Microhydrodynamics Principles and Selected Applications

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ISBN: 9780486442198 | 0486442195
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Copyright: 6/17/2005

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This text focuses on determining the motion of particles through a viscous fluid in bounded and unbounded flow. Its central theme is the mobility relation between particle motion and forces, and it functions as a manual that explains methods for solving particulate flows. 99 figures. 47 tables. 1991 edition.

Preface

Organization Scheme

xvii

I. Governing Equations and Fundamental Theorems

Microhydrodynamic Phenomena

(12)

Objective and Scope

(5)

The Governing Equations

(6)

The Equation of Continuity

(1)

The Momentum Balance

(2)

The Stokes Equations

(1)

Boundary Conditions for Fluid Flows

(1)

The Energy Balance

(1)

Colloidal Forces on Particles

(1)

General Properties and Fundamental Theorems

(34)

Introduction

(1)

Energy Dissipation Theorems

(5)

Uniqueness

(1)

Minimum Energy Dissipation

(1)

Lower Bounds on Energy Dissipation

(1)

Energy Dissipation in Particulate Flows

(2)

Energy Dissipation in Mobility Problems

(1)

Lorentz Reciprocal Theorem

(1)

Integral Representations

(7)

The Green's Function for Stokes Flow

(3)

Integral Representation with Single and Double Layer Potentials

(2)

Representation of Flows Outside a Rigid Particle

(2)

The Multipole Expansion

(20)

Exercises

(10)

References

(6)

II. Dynamics of a Single Particle

The Disturbance Field of a Single Particle in a Steady Flow

(36)

Introduction

(1)

The Far Field Expansion: Rigid Particles and Drops

(1)

Singularity Solutions

(18)

Singularity System for Spheres

(2)

The Spherical Drop and Interior Flows

(1)

Singularity System for Ellipsoids

(8)

Singularity System for Prolate Spheroids

(5)

Singularity System for Oblate Spheroids

(1)

Slender Body Theory

(6)

Faxen Laws

(10)

Ellipsoids and Spheroids

(2)

The Spherical Drop

(2)

Exercises

(3)

Solutions in Spherical Coordinates

(24)

Introduction

(1)

Lamb's General Solution

(10)

The Connection with the Multipole Expansion

(2)

Force, Torque, and Stresslet

(1)

Matching of Boundary Conditions

(5)

The Adjoint Method

(2)

An Orthonormal Basis for Stokes Flow

(2)

The Stokes Streamfunction

(10)

Relation to the Vector Potential

(1)

The Stokes Equations and the Streamfunction

(1)

Boundary Conditions for the Streamfunction

(1)

The Axisymmetric Stokeslet and the Drag on a Body

(1)

Separation in Spherical Coordinates

100

(2)

Exercises

102

(5)

Resistance and Mobility Relations

107

(40)

Introduction

107

(1)

The Resistance Tensor

108

(7)

Symmetry and Positive Definiteness

110

(2)

The Hydrodynamic Center of Resistance

112

(3)

Translation Theorems for Resistance Tensors

115

(1)

The Mobility Tensor

115

(3)

Translation Theorems for Mobility Tensors

117

(1)

The Hydrodynamic Center of Mobility

117

(1)

Relations Between the Resistance and Mobility Tensors

118

(3)

Axisymmetric Particles

121

(5)

General Resistance Formulation in a Linear Field

121

(2)

A Torque-Free Axisymmetric Particle in a Linear Field: The Jeffery Orbits

123

(3)

Rheology of a Dilute Suspension of Spheroids

126

(5)

Electrophoresis

131

(16)

Particles with Finite Electric Double Layers

135

(6)

Nonuniform Surface Potentials

141

(2)

Exercises

143

(4)

Transient Stokes Flows

147

(28)

Time Scales

147

(2)

The Fundamental Solution

149

(5)

The Oscillating Sphere

150

(1)

Sphere Released from Rest

151

(2)

Oscillatory Rotation of a Sphere

153

(1)

Reciprocal Theorem and Applications

154

(7)

Integral Representations

154

(1)

The Faxen Law: Particles of Arbitrary Shape

155

(1)

The Faxen Law: Force on a Rigid Sphere

156

(1)

The Faxen Law: Viscous Drop

157

(1)

The Oscillating Spherical Drop

158

(2)

The Faxen Law: Force on a Spherical Drop

160

(1)

The Low-Frequency Limit

161

(14)

Exercises

162

(3)

References

165

(10)

III. Hydrodynamic Interactions

General Formulation of Resistance and Mobility Relations

175

(12)

Introduction

175

(3)

Resistance and Mobility Relations

178

(9)

The Resistance Matrix

178

(1)

The Mobility Matrix

179

(1)

Relations Between the Resistance and Mobility Tensors

180

(1)

Axisymmetric Geometries

180

(3)

Exercises

183

(4)

Particles Widely Separated: The Method of Reflections

187

(32)

The Far Field

187

(2)

Resistance Problems

189

(7)

Mobility Problems

196

(6)

Renormalization Theory

202

(4)

Multipole Expansions for Two Spheres

206

(7)

Translations Along the Axis

209

(4)

Electrophoresis of Particles with Thin Double Layers

213

(6)

Hydrodynamic Interaction Between Spheres

213

(2)

Multiple Ellipsoids

215

(1)

Exercises

215

(4)

Particles Near Contact

219

(20)

Overview

219

(1)

Shearing Motions of Rigid Surfaces

220

(6)

Squeezing Motions of Rigid Surfaces

226

(5)

Squeezing Flow Between Viscous Drops

231

(6)

Nearly Rigid Drops

233

(1)

Fully Mobile Interfaces

233

(1)

Solution in Bispherical Coordinates

233

(2)

Nearly Touching Bubbles

235

(2)

Shearing Flow Between Viscous Drops

237

(2)

Exercises

238

(1)

Interactions Between Large and Small Particles

239

(32)

Multiple Length Scales

239

(1)

Image System for the Stokeslet Near a Rigid Sphere

240

(14)

The Axisymmetric Stokeslet

242

(2)

The Transverse Stokeslet

244

(2)

Singularity Solutions for the Image of a Stokeslet

246

(3)

Image for the Stokeslet Near a Viscous Drop

249

(5)

Image Systems for Stokes Dipoles

254

(5)

The Axisymmetric Stresslet

255

(1)

The Stresslet in Hyperbolic Flow (m = 1)

255

(3)

The Stresslet in Hyperbolic Flow (m = 2)

258

(1)

Image System for the Degenerate Stokes Quadrupole

259

(2)

The Axisymmetric Problem

260

(1)

The Transverse Problem

261

(1)

Hydrodynamic Interactions Between Large and Small Spheres

261

(5)

Mobility Functions x12 and xa22

262

(3)

Mobility Functions x11 and xa21

265

(1)

Hydrodynamic Interactions Between Large and Small Drops

266

(5)

Exercises

268

(3)

The Complete Set of Resistance and Mobility Functions for Two Rigid Spheres

271

(40)

Regimes of Interaction

271

(1)

Examples of the Usage of Resistance and Mobility Functions

272

(5)

Two Spheres Moving in Tandem Along their Line of Centers

272

(1)

Two Spheres Approaching Each Other

273

(1)

Heavy Sphere Falling Past a Neutrally Buoyant Sphere

273

(2)

Two Almost-Touching Spheres in Rigid-Body Rotation

275

(1)

Two Force-Free and Torque-Free Spheres in a Linear Field

276

(1)

Tables of the Resistance and Mobility Functions

277

(34)

Particle-Wall Interactions

311

(10)

The Lorentz Image

311

(1)

Stokeslet Near a Rigid Wall

312

(5)

A Drop Near a Fluid-Fluid Interface

317

(4)

Exercises

319

(2)

Boundary-Multipole Collocation

321

(34)

Introduction

321

(2)

Two-Sphere Problems

323

(6)

Equal Spheres

326

(1)

Resistance Functions

327

(1)

Collocation Schemes

328

(1)

Error Analysis for Spheres

329

(10)

Axisymmetric Stokeslet

330

(4)

The Transverse Stokeslet

334

(5)

Error Analysis for Spheroids

339

(16)

Stokeslet on the Axis

341

(2)

Stokeslet on the Equatorial Plane

343

(2)

Exercises

345

(2)

References

347

(8)

IV. Foundations of Parallel Computational Microhydrodynamics

The Boundary Integral Equations for Stokes Flow

355

(20)

The Setting for Computational Microhydrodynamics

355

(3)

Integral Operators and Integral Equations

358

(9)

Motivation

358

(1)

First-and Second-Kind Equations

358

(1)

Weakly Singular Kernels

359

(1)

Ill- and Well-Posedness

359

(1)

Compact Operators

360

(3)

Numerical Solution Methods

363

(2)

Applications to Boundary Integral Equations

365

(2)

Notation and Definitions

367

(1)

The Boundary Integral Equation in the Primary Variables

368

(4)

Physically Occurring Boundary Conditions

371

(1)

Difficulties with the BIE in the Primary Variables

372

(1)

On Solving Problems with Velocity BCs

372

(3)

Exercises

374

(1)

Odqvist's Approach for a Single Particle Surface

375

(14)

Smoothness of the Boundary Surfaces

375

(1)

Single and Double Layer Potentials, and Some of Their Properties

376

(4)

Results for a Single Closed Surface

380

(4)

The Completion Method of Power and Miranda for a Single Particle

384

(5)

Exercises

385

(4)

Multiparticle Problems in Bounded and Unbounded Domains

389

(20)

The Double Layer on Multiple Surfaces

389

(4)

The Lyapunov-Smooth Container

393

(3)

The Canonical Equations

396

(7)

The Bordering Method in General

396

(1)

On Removing the Null Functions

397

(1)

Additions to the Range Space

398

(1)

Summary of the Canonical Equations for Resistance and Mobility Problems

398

(5)

RBM-Tractions from the Riesz Representation Theorem

403

(4)

The Riesz Representation Theorem

403

(1)

Force and Torque from the Lorentz Reciprocal Theorem

404

(1)

Tractions for Rigid-Body Motion

405

(2)

The Stresslet

407

(2)

Exercises

407

(2)

Iterative Solutions for Mobility Problems

409

(40)

Conditions for Successful Direct Iteration

409

(1)

The Spectrum of the Double Layer Operator

410

(18)

The Spectrum for the Sphere

412

(6)

Double Layer Eigenfunctions for Ellipsoids in Rate-of-Strain Fields

418

(7)

The Spectrum for Two Spheres: HI-Induced Spectral Splitting

425

(3)

Wielandt's Deflation

428

(3)

Deflation for a Single Particle

431

(3)

Deflation for a Container

434

(1)

Multiparticle Problems in Bounded and Unbounded Domains

435

(5)

Iterative Solution of the Tractions for a Mobility Problem

440

(9)

Exercises

444

(5)

Fourier Analysis for Axisymmetric Boundaries

449

(22)

How the Components Separate in Wave-number

449

(1)

Another Symmetry Argument for the Fourier Decomposition

450

(1)

Analytical Fourier Decomposition of the Kernel with Toroidals

451

(1)

Numerical Computation of the Toroidal Functions

452

(1)

The Numerical Solution Procedure

453

(2)

The Choice of the Numerical Method

453

(1)

Theory of Singularity Subtraction

454

(1)

Axial Torque as an Example

455

(3)

The Azimuthal Integrations

455

(1)

Discretizing with Quadrature and Singularity Subtraction

456

(2)

Transverse Force and Torque

458

(3)

Fourier Decomposition of the Double Layer Kernel

459

(2)

Other Details of Implementation

461

(1)

Limitations of the Fourier Analysis Approach

461

(1)

Results from the Axisymmetric Codes

462

(7)

Prolate Spheroids; Comparison of Surface Tractions with Known Analytical Results

462

(1)

Mesh Effects: Grooved Particles

462

(3)

The Effect of Sharp Edges: Finite Circular Cylinder

465

(3)

A Container Problem

468

(1)

Possibilities for Improvement and Generalization

469

(2)

Exercises

470

(1)

Three-Dimensional Numerical Results

471

(24)

Discretization with Constant Elements

472

(4)

Resistance and Mobility of Spheres

476

(1)

Sedimentation of Platonic Solids

477

(4)

Benchmarks

481

(1)

CDL-BIEM and Parallel Processing

482

(3)

Reducing Communication Between Processors

485

(10)

Asynchronous Iterations

485

(3)

Compactification of Distant Information

488

(1)

Exercises

488

(1)

References

489

(6)

Notation

495

(8)

Index

503

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Microhydrodynamics Principles and Selected Applications

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Table of Contents

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