You are here

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Publisher: 
Springer Verlag
Number of Pages: 
480
Price: 
69.95
ISBN: 
0387908196
Date Received: 
Thursday, April 27, 2006
Reviewable: 
Yes
Include In BLL Rating: 
Yes
John Guckenheimer and Philip Holmes
Series: 
Applied Mathematical Sciences 42
Publication Date: 
2002
Format: 
Hardcover
Category: 
Monograph
BLL Committee
01/19/2009
BLL Rating: 

Foreword by Chih-Ming Ho v

Preface vii

1 Basic Concepts and Technologies 1

1.1 New Flow Regimes in Microsystems . . . . . . . . . . . . . 1

1.2 The Continuum Hypothesis . . . . . . . . . . . . . . . . . . 8

1.2.1 Molecular Magnitudes . . . . . . . . . . . . . . . . . 13

1.2.2 Mixed Flow Regimes . . . . . . . . . . . . . . . . . . 18

1.2.3 Experimental Evidence . . . . . . . . . . . . . . . . 19

1.3 The Pioneers . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4 Modeling of Microflows . . . . . . . . . . . . . . . . . . . . 30

1.5 Modeling of Nanoflows . . . . . . . . . . . . . . . . . . . . . 34

1.6 Numerical Simulation at All Scales . . . . . . . . . . . . . . 37

1.7 Full-System Simulation of Microsystems . . . . . . . . . . . 38

1.7.1 Reduced-Order Modeling . . . . . . . . . . . . . . . 40

1.7.2 Coupled Circuit/Device Modeling . . . . . . . . . . 41

2 Governing Equations and Slip Models 51

2.1 The Basic Equations of Fluid Dynamics . . . . . . . . . . . 51

2.1.1 Incompressible Flow . . . . . . . . . . . . . . . . . . 54

2.1.2 Reduced Models . . . . . . . . . . . . . . . . . . . . 56

2.2 Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . 57

2.2.1 First-Order Models . . . . . . . . . . . . . . . . . . . 59

xvi Contents

2.2.2 The Role of the Accommodation Coefficients . . . . 61

2.3 High-Order Models . . . . . . . . . . . . . . . . . . . . . . . 66

2.3.1 Derivation of High-Order Slip Models . . . . . . . . 67

2.3.2 General Slip Condition . . . . . . . . . . . . . . . . . 70

2.3.3 Comparison of Slip Models . . . . . . . . . . . . . . 74

3 Shear-Driven Flows 79

3.1 Couette Flow: Slip Flow Regime . . . . . . . . . . . . . . . 79

3.2 Couette Flow: Transition and Free-Molecular

Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.2.1 Velocity Model . . . . . . . . . . . . . . . . . . . . . 83

3.2.2 Shear Stress Model . . . . . . . . . . . . . . . . . . . 86

3.3 Oscillatory Couette Flow . . . . . . . . . . . . . . . . . . . 90

3.3.1 Quasi-Steady Flows . . . . . . . . . . . . . . . . . . 91

3.3.2 Unsteady Flows . . . . . . . . . . . . . . . . . . . . . 96

3.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . 109

3.4 Cavity Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.5 Grooved Channel Flow . . . . . . . . . . . . . . . . . . . . . 112

4 Pressure-Driven Flows 117

4.1 Slip Flow Regime . . . . . . . . . . . . . . . . . . . . . . . . 117

4.1.1 Isothermal Compressible Flows . . . . . . . . . . . . 118

4.1.2 Adiabatic Compressible Flows – Fanno Theory . . . 126

4.1.3 Validation of Slip Models with DSMC . . . . . . . . 131

4.1.4 Effects of Roughness . . . . . . . . . . . . . . . . . . 136

4.1.5 Inlet Flows . . . . . . . . . . . . . . . . . . . . . . . 137

4.2 Transition and Free-Molecular Regimes . . . . . . . . . . . 140

4.2.1 Burnett Equations . . . . . . . . . . . . . . . . . . . 144

4.2.2 A Unified Flow Model . . . . . . . . . . . . . . . . . 146

4.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . 166

5 Thermal Effects in Microscales 167

5.1 Thermal Creep (Transpiration) . . . . . . . . . . . . . . . . 167

5.1.1 Simulation Results . . . . . . . . . . . . . . . . . . . 169

5.1.2 A Thermal Creep Experiment . . . . . . . . . . . . . 173

5.1.3 Knudsen Compressors . . . . . . . . . . . . . . . . . 174

5.2 Other Temperature-Induced Flows . . . . . . . . . . . . . . 175

5.3 Heat Conduction and the Ghost Effect . . . . . . . . . . . . 177

5.4 Heat Transfer in Poiseuille Microflows . . . . . . . . . . . . 179

5.4.1 Pressure-Driven Flows . . . . . . . . . . . . . . . . . 179

5.4.2 Force-Driven Flows . . . . . . . . . . . . . . . . . . . 186

5.5 Heat Transfer in Couette Microflows . . . . . . . . . . . . . 188

6 Prototype Applications of Gas Flows 195

6.1 Gas Damping and Dynamic Response of Microsystems . . . 196

Contents xvii

6.1.1 Reynolds Equation . . . . . . . . . . . . . . . . . . . 199

6.1.2 Squeezed Film Effects in Accelerometers . . . . . . . 210

6.2 Separated Internal Flows . . . . . . . . . . . . . . . . . . . . 214

6.3 Separated External Flows . . . . . . . . . . . . . . . . . . . 221

6.4 Flow Past a Sphere: Stokes Flow Regime . . . . . . . . . . . 224

6.4.1 External Flow . . . . . . . . . . . . . . . . . . . . . 224

6.4.2 Sphere-in-a-Pipe . . . . . . . . . . . . . . . . . . . . 225

6.5 Microfilters . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

6.5.1 Drag Force Characteristics . . . . . . . . . . . . . . 232

6.5.2 Viscous Heating Characteristics . . . . . . . . . . . . 234

6.5.3 Short Channels and Filters . . . . . . . . . . . . . . 234

6.5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . 239

6.6 Micropropulsion and Micronozzle Flows . . . . . . . . . . . 239

6.6.1 Micropropulsion Analysis . . . . . . . . . . . . . . . 240

6.6.2 Rarefaction and Other Effects . . . . . . . . . . . . . 245

7 Electrokinetic Flows 255

7.1 Electrokinetic Effects . . . . . . . . . . . . . . . . . . . . . . 256

7.2 The Electric Double Layer (EDL) . . . . . . . . . . . . . . . 258

7.2.1 Near-Wall Potential Distribution . . . . . . . . . . . 261

7.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . 263

7.4 Electroosmotic Flows . . . . . . . . . . . . . . . . . . . . . . 266

7.4.1 Channel Flows . . . . . . . . . . . . . . . . . . . . . 266

7.4.2 Time-Periodic and AC Flows . . . . . . . . . . . . . 272

7.4.3 EDL/Bulk Flow Interface Velocity

Matching Condition . . . . . . . . . . . . . . . . . . 279

7.4.4 Slip Condition . . . . . . . . . . . . . . . . . . . . . 280

7.4.5 A Model for Wall Drag Force . . . . . . . . . . . . . 281

7.4.6 Joule Heating . . . . . . . . . . . . . . . . . . . . . . 282

7.4.7 Applications . . . . . . . . . . . . . . . . . . . . . . 283

7.5 Electrophoresis . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.5.1 Governing Equations . . . . . . . . . . . . . . . . . . 294

7.5.2 Classification . . . . . . . . . . . . . . . . . . . . . . 295

7.5.3 Taylor Dispersion . . . . . . . . . . . . . . . . . . . . 297

7.5.4 Charged Particle in a Pipe . . . . . . . . . . . . . . 302

7.6 Dielectrophoresis . . . . . . . . . . . . . . . . . . . . . . . . 302

7.6.1 Applications . . . . . . . . . . . . . . . . . . . . . . 304

8 Surface Tension-Driven Flows 311

8.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.2 General Form of Young’s Equation . . . . . . . . . . . . . . 317

8.3 Governing Equations for Thin Films . . . . . . . . . . . . . 319

8.4 Dynamics of Capillary Spreading . . . . . . . . . . . . . . . 321

8.5 Thermocapillary Pumping . . . . . . . . . . . . . . . . . . . 324

8.6 Electrocapillary . . . . . . . . . . . . . . . . . . . . . . . . . 328

xviii Contents

8.6.1 Generalized Young–Lippmann Equation . . . . . . . 333

8.6.2 Optoelectrowetting . . . . . . . . . . . . . . . . . . . 335

8.7 Bubble Transport in Capillaries . . . . . . . . . . . . . . . . 337

9 Mixers and Chaotic Advection 343

9.1 The Need for Mixing at Microscales . . . . . . . . . . . . . 344

9.2 Chaotic Advection . . . . . . . . . . . . . . . . . . . . . . . 346

9.3 Micromixers . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

9.4 Quantitative Characterization of Mixing . . . . . . . . . . . 357

10 Simple Fluids in Nanochannels 365

10.1 Atomistic Simulation of Simple Fluids . . . . . . . . . . . . 366

10.2 Density Distribution . . . . . . . . . . . . . . . . . . . . . . 368

10.3 Diffusion Transport . . . . . . . . . . . . . . . . . . . . . . . 375

10.4 Validity of the Navier–Stokes Equations . . . . . . . . . . . 381

10.5 Boundary Conditions at Solid–Liquid Interfaces . . . . . . . 387

10.5.1 Experimental and Computational Results . . . . . . 387

10.5.2 Conceptual Models of Slip . . . . . . . . . . . . . . . 396

10.5.3 Reynolds–Vinogradova Theory for

Hydrophobic Surfaces . . . . . . . . . . . . . . . . . 401

11 Water in Nanochannels 407

11.1 Definitions and Models . . . . . . . . . . . . . . . . . . . . . 407

11.1.1 Atomistic Models . . . . . . . . . . . . . . . . . . . . 409

11.2 Static Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 416

11.2.1 Density Distribution and Dipole Orientation . . . . 417

11.2.2 Hydrogen Bonding . . . . . . . . . . . . . . . . . . . 422

11.2.3 Contact Angle . . . . . . . . . . . . . . . . . . . . . 427

11.2.4 Dielectric Constant . . . . . . . . . . . . . . . . . . . 429

11.3 Dynamic Behavior . . . . . . . . . . . . . . . . . . . . . . . 430

11.3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . 430

11.3.2 Diffusion Transport . . . . . . . . . . . . . . . . . . 435

11.3.3 Filling and Emptying Kinetics . . . . . . . . . . . . 437

12 Electroosmotic Flow in Nanochannels 447

12.1 The Need for Atomistic Simulation . . . . . . . . . . . . . . 447

12.2 Ion Concentrations . . . . . . . . . . . . . . . . . . . . . . . 452

12.2.1 Modified Poisson–Boltzmann Equation . . . . . . . . 455

12.3 Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . 457

12.4 Slip Condition . . . . . . . . . . . . . . . . . . . . . . . . . 461

12.5 Charge Inversion and Flow Reversal . . . . . . . . . . . . . 464

13 Functional Fluids and Functionalized Nanotubes 471

13.1 Colloidal Particles and Self-Assembly . . . . . . . . . . . . . 472

13.1.1 Magnetorheological (MR) Fluids . . . . . . . . . . . 475

Contents xix

13.1.2 Electrophoretic Deposition . . . . . . . . . . . . . . 486

13.2 Electrolyte Transport Through Carbon Nanotubes . . . . . 490

13.2.1 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . 491

13.2.2 Ion Channels in Biological Membranes . . . . . . . . 493

13.2.3 Transport Through Unmodified Nanotubes . . . . . 495

13.2.4 Transport Through Nanotubes with Charges at

the Ends . . . . . . . . . . . . . . . . . . . . . . . . 497

13.2.5 Transport Through Functionalized Nanotubes . . . . 498

13.2.6 Anomalous Behavior . . . . . . . . . . . . . . . . . . 499

14 Numerical Methods for

Continuum Simulation 509

14.1 Spectral Element Method: The μFlow Program . . . . . . . 510

14.1.1 Incompressible Flows . . . . . . . . . . . . . . . . . . 514

14.1.2 Compressible Flows . . . . . . . . . . . . . . . . . . 517

14.1.3 Verification Example: Resolution of the Electric

Double Layer . . . . . . . . . . . . . . . . . . . . . . 524

14.1.4 Moving Domains . . . . . . . . . . . . . . . . . . . . 525

14.2 Meshless Methods . . . . . . . . . . . . . . . . . . . . . . . 531

14.2.1 Domain Simulation . . . . . . . . . . . . . . . . . . . 532

14.2.2 Boundary-Only Simulation . . . . . . . . . . . . . . 537

14.3 Particulate Microflows . . . . . . . . . . . . . . . . . . . . . 542

14.3.1 Hydrodynamic Forces on Spheres . . . . . . . . . . . 543

14.3.2 The Force Coupling Method (FCM) . . . . . . . . . 547

15 Multiscale Modeling of Gas Flows 559

15.1 Direct Simulation Monte Carlo (DSMC) Method . . . . . . 560

15.1.1 Limitations and Errors in DSMC . . . . . . . . . . . 562

15.1.2 DSMC for Unsteady Flows . . . . . . . . . . . . . . 567

15.1.3 DSMC: Information-Preservation Method . . . . . . 569

15.2 DSM: Continuum Coupling . . . . . . . . . . . . . . . . . . 572

15.2.1 The Schwarz Algorithm . . . . . . . . . . . . . . . . 575

15.2.2 Interpolation Between Domains . . . . . . . . . . . . 577

15.3 Multiscale Analysis of Microfilters . . . . . . . . . . . . . . 578

15.3.1 Stokes/DSMC Coupling . . . . . . . . . . . . . . . . 579

15.3.2 Navier–Stokes/DSMC Coupling . . . . . . . . . . . . 584

15.4 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . 588

15.4.1 Classical Solutions . . . . . . . . . . . . . . . . . . . 592

15.4.2 Sone’s Asymptotic Theory . . . . . . . . . . . . . . . 596

15.4.3 Numerical Solutions . . . . . . . . . . . . . . . . . . 606

15.4.4 Nonisothermal Flows . . . . . . . . . . . . . . . . . . 611

15.5 Lattice–Boltzmann Method (LBM) . . . . . . . . . . . . . . 611

15.5.1 Boundary Conditions . . . . . . . . . . . . . . . . . 618

15.5.2 Comparison with Navier–Stokes Solutions . . . . . . 618

15.5.3 LBM Simulation of Microflows . . . . . . . . . . . . 620

xx Contents

16 Multiscale Modeling of Liquid Flows 625

16.1 Molecular Dynamics (MD) Method . . . . . . . . . . . . . . 626

16.1.1 Intermolecular Potentials . . . . . . . . . . . . . . . 628

16.1.2 Calculation of the Potential Function . . . . . . . . 634

16.1.3 Thermostats . . . . . . . . . . . . . . . . . . . . . . 638

16.1.4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . 640

16.1.5 Practical Guidelines . . . . . . . . . . . . . . . . . . 646

16.1.6 MD Software . . . . . . . . . . . . . . . . . . . . . . 648

16.2 MD-Continuum Coupling . . . . . . . . . . . . . . . . . . . 648

16.3 Embedding Multiscale Methods . . . . . . . . . . . . . . . . 656

16.3.1 Application to the Poisson–Boltzmann Equation . . 657

16.3.2 Application to Navier–Stokes Equations . . . . . . . 659

16.4 Dissipative Particle Dynamics (DPD) . . . . . . . . . . . . 663

16.4.1 Governing Equations . . . . . . . . . . . . . . . . . . 665

16.4.2 Numerical Integration . . . . . . . . . . . . . . . . . 668

16.4.3 Boundary Conditions . . . . . . . . . . . . . . . . . 673

17 Reduced-Order Modeling 677

17.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . 677

17.1.1 Quasi-Static Reduced-Order Modeling . . . . . . . . 678

17.1.2 Dynamical Reduced-Order Modeling . . . . . . . . . 679

17.2 Generalized Kirchhoffian Networks . . . . . . . . . . . . . . 680

17.2.1 Equivalent Circuit Representation . . . . . . . . . . 681

17.2.2 Description Languages . . . . . . . . . . . . . . . . . 689

17.3 Black Box Models . . . . . . . . . . . . . . . . . . . . . . . 695

17.3.1 Nonlinear Static Models . . . . . . . . . . . . . . . . 695

17.3.2 Linear Dynamic Models . . . . . . . . . . . . . . . . 697

17.3.3 Nonlinear Dynamic Models . . . . . . . . . . . . . . 701

17.4 Galerkin Methods . . . . . . . . . . . . . . . . . . . . . . . 705

17.4.1 Linear Galerkin Methods . . . . . . . . . . . . . . . 705

17.4.2 Nonlinear Galerkin Methods . . . . . . . . . . . . . 717

18 Reduced-Order Simulation 721

18.1 Circuit and Device Models for Lab-on-a-Chip Systems . . . 721

18.1.1 Electrical Model . . . . . . . . . . . . . . . . . . . . 723

18.1.2 Fluidic Model . . . . . . . . . . . . . . . . . . . . . . 725

18.1.3 Chemical Reactions: Device Models . . . . . . . . . 730

18.1.4 Separation: Device Model . . . . . . . . . . . . . . . 731

18.1.5 Integration of the Models . . . . . . . . . . . . . . . 733

18.1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . 733

18.2 Macromodeling of Squeezed Film Damping . . . . . . . . . 745

18.2.1 Equivalent Circuit Models . . . . . . . . . . . . . . . 747

18.2.2 Galerkin Methods . . . . . . . . . . . . . . . . . . . 749

18.2.3 Mixed-Level Simulation . . . . . . . . . . . . . . . . 751

18.2.4 Black Box Models . . . . . . . . . . . . . . . . . . . 752

Contents xxi

18.3 Compact Model for Electrowetting . . . . . . . . . . . . . . 753

18.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754

Bibliography 757

Index 808

Publish Book: 
Modify Date: 
Monday, January 19, 2009

Dummy View - NOT TO BE DELETED