Computational Rheology

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ISBN: 9781860941863 | 1860941869
Cover: Hardcover
Copyright: 7/1/2002

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Modern day high-performance computers are making available to 21st-century scientists solutions to theological flow problems of ever-increasing complexity. Computational rheology is a fast-moving subject -- problems which only 10 years ago were intractable, such as 3D transient flows of polymeric liquids, non-isothermal non-Newtonian flows or flows of highly elastic liquids through complex geometries, are now being tackled owing to the availability of parallel computers, adaptive methods and advances in constitutive modelling.

Techniques of Computational Rheology traces the development of numerical methods for non-Newtonian flows from the late 1960's to the present day. It begins with broad coverage of non-Newtonian fluids, including their mathematical modelling and analysis, before specific computational techniques are discussed. The application of these techniques to some important rheological flow problems of academic and industrial interest is then treated in a detailed and up-to-date exposition. Finally, the reader is kept abreast of ideas at the cutting

Introduction

(18)

Everything Flows

(7)

The Maxwell model and the relaxation time

(3)

The Kelvin model and the retardation time

(1)

Non-Newtonian Fluids

(8)

Non-Newtonian fluids in shear flow

(5)

Non-Newtonian fluids in uniaxial extensional flow

(2)

Numerical Simulation of Non-Newtonian Flow

(3)

Early work

(1)

The high Weissenberg number problem

(1)

Numerical work since the mid 1980's

(1)

Fundamentals

(30)

Some Important Vectors

(2)

Fluid velocity and acceleration

(1)

Forces and the stress vector

(1)

Conservation Laws and the Stress Tensor

(3)

Conservation of mass

(1)

Conservation of linear momentum

(1)

Conservation of angular momentum

(1)

The stress tensor

(1)

Conservation of energy

(1)

The Newtonian Fluid

(1)

The Generalized Newtonian Fluid

(2)

Derivation of the stress tensor

(1)

The Order Fluids and the CEF Equation

(6)

Strain tensors

(1)

Order fluids

(2)

Limitations of order fluid models

(2)

The CEF equation

(1)

More Complicated Constitutive Relations

(16)

Differential constitutive models

(12)

Integral constitutive models

(4)

Mathematical Theory of Viscoelastic Fluids

(24)

Introduction

(2)

Existence and Uniqueness

(3)

Properties of the Differential Systems

(10)

Loss of evolution and Hadamard instability

(3)

Classification

(2)

Change of type in steady flow

(1)

Characteristic variables

(2)

Boundary Conditions

(1)

Singularities

(8)

Elliptic problems

(3)

Viscoelastic flows

(3)

Numerical investigations

(2)

Parameter Estimation in Continuum Models

(22)

Introduction

(3)

Determination of Viscosity

(7)

Shear viscosity

(3)

Dependence of viscosity on temperature

(2)

Dependence of viscosity on pressure

(1)

Extensional viscosity

(1)

Determination of the Relaxation Spectrum

(12)

Dynamic experiments

(1)

Mathematical problems

(1)

Linear regression techniques

(2)

Nonlinear regression techniques

(1)

Examples

(2)

Sampling localization

(4)

From the Continuous to the Discrete

(40)

Introduction

(2)

Finite Difference Approximations

(7)

Finite differences for viscoelastic flows

100

(4)

Finite Element Approximations

104

(9)

Finite elements in one dimension

105

(2)

Finite elements in two dimensions

107

(3)

Finite elements for viscoelastic flows

110

(3)

Finite Volume Methods

113

(8)

Finite volumes for viscoelastic flows

117

(4)

Spectral Methods

121

(8)

Spectral methods in one dimension

121

(4)

Spectral methods in two dimensions

125

(2)

Spectral methods for viscoelastic flows

127

(2)

Spectral Element Methods

129

(6)

Spectral elements for viscoelastic flows

130

(5)

Numerical Algorithms for Macroscopic Models

135

(38)

Introduction

135

(1)

From Picard to Newton

136

(1)

Differential Models: Steady Flows

137

(12)

Direct methods

138

(2)

Iterative methods

140

(9)

Differential Models: Transient Flows

149

(14)

Projection methods

150

(3)

The influence matrix method

153

(2)

Taylor-Galerkin methods

155

(2)

The θ method

157

(3)

Lagrangian methods

160

(3)

Computing with Integral Models

163

(1)

Integral Models: Steady Flows

164

(3)

Integral Models: Transient Flows

167

(6)

Lagrangian techniques

167

(3)

Eulerian techniques

170

(3)

Defeating the High Weissenberg Number Problem

173

(28)

Introduction

173

(4)

Discretization of Differential Constitutive Equations

177

(10)

Streamline upwinding,- SU and SUPG

177

(5)

Discontinuous Galerkin methods

182

(5)

Discretization of the Coupled Governing Equations

187

(14)

Compatible approximation spaces

187

(3)

EVSS-type methods

190

(5)

Change of type and loss of evolution

195

(6)

Benchmark Problems I: Contraction Flows

201

(46)

Vortex Growth Dynamics

202

(14)

Boger fluids - observed flow transitions

202

(4)

Boger fluids - effects of changes of geometry

206

(3)

Concentrated polymer solutions and melts - observed flow transitions

209

(4)

Concentrated polymer solutions and melts - effects of change of geometry

213

(3)

Vortex Growth Mechanisms

216

(9)

Experimental work

216

(6)

Numerical studies

222

(3)

Numerical Simulation

225

(22)

Comparisons with experiments

226

(5)

Benchmarking numerical methods

231

(16)

Benchmark Problems II

247

(58)

Flow Past a Cylinder in a Channel

247

(20)

Streamline patterns and drag: unbounded flows

247

(4)

Streamline patterns and drag: cylinders in channels

251

(1)

Comparison of numerical and experimental results

252

(2)

Purely elastic instabilities

254

(3)

Comparison of numerical methods

257

(6)

Mesocopic calculations

263

(4)

Flow Past a Sphere in a Tube

267

(20)

Drag coefficient

268

(6)

Benchmarking numerical methods

274

(3)

Negative wakes: steady flow

277

(4)

Velocity overshoots - transient flow calculations

281

(3)

Comparison between experimental and numerical results

284

(3)

Flow between Eccentrically Rotating Cylinders

287

(18)

The Taylor-Couette problem

288

(3)

Lubrication theory

291

(2)

Statically loaded journal bearing

293

(4)

Dynamically loaded journal bearing

297

(8)

Error Estimation and Adaptive Strategies

305

(22)

Introduction

305

(3)

Problem Description

308

(3)

Weak formulation

308

(3)

Discretization and Error Analysis (Galerkin method)

311

(5)

An error indicator

315

(1)

Adaptive Strategies

316

(11)

Numerical example: flow past a sphere in a tube

317

(10)

Contemporary Topics in Computational Rheology

327

(34)

Advances in Mathematical Modelling

327

(3)

Dynamics of Dilute Polymer Solutions

330

(5)

The Rouse model

331

(4)

Dumbbell models

335

(1)

Closure Approximations

335

(3)

Stochastic Differential Equations

338

(9)

The Connffessit approach

341

(2)

Variance reduction technique

343

(2)

Lagrangian particle method

345

(1)

Brownian configuration fields

346

(1)

Dynamics of Polymer Melts

347

(10)

The Doi-Edwards model

347

(7)

The pom-pom model

354

(3)

Lattice Boltzmann Methods

357

(2)

Closing Comments

359

(2)

A Some Results about Tensors

361

(4)

Existence and Symmetry of the Stress Tensor

361

(3)

Small Displacement Gradient Limit of γ[0] (x,t,t')

364

(1)

Partial Time Derivative of the Deformation Gradient Tensor F(x, t, t')

364

(1)

B Governing Equations in Orthogonal Curvilinear Coordinates

365

(1)

Differential Relations and Identities

365

(1)

Differential Operators in Orthogonal Curvilinear Coordinates

366

(1)

Rectangular coordinates (x1, x2, x3) = (x, y, z)

366

(2)

Cylindrical polar coordinates (x1, x2, x3) = (r, θ, z)

368

(1)

Spherical polar coordinates (x1, x2, x3) = (r, θ, φ)

369

(2)

Conservation Equations

371

(1)

Rectangular coordinates (x, y, z)

371

(1)

Cylindrical polar coordinates (r, θ, z)

371

(1)

Spherical polar coordinates (r, θ, φ)

372

(1)

Some Important Theorems in Vector and Tensor Calculus

373

(1)

The Reynolds transport theorem

373

(1)

The divergence theorem

373

(1)

Stokes's theorem

373

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Computational Rheology

Summary

Table of Contents

Supplemental Materials

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