Table of contents for Micro flows : fundamentals and simulation / George Em Karniadakis, Ali Beskok.


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1 Basic Concepts and Technologies                          1
1.1 New Flow Regimes in MEMS ............   ..     ...   1
1.2 The Continuum Hypothesis ........      .........     7
1.2.1 Molecular Magnitudes ................. 11
1.2.2  Mixed Flow Regimes .....................     16
1.2.3  Experimental Evidence .  ...... .........    17
1.3 The Pioneers .........         ...............   .  21
1.4 Full-System Simulation of MEMS ...............      24
1.5 Modeling of Micro Flows ........     ............ 31
2 Governing Equations and Slip Models                     39
2.1 The Basic Equations of Fluid Dynamics ...........   39
2.1.1 Incompressible Flow ....  ...................  42
2.1.2 Reduced Models ............... 44
2.2 Compressible Flow .......       ...............     45
2.2.1 First-Order Models .   ....   ..    ......    47
2.2.2 The Role of the Accommodation Coefficients  .... 49
2.3 High-Order Models ......................            53
2.3.1 Derivation of High-Order Slip Models .... ... 54
2.3.2  General Slip Condition ...............  ..   57



2.3.3  Comparison of Slip Models .............. 61
3 Shear-Driven and Separated
Micro Flows                                             63
3.1 Couette Flow ....    ...............       .......  63
3.2  Cavity Flow  ............  .....   ............     67
3.3 Grooved Channel Flow ..........       .........      68
3.4 Separated Internal Flows .................... 71
3.4.1  Validation of Slip Models with DSMC ........ 77
3.5 Separated External Flows ................... 83
4 Pressure-Driven Micro Flows:
Slip Flow Regime                                        87
4.1 Isothermal Compressible Flows ................      87
4.2 Adiabatic Compressible Flows - Fanno Theory .......  95
4.3 Inlet Flows .......................... 101
4.4 Validation of Slip Models with DSMC ............ 102
4.5 Effects of Roughness ...............      ....... 108
5 Pressue-Driven Micro Flows: Transition and
Free-Molecular Regimes                                 113
5.1 Transition and Free-Molecular Flow Regimes ....... 113
5.2 Burnett Equations in Micro Channels .... ....... 117
5.3 A Unified Flow Model ............       ....... 119
5.3.1 Velocity Scaling.................... 119
5.3.2  Flowrate Scaling ...........  ......... 122
5.3.3  Model for Pipe and Duct Flows ............ 127
6 Thermal Effects in Micro Scales                         139
6.1 Thermal Creep (Transpiration)  .....    .......    139
6.1.1  Simulation Results ................... 141
6.1.2  A Thermal Creep Experiment ..    ......  .. 145
6.1.3  Knudsen Compressors ................. 146
6.1.4  Other Temperature-Induced Flows .......... 147
6.1.5  Heat Conduction and the Ghost Effect ........ 149
6.2 Heat Transfer in Micro Poiseuille Flows ........... 151
6.3 Heat Transfer in Micro Couette Flows ............ 158
7 Prototype Applications of Gas
Micro Flows                                            163
7.1 Gas Damping and Dynamic Response of MEMS  ...... 163
7.1.1  Reynolds Equation ................... 166
7.1.2  Squeezed Film Effects in Accelerometers ....... 173
7.2 Micro Propulsion and Micro Nozzle Flows ......... 178
7.2.1 Micro Propulsion Analysis .... .......... 180



7.2.2  Rarefaction and Other Effects ............  . 184
8 Electrokinetically Driven Liquid
Micro Flows                                               193
8.1 Electrokinetic Effects - Review .......      ...... 194
8.2  The Electric Double Layer ................... 195
8.3  Near-Wall Potential Distribution . ............. 197
8.4  Governing Equations for Electroosmotic Flows ....... 199
8.4.1  Numerical Formulation and Validation ........ 200
8.5 Electrokinetic Micro Channel Flows . .....     ..... . 202
8.6 EDL/Bulk Flow Interface Velocity
Matching Condition .       ................ ....   . 207
8.7 Electroosmotic Slip Condition .  .............        208
8.7.1 Approximate Evaluation of Drag Force due to
Electroosmotic Effects ................. 209
8.8  Complex Geometry Flows .      ..................... 210
8.8.1  Cross-Flow Junctions ................. 211
8.8.2  Array of Circular and Square Posts .......... 213
8.9  Dielectrophoresis .     .......................      215
9 Numerical Methods for
Continuum Simulation                                      223
9.1 A High-Order Numerical Method: The Flow Code ... . 224
9.1.1  Formulation for Incompressible Micro Flows ..... 228
9.1.2  Formulation for Compressible Micro Flows  ..... 231
9.1.3 Implementation of Slip Boundary Conditions ... 236
9.1.4  Validation Problems .................. 237
9.2 A Meshless Numerical Method .     .................. 239
9.3  The Force Coupling Method for Particulate
Micro Flows ..................     .   ........ 246
10 Numerical Methods for
Atomistic Simulation                                      257
10.1 Molecular Dynamics (MD) Method ............       .. 257
10.1.1 MD-Continumm Coupling .......         ....... 263
10.2 Direct Simulation Monte Carlo (DSMC) Method  ..... 266
10.2.1 Limitations and Errors in DSMC ........... 269
10.2.2 DSMC-Information Preservation Method  ..... . 274
10.2.3 DSMC-Continuum Coupling   ............. 275
10.3 The Boltzmann Equation ................... 279
10.3.1 General Theory .............      .......   . 279
10.3.2 Classical Solutions of Boltzmann Equation  ..... 284
10.3.3 Sone's Asymptotic Theory of
Boltzmann Equation ..............              288
10.3.4 Numerical Solutions of Boltzmann Equation ..... 297


10.3.5 Non-Isothermal Flows ................     . 301
10.4 Lattice-Boltzmann Method (LBM) .............    . 303
10.4.1 Comparison with Navier-Stokes Solution  ...... 306
10.4.2 LBM Simulation of Micro Flows ........... 308
Bibliography                                              311
Index                                                     335