Microhydrodynamics Principles and Selected Applications

by Kim, Sangtae; Karrila, Seppo J.

• ISBN13: 9780486442198
• ISBN10: 0486442195
• Format: Paperback
• Copyright: 2005-06-17
• Publisher: Dover Publications

Summary

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.

Table of Contents

Preface	xv
Organization Scheme	xvii
I. Governing Equations and Fundamental Theorems
Microhydrodynamic Phenomena	1	(12)
Objective and Scope	1	(5)
The Governing Equations	6	(6)
The Equation of Continuity	6	(1)
The Momentum Balance	7	(2)
The Stokes Equations	9	(1)
Boundary Conditions for Fluid Flows	10	(1)
The Energy Balance	11	(1)
Colloidal Forces on Particles	12	(1)
General Properties and Fundamental Theorems	13	(34)
Introduction	13	(1)
Energy Dissipation Theorems	14	(5)
Uniqueness	14	(1)
Minimum Energy Dissipation	15	(1)
Lower Bounds on Energy Dissipation	15	(1)
Energy Dissipation in Particulate Flows	16	(2)
Energy Dissipation in Mobility Problems	18	(1)
Lorentz Reciprocal Theorem	19	(1)
Integral Representations	20	(7)
The Green's Function for Stokes Flow	20	(3)
Integral Representation with Single and Double Layer Potentials	23	(2)
Representation of Flows Outside a Rigid Particle	25	(2)
The Multipole Expansion	27	(20)
Exercises	31	(10)
References	41	(6)
II. Dynamics of a Single Particle
The Disturbance Field of a Single Particle in a Steady Flow	47	(36)
Introduction	47	(1)
The Far Field Expansion: Rigid Particles and Drops	48	(1)
Singularity Solutions	49	(18)
Singularity System for Spheres	50	(2)
The Spherical Drop and Interior Flows	52	(1)
Singularity System for Ellipsoids	53	(8)
Singularity System for Prolate Spheroids	61	(5)
Singularity System for Oblate Spheroids	66	(1)
Slender Body Theory	67	(6)
Faxen Laws	73	(10)
Ellipsoids and Spheroids	76	(2)
The Spherical Drop	78	(2)
Exercises	80	(3)
Solutions in Spherical Coordinates	83	(24)
Introduction	83	(1)
Lamb's General Solution	83	(10)
The Connection with the Multipole Expansion	86	(2)
Force, Torque, and Stresslet	88	(1)
Matching of Boundary Conditions	88	(5)
The Adjoint Method	93	(2)
An Orthonormal Basis for Stokes Flow	95	(2)
The Stokes Streamfunction	97	(10)
Relation to the Vector Potential	97	(1)
The Stokes Equations and the Streamfunction	98	(1)
Boundary Conditions for the Streamfunction	99	(1)
The Axisymmetric Stokeslet and the Drag on a Body	99	(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