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More About This Title Rarefied Gas Dynamics - Fundamentals for Researchand Practice
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Aimed at both researchers and professionals who deal with this topic in their routine work, this introduction provides a coherent and rigorous access to the field including relevant methods for practical applications. No preceding knowledge of gas dynamics is assumed.
English
English
Preface XIII
List of Symbols XV
List of Acronyms XXI
1 Molecular Description 1
1.1 Mechanics of Continuous Media and Its Restriction 1
1.2 Macroscopic State Variables 2
1.3 Dilute Gas 3
1.4 Intermolecular Potential 4
1.4.1 Definition of Potential 4
1.4.2 Hard Sphere Potential 4
1.4.3 Lennard-Jones Potential 5
1.4.4 Ab initio Potential 5
1.5 Deflection Angle 7
1.6 Differential Cross Section 8
1.7 Total Cross Section 9
1.8 Equivalent Free Path 10
1.9 Rarefaction Parameter and Knudsen Number 10
2 Velocity Distribution Function 13
2.1 Definition of Distribution Function 13
2.2 Moments of Distribution Function 15
2.3 Entropy and Its Flow Vector 18
2.4 Global Maxwellian 18
2.5 Local Maxwellian 20
3 Boltzmann Equation 23
3.1 Assumptions to Derive the Boltzmann Equation 23
3.2 General Form of the Boltzmann Equation 23
3.3 Conservation Laws 25
3.4 Entropy Production due to Intermolecular Collisions 27
3.5 Intermolecular Collisions Frequency 27
4 Gas–Surface Interaction 31
4.1 General form of Boundary Condition for Impermeable Surface 31
4.2 Diffuse–Specular Kernel 33
4.3 Cercignani–Lampis Kernel 34
4.4 Accommodation Coefficients 34
4.5 General form of Boundary Condition for Permeable Surface 37
4.6 Entropy Production due to Gas–Surface Interaction 38
5 Linear Theory 43
5.1 Small Perturbation of Equilibrium 43
5.2 Linearization Near Global Maxwellian 43
5.3 Linearization Near Local Maxwellian 46
5.4 Properties of the Linearized Collision Operator 47
5.5 Linearization of Boundary Condition 48
5.5.1 Impermeable Surface Being at Rest 48
5.5.2 Impermeable Moving Surface 49
5.5.3 Permeable Surface 50
5.5.4 Linearization Near Reference Maxwellian 50
5.5.5 Properties of Scattering Operator 50
5.5.6 Diffuse Scattering 51
5.6 Series Expansion 51
5.7 Reciprocal Relations 53
5.7.1 General Definitions 53
5.7.2 Kinetic Coefficients 54
6 Transport Coefficients 57
6.1 Constitutive Equations 57
6.2 Viscosity 58
6.3 Thermal Conductivity 59
6.4 Numerical Results 61
6.4.1 Hard Sphere Potential 61
6.4.2 Lennard-Jones Potential 61
6.4.3 Ab Initio Potential 62
7 Model Equations 65
7.1 BGK Equation 65
7.2 S-Model 67
7.3 Ellipsoidal Model 69
7.4 Dimensionless Form of Model Equations 70
8 Direct Simulation Monte Carlo Method 73
8.1 Main Ideas 73
8.2 Generation of Specific Distribution Function 74
8.3 Simulation of Gas–Surface Interaction 75
8.3.1 Kernel Decomposition 75
8.3.2 Diffuse Scattering 75
8.3.3 Cercignani–Lampis Scattering 76
8.4 Intermolecular Interaction 77
8.5 Calculation of Post-Collision Velocities 78
8.6 Calculation of Macroscopic Quantities 80
8.7 Statistical Scatter 81
9 Discrete Velocity Method 83
9.1 Main Ideas 83
9.2 Velocity Discretization 85
9.2.1 Onefold Integral 85
9.2.2 Twofold Integral 86
9.3 Iterative Procedure 87
9.4 Finite Difference Schemes 88
9.4.1 Main Principles 88
9.4.2 One-Dimensional Planar Flows 89
9.4.3 Two-Dimensional Planar Flows 90
9.4.4 One-Dimensional Axisymmetric Flows 93
9.4.5 Full Kinetic Equation 96
10 Velocity Slip and Temperature Jump Phenomena 97
10.1 General Remarks 97
10.2 Viscous Velocity Slip 98
10.2.1 Definition and Input Equation 98
10.2.2 Velocity and Heat Flow Profiles 100
10.2.3 Numerical and Experimental Data 101
10.3 Thermal Velocity Slip 104
10.3.1 Definition and Input Equation 104
10.3.2 Velocity and Heat Flow Profiles 106
10.3.3 Numerical and Experimental Data 107
10.4 Reciprocal Relation 108
10.5 Temperature Jump 110
10.5.1 Definition and Input Equation 110
10.5.2 Temperature Profile 112
10.5.3 Numerical Data 112
11 One-Dimensional Planar Flows 115
11.1 Planar Couette Flow 115
11.1.1 Definitions 115
11.1.2 Free-Molecular Regime 116
11.1.3 Velocity Slip Regime 117
11.1.4 Kinetic Equation 117
11.1.5 Numerical Scheme 119
11.1.6 Numerical Results 120
11.2 Planar Heat Transfer 121
11.2.1 Definitions 121
11.2.2 Free-Molecular Regime 122
11.2.3 Temperature Jump Regime 123
11.2.4 Kinetic Equation 124
11.2.5 Numerical Scheme 126
11.2.6 Numerical Results 127
11.3 Planar Poiseuille andThermal Creep Flows 128
11.3.1 Definitions 128
11.3.2 Slip Solution 130
11.3.3 Kinetic Equation 131
11.3.4 Reciprocal Relation 133
11.3.5 Numerical Scheme 133
11.3.6 Splitting Scheme 134
11.3.7 Free-Molecular Limit 137
11.3.8 Numerical Results 137
12 One-Dimensional Axisymmetrical Flows 145
12.1 Cylindrical Couette Flow 145
12.1.1 Definitions 145
12.1.2 Slip Flow Regime 146
12.1.3 Kinetic Equation 147
12.1.4 Free-Molecular Regime 148
12.1.5 Numerical Scheme 149
12.1.6 Splitting Scheme 150
12.1.7 Results 152
12.2 Heat Transfer between Two Cylinders 153
12.2.1 Definitions 153
12.2.2 Temperature Jump Solution 154
12.2.3 Kinetic Equation 155
12.2.4 Free-Molecular Regime 156
12.2.5 Numerical Scheme 157
12.2.6 Splitting Scheme 158
12.2.7 Numerical Results 159
12.3 Cylindrical Poiseuille andThermal Creep Flows 161
12.3.1 Definitions 161
12.3.2 Slip Solution 163
12.3.3 Kinetic Equation 163
12.3.4 Reciprocal Relation 165
12.3.5 Free-Molecular Regime 165
12.3.6 Numerical Scheme 166
12.3.7 Results 168
13 Two-Dimensional Planar Flows 173
13.1 Flows Through a Long Rectangular Channel 173
13.1.1 Definitions 173
13.1.2 Slip Solution 174
13.1.3 Kinetic Equation 175
13.1.4 Free-Molecular Regime 177
13.1.5 Numerical Scheme 177
13.1.6 Numerical Results 178
13.2 Flows Through Slits and Short Channels 180
13.2.1 Formulation of the Problem 180
13.2.2 Free-Molecular Regime 181
13.2.3 Small Pressure and Temperature Drops 183
13.2.3.1 Definitions 183
13.2.3.2 Kinetic Equation 184
13.2.3.3 Hydrodynamic Solution 186
13.2.3.4 Numerical Results 186
13.2.4 Arbitrary Pressure Drop 189
13.2.4.1 Definition 189
13.2.4.2 Kinetic Equation 189
13.2.4.3 Numerical Results 190
13.3 End Correction for Channel 194
13.3.1 Definitions 194
13.3.2 Kinetic Equation 196
13.3.3 Numerical Results 197
14 Two-Dimensional Axisymmetrical Flows 201
14.1 Flows Through Orifices and Short Tubes 201
14.1.1 Formulation of the Problem 201
14.1.2 Free-Molecular Flow 202
14.1.3 Small Pressure Drop 203
14.1.3.1 Definitions 203
14.1.3.2 Kinetic Equations 204
14.1.3.3 Hydrodynamic Solution 205
14.1.3.4 Numerical Results 205
14.1.4 Arbitrary Pressure Drop 206
14.2 End Correction for Tube 210
14.2.1 Definitions 210
14.2.2 Numerical Results 212
14.3 Transient Flow Through a Tube 213
15 Flows Through Long Pipes Under Arbitrary Pressure and Temperature Drops 219
15.1 Stationary Flows 219
15.1.1 Main Equations 219
15.1.2 Isothermal Flows 221
15.1.3 Nonisothermal Flows 223
15.2 Pipes with Variable Cross Section 224
15.3 Transient Flows 226
15.3.1 Main Equations 226
15.3.2 Approaching to Equilibrium 227
16 Acoustics in Rarefied Gases 231
16.1 General Remarks 231
16.1.1 Description ofWaves in Continuous Medium 231
16.1.2 Complex Perturbation Function 232
16.1.3 One-Dimensional Flows 233
16.2 Oscillatory Couette Flow 234
16.2.1 Definitions 234
16.2.2 Slip Regime 235
16.2.3 Kinetic Equation 237
16.2.4 Free-Molecular Regime 238
16.2.5 Numerical Scheme 239
16.2.6 Numerical Results 241
16.3 LongitudinalWaves 242
16.3.1 Definitions 242
16.3.2 Hydrodynamic Regime 244
16.3.3 Kinetic Equation 246
16.3.4 Reciprocal Relation 249
16.3.5 High-Frequency Regime 250
16.3.6 Numerical Results 252
A Constants and Mathematical Expressions 257
A.1 Physical Constants 257
A.2 Vectors and Tensors 257
A.3 Nabla Operator 259
A.4 Kronecker Delta and Dirac Delta Function 259
A.5 Some Integrals 260
A.6 Taylor Series 260
A.7 Some Functions 260
A.8 Gauss–Ostrogradsky’sTheorem 262
A.9 Complex Numbers 262
B Files and Listings 263
B.1 Files with Nodes andWeights of Gauss Quadrature 263
B.1.1 Weighting Function (9.16) 263
B.1.1.1 File cw4.dat, Nc = 4 263
B.1.1.2 File cw6.dat, Nc = 6 263
B.1.1.3 File cw8.dat, Nc = 8 263
B.1.2 Weighting Function (9.22) 264
B.1.2.1 File cpw4.dat, Nc = 4 264
B.1.2.2 File cpw6.dat, Nc = 6 264
B.1.2.3 File cpw8.dat, Nc = 8 264
B.2 Files for Planar Couette Flow 264
B.2.1 Listing of Program “couette_planar.for” 264
B.2.2 Output File with Results “Res_couette_planar.dat” 266
B.3 Files for Planar Heat Transfer 266
B.3.1 Listing of Program “heat_planar.for” 266
B.3.2 Output File with Results “Res_heat_planar.dat” 268
B.4 Files for Planar Poiseuille and Creep Flows 268
B.4.1 Listing of Program “poiseuille_creep_planar.for” 268
B.4.2 Output File “Res_pois_cr_pl.dat” with Results 272
B.5 Files for Cylindrical Couette Flows 272
B.5.1 Listing of Program “couette_axisym.for” 272
B.5.2 Output File “Res_couet_axi.dat” with Results 275
B.6 Files for Cylindrical Heat Transfer 276
B.6.1 Listing of Program “heat_axisym.for” 276
B.6.2 Output File “Res_heat_axi.dat” with Results 280
B.7 Files for Axi-Symmetric Poiseuille and Creep Flows 280
B.7.1 Listing of Program “poiseuille_creep_axisym.for” 280
B.7.2 Output File “Res_pois_cr_axi.dat” with Results 284
B.8 Files for Poiseuille and Creep FlowsThrough Channel 284
B.8.1 Listing of Program “poiseuille_creep_chan.for” 284
B.8.2 Output File “Res_pois_cr_ch.dat” with Results 287
B.9 Files for Oscillating Couette Flow 287
B.9.1 Listing of Program “couette_osc.for” 287
B.9.2 Output File “Res_couette_osc.dat” with Results 290
References 291
Index 303
