Viscoelasticity and Rheology (eBook)

Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin-Madison, October 16-18, 1984

Arthur S. Lodge, John A. Nohel, Michael Renardy (Herausgeber)

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2014 | 1. Auflage
456 Seiten
Elsevier Science (Verlag)
978-1-4832-6335-9 (ISBN)

Infos

Viscoelasticity and Rheology covers the proceedings of a symposium by the same title, conducted by the Mathematics Research Center held at the University of Wisconsin-Madison on October 16-18, 1984. The contributions to the symposium are divided into four broad categories, namely, experimental results, constitutive theories, mathematical analysis, and computation. This 16-chapter work begins with experimental topics, including the motion of bubbles in viscoelastic fluids, wave propagation in viscoelastic solids, flows through contractions, and cold-drawing of polymers. The next chapters covering constitutive theories explore the molecular theories for polymer solutions and melts based on statistical mechanics, the use and limitations of approximate constitutive theories, a comparison of constitutive laws based on various molecular theories, network theories and some of their advantages in relation to experiments, and models for viscoplasticity. These topics are followed by discussions of the existence, regularity, and development of singularities, change of type, interface problems in viscoelasticity, existence for initial value problems and steady flows, and propagation and development of singularities. The remaining chapters deal with the numerical simulation of flow between eccentric cylinders, flow around spheres and bubbles, the hole pressure problem, and a review of computational problems related to various constitutive laws. This book will prove useful to chemical engineers, researchers, and students.

Inhalt

Front Cover 1
Viscoelasticity and Rheology 4
Copyright Page 5
Table of Contents 6
Contributors 8
Preface 10
CHAPTER 1. THE MOTION OF VISCOELASTIC FLUIDS AROUND SPHERES AND BUBBLES 12
INTRODUCTION 12
2. EXPERIMENTAL EVIDENCE 12
3. LOW DEBORAH NUMBER FLOW 14
4. NUMERICAL SIMULATIONS AT SMALL AND INTERMEDIATE DEBORAH NUMBERS 17
REFERENCES 22
CHAPTER 2. WAVE PROPAGATION IN VISCOELASTIC SOLIDS 24
1. INTRODUCTION 24
2. KINEMATICS OF WAVES 25
3. SIMPLE MATERIALS WITH FADING MEMORY 27
4. STEADY WAVES 29
5. SHOCK WAVES 36
6. ACCELERATION WAVES 43
7. THERMODYNAMIC INFLUENCES 52
REFERENCES 54
CHAPTER 3. 
58 
1. INTRODUCTION 58
2. BEHAVIOUR IN SIMPLE FLOWS (RHEOMETRY) 60
3. BEHAVIOUR IN COMPLEX FLOWS 64
4. THEORETICAL SIMULATION OF OBSERVED BEHAVIOUR IN COMPLEX FLOWS 79
5. COMPARISON OF THEORY AND EXPERIMENT 86
REFERENCES 88
ACKNOWLEDGMENT 90
CHAPTER 4. 
92 
1. Introduction 92
2. Theoretical Considerations 93
3. Experimental Procedures 95
4. Experimental Results 96
5. Discussion 105
REFERENCES 114
CHAPTER 5. POLYMERIC LIQUIDS: FROM MOLECULAR MODELS TO CONSTITUTIVE EQUATIONS 116
1. INTRODUCTION 116
2. MOLECULAR MODELS 117
3. KINETIC THEORY FOR DILUTE POLYMER SOLUTIONS, ILLUSTRATED WITH THE ELASTIC DUMBBELL MODEL [DPL, Chapter 11] 119
4. DISTRIBUTION FUNCTION AND CONSTITUTIVE EQUATION FOR HOOKEAN DUMBBELLS [DPL, §10.4] 122
5. AN APPROXIMATE CONSTITUTIVE EQUATION FOR FINITELY-EXTENSIBLE NONLINEAR ELASTIC (FENE) DUMBBELLS [DPL, §10.5] 124
6. MODEL OF INTERACTING KRAMERS CHAINS AS A MODEL FOR A POLYMER MELT 126
7. USES OF THE KINETIC THEORY RESULTS 128
REFERENCES 133
ACKNOWLEDGMENTS 134
CHAPTER 6. ON SLOW-FLOW APPROXIMATIONS TO FLUIDS WITH FADING MEMORY 136
1. INTRODUCTION 136
2. FADING MEMORY 141
3. ON INFINITESIMAL DEFORMATIONS 146
4. ORIGIN AND PROPERTIES OF SECOND-ORDER FLUIDS 149
5. A CONDITION SUFFICIENT FOR FAILURE OF THE SECOND-ORDER APPROXIMATION 159
REFERENCES 164
CHAPTER 7. A COMPARISON OF MOLECULAR AND NETWORK-CONSTITUTIVE THEORIES FOR POLYMER FLUIDS 168
1. INTRODUCTION 168
2. COMPARISON OF THE SIMPLEST REPRESENTATIVES OF MOLECULAR AND TRANSIENT NETWORK MODELS 169
3. MULTI-MODE MOLECULAR AND NETWORK MODELS 174
4. MOLECULAR MODELS WITH HYDRODYNAMIC INTERACTION AND NETWORK MODELS WITH NON-AFFINE MOTION 176
5. NETWORK MODELS WITH TIME DEPENDENT JUNCTION DENSITY AND MOLECULAR MODELS WITH FUNCTIONAL-TYPE CONFIGURATION-DEPENDENT MOBILITY 182
6. CONCLUSIONS 187
REFERENCES 189
CHAPTER 8. ON USING RUBBER AS A GUIDE FOR UNDERSTANDING POLYMERIC LIQUID BEHAVIOR 192
1. INTRODUCTION 192
2. PERFECTLY ELASTIC SOLIDS 195
3. POLYMERIC LIQUIDS 198
4. THE ONE-STEP SHEAR EXPERIMENT 201
5. SOME CLASS I EQUATIONS 205
6. APPENDIX: DEFINITIONS 209
REFERENCES 217
CHAPTER 9. 
220 
1. General Considerations 220
2. A specific model: general description 223
3. Construction of the model 224
4. Characterization 225
5. A three-dimensional extension 229
REFERENCES 231
Acknowledgement 231
CHAPTER 10. 
232 
1. INTRODUCTION 232
2. STATEMENT OF RESULTS 234
3. PROOF OF THEOREMS 236
REFERENCES 244
CHAPTER 11. HYPERBOLIC PHENOMENA IN THE FLOW OF VISCOELASTIC FLUIDS 246
ABSTRACT 246
1. INTRODUCTION 247
2. RATE EQUATIONS FOR FLUIDS WITH INSTANTANEOUS ELASTICITY 250
3. WAVE SPEEDS I, THEORETICAL 254
4. WAVE SPEEDS II, PHYSICAL 255
5. VORTICITY 256
6. SPECIAL MODELS 258
7. CLASSIFICATION OF TYPE IN STEADY PLANE FLOW 262
8. CONDITIONS FOR A CHANGE OF TYPE. PROBLEMS OF NUMERICAL SIMULATION 267
9. LINEARIZED PROBLEMS OF CHANGE OF TYPE 270
10. CHANNEL FLOWS WITH WAVY WALLS 275
11. PROBLEMS ASSOCIATED WITH THE FLOW OF VISCOELASTIC FLUIDS AROUND BODIES 287
12. FLOW OVER A FLAT PLATE 295
13. NONLINEAR WAVE PROPAGATION AND SHOCKS 305
REFERENCES 317
APPENDIX A: BREAKDOWN OF SMOOTH SHEARING FLOW IN VISCOELASTIC FLUIDS FOR TWO CONSTITUTIVE RELATIONS: THE VORTEX SHEET VS. THE VORTEX SHOCK 320
A0. INTRODUCTION 320
A1. RECTILINEAR SHEARING FLOWS 321
A2. CONSTITUTIVE ASSUMPTIONS 322
A3. SHEARING PERTURBATION OF A STEADY SHEARING FLOW 324
A4. ANALYSIS OF FLOW WITH FIRST CONSTITUTIVE ASSUMPTION: STRESS NON-LINEAR FUNCTION OF A LINEAR FUNCTIONAL OF SHEAR RATE 325
A5. ANALYSIS OF FLOW WITH SECOND CONSTITUTIVE ASSUMPTION: STRESS LINEAR FUNCTIONAL OF A NON-LINEAR FUNCTION OF SHEAR RATE 327
A6. A BREAKDOWN RESULT 329
7. PHYSICAL IMPLICATIONS OF BREAKDOWN OF SMOOTH SOLUTIONS 330
REFERENCES 332
CHAPTER 12. ABSORBING BOUNDARIES FOR VISCOELASTICITY 334
I. Introduction 334
II. Elastic Bars 336
III. Linear, Homogeneous, Viscoelastic Bars 339
IV. Approximate Boundary Conditions 342
V. Inhomogeneous Elastic Bars 350
VI. Remarks on Two-Dimensional Problems 353
REFERENCES 355
CHAPTER 13. RECENT DEVELOPMENTS AND OPEN PROBLEMS IN THE MATHEMATICAL THEORY OF VISCOELASTICITY 356
1. EXISTENCE RESULTS FOR INITIAL VALUE PROBLEMS 356
2. PROPAGATION AND DEVELOPMENT OF SINGULARITIES 360
3. STEADY FLOWS OF VISCOELASTIC FLUIDS 364
REFERENCES 368
CHAPTER 14. EVALUATION OF CONSTITUTIVE EQUATIONS: MATERIAL FUNCTIONS AND COMPLEX FLOWS OF VISCOELASTIC FLUIDS 372
I. Introduction 372
2. Six Differential Constitutive Equations 373
3. Shear and Shear-Free Flow Material Functions 375
4. Material Functions for the Six Constitutive Equations 382
5. An Example Complex Flow: The Journal Bearing 385
6. Conclusions 398
REFERENCES 400
CHAPTER 15. FINITE ELEMENT METHODS FOR VISCOELASTIC FLOW 402
ABSTRACT 402
INTRODUCTION 403
CONSTITUTIVE EQUATIONS 404
CHARACTERISTICS AND NUMERICAL METHODS 409
COMPUTATION WITH A SINGLE-INTEGRAL MODEL 415
A MODEL PROBLEM 420
CONCLUSIONS 428
REFERENCES 428
CHAPTER 16. CONSTITUTIVE EQUATIONS FOR THE COMPUTING PERSON 432
1. INTRODUCTION 432
2. FAMILIES OF CONSTITUTIVE EQUATIONS 434
3. STABILITY CONSIDERATIONS 438
4. SOME RESULTS FOR LEONOV AND PTT MODELS 442
6. CONCLUSION 447
REFERENCES 449
Index 452

Textauszug

THE MOTION OF VISCOELASTIC FLUIDS AROUND SPHERES AND BUBBLES 

Ole Hassager,     Danmarks Tekniske HØjskole, Lyngby, Denmark and Mathematics Research Center and Chemical Engineering Department, University of Wisconsin, Madison, Wisconsin

Publisher Summary

This chapter discusses the translation of bubbles and solid spheres in a viscoelastic fluid. The motion is assumed to be caused by a gravitational field or an imposed force. The chapter presents some experimental observations of bubble shapes, velocity fields, and friction coefficients. One of the most striking features of translating air bubbles in viscoelastic fluids is that they develop a cusp at the rear pole. Often this effect may be observed in an almost full shampoo bottle by rapidly turning it upside down and watching an air bubble rise. More controlled experiments by Astarita and Apuzzo with a series of bubbles of increasing volume show a transition from a spherical bubble shape to an elongated ellipsoidal shape and the development of the cusp at the rear pole. They also measured the rise velocity of the bubbles as a function of the volume, and showed that for some viscoelastic fluids there is a critical volume at which the rise velocity appears to have a discontinuous increase when plotted as function of the volume.

INTRODUCTION

This paper is concerned with the translation of bubbles and solid spheres in a viscoelastic fluid. The motion is assumed to be caused by a gravitational field or an imposed force, and we will consider the situations in which the motion takes place in an unbounded fluid, quiescent far from the object as well as the situation in which the motion takes place in a finite container (a cylinder). First we will review some experimental observations of bubble shapes, velocity fields and friction coefficients. In the next section we then consider perturbation solutions for motions that are so slow that the viscoelastic fluid behaves almost as a Newtonian fluid. In this so-called low Deborah number limit analytical solutions may be obtained for bubble shapes, friction factors and velocity fields by perturbation methods. In the last section we consider some numerical simulations that apply also for intermediate Deborah numbers.

2 EXPERIMENTAL EVIDENCE

One of the most striking features of translating air bubbles in viscoelastic fluids is that they develop a cusp at the rear pole. Often this effect may be observed in an almost full shampoo bottle by rapidly turning it upside down and watching an air bubble rise. More controlled experiments by Astarita and Apuzzo (1965) with a series of bubbles of increasing volume show a transition from a spherical bubble shape to an elongated ellipsoidal shape and the development of the cusp at the rear pole. Astarita and Apuzzo also measured the rise velocity of the bubbles as a function of the volume, and showed that for some viscoelastic fluids there is a critical volume at which the rise velocity appears to have a discontinuous increase when plotted as function of the volume. Similar measurements have been performed by Calderbank, Johnson and Loudon (1970) and Leal, Skoog and Acrivos (1971) who also documented discontinuous jumps in the rise velocity. The actual value of the jump depends on the particular polymer/solvent system as well as the temperature. The above investigators have reported jumps by factors in the range of 2–10, and the effect is in fact quite remarkable. It is possible that the velocity discontinuity is related to the development of the cusp at the rear pole. It is proposed that the “cusped” bubbles are really not closed surfaces, but rather open surfaces in which the “cusps” continue into thin gas filaments that eventually dissolve in the liquid. In some fluids (Hassager (1979)) the “cusps” lose rotational symmetry and take the form of a knife edge. In these circumstances the knife edge appears to continue into a thin sheet of air that supposedly eventually dissolves in the liquid. Both in the situation where the bubbles extend into filaments or into sheets there must be a critical total volume at which the boundary condition at the rear pole changes from one involving a stagnation point into another condition with no stagnation point. This could certainly give a discontinuous change in rise velocity and one may argue qualitatively that at least two mechanisms that would retard the bubble when it has a rear stagnation point will not be present when the bubble does not close at the rear pole. First as long as the bubble has a rear stagnation point there may be an accumulation of surface active impurities near that point that would cause immobilization of the surface. Second with the stagnation point present there will be an elongational flow near the rear pole that could account for much of the drag on the bubble.

More detailed information on the flow around bubbles and spheres may be obtained by laser Doppler measurements of the fluid velocity fields. The technique, unfortunately, is limited to fluid velocities well above those at which the velocity discontinuity takes place. We will refer to this region as the high Deborah number region. In this region the following two phenomena have been found and documented by laser Doppler anemometry:

First, in the wake region behind the bubble the fluid velocity as seen by an observer stationary with respect to the fluid far from the bubble is in the opposite direction to that in which the bubble is moving (Hassager (1979)). This wake flow is strikingly different from wake flow in Newtonian fluids where fluid is always pulled with the bubble, and has been termed “negative wake”. It has been demonstrated also by flow visualization by Contanceau and Hajjam (1982).

Second, in the wake flow region the fluid velocity field (referred to an observer on the bubble) is not steady even at Reynolds numbers much less than unity (Bisgaard 1983). This observation however is currently limited to one particular polymer solution.

The above described two phenomena in the wake behind bubbles in viscoelastic fluids have been observed also in wakes behind solid spheres, whereas the velocity discontinuity at low Deborah numbers does not occur for spheres.

3 LOW DEBORAH NUMBER FLOW

For intermediate and high Deborah numbers there is considerable ambiguity as to the correct constitutive equation to be used for incompressible viscoelastic fluids. However in the limit as the Deborah number tends to zero the correct expression for the stress tensor is the retarded motion expansion, which through terms of third order may be written:

__=−b1[γ__(1)+B2γ__(2)+B11γ__(1)·γ__(1)+B3γ__(3)+B12(γ__(1)·γ__(2)+γ__(2)·γ__(1))+B1:11γ__(1)(γ__(1):γ__(1))+…] (3.1)

(3.1)

where

__(1)=▽v_+(▽v_)†  andγ__(n+1)=DDtγ__(n)−{(▽v_)†·γ__(n)+γ__(n)·(▽v_)} (3.2)

(3.2)

Here b1 is the zero shear-rate viscosity, B2 and B11 are constants with dimension of time and B3, B12 and B1:11 are constants with dimension of time squared. The B2 and B11 are related to the zero-shear-rate first and second normal stress coefficients Ψ1,0 and Ψ2,0 by Ψ1,0 = −2b1 B2 and Ψ2,0 = b1 B11. Values of the parameters for various molecular models may be obtained from Table 1 of Bird (1984).

TABLE 1

Simulation Results for K(De, Rs/Rc) for the Single Time Constant Lodge Rubberlike...

Produkt-Details

• Erscheint lt. Verlag: 28.6.2014
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 1-4832-6335-5 / 1483263355
• ISBN-13: 978-1-4832-6335-9 / 9781483263359