Rarefied gas dynamics: fundamentals for research and practice

315 Pages·2015·5.549 MB·English

by  Sharipov, Felix

FelixSharipov
RarefiedGasDynamics
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FelixSharipov
Rarefied Gas Dynamics
FundamentalsforResearchandPractice
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V
Contents
Preface XIII
ListofSymbols XV
ListofAcronyms XXI
1 MolecularDescription 1
1.1 MechanicsofContinuousMediaandItsRestriction 1
1.2 MacroscopicStateVariables 2
1.3 DiluteGas 3
1.4 IntermolecularPotential 4
1.4.1 DefinitionofPotential 4
1.4.2 HardSpherePotential 4
1.4.3 Lennard-JonesPotential 5
1.4.4 AbinitioPotential 5
1.5 DeflectionAngle 7
1.6 DifferentialCrossSection 8
1.7 TotalCrossSection 9
1.8 EquivalentFreePath 10
1.9 RarefactionParameterandKnudsenNumber 10
2 VelocityDistributionFunction 13
2.1 DefinitionofDistributionFunction 13
2.2 MomentsofDistributionFunction 15
2.3 EntropyandItsFlowVector 18
2.4 GlobalMaxwellian 18
2.5 LocalMaxwellian 20
3 BoltzmannEquation 23
3.1 AssumptionstoDerivetheBoltzmannEquation 23
3.2 GeneralFormoftheBoltzmannEquation 23
3.3 ConservationLaws 25
3.4 EntropyProductionduetoIntermolecularCollisions 27
3.5 IntermolecularCollisionsFrequency 27
VI Contents
4 Gas–SurfaceInteraction 31
4.1 GeneralformofBoundaryConditionforImpermeableSurface 31
4.2 Diffuse–SpecularKernel 33
4.3 Cercignani–LampisKernel 34
4.4 AccommodationCoefficients 34
4.5 GeneralformofBoundaryConditionforPermeableSurface 37
4.6 EntropyProductionduetoGas–SurfaceInteraction 38
5 LinearTheory 43
5.1 SmallPerturbationofEquilibrium 43
5.2 LinearizationNearGlobalMaxwellian 43
5.3 LinearizationNearLocalMaxwellian 46
5.4 PropertiesoftheLinearizedCollisionOperator 47
5.5 LinearizationofBoundaryCondition 48
5.5.1 ImpermeableSurfaceBeingatRest 48
5.5.2 ImpermeableMovingSurface 49
5.5.3 PermeableSurface 50
5.5.4 LinearizationNearReferenceMaxwellian 50
5.5.5 PropertiesofScatteringOperator 50
5.5.6 DiffuseScattering 51
5.6 SeriesExpansion 51
5.7 ReciprocalRelations 53
5.7.1 GeneralDefinitions 53
5.7.2 KineticCoefficients 54
6 TransportCoefficients 57
6.1 ConstitutiveEquations 57
6.2 Viscosity 58
6.3 ThermalConductivity 59
6.4 NumericalResults 61
6.4.1 HardSpherePotential 61
6.4.2 Lennard-JonesPotential 61
6.4.3 AbInitioPotential 62
7 ModelEquations 65
7.1 BGKEquation 65
7.2 S-Model 67
7.3 EllipsoidalModel 69
7.4 DimensionlessFormofModelEquations 70
8 DirectSimulationMonteCarloMethod 73
8.1 MainIdeas 73
8.2 GenerationofSpecificDistributionFunction 74
8.3 SimulationofGas–SurfaceInteraction 75
8.3.1 KernelDecomposition 75
Contents VII
8.3.2 DiffuseScattering 75
8.3.3 Cercignani–LampisScattering 76
8.4 IntermolecularInteraction 77
8.5 CalculationofPost-CollisionVelocities 78
8.6 CalculationofMacroscopicQuantities 80
8.7 StatisticalScatter 81
9 DiscreteVelocityMethod 83
9.1 MainIdeas 83
9.2 VelocityDiscretization 85
9.2.1 OnefoldIntegral 85
9.2.2 TwofoldIntegral 86
9.3 IterativeProcedure 87
9.4 FiniteDifferenceSchemes 88
9.4.1 MainPrinciples 88
9.4.2 One-DimensionalPlanarFlows 89
9.4.3 Two-DimensionalPlanarFlows 90
9.4.4 One-DimensionalAxisymmetricFlows 93
9.4.5 FullKineticEquation 96
10 VelocitySlipandTemperatureJumpPhenomena 97
10.1 GeneralRemarks 97
10.2 ViscousVelocitySlip 98
10.2.1 DefinitionandInputEquation 98
10.2.2 VelocityandHeatFlowProfiles 100
10.2.3 NumericalandExperimentalData 101
10.3 ThermalVelocitySlip 104
10.3.1 DefinitionandInputEquation 104
10.3.2 VelocityandHeatFlowProfiles 106
10.3.3 NumericalandExperimentalData 107
10.4 ReciprocalRelation 108
10.5 TemperatureJump 110
10.5.1 DefinitionandInputEquation 110
10.5.2 TemperatureProfile 112
10.5.3 NumericalData 112
11 One-DimensionalPlanarFlows 115
11.1 PlanarCouetteFlow 115
11.1.1 Definitions 115
11.1.2 Free-MolecularRegime 116
11.1.3 VelocitySlipRegime 117
11.1.4 KineticEquation 117
11.1.5 NumericalScheme 119
11.1.6 NumericalResults 120
11.2 PlanarHeatTransfer 121
VIII Contents
11.2.1 Definitions 121
11.2.2 Free-MolecularRegime 122
11.2.3 TemperatureJumpRegime 123
11.2.4 KineticEquation 124
11.2.5 NumericalScheme 126
11.2.6 NumericalResults 127
11.3 PlanarPoiseuilleandThermalCreepFlows 128
11.3.1 Definitions 128
11.3.2 SlipSolution 130
11.3.3 KineticEquation 131
11.3.4 ReciprocalRelation 133
11.3.5 NumericalScheme 133
11.3.6 SplittingScheme 134
11.3.7 Free-MolecularLimit 137
11.3.8 NumericalResults 137
12 One-DimensionalAxisymmetricalFlows 145
12.1 CylindricalCouetteFlow 145
12.1.1 Definitions 145
12.1.2 SlipFlowRegime 146
12.1.3 KineticEquation 147
12.1.4 Free-MolecularRegime 148
12.1.5 NumericalScheme 149
12.1.6 SplittingScheme 150
12.1.7 Results 152
12.2 HeatTransferbetweenTwoCylinders 153
12.2.1 Definitions 153
12.2.2 TemperatureJumpSolution 154
12.2.3 KineticEquation 155
12.2.4 Free-MolecularRegime 156
12.2.5 NumericalScheme 157
12.2.6 SplittingScheme 158
12.2.7 NumericalResults 159
12.3 CylindricalPoiseuilleandThermalCreepFlows 161
12.3.1 Definitions 161
12.3.2 SlipSolution 163
12.3.3 KineticEquation 163
12.3.4 ReciprocalRelation 165
12.3.5 Free-MolecularRegime 165
12.3.6 NumericalScheme 166
12.3.7 Results 168
13 Two-DimensionalPlanarFlows 173
13.1 FlowsThroughaLongRectangularChannel 173
13.1.1 Definitions 173
Contents IX
13.1.2 SlipSolution 174
13.1.3 KineticEquation 175
13.1.4 Free-MolecularRegime 177
13.1.5 NumericalScheme 177
13.1.6 NumericalResults 178
13.2 FlowsThroughSlitsandShortChannels 180
13.2.1 FormulationoftheProblem 180
13.2.2 Free-MolecularRegime 181
13.2.3 SmallPressureandTemperatureDrops 183
13.2.3.1 Definitions 183
13.2.3.2 KineticEquation 184
13.2.3.3 HydrodynamicSolution 186
13.2.3.4 NumericalResults 186
13.2.4 ArbitraryPressureDrop 189
13.2.4.1 Definition 189
13.2.4.2 KineticEquation 189
13.2.4.3 NumericalResults 190
13.3 EndCorrectionforChannel 194
13.3.1 Definitions 194
13.3.2 KineticEquation 196
13.3.3 NumericalResults 197
14 Two-DimensionalAxisymmetricalFlows 201
14.1 FlowsThroughOrificesandShortTubes 201
14.1.1 FormulationoftheProblem 201
14.1.2 Free-MolecularFlow 202
14.1.3 SmallPressureDrop 203
14.1.3.1 Definitions 203
14.1.3.2 KineticEquations 204
14.1.3.3 HydrodynamicSolution 205
14.1.3.4 NumericalResults 205
14.1.4 ArbitraryPressureDrop 206
14.2 EndCorrectionforTube 210
14.2.1 Definitions 210
14.2.2 NumericalResults 212
14.3 TransientFlowThroughaTube 213
15 FlowsThroughLongPipesUnderArbitraryPressureandTemperature
Drops 219
15.1 StationaryFlows 219
15.1.1 MainEquations 219
15.1.2 IsothermalFlows 221
15.1.3 NonisothermalFlows 223
15.2 PipeswithVariableCrossSection 224
15.3 TransientFlows 226
X Contents
15.3.1 MainEquations 226
15.3.2 ApproachingtoEquilibrium 227
16 AcousticsinRarefiedGases 231
16.1 GeneralRemarks 231
16.1.1 DescriptionofWavesinContinuousMedium 231
16.1.2 ComplexPerturbationFunction 232
16.1.3 One-DimensionalFlows 233
16.2 OscillatoryCouetteFlow 234
16.2.1 Definitions 234
16.2.2 SlipRegime 235
16.2.3 KineticEquation 237
16.2.4 Free-MolecularRegime 238
16.2.5 NumericalScheme 239
16.2.6 NumericalResults 241
16.3 LongitudinalWaves 242
16.3.1 Definitions 242
16.3.2 HydrodynamicRegime 244
16.3.3 KineticEquation 246
16.3.4 ReciprocalRelation 249
16.3.5 High-FrequencyRegime 250
16.3.6 NumericalResults 252
A ConstantsandMathematicalExpressions 257
A.1 PhysicalConstants 257
A.2 VectorsandTensors 257
A.3 NablaOperator 259
A.4 KroneckerDeltaandDiracDeltaFunction 259
A.5 SomeIntegrals 260
A.6 TaylorSeries 260
A.7 SomeFunctions 260
A.8 Gauss–Ostrogradsky’sTheorem 262
A.9 ComplexNumbers 262
B FilesandListings 263
B.1 FileswithNodesandWeightsofGaussQuadrature 263
B.1.1 WeightingFunction(9.16) 263
B.1.1.1 Filecw4.dat,N =4 263
c
B.1.1.2 Filecw6.dat,N =6 263
c
B.1.1.3 Filecw8.dat,N =8 263
c
B.1.2 WeightingFunction(9.22) 264
B.1.2.1 Filecpw4.dat,N =4 264
c
B.1.2.2 Filecpw6.dat,N =6 264
c
B.1.2.3 Filecpw8.dat,N =8 264
c
B.2 FilesforPlanarCouetteFlow 264