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 |  موضوع: كتاب Rarefied Gas Dynamics - Fundamentals for Research and Practice الأربعاء 02 مارس 2022, 7:17 am | |
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أخواني في الله
أحضرت لكم كتاب
Rarefied Gas Dynamics - Fundamentals for Research and Practice
Felix Sharipov
و المحتوى كما يلي :
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 27VI Contents
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 75Contents VII
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 121VIII Contents
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 and Thermal 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 and Thermal 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 173Contents IX
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 226X Contents
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 of Waves 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 Longitudinal Waves 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’s Theorem 262
A.9 Complex Numbers 262
B Files and Listings 263
B.1 Files with Nodes and Weights 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 264Contents XI
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 Flows Through 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
Index
a
acceptance – rejection method 74
accommodation coefficient
– definition 34
– of energy of normal motion 34
– of tangential momentum 34
– values 37, 103
acoustics 231–255
amplitude of oscillation 232
atomic weight 2, 257
attenuation 231
average value
– per mass unity 15
– per particle 15
– per volume unity 15
Avogadro number 2, 257
b
BGK, see model equation
Boltzmann constant 3, 257
Boltzmann equation
– full 23–27
– linearized 43–47
bulk velocity 15
c
Chapman – Enskog method 57
chemical potential 39
collision integral
– full 24
– linearized 45
condensation coefficient 38
conservations laws 25
constitutive equations 57
Couette flow
– cylindrical 145–153
– oscillatory 234–242
– planar 115–121
cross section
– differential 8
– total 9
cumulative function 74
d
derivative approximation
– centered 88, 90, 95, 96, 177, 240
– one-sided 88–92
diffuse reflection see scattering kernel
Dirac delta function 259
direct simulation Monte Carlo 73–82
discrete velocity method 83–96
e
effusion 181
ellipsoidal model see model equation
end correction
– channel 194–198
– long pipe 222
– tube 210–213
entropy 18
entropy production
– due to collisions 27
– due to gas-surface interaction, 38
equivalent free path 10
error function 260
f
flow through
– long pipe 219–230
– orifice 201–210
– short channel 189–193
– short tube 201–210
Index
flow through (contd.)
– slit 189–193
flow vector of
– energy 16
– entropy 18
– heat 17
– mass 16
– particles 16
flows
– one-dimensional axisymmetric 93, 145
– one-dimensional planar 89, 115
– two-dimensional axisymmetric 201
– two-dimensional planar 90, 173
Fourier’s law 57, 60, 105, 123, 131, 154
free-molecular flow 116, 122, 137, 148, 156,
165, 177, 181, 202, 238
g
gas-surface interaction 31–40
Gauss-Ostrogradsky’s theorem 262
h
H-theorem 27, 40, 66, 68, 70
heat transfer
– cylindrical 153–161
– planar 121–128
hydrodynamic velocity see bulk velocity
i
intermolecular collisions frequency 27
intermolecular potential
– ab initio 5, 62, 120, 127, 207
– definition 4
– hard sphere 4, 27, 61, 79, 101, 113, 120,
138, 169, 190, 207, 208, 242
– Lennard-Jones 5, 8, 9, 61, 62, 68, 102, 107,
113, 138
internal energy 3
iterative procedure 87
j
jump boundary condition see temperature
jump coefficient
k
kinetic coefficients 53, 54, 109, 133, 165, 185,
249
Knudsen layer 98, 100
Knudsen number 10
Kronecker delta 259
l
linearization of
– Boltzmann equation
– near global Maxwellian 43
– near local Maxwellian 46
– boundary condition 48
Loschmidt constant 12, 13, 257
m
Mach belt 193
Mach disk 210
Maxwellian distribution function
– global 18
– local 20
– reference 46
– wall 38
model equation
– BGK 65, 70, 99, 112, 118, 120, 147, 189,
193, 197, 204, 213, 237
– ellipsoidal model 69, 155
– general form 71
– linearized 71
– S-model 67, 99, 110, 124, 138, 155, 163,
175, 184, 207, 217, 247
molar gas constant 3, 257
moments of distribution function 15, 44, 47,
84, 110, 116, 129
momentum flux tensor 16
n
Navier-Stokes equations 57, 146, 174, 205,
212, 235
Newton’s law 57, 186, 235
no time counter method 78
number density 2, 15
o
oscillation parameter 233
p
peculiar velocity 16
phase of oscillation 232
Poiseuille flow
– cylindrical 161–171
– planar 128–142
– through finite channel 183–188
– through rectangular channel 173–180
– through slit 183–188
polar coordinates 75, 86, 92
Prandtl number 58Index 305
pressure 3, 17
pressure tensor 17
q
quadrature rule
– Gauss 85, 119, 134, 149, 263
– Simpson 85, 86, 136, 150, 178
r
rarefaction parameter 10, 71
reciprocal relations 53, 108, 133, 165, 249
s
S-model, see model equation
scattering kernel
– Cercignani-Lampis 34, 76
– definition 31
– diffuse 33, 75
– diffuse-specular 33
– specular 33
signum function 261
slip boundary condition, see velocity slip
coefficient
slip regime flow 130, 146, 163, 174, 235
source term
– bulk 47
– surface 49
specular reflection, see scattering kernel
standard atmosphere 257
state equation 3
statistical scatter 81
t
temperature 16
temperature jump coefficient 110–113
thermal conductivity coefficient 59
thermal creep
– cylindrical 161–171
– planar 128–142
– through finite channel 183–188
– through rectangular channel 173–180
– through slit 183–188
transient flow through
– long tube 226–230
– orifice and short tube 213–217
transmission probability through
– channel 182
– tube 203
v
velocity distribution function 13–18
velocity slip coefficient
– thermal 104–108
– viscous 98–103
viscosity coefficient 58
w
wave number 231
wave propagation
– longitudinal 242–255
– transversal 234–242
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