# Numerical Simulation of Non-Newtonian Flow

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Numerical Simulation of Non-Newtonian Flow focuses on the numerical simulation of non-Newtonian flow using finite difference and finite element techniques. Topics range from the basic equations governing non-Newtonian fluid mechanics to flow classification and finite element calculation of flow (generalized Newtonian flow and viscoelastic flow). An overview of finite difference and finite element methods is also presented. Comprised of 11 chapters, this volume begins with an introduction to non-Newtonian mechanics, paying particular attention to the rheometrical properties of non-Newtonian fluids as well as non-Newtonian flow in complex geometries. The role of non-Newtonian fluid mechanics is also considered. The discussion then turns to the basic equations governing non-Newtonian fluid mechanics, including Navier Stokes equations and rheological equations of state. The next chapter describes a flow classification in which the various flow problems are grouped under five main headings: flows dominated by shear viscosity, slow flows (slightly elastic liquids), small deformation flows, nearly-viscometric flows, and long-range memory effects in complex flows. The remainder of the book is devoted to numerical analysis of non-Newtonian fluids using finite difference and finite element techniques. This monograph will be of interest to students and practitioners of physics and mathematics.

## Table of Contents

Preface

Section 1 : Non-Newtonian Fluid Mechanics

1. General Introduction

1.1 Introduction

1.2 Rheometrical Properties of Non-Newtonian Fluids

1.3 Non-Newtonian Flow in Complex Geometries

1.4 The Role of Non-Newtonian Fluid Mechanics

2. Basic Equations

2.1 Introduction

2.2 Field Equations

2.3 Navier Stokes Equations

2.4 Rheological Equations of State. Formulation Principles

2.5 The Simple Fluid

2.6 Approximate Constitutive Equations

2.7 A Pragmatic Approach to Constitutive Equations

2.8 Constraints on Rheological Equations of State

2.9 Boundary Conditions

Appendix I

3. Flow Classification

3.1 Introduction

3.2 Flows Dominated by Shear Viscosity

3.3 Slow Flow (Slightly Elastic Liquids)

3.4 Small-Deformation Flows

3.5 Nearly-Viscometric Flows

3.6 Highly Elastic Liquids Flowing in Complex Geometries

3.7 General Comments Concerning Flows Involving Abrupt Changes in Geometry

3.8 Some Remarks on Non-Dimensional Parameters

3.9 Basic Equations for the Flow of a Maxwell Fluid

4. An Overview of Numerical Simulation

4.1 Introduction

4.2 Step 1 : Formulating the Governing Equations and Boundary Conditions

4.3 Step 2 : Time Discretization

4.4 Step 3 : Space Discretization

4.5 Step 4 : Linearization

4.6 Step 5 : Solution of the Linearization Equation

4.7 Step 6 : Termination of the Nonlinear Iteration Loop

Section 2 : Finite Difference Techniques

5. Introduction to Finite Differences

5.1 Boundary Value Problems in One and Two Space Dimensions

5.2 Finite Difference Solution of Two-Point Boundary Value Problems: The Linear Case

5.3 Finite Difference Solution of Two-Point Boundary Value Problems: The Nonlinear Case

5.4 Finite Difference Solution of Elliptic Boundary Value Problems: Poisson's Equation

6. Finite Difference Simulation : Differential Models

6.1 Introduction

6.2 Discretization

6.3 Solution of Linear Equations

6.4 Solution of Coupled Systems

6.5 Examples

6.6 Miscellaneous Topics

7. Finite Difference Simulation ; Time Dependence

7.1 Introduction

7.2 Unsteady Flows

7.3 Integral Constitutive Models

Section 3 : Finite Element Techniques

8. Introduction to Finite Elements

8.1 Introduction

8.2 Finite Element Representation

8.3 The Finite Element Method

8.4 Method of Weighted Residuals

8.5 Construction of the Algebraic System

8.6 Solution of the Algebraic System

8.7 Examples

8.8 Two-Dimensional Problems. Triangular and Rectangular Elements

8.9 Isoparametric Elements

8.10 Method of Weighted Residuals

8.11 Numerical Integration

8.12 Example. Convergence of the Finite Element Method

9. Finite Element Calculation of Generalized Newtonian Flow

9.1 Introduction

9.2 A Variational Theorem for Creeping Generalized Newtonian Flow

9.3 Galerkin Formulation of the Equations of Motion; Plane Flow

9.4 Galerkin Formulation of the Equations of Motion; Axisymmetric Flow

9.5 Finite Elements for Solving the Navier-Stokes Equations

9.6 Penalty Formulation for Solving the Navier-Stokes Equations

9.7 Calculation of the Stream Function

9.8 Solving the Generalized Newtonian Flow

9.9 Entry Flow in a Tubular Contraction

9.10 Die Swell of a Generalized Newtonian Fluid

9.11 The Flow of a Power-Law Fluid Around a Sphere

10. Finite Element Calculation of Viscoelastic Flow

10.1 Introduction

10.2 Another Variational Theorem for Creeping Newtonian Flow

10.3 A Mixed Method for Solving the Stokes Equations

10.4 A Mixed Method for Solving the Flow of a Maxwell Fluid (MIX1)

10.5 A Second Mixed Method for Solving the Flow of a Maxwell Fluid (MIX2, MIX3)

10.6 Axisymmetric Flow

10.7 Problems with the Mixed Methods

10.8 The Oldroyd-B Fluid and Related Models

10.9 A Third Method for Solving the Flow of a Maxwell Fluid (MIX4)

10.10 The Flow of Viscoelastic Fluids of the Integral Type

10.11 Example of the General Development - Entry Flow in a Tubular Contraction

10.12 Example of the General Development - Die Swell of a Viscoelastic Fluid

10.13 Related Problems

Section 4 : Epilogue

11. Outstanding Problems. Future Trends

11.1 General

11.2 Numerical Simulation Breakdown

11.3 Possible Reasons for Breakdown : An Evaluation

11.4 Concluding Remarks

References

Author Index

Subject Index

## Product details

- Language: English
- Copyright: © Elsevier Science 1984
- Published: February 1, 1984
- Imprint: Elsevier Science
- eBook ISBN: 9780444598554

## About the Authors

### M.J. Crochet

### A.R. Davies

### K. Walters

#### Affiliations and Expertise

University of Wales, Aberystwyth, UK

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