Rarefied Gas Dynamics: Fundamentals for Research and Practice

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.

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