of drug delivery or gene therapy to the heart. Reproduced with permission from Unger et al. (2001). © John Wiley and Sons

be used to deliver an antibiotic (Tiukinhoy et al. 2004) or a radionuclide (van Wamel et al. 2004). Studies of the interaction of microbubbles contrast agents with skeletal muscle are also pertinent to cardiovascular treatment because of the similarity of cardiac and skeletal muscle as target tissues. Two different strategies have been described to transfect skeletal muscle. Direct injection of microbubbles and green fluorescent protein encoding plasmids into the skeletal muscle with ultrasound application increased green fluorescent protein expression compared with intra-muscular naked plasmid injection alone and, at the same time, reduced muscle damage (Lu et al. 2003). This study also demonstrated an enhanced transfection of DNA by microbubbles without ultrasound, although the mechanism for this finding was not elucidated. In a second approach, intravascular infusion of cytomegalovirus-luciferase encoding plasmids bound to microbubbles with ultrasound was able to achieve luciferase expression in rat skeletal muscle, with intra-arterial application more efficient than intravenous infusion (Christiansen et al. 2003). Taniyama et al. (2002b) demonstrated increased capillary density in rabbit skeletal muscle using hepatocyte growth factor plasmid combined with microbubble contrast agents. Gene delivery to the myocardium of rats was obtained with harmonic mode diagnostic ultrasound, a microbubble contrast agent and a viral β-galactosidase

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vector (Shohet et al. 2000). Three frames from a 1.3 MHz transducer destroyed the microbubbles evident in the second-harmonic image, and three frame bursts were triggered to allow refill of the tissue between scans. Expression of the reporter gene was assayed in histological sections and by measurement of enzyme activity. Staining and enzyme activity was detected in the myocardium after echocardiographic destruction of the microbubbles mixed with the viral vector at about ten times the levels found in controls (bubbles plus ultrasound, no vector; bubbles plus vector, no ultrasound; vector alone, no bubbles, no ultrasound). Cavitation activity was clearly responsible for the effect because the procedure involved destruction of contrast agent microbubbles. However, it is uncertain whether the viral vector was delivered by sonoporation or by some other process. Echocardiographic microbubble destruction followed by vector infusion generated about twice the gene expression of controls, indicating that disruption of the endothelial barrier during microbubble destruction might be a factor in the enhanced viral transduction. The potential application of microbubble contrast agents as an adjuvant to thrombolytic therapy is also promising. Ultrasound at frequencies ranging from 20 to 3 MHz has been shown to enhance the thrombolytic efficacy of urokinase and tissue plasminogen activator (Lauer et al. 1992; Francis et al. 1992; Tachibana and Tachibana 1995; Porter et al. 1996). Acceleration of thrombolysis with ultrasound is probably due to local cavitation that may weaken the clot surface and/or improve clot penetration by the fibrinolytic agents. This process can be enhanced greatly by the presence of microbubbles. In vitro studies have shown that ultrasound energy at high acoustic pressures combined with microbubble administration enhances the thrombolytic efficacy of urokinase from 1.5- to over 3-fold (Tachibana and Tachibana 1995; Porter et al. 1996), and can even result in efficient clot lysis in the absence of thrombolytic therapy (Porter et al. 1996). The mechanisms by which ultrasound-contrast agent interactions induce an increase in cell and microvessel permeability are poorly understood, although several hypotheses exist. Postema et al. (2004) describe the different effects of ultrasound on microbubbles and demonstrate these effects by experiments using high-speed photography. Depending on the applied ultrasound amplitude and frequency, effects such as stable oscillation of microbubbles, inertial cavitation, coalescence, fragmentation, ultrasound induced damage of the shell causing gas to escape from microbubbles (sonic cracking), and jetting are ascribed. Sustained oscillatory motion of bubbles (stable cavitation) induces fluid velocities and exert shear forces on the surrounding tissues and cells (Suslick 1988). In the presence of a high-power, low-frequency ultrasound beam, microbubble contrast agents expand and contract nonlinearly, a phenomenon known as inertial cavitation, which often leads to bubble fragmentation (Chomas et al. 2000; de Jong et al. 2000; Boukaz et al. 2005; Boukaz and de Jong 2007). An example of microbubble fragmentation is shown in Fig. 6.8 for the case of the experimental contrast agent MP1950 containing C4 F10 encapsulated by a phospholipid shell (Chomas et al. 2000). The effects of various factors including the ultrasound driving frequency, pulse length, peak negative pressure, bubble size and shell properties on the fragmentation of microbubbles were investigated by Chomas et al. (2001) and Bloch et al. (2004).

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Fig. 6.8 Optical frame images corresponding to the oscillation and fragmentation of a contrast agent microbubble. The initial diameter of the microbubble is 3 μm (frame (a)). The streak image in frame (h) shows the diameter of the bubble as a function of time, and dashed lines indicate the times at which the two-dimensional frame images in frames (a)–(g) were acquired relative to the streak image. Reproduced with permission from Chomas et al. (2000). © American Institute of Physics

The constrained boundary also has a significant effect on microbubble fragmentation. Zheng et al. (2007) demonstrated that microbubbles within smaller tubes have a higher fragmentation which may result from the decreased radial oscillation, and decreased wall velocity and acceleration within the small tube. The collapse of inertial cavitation bubbles generates shock waves with amplitude exceeding 5 GPa (Pecha and Gompf 2000; Brujan et al. 2008). Although cavitation-induced shock waves persist for a very short period of time, the large spatio-temporal pressure gradients associated with shock waves can disrupt tissue. Rapid collapse of microbubbles near a boundary will lead to asymmetric movements that can form high velocity fluid microjets (Brujan 2004; Brujan et al. 2005). Microjet formation during collapse of OptisonTM microbubbles in the vicinity of a boundary was experimentally observed by Prentice et al. (2005) (Fig. 6.9). They also noted that the jetting and the microbubble translation towards the boundary are dependent on the relative distance between microbubble and boundary. This jetting is associated with high pressures at the tip of the jet that are sufficient to penetrate any cell membrane. It is widely proposed that jetting is responsible for the transient nanopores which were observed in cell membranes by electron microscopy immediately after destruction of microbubbles (Tachibana et al. 2002; Miller et al. 2002). Translation of microbubbles was also observed by Zheng et al. (2007) (Fig. 6.10). Cavitating microbubble contrast agents may also induce significant but transient thermal fluctuations (Wu 1998) as well as toxic chemical production (Kondo et al. 1998). During the collapse, the temperature of the bubble core can increase by more than 1,000 K and induce chemical changes in the surrounding medium, an effect termed sonochemistry (Suslick 1988). Of particular

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Fig. 6.9 Illustrating the formation of a microjet during the collapse of a microbubble contrast agent near a solid boundary. The microjet is indicated by the white arrow. The initial position of the microbubble is indicated by the black arrow. It is 26.5 μm in the top sequence and 19 μm in the bottom sequence. Frame size is 163 μm × 110 μm. Reproduced with permission from Prentice et al. (2005). © Macmillan Publishers Ltd

Fig. 6.10 Microbubble translation under high-pulse repetition frequency ultrasound within microtubes observed by a microscope video camera system at 240 frames/s. The microbubble with an initial radius of 1.2 μm is moving fom the center of the microtube to the wall. Reproduced with permission from Zheng et al. (2007). © Elsevier B.V.

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importance is the generation of highly reactive species, such as free radicals, that can induce chemical transformations in the medium which may be involved in the enhancement of permeability of endothelial cell layers. A significant increase in free radical production in endothelial cells after exposure to ultrasound was demonstrated by Basta et al. (2003). Finally, another mechanism by which the use of encapsulated microbubbles could facilitate deposition of drugs and genes in a cell is fusion of the phospholipid microbubble coating with the bilayer of the cell membrane. This could result in delivery of the microbubble substances directly into the cytoplasm of the cell (Dijkmans et al. 2004). All or a combination of these events may alter, displace, or destroy cells, possibly resulting in cell microporation (Deng et al. 2004) and gaps between neighboring cells. For example, Ohl et al. (2006) demonstrated that the collapse of microbubbles cause membrane poration to cells plated on a substrate through a complex sequence of events. When the jet developed during bubble collapse impacts onto the boundary, it spreads out radially along the substrate causing a strong gradient in the velocity component parallel with the substrate. The resulting shear stress leads to the detachment of cells. Cells at the edge of the area of detachment were found to be permanently porated, whereas cells at some distance from the detachment area undergo viable cell membrane poration. The high shear stress caused by violent microstreaming or microjets developed during microbubble collapse may explain the maximum transfection efficiency and lowest cell viability obtained at high ultrasound pressures (Wu 2002; Wu et al. 2002). Several hypotheses on the mechanism of blood-brain barrier disruption with microbubbles and ultrasound have been proposed (Sheikov et al. 2004). Since an ultrasound wave causes microbubbles to expand and contract in the capillaries, the expansion of larger microbubbles could fill the entire capillary lumen, resulting in a mechanical stretching of the vessel wall which, in turn, could result in the opening of the tight junctions. This interaction could create a change in the pressure in the capillary to evoke biochemical reactions that trigger the opening of the bloodbrain barrier. Moreover, bubble oscillation may also reduce the local blood flow and induce transient ischemia, which could also trigger blood-brain barrier opening. Extensive reviews of the therapeutic applications ultrasound-targeted microbubble destruction, including ultrasound–microbubble interactions, are currently available in literature (Lindner 2004; Unger et al. 2004; Liu et al. 2006; Chappell and Price 2006; Bekeredjian et al. 2005b, 2006; Ferrara et al. 2007; Shengping et al. 2009). 6.1.2.4 Collateral Effects Induced by Cavitation The collateral effects induced by microbubble contrast agents in the cardiovascular applications of ultrasound have been recently summarized by Dalecki (2007). Diagnostic ultrasound can produce premature cardiac contractions in laboratory animals and humans when microbubble contrast agents are present in the blood with end-systolic triggering. Myocardial damage in humans has not been reported to result from the interaction of ultrasound and contrast agents. Premature atrial contractions, ventricular contractions and ventricular tachycardia were observed in

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animals exposed to ultrasound in the presence of microbubble contrast agents. The interaction of diagnostic ultrasound and microbubble contrast agents can produce damage to the vasculature in kidneys. Lower ultrasound frequencies produce more damage than higher frequencies. Diagnostic ultrasound imaging devices were also reported to produce capillary damage in muscle in laboratory animals when contrast agents are present in the blood.

6.2 Cavitation in Laser Surgery Whenever laser pulses are used to ablate, cut, or disrupt tissue inside the human body, cavitation bubbles are produced that interact with the tissue. In cardiovascular laser applications, this situation is encountered in myocardial laser revascularization and laser angioplasty.

6.2.1 Transmyocardial Laser Revascularization 6.2.1.1 The Basic Principles of Transmyocardial Laser Revascularization Transmyocardial laser revascularisation is used to treat patients with severe coronary disease. Although surgical procedures such as coronary angioplasty and coronary artery bypass grafting are proven methods of treating heart disease, many patients have conditions that are not amenable to these therapies. Transmyocardial laser revascularisation was proposed as a means of bypassing the coronary circulation altogether, instead perfusing the myocardium with oxygenated blood directly from the left ventricular chamber, in a similar manner to the embryonic and reptilian cardiac circulation. Up to 50 narrow channels are drilled in the left ventricular myocardium, which are closed at the epicardial surface and open to the left ventricular cavity at the endocardial surface. These channels are typically about 1 mm in diameter and are created approximately 1 cm apart (Horvath et al. 1995). How long these channels remain open and to what extent the blood flows through them to contribute to angina relief remains a matter of controversy. The types of lasers currently used for transmyocardial revascularisation are mainly the carbon dioxide (CO2 ) which delivers light pulses at 10.6 μm wavelength with 20–90 ms duration and energies of up to 40 J, and the Holmium-Yag (Ho:Yag) which emits light pulses at 2.1 μm wavelength with 100–500 μs pulse duration and energies up to 30 J. Another type of laser is the Excimer laser (XeCl) emitting shorter light pulses (150 ns) at a wavelength of 308 nm with energies between 20 and 40 mJ. The CO2 and Ho:Yag lasers are infrared lasers exerting their effect by vaporising water molecules. These lasers have frequencies similar to the vibrational frequency of water and absorption of laser energy by water molecules results in heating, evaporation, and tissue ablation. The XeCl laser, on the other hand, operates in the ultraviolet spectrum and exerts its effect by dissociating the dipeptide bonds of

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proteins. Because the myocardium is composed predominantly of water and proteins, these types of lasers are the most used for creating transmyocardial channels (Cooley et al. 1996; Klein et al. 1998; Lange and Hillis 1999). With CO2 laser, a single light pulse is used to create the laser channel. The laser light is delivered to the beating heart through an intercostal incision 15–25 cm long using an articulated mirror arm and long focusing optics. With the Ho:Yag and XeCl lasers, multiple pulses guided by a silica fibre are required. The fiber is directed to the myocardium through a very small incision or even percutaneously through the femoral artery, as in balloon angioplasty. The laser energy causes tissue ablation and vaporisation that can be detected as a puff of smoke on transesophageal echocardiogram when the laser transverses the free wall of the left ventricle (Horvath et al. 1996). 6.2.1.2 Collateral Effects Induced by Cavitation The expansion of gaseous products produced during tissue ablation creates a cavitation bubble in the medium surrounding the ablation site (Duco Jansen et al. 1996; Brinkmann et al. 1999; Vogel and Venugopalan 2003). When the optical fibre is not in contact with tissue, a cavitation bubble is formed by absorption of infrared laser radiation in the liquid separating the fibre tip and the tissue surface. This bubble is essential for the transmission of the optical energy to the tissue. A similar event occurs during ultraviolet ablation when the surrounding fluid is blood, because hemoglobin and tissue proteins absorb strongly in the ultraviolet range (van Leeuwen et al. 1992). When the fibre tip is placed in contact with the tissue surface, the ablation products are even more strongly confined than when they are surrounded by liquid alone, resulting in considerably higher temperatures and pressures within the tissue. The cavitation bubble dynamics influence the ablation efficiency in two ways. First, the bubble creates a transmission channel for the laser radiation. Second, the forces exerted on the tissue as a consequence of the bubble dynamics may also contribute to the material removal. The dynamics of channel formation within tissue has been studied in the context of transmyocardial laser revascularisation by Brinkmann et al. (1999). They noted that the shape and lifetime of the transmitted channel depend on the laser pulse duration and the optical penetration depth. For example, a 2.2 ms pulse of a Ho:Yag laser, transmitted through an optical fibre at the surface of a tissue phantom, creates an elongated bubble that partially collapses during the laser pulse, such that the light path from the fibre to the target is partially blocked. Tissue was ablated at the bubble wall opposite to the fibre tip but even at the largest value of the pulse energy used in their experiment the channel to the surface of the tissue phantom is almost closed at the end of the laser pulse (Fig. 6.11). By contrast, a 15 ms CO2 laser pulse generates an oscillating vapour channel that remains opened at the end of the laser pulse (Fig. 6.12). The confinement of the ablation products by the ablation channel leads to an increase of the collateral damage because of the high pressure and heat contained in the ablation products. This effect is clearly visible in Fig. 6.12 and is manifested by the formation of a large cavity at a depth of about 10 mm from the surface of the tissue phantom. Cavitation can thus

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Fig. 6.11 A series of pictures illustrating the interaction of a holmium laser pulse with a polyacrylamide sample in water environment. The energy of the laser pulse is 12 J and the pulse duration is 2.2 ms. Times indicated are delay times of the photograph relative to the onset of the laser pulse. Reproduced with permission from Brinkmann et al. (1999). © IEEE

Fig. 6.12 A series of pictures illustrating the interaction of a CO2 laser pulse with a polyacrylamide sample in water environment. The power of the laser pulse is 800 W and the pulse duration is 15 ms. Times indicated are delay times of the photograph relative to the onset of the laser pulse. Reproduced with permission from Brinkmann et al. (1999). © IEEE

lead to a structural deformation of the tissue adjacent to the ablation site that is much more pronounced than the ablative tissue effect itself and compromises the high precision of the original ablation. In addition, the authors concluded, from experiments on porcine heart tissue, that the orientation of the myocardial fibrils significantly influences the dynamics of cavitation bubbles, the shape of the ablated cavities, and

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the thermo-mechanical collateral damage areas. Deep and straight channels were found for fibrils running perpendicular to the endocardium, while much smaller but widely dissected tissue was found for horizontal channels. The fragmentation of cavitation bubbles during implosion may result in many gaseous microemboli that may persist briefly in the circulation. The formation of such microemboli, during transmyocardial laser revascularisation on human subjects, was observed in the middle cerebral artery by von Knobelsdorff et al. (1997). However, none of the patients exhibited major neurological deficits on the first day after surgery, indicating that transmyocardial laser revascularisation does not cause significant cerebral ischemia. The authors explained this result by the very small size of the induced microemboli. Indeed, Feinstein et al. (1984) found that only arterial emboli larger than 15 μm lead to temporary oclusion of more than 1 min, whereas emboli of less than 10 μm in diameter pass the capillary vasculature unrestricted. Multiple microembolic signals were also detected in the ophthalmic artery during transmyocardial laser revascularisation in pigs by Gerriets et al. (2004). They demonstrated that the microembolic load can be reduced by ventilation with 100% oxygen and by decreasing the laser pulse energy.

6.2.2 Laser Angioplasty 6.2.2.1 The Basic Principles of Coronary Angioplasty The main goal of coronary angioplasty is to recanalize the blood vessels that are obstructed by fatty or artheroscopic plaque. Angioplasty is designed to relieve the chest pain a person usually feels when the heart is not getting enough blood and oxygen. Percutaneous transluminal coronary angioplasty or ballon angioplasty is the most frequently applied interventional technique for treatment of coronary artery disease (Bittl 1996). Plastic deformation of the obstructive plaque with the creation of splits, intimal tears and dissections is the main mechanism of percutaneous transluminal coronary angioplasty for lumen widening. Limiting dissections and acute vessel closure can unpredictably occur resulting in myocardial infarction and urgent bypass surgery. Moreover, long-term success of percutaneous transluminal coronary angioplasty is limited by restenosis. In order to overcome these limitations, alternative interventional techniques were developed. These techniques include directional angioplasty (Bittl 1996), ultrasound angioplasty (Rosenschein et al. 1991), laser angioplasty (Lee and Mason 1992), and high-speed rotational angioplasty (Safian et al. 1993). During directional coronary atherectomy, artherosclerotic tissue is extracted from the coronary artery with a cutting blade spinning at 5,000 rpm in the tip of the atherectomy device. In ultrasound angioplasty, direct mechanical contact between an oscillating tip and vascular plaque results in fragmentation and ablation of material into microscopic particles. Flexible biological materials such as healthy arterial wall or skin easily distend with the oscillation of the distal-tip. In contrast, the rigid calcium plaque matrix lacks flexibility and is disrupted (Demer et al. 1991). During excimer-laser angioplasty, short light pulses (<200 ns) at a wavelength of

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308 nm emitted from an optical fibre at the catheter tip vaporizes atheromatous tissue. This technique requires direct contact of the fibre tip to the obstructive plaque. Main indications for pulsed laser angioplasty are diffuse and long coronary lesions and total coronary occlusions (Haude et al. 1997). Rotational atherectomy is another approach for removing atheromatous plaque from coronary arteries. This technique uses a diamond-studded burr spinning at 50,000–200,000 rpm to excavate calcified or fibrotic plaque, allowing microscopic debris to embolize to the coronary capillary bed. Although directional atherectomy and excimer-laser angioplasty usually result in larger lumen diameters than percutaneous transluminal coronary angioplasty, these treatments have not reduced the rates of acute complications or restenosis after coronary angioplasty (Topol et al. 1993; Appelman et al. 1996). Furthermore, no multi-center, randomized trials proving the superiority of rotational angioplasty over percutaneous transluminal coronary angioplasty have been reported. 6.2.2.2 Collateral Effects Induced by Cavitation The failure of excimer-laser angioplasty to achieve better clinical outcomes than percutaneous transluminal coronary angioplasty was attributed to inadequate tissue removal (Mintz et al. 1995), along with an increased risk of vessel dissection (Baumbach et al. 1994) and perforation (Bittl et al. 1993) from the generation of pressure transients with large amplitude and the formation of intraluminal cavitation bubbles in blood (van Leeuwen et al. 1993). The expansion of cavitation bubble induces a fast dilation of the vessel wall, while the subsequent bubble collapse leads to an invagination of the vessel wall (Fig. 6.13). Both effects are responsible for the observed structural deformation of the adjacent tissue. Several strategies have been proposed to minimize the negative collateral effects generated by cavitation. The incidence of dissection may be reduced by pulse multiplexing (Oberhoff et al. 1992; Haase et al. 1997) or by infusing saline through the guide catheter during excimerlaser angioplasty (Litvack et al. 1993; van Leeuwen et al. 1996). Oberhoff et al. (1992) suggested to divide the energy of one XeCl laser pulse both spatially and in time into eight smaller pulses (pulse multiplexing) which diminishes the area radiated with each pulse and thus cavitation effects. In a later study, Haase et al. (1997) conducted experiments on pressure wave propagation during pulsed laser ablation using conventional and experimental XeCl lasers emitting light at a wavelength of 308 nm and pulse duration of 115 ns. The experimental XeCl laser divides the laser beam into several areas of uniform fluence by scanning the beam from one section to the other using the intermission between two laser discharges. Peak pressures of the order of 1 MPa were measured during ablation of pure blood. They demonstrated that the maximum amplitude of the pressure waves emitted during laser ablation was diminished by pulse multiplexing or when saline was flushed during laser pulse delivery. The authors noted, however, that high concentrations of saline solution are necessary to achieve a significant reduction of the peak pressures. Van Leeuwen et al. (1998) also investigated the effect of flushing saline during pulse delivery on the arterial wall damage. At a flow rate of 0.2 m/s they found that saline significantly

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Fig. 6.13 Dilatation and invagination of a silicone tube after a dye-laser pulse. Pulse energy 70 mJ, pulse duration 3 μs, initial diameter of the tube 5 mm. Pictures taken with 50,000 frames/s. Reproduced with permission from Vogel et al. (1996). © Springer Science + Business Media

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reduces the incidence of arterial wall ruptures and prevents intimal hyperplasia formation. Both concepts can, however, only lead to a gradual decrease of cavitation effects, because they do not avoid ablation taking place in a liquid environment. To avoid this limitation, Vogel and co-workers (1996, 2001, 2002) introduced a technique for the reduction of cavitation effects produced by short laser pulses. The laser pulse is divided into a pre-pulse with low energy and an ablation pulse with higher energy, separated by time intervals of 50–100 μs. The pre-pulse creates a small bubble at the application site, and the ablation pulse is applied when this bubble is maximally expanded and can be filled by the ablation products of the main pulse. With a suitable energy ratio between pulses, the ablation products will not enlarge the bubble generated by the pre-pulse, and the maximal bubble size remains much smaller than after a single ablation pulse. In this manner, no additional cavitation effects are induced, and tissue tearing and other mechanical side effects are minimal. The reduction of the cavitation bubble size by a pre-pulse is illustrated in Fig. 6.14. In addition, the transiently empty space created by the pre-pulse between

Fig. 6.14 Illustrating the concept of double pulses for reduction of cavitation effects in a silicone tube. (a) Pre-pulse alone, (b) pre-pulse (delivered in frame 1) and ablation pulse (delivered in frame 6). The initial diameter of the tube is 5 mm. Pictures taken with 50,000 frames/s. Reproduced with permission from Vogel et al. (1996). © Springer Science + Business Media

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the optical transmission and the artherosclerotic plaque surface improves the optical transmission to the target and thus increase the ablation efficiency. The occurrence of cavitation was also observed during high-speed rotational angioplasty (Zotz et al. 1992). In an in vitro study with a burr of 1.25 mm diameter spinning at 160,000 rpm the authors observed that the mean size of the cavitation bubbles in distilled water was 90 ± 33 (between 52 and 145) μm. They also noted that the production of bubbles was more pronounced in fresh human blood than in distilled water, probably because red blood cells act as cavitation nuclei.

6.3 Cavitation in Mechanical Heart Valves Implantation of a mechanical heart valve has been used as a surgical treatment for various heart valve diseases, where the genuine valve, due to stenosis or insufficiency, compromises ventricular function. Despite the usual success of this surgical therapy, patients still face the risks of blood cell damage, thromboembolic events, and material failure of the prosthetic device (Johansen 2004). Many different mechanical valves are or were available for implantation which can be classified into three main groups: cage ball valve (Starr-Edwards, Smeloff-Cutter), tilting disk valve (Bjork-Shiley, Medtronic-Hall), and bileaflet valve (Carbomedics, CarpentierEdwards). In 1976, cavitation was first suggested to be related to erosion of titanium struts of a caged ball mechanical heart valve (Zubarev et al. 1976) but it was only in the mid 1980s that cavitation has clinically been found to cause valve failure and valve fragment embolization (Deuvaert et al. 1989; Alvarez and Deal 1990). An example of cavitation pattern is given in Fig. 6.15 for the case of a rigidly mounted bileaflet valve (Rambod et al. 1999). Clouds of bubbles occur in various locations along the peripherial circumference of the leaflets and along the gap between the leaflets.

6.3.1 Detection of Cavitation in Mechanical Heart Valves Graf et al. (1991) were among the first to visualize cavitation near mechanical heart valves. Cavitation appeared during valve closure as growth and subsequent collapse of bubbles at the impact between the occluder and the valve housing. As an indicator for the valve load during closure, Graf and coworkers evaluated the temporal rate of change of the left ventricular pressure, measured as the slope of the ventricular pressure curve (dP/dt). In a later study, Kingsbury et al. (1993) reconfirmed the findings reported by Graf et al. by visually observing cavitation at dP/dt levels within the physiological range. Kafesjian et al. (1994) observed that the locations on the valve where cavitation occurred correspond to the areas where microscopic pitting and erosion was found. They also found that highly polished surfaces reduced the risk of erosion, presumably due to the fewer nucleation sites. Chahine (1996) indicated that the characteristic size of the bubbles are in the range of tens to hundreds of

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Fig. 6.15 The sequence of microbubbles formation in a 29 mm bileaflet valve at closure. (a) Enddiastolic period (<1 ms prior to complete closure, (b) the burst at the instant of closure and formation of clouds of microbubbles (t0 ), (c) dissipation of clouds at about t0 + 5 ms, (d) and (e) pressure recovery and growth of persisting microbubbles at approximately t0 + 8 ms. Reproduced with permission from Rambod et al. (1999). © Springer Science + Business Media

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micrometers, while Hwang (1998) noted that the oscillation period of the cavitation bubbles typically last a few microseconds. Recently, Takiura et al. (2003) demonstrated that the emitted light from cavitation bubble collapse is in the same locations where bubbles have been seen to collapse. All of the above-mentioned in vitro studies on cavitation in mechanical heart valves have used the valve fixed in a rigid mounting, which does not allow motion of the valve housing. Obviously, this experimental model does not describe accurately the in vivo conditions. Wu et al. (1995) examined how the compliance in mounting a mechanical heart valve influences the flow field around the valve. They compared a rigid and a flexible mounting of the valve in the mitral position of a pulsatile mock flow loop. They found that the velocity of the occluder as it approaches the housing was similar in both cases, but the rebound of the occluder was much stronger when the valve was mounted rigidly than when it was mounted in a flexible material. Valve holder flexibility was also investigated by Chandran and Aluri (1997). In contrast to the results reported by Wu et al., they found that the flexibility of the valve holder did not affect pressure field close to the inflow surface of the leaflet at the instant of valve closure. Moreover, they found that leaflet closing velocity is an important factor for cavitation initiation if magnitudes are compared only for the same valve model. In vivo investigations on cavitation in mechanical heart valves were conducted by Zapanta et al. (1996) who measured high-frequency pressure fluctuations in valves that had undergone implantation of a left ventricular assist device equipped with a mechanical heart valve. These pressure fluctuations, which are a consequence of the transient bubble collapse, provide information on the intensity and duration of the cavitation phenomenon (Garrison et al. 1994). The most interesting conclusion of their work is that the pressure signals observed in vivo were similar to those observed in vitro where cavitation was visually demonstrated. Dexter et al. (1999) found no transient negative pressure spikes in goats with pericardial valves. However, transient pressures below the vapour pressure of blood were detected in goats in which mechanical heart valves were implanted. Paulsen et al. (1999) measured high-frequency pressure fluctuations, in the frequency range 35–150 kHz, in the first intraoperative study of patients. They also found high-frequency pressure fluctuations in patients with mechanical heart valve implant and no signals in the patients with normal or bio-prosthetic heart valves. Similar results were reported by Andersen et al. (2003).

6.3.2 Mechanisms of Cavitation Inception in Mechanical Heart Valves Several investigations were conducted in order to understand the mechanisms of cavitation inception and the factors that influence cavitation intensity. Wu et al. (1994) indicated, as a mechanism of cavitation occurrence, that during valve closure the closing disk could rebound and thus cause repetitive inception of cavitation.

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When the occluder approaches the housing during closure fluid is squeezed out into the space between occluder surface and the housing surface forming a low pressure region where cavitation occurs (Bluestein et al. 1994; Zhang et al. 2006; Lee and Taenaka 2006). Later, Zapanta et al. (1998) found that the cavitation intensity is influenced by several features that take place during valve closure. They noted that occluder closing velocity, deceleration time, occluder density, and valve geometry could affect the likelihood and intensity of cavitation. Their findings suggest that a rigid occluder, closing at high velocity and decelerating rapidly, is more prone to cause cavitation than a flexible leaflet that is slowly decelerated. Cavitation may be also generated in the core of the vortices where significant pressure decrease occurs towards the vortex core. During mechanical heart valve operation such vortices may form at the edge of the valve occluder at closure (Avrahami et al. 2000; Manning et al. 2003; Rambod et al. 2007). Both squeeze flow and vortex cavitation are seen as a cloud of bubbles at the circumferential lip of the occluder. Another mechanism of cavitation inception is given by the sudden stop of the valve occluder as it reaches the valve housing. In this case, blood is subjected to tension and the pressure may drop sufficiently to induce the formation cavitation bubbles on the occluder orifice (Hwang 1998).

6.3.3 Collateral Effects Induced by Cavitation The collapse of cavitation bubbles can lead to damage of the mechanical heart valves structures and to some biological unwanted collateral effects such as formation of microemboli (microbubbles), thromboembolic complications, and blood cell damage. Scanning electron microscopy of the explanted valves revealed confined areas of pitting and erosion on the leaflet and the housing (Klepetko et al. 1989). On the other hand, clinical studies using transcranial Doppler ultrasonography have shown the presence of microemboli in the cranial circulation of some mechanical heart valve patients (Georgiadis et al. 1997; Deklunder et al. 1998; Nötzold et al. 2006). Transesophagial echocardiography of mechanical heart valve patients has shown images of bright, mobile signals, also considered to be gas bubbles, near the valve (Levy et al. 1999; Girod et al. 2002). In vitro studies performed to investigate the relationship between dissolved gas concentration and the incidence of bubble formation after valve closure (Biancucci et al. 1999) indicate that stable gas microbubbles can form during mechanical heart valve operation. The microbubble likely form from the combined effects of gaseous nuclei formed by cavitation, low-pressure regions associated with regurgitant flow, and the presence of CO2 , a highly soluble blood gas. Levy et al. (1999) suggested cavitation to be responsible for the formation of microbubbles. They observed such microbubbles in patients with mechanical heart valves implants but not in patients with biological valve implants. Furthermore, they observed that increased systolic ventricular function increased the number of the detected microbubbles. Milo et al. (2003) demonstrated microbubble formation in vortices under physiological conditions during in vitro

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experiments with mechanical mitral valve prostheses. Large cavitation bubbles disintegrated rapidly into microbubbles with a stability of several cycles in the system. Bachmann et al. (2002) demonstrated in vitro that stable microbubble formation occurs in the core of vortices at the closing rim of Bjork-Shiley valves. They also observed cavitation in these vortices but stable microbubbles only appeared once the cavitation bubbles collapsed. A similar observation was reported by Lin et al. (2000). Potthast et al. (2000) also supported the idea that cavitation is the key factor in the appearance of gaseous microemboli at heart valve prostheses as they found a relationship between dP/dt and the number of stable microbubbles detected by ultrasonography as high-intensity transient signals. However, Girod et al. (2002) reported that the microbubbles observed in the left atrium of mechanical mitral valve patients are most likely induced by degassing of CO2 in blood rather than a cavitation phenomenon. A third type of collateral effects induced by cavitation is the release of cell content into the blood as a consequence of the destruction of blood cells by cavitation bubble collapse (Johansen 2004). One important released agent is the tissue factor from ruptured monocytes and platelets. It is well known that the tissue factor is the primary initiator of blood coagulation and thereby plays an important role in thrombogenesis and thromboembolic complications (Rapaport and Rao 1995; Sambola et al. 2003). The release of tissue factor into the bloodstream may be responsible for thrombus formation seen in patients with mechanical heart valve implants (Johansen 2004).

6.4 Gas Embolism Gas embolism occurs when gas enters vascular structures and can result in morbidity or death (Murphy et al. 1985; Bull 2005). The most common gas embolism is air embolism, but the medical use of other gases, such as nitrogen, carbon dioxide, and nitrous oxide, may also result in the formation of cardiovascular bubbles consisting of other gases. Emboli may enter the cardiovascular system during neurosurgical procedures (Porter et al. 1999), cardiac surgical procedures using extracorporeal bypass (Ziser et al. 1999), through central venous (Halliday et al. 1994) and hemodialysis catheters (Yu and Levy 1997), or can be produced by the disintegration of cavitation bubbles generated during transmyocardial laser revascularization, laser angioplasty and mechanical heart valve operation, as we discussed previously. The mechanisms underlying gas-embolism-induced injuries are not completely understood. One explanation is that the microbubbles lodge in the microcirculation, occluding flow and inducing transient local ischemia (Herren et al. 1998; Kort and Kronzon 1982). Another possible explanation is that injury is primarily due to thrombo-inflammatory response activated by microbubble damage to the endothelium or interactions of blood proteins or platelets with the microbubble interface (Ryu et al. 1996; Helps et al. 1990). This likely occurs over a longer time scale than transient ischemia, and both mechanisms may play important roles in gas embolism.

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6.4.1 Treatment Strategies for Gas Embolism Hyperbaric oxygen therapy is currently the standard treatment for gas embolism. It reduces the size of microemboli by raising the ambient pressure which, in turn, induces diffusion gradients for oxygen transport into the microbubble and nitrogen out of the microbubble (Muth and Shank 2000; Dutka 1985). On the other hand, Cavanagh and Eckmann (2002) found that even a slight reduction of the interfacial tension by surfactants dramatically altered the microbubble shape and could lead to translation of the microbubble or microbubble detachment from a wall. Similar results were reported in a later study by Eckmann and Cavanagh (2003) and a computational study by DeBisschop et al. (2002). The results of these studies demonstrate that soluble surfactants have the potential to dislodge gas bubbles adhering to a tube wall by intentional manipulation of interfacial shape and wetting properties and suggest that a similar approach may be effective for dislodging gas emboli from vessel walls (Eckmann and Cavanagh 2003). Modification of emboli interfacial tension and shape by addition of surfactants is thus a potential method for treating or preventing gas embolism. In a recent study, Eckmann and Lomivorotov (2003) suggested that fluorocarbon administration is a potential treatment for vascular gas embolism. They demonstrated that the injection of fluorocarbon into the bloodstream affects initial bubble conformation, increases bubble dislodgement, and results in bubble displacement further into the periphery of the microcirculation network.

6.4.2 Gas Embolotherapy In a number of physiological situations, gas microbubbles are intentionally placed in the bloodstream. The flow and transport of microbubbles in the cardiovascular system can have significant therapeutic effects. Embolotherapy, which occludes vessels using foreign bodies for therapeutic applications, is a potential means of treating a variety of cancers, such as heptatocellular carcinoma and renal carcinoma. Gas embolotherapy uses site-activated gas emboli to occlude arteries and capillary beds (Ye and Bull 2004; Kripfgans et al. 2000). Dodecafluoropentane (C5 F12 ) droplets coated by an albumin shell are injected into the vascular system upstream from the tumor. The droplets are imaged using standard ultrasound, and high-intensity ultrasound is used to initiate acoustic droplet vaporization to form bubbles near the desired occlusion site (Figs. 6.16 and 6.17). Although the droplets are small enough to pass through capillaries, the resulting bubbles are large enough to become stuck in the tumor vasculature. These bubbles persist for a long time in blood, ranging from over 30 min when exposed to flow to about 2 h when the flow is absent. The long persistence time of such bubbles is sufficient to cause tumor necrosis. The main advantage of gas embolotherapy is that it is minimally invasive and provides highly selective delivery of the emboli to the tumor, minimizing the risk of damage to healthy tissue.

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Fig. 6.16 A schematic illustration of the gas embolotherapy technique to kill tumors. A stream of encapsulated superheated dodecafluoropentane liquid droplets goes into the body by way of an intravenous injection. The droplets are small enough so that they do not lodge in vessels. Once the droplets reach their destination, they are hit with high intensity ultrasound and expand into a gas bubbles that lodge in the blood vessel

Several studies have been performed on the threshold and efficiency of vaporization using different acoustic parameters. It is currently hypothesized that there are two main methods for vaporizing droplets. The first mechanism is vaporization within the core of the droplet resulting from the transmitted ultrasonic field interacting with the dispersed medium. Vaporization via this mechanism seems to be supported from high-speed photography images taken by Kripfgans et al. (2004) (see, for example, Fig. 6.17). The second mechanism is vaporization due to inertial cavitation in or near the droplet. During the collapse of an inertial cavitation bubble a secondary shock-wave is emitted which causes vaporization of the droplet. Support for multiple mechanisms is further assisted when comparing the acoustic pressure thresholds that induce vaporization. Using short pulses Kripfgans et al. (2000) found the vaporization threshold to decrease with increasing frequency. On the other hand, Giesecke and Hynynen (2003) found the vaporization threshold to increase with increasing frequency, similar to an inertial cavitation threshold curve. In a recent experimental study, Calderon et al. (2006) investigated the evolution of cardiovascular gas bubbles in a microfluidic model bifurcation. Their work was motivated by the potential treatment of cancer by tumor infarction using gas embolotherapy. The interesting result of their investigations is that the critical driving pressure below which a bubble will lodge in a bifurcation is significantly less than the driving pressure required to dislodge it. The authors estimated that gas bubbles from embolotherapy can lodge in vessels 20 μm or smaller in diameter, and concluded that bubbles may potentially be used to reduce blood flow to tumor microcirculation. There are two states in which bubbles lodged at the bifurcation

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Fig. 6.17 Acoustic vaporization of an 18-μm dodecafluoropentane encapsulated with albumin droplet. Two sequences are shown. (a) Images 1–7 are full frame pictures of the droplet in the flow tube. The eighth frame is a streak image. The acoustic pressure in the first sequence is not sufficient to vaporize the droplet (ultrasound frequency 3 MHz, 10 cycle). (b) Increasing the acoustic pressure to 6.5 MPa leads to the vaporization of the droplet after 8 cycles. Reproduced with permission from Kripfgans et al. (2004). © Acoustical Society of America

(Fig. 6.18). Bubbles usually lodged when the bubble’s front and back menisci contacted the wall (Fig. 6.18a). Most of the bubbles lodged in this state originally, and as the pressure was increased, the bubble either dislodged or lodged again at a higher pressure in a different state. In Fig. 6.18b the rear meniscus appears to stretch and eventually prevents the bubble from moving. This lodging state is more common in the microchannels with the bifurcation angle of 110◦ , in which the lodged bubble could withstand higher pressure than in the 78◦ bifurcation. In some instances, when the driving pressure was increased to dislodge the bubble, the shorter portion of the bubble in the daughter branch returned and the rear meniscus bulged such that the bubble lodged in just one branch (Fig. 6.18c). Acoustic droplet vaporization is a fast process (of the order of microseconds), and the phase change of droplets to bubbles and their subsequent rapid expansion in a confined space may potentially result in sufficiently large normal and shear stresses to rupture blood vessels and damage the vessel endothelium. The acoustic vaporization procedure was numerically investigated by Ye and Bull (2004), in the case of a rigid-walled tube, in order to assess the risk of vessel damage during gas embolotherapy. They demonstrated that, for a given initial bubble size, the maximum pressure at the vessel wall increases with the initial bubble pressure, and that the maximum occurs early in the bubble expansion. The vessel wall is therefore

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Fig. 6.18 Various states of bubble lodging in a bifurcation model. Reproduced with permission from Calderon et al. (2006). © American Institute of Physics

exposed to the highest risk of rupture at the beginning of the bubble expansion. The authors also noted that the initial bubble size has the largest effect on shear stress. To minimize the potential of damaging the endothelium, smaller bubbles relative to the diameter of the blood vessel are favored. The overall magnitude of the shear stress is much less than that of the pressure. In a subsequent study, Ye and Bull (2006) indicated that wall flexibility can significantly influence the wall stresses that result from acoustic vaporization of intravascular perfluorocarbon droplets, and suggests that acoustic activation of droplets in larger more flexible vessels may be less likely to damage or rupture vessels than activation in smaller and stiffer vessels.

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Chapter 7

Nanocavitation for Cell Surgery

In the previous chapters we discussed the dynamic behaviour of cavitation bubbles with micrometer-order sizes. These bubbles are generated during picosecond or nanosecond laser surgery as well as in the diagnostic and therapeutic applications of ultrasound in the cardiovascular system or in the hydrodynamic cavitation. In the search for new methods of creating very localized effects with minimal collateral damage, femtosecond laser ablation is emerging as an exquisite tool to perform non-invasive, submicrometer-sized surgeries in biological cells. One important application is the separation of individual cells or other small amounts of biomaterial from heterogeneous tissue samples for subsequent genomic or proteomic analysis. Other applications of cell surgery include selective disruption of individual chromosomes and organelles, ablative severing of cytoskeletal fibers, and membrane puncture for transfection. However, the most important application of femtosecond laser ablation is the treatment of cancer at the level of single cells. At present, the most common cancer treatment methods are based on chemotherapy, radiation therapy, or surgery, all of which can be successful, but have substantial disadvantages. Chemotherapy often induces severe side effects and can cause damage to healthy cells, while radiation is only useful on localized, well-defined tumors. Surgical removal also requires that the tumor be well localized, and is often impossible if the tumor is surrounded by sensitive tissues such as the brain. Hyperthermic treatment has been tried in many forms, but conventional techniques tend to cause substantial damage to surrounding tissues. Nanoparticle-based techniques have the potential to offer many advantages over more conventional forms of cancer treatment. Modern nanotechnology offers the possibility of materials that selectively bind to particular types of cancer cells, sensitizing them to light without affecting surrounding healthy tissues. In this chapter we shall focus on the dynamics of cavitation nanobubbles generated during micromanipulation of biological cells and plasmonic photothermal therapy of cancer cells.

E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_7,

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7.1 Cavitation Induced by Femtosecond Laser Pulses The cavitation bubbles from single ultra-short laser pulses are produced by thermoelastically induced tensile stress. Because the region subjected to large tensile stress amplitudes is very small, the presence of inhomogeneous nuclei that could facilitate bubble formation is unlikely, and only the tensile strength of the liquid must be considered in order to estimate the bubble formation threshold. This is in contrast to simulations of heterogeneous cavitation where pre-existing gas nuclei interact with a time-varying pressure wave (see, for example, Paltauf and Schmidt-Kloiber 1996). Extensive investigations on the dynamics of cavitation bubbles generated by femtosecond laser pulses have been conducted by Vogel and co-workers (2005, 2008).

7.1.1 Numerical Simulations In femtosecond optical breakdown, laser pulse duration and thermalization time of the energy of the free electrons are much shorter than the acoustic transit time from the center of the focus to its periphery. Therefore, no acoustic relaxation is possible during the thermalization time, and the thermo-elastic stresses caused by the temperature rise stay confined in the focal volume, leading to a maximum pressure rise (Vogel and Venugopalan 2003; Paltauf and Dyer 2003). Conservation of momentum requires that the stress wave emitted from the focal volume must contain both compressive and tensile components such that the integral of the stress over time vanishes. In biological cells, the tensile stress wave will cause the formation of a cavitation bubble when the rupture strength of the liquid is exceeded. For cell surgery, in the single pulse regime, the threshold for bubble formation defines the onset of disruptive mechanisms contributing to dissection. To determine the evolution of the thermo-elastic stress distribution in the vicinity of the laser focus, Vogel et al. (2005) solved the three-dimensional thermo-elastic wave equation arising from the temperature distribution at the end of a single femtosecond laser pulse. This temperature distribution is characterized by Tmax , the temperature in the center of the focal volume. From this temperature distribution the initial thermo-elastic pressure was calculated using T2 (r)

p(r) =

β(T) dT, K(T)

(7.1)

T1

where T1 is the temperature before the laser pulse, T2 (r) the temperature of the plasma after the laser pulse, which depends on the location within the focal volume, β the thermal expansion coefficient, and K the liquid compressibility. The timeand space-dependent pressure distribution p(r, t) due to the relaxation of the initial

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Cavitation Induced by Femtosecond Laser Pulses

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thermoelastic pressure was calculated using a k-space (spatial frequency) domain propagation model (Köstli et al. 2001). They used the crossing of the kinetic spinodal as defined by Kiselev (1999) as threshold criterion for bubble formation. In the case of water, for a temperature of 20◦ C, this occurs at T= 132◦ C, and p = –71.5 MPa. For superthreshold pulse energies, the size of the bubble nucleus was identified with the extent of the region in which the negative pressure exceeds the kinetic spinodal limit. The initial radius of a spherical bubble with the same volume was taken as the starting nucleus for the cavitation bubble. As driving force for the expansion only the negative part of the time-dependent stress in the center of the focal volume was considered, because the nucleus does not exist before the tensile stress arrives. The temporal evolution of the bubble can then be calculated by using a mathematical model of bubble dynamics (see Chap. 3), considering the tensile stress around the bubble and the vapour pressure inside the bubble as driving forces for the bubble expansion. In the numerical calculations, Vogel et al. (2005) considered two limiting cases depending on the relative size of the bubble and focal volume of the laser. When the size of the bubble is much smaller than the focal volume, the bubble initially expands adiabatically until the average temperature of the bubble content has fallen to the temperature of the liquid at the nucleus wall. Afterwards, heat flow from the surrounding liquid maintains the temperature of the bubble content at the same level as that of the surrounding liquid. The bubble pressure thus equals the equilibrium vapour pressure corresponding to the temperature at the nucleus wall. When the size of the nucleus becomes comparable to the size of the focal volume, the bubble expands more rapidly. The heated liquid shell surrounding the bubble is rapidly thinned, which leads to an accelerated heat dissipation into adjacent liquid. The heat flow into the bubble is assumed to be small compared with the amount of heat contained in the material within the bubble nucleus and, therefore, the entire bubble dynamics is modeled as an adiabatic expansion. Examples of the temporal evolution of the bubble radius after a temperature rise from room temperature to 200◦ C, for the case of water, are given in Fig. 7.1. The most prominent feature of the transient bubbles produced close to the threshold of femtosecond optical breakdown is their small size and short lifetime. The bubble radius amounts to only about 200 nm in water, and will be even smaller in a viscoelastic medium such as the cytoplasm. This makes a dissection mechanism associated with bubble formation compatible with intracellular nanosurgery, in contrast to nanosecond optical breakdown where the minimum bubble radius in water observed for a numerical aperture NA = 0.9 was Rmax = 45 μm (Venugopalan et al. 2002). We conclude this section with the observation that during high-repetition-rate pulse series accumulative thermal effects and chemical dissociation of biomolecules come into play that can produce long-lasting bubbles that are easily observable under the microscope (König et al. 2002; Supatto et al. 2005). Dissociation of biomolecules may provide inhomogeneous nuclei that lower the bubble-formation threshold below the superheat limit defined by the kinetic spinodal.

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Fig. 7.1 Radius-time curve of the cavitation bubble produced by a single femtosecond laser pulse focused at a numerical aperture NA = 1.3 that leads to a peak temperature of Tmax = 200◦ C at the focus center. The radius of the bubble nucleus is R0 = 91.1 nm. The temperature at the wall of the nucleus is Twall = 145◦ C and the mean temperature averaged over all volume elements within the bubble nucleus is Tmean = 168◦ C. The radius versus time curve in (a) was calculated under the assumption that the vapour pressure within the bubble is given by the mean temperature within the nucleus and decays due to heat diffusion (the size of the bubble is much smaller than the focal volume). The curve in (b) was calculated assuming that the vapour pressure drops adiabatically during bubble expansion (the size of the nucleus is comparable to the size of the focal volume). In this figure τ represents the oscillation time of the bubble. Reproduced with permission from Vogel et al. (2005). © Springer Science + Business Media

7.1.2 Experimental Results Time resolved investigations of the effects of transient femtosecond laser induced cavitation bubbles on biological cells are not yet available. Even in the case of water the experimental work is difficult due to the short lifetime of the bubble and small bubble radius. These bubbles can only be detected by very fast measurement schemes. Vogel et al. (2008) investigated the effect of laser pulse energy, EL , on the maximum bubble radius, Rmax , for near ultraviolet, visible and infrared wavelengths. They focused either the fundamental wavelength (1,040 nm) or the 2nd or 3rd harmonic (520 or 347 nm) of an amplified Yb:glass laser with a pulse width of 340 fs through long-distance water immersion objectives built into the wall of a waterfilled cell. Figure 7.2 shows the nanocavitation range for all investigated parameters. The maximum bubble radius increases with increasing the laser wavelength and decreasing the numerical aperture, NA. A major conclusion of their study is that near ultraviolet wavelengths are best suited for nanosurgery because the maximum bubble radius is smallest at threshold (around 200 nm) and increases most slowly with laser pulse energy. The Rmax (EL ) dependence is generally quite strong, and for EL = 2Eth , where Eth is the threshold energy for bubble formation, Rmax already resembles the size of a biological cell. This study also demonstrates that, for sufficiently large pulse energies, bubble expansion can cause effects far beyond the focal volume, which lead to cell death. To avoid unwanted collateral effects, irradiances should be used that are only slightly above the bubble formation threshold.

7.2

Cavitation During Plasmonic Photothermal Therapy

229

Fig. 7.2 Bubble radius Rmax as a function of dimensionless pulse energy EL /Eth for different wavelengths at numerical apertures NA = 0.9 (•) and NA = 0.8 (). Reproduced with permission from Vogel et al. (2008). © American Physical Society

7.2 Cavitation During Plasmonic Photothermal Therapy Historically, the first method used for tumor therapy is the photodynamic therapy, also known as photochemotherapy (Wilson 1986; Henderson and Dougherty 1992). This method involves cell destruction caused by means of toxic singlet oxygen and other free radicals that are produced from a sequence of photochemical and photobiological processes. These processes are induced by the reaction of a photosensitizer with tissue oxygen upon exposure to a specific wavelength of light in the visible or near-infrared region. Although many chemicals have been reported for photochemical therapy, porphyrin-based sensitizers (Moan 1986; Vicente 2001) lead the role in clinical applications because of their preferential retention in cancer tissues and due to the high quantum yields of singlet oxygen produced. Porphyrin-based therapy can only be used for tumors on or just under the skin or on the lining of internal organs or cavities because it absorbs light shorter than 640 nm in wavelength. For deep-seated tumors, second generation sensitizers, which have absorbance in the near-infrared region, such as core-modified porphyrins (Stilts et al. 2000), chlorins (Spikes 1990), or naphthalocyanine (Bonnett 1995), have been introduced. A wellknown limitation of photodynamic therapy is that it requires tissue oxygenation for production of singlet oxygen as a toxic agent, while malignant tissues are usually hypoxic (Wilson 2002). Also, due to nonspecific accumulation of the photochemical agents in normal cells, photodynamic therapy may cause toxic side-effects. An alternative to photodynamic therapy is the photothermal therapy in which photothermal agents are employed to induce local heating of cellular structures (Anderson and Parrish 1983; Jori and Spikes 1990). Tumors are selectively

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destroyed in this case because of their reduced heat tolerance compared to normal tissue, which is due to their poor blood supply. Hyperthermia causes irreversible cell damage by loosening cell membranes and denaturing proteins. When the photothermal therapy agents absorb light, electrons make transitions from the ground state to the excited state. The electronic excitation energy subsequently relaxes through non-radiative decay channels. This results in the increase in the kinetic energy leading to the overheating of the local environment around the light absorbing species. The photoabsorbing agents can be natural chromophores in the tissue (Morelli et al. 1986) or externally added dye molecules such as indocyanine green (Chen et al. 1995) or naphthalocyanines (Jori et al. 1996). The choice of the exogenous photothermal agents is made on the basis of their strong absorption cross sections and highly efficient light-to-heat conversion. This greatly reduces the amount of laser energy required to achieve the local damage of the diseased cells, rendering the therapy method less invasive. It was recently established that laser-induced local heating of cellular structures using either pulsed or continuous laser radiation and mediated by light-absorbing nanoparticles may provide precisely localized damage that can be limited to single cells (see, for example, West and Halas 2003). Accumulation of light-absorbing

Fig. 7.3 Schematic representation of plasmonic photothermal theray. (a) Cell membrane targeting with nanoparticles, (b) Clusterization of nanoparticles, (c) Laser-induced cavitation bubbles and cell damage. Reproduced with permission from Lapotko et al. (2006a). © John Wiley and Sons

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nanoparticles in relatively transparent cells may enhance their optical absorption up to several orders of magnitude (Oraevsky et al. 2001). This technique is sometimes referred to as plasmonic photothermal therapy. Using this method, near infrared radiation from a laser may be used to penetrate through the skin to a deeper extent because the light will undergo less absorption from tissue chromophore and water. Heat absorbed from the radiation will cause thermal denaturation and coagulation of affected cells. In addition, heating will cause vaporization of surrounding fluid, producing cavitation bubble formation. This sudden formation of bubbles causes mechanical stress on the cells which lead to cell destruction. The method for selective killing of target cells includes two main steps: the delivery of nanoparticles into the cells and laser treatment (Fig. 7.3). The purpose of the first step is to create light absorbing clusters of nanoparticles that can be activated to generate vapour microbubbles that can damage target cells. Desirably, accumulation of nanoparticles occurs only in target cells. Even more desirable is to form clusters of nanoparticles in the target cells in order to reduce the laser fluence thresholds for generation of vapour bubbles. The purpose of the laser treatment is to activate nanoparticle clusters and generate bubbles in target cells using minimal laser fluence, so that normal cells will not be damaged. It has been proposed, in some studies, that clusters of light-absorbing nanoparticles could serve as centers of heat deposition for effective generation of vapour microbubbles that kill individual cells (Lapotko et al. 2006b). Advantage of the clusters relative to single nanoparticles is that clusters allow significant decrease of optical energy required for bubble generation and produce bigger bubbles (Lapotko et al. 2006c).

7.2.1 Nanoparticles and Surface Plasmon Resonance The key idea of plasmonic photothermal therapy is that nanoparticles locally convert optical energy into thermal energy through their unique mechanism of plasmon resonance. When a spherical nanoparticle, with a diameter much smaller than the wavelength of light, is irradiated with an electromagnetic field at a certain frequency, a resonant oscillation of the metal free electrons across the nanoparticle is induced. This oscillation is known as the surface plasmon resonance (Kreibig and Vollmer 1995). The frequency of surface plasmon resonance absorption is dependent on the metal composition, nanoparticle size and shape, and dielectric properties of the surrounding medium. The resonance lies at visible frequencies for the noble metals gold and silver (Link and El-Sayed 2003). Gold is, however, the metal of choice for biomedical applications. The most important factor motivating the use of gold nanoparticles is their facile bioconjugation and biomodification (Katz and Willner 2004). Further, spherical gold colloids can easily be made in a wide range of sizes by facile chemistry involving the reduction of gold ions in solution (Turkevich et al. 1951). The ability of gold nanoparticles to convert strongly absorbed light efficiently into heat can be exploit for the selective photothermal therapy of cancer (Khlebtsov et al. 2006), and bacterial (Zharov and Kim 2006) and viruses infection (Sain et al.

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2006). In vivo photothermal therapy applications require light in the near-infrared region where tissue has the highest transmissivity (Weissleder 2001). For example, the penetration depth of light can be up to a few centimeters in the spectral region 650–900 nm, also known as the biological near-infrared region. While changing the size of gold nanospheres shows limited tunability of the surface plasmon resonance wavelength, changing the nanoparticle shape and composition offers dramatic variation in surface plasmon resonance absorption properties. Several nanostructures, such as silica-gold nanoshells, gold nanorods, and gold nanocages show optical tenability in the near infrared region suitable for in vivo applications. The surface plasmon resonace of gold nanoparticles is followed by the rapid conversion (picosecond time scale) of the absorbed light into heat. Depending on the gold nanoparticle temperature, T, the following effects can occur (Letfullin et al. 2006): • For T < TLV , where TLV is the liquid vaporization temperature, only acoustic waves are generated; • For TLV ≤ T < TNPM , where TNPM is the nanoparticle melting point (for gold ~1,060◦ C), shock wave emission and formation of a cavitation bubble are the main feaures of the interaction (Fig. 7.4); • For TNPM ≤ T < TNPB , where TNPB is the nanoparticle boiling point (for gold ~2,700◦ C), melting of the nanoparticle occurs, while • For T > TNPB , gold vapour around liquid drops are generated. Thus, the therapeutic effect of laser-induced explosion of nanoparticles can be reached owing to several phenomena, such as protein inactivation around hot nanoparticles (Huttmann et al. 2003), generation of acoustic and shock waves (Vogel and Venugopalan 2003), cavitation bubble formation (Zharov et al. 2005a), and interaction with nanoparticle fragments and atoms (Zharov et al. 2003). As noted before, cavitation has the dominant role in the therapeutic effect. The damage

Fig. 7.4 Schematic representation of shock wave emission and cavitation bubble formation following laser irradiation of gold nanoparticles

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of cellular structures depends on nanoparticle parameters (composition, size, and shape), laser parameters (wavelength, fluence, pulse duration), and properties of the surrounding media (viscoelastic properties of cytoplasm and cytoskeleton).

7.2.2 Bubble Dynamics Despite the extensive studies of the photothermal properties of nanoparticles, the generation of photothermal cavitation bubbles around them remains an underrecognized phenomenon. Several experimental and theoretical studies investigated the dynamics cavitation bubbles around laser-heated plasmonic nanoparticles, but they are all restricted to the case of bubbles situated in water. The effect of viscoelastic properties of cytoplasm and cytoskeleton still needs to be investigated. 7.2.2.1 Experimental Studies Studies by Kotaidis and Plech Kotaidis and Plech (2005) were the first to investigate the evolution of cavitation bubbles generated around laser-heated nanoparticles. When gold nanoparticles with 4.5-nm radius, situated in water, were irradiated by 400-nm, 50-fs pulses, bubbles of up to 20-nm radius were observed by means of X-ray scattering techniques. Figure 7.5 illustrates the maximum bubble radius as a function of laser power. The small size of these bubbles, which is one order of magnitude less than for those produced by focused femtosecond laser pulses, is consistent with the fact that the collective action of a large number of nanoparticles is required to produce the desired surgical effect. Kotaidis and Plech (2005) also investigated the temporal evolution of the bubble radius. They used the Rayleigh-Plesset equation (see also Chap. 3) in the form ρ(RR¨ +

3 2 R˙ ) = pi,eq 2

Fig. 7.5 Maximum bubble radius as a function of laser power. The maximum bubble radius was determined using a liquid scattering (LS) and small angle X-ray scattering (SAXS) methods. No bubbles were detected below a fluence of 290 J/m2 . Reproduced with permission from Kotaidis and Plech (2005). © American Institute of Physics

R0 R

3κ − p∞ −

R˙ 2σ − 4η , R R

(7.2)

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where R is the bubble radius, ρ the liquid density, κ the polytropic exponent, and the term with pi,eq describes the pressure evolution inside the bubble for the varying radius. The authors performed the numerical integration of this equation to fit the measured values of the bubble radius as function of time. An example is shown in Fig. 7.6 for a value of the laser power of 1,300 J/m2 . In a subsequent study, Kotaidis et al. (2006) investigated the evolution of cavitation bubbles generated by a 50-ns, 400-nm laser excitation of 9- and 39-nm gold nanoparticles in water. Their results are given in Fig. 7.7 where the bubble volume is plotted versus the laser fluence. An extrapolation of the results reveals a threshold below which no bubble is generated. The threshold value is 300 J/m2 for a particle diameter of 9 nm and 80 J/m2 for a particle diameter of 39 nm.

Fig. 7.6 Bubble radius and pressure transients of the water vapour inside the bubble as calculated from the Rayleigh-Plesset equation together with the measured radii. The first maximum in pressure at 650 ps marks the bubble collapse, the following modulations are only expected for oscillatory bubble motion. The parameters used in the numerical calculations are R0 = 7.2 nm, pi,eq = 0.3 GPa, η = 0.15×10–3 kg/m·s, while the initial values are R = 4.5 nm and R˙ = 90 m/s. Reproduced with permission from Kotaidis and Plech (2005). © American Institute of Physics

Fig. 7.7 Ratio of bubble volume to nanoparticle volume as a function of laser fluence for particle sizes of 39 and 9 nm at the delay of maximum bubble radius (650 ps for 39 nm particles, 300 ps for 9 nm particles). The dashed vertical lines indicate the threshold. Reproduced with permission from Kotaidis et al. (2006). © American Institute of Physics

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Studies by Lapotko Laser-induced generation of cavitation bubbles in water around plasmonic nanoparticles was also studied by Lapotko (2009a) using optical scattering methods. The behaviour of cavitation bubbles was investigated for values of the laser pulse duration, τp , of 0.5 and 10 ns at a wavelength λ = 532 nm. All experiments were performed in a response mode for a single laser pulse. Several types of gold nanoparticles and their clusters were used as plasmonic heat sources: spherical gold nanoparticles with diameters, d, of 30 and 100 nm (with a plasmon resonance peak at 530 nm), gold nanorods with dimensions 14 nm × 45 nm (with plasmon resonance peaks at 532 nm and 750 nm), and silica-gold shells with outer diameters of 60 and 170 nm and with broad extinction spectra. All samples were studied as water suspensions in closed micro-volumes confined to a glass sample chamber with a diameter of 9 mm and variable height from 10 to 1,000 μm. The concentration of nanoparticles was adjusted so as to provide a mean interparticle distance of 8 μm. In addition to single nanoparticles, aggregates of nanoparticles were prepared by adding 40% of acetone and re-suspending the nanoparticles in water. The nanoparticle cluster was defined as the aggregate with the interparticle distance smaller than the nanoparticle size and containing at least several nanoparticles. Both single nanoparticles and their clusters generated bubbles starting from a specific threshold of the laser fluence. For τ p = 0.5 ns, λ = 532 nm, and spherical nanoparticles, this fluence level was almost one order of magnitude lower for nanoparticle clusters (0.088 J/cm2 ) relative to that for single nanoparticles (0.72 J/cm2 ). This is probably because clusters generate bubbles at a much lower initial laser-induced temperature, and therefore the clusterization of nanoparticles significantly improves the efficacy of bubble generation. A 20-fold increase of the laser pulse length caused an increase in the bubble generation threshold fluence in all the nanoparticles and their clusters studied. The bubble threshold ratio for 10 ns/0.5 ns pulses varied in the range from 13 (for single 30-nm nanoparticles) to 24 (for single 100-nm nanoparticles). This means that despite the significantly decreased intensity of the long laser pulse the efficacy of the bubble generation turned out to be relatively small. This could be due to the increased thermal losses in the case of the long pulse. It may also be possible that the bubble scatters the incident long pulse thus decreasing its actual fluence. The increase of the spherical nanoparticle diameter from 30 to 100 nm resulted in a several-fold decrease on the bubble threshold fluence. For example, for τp = 0.5 ns the bubble threshold fluence is about 1.1 J/cm2 for d = 30 nm and only 0.4 J/cm2 for d = 100 nm. Even a stronger effect was observed for gold nanorods (a bubble threshold fluence of about 0.35 J/cm2 for τ p = 0.5 ns) despite the fact that the 532 nm band does not provide maximal optical absorbance for this geometry. Minimal thresholds were achieved with nanoparticle clusters regardless the type of the nanoparticles (spheres and two different types of shells). The increase of the cluster size lowered the bubble generation threshold. Gold 60-nm nanoshell clusters of 500–700 nm yielded the lowest bubble threshold fluence of 12 mJ/cm2 . Therefore, nanoparticle clusters may be considered as the best solution for minimizing the vaporization thresholds fluence and temperature.

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As the bubble oscillation time is much longer than the duration of a short laser pulse, Lapotko also investigated the effect exerted by a pulse train with that made by a single pulse. He exposed single spherical nanoparticles with a diameter of 30 nm to paired short pulses with a variable interval: 1 and 6.5 ns. For an interval of 6.5 ns, two pulses did not influence the threshold fluence levels and the oscillation time of the bubbles. At a shorter interval of 1 ns the bubble generation threshold decreased by a factor of 1.7, and the bubble oscillation time increased by a factor of 1.5 relative to the values obtained with a single pulse. Therefore the pulse train mode may additionally increase the efficacy of the bubble generation and may also lead to a decrease in the initial laser-induced temperature of a nanoparticle. Although it was not possible to directly measure a maximal diameter of bubbles, it can still be estimated by the measuring the bubble oscillation time and using the Rayleigh formula (Eq. (3.17)) to convert the oscillation time into maximum bubble radius. Whereas in the case of single 30-nm nanoparticles the bubble oscillation time is about 18 ns, the corresponding value obtained in the case of clusters of 30nm of nanoparticles is about 420 ns. The diameter of bubbles, generated from single nanoparticles, with a minimal oscillation time (15–20 ns) was estimated to be at about 180 nm. Nanoparticle clusters generate a much bigger cavitation bubble with a maximum diameter of about 4 μm. It is also interesting to note that bubbles with an oscillation time shorter than 15 ns were never detected, irrespective of the type of nanoparticles and laser pulse fluence. In a second experiment, Lapotko (2009b) studied the influence of the laser pulse duration on bubble evolution in uniformly absorbing micro- and macrosamples: single red blood cells and in a solution of hemoglobin. The influence of the pulse duration was similar to that found for gold nanoparticles, although the increase of the bubble threshold fluence for the10-ns pulse was smaller than that in the case of nanoparticles. For the red blood cells it was 8.4 times smaller while for the homogeneous solution of hemoglobin it was 2.9 times smaller. The results obtained by Lapotko (2009a, b) clearly indicate that the mechanism of bubble generation depends on many factors related to the laser pulse duration, nanoparticle size and shape, and nanoparticles aggregation state. An increase in the size of nanoparticles as well as the shortening of the laser pulse duration leads to a significant decrease in the threshold fluence. Clusterization of nanoparticles significantly improves the conditions of bubble generation by decreasing its threshold fluence and increasing its oscillation time, and hence the maximal size of the bubble. Vasiliev et al. (2009) studied the evolution of cavitation bubbles generated by three types of gold nanoparticles: spheres with a diameter of 30 nm (plasmon resonance at 520 nm), gold rods with a diameter of 14 nm and length of 45 nm (transverse plasmon resonance at 530 nm and the main longitudinal plasmon resonance at 750 nm) and shells consisting of the silica core and gold shell with the outer diameter of 170 nm. All nanoparticles were prepared as water suspension at a concentration of 2.5 × 1011 /ml which provide an average inter-particle distance of 1.7 μm. The nanoparticles were irradiated with a 10-ns, 532-nm laser pulse and the detection of the cavitation bubbles in the sample liquid (water) was performed using a photothermal microscope. They found that the energy threshold for

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bubble generation is 8 J/cm2 for gold nanorods, 1.8 J/cm2 for gold nanospheres and 70 mJ/cm2 for gold nanoshells. The latest value is 25 times lower than the threshold obtained for an homogeneous model (without nanoparticles). The authors also noted that, at a fluence of 63 mJ/cm2 , the oscillation time of the bubble is 50 ns for the gold nanorods, 20 ns for gold nanospheres, and 200 ns for the gold nanoshells. According to the Rayleigh formula these values correspond to a maximum bubble diameter of about 500 nm in the case of gold rods, 200 nm in the case of gold spheres, and 2 μm for gold shells. Similar results were reported by Lapotko (2009b). Other Experimental Studies Optical microscope images of the cavitation bubbles generated by laser excitation of gold nanoparticles were presented by Liu et al. (2010). In their study, a colloidal suspension of 20 nm diameter gold nanoparticles was irradiated with a 532-nm solid state laser. The resonance wavelength of the gold colloidal nanoparticle suspension occurs at approximately 520 nm, which is close to the wavelength of laser excitation. Figure 7.8 shows images of a silicon microchannel containing a colloidal nanoparticle suspension before and after 30 s and 2 min exposures, at a laser power of 30 mW·μm−2 . In Fig. 7.8b, c, the bubble is seen to grow from 15 to 60 μm in diameter, respectively. The dark ring surrounding the bubble corresponds to the

Fig. 7.8 Optical microscope images of gold nanoparticle aggregation and bubble formation induced by laser irradiation (a–c). Bubble diameter plotted as a function of time after the laser is turned off (d). Reproduced with permission from Liu et al. (2010). © IOP Publishing Ltd

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optical absorption of the gold nanoparticles, which aggregate at the bubble interface. When the laser is turned off, the bubble shrinks during the first few minutes as the temperature decreases, as shown in Fig. 7.8d. The bubble then remains constant in size once the temperature equilibrates with room temperature. Without gold nanoparticles in the suspension, no bubble formation was observed even for laser exposures of 10 min. Dayton et al. (2001) investigated the oscillations of bubbles with a maximum radius of 1.5 μm that were phagocytosed by leukocytes and stimulated by a rarefaction-first, one-cycle acoustic pulse with 440 ns duration. By means of streak photography and high-speed photography with 100 million frames/s they observed that phagocytosed bubbles expanded about 20–45% less than free microbubbles in response to a single acoustic pulse of the same intensity. The difference is most likely due to the viscoelasticity of the cytoplasm and cytoskeleton. 7.2.2.2 A Mathematical Formulation Egerev et al. (2009) presented a mathematical formulation of the dynamics of a cavitation bubble generated around a spherical gold nanoparticle. They assumed the existence of a threshold value of the incident laser fluence for the generation of the bubble. The critical laser fluence together with the absorption cross section of the nanoparticle determines the initial bubble radius. The critical temperature and pressure of the medium surrounding the bubble have been chosen as a criterion for bubble formation. Consider a spherical nanoparticle of radius Rnp suspended in a liquid of infinite extent and irradiated by a laser pulse with duration τL at a wavelength λ (Fig. 7.9). Neglecting the liquid compressibility, the equation of motion for the bubble radius, Rb , reads as (see Example 3.2)

Fig. 7.9 Schematic representation of a cavitation bubble generated from a spherical nanoparticle

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∞ 3 τrr 3 2 1 2σ pb (t) − − p∞ − dr, Rb R¨ b + R˙ b = 2 ρl R ρl r

(7.3)

Rb

with pb (t) =

kTb (t) a − 2 , Vb (t) − b Vb (t)

(7.4)

where kB is the Boltzmann constant, a and b are the constants of the van der Waals equation of state, and Tb (t) and Vb (t) are defined as Tb (t) = Tcl (t)

Vb (t) − b Vcw − b

Rg /cv ,

(7.5)

and Vb (t) =

4 3 π Rb − R3np , 3

(7.6)

where Rg is the universal gas constant and cv is the specific heat. The initial values of the temperature and pressure inside the bubble are those at the critical point of the liquid surrounding the nanoparticle, Tcl and pcl , respectively. The initial value of the bubble radius, Rb,0 , is obtained by assuming that the excessive light energy absorbed by the nanoparticle during the laser pulse is spent on boiling at the critical point. The amount of evaporated water can be estimated as

mb = 4π R3b,0 − R3np ρcl /3 = (F − Fc ) σabs /Ecl ,

(7.7)

and thus 2 Rb,0 = 3

3 4πρcl

(F − Fc ) σabs + R3np . Ecl

(7.8)

In the above equations, Ecl is the internal energy of the liquid at the critical point, ρ cl is the critical density of the liquid, F is the laser fluence, and Fc is the critical laser fluence required to heat the nanoparticle to the boiling temperature Tboil , which in the case of short laser pulses (χl τL << R2np , where χ l is the thermal diffusivity of the liquid) is given by (Egerev et al. 2009) Fc = Vnp cnp ρnp Tboil /σabs .

(7.9)

For long laser pulses (χl τL >> R2np ), the expression for the critical laser fluence becomes (Egerev et al. 2009)

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Fc = 4π Rnp cl ρl χl τL Tboil /σabs .

(7.10)

Here, Vnp , cnp , and ρnp are the volume, specific heat capacity, and density of the nanoparticle. The term σabs represents the absorption cross section given by ( (2 ( (2

2π (2n + 1) (an ( Cn + (bn ( (n + 1)Cn−1 + nCn+1 , 2 k0 |ε| n=1 ∞

σabs =

(7.11)

with Cn = Im

-√ √ . √ ε∗ j n k0 Rnp ε jn−1 k0 Rnp ε∗ ,

(7.12)

where an and bn are the Mie coefficients for the transmitted field, k0 = 2π/λ is the wave number, jn is the Riccati-Bessel function, and the superscript ∗ indicates the operation of complex conjugation. The mathematical formulation proposed by Egerev et al. (2009) requires the knowledge of the optical absorption cross section for the determination of the initial bubble radius. It is well known that the optical absorption and scattering properties of gold nanoparticles can be tuned by changing their size and shape (Jain et al. 2006). For example, gold nanospheres with a diameter of 20 nm show essentially only surface plasmon enhanced absorption with negligible scattering (Jain et al. 2006). However, when the nanoparticle diameter is increased from 20 to 80 nm, the relative contribution of surface plasmon scattering to the total extinction of the nanoparticle increases. Thus, larger nanoparticles are more suitable for light-scattering-based applications. Calculations show that, for gold nanoparticles irradiated at λ = 532 nm, the maximum absorption corresponds to the nanoparticles with radius of 10–40 nm (Jain et al. 2006). This is the optimal size of a gold nanosphere to achieve maximum energy absorption per unit volume for the specific incident laser wavelength. For a gold nanosphere with a radius of 10 nm the calculated optical cross-section equals the geometrical cross-section. The calculated optical cross-section increases with the diameter of the nanospheres and is about three times larger than the geometrical cross-section for a nanoparticle radius of 40 nm (Jain et al. 2006). Another interesting property of gold nanoparticle surface plasmon resonance is its sensitivity to the local refractive index or dielectric constant of the environment surrounding the nanoparticle surface. The nanosphere plasmon resonance shifts to higher wavelengths with increasing refractive index of the surrounding medium (Lee and El-Sayed 2006). The nanoparticle surface plasmon resonance can also be red-shifted by the self-assembly or aggregation of nanoparticles (Sönnichsen et al. 2005). Oraevsky (2008) has indicated that for a nanoparticle inside a bubble irradiated at the wavelength close to its plasmon resonance optical absorption, the absorption cross section can be approximately estimated as equal to its geometric cross section.

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It is also worth noting here that, for very short laser pulses, the critical laser fluence is independent of the pulse shape. Furthermore, for a specific value of the pulse duration there is an optimal particle size, which has the minimal value of Fc . For large particles, heat exchange with the surroundings is negligible, and Fc ∝ Rnp . In contrast, for small particles (χl τL >> R2np ), Fc ∝ R−1 np (Egerev et al. 2009).

7.2.3 Biological Effects of Cavitation The first thorough study using pulsed laser radiation and gold nanospheres was performed in 2003 by Lin and co-workers for selective and highly localized photothermolysis of targeted lumphocytes cells (Pitsillides et al. 2003). Lumphocytes incubated with gold nanoparticles conjugated to antibodies were exposed to nanosecond laser pulses (Q-switched Nd:YAG laser, 565 nm wavelength, 20 ns duration) showed cell death with 100 laser pulses at an energy of 0.5 J/cm2 . Adjacent cells just a few micrometers away without nanoparticles remained viable. Their numerical calculations showed that the peak temperature lasting for nanoseconds under a single pulse exceeds 2,000 K at a fluence of 0.5 J/cm2 with a heat fluid layer of 15 nm. The cell death was attributed mainly to the cavitation damage induced by the generated cavitation bubbles around the nanoparticles. In the same year, Zharov et al. (2003) performed similar studies on the photothermal destruction of K562 cancer cells. They further detected the laserinduced bubbles and studied their dynamics during the treatment using a pump– probe photothermal imaging technique. Later they demonstrated the technique in vitro on the treatment of some other type of cancer cells such as cervical and breast cancer using the laser induced-bubbles under nanosecond laser pulses (Zharov et al. 2004, 2005b, c). Recent work has demonstrated the treatment modality for in vivo tumor ablation in a rat (Hleb et al. 2008). Intracelullar bubble formation resulted in individual tumor cell damage. The formation of cavitation bubbles around nanoparticles also caused physical damage to the Staphylococcus aureus bacterium as confirmed by the images presented by Zharov et al. (2006) (Figs. 7.10 and 7.11). At relatively low laser energies, they observed a very slight penetration of nanoparticles in the cell wall (Fig. 7.11b) compared to the control without laser exposure (Fig. 7.11a). Higher laser energies, or the formation of nanoparticle clusters, led to a deeper penetration of nanoparticles inside the bacterial wall (Fig. 7.11c). High laser energy and/or formation of nanoclusters coupled with multi-pulse exposure produced local cell-wall damage (Fig. 7.11d) and finally complete bacterial disintegration (Fig. 7.11e, shows fragmented bacteria). These data demonstrate that, despite the relatively high thickness and density of the bacterial cell wall, bubble formation around nanoparticles may potentially cause irreparable damage to bacteria. The photothermolysis of living EMT-6 breast tumor cells triggered by gold nanorods was investigated by Chen et al. (2010). In the absence of gold nanoparticles, the cells survived under the excited energy fluence of 93 mJ/cm2 . However, cell mortality was observed at 113 mJ/cm2 energy fluence. Results of

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Fig. 7.10 Images of Staphylococcus aureus with attached gold nanoparticles: (a) phase contrast image; (b) photothermal image of bacteria alone; (c) photothermal images of bacteria with 40-nm gold particles irradiated at a laser fluence of 0.4 J/cm2 ; and (d) photothermal images of bacteria with 40-nm gold particles irradiated at a laser fluence of 2 J/cm2 . Dashed lines represent the bacterial boundary in (c) and (d). Arrows in (d) indicate photothermal images of single nanoparticles, whereas the arrowhead shows a bubble around one nanocluster. Reproduced with permission from Zharov et al. (2006). © Elsevier B.V.

Fig. 7.11 Images of Staphylococcus aureus conjugated with gold nanoparticles before (a) and after (b–e) multilaser exposure of 100 pulses, wavelength of 532 nm, and pulse duration of 12 ns at a different conditions: (b) laser fluence of 0.5 J/cm2 and no clusters; (c) laser fluence of 0.5 J/cm2 with clustered nanoparticles; and (d) laser fluence of 3 J/cm2 at one and several (e) nanocluster numbers. A dashed line represents the bacterial boundary in (e). Arrows in (b) and (c) indicate penetration of nanoparticles into the wall, and in (d), arrows indicate local cell-wall damage. Reproduced with permission from Zharov et al. (2006). © Elsevier B.V.

the cells with gold nanoparticles, under excitation at energy fluences of 113 and 93 mJ/cm2 , are shown in the series of images in Fig. 7.12; the images were taken within a period of 60 s. Upon reaching an energy fluence of 113 mJ/cm2 , the whole cell was seriously destroyed (Fig. 7.12a–d). At an energy fluence of 93 mJ/cm2 , a discernible internal explosion phenomenon occurred upon excitation (Fig. 7.12e–h). Meanwhile, the formation of characteristic cavities (shadows indicated by arrows) was especially pronounced at nanoparticle cluster locations (cluster size between 2 and 3 μm). The diameter of the cavities can reach as large as 10 μm. The results showed that localized photothermal effect of gold nanoparticles was large enough to trigger a considerable explosion, resulting in the formation of cavitation bubbles inside cells. These bubbles are responsible for the perforation or sudden rupture of plasma membrane. Their study also indicates that the energy threshold for cell therapy depends significantly on the number of nanoparticles taken up per cell. For an ingested gold nanoparticle cluster quantity N ∼ 10–30 per cell, it was found that energy fluences larger than 93 mJ/cm2 led to effective cell destruction within

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Fig. 7.12 Photothermolysis of the EMT-6 tumor cell triggered by gold nanoparticles under different energy fluences. (a–d) 113 mJ/cm2 ; (e–h) 93 mJ/cm2 . The shadows indicated by arrows are attributed to the formation of transient cavitation bubbles. The gold nanoparticles inside the cell can be seen in (a) and (e). Reproduced with permission from Chen et al. (2010). © Elsevier B.V.

a very short period. As for a lower energy level (18 mJ/cm2 ) with N ∼ 60–100, a non-instant, but progressive cell deterioration, was observed. The photothermolysis of lung carcinoma cells (A549) triggered by gold nanospheres with a diameter of 50 nm was investigated by Lukianova-Hleb et al. (2010). They found that at laser fluences below the bubble generation threshold, the nanoparticles in cells still were significantly heated by the laser pulse but did not cause detectable damage to the cells. Also, the exposure of the cell to 16 pump laser pulses (at 15 Hz frequency), instead of a single pulse, did not influence the cell viability and the level of the damage threshold fluence, which suggests that the cell damage results from a single event rather than from an accumulative effect of the sequence of the bubbles. They concluded that the bubble damage mechanism is mechanical and non-thermal: a single laser pulse induces an expanding bubble that disrupts the cellular cytoskeleton and plasma membrane causing visible membrane blebs. Blebbing of the plasma membrane for various cell types was also observed by Tong et al. (2007) (Fig. 7.13). The authors noted that bleb formation could not be the direct product of cavitation, as the rates of growth were several orders of magnitude slower than the timescale for microbubble expansion. They hypothesized that the blebbing response was due to the disruption of actin filaments, which form a dense three dimensional network beneath the cell membrane to provide mechanical support and sustain cell shape. However, an important conclusion of their study is that the cell death is attributed to the disruption of the plasma membrane as a

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Fig. 7.13 Photothermolysis mediated by gold nanorods with longitudinal plasmon resonances centered at 765 nm. Cells were irradiated at 765 nm using a Ti:sapphire laser which could be switched between fs-pulsed and cw mode. (a, b) Cells with membrane-bound gold nanorods exposed to cw near infrared laser irradiation experienced membrane perforation and blebbing at 6 mW power. The loss of membrane integrity was indicated by EB staining (light grey, yellow online). (c, d) Cells with internalized gold nanorods required 60 mW to produce a similar level of response. (e, f) Gold nanorods internalized in KB cells labeled by folate-Bodipy (lighter grey, green online) were exposed to laser irradiation at 60 mW, resulting in both membrane blebbing and disappearance of the gold nanorods. (g, h) NIH-3T3 cells were unresponsive to gold nanorods, and did not suffer photoinduced damage upon 60 mW laser irradiation. (i, j) Cells with membranebound gold nanorods exposed to fs-pulsed laser irradiation produced membrane blebbing at 0.75 mW. (k, l) Cells with internalized gold nanorods remained viable after fs-pulsed irradiation at 4.50 mW, as indicated by a strong calcein signal (grey, green online). Reproduced with permission from Tong et al. (2007). © Wiley-VCH Verlag GmbH & Co. KGaA

consequence of gold nanoparticles mediated cavitation. Membrane perforation led to an influx of extracellular Ca2+ followed by degradation of the actin network, producing a dramatic blebbing response. Lin et al. investigated the thresholds for cell death produced by cavitation induced around absorbing microparticles irradiated by nanosecond laser pulses (Lin et al. 1990; Leszczynski et al. 2001). They observed that an energy of 3 nJ absorbed by a single particle of 1-μm diameter produced sufficiently strong cavitation to kill a trabecular meshwork cell after irradiation with a single laser pulse. Pulses with 1-nJ absorbed energy produced lethality after several exposures (Lin et al. 1990). Viability was lost even when no morphological damage was apparent immediately after the collapse of a transient bubble with a maximum radius of about 6 μm.

References

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It is clear that the laser-induced cavitation bubbles represent an important damaging factor in plasmonic photothermal therapy. The generation of cavitation bubbles may occur simultaneously in several locations inside the cell volume. The most effective cell killing occurs when the nanoparticles are located on or inside the cell membrane to provide membrane rupture. When the bubble reaches the size comparable to that of the cell it would definitely damage cellular membrane causing necrosis and lysis. Smaller bubbles may also induce apoptosis without rupturing the membrane. However, nanometer-sized cavitation bubbles that emerge around nanoparticles located at a distance from the cell membrane do not damage the cells due to their limited diameter of less than a micrometer. The threshold of pulsed laser interaction with clusters of nanoparticles is significantly lower than that for a single nanoparticle. Superheating of the nanoparticle clusters generates a much larger cavitation bubble capable of damaging even large cells. Thus, the creation of nanoclusters, consisting of many small nanoparticles, on the cell membrane or inside the cell is one potential way to overcome the limitations of using single nanoparticles which are due to the lower efficiency of bubble formation in the case of small nanoparticles or to the difficulties with their selective delivery to the target in the case of large nanoparticles. More effective bubble formation in a cluster of gold nanoparticles is associated with optical and thermal amplification effects and, especially, with overlapping nanobubbles from a single nanoparticle as separate nucleation centers or the generation of one large bubble around a gold nanoparticles cluster as a single nucleation center due to rapid heat redistribution between very closely located gold nanoparticles within a cluster (Zharov et al. 2005a). An alternative damage mechanism that should be considered is the mechanical destruction of cell structures by high tensile stresses. The numerical results presented by Volkov et al. (2007) indicate that the pressure waves emitted from the nanoparticles do not have any significant tensile stress component. However, particle reflection of the compressive pressure wave from internal cellular structures may result in the generation of the tensile stresses and associated cell damage. They estimated that, for a laser pulse duration of 200 fs, the maximum amplitude of the tensile stress exceeds 1 MPa for particles larger than 25 nm and laser fluences larger than 20 J/m2 . Additionally, the effective therapeutic effect for cancer cell killing may be achieved owing to nonlinear phenomena that accompany the thermal explosion of the gold nanoparticles, such as the generation of nanoparticle explosion products with high kinetic energy as well as strong shock waves with supersonic expansion in the cell volume.

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Volkov, A.N., Sevilla, C., Zhigilev, L.V. 2007 Numerical modeling of short pulse laser interaction with Au nanoparticle surrounded by water. Appl. Surf. Sci. 253, 6394–6399. Vogel, A., Venugopalan, V. 2003 Mechanisms of pulsed laser ablation of biological tissues. Chem. Rev. 103, 577–644. Vogel, A., Noack, J. Hüttman, G., Paltauf, G. 2005 Mechanisms of femtosecond laser nanosurgery of cells and tissues. Appl. Phys. B 81, 1015–1047. Vogel, A., Linz, N., Freiank, S., Paltauf, G. 2008 Femtosecond-laser-induced nanocavitation in water: implications for optical breakdown threshold and cell surgery. Phys. Rev. Lett. 100, 038102. Weissleder, R. 2001 A clearer vision for in vivo imaging. Nat. Biotechnol. 19, 316–317. West, J.L., Halas, N.J. 2003 Engineered nanomaterials for biophotonics applications: improving sensing, imaging, and therapeutics. Annu. Rev. Biomed. Eng. 5, 285–292. Wilson, B.C. 1986 The physics of photodynamic therapy. Phys. Med. Biol. 31, 327–360. Wilson, B.C. 2002 Photodynamic therapy for cancer: principles. Can. J. Gastroenterol. 16, 393–396. Zharov, V.P., Kim, J.W. 2006 Amplified laser-nanocluster interaction in DNA, viruses, bacteria, aand cancer cells: potential for nanodiagnostic and nanotherapy. Lasers Surg. Med. 18, 16–17. Zharov, V.P., Galitovsky, V., Viegas, M. 2003 Photothermal detection of local thermal effects during selective nanophotothermolysis. Appl. Phys. Lett. 83, 4897–4899. Zharov, V.P., Galitovskaya, E., Viegas, M. 2004 Photothermal guidance for selective photothermolysis with nanoparticles. Proc. SPIE 5319, 291–300. Zharov, V.P., Letfullin, R.R., Galitovskaya, E.N. 2005a Microbubbles-overlaping mode for laser killing of cancer cells with absorbing nanoparticle clusters. J. Phys. D Appl. Phys. 38, 2571–2581. Zharov, V.P., Galitovskaya, E.N., Johnson, C., Kelly, T. 2005b Synergistic enhancement of selective nanophotothermolysis with gold nanoclusters: potential for cancer therapy. Lasers Surg. Med. 37, 219–226. Zharov, V.P., Kim, J.W., Curiel, D.T., Everts, M. 2005c Self-assembling nanoclusters in living systems: application for integrated photothermal nanodiagnostics and nanotherapy. Nanomed. Nanotechnol. Biol. Med. 1, 326–345. Zharov, Z.P., Mercer, K.E., Galitovskaya, E.N., Smeltzer, M.S. 2006 Photothermal nanotherapeutics and nanodiagnostics for selective killing of bacteria targeted with gold nanoparticles. Biophys. J. 90, 619–627.

Chapter 8

Cavitation in Other Non-Newtonian Biological Fluids

In the previous chapters we have described the effects of cavitation in the cardiovascular system and cell surgery. There are an increasing number of biomedical contexts where cavitation takes place in other non-Newtonian biological fluids, such as saliva or synovial fluid. In saliva, cavitation occurs during some medical applications of lasers and ultrasound. In synovial liquid, cavitation is responsible for the cracking noise emitted from joints and may also damage the articular cartilage. In this chapter, we provide a qualitative description of cavitation and some of its associated bioeffects encountered in clinical applications. The archival literature in these cases is not as impressive as in the case of blood. Threshold conditions for the onset of cavitation in various biological fluids require more precise definition, preferably mathematical models underpinned by an extensive body of experimental evidence. The conditions associated with the onset of morphological damage also merit a more precise description. Nevertheless, we invite the reader to appreciate how cavitational activity can help address some of the present therapeutic challenges in several non-Newtonian biological fluids.

8.1 Cavitation in Saliva In some dentistry applications, an ultrasonically vibrating probe is placed in close proximity to the biological tissue or rigid material. The cavitation induced at the tip of this probe (or around the probe) creates the desired effect when it is placed close to the tissue or rigid material. Cleaning the teeth by dislodging plaque is one of the earliest applications of such an ultrasonic probe in dentistry. Other current applications include passive irrigation of the root canal and orthognathic surgery of the mandible.

8.1.1 Cavitation During Ultrasonic Plaque Removal The old-fashioned technique of plaque removal is the hand instrumentation. In this case, a curette must be placed below the deposit to be effective. When deep calculus approaches the bottom of the pocket, positioning the instrument may damage E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_8,

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the periodontal attachment. In the attempt to create smooth roots, free of any deep accretions, the hand instruments tear into the fragile periodontal ligament and scrape off tooth structure. Unlike the curette, an ultrasonic scaler tip works from the top of the deposit downward, so there is no need to violate the attachment. Thus, ultrasonic instrumentation is now the first choice over hand instrumentation for most patients. The ultrasonic scaler has been used for about 30 years. In most practices it is used primarily for gross removal of supra-gingival calculus. New super-thin tips are now available that fit into deep pockets and small furcal areas where a standard curette is ineffective. Some manufacturers have designed machines with a far wider power range, so they can create effective cavitation at the low power settings needed for sub-gingival use (O’Leary et al. 1997). Ultrasonic scalers are now the preferred method for sub-gingival debridement. Recent research has shown that sub-gingival ultrasonic scaling not only removes calculus as well as traditional hand instrumentation, but that it also kills bacteria and reduces the level of endotoxins. Back in the early 70’s researchers noticed that the ultrasonic scaler cleaning ability dropped significantly when the water flow was interrupted. They speculated that this was due to the irrigating effect of the water (Clark 1969). Later research indicated that no matter which tip was used, and no matter at what angle it touched the tooth, the amount of plaque-free surface increased by 500–800% when the water was turned on (Walmsley et al. 1988). They noted that the dry tip removed plaque only where it contacted the tooth. However, when a water cooled tip was used they observed that surfaces as much as a half millimeter away from the tip were completely plaque-free. The tip’s high-frequency vibrations create cavitation bubbles. When the energized spray from the hand-piece contacts the tooth surface, these bubbles collapse and release short bursts of energy which literally blast the plaque from the surface and tear apart bacterial cell membranes in the process (Walmsley et al. 1988; Lea et al. 2005; Felver et al. 2009). The ultrasound field generated by the scaler is comprised of a series of compressions and rarefactions (regions of high and low pressure) which cause small cavitation nuclei to expand and contract (Fig. 8.1a). Inertial cavitation bubbles oscillate violently and may expand to many times their original size before imploding. The collapse of such a bubble can result in shock

Fig. 8.1 Cavitation and acoustic streaming around an ultrasonic scaler. (a) The oscillating pressure field causes a cavitation nucleus to expand and contract. (b) When the cavitation bubble is located close to the scaler, it may collapse accompanied by the formation of a liquid jet

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waves associated with massive temperatures and pressures. If the bubble collapses or implodes near to the surface of a tooth or a scaler tip then the collapse is asymmetrical resulting in an inrushing jet of liquid targeted at the surface (Fig. 8.1b). This jet of liquid is powerful enough to potentially remove calculus and other materials from the tooth surface (Walmsley et al. 1984, 1988). Furthermore, the force of these jets is enough to visibly roughen the metallic surface of the ultrasonic scaler tip. A clear visualization of the spatial distribution of cavitation bubbles around three scaler tips, observed using sonochemiluminescence from a luminol solution, is given in Fig. 8.2 (Felver et al. 2009). The highest levels of cavitation activity were observed around vibration antinodes close to the bend in each tip. Surprisingly,

Fig. 8.2 Luminol photography of three scaler tips (Piezon miniMaster, Electro Medical Systems). Light regions indicate areas of high cavitation activity, with dark regions indicating little or no activity. Reproduced with permission from Felver et al. (2009). © Elsevier B.V.

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while the displacement amplitude was greatest at the free end of the tip, little to no cavitation was observed at the free end using luminol photography. It is also interesting to note here that cavitation does not occur around powered tooth brushes. Lea et al. (2004) tested five commercial brushes and monitored the formation of the hydroxyl radical that occurs during cavitation bubble collapse. Operating the toothbrushes for periods up to 20 min resulted in no cavitational activity being detected.

8.1.2 Cavitation During Passive Ultrasonic Irrigation of the Root Canal The goal of ultrasonic irrigation of the root canal is to remove pulp tissue and microorganisms from the root canal system as well as smear layer and dentim debris that occur following instrumentation of the root canal (van der Sluis et al. 2007). Passive (non-cutting) ultrasonic irrigation is based on the transmission of energy from an ultrasonically oscillating instrument to an irrigant in the root canal (van der Sluis et al. 2005). After the root canal has been shaped to the master apical file, a small file (or wire) is inserted in the centre of the root canal, as far as the apical region. The root canal is then filled with an irrigant, usually a sodium hypochlorate solution, and the ultrasonically vibrating file activates the irrigant in order to penetrate more easily into the apical part of the root canal system (Krell et al. 1988). The file is driven to operate in transverse mode at frequencies of 25–30 kHz. During passive ultrasonic irrigation, acoustic microstreaming and cavitation can occur which cause a streaming pattern within the root canal from the apical to the coronal (Ahmad et al. 1987; Roy et al. 1994). Because of this microstreaming, more dentine debris can be removed from the root canal compared with syringe delivery of the irrigant (Lea et al. 2004), even from remote places in the root canal (Goodman et al. 1985). A detailed investigation on the behaviour of cavitation bubbles generated during passive ultrasonic irrigation of the root canal was conducted by Roy et al. (1994). They indicated that transient cavitation bubbles only occur when the file can vibrate freely in the canal or when the file touches un-intentionally (or for a short duration) the canal wall (Fig. 8.3) (see also Lumley et al. 1993). Intentional (long duration) contact with the canal wall suppresses the formation of transient cavitation bubbles. The authors also noted that a smooth file with sharp edges and a square cross-section produced significantly more transient cavitation than a normal file. The transient cavitation was visible at the apical end and along the length of the file. When the file came in contact with the canal wall, stable cavitation was affected less than transient cavitation and was mainly seen at the midpoint of the file. A pre-shaped file brought into a curved canal is more likely to produce transient cavitation rather than a straight file. Other researchers claim that cavitation provides only minor benefit in ultrasonic irrigation, or that it does not occur at all (Walmsley 1987; Ahmad et al. 1988; Lumley et al. 1988).

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Fig. 8.3 A glass root canal model showing the file at rest (left) and file in operation displaying cavitation bubbles. Reproduced with permission from Roy et al. (1994). © John Wiley and Sons

Cavitation bubbles can also be used for the delivery of antibacterial nanoparticles into dentinal tubules. Persistent root canal infection has been associated with bacterial presence in the dentinal tubules. Studies have shown that bacteria can penetrate into dentinal tubules, and the depth of penetration varies from 300 to 1,500 μm (Love and Jenkinson 2002). However, the bacteria within the dentinal tubules are inaccessible to the conventional root canal irrigants, medicaments, and sealers because they have limited penetrability into the dentinal tubules. Although the application of ultrasound produces better results compared with syringe irrigation in cleaning and delivering irrigants into the anatomic complexities, ultrasonic irrigation does not debride the root canal system completely. In a very recent study, Shrestha et al. (2009) have indicated that the collapsing cavitation bubbles treatment using high-intensity focused ultrasound can result in a significant penetration up to 1,000 μm of antibacterial nanoparticles into the dentinal tubules. The cavitation bubbles produced using high-intensity focused ultrasound can be used as a potential method to deliver antibacterial nanoparticles into the dentinal tubules to enhance root canal disinfection. The mechanism responsible for the delivery of antibacterial nanoparticles is illustrated in Fig. 8.4 in the case of a spark-generated cavitation bubble. The bubble grows to a maximum size (with maximum radius of 3.3 mm) in a time of 0.46 ms (Fig. 8.4b). The collapse of cavitation bubble (Fig. 8.4c–g) generates a high-speed jet, which moved toward the channel at about 68 m/s. This jet delivers the bead of plaster (with a mass of approximately 6 mg), which was originally placed about 2 mm from the top of the channel, into the whole length of the channel (Fig. 8.4e–g).

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Fig. 8.4 The collapse of a cavitation bubble with a maximum bubble radius of 3.3 mm on top of a tubular channel of 3.3-mm diameter. The time is indicated at the bottom right. The bubble collapses at time t = 0.73 ms with a jet speed of about 68 m/s. The rubber plaster balls are centered initially 2 mm from the top of the channel opening. It can be seen from frames corresponding to times t = 1.3 to t = 5.8 ms that the rubber plaster is pushed by the flow down the entire length of the channel. Reproduced with permission from Shrestha et al. (2009). © Elsevier B.V.

8.1.3 Cavitation During Laser Activated Irrigation of the Root Canal The first laser use in endodontics was reported by Weichman and Johnson (1971) who attempted to seal the apical foramen in vitro by means of a high power-infrared (CO2 ) laser. Although their goal was not achieved, sufficient relevant and interesting data were obtained to encourage further study. Subsequently, attempts have been made to seal the apical foramen using the Nd:YAG laser (Weichman et al. 1972). Although more information regarding this laser interaction with dentine was obtained, the use of the laser in endodontics was not feasible at that time. Since then, many papers on laser applications in dentistry have been published (see, for example, Wigdor et al. 1995 and the references therein). Nevertheless, in dentistry and in

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endodontics in particular, acceptance of this technology by clinicians has remained limited, perhaps partly due to the fact that this technology blurs the border between technical, biological, and dental research. Lasers, such as the Er,Cr:YSGG laser, have been also proposed as an alternative for the conventional approach in cleaning, disinfecting and even shaping of the root canal or as an adjuvant to conventional chemo-mechanical preparation in order to enhance debridement and disinfection (Kimura et al. 2000; Stabholz et al. 2004). The high-speed recordings obtained by Blanken et al. (2009) have demonstrated that vaporization of the liquid inside a root canal model will result in the formation of cavitation bubbles, which expand and implode with secondary cavitation effects. At the beginning of the laser pulse, the energy is absorbed in a thin liquid layer that is instantly heated to boiling temperature at high pressure and turned into vapour. This vapour at high pressure starts to expand at high speed leading to the formation of a cavitation bubble. A free expansion of the bubble laterally is not possible in the root canal model, and hence the liquid is pushed forward and backward in the canal. The forward pressure can be easily observed in the Fig. 8.5 showing an air bubble, present in the canal, being compressed to a flat disk. As the energy source stops, the vapour cools and starts condensing, while the momentum of expansion creates a lower pressure inside the bubble. Liquid surrounding the bubble is accelerated to fill in the gap. Secondary cavitation bubbles are also be induced at irregularities along the root canal wall. The implosion of the primary and secondary bubbles creates microjets in the fluid aimed at the wall with very high forces locally. This mechanism might also contribute to the disruption of cells and the smear layer at the wall.

Fig. 8.5 An air inclusion being compressed when a laser-induced cavitation bubble grows and expands in an artificial root canal. The maximum compression of the air inclusion is visible in the third frame. Reproduced with permission from Blanken et al. (2009). © John Wiley and Sons

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8.1.4 Cavitation During Orthognathic Surgery of the Mandible Ultrasonic devices might be also effective in minimizing the hazard of surgical trauma in maxillofacial surgery. Particularly in elective orthognatic surgery of the mandible protection of the inferior alveolar nerve is important to reduce surgical morbidity. Gruber et al. (2005) have recently presented some preliminary results on using an ultrasonic bone cutting device in bilateral sagittal split osteotomies of the mandible. They noted that the cavitation phenomenon is responsible for the good visibility of the surgical site due to a cleaning effect of the microstream towards the rigid boundary of the surface of the bone. The effect of cavitation on cells of the adjacent tissue such as periosteal cells or bone cells is still not fully understood.

8.2 Cavitation in Synovial Liquid Cavitation in human joints has been linked with the sharp cracking noise emitted from some joints, particularly from the metacarpophalangeal joint (Unsworth et al. 1971). When a synovial joint is distracted, the pressure in the synovial liquid can drop below its vapour pressure (approximately 6,500 Pa), and the fluid evaporates spontaneously forming a bubble in the joint space (Fig. 8.6) (Unsworth et al. 1971).

Fig. 8.6 Roentgenogram of a metacarpophalangeal joint after cracking showing the bubble present in the joint space. Reproduced with permission from Unsworth et al. (1971). © BMJ Publishing Group Ltd

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A very nice visualisation of bubble formation in the metacarpophalangeal joint is also given by Watson et al. (1989). This phenomenon is known as viscous adhesion or tribonucleation (Campbell 1968). It causes bubble formation as a result of the large negative pressure generated by viscous adhesion between surfaces separating in liquid. It occurs when two closely opposed surfaces separated by a thin film of viscous liquid are pulled rapidly apart. Viscosity prevents the liquid from filling the widening gap, resulting in negative pressure. Cavitation may also occur when the articular surfaces are separated through the elastic recoil of the synovial fluid above a critical velocity, causing the synovial fluid to fracture like a solid. A proportion of the cracking noise during cavitation of synovial fluid may therefore be considered as synonymous with the inception of the cavitation (Chen and Israelachvili 1991; Chen et al. 1992). Cavitation is not the only mechanism of all cracking noises emitted from joints. Some sounds are produced by patellofemoral crepitus (a fine crunching noise, usually on bending the knee from standing, which is said to be due the kneecap cartilage rubbing against the underlying cartilage of the femur) or when the plica (a thin wall of fibrous tissue that are extensions of the synovial capsule of the knee) snaps over the end of the femur (Beverland et al. 1986; McCoy et al. 1987). Other studies provide clear evidence that the anatomic source of the cracking sound associated with spinal high-velocity low-amplitude thrust manipulations is associated with cavitation of the synovial fluid (Watson and Mollan 1990; Evans 2002). The audible “crack” is often viewed as signifying a successful manipulation. Several authors suggested that cavitation during in vivo conditions can take two forms: (a) that which produces the familiar cracking noise (called macrocavitation), and (b) microbubble activity that may be occurring because of the bubbles remaining in the synovial fluid after the crack (called microcavitation) (Watson et al. 1989; Unsworth et al. 1971). The existence of gas bubbles in synovial joints (after macrocavitation) has been demonstrated by radiography as a dark, intra-articular radiolucent region since early in the twentieth century (Unsworth et al. 1971; Fuiks and Grayson 1950; Kramer 1990). Damage of the articular cartilage is a possible consequence of cavitation in the synovial liquid. Watson et al. (1989) investigated the effects of cavitation on bovine knee joint articular cartilage. Cavitation was generated using a vibrating tip operating at 20 kHz with a maximum amplitude of 0.127 mm. During the first 20 s of exposure to cavitation, no significant damage of the specimen was observed. The specimen exposed to cavitation for 1 min presented shallow depressions, approximately 20 μm in diameter, covering the surface. After 10 min of exposure to cavitation, the specimen displayed considerable surface disruption with distinct craters that have approximately 100 μm in diameter. Obviously, such large collateral effects are unlikely in an in vivo situation but they emphasize a possible role of cavitation in damaging the articular cartilage. Watson et al. (1989) noted that the cumulative effects of cavitation may be, over a period of time, sufficient to damage the articular cartilage. Although these authors proposed this theory as a cause of direct damage to the joint cartilage, there is, so far, little clinical evidence to support this mechanism. In a recent in vivo study, the effect of extracorporeal shock waves on joint cartilage was evaluated in 24 rabbits

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(Väterlein et al. 2000). It is well known that the combined effects of the shock waves and cavitational collapse induce harmful side effects on the adjacent biological structures, such as that observed in lithotripsy (Delius et al. 1988). However, macroscopical radiological and histological analysis at 0, 3, 12 and 24 weeks after treatment showed no pathological changes in the joint cartilage. Arthroscopic cartilage ablation (see, for example, Smith 1993) is another medical application where cavitation takes place in the synovial liquid. The dynamics of cavitation bubbles generated by a pulsed holmium laser radiation (wavelength 2.12 mm, pulse duration between 100 and 1,000 ps), transmitted through an optical fiber, and their impact on medical laser use for cartilage ablation have been investigated by Asshauer (1996). Shock waves were observed at the bubble collapse several hundred microseconds after the start of the laser pulse and peak pressures up to several kilobars were measured. The observed complex bubble dynamics and pressure transient generation was explained by a two stage model: in the first stage of the bubble formation process, a water volume at the fiber tip is superheated by the laser radiation, until an explosive vaporization induces an isotropic vapour bubble expansion. In the second stage, a quasi-continuous ablation through the bubble takes place. The relative importance of the second stage increases for higher fluences and longer pulse durations, perturbing the initial nearly spherical symmetry of the bubbles. The angle of incidence of the laser radiation was identified as an important additional parameter for cartilage ablation. It was shown that shallow angles of incidence reduce pressure transient amplitudes as well as thermal side effects of cartilage ablation. Ultrasonically induced cavitation may also have a clinical benefit to control synovial proliferation and inflammation or some other disorders of joints. Nakaya et al. (2005) investigated the effect of a microbubble-enhanced ultrasound treatment on the delivery of methotrexate (an antimetabolite and antifolate drug used in treatment of cancer and autoimmune diseases) into synovial cells. They found that the methotrexate concentration in synovial tissue was significantly higher in the presence microbubbles while the synovial inflamation was less prominent. Saito et al. (2007) reported that the expression of plasmid DNA and small interfering RNA in the synovium was significantly enhanced by ultrasound in combination with microbubbles. In a more recent study, Nakamura et al. (2008) observed that ultrasound treatment in combination with microbubbles increased cellular uptake of enzymes (histone deacetylase) into human rheumatoid synovial cells.

8.3 Cavitation in Aqueous Humor An interesting application of cavitation in non-Newtonian fluids is encountered in ultrasound phacoemulsification. In cataract surgery, the turbid nucleus and cortex of the lens of the eye are removed and an artificial lens is implanted into the capsular bag to restore vision. After the cornea and the anterior lens capsule are surgically opened, an ultrasound tip similar to a Mason-horn, operating at frequencies between

8.3

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259

20 and 40 kHz, is used to emulsify or fragment the lens nucleus (whether it is emulsification or fragmentation depends on the hardness of the nucleus). The fragments are then removed by means of an irrigation suction system that is integrated into the phaco-tip. After the lens capsule is cleaned, the artificial lens is implanted and the eye is closed again. Manual extraction of the nucleus demands a large wound of approximately 9 mm chord length. Phacoemulsification can be done through a smaller wound of approximately 3 mm length. Wound length is also governed by the size of the intraocular lens that is inserted. Conventional polymethylmethacrylate lenses require the phacoemulsification incision to be enlarged to 6 mm to allow their insertion. Intraocular lenses made from different materials such as hydroxymethyl methacrylate or silicone can be folded to allow their insertion. This further facilitates the use of small incisions. Besides the intended surgical effect, some unwanted collateral effects are observed, such as damage of the corneal endothelium (Walkow et al. 2000), rupture of the posterior capsule (Martin and Burton 2000) and damage of the phaco-tip itself with metal particulate often left in the eye after surgery (Gimbel 1990; Kreiler et al. 1992). A typical lesion on human corneal endothelium is shown in Fig. 8.7. The most serious ocular complication of phacoemulsification lens extraction is dropping the nucleus into the vitreous cavity. This may result in visual loss due to inflammation and retinal detachment. Fortunately, this complication is unusual in experienced surgeons and sight loss can be prevented by vitrectomy and nucleus removal. Phacoemulsification predominates as the procedure of choice for cataract extraction. The reasons are rooted in improved outcome for the patient. The main advantage is reduced corneal astigmatism after cataract surgery. Since the cornea is the major refracting surface of the eye, minor disturbances to its shape may result in marked astigmatism with serious consequences for vision. All corneal surgery has the tendency to produce astigmatism with less intervention producing less distortion than the more disruptive procedures. However, small cataract incisions produce less astigmatism than large incision cataract surgery.

Fig. 8.7 Scanning-electron micrograph of human endothelium lesion resulting from a 5-min exposure to ultrasound. Bar marker: 20 μm. Reproduced with permission from Olson et al. (1978). © BMJ Publishing Group Ltd

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The phaco-tip vibrations are strong enough to generate both transient and stable cavitation bubbles which probably produce most of the fragmentation and emulsification of the lens and which may also cause a removal of material from the phaco-tip. An example of cavitation bubble formation as a result of phaco-tip vibrations is illustrated in Fig. 8.8. Several authors cite the formation of free radicals as evidence of cavitation during phacoemulsification. These species are thought to be generated when the heat from the implosion of cavitation bubbles causes the decomposition of water (Augustin and Dick 2004; Shimmura et al. 1992; Takahashi et al. 2002). Holst et al. (1993) used a single photon counting apparatus and luminol in rabbit eyes to demonstrate chemoluminescence secondary to the production of free radicals during phacoemulsification. They also obtained data correlating the amount of free radicals produced with the amount of ultrasonic power used. Topaz et al. (2002) demonstrated sonoluminescence under simulated phacoemulsification in aqueous medium using electron paramagnetic resonance spectroscopy and photon detection. They also noted reduction of cavitation intensity and elimination of sonoluminescence by saturation of the solution with carbon dioxide. Cavitation around the phaco tip was also observed by Zacharias (2008). However, his study found strong evidence that cavitation plays no role in phacoemulsification, leaving the jackhammer effect as the most important mechanism responsible for the lens-disrupting power of phacomeulsification. Current surgical procedures, particularly ultrasound phacoemulsification for cataract surgery and other operations involving the anterior chamber of the eye, have benefited from the use of ophthalmic viscoelastic substances (Silver et al. 1992; Behndig and Lundberg 2002). The primary goal of these substances is to protect the corneal endothelium during surgical procedures. The viscoelastic substances should offer minimal thixotropy in order to aid retention within the eye and yet, following implantation, should possess high-equilibrium viscosity to ensure that there is an appropriate maintenance of the ocular space (Andrews et al. 2005). The viscoelastic properties of these substances may also reduce the cavitation intensity and, thus, the

Fig. 8.8 Ultrasonic phaco-tip showing wave propagation and presence of presumed cavitation bubbles. Reproduced with permission from Packer et al. (2005). © Elsevier B.V.

References

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addition of a suitably viscoelastic substance to the eye seems to be a potential way of preventing or mitigating the negative collateral effects induced by cavitation in ultrasound phacoemulsification. Although no direct evidence is available in literature, numerous experimental results indicate the reduction of cavitation damage in viscoelastic liquids for conditions similar to those encountered during ultrasound phacoemusification (see, for a detailed list of references, Chap. 3).

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Holst, A., Rolfsen, W., Svensson, B., Ollinger, K., Lundgren, B. 1993 Formation of free radicals during phacoemulsification. Curr. Eye Res. 12, 359–365. Kimura, Y., Wilder-Smith, P., Matsumoto, K. 2000 Lasers in endodontics: a review. Int. Endod. J. 33, 173–185. Kramer, J. 1990 Intervertebral Disk Diseases: Causes, Diagnosis, Treatment and Prophylaxis. George Thieme Verlag, New York. Kreiler, K.R., Mortensen, S.W., Mamalis, N. 1992 Endothelial cell loss following modern phacoemulsification by a senior resident. Ophthalmic Surg. 23, 158–160. Krell, K.V., Johnson, R.J., Madison, S. 1988 Irrigation patterns during ultrasonic canal instrumentation. Part I: K-type files. J. Endod. 14, 65–68. Lea, S.C., Price, G.J., Walmsley, A.D. 2004 Does cavitation occur around powered toothbrushes? J. Clin. Periodontol. 31, 77–78. Lea, S.C., Price, G.J., Walmsley, A.D. 2005 A study to determine whether cavitation occurs around dental ultrasonic scaling instruments. Ultrasonics Sonochem. 12, 233–236. Love, R.M., Jenkinson, H.F. 2002 Invasion of dentinal tubules by oral bacteria. Crit. Rev. Oral Biol. Med. 13, 171–83. Lumley, P.J., Walmsley, A.D., Laird, W.R.E. 1988 An investigation into cavitational activity occurring in endosonic instrumentation. J. Dent. 16, 120–122. Lumley, P.J., Walmsley, A.D., Walton, R.E., Rippin, J.W. 1993 Cleaning of oval canals using ultrasonic or sonic instrumentation. J. Endod. 19, 453–457. Martin, K.R.G., Burton, R.L. 2000 The phacoemulsification learning curve: per-operative complications in the first 3000 cases of an experienced surgeon. Eye 14, 190–195. McCoy, G.F., McCrea, J.D., Beverland, D.E., Kernohan, W.G., Mollam, R.A.B. 1987 Vibration arthrography as a diagnostic-aid diseases of the knee – a preliminary report. J. Bone Joint Surg. 69, 288–293. Nakamura, C., Matsushita, I., Kosake, E., Kondo, T., Kimura, T. 2008 Anti-arthritic effects of combined treatment with histone deacetylase inhibitor and low-intensity ultrasound in the presence of microbubbles in human rheumatoid synovial cells. Rheumatology 47, 418–424. Nakaya, H., Shimizu, T., Isobe, K., Tensho, K., Okabe, T., Nakamura, Y., Nawata, M., Yoshikawa, H., Takaoka, K., Wakitani, S. 2005 Microbubble-enhanced ultrasound exposure promotes uptake of methotrexate into synovial cells and enhanced antiinflammatory effects in the knees of rabbits with antigen-induced arthritis . Arthritis Rheum. 52, 2559–2566. O’Leary, R., Sved, A.M., Davies, E.H., Leighton, T.G., Wilson, M., Kieser, J.B. 1997 The bactericidal effects of dental ultrasound on Actinobacillus actinomycetemcomitans and Porphyromonas gingivalis – An in vitro investigation. J. Clin. Periodontol. 24, 432–439. Olson, L.E., Marshall, J., Rice, N.S., Andrews, R. 1978 Effects of ultrasound on the corneal endothelium: I. The acute lesion. Br. J. Ophthalmol. 62, 134–144. Packer, M., Fishkind, W.J., Fine, H., Seibel, B.S., Hoffman, R.S. 2005 The physics of phaco: a review. J. Cataract Refract. Surg. 31, 424–431. Roy, R.A., Ahmad, M., Crum, L.A. 1994 Physical mechanisms governing the hydrodynamic response of an oscillating ultrasonic file. Int. Endod. J. 27, 197–207. Saito, M., Mazda, O., Takahashi, K.A., Arai, Y., Kishida, T., Shin-Ya, M., Inoue, A., Tonomura, H., Sakao, K., Morihara, T., Imanishi, J., Kawata, M., Kubo, T. 2007 Sonoporation mediated transduction of pDNA/siRNA into joint synovium in vivo. J. Orthop. Res. 25, 1308–1316. Shimmura, S., Tsubota, K., Oguchi, Y., Fukumura, D., Suematsu, M., Tsuchiya, M. 1992 Oxiradical-dependent photoemission induced by a phacoemulsification probe. Invest. Ophthalmol. Vis. Sci. 33, 2904–2907. Shrestha, A., Fong, S.W., Khoo, B.C., Kishen, A. 2009 Delivery of antibacterial nanoparticles into dentinal tubules using high-intensity focused ultrasound. J. Endod. 35, 1028–1033. Silver, F.H., LiBrizzi, J., Benedetto, D. 1992 Use of viscoelastic solution in ophthalmology: a review of physical properties and long term effects. J. Long Term Effects Med. Implants 2, 49–66. Smith, C.F. 1993 Lasers in orthopedic surgery. Orthopedics 16, 531–534.

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Index

A Ablation products, 200, 205 Absorption cross section, 230, 238, 240 Acoustic droplet vaporization, 211, 213 Acoustic power, 188 Acoustic pressure, 133–134, 179–180, 183, 186, 191, 195, 212–213 Added mass, 183 Added pressure, 149 Angioplasty laser, 175, 199, 202–206, 210 percutaneous transluminal, 202–203 rotational, 202–203, 206 ultrasound, 202 Annular flow, 98, 102, 105 Artheroscopic plaque, 202 Arthroscopic cartilage ablation, 258 Articular cartilage, 37–38, 249, 257 Attenuation stage, 155–156 B Berthelot tube, 57, 59 Bifurcation, 89–91, 212–214 Bjerknes force, 102, 104 Blood density, 34, 37 elasticity, 34, 37 infinite-shear viscosity, 35 sound speed, 37 structure, 35, 37 surface tension, 37 thixotropy, 36 zero-shear viscosity, 35 Blood-brain barrier, 193, 198 Blunt bodies, 129–134, 148 Boger fluid, 19, 146–147 Boiling temperature, 239, 255 Boltzmann constant, 51, 239 Boundary integral methods, 97

Boundary layer transition, 126, 131 Bubble cloud, 129, 166–167, 177, 206–207, 209 Bubble splitting, 98, 101–105 Bullet-piston method, 58 C Capillary rheometer, 28–29 Cataract surgery, 258–260 Cavitation erosion, 63, 92, 101, 155–172 fixed, 117 hydrodynamic, 117–150 incipient, 117, 127, 142–143, 149 inertial, 195–196, 212, 250 jet, 126–129 noise, 128 number, 124–128, 130–131, 133, 135, 137–139, 142–145, 149 tip vortex, 117, 134–142, 148 traveling, 117 vortex, 117, 134–142, 148, 209 Cavitation damage mechanisms polymer solutions, 54, 84–85 water, 54, 84–85 Cavitation nanobubbles, 225 Cavitation susceptibility meter, 53, 56 Cell constituents, 34, 41–43 cytoplasmic viscosity, 42 cytoskeleton rheology, 41 Chaotic oscillations, 89–91 Circulation, 56, 123, 135, 138–139, 148, 177, 199, 202, 209 Clot lysis, 175, 195 Concentric cylinder rheometer, 25, 27 Cone and plate rheometer, 26–27 Confined space, 144–145, 149, 213 Constitutive equation

E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3,

265

266 Carreau, 16 Casson, 16 elastic dumbbell, 19–20 Giesekus, 18 Jeffreys, 19 KBKZ, 20–21 linear Oldroyd, 71 Maxwell, 17–18 Oldroyd-B, 18–19 Phan-Thien-Tanner, 18 power law, 15–16 rigid dumbell, 20 upper convective Maxwell, 23, 25 Williamson, 70–71, 76 Contrast particles, 177 Convected time derivative, 17–18, 81 Corneal endothelium, 259–260 Coulter counter, 53–54 Counterjet, 92 Cracking noise, 117, 249, 256–257 Creep, 6, 32–33, 97 Critical break-up tension, 182 Critical laser fluence, 238–239, 241 Critical nucleus, 51, 53 D Damaged area, 159–161 Dentinal tubules, 253 Depolymerization, 42, 157 Desinent cavitation number, 127–128, 138–139 Diagnostic ultrasound, 175, 193–194, 198–199 Dilatational rate, 184 Dimensionless number Deborah, 13–14, 76, 79, 81, 89–90 elasticity, 13, 15 Reynolds, 13–15, 75, 79, 81, 85, 89, 118–119, 122–124, 127–128, 131–134, 143–145, 148–149, 183 Weissenberg, 13–14 Dirac function, 73 Drag coefficient, 183 Drag reduction, 118–121, 123, 144, 148 Dynamic rigidity, 9 E Elastic boundary, 101–107 Elastic compression modulus, 182 Elastic modulus, 33, 42, 102–108, 120, 168, 184 Elastic solid, 3, 9, 14 Encapsulated microbubbles buckling radius, 182 mathematical formulations, 238–241

Index translational motion, 183 Endodontics, 254–255 Enthalpy, 66, 68 Equation of state, 64, 169, 239 Equilibrium radius, 50, 69, 71, 86, 182 Erosion pattern, 158, 162 Extensional rheometry, 29–31 Extensional viscosity, 11–12, 17–20, 24, 29–31, 96, 110–112, 118–119, 127, 132, 148–150, 171 estimation, 110–112, 149–150 Extra stress tensor, 4, 7, 10, 15, 18, 21–23, 64–65, 80 F Fahreus effect, 35 Femtosecond optical breakdown, 226–227 Filament stretching rheometer, 31 Flow biaxial extensional, 11 oscillatory shear, 8–10 planar extensional, 12 simple shear, 3–4, 7–8, 17, 21 uniaxial extensional, 10–11, 23–24, 31, 41, 97 Flow time scale, 14 Fluid ideal, 1, 3 Newtonian, 1–4, 11–12, 15–16, 25, 27–28, 30, 35, 58, 66, 68, 71, 76, 97, 118, 137, 144, 146, 150, 163, 171, 186 non-Newtonian, 1–43, 63, 97, 118, 124, 137, 144, 148–149, 156–163, 258 real, 1, 5 rheopectic, 5–6 shear-thickening, 5, 16 shear-thinning, 5, 28–29, 35, 70–71, 79, 146 thixotropic, 5–6 viscoelastic, 5–6, 8, 16–21, 26, 30, 33, 37, 43, 78, 98, 120, 137, 146–148 viscoplastic, 5 Fluorescence correlation spectroscopy, 33 Free radicals, 198, 229, 260 Free-stream turbulence, 125–126 Frequency response curve, 86–89 G Gas content, 54, 125, 127–128, 135 embolism, 210–214 embolotherapy, 211–214 Geometric focusing effects, 168

Index Gibbs equations, 52 Gilmore equation, 67, 74 Green fluorescent protein, 194 H Harmonic resonance, 87–88, 90 Hookean relaxation time, 20 Huggins slope constant, 12–13 Hyperbaric oxygen therapy, 211 Hysteresis, 6, 51 I Imaging techniques, 179, 241 Inception cavitation number, 127, 131, 135, 137, 145, 149 Incubation stage, 155 Internal energy, 239 Intracorporeal stones, 165 Irrigant, 252–253 J Jet formation, 91, 100–101, 104, 164, 166, 168, 196 Jet velocity, 92–93, 96–97, 104–105, 121, 127, 166 K Keller-Herring equation, 67 Kinetic spinodal, 227 L Lamb vortex, 135 Laminar separation point, 126 Laplace transform, 73 Laser fluence threshold, 231 Loss modulus, 9, 33, 39, 43 Loss tangent, 9 Lumley hypothesis, 120 M Magnetic tweezer, 33 Mark-Houwink equation, 13 Mean-square displacement, 32–33 Mechanical heart valve, 206–210 Membrane blebbing, 244 Metacarpophalangeal joint, 256–257 Microemboli, 202, 209–211 Microrheology active methods, 32–34 passive methods, 32–33 Mie coefficients, 240

267 N Nanoparticle, 225, 230–245, 253 Normal stress coefficients, 8 Nucleation barrier, 51–52 heterogeneous, 49–50 homogeneous, 49, 51–53 rate, 51 Nuclei distribution blood, 55–57 polymer solutions, 54–55 water, 53–54 stabilization mechanisms, 50 Numerical methods, 11–12, 14, 17, 78, 80–82, 85, 87–90, 98, 105–106, 110, 119, 142, 148, 167, 181, 191, 213, 226–229, 234, 241, 245 O Opposed jet rheometer, 29–30 Optical tweezer, 32–33 Orifice flow, 121–123, 126, 148 Orthognatic surgery, 256 P Period-doubling cascade, 90 Perturbation approach, 97 Photodynamic therapy, 229 Photothermal therapy, 225, 229–245 Plasma, 34–38, 226, 242–243 Plasmonic photothermal therapy, 225, 229–245 Plastic flow stress, 168, 170–171 Polymer injection, 142 solutions dilute, 15, 82, 85, 110, 131, 149 semi-dilute, 13, 138, 143 ultrasonic degradation, 156–157 Polytropic index, 64 Pressure attenuation, 78–79 coefficient, 124, 135 drop, 28, 118, 121–123, 139, 143, 228 gradient, 91–92, 102, 118, 137, 144, 178, 196 Protein inactivation, 232 R Rankine vortex, 142 Rate of deformation tensor, 4, 11, 18, 23, 70 invariants, 4, 65 Rayleigh-Plesset equation, 180, 184, 233, 234 Red cell aggregation, 35

268 Relaxation time, 1, 14, 17–18, 20–21, 41, 71–72, 88–89, 146–147, 186, 189 Resonance frequency, 86, 187–190 Retardation time, 20, 71, 79, 81, 88–89, 189 Riccati-Bessel function, 240 Root canal, 249, 252–255 infection, 253 S Saddle-node bifurcation, 90 Saliva elasticity, 40–41 relaxation time, 41 structure, 40 viscosity, 40–41 Saturation pressure, 52, 144 Scattering cross section, 188, 191 Schiebe body, 131 Secondary flow, 27–28 Shear rheometry, 25–29 Shear waves, 120 Shock -induced collapse, 99 -induced jet, 99–101, 171 pressure, 96, 128, 164, 166 wave, 78, 82–83, 93–95, 99–101, 109, 164, 166–168, 171–172, 196, 212, 232, 245, 257–258 emission, 167, 232 Sonoporation, 192, 195 Sonothrombolysis, 175–177 Specific heat capacity, 240 Spherical acoustic wave, 78 Spherical bubble collapse time, 66, 76, 83–85, 99–100 dimensionless variables, 74 energy, 63–68 natural frequency, 71, 187 pressure distribution, 71, 78, 147, 226 scaling laws, 84–85, 108–109 thermal effects, 81, 227 Spherical bubble dynamics compressible formulation, 77, 79–80 general equations, 63–65 incompressible formulation, 66–67, 77, 79–80, 86 Splash effect, 165 Squeeze flow, 209 Stagnation point, 29, 131–132 Stagnation pressure, 170 Stokes-Einstein relation, 32 Storage modulus, 9, 19, 33, 39, 43 Strange attractor, 89–90

Index Streamwise velocity fluctuations, 120 Stress relaxation, 6, 21 tensor, 4, 7, 10, 18–19, 22–23, 64–65, 80 Subharmonic resonance, 87–88, 90 Surface dilatational viscosity, 65, 182, 184 plasmon resonance, 231–233, 240 Surfactant, 50, 57, 65, 97, 178, 183, 211 Synovial fluid density, 38 elasticity, 38–39 rheopexy, 38–39 structure, 39 viscosity, 38–39 Synovial proliferation, 258 T Tensile strength polymer solutions, 58–59 water, 58–59 Tensile stress, 57, 107, 111, 147, 166, 226–227, 245 Therapeutic ultrasound, 175 Thermal diffusivity, 239 Thermal expansion coefficient, 226 Thermo-elastic stress, 226 Threshold fluence, 235–236, 243 Thrombogenesis, 210 Thrombolitic agents, 176 Tissue ablation, 199–200 oxygenation, 229 plasminogen activator, 176, 195 Tooth brush, 252 Transmyocardial laser revascularisation, 199–200, 202 Tribonucleation, 257 Trouton ratio, 11–12, 30, 94–95, 148–150, 172 U Ultraharmonic resonance, 87, 90 Ultrasonic irrigation, 252–254 Ultrasonic scaler, 250–251 Ultrasound contrast agents, 106, 177–199 Ultrasound phacoemulsification, 258, 260–261 V Van der Waals equation, 239 Vascular endothelial growth factor, 193 Vena contracta, 121 Ventricular pressure, 206 Vibratory devices, 156 Viscoelastic effects

Index die swell, 6 Uebler, 6 Weissenberg, 6, 13–14, 28 Viscoelastic shell, 186 Viscometric functions, 8 Viscosity apparent, 4, 27–28, 35–36, 38–40, 42–43, 76 biaxial extensional, 11–12 dynamic, 2, 9 infinite-shear, 5, 16, 35, 70, 76, 87 intrinsic, 12–13, 20, 29 kinematic, 3, 146, 183 molecular weight, 13 planar extensional, 12 shear, 1–2, 5, 11–12, 16–17, 29–30, 35, 38, 65, 70, 76, 87, 90, 100–101, 112, 120, 127, 144–146, 149, 184, 186 specific, 12

269 uniaxial extensional, 11, 17, 19, 24, 31, 111 volume, 4 zero-shear, 5, 12, 16, 29, 35, 70, 146 Viscous adhesion, 257 Viscous sublayer, 119 Vortex chamber, 142–143 inhibition, 118, 123 strength, 135 W Wall turbulence, 120 Wave number, 240 Weight loss, 155–160, 163 Weissenberg-Rabinowitsch equation, 28 Y Yield strength, 107, 165, 168

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