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Rarefied Gas Dynamics: Fundamentals for Research and Practice

ISBN: 978-3-527-41326-3
328 pages
February 2016
Rarefied Gas Dynamics: Fundamentals for Research and Practice (352741326X) cover image

Description

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.

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Table of Contents

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

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Author Information

Professsor Felix Sharipov graduated from the Moscow University of Physics and Technology, Faculty of Aerophysics and Space Research, and the Ural State Technical University. Since 1988 he is active in rarefied gas dynamics, since 1992 at the Federal University of Parana in Brazil. His research interests are numerical methods of rarefied gas dynamics applied to microfluidics, vacuum technology and aerothermodynamics. His group develops both probabilistic and deterministic approaches. Prof. Sharipov was organizer of numerous vacuum gas dynamics meetings, and published over a hundred journal articles, conference papers, and book chapters. He is a member of editorial board of international journal ?Vacuum?
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