Microhydrodynamics

Microhydrodynamics: Principles and Selected Applications presents analytical and numerical methods for describing motion of small particles suspended in viscous fluids.
The text first covers the fundamental principles of low-Reynolds-number flow, including the governing equations and fundamental theorems; the dynamics of a single particle in a flow field; and hydrodynamic interactions between suspended particles. Next, the book deals with the advances in the mathematical and computational aspects of viscous particulate flows that point to innovations for large-scale simulations on parallel computers. The book will be of great use to students in engineering and applied mathematics. Students and practitioners of chemistry will also benefit from this book.

Preface

Organization Scheme

I Governing Equations and Fundamental Theorems

1 Microhydrodynamic Phenomena

1.1 Objective and Scope

1.2 The Governing Equations

1.3 Colloidal Forces on Particles

2 General Properties and Fundamental Theorems

2.1 Introduction

2.2 Energy Dissipation Theorems

2.3 Lorentz Reciprocal Theorem

2.4 Integral Representations

2.5 The Multipole Expansion

Exercises

References

II Dynamics of a Single Particle

3 The Disturbance Field of a Single Particle in a Steady Flow

3.1 Introduction

3.2 The Far Field Expansion: Rigid Particles and Drops

3.3 Singularity Solutions

3.4 Slender Body Theory

3.5 Faxen Laws

Exercises

4 Solutions in Spherical Coordinates

4.1 Introduction

4.2 Lamb's General Solution

4.3 The Adjoint Method

4.4 An Orthonormal Basis for Stokes Flow

4.5 The Stokes Streamfunction

Exercises

5 Resistance and Mobility Relations

5.1 Introduction

5.2 The Resistance Tensor

5.3 The Mobility Tensor

5.4 Relations between the Resistance and Mobility Tensors

5.5 Axisymmetric Particles

5.6 Rheology of a Dilute Suspension of Spheroids

5.7 Electrophoresis

Exercises

6 Transient Stokes Flows

6.1 Time Scales

6.2 The Fundamental Solution

6.3 Reciprocal Theorem and Applications

6.4 The Low-Frequency Limit

Exercises

References

III Hydrodynamic Interactions

7 General Formulation of Resistance and Mobility Relations

7.1 Introduction

7.2 Resistance and Mobility Relations

Exercises

8 Particles Widely Separated: The Method of Reflections

8.1 The Far Field

8.2 Resistance Problems

8.3 Mobility Problems

8.4 Renormalization Theory

8.5 Multipole Expansions for Two Spheres

8.6 Electrophoresis of Particles with Thin Double Layers

Exercises

9 Particles Near Contact

9.1 Overview

9.2 Shearing Motions of Rigid Surfaces

9.3 Squeezing Motions of Rigid Surfaces

9.4 Squeezing Flow between Viscous Drops

9.5 Shearing Flow between Viscous Drops

Exercises

10 Interactions between Large and Small Particles

10.1 Multiple Length Scales

10.2 Image System for the Stokeslet Near a Rigid Sphere

10.3 Image Systems for Stokes Dipoles

10.4 Image System for the Degenerate Stokes Quadrupole

10.5 Hydrodynamic Interactions between Large and Small Spheres

10.5.1 Mobility Functions x12 and x22a

10.5.2 Mobility Functions x11 and x21a

10.6 Hydrodynamic Interactions between Large and Small Drops

Exercises

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

11.1 Regimes of Interaction

11.2 Examples of the Usage of Resistance and Mobility Functions

11.3 Tables of the Resistance and Mobility Functions

12 Particle-Wall Interactions

12.1 The Lorentz Image

12.2 Stokeslet Near a Rigid Wall

12.3 A Drop Near a Fluid-Fluid Interface

Exercises

13 Boundary-Multipole Collocation

13.1 Introduction

13.2 Two-Sphere Problems

13.3 Error Analysis for Spheres

13.4 Error Analysis for Spheroids

Exercises

References

IV Foundations of Parallel Computational Microhydrodynamics

14 The Boundary Integral Equations for Stokes Flow

14.1 The Setting for Computational Microhydrodynamics

14.2 Integral Operators and Integral Equations

14.3 Notation and Definitions

14.4 The Boundary Integral Equation in the Primary Variables

14.5 On Solving Problems with Velocity BCs

Exercises

15 Odqvist's Approach for a Single Particle Surface

15.1 Smoothness of the Boundary Surfaces

15.2 Single and Double Layer Potentials, and Some of Their Properties

15.3 Results for a Single Closed Surface

15.4 The Completion Method of Power and Miranda for a Single Particle

Exercises

16 Multiparticle Problems in Bounded and Unbounded Domains

16.1 The Double Layer on Multiple Surfaces

16.2 The Lyapunov-Smooth Container

16.3 The Canonical Equations

16.4 RBM-Tractions from the Riesz Representation Theorem

16.5 The Stresslet

Exercises

17 Iterative Solutions for Mobility Problems

17.1 Conditions for Successful Direct Iteration

17.2 The Spectrum of the Double Layer Operator

17.3 Wielandt's Deflation

17.4 Deflation for a Single Particle

17.5 Deflation for a Container

17.6 Multiparticle Problems in Bounded and Unbounded Domains

17.7 Iterative Solution of the Tractions for a Mobility Problem

Exercises

18 Fourier Analysis for Axisymmetric Boundaries

18.1 How the Components Separate in Wave-number

18.2 Another Symmetry Argument for the Fourier Decomposition

18.3 Analytical Fourier Decomposition of the Kernel with Toroidals

18.4 Numerical Computation of the Toroidal Functions

18.5 The Numerical Solution Procedure

18.6 Axial Torque as an Example

18.7 Transverse Force and Torque

18.8 Other Details of Implementation

18.9 Limitations of the Fourier Analysis Approach

18.10 Results from the Axisymmetric Codes

18.11 Possibilities for Improvement and Generalization

Exercises

19 Three-Dimensional Numerical Results

19.1 Discretization with Constant Elements

19.2 Resistance and Mobility of Spheres

19.3 Sedimentation of Platonic Solids

19.4 Benchmarks

19.5 CDL-BIEM and Parallel Processing

19.6 Reducing Communication between Processors

Exercises

References

Notation

Index