Microhydrodynamics

Principles and Selected Applications

By: SANGTAE KIM

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Paperback | 17 June 2005

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507 Pages

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""This book is well organised and comprehensive . . . an eloquent and enduring statement of significant hydrodynamic principles."" - AIChE JournalMicrohydrodynamics concerns the flow and related phenomena pertinent to the motion of small particles suspended in viscous fluids. This text focuses on determining the motion of a particle or particles through a viscous fluid in bounded and unbounded flow. Its central theme is the mobility relation between particle motion and forces.Microhydrodynamics: Principles and Selected Applications functions as a manual that explains methods for solving particulate flows at low-Reynolds number, from analytical to computational methods. The ever-increasing growth in computational power has resulted in a similar growth in the range of solvable problems in microhydrodynamics. Suitable for graduate students in engineering and applied mathematics, this text treats the mathematical foundations and highlights the interplay of both mathematical and physical insights, guiding readers through single particle theory and problems related to multiparticle analyses.

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Non-Fiction Engineering & Technology Mechanical Engineering & Materials Materials Science Mechanics of Fluids Hydraulics & Pneumatics Science Physics Classical Mathematics Fluid Mechanics Science in General

Preface	p. xv
Organization Scheme	p. xvii
Governing Equations and Fundamental Theorems
Microhydrodynamic Phenomena	p. 1
Objective and Scope	p. 1
The Governing Equations	p. 6
The Equation of Continuity	p. 6
The Momentum Balance	p. 7
The Stokes Equations	p. 9
Boundary Conditions for Fluid Flows	p. 10
The Energy Balance	p. 11
Colloidal Forces on Particles	p. 12
General Properties and Fundamental Theorems	p. 13
Introduction	p. 13
Energy Dissipation Theorems	p. 14
Uniqueness	p. 14
Minimum Energy Dissipation	p. 15
Lower Bounds on Energy Dissipation	p. 15
Energy Dissipation in Particulate Flows	p. 16
Energy Dissipation in Mobility Problems	p. 18
Lorentz Reciprocal Theorem	p. 19
Integral Representations	p. 20
The Green's Function for Stokes Flow	p. 20
Integral Representation with Single and Double Layer Potentials	p. 23
Representation of Flows Outside a Rigid Particle	p. 25
The Multipole Expansion	p. 27
Exercises	p. 31
References	p. 41
Dynamics of a Single Particle
The Disturbance Field of a Single Particle in a Steady Flow	p. 47
Introduction	p. 47
The Far Field Expansion: Rigid Particles and Drops	p. 48
Singularity Solutions	p. 49
Singularity System for Spheres	p. 50
The Spherical Drop and Interior Flows	p. 52
Singularity System for Ellipsoids	p. 53
Singularity System for Prolate Spheroids	p. 61
Singularity System for Oblate Spheroids	p. 66
Slender Body Theory	p. 67
Faxen Laws	p. 73
Ellipsoids and Spheroids	p. 76
The Spherical Drop	p. 78
Exercises	p. 80
Solutions in Spherical Coordinates	p. 83
Introduction	p. 83
Lamb's General Solution	p. 83
The Connection with the Multipole Expansion	p. 86
Force, Torque, and Stresslet	p. 88
Matching of Boundary Conditions	p. 88
The Adjoint Method	p. 93
An Orthonormal Basis for Stokes Flow	p. 95
The Stokes Streamfunction	p. 97
Relation to the Vector Potential	p. 97
The Stokes Equations and the Streamfunction	p. 98
Boundary Conditions for the Streamfunction	p. 99
The Axisymmetric Stokeslet and the Drag on a Body	p. 99
Separation in Spherical Coordinates	p. 100
Exercises	p. 102
Resistance and Mobility Relations	p. 107
Introduction	p. 107
The Resistance Tensor	p. 108
Symmetry and Positive Definiteness	p. 110
The Hydrodynamic Center of Resistance	p. 112
Translation Theorems for Resistance Tensors	p. 115
The Mobility Tensor	p. 115
Translation Theorems for Mobility Tensors	p. 117
The Hydrodynamic Center of Mobility	p. 117
Relations Between the Resistance and Mobility Tensors	p. 118
Axisymmetric Particles	p. 121
General Resistance Formulation in a Linear Field	p. 121
A Torque-Free Axisymmetric Particle in a Linear Field: The Jeffery Orbits	p. 123
Rheology of a Dilute Suspension of Spheroids	p. 126
Electrophoresis	p. 131
Particles with Finite Electric Double Layers	p. 135
Nonuniform Surface Potentials	p. 141
Exercises	p. 143
Transient Stokes Flows	p. 147
Time Scales	p. 147
The Fundamental Solution	p. 149
The Oscillating Sphere	p. 150
Sphere Released from Rest	p. 151
Oscillatory Rotation of a Sphere	p. 153
Reciprocal Theorem and Applications	p. 154
Integral Representations	p. 154
The Faxen Law: Particles of Arbitrary Shape	p. 155
The Faxen Law: Force on a Rigid Sphere	p. 156
The Faxen Law: Viscous Drop	p. 157
The Oscillating Spherical Drop	p. 158
The Faxen Law: Force on a Spherical Drop	p. 160
The Low-Frequency Limit	p. 161
Exercises	p. 162
References	p. 165
Hydrodynamic Interactions
General Formulation of Resistance and Mobility Relations	p. 175
Introduction	p. 175
Resistance and Mobility Relations	p. 178
The Resistance Matrix	p. 178
The Mobility Matrix	p. 179
Relations Between the Resistance and Mobility Tensors	p. 180
Axisymmetric Geometries	p. 180
Exercises	p. 183
Particles Widely Separated: The Method of Reflections	p. 187
The Far Field	p. 187
Resistance Problems	p. 189
Mobility Problems	p. 196
Renormalization Theory	p. 202
Multipole Expansions for Two Spheres	p. 206
Translations Along the Axis	p. 209
Electrophoresis of Particles with Thin Double Layers	p. 213
Hydrodynamic Interaction Between Spheres	p. 213
Multiple Ellipsoids	p. 215
Exercises	p. 215
Particles Near Contact	p. 219
Overview	p. 219
Shearing Motions of Rigid Surfaces	p. 220
Squeezing Motions of Rigid Surfaces	p. 226
Squeezing Flow Between Viscous Drops	p. 231
Nearly Rigid Drops	p. 233
Fully Mobile Interfaces	p. 233
Solution in Bispherical Coordinates	p. 233
Nearly Touching Bubbles	p. 235
Shearing Flow Between Viscous Drops	p. 237
Exercises	p. 238
Interactions Between Large and Small Particles	p. 239
Multiple Length Scales	p. 239
Image System for the Stokeslet Near a Rigid Sphere	p. 240
The Axisymmetric Stokeslet	p. 242
The Transverse Stokeslet	p. 244
Singularity Solutions for the Image of a Stokeslet	p. 246
Image for the Stokeslet Near a Viscous Drop	p. 249
Image Systems for Stokes Dipoles	p. 254
The Axisymmetric Stresslet	p. 255
The Stresslet in Hyperbolic Flow (m = 1)	p. 255
The Stresslet in Hyperbolic Flow (m = 2)	p. 258
Image System for the Degenerate Stokes Quadrupole	p. 259
The Axisymmetric Problem	p. 260
The Transverse Problem	p. 261
Hydrodynamic Interactions Between Large and Small Spheres	p. 261
Mobility Functions x[subscript 12] and x[superscript a subscript 22]	p. 262
Mobility Functions x[subscript 11] and x[superscript a subscript 21]	p. 265
Hydrodynamic Interactions Between Large and Small Drops	p. 266
Exercises	p. 268
The Complete Set of Resistance and Mobility Functions for Two Rigid Spheres	p. 271
Regimes of Interaction	p. 271
Examples of the Usage of Resistance and Mobility Functions	p. 272
Two Spheres Moving in Tandem Along their Line of Centers	p. 272
Two Spheres Approaching Each Other	p. 273
Heavy Sphere Falling Past a Neutrally Buoyant Sphere	p. 273
Two Almost-Touching Spheres in Rigid-Body Rotation	p. 275
Two Force-Free and Torque-Free Spheres in a Linear Field	p. 276
Tables of the Resistance and Mobility Functions	p. 277
Particle-Wall Interactions	p. 311
The Lorentz Image	p. 311
Stokeslet Near a Rigid Wall	p. 312
A Drop Near a Fluid-Fluid Interface	p. 317
Exercises	p. 319
Boundary-Multipole Collocation	p. 321
Introduction	p. 321
Two-Sphere Problems	p. 323
Equal Spheres	p. 326
Resistance Functions	p. 327
Collocation Schemes	p. 328
Error Analysis for Spheres	p. 329
Axisymmetric Stokeslet	p. 330
The Transverse Stokeslet	p. 334
Error Analysis for Spheroids	p. 339
Stokeslet on the Axis	p. 341
Stokeslet on the Equatorial Plane	p. 343
Exercises	p. 345
References	p. 347
Foundations of Parallel Computational Microhydrodynamics
The Boundary Integral Equations for Stokes Flow	p. 355
The Setting for Computational Microhydrodynamics	p. 355
Integral Operators and Integral Equations	p. 358
Motivation	p. 358
First- and Second-Kind Equations	p. 358
Weakly Singular Kernels	p. 359
Ill- and Well-Posedness	p. 359
Compact Operators	p. 360
Numerical Solution Methods	p. 363
Applications to Boundary Integral Equations	p. 365
Notation and Definitions	p. 367
The Boundary Integral Equation in the Primary Variables	p. 368
Physically Occurring Boundary Conditions	p. 371
Difficulties with the BIE in the Primary Variables	p. 372
On Solving Problems with Velocity BCs	p. 372
Exercises	p. 374
Odqvist's Approach for a Single Particle Surface	p. 375
Smoothness of the Boundary Surfaces	p. 375
Single and Double Layer Potentials, and Some of Their Properties	p. 376
Results for a Single Closed Surface	p. 380
The Completion Method of Power and Miranda for a Single Particle	p. 384
Exercises	p. 385
Multiparticle Problems in Bounded and Unbounded Domains	p. 389
The Double Layer on Multiple Surfaces	p. 389
The Lyapunov-Smooth Container	p. 393
The Canonical Equations	p. 396
The Bordering Method in General	p. 396
On Removing the Null Functions	p. 397
Additions to the Range Space	p. 398
Summary of the Canonical Equations for Resistance and Mobility Problems	p. 398
RBM-Tractions from the Riesz Representation Theorem	p. 403
The Riesz Representation Theorem	p. 403
Force and Torque from the Lorentz Reciprocal Theorem	p. 404
Tractions for Rigid-Body Motion	p. 405
The Stresslet	p. 407
Exercises	p. 407
Iterative Solutions for Mobility Problems	p. 409
Conditions for Successful Direct Iteration	p. 409
The Spectrum of the Double Layer Operator	p. 410
The Spectrum for the Sphere	p. 412
Double Layer Eigenfunctions for Ellipsoids in Rate-of-Strain Fields	p. 418
The Spectrum for Two Spheres: HI-Induced Spectral Splitting	p. 425
Wielandt's Deflation	p. 428
Deflation for a Single Particle	p. 431
Deflation for a Container	p. 434
Multiparticle Problems in Bounded and Unbounded Domains	p. 435
Iterative Solution of the Tractions for a Mobility Problem	p. 440
Exercises	p. 444
Fourier Analysis for Axisymmetric Boundaries	p. 449
How the Components Separate in Wave-number	p. 449
Another Symmetry Argument for the Fourier Decomposition	p. 450
Analytical Fourier Decomposition of the Kernel with Toroidals	p. 451
Numerical Computation of the Toroidal Functions	p. 452
The Numerical Solution Procedure	p. 453
The Choice of the Numerical Method	p. 453
Theory of Singularity Subtraction	p. 454
Axial Torque as an Example	p. 455
The Azimuthal Integrations	p. 455
Discretizing with Quadrature and Singularity Subtraction	p. 456
Transverse Force and Torque	p. 458
Fourier Decomposition of the Double Layer Kernel	p. 459
Other Details of Implementation	p. 461
Limitations of the Fourier Analysis Approach	p. 461
Results from the Axisymmetric Codes	p. 462
Prolate Spheroids; Comparison of Surface Tractions with Known Analytical Results	p. 462
Mesh Effects: Grooved Particles	p. 462
The Effect of Sharp Edges: Finite Circular Cylinder	p. 465
A Container Problem	p. 468
Possibilities for Improvement and Generalization	p. 469
Exercises	p. 470
Three-Dimensional Numerical Results	p. 471
Discretization with Constant Elements	p. 472
Resistance and Mobility of Spheres	p. 476
Sedimentation of Platonic Solids	p. 477
Benchmarks	p. 481
CDL-BIEM and Parallel Processing	p. 482
Reducing Communication Between Processors	p. 485
Asynchronous Iterations	p. 485
Compactification of Distant Information	p. 488
Exercises	p. 488
References	p. 489
Notation	p. 495
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Microhydrodynamics

Principles and Selected Applications

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Paperback

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