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

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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

v vi 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 presenta- tion of the fundamental processes involved in bubble dynamics and cavitation in non-Newtonian fluids has not appeared in the scientific literature. This is not sur- prising, 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 oscillat- ing in non-Newtonian fluids are examined using various mathematical models. One main message here is that the introduction of ideas from theoretical studies of non- linear 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 pro- duced 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 inter- est, 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 vii

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 Emil-Alexandru Brujan June 2010 Contents

1 Non-Newtonian Fluids ...... 1 1.1Deﬁnitions...... 1 1.1.1NewtonianFluids...... 1 1.1.2Non-NewtonianFluids...... 4 1.2Non-NewtonianFluidBehaviour...... 7 1.2.1SimpleFlows...... 7 1.2.2IntrinsicViscosityandSolutionClassiﬁcation...... 12 1.2.3DimensionlessNumbers...... 13 1.2.4 Constitutive Equations ...... 15 1.3 Rheometry ...... 25 1.3.1 Shear Rheometry ...... 25 1.3.2 Extensional Rheometry ...... 29 1.3.3 Microrheology Measurement Techniques ...... 32 1.4ParticularNon-NewtonianFluids...... 34 1.4.1Blood...... 34 1.4.2 Synovial Fluid ...... 37 1.4.3Saliva...... 40 1.4.4 Cell Constituents ...... 41 1.4.5 Other Viscoelastic Biological Fluids ...... 43 References ...... 43 2 Nucleation ...... 49 2.1 Nucleation Models ...... 49 2.2NucleiDistribution...... 53 2.2.1DistributionofCavitationNucleiinWater...... 53 2.2.2 Distribution of Cavitation Nuclei in Polymer Solutions . . 54 2.2.3CavitationNucleiinBlood...... 55 2.3 Tensile Strength ...... 57 References ...... 59 3 Bubble Dynamics ...... 63 3.1 Spherical Bubble Dynamics ...... 63 3.1.1 General Equations of Bubble Dynamics ...... 63

ix x Contents

3.1.2 The Equations of Motion for the Bubble Radius ...... 65 3.1.3 Heat and Mass Transfer Through the Bubble Wall . .... 81 3.1.4 Experimental Results ...... 82 3.1.5 Bubbles in a Sound-Irradiated Liquid ...... 86 3.2 Aspherical Bubble Dynamics ...... 91 3.2.1 Bubbles Near a Rigid Wall ...... 92 3.2.2 Bubbles Between Two Rigid Walls ...... 98 3.2.3 Bubbles in a Shear Flow ...... 98 3.2.4 Shock-Wave Bubble Interaction ...... 99 3.3 Bubbles Near an Elastic Boundary ...... 101 3.4 Bubbles in Tissue Phantoms ...... 107 3.5 Estimation of Extensional Viscosity ...... 110 References ...... 112 4 Hydrodynamic Cavitation ...... 117 4.1Non-cavitatingFlows...... 118 4.1.1 Drag Reduction ...... 118 4.1.2 Reduction of Pressure Drop in Flows Through Oriﬁces . . 121 4.1.3 Vortex Inhibition ...... 123 4.2CavitatingFlows...... 123 4.2.1CavitationNumber...... 124 4.2.2JetCavitation...... 126 4.2.3 Cavitation Around Blunt Bodies ...... 129 4.2.4VortexCavitation...... 134 4.2.5 Cavitation in Conﬁned Spaces ...... 143 4.2.6 Mechanisms of Cavitation Suppression byPolymerAdditives...... 148 4.3 Estimation of Extensional Viscosity ...... 149 References ...... 150 5 Cavitation Erosion ...... 155 5.1CavitationErosioninNon-NewtonianFluids...... 156 5.2 Mechanisms of Cavitation Damage in Newtonian Fluids . .... 163 5.3 Reduction of Cavitation Erosion in Polymer Solutions ...... 171 References ...... 172 6 Cardiovascular Cavitation ...... 175 6.1CavitationforUltrasonicSurgery...... 175 6.1.1 Sonothrombolysis ...... 175 6.1.2 Ultrasound Contrast Agents ...... 177 6.2CavitationinLaserSurgery...... 199 6.2.1 Transmyocardial Laser Revascularization ...... 199 6.2.2LaserAngioplasty...... 202 6.3 Cavitation in Mechanical Heart Valves ...... 206 6.3.1 Detection of Cavitation in Mechanical Heart Valves .... 206 Contents xi

6.3.2 Mechanisms of Cavitation Inception in Mechanical HeartValves...... 208 6.3.3 Collateral Effects Induced by Cavitation ...... 209 6.4GasEmbolism...... 210 6.4.1TreatmentStrategiesforGasEmbolism...... 211 6.4.2GasEmbolotherapy...... 211 References ...... 214 7 Nanocavitation for Cell Surgery ...... 225 7.1 Cavitation Induced by Femtosecond Laser Pulses ...... 226 7.1.1NumericalSimulations...... 226 7.1.2 Experimental Results ...... 228 7.2 Cavitation During Plasmonic Photothermal Therapy ...... 229 7.2.1 Nanoparticles and Surface Plasmon Resonance ...... 231 7.2.2 Bubble Dynamics ...... 233 7.2.3BiologicalEffectsofCavitation...... 241 References ...... 245 8 Cavitation in Other Non-Newtonian Biological Fluids ...... 249 8.1CavitationinSaliva...... 249 8.1.1 Cavitation During Ultrasonic Plaque Removal ...... 249 8.1.2 Cavitation During Passive Ultrasonic Irrigation of the Root Canal ...... 252 8.1.3 Cavitation During Laser Activated Irrigation of the Root Canal ...... 254 8.1.4 Cavitation During Orthognathic Surgery of the Mandible ...... 256 8.2 Cavitation in Synovial Liquid ...... 256 8.3 Cavitation in Aqueous Humor ...... 258 References ...... 261 Index ...... 265 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, 1 DOI 10.1007/978-3-642-15343-3_1, C Springer-Verlag Berlin Heidelberg 2011 2 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

AU F ∼ . (1.1) Y 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

F U du τ = = η = η . (1.2) A Y dy

In transposed form it serves to define the proportionality constant τ η = , (1.3) du/dy which is called the dynamic coefficient of viscosity.Thetermdu/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 η v = 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 + λvekkδij, (1.5)

Fig. 1.2 Rheological behaviour of materials 4 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 CauchyÐGreen 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 ∂ui ∂uj eij = + (1.7) 2 ∂xj ∂xi or

1 E(x, t) = (∇u) + (∇u)T , (1.8) 2 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:

1 I = eii, I = (eiiejj − eijeij), I = det(eij). (1.10) 1 2 2 3

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 6 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 stressÐstrain 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 non- Newtonian 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 analy- sis 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 , (1.13) 00τzz 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 atten- tion 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:

2 τxx − τyy = ψ1(γ˙)γ˙ , (1.17) and

2 τyy − τzz = ψ2(γ˙)γ˙ , (1.18) 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 harmon- ically 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) and

γ˙(t) = γ0ω cos(ωt) =˙γ0 cos(ωt), (1.20) 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, τ δ = π − φ 0, and phase shift, 2 , as follows:

τxy(t) = τ0 sin(ωt + δ), (1.21) and

τxy(t) = τ0ω cos (ωt − φ). (1.22)

These equations may be expanded and rewritten in terms of the in-phase and out- of-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) = γ0ω 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

η = G ω , (1.25) η = G ω . (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, 2 η approaches η0 and 2G /ω approaches ψ1,0 (the zero-shear-rate value of ψ1). Correspondingly, the loss modulus is asymptotic to η0ω as ω → 0. A method of comparing the storage and loss modulus is made by the calculation of the loss tangent deﬁned as

G tan δ = , (1.27) G and represents the phase angle between stress and strain. 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 non- Newtonian 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: ε˙ ε˙ ux =˙εx, uy =− y, uz =− z, (1.28) 2 2 where ε˙ = dux/dx is a constant strain rate, and the corresponding extra stress tensor is ⎛ ⎞ τxx 00 ⎝ ⎠ τ = 0 τyy 0 . (1.29) 00τzz

Fig. 1.5 Extensional ﬂow ﬁelds: (a) uniaxial, (b)biaxial,(c)planar 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 extensional- thinning if ηE decreases with increasing ε˙. They are considered extensional- thickening 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 3ε˙. 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 drasti- cally but they change only in two directions. The velocity field in simple biaxial extensional flow is given by

ux =˙εBx, uy =˙εBy, uz =−2ε˙Bz. (1.32)

The corresponding stress distribution is

τxx − τzz = τyy − τzz = ηEB(ε˙B)ε˙B, (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 12ε˙.Fora 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 =˙εPx, uy =−˙εPy, uz = 0. (1.34)

In this case, the stress distribution is given as

τxx − τyy = ηEP (ε˙P) 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 2ε˙. 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 non- Newtonian 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:

η0 − ηs [η] = lim ηred = lim . (1.36) c→0 c→0 cηs Note that the intrinsic viscosity has dimensions of reciprocal concentration. The intrinsic viscosity is determined graphically by plotting ηred versus c and extrapo- lating to zero concentration. It is also found that extrapolation to zero concentration η = 1 η + of the inherent viscosity, inh c ln( sp 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]2c, (1.37) c 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]2c, (1.38) c c 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)

α [η] = kMη , (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)orcNAV/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 = π 3/ to occupy a spherical region in solution such that V 4 Rh 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 = πL3/6, where L is the length of the molecule. The length L can be determined using relations given by Broersma (1960 ) and Young et al. (1978)for 2 rigid molecules, L = Rh 2δ − 0.19 − (8.24/δ) + 12/δ , 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:

ρ = LU = LU Re 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 = τfluide˙ or τfluidγ˙, (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 equilib- rium 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

τﬂuid De = . (1.42) τ ﬂow

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

τfluidη El = De/Re = , (1.43) ρL2 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 = Kγ˙ 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 shear- thinning, 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 , (1.45) 1 + (λcγ˙) 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

τ ηp η = 0 + , (1.46) γ˙ N 1 + (λcγ˙) 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

√ √ τ− τ 2 √ 0 τ ≥ τ γ˙ = η ,for 0 , (1.47) 0,forτ<τ0 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 τ + λ = 2ηE, (1.48) ∂t 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) . (1.50) ∂t For steady simple shear ﬂow, the Maxwell relaxation time is

N ψ λ = 1 = 1 , (1.51) 2ηγ˙ 2 2η while in small-amplitude oscillatory ﬂow, the viscoelastic properties for this model are given by

2 ληω G = , (1.52) 1 + λ2ω2 and

η η = . (1.53) 1 + λ2ω2 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

1 ηE = 3η . (1.54) (1 + λε˙)(1 − 2λε˙) 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 αλ Y = I + τ η , (1.56) and the Phan-Thien-Tanner model, for which ελ Y = τ I exp η tr( ) . (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 n τ = τi, (1.58) i=1 where each τi is described by

∇ τι + λi τi = 2ηiE. (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

2 (λ − λ )ηω G = 1 2 , (1.62) + λ2ω2 1 1 and 2 1 + λ1λ2ω η η = . (1.63) + λ2ω2 1 1 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

1 − λ ε˙ − 2λ λ ε˙2 η = 3η 2 1 2 , (1.64) E − λ ε˙ − λ2ε˙2 1 1 2 1 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 τ + λ = 2η E + λ . (1.65) 1 ∂t 2 ∂t 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 = nkBTλHE, (1.66)