Cavitation in Non-Newtonian Fluids...non-Newtonian ﬂuids are described in this book. It is essential to understand that the effects of non-Newtonian properties on bub-ble dynamics

Cavitation in Non-Newtonian Fluids

Emil-Alexandru Brujan

Cavitation in Non-NewtonianFluids

With Biomedical and BioengineeringApplications

123

Emil-Alexandru BrujanUniversity Politechnica of BucharestDepartment of HydraulicsSpl. Independentei 313, sector 6060042 [email protected]

ISBN 978-3-642-15342-6 e-ISBN 978-3-642-15343-3DOI 10.1007/978-3-642-15343-3Springer Heidelberg Dordrecht London New York

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Preface

Cavitation is the formation of voids or bubbles containing vapour and gas in anotherwise homogeneous fluid in regions where the pressure falls locally to that ofthe vapour pressure corresponding to the ambient temperature. The regions of lowpressure may be associated with either a high fluid velocity or vibrations. Cavitationis an important factor in many areas of science and engineering, including acous-tics, biomedicine, botany, chemistry and hydraulics. It occurs in many industrialprocesses such as cleaning, lubrication, printing and coating. While much of theresearch effort into cavitation has been stimulated by its occurrence in pumps andother fluid mechanical devices involving high speed flows, cavitation is also animportant factor in the life of plants and animals, including humans.

Several books and review articles have addressed general aspects of bubbledynamics and cavitation in Newtonian fluids but there is, at present, no book devotedto the elucidation of these phenomena in non-Newtonian fluids. The proposed bookis intended to provide such a resource, its significance being that non-Newtonianfluids are far more prevalent in the rapidly emerging fields of biomedicine and bio-engineering, in addition to being widely encountered in the process industries. Theobjective of this book is to present a comprehensive perspective of cavitation andbubble dynamics from the stand point of non-Newtonian fluid mechanics, physics,chemical engineering and biomedical engineering. In the last three decades thisfield has expanded tremendously and new advances have been made in all fronts.Those that affect the basic understanding of cavitation and bubble dynamics innon-Newtonian fluids are described in this book.

It is essential to understand that the effects of non-Newtonian properties on bub-ble dynamics and cavitation are fundamentally different from those of Newtonianfluids. Arguably the most significant effect arises from the dramatic increase in vis-cosity of polymer solutions in an extensional flow, such as that generated about aspherical bubble during its growth or collapse phase. Specifically, polymers, whichare randomly-oriented coils in the absence of an imposed flow-field, are pulled apartand may increase their length by three orders of magnitude in the direction of exten-sion. As a result, the solution can sustain much greater stresses, and pinching isstopped in regions where polymers are stretched. Furthermore, many biological flu-ids, 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 topicbecause cavitation is playing an increasingly important role in the development ofmodern 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 innon-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 arewide-ranging. Consequently, researchers who investigate cavitation phenomenon innon-Newtonian fluids originate from several disciplines. Moreover, the resulting sci-entific reports are often narrow in scope and scattered in journals whose foci rangefrom the physical sciences and engineering to medical sciences. The purpose of thisbook is to provide, for the first time, an improved mechanistic understanding ofbubble dynamics and cavitation in non-Newtonian fluids.

The book starts with a concise but readable introduction into non-Newtonianfluids with a special emphasis on biological fluids (blood, synovial liquid, saliva,and cell constituents). A distinct chapter is devoted to nucleation and its role oncavitation inception. The dynamics of spherical and non-spherical bubbles oscillat-ing in non-Newtonian fluids are examined using various mathematical models. Onemain 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 inour understanding of this topic. Two chapters are devoted to hydrodynamic cavita-tion and cavitation erosion, with special emphasis on the mechanisms of cavitationerosion in non-Newtonian fluids.

The second part of the book describes the role of cavitation and bubbles in thetherapeutic applications of ultrasound and laser surgery. Whenever laser pulses areused to ablate or disrupt tissue in a liquid environment, cavitation bubbles are pro-duced which interact with the tissue. The interaction between cavitation bubbles andtissue may cause collateral damage to sensitive tissue structures in the vicinity of thelaser focus, and it may also contribute in several ways to ablation and cutting. Thesesituations are encountered in laser angioplasty and transmyocardial laser revascu-larization. Cavitation is also one of the most exploited bioeffects of ultrasound fortherapeutic advantage. In both cases, the violent implosion of cavitation bubblescan lead to the generation of shock waves, high-velocity liquid jets, free radicalspecies, and strong shear forces that can damage the nearby tissue. Knowledge ofthese physical mechanisms is therefore of vital importance and would provide aframework wherein novel and improved surgical techniques can be developed.

This field is as interdisciplinary as any, and the numerous disciplines involvedwill continue to overlook and reinvent each others’ work. My hope in this bookis to attempt to bridge the various communities involved, and to convey the inter-est, elegance, and variety of physical phenomena that manifest themselves on themicrometer and microsecond scales. This book is offered to mechanical engineers,chemical engineers and biomedical engineers; it can be used for self study, as wellas in conjunction with a lecture course.

I would like to gratefully acknowledge the advice and help I received fromProfessor 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 ofBirmingham). I also appreciate fruitful conversations with and kind help I receivedfrom Professor Werner Lauterborn (Göttingen University), Dr. Teiichiro Ikeda(Hitachi Ltd), Dr. Kester Nahen (Heidelberg Engineering GmbH), and PeterSchmidt.

Bucharest, Romania Emil-Alexandru BrujanJune 2010

Contents

1 Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . 4

1.2 Non-Newtonian Fluid Behaviour . . . . . . . . . . . . . . . . . . 71.2.1 Simple Flows . . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Intrinsic Viscosity and Solution Classification . . . . . . . 121.2.3 Dimensionless Numbers . . . . . . . . . . . . . . . . . . 131.2.4 Constitutive Equations . . . . . . . . . . . . . . . . . . . 15

1.3 Rheometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.1 Shear Rheometry . . . . . . . . . . . . . . . . . . . . . . 251.3.2 Extensional Rheometry . . . . . . . . . . . . . . . . . . . 291.3.3 Microrheology Measurement Techniques . . . . . . . . . . 32

1.4 Particular Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . 341.4.1 Blood . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.4.2 Synovial Fluid . . . . . . . . . . . . . . . . . . . . . . . . 371.4.3 Saliva . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.4.4 Cell Constituents . . . . . . . . . . . . . . . . . . . . . . 411.4.5 Other Viscoelastic Biological Fluids . . . . . . . . . . . . 43References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.1 Nucleation Models . . . . . . . . . . . . . . . . . . . . . . . . . 492.2 Nuclei Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2.1 Distribution of Cavitation Nuclei in Water . . . . . . . . . 532.2.2 Distribution of Cavitation Nuclei in Polymer Solutions . . 542.2.3 Cavitation Nuclei in Blood . . . . . . . . . . . . . . . . . 55

2.3 Tensile Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 57References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Bubble Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.1 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 . . . . . . 653.1.3 Heat and Mass Transfer Through the Bubble Wall . . . . . 813.1.4 Experimental Results . . . . . . . . . . . . . . . . . . . . 823.1.5 Bubbles in a Sound-Irradiated Liquid . . . . . . . . . . . . 86

3.2 Aspherical Bubble Dynamics . . . . . . . . . . . . . . . . . . . . 913.2.1 Bubbles Near a Rigid Wall . . . . . . . . . . . . . . . . . 923.2.2 Bubbles Between Two Rigid Walls . . . . . . . . . . . . . 983.2.3 Bubbles in a Shear Flow . . . . . . . . . . . . . . . . . . 983.2.4 Shock-Wave Bubble Interaction . . . . . . . . . . . . . . 99

3.3 Bubbles Near an Elastic Boundary . . . . . . . . . . . . . . . . . 1013.4 Bubbles in Tissue Phantoms . . . . . . . . . . . . . . . . . . . . 1073.5 Estimation of Extensional Viscosity . . . . . . . . . . . . . . . . 110

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4 Hydrodynamic Cavitation . . . . . . . . . . . . . . . . . . . . . . . . 1174.1 Non-cavitating Flows . . . . . . . . . . . . . . . . . . . . . . . . 118

4.1.1 Drag Reduction . . . . . . . . . . . . . . . . . . . . . . . 1184.1.2 Reduction of Pressure Drop in Flows Through Orifices . . 1214.1.3 Vortex Inhibition . . . . . . . . . . . . . . . . . . . . . . 123

4.2 Cavitating Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.2.1 Cavitation Number . . . . . . . . . . . . . . . . . . . . . 1244.2.2 Jet Cavitation . . . . . . . . . . . . . . . . . . . . . . . . 1264.2.3 Cavitation Around Blunt Bodies . . . . . . . . . . . . . . 1294.2.4 Vortex Cavitation . . . . . . . . . . . . . . . . . . . . . . 1344.2.5 Cavitation in Confined Spaces . . . . . . . . . . . . . . . 1434.2.6 Mechanisms of Cavitation Suppression

by Polymer Additives . . . . . . . . . . . . . . . . . . . . 1484.3 Estimation of Extensional Viscosity . . . . . . . . . . . . . . . . 149

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5 Cavitation Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.1 Cavitation Erosion in Non-Newtonian Fluids . . . . . . . . . . . . 1565.2 Mechanisms of Cavitation Damage in Newtonian Fluids . . . . . 1635.3 Reduction of Cavitation Erosion in Polymer Solutions . . . . . . . 171

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6 Cardiovascular Cavitation . . . . . . . . . . . . . . . . . . . . . . . 1756.1 Cavitation for Ultrasonic Surgery . . . . . . . . . . . . . . . . . . 175

6.1.1 Sonothrombolysis . . . . . . . . . . . . . . . . . . . . . . 1756.1.2 Ultrasound Contrast Agents . . . . . . . . . . . . . . . . . 177

6.2 Cavitation in Laser Surgery . . . . . . . . . . . . . . . . . . . . . 1996.2.1 Transmyocardial Laser Revascularization . . . . . . . . . 1996.2.2 Laser Angioplasty . . . . . . . . . . . . . . . . . . . . . . 202

6.3 Cavitation in Mechanical Heart Valves . . . . . . . . . . . . . . . 2066.3.1 Detection of Cavitation in Mechanical Heart Valves . . . . 206

Contents xi

6.3.2 Mechanisms of Cavitation Inception in MechanicalHeart Valves . . . . . . . . . . . . . . . . . . . . . . . . . 208

6.3.3 Collateral Effects Induced by Cavitation . . . . . . . . . . 2096.4 Gas Embolism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

6.4.1 Treatment Strategies for Gas Embolism . . . . . . . . . . 2116.4.2 Gas Embolotherapy . . . . . . . . . . . . . . . . . . . . . 211References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

7 Nanocavitation for Cell Surgery . . . . . . . . . . . . . . . . . . . . 2257.1 Cavitation Induced by Femtosecond Laser Pulses . . . . . . . . . 226

7.1.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . 2267.1.2 Experimental Results . . . . . . . . . . . . . . . . . . . . 228

7.2 Cavitation During Plasmonic Photothermal Therapy . . . . . . . . 2297.2.1 Nanoparticles and Surface Plasmon Resonance . . . . . . 2317.2.2 Bubble Dynamics . . . . . . . . . . . . . . . . . . . . . . 2337.2.3 Biological Effects of Cavitation . . . . . . . . . . . . . . . 241References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

8 Cavitation in Other Non-Newtonian Biological Fluids . . . . . . . . 2498.1 Cavitation in Saliva . . . . . . . . . . . . . . . . . . . . . . . . . 249

8.1.1 Cavitation During Ultrasonic Plaque Removal . . . . . . . 2498.1.2 Cavitation During Passive Ultrasonic Irrigation

of the Root Canal . . . . . . . . . . . . . . . . . . . . . . 2528.1.3 Cavitation During Laser Activated Irrigation

of the Root Canal . . . . . . . . . . . . . . . . . . . . . . 2548.1.4 Cavitation During Orthognathic Surgery

of the Mandible . . . . . . . . . . . . . . . . . . . . . . . 2568.2 Cavitation in Synovial Liquid . . . . . . . . . . . . . . . . . . . . 2568.3 Cavitation in Aqueous Humor . . . . . . . . . . . . . . . . . . . 258

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Chapter 1Non-Newtonian Fluids

A fluid can be defined as a material that deforms continually under the application ofan external force. In other words, a fluid can flow and has no rigid three-dimensionalstructure. An ideal fluid may be defined as one in which there is no friction. Thusthe forces acting on any internal section of the fluid are purely pressure forces, evenduring motion. In a real fluid, shearing (tangential) and extensional forces alwayscome into play whenever motion takes place, thus given rise to fluid friction, becausethese forces oppose the movement of one particle relative to another. These frictionforces are due to a property of the fluid called viscosity. The friction forces in fluidflow result from the cohesion and momentum interchange between the moleculesin the fluid. The viscosity of most of the fluids we encounter in every day life isindependent of the applied external force. There is, however, a large class of fluidswith a fundamental different behaviour. This happens, for example, whenever thefluid contains polymer macromolecules, even if they are present in minute concen-trations. Two properties are responsible for this behaviour. On one hand, polymerschange the viscosity of the suspension by changing their shape depending on thetype of flow. On the other hand, polymer have long relaxation times associated withthem, which are on same order as the time scale of the flow, and allow the polymersto respond to the flow with a corresponding time delay. Other complex systemsconsisting of several phases, such as suspensions or emulsions and most of the bio-logical fluids, behave in a similar manner. In the following, we will focus on someof 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 becauseit relates the local stresses in a moving fluid to the rate of deformation of the fluidelement. When a fluid is sheared, it begins to move at a rate of deformation inverselyproportional to viscosity. To better understand the concept of shear viscosity weassume the model illustrated in Fig. 1.1. Two solid parallel plates are set on the top

1E-A. Brujan, Cavitation in Non-Newtonian Fluids,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 atrest, 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 liq-uid molecules into motion; this layer transmits the action to the subsequent layersunderneath it because of the intermolecular forces between the liquid molecules. Ina steady state, the velocities of these layers range from U (the layer closest to themoving 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 ofthe film is called shear strain (�x/L), and the shear strain per unit time is called theshear rate (U/Y). If the distance Y is not too large or the velocity U too high, thevelocity 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 thevelocity gradient du/dy. If a constant of proportionality η is now introduced, theshearing stress between any two thin sheets of fluid may be expressed by

τ = F

A= η

U

Y= η

du

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 calledthe shear rate. The dimensions of dynamic viscosity are force per unit area dividedby velocity gradient or shear rate. In the metric system the dimensions of dynamicviscosity 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 frequentlya more convenient unit. It has a further advantage that the dynamic viscosity of waterat 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 involvingviscosity there frequently appears the value of viscosity divided by density. This isdefined as kinematic viscosity, ν, so called because force is not involved, the onlydimensions 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 systemthe common units are cm2/s, also called the stoke (St). The centistoke (cSt) (0.01St) 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 notchange with rate of deformation is said to be a Newtonian fluid and can be rep-resented by a straight line in Fig. 1.2. The slope of this line is determined by theviscosity. 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 whichsustains a certain amount of stress before suffering a plastic flow can be shown by astraight line intersecting the vertical axis at the yield stress.

The relationship between stress and deformation rate given in Eq. (1.3) representsa constitutive equation of the fluid in a simple shear flow. We can generalize thisresult by saying that, in simple fluids, the stress on a material is determined by thehistory of the deformation involving only gradients of the first order or more exactlyby the relative deformation tensor as every fluid is isotropic.

A general constitutive equation which describes the mechanics of materials inclassical fluid mechanics can be written as:

tij = −pδij + τij + λvekkδij, (1.5)

Fig. 1.2 Rheological behaviour of materials


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 andtime t, p(�x, t) is the pressure in the fluid, τ is the extra stress tensor, λv is the volumeviscosity, and E is the rate of deformation tensor of the velocity field �u(�x, t):

eij = 1

2

(∂ui

∂xj+ ∂uj

∂xi

)(1.7)

or

E(�x, t) = 1

2

((∇�u) + (∇�u)T) , (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, secondand third invariants of the rate of deformation tensor:

I1 = eii, I2 = 1

2(eiiejj − eijeij), I3 = det(eij). (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 thenis a function of the second invariant I2 alone, and Eq. (1.9) can be written in asimplified form as

τ = η(I2)E. (1.11)

If the fluid does not undergo a volume change, i.e. it is incompressible, then the lastterm on the right-hand side of Eq. (1.6) drops out and the volume viscosity has norole to play.

1.1.2 Non-Newtonian Fluids

There is a certain class of fluids, called non-Newtonian fluids, in which the viscosityη varies with the shear rate. A particular feature of many non-Newtonian flu-ids is the retention of a fading “memory” of their flow history which is termedelasticity. Typical representatives of non-Newtonian fluids are liquids which areformed either partly or wholly of macromolecules (polymers), or two phase

1.1 Definitions 5

materials, like, for example, high concentration suspensions of solid particles ina liquid carrier solution.

There are various types of non-Newtonian fluids. Pseudoplastic fluids are thosefluids for which viscosity decreases with increasing shear rate and hence are oftenreferred to as shear-thinning fluids. These fluids are found in many real fluids, suchas polymer melts and solutions or glass melt. When the viscosity increases withshear rate the fluids are referred to as dilatant or shear-thickening fluids. These flu-ids are less common than with pseudoplastic fluids. Dilatant fluids have been foundto closely approximate the behaviour of some real fluids, such as starch in waterand 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 shearstress, and the duration of shear does not play any role so far as the viscosity isconcerned. Many of these fluids exhibits a constant viscosity at very small shearrates (referred to as zero-shear viscosity, η0) and at very large shear rates (referredto as infinite-shear viscosity, η∞). Some fluids do not flow unless the stress appliedexceeds a certain value referred to as the yield stress. These fluids are termed fluidswith yield stress or viscoplastic fluids. The variation of the shear stress with shearrate for pseudoplastic and dilatant fluids with and without yield stress is shown inFig. 1.3. Viscoelastic fluids are those fluids that possess the added feature of elastic-ity apart from viscosity. These fluids have a certain amount of energy stored insidethem as strain energy thereby showing a partial elastic recovery upon the removalof a deforming stress. In the case of thixotropic fluids, the shear stress decreaseswith time at a constant shear rate. An example of a thixotropic material is non-drippaint, which becomes thin after being stirred for a time, but does not run on the wallwhen it is brushed on. By contrast, when the shear stress increases with time at aconstant shear rate the fluids are referred to as rheopectic fluids. Some clay suspen-sions exhibit rheopectic behaviour. Figure 1.4 shows a schematic of the thixotropic

Fig. 1.3 Rheological behaviour of non-Newtonian fluids


Fig. 1.4 Rheopectic and thixotropic fluids

and rheopectic fluid behaviour. In the case of thixotropic and rheopectic fluids, theshear rate is a function of the magnitude and duration of shear, and the time lapsebetween consecutive applications of shear stress.

Viscoelastic fluids have some additional features. When a viscoelastic fluid issuddenly strained and then the strain is maintained constant afterward, the corre-sponding stresses induced in the fluid decrease with time. This phenomenon is calledstress relaxation. If the fluid is suddenly stressed and then the stress is maintainedconstant afterward, the fluid continues to deform, and the phenomenon is calledcreep. If the fluid is subjected to a cycling loading, the stress–strain relationship inthe 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 flowbehaviour may seem abnormal or even paradoxical. For example, when a rod isrotated in an elastic non-Newtonian fluid, the fluid climbs up the rod against theforce of gravity. This is because the rotational force acting in a horizontal planeproduces a normal force at right angles to that plane. The tendency of a fluidto flow in a direction normal to the direction of shear stress is known as theWeissenberg effect. Another effect caused by viscoelasticity is the die swell effectof the fluid as it leaves a die exit. This expansion is an elastic response of thefluid to energy stored when its shape changes while entering the die. This energyis released as the fluid leaves the die and causes a swelling effect normal to thedirection of flow in the die. It has been also observed that, when a viscoelasticfluid flows in a tube with a sudden contraction, bubbles with a certain diametercome to a sudden stop right at the entrance of the contraction along the centerlinebefore finally passing through after a hold time. This behaviour has been termed theUebler 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 internalstress to the applied strain. For non-Newtonian fluids, more elaborate constitu-tive equations, containing several material parameters, are needed to describe theresponse of these fluids to complex, time-dependent flows. There exists no generalmodel, i.e., no universal constitutive equation that describes all non-Newtonian fluidbehaviour. Currently successful theories are either restricted to very specific, sim-ple flows, especially generalizations of simple shear flow and extensional flow, forwhich rheological data can be used to develop empirical models, or to very dilutesolutions for which the microscale dynamics is dominated by the motion of simple,isolated macromolecules. This section deals with the description of the nature anddiversity of material response to simple shearing and extensional flows. The analy-sis of experimental methods for measuring these quantities is presented in the nextsection.

1.2.1 Simple Flows

We shall now examine some simple flow fields of fluids. Simple flow fields arerequired to determine the material properties of the fluids and these are separated inthree 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 rectangularCartesian 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 rateis independent of time; it is presumed that the shear rate has been constant for sucha long time that all the stresses in the fluid are time-independent. The extra stresstensor in such a flow is thus defined by

τ =⎛⎝ τxx τxy 0τyx τyy 00 0 τzz

⎞⎠ , (1.13)

where τxy = τyx are called the shear stress components, and τxx, τyy, and τ zz arecalled the normal stress components. The corresponding stress distribution for anon-Newtonian fluid can be written in the form

τxy = τ (γ ) = η(γ )γ , (1.14)


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 Newtonianfluid, η is a constant and N1 and N2 are zero. The variation of η with shear rateand non-zero values of N1 and N2 are manifestations of non-Newtonian viscoelasticbehaviour. The second normal stress difference N2, however, receives less atten-tion due to difficulties in its measurement and for the smallness of its value. Formany non-Newtonian fluids, the value of N2 would be usually an order of magnitudesmaller than that of N2.

The viscosity function η, the primary and secondary normal stress coefficientsψ1, and ψ2, respectively, are the three parameters which completely determine thestate of stress in any steady simple shear flow. They are often referred to as theviscometric functions. The normal stress coefficients are defined as follows:

τxx − τyy = ψ1(γ )γ 2, (1.17)

and

τyy − τzz = ψ2(γ )γ 2, (1.18)

and are also functions of the magnitude of the strain rate. The first and second nor-mal stress coefficients do not change in sign when the direction of the strain ratechanges. The primary normal stress coefficient is used to characterize the elasticityof a non-Newtonian fluid. A constant primary normal stress coefficient is obtainedwhen 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 vis-coelastic fluid. The oscillatory tests belong to the general framework of dynamiccharacterization of viscoelastic fluids in which both stress and strain vary harmon-ically with time. The dynamic properties of viscoleastic fluids are of considerableimportance because they can be directly related to the viscous and elastic parametersderived from such measurements.

Oscillatory tests involve the measurement of the response of the fluid to a smallamplitude 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)


where γ 0 is the amplitude of the applied strain, γ0 is the shear rate amplitude, andω is the frequency. The resulting shear stress may be given in terms of amplitude,τ0, and phase shift, δ = π

2 − φ, 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 materialfunctions 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 sinu-soidal shear deformation divided by the strain and is a measure of the elastic energystored in the system at a particular frequency. G′ represents the solid like response ofa 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, theloss modulus, G′′, is defined as the stress 90◦ out-of-phase with the strain divided bythe 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 defor-mation. The dynamic viscosity, η′, and dynamic rigidity, η′′, are related to G′′ andG′′ 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 viscoelasticproperties because they are determined from the shear stress which is linear in strainfor 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 η0ω as ω → 0.

A method of comparing the storage and loss modulus is made by the calculationof the loss tangent defined as

tan δ = G′′

G′ , (1.27)

and represents the phase angle between stress and strain.


For more detail on these and other linear viscoelastic properties, standardreferences 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 exten-sion or extension-like deformations. An extensional flow is one in which fluidelements are stretched or extended without being rotated or sheared. Extensionalflow can be visualized as that occurring when a material is longitudinally stretchedas in fiber spinning. In this case, the extension occurs in a single direction and therelated flow is termed uniaxial extension. Extension of material takes place in pro-cessing operation as well, such as film blowing and flat-film extrusion. Here, theextension occurs in two directions and the flow is referred to as biaxial extensionin one case and planar extension in the other. In biaxial extension, the material isstretched in two directions and compressed in the other. In planar extension, thematerial is stretched in one direction, held to the same dimension in a second, andcompressed in the third. A schematic representation of the three types of extensionalflow fields is shown in Fig. 1.5.

Uniaxial Extensional Flow

In a uniaxial extensional flow, the dimension of the fluid elements changes in onlyone direction. The velocity components are:

ux = εx, uy = − ε2

y, uz = − ε2

z, (1.28)

where ε = dux/dx is a constant strain rate, and the corresponding extra stresstensor is

τ =⎛⎝ τxx 0 0

0 τyy 00 0 τzz

⎞⎠ . (1.29)

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


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 andshear-thickening used to describe changes in viscosity with shear rate. The uniax-ial extensional viscosity is frequently qualitatively different from shear viscosity.For example, highly elastic polymer solutions that posses a shear viscosity thatdecreases in shear often exhibit uniaxial extensional viscosity that increases withstrain rate.

In most applications, the extensional viscosity is presented in terms of a Troutonratio which is defined conveniently to be the ratio of extensional viscosity to theshear viscosity, Tr = ηE/η. For calculating Trouton ratio in uniaxial extensionalflow, 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 equalvalues 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 inelasticliquids, 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 biaxialextensional flow is given by

ux = εBx, uy = εBy, uz = −2εBz. (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, theshear viscosity should be evaluated at a shear rate numerically equal to

√12ε. For a

Newtonian fluid, TrEB = 6.


Planar Extensional Flow

Planar extensional flow is the type of flow where there is no deformation in onedirection. 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 (εP) εP, (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 vis-cosity should be evaluated at a shear rate numerically equal to 2ε. For a Newtonianfluid, TrP = 4.

It is difficult to generate planar extensional flow and experimental tests of thistype 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 asthe zero concentration limit of the reduced viscosity, ηred = ηsp/c, where c is thepolymer concentration and ηsp is the specific viscosity. The specific viscosity isdefined as the relative polymer contribution to viscosity ηsp = (η0 − ηs)/μs, whereη0 is the zero-shear viscosity and ηs is the solvent viscosity. The intrinsic viscositycan thus be expressed as:

[η] = limc→0

ηred = limc→0

η0 − ηs

cηs. (1.36)

Note that the intrinsic viscosity has dimensions of reciprocal concentration. Theintrinsic viscosity is determined graphically by plotting ηred versus c and extrapo-lating to zero concentration. It is also found that extrapolation to zero concentrationof the inherent viscosity, ηinh = 1

c ln(ηsp + 1), can also be used to determine theintrinsic viscosity and the same result for [η] must be achieved.

The most common relation between specific viscosity and polymer concentrationis that of Huggins (1942),

ηsp

c= [η] + k′[η]2c, (1.37)

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


1

cln(ηsp

c

)= [η] − k′′[η]2c, (1.38)

where k′′ is the Kraemer constant, may also be used. Huggins slope constant andKraemer 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] = kMαη , (1.39)

where k and α are determined from a double logarithmic plot of intrinsic viscosityand molecular weight. These parameters have been published for many systems byBandrup and Immergut (1975).

The polymer solutions are regarded as dilute when there is no interaction betweenmolecules. A standard method to evaluate whether a polymer solution is dilute is todetermine a dimensionless concentration of polymer solution which can be givenby either [η]c (Flory 1953) or cNAV/Mw (Doi and Edwards 1986), where c is thepolymer concentration, NA is the Avogadro’s number, V is the volume occupied by apolymer molecule, and Mw is the average molecular weight. Flexible polymers tendto occupy a spherical region in solution such that V = 4πR3

h/3. In the case of rigidmolecules, the spherical region required such that the large aspect ratio molecule canfreely rotate without interaction with its neighbours is calculated from the moleculelength such that V = πL3/6, where L is the length of the molecule. The length L canbe determined using relations given by Broersma (1960) and Young et al. (1978) forrigid molecules, L = Rh

[2δ − 0.19 − (8.24/δ)+ (

12/δ2)]

, where δ = ln (L/r) isthe aspect ratio of a rod and r is the radius of the rigid rod. The polymer solutionis regarded as dilute when both dimensionless concentrations are less than unity.When one of the dimensionless concentrations is larger than 1, the polymer solutionis considered semi-dilute.

1.2.3 Dimensionless Numbers

Fluid dynamics is parametrized by a series of dimensionless numbers expressingthe relative importance of various physical phenomena. These include, for example,the Reynolds number, addressing inertial effects, the Froude number, describinggravity-driven flows, the Weber number, addressing the importance of surfacetension forces, the Grashof number, addressing buoyancy effects, or the Mach num-ber, describing the importance of liquid compressibility. In the specific case ofnon-Newtonian fluids, three additional sets of non-dimensional parameters are gen-erated, namely the Weissenberg number, the Deborah number, and the elasticitynumber, describing elastic effects. The dimensionless numbers are particularly use-ful for scaling arguments, for consolidating experimental, analytical, and numericalresults into a compact form, and also for cataloging various flow regimes.


1.2.3.1 Reynolds Number

Of all dimensionless numbers encountered in fluid dynamics, the Reynolds num-ber is the one most often mentioned in connection with non-Newtonian fluids. TheReynolds number represents the ratio of inertia forces to viscous forces and has theexpression:

Re = LU

ν= ρLU

η, (1.40)

where L is a linear dimension that may be any length that is significant in the flowpattern and U is the flow velocity. For example, for a pipe completely filled, Lmight be either the diameter or the radius, and the numerical value of Re will varyaccordingly.

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 deformationtime, either inverse extension rate 1/ε or shear rate 1/γ . When Wi is small, theliquid 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 isdeformed 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 timescale τflow = L0/U0 required for a liquid to transverse it with velocity U0. The flowtime 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, wehave a solid-like response. In the limit, when De = 0 one has a Newtonian fluid, andwhen De = ∞, an elastic solid. The usage of De and Wi can vary. Some referencesuse Wi exclusively to describe shear flows and use De for the general case, whereasothers use Wi for local flow time scales due to a local shear and De for global flowtime scales due to residence time in flow.


1.2.3.4 Elasticity Number

As the flow velocity increases, elastic effects become stronger and De and Weincrease. However, the Reynolds number Re increases in the same way, so thatinertial effects become more important as well. The elasticity number

El = De/Re = τfluidη

ρL2, (1.43)

where L is a dimension setting the shear rate, expresses the relative importance ofelastic to inertial effects. Significantly, El depends only on the geometry and mate-rial properties of the fluid, and is independent of flow rate. For example, extrusion ofpolymer melts corresponds to El >> 1, whereas processing flows for dilute polymersolutions (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 governsthe motion of a non-Newtonian fluid. Numerous constitutive equations have beenproposed to describe various classes of non-Newtonian fluids and a few of the sim-plest 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) canbe 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 describesthe 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. Forthe special case of a Newtonian fluid (n = 1), the consistency index K is identicallyequal 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 themost well-known and widely-used empiricism in engineering work, because a widevariety of flow problems have been solved analytically for it. One can often get arough estimate of the effect of the non-Newtonian viscosity by making a calculationbased on the power-law model. One shortcoming of the power law model is thatit does not describe the low shear and high shear rate constant viscosity data of


shear-thinning or shear-thickening fluids. For n < 1, this model presents a problemwhen 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

η = η∞ + η0 − η∞[1 + (λcγ )

]N , (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. Forexample, the Carreau model with a yield term given by

η = τ0

γ+ ηp[

1 + (λcγ )]N , (1.46)

where τ0 is the yield stress and ηp is the plateau viscosity, was employed in the studyof the rheological behaviour of glass-filled polymers (Poslinski et al. 1988).

The Casson Model

The Casson model given by

γ ={(√

τ−√τ0√

η

)2, for τ ≥ τ0

0 , for τ < τ0

, (1.47)

where τ0 is the yield stress, captures both the yield stress and shear dependent vis-cosity 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 yieldstress results in a plug flow and the velocity distribution shaped like a bluntedparabola that is so typical of blood flow in small diameter vessels. The Casson modelwas originally developed to describe the flow of printing ink through capillaries andwas later applied to other fluids containing chain like particles. The Casson equa-tion has also proven useful for the description of the flow of blood on both glass andfibrin surfaces.

1.2.4.2 Viscoelastic Fluids

A large number of constitutive equations have been proposed to describe theviscoelastic behaviour of non-Newtonian fluids. The Maxwell and Oldroyd-B mod-els have had a popularity far beyond expectation and anticipation. Their relativesimplicity has obviously been an attraction, especially in the case of numerical


simulation of viscoelastic flows, where simple models have been essential in thedevelopment of numerical strategies. Other important viscoelastic models that havebeen 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 modelwhich considers the fluid as being both viscous and elastic. The Maxwell equationis given by:

τ + λ∂τ

∂t= 2ηE, (1.48)

where λ is the relaxation time and η is the constant shear viscosity. For steady-statemotions this equation simplifies to the Newtonian fluid with viscosity η.

By replacing the time derivative with the convected time derivative, the upperconvected 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

∂t+ (�u · ∇)τ − (∇�u)T τ − τ (∇�u) . (1.50)

For steady simple shear flow, the Maxwell relaxation time is

λ = N1

2ηγ 2= ψ1

2η, (1.51)

while in small-amplitude oscillatory flow, the viscoelastic properties for this modelare given by

G′ = ληω2

1 + λ2ω2, (1.52)

and

η′ = η

1 + λ2ω2. (1.53)

At low frequency, G′ is predicted to vary quadratically with frequency while itapproaches a constant value at high frequencies.

The uniaxial extensional viscosity for the upper convected Maxwell model is

ηE = 3η1

(1 + λε) (1 − 2λε). (1.54)


This model predicts strain rate thickening behaviour, but the predicted extensionalviscosity asymptotes to infinity when ε = 1

/(2λ) .

The upper convective Maxwell model exhibits many of the qualitative behavioursof viscoelastic fluids, including normal stresses in shear, extension thickening, andelastic recovery. However, it does not exhibit shear thinning. To get a reasonablematch to viscoelastic behaviour, one must introduce some additional nonliniaritiesby 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

ητ, (1.56)

and the Phan-Thien-Tanner model, for which

Y = exp

(ελ

ηtr(τ)

)I. (1.57)

Each of these models adds another dimensionless parameter, α or ε, that control thenonlinearity.

A multi-mode Maxwell model may also be used to allow the material functionsto be predicted more accurately by adjusting the parameters in each mode. Theextra stress tensor is expressed, in this case, as a combination of several relaxationtimes as

τ =n∑

i=1

τi, (1.58)

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 equationby including the convected time derivative of the rate of deformation tensor. Thisway 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 viscosityhas also a constant value. We observe that, by setting λ2 = 0, the above equation


reduces to the upper convected Maxwell model. The Oldroyd-B model qualitativelydescribes many features of the so-called Boger fluids (elastic fluids with almostconstant 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 + λ21ω

2, (1.62)

and

η′ =(1 + λ1λ2ω

2)η

1 + λ21ω

2. (1.63)

As in the case of Maxwell model, the Oldryod-B model predicts that at low fre-quencies the storage modulus varies quadratically with frequency while at highfrequencies a constant value is obtained.

The equation for the uniaxial extensional viscosity is given by

ηE = 3η1 − λ2ε − 2λ1λ2ε

2

1 − λ1ε − 2λ21ε

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 modelwhich is given by

τ + λ1∂τ

∂t= 2η

(E + λ2

∂E∂t

). (1.65)

It is interesting to note that this equation was originally proposed for the study ofwave propagation in the earth’s mantle (Jeffreys 1929).

The Dumbbell Model

In elastic dumbbell models a polymer is described as two beads connected by aHookean spring. The beads represent the ends of the molecule and their separationis 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 dueto the spring connecting one bead to the other. It can be further assumed that thepolymer solution is sufficiently dilute that the polymer molecules do not interactwith one another. The polymer contribution to the stress tensor is

τp + λH∇τp = nkBTλHE, (1.66)

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