Lodge / Renardy / Nohel Viscoelasticity and Rheology

Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin-Madison, October 16-18, 1984
1. Auflage 2014
ISBN: 978-1-4832-6335-9
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 456 Seiten

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.

Autoren/Hrsg.

Lodge, Arthur S.

Renardy, Michael

Nohel, John A.

Inhaltsverzeichnis

1;Front Cover;1
2;Viscoelasticity and Rheology;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;8
6;Preface;10
7;CHAPTER 1. THE MOTION OF VISCOELASTIC FLUIDS AROUND SPHERES AND BUBBLES;12
7.1;INTRODUCTION;12
7.2;2. EXPERIMENTAL EVIDENCE;12
7.3;3. LOW DEBORAH NUMBER FLOW;14
7.4;4. NUMERICAL SIMULATIONS AT SMALL AND INTERMEDIATE DEBORAH NUMBERS;17
7.5;REFERENCES;22
8;CHAPTER 2. WAVE PROPAGATION IN VISCOELASTIC SOLIDS;24
8.1;1. INTRODUCTION;24
8.2;2. KINEMATICS OF WAVES;25
8.3;3. SIMPLE MATERIALS WITH FADING MEMORY;27
8.4;4. STEADY WAVES;29
8.5;5. SHOCK WAVES;36
8.6;6. ACCELERATION WAVES;43
8.7;7. THERMODYNAMIC INFLUENCES;52
8.8;REFERENCES;54
9;CHAPTER 3.
OVERVIEW OF MACROSCOPIC VISCOELASTIC FLOW;58
9.1;1. INTRODUCTION;58
9.2;2. BEHAVIOUR IN SIMPLE FLOWS (RHEOMETRY);60
9.3;3. BEHAVIOUR IN COMPLEX FLOWS;64
9.4;4. THEORETICAL SIMULATION OF OBSERVED BEHAVIOUR IN COMPLEX FLOWS;79
9.5;5. COMPARISON OF THEORY AND EXPERIMENT;86
9.6;REFERENCES;88
9.7;ACKNOWLEDGMENT;90
10;CHAPTER 4.
NECKING PHENOMENA AND COLD DRAWING;92
10.1;1. Introduction;92
10.2;2. Theoretical Considerations;93
10.3;3. Experimental Procedures;95
10.4;4. Experimental Results;96
10.5;5. Discussion;105
10.6;REFERENCES;114
11;CHAPTER 5. POLYMERIC LIQUIDS: FROM MOLECULAR MODELS TO CONSTITUTIVE EQUATIONS;116
11.1;1. INTRODUCTION;116
11.2;2. MOLECULAR MODELS;117
11.3;3. KINETIC THEORY FOR DILUTE POLYMER SOLUTIONS, ILLUSTRATED WITH THE ELASTIC DUMBBELL MODEL [DPL, Chapter 11];119
11.4;4. DISTRIBUTION FUNCTION AND CONSTITUTIVE EQUATION FOR HOOKEAN DUMBBELLS [DPL, §10.4];122
11.5;5. AN APPROXIMATE CONSTITUTIVE EQUATION FOR FINITELY-EXTENSIBLE NONLINEAR ELASTIC (FENE) DUMBBELLS [DPL, §10.5];124
11.6;6. MODEL OF INTERACTING KRAMERS CHAINS AS A MODEL FOR A POLYMER MELT;126
11.7;7. USES OF THE KINETIC THEORY RESULTS;128
11.8;REFERENCES;133
11.9;ACKNOWLEDGMENTS;134
12;CHAPTER 6. ON SLOW-FLOW APPROXIMATIONS TO FLUIDS WITH FADING MEMORY;136
12.1;1. INTRODUCTION;136
12.2;2. FADING MEMORY;141
12.3;3. ON INFINITESIMAL DEFORMATIONS;146
12.4;4. ORIGIN AND PROPERTIES OF SECOND-ORDER FLUIDS;149
12.5;5. A CONDITION SUFFICIENT FOR FAILURE OF THE SECOND-ORDER APPROXIMATION;159
12.6;REFERENCES;164
13;CHAPTER 7. A COMPARISON OF MOLECULAR AND NETWORK-CONSTITUTIVE THEORIES FOR POLYMER FLUIDS;168
13.1;1. INTRODUCTION;168
13.2;2. COMPARISON OF THE SIMPLEST REPRESENTATIVES OF MOLECULAR AND TRANSIENT NETWORK MODELS;169
13.3;3. MULTI-MODE MOLECULAR AND NETWORK MODELS;174
13.4;4. MOLECULAR MODELS WITH HYDRODYNAMIC INTERACTION AND NETWORK MODELS WITH NON-AFFINE MOTION;176
13.5;5. NETWORK MODELS WITH TIME DEPENDENT JUNCTION DENSITY AND MOLECULAR MODELS WITH FUNCTIONAL-TYPE CONFIGURATION-DEPENDENT MOBILITY;182
13.6;6. CONCLUSIONS;187
13.7;REFERENCES;189
14;CHAPTER 8. ON USING RUBBER AS A GUIDE FOR UNDERSTANDING POLYMERIC LIQUID BEHAVIOR;192
14.1;1. INTRODUCTION;192
14.2;2. PERFECTLY ELASTIC SOLIDS;195
14.3;3. POLYMERIC LIQUIDS;198
14.4;4. THE ONE-STEP SHEAR EXPERIMENT;201
14.5;5. SOME CLASS I EQUATIONS;205
14.6;6. APPENDIX: DEFINITIONS;209
14.7;REFERENCES;217
15;CHAPTER 9.
ON VISCOPLASTIC MODELS;220
15.1;1. General Considerations;220
15.2;2. A specific model: general description;223
15.3;3. Construction of the model;224
15.4;4. Characterization;225
15.5;5. A three-dimensional extension;229
15.6;REFERENCES;231
15.7;Acknowledgement;231
16;CHAPTER 10.
DISSIPATION IN MATERIALS WITH MEMORY;232
16.1;1. INTRODUCTION;232
16.2;2. STATEMENT OF RESULTS;234
16.3;3. PROOF OF THEOREMS;236
16.4;REFERENCES;244
17;CHAPTER 11. HYPERBOLIC PHENOMENA IN THE FLOW OF VISCOELASTIC FLUIDS;246
17.1;ABSTRACT;246
17.2;1. INTRODUCTION;247
17.3;2. RATE EQUATIONS FOR FLUIDS WITH INSTANTANEOUS ELASTICITY;250
17.4;3. WAVE SPEEDS I, THEORETICAL;254
17.5;4. WAVE SPEEDS II, PHYSICAL;255
17.6;5. VORTICITY;256
17.7;6. SPECIAL MODELS;258
17.8;7. CLASSIFICATION OF TYPE IN STEADY PLANE FLOW;262
17.9;8. CONDITIONS FOR A CHANGE OF TYPE. PROBLEMS OF NUMERICAL SIMULATION;267
17.10;9. LINEARIZED PROBLEMS OF CHANGE OF TYPE;270
17.11;10. CHANNEL FLOWS WITH WAVY WALLS;275
17.12;11. PROBLEMS ASSOCIATED WITH THE FLOW OF VISCOELASTIC FLUIDS AROUND BODIES;287
17.13;12. FLOW OVER A FLAT PLATE;295
17.14;13. NONLINEAR WAVE PROPAGATION AND SHOCKS;305
17.15;REFERENCES;317
17.16;APPENDIX A: BREAKDOWN OF SMOOTH SHEARING FLOW IN VISCOELASTIC FLUIDS FOR TWO CONSTITUTIVE RELATIONS: THE VORTEX SHEET VS. THE VORTEX SHOCK;320
17.16.1;A0. INTRODUCTION;320
17.16.2;A1. RECTILINEAR SHEARING FLOWS;321
17.16.3;A2. CONSTITUTIVE ASSUMPTIONS;322
17.16.4;A3. SHEARING PERTURBATION OF A STEADY SHEARING FLOW;324
17.16.5;A4. ANALYSIS OF FLOW WITH FIRST CONSTITUTIVE ASSUMPTION: STRESS NON-LINEAR FUNCTION OF A LINEAR FUNCTIONAL OF SHEAR RATE;325
17.16.6;A5. ANALYSIS OF FLOW WITH SECOND CONSTITUTIVE ASSUMPTION: STRESS LINEAR FUNCTIONAL OF A NON-LINEAR FUNCTION OF SHEAR RATE;327
17.16.7;A6. A BREAKDOWN RESULT;329
17.16.8;7. PHYSICAL IMPLICATIONS OF BREAKDOWN OF SMOOTH SOLUTIONS;330
17.16.9;REFERENCES;332
18;CHAPTER 12. ABSORBING BOUNDARIES FOR VISCOELASTICITY;334
18.1;I. Introduction;334
18.2;II. Elastic Bars;336
18.3;III. Linear, Homogeneous, Viscoelastic Bars;339
18.4;IV. Approximate Boundary Conditions;342
18.5;V. Inhomogeneous Elastic Bars;350
18.6;VI. Remarks on Two-Dimensional Problems;353
18.7;REFERENCES;355
19;CHAPTER 13. RECENT DEVELOPMENTS AND OPEN PROBLEMS IN THE MATHEMATICAL THEORY OF VISCOELASTICITY;356
19.1;1. EXISTENCE RESULTS FOR INITIAL VALUE PROBLEMS;356
19.2;2. PROPAGATION AND DEVELOPMENT OF SINGULARITIES;360
19.3;3. STEADY FLOWS OF VISCOELASTIC FLUIDS;364
19.4;REFERENCES;368
20;CHAPTER 14. EVALUATION OF CONSTITUTIVE EQUATIONS: MATERIAL FUNCTIONS AND COMPLEX FLOWS OF VISCOELASTIC FLUIDS;372
20.1;I. Introduction;372
20.2;2. Six Differential Constitutive Equations;373
20.3;3. Shear and Shear-Free Flow Material Functions;375
20.4;4. Material Functions for the Six Constitutive Equations;382
20.5;5. An Example Complex Flow: The Journal Bearing;385
20.6;6. Conclusions;398
20.7;REFERENCES;400
21;CHAPTER 15. FINITE ELEMENT METHODS FOR VISCOELASTIC FLOW;402
21.1;ABSTRACT;402
21.2;INTRODUCTION;403
21.3;CONSTITUTIVE EQUATIONS;404
21.4;CHARACTERISTICS AND NUMERICAL METHODS;409
21.5;COMPUTATION WITH A SINGLE-INTEGRAL MODEL;415
21.6;A MODEL PROBLEM;420
21.7;CONCLUSIONS;428
21.8;REFERENCES;428
22;CHAPTER 16. CONSTITUTIVE EQUATIONS FOR THE COMPUTING PERSON;432
22.1;1. INTRODUCTION;432
22.2;2. FAMILIES OF CONSTITUTIVE EQUATIONS;434
22.3;3. STABILITY CONSIDERATIONS;438
22.4;4. SOME RESULTS FOR LEONOV AND PTT MODELS;442
22.5;6. CONCLUSION;447
22.6;REFERENCES;449
23;Index;452

Leseproben

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[?__(1)+B2?__(2)+B11?__(1)·?__(1)+B3?__(3)+B12(?__(1)·?__(2)+?__(2)·?__(1))+B1:11?__(1)(?__(1):?__(1))+…] (3.1)

(3.1)

where

__(1)=?v_+(?v_)†  and?__(n+1)=DDt?__(n)-{(?v_)†·?__(n)+?__(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 ?1,0 and ?2,0 by ?1,0 = -2b1 B2 and ?2,0 = 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...