Rational Extended Thermodynamics beyond the Monatomic Gas (eBook)

Preface 8
Acknowledgements 10
Contents 12
List of Symbols 22
1 Introduction 26
1.1 Thermodynamics of Irreversible Processes and the Laws of Navier-Stokes and Fourier 27
1.1.1 Dawn of Thermodynamics 27
1.1.2 TIP of One-Component Viscous and Heat-Conducting Fluids 28
1.1.3 Laws of Navier-Stokes and Fourier 30
1.1.4 Parabolic Structure and the Prediction of Infinite Speed of Waves in TIP 31
1.1.5 Cattaneo Equation 32
1.1.5.1 Classical Cattaneo Equation 32
1.1.5.2 Generalized Cattaneo Equation and the Second Sound 33
1.2 First Tentative of Extended Thermodynamics and Rational Extended Thermodynamics 34
1.3 Rational Thermodynamics and the Entropy Principle 34
1.4 Other Approaches 35
1.5 Rational Extended Thermodynamics and the Kinetic Theory 36
1.5.1 Boltzmann Equation and the Moments 37
1.5.2 Closure of RET 38
1.5.3 Macroscopic Approach of RET with 13 Fields 39
1.5.4 Grad Distribution 40
1.5.5 Closure via the Maximum Entropy Principle and Molecular RET of Monatomic Gases 40
1.6 New Approach to Polyatomic Gas and Dense Gas 41
1.6.1 Macroscopic Approach with 14 Fields 42
1.6.2 Singular Limit from Polyatomic to Monatomic Gas 43
1.6.3 MEP Closure and the Molecular Approach for the 14-Moment Theory 44
1.6.4 Applications of the 14-Field RET Theory 45
1.6.4.1 Dispersion Relation for Sound in Rarefied Diatomic Gases 45
1.6.4.2 Shock Wave Structure in a Rarefied Polyatomic Gas 45
1.6.4.3 Some Other Applications 46
1.6.5 Molecular RET of Polyatomic Gases 47
1.6.6 6-Field RET Theory and Comparison with the Meixner Theory of Relaxation Processes 49
1.6.7 Nonequilibrium Temperature 50
1.7 Mixture of Gases with Multi-Temperature 50
1.8 Qualitative Analysis 51
1.9 About this Book 52
References 53
Part I Mathematical Structure and Waves 58
2 Mathematical Structure 59
2.1 System of Balance Laws 59
2.1.1 Hyperbolicity in the t-Direction 60
2.1.2 Symmetric Hyperbolic System 60
2.2 Axioms of Rational Extended Thermodynamics 61
2.3 Entropy Principle and Symmetric Systems 62
2.3.1 General Discussions 62
2.3.2 Symmetric System of Euler Fluids 64
2.4 Principal Subsystems 65
2.5 Conservation and Balance Laws, and Equilibrium Subsystem 66
2.6 Qualitative Analysis 68
2.6.1 Competition Between Hyperbolicity and Dissipation 68
2.6.1.1 A Simple Example: Burgers' Equation 68
2.6.2 Shizuta Kawashima K-Condition 70
2.6.3 Global Existence and Stability of Constant State 71
2.6.3.1 An Example: Global Existence without the K-Condition 71
2.7 Galilean Invariance 72
2.7.1 Field Equations in Terms of Intrinsic Quantities 74
2.7.2 Diagonal Structure in RET 74
References 76
3 Waves in Hyperbolic Systems 78
3.1 Linear Wave 78
3.1.1 Plane Harmonic Waves and the Dispersion Relation 78
3.1.2 High Frequency Limit 79
3.2 Acceleration Wave 80
3.3 Shock Wave 85
3.3.1 Rankine-Hugoniot Relations 85
3.3.2 Admissibility of Shock Waves 86
3.3.2.1 Lax, Entropy Growth, and Liu Admissibility Conditions 87
3.4 Shock Structure 89
3.4.1 Shock Wave Structure and Sub-shock Formation 90
3.4.2 Non-existence of Smooth Shocks When s> ?max(uo)
3.5 Riemann Problem for Balance Laws 93
3.5.1 Riemann Problem with Structure 93
3.5.2 Conjecture Concerning Large-Time Asymptotic Behavior of Shock Structure 94
References 97
Part II Survey of Rational Extended Thermodynamics of Monatomic Gas 100
4 RET of Rarefied Monatomic Gas 101
4.1 Extended Thermodynamics with 13 Fields and Subsystems 101
4.1.1 10-Field Principal Subsystem 102
4.1.2 Euler 5-Field Principal Subsystem 103
4.1.3 4-Field Principal Subsystem 103
4.1.4 1-Field Principal Subsystem 104
4.2 Bounded Domain: Heat Conduction and Problematic Boundary Data 104
4.2.1 Heat Conduction Analyzed by the 13-Moment ET Theory 105
4.2.1.1 One Dimensional Solutions and Their Comparison with the Solutions Derived from the Navier-Stokes Fourier Theory 106
4.2.1.2 Solution of a Boundary Value Problem 109
4.2.2 Difficulty in the ET Theory in a Bounded Domain When the Number of Fields is More Than 13 110
4.3 Molecular RET for Large Number of Moments 111
4.3.1 Closure via the Entropy Principle 113
4.3.2 Closure via the Maximum Entropy Principle 115
4.4 Maximum Characteristic Velocity in the Classical Theory 116
4.4.1 Lower Bound Estimate and Characteristic Velocities for Large Number of Moments 117
4.5 Convergence Problem and a Theory Near Equilibrium State 119
4.6 Comparison with Experimental Data: Sound Waves and Light Scattering 120
4.7 Relativistic Theory and the Limit of Maximum Characteristic Velocity 121
4.7.1 Finite System of Moment Equations and Its Closure 123
4.7.1.1 Propagation in an Equilibrium State and the Maximum Characteristics Velocity 123
4.7.2 The Macroscopic Relativistic 14-Field Theory 125
4.7.2.1 Remark on the Einstein Equation 127
References 127
Part III RET of Polyatomic Gas and Dense Gas with 14 Fields 129
5 RET 14-Field Theory of Polyatomic Gas and Dense Gas 130
5.1 Previous Tentatives 130
5.2 Binary Hierarchy in ET of Polyatomic Gas and Dense Gas: Heuristic Viewpoint 131
5.3 ET 14-Field Theory 132
5.3.1 Exploitation of the Galilean Invariance 133
5.3.2 Exploitation of the Entropy Principle 135
5.3.2.1 Equilibrium State 135
5.3.2.2 Entropy 136
5.3.2.3 Constitutive Equations Near Equilibrium 136
5.3.2.4 Productions 140
5.3.2.5 Linear Constitutive Equations Expressed by the Usual Fields 141
5.3.3 Convexity of the Entropy Density 142
5.3.4 Main Field 143
5.3.5 Field Equations 144
5.3.6 Relationship Between ET and Navier-Stokes Fourier Theories 146
5.4 Rarefied-Gas Limit 147
5.4.1 Non-polytropic Gas 147
5.4.2 Polytropic Gas 149
5.5 Models of Dense Gas 151
5.5.1 Gas with the Virial Equations of State 151
5.5.2 Hard-Sphere System 152
5.5.3 van der Waals Fluid 154
5.5.4 Remark 156
5.6 Singular Limit from Polyatomic to Monatomic Gas 156
References 158
6 Maximum Entropy Principle for Rarefied Polyatomic Gas 159
6.1 Generalized Distribution Function for Rarefied Polyatomic Gases 159
6.2 Equilibrium Distribution Function for Polyatomic Gases and the Euler System 160
6.3 Justification of the Binary Hierarchy of the Moment-Equations for Polyatomic Gases 165
6.3.1 MEP for the 14-Moment Systemfor Polyatomic Gases 166
6.3.2 Non-convective Fluxes and Productions 168
6.3.3 Entropy and Entropy Flux 170
6.3.4 Remark 170
References 171
Part IV Applications of the ET14 Theory 172
7 Linear Wave in a Polyatomic Gas 173
7.1 Basic Equations 173
7.2 Dispersion Relation for Sound 175
7.2.1 Dispersion Relation, Phase Velocity and Attenuation Factor 175
7.2.2 High Frequency Limit of the Phase Velocity and the Attenuation Factor 176
7.3 Comparison with Experimental Data 177
7.3.1 Preliminary Calculations 178
7.3.1.1 Specific Heat 178
7.3.2 Relaxation Times 180
7.3.3 Experimental Data and Theoretical Predictions for the Dispersion Relation 180
7.3.3.1 Hydrogen Gases: n-H2 and p-H2 180
7.3.3.2 Deuterium Gases: n-D2 and o-D2 183
7.3.3.3 Hydrogen Deuteride Gases: HD 183
7.3.4 Remarks 184
7.4 Conclusion 187
References 188
8 Shock Wave in a Polyatomic Gas 190
8.1 Introduction 190
8.2 Basic Equations 192
8.2.1 Equations of State, Internal Energy,and Sound Velocity 192
8.2.2 Balance Equations 192
8.3 Setting of the Problem 194
8.3.1 Dimensionless Form of the Field Equations 194
8.3.2 Boundary Conditions: Rankine-Hugoniot Conditions for the System of the Euler Equations 196
8.3.3 Parameters 196
8.3.4 Numerical Methods 198
8.4 Navier-Stokes Fourier Theory 204
8.5 Shock Wave Structure 205
8.5.1 Type A: Nearly Symmetric Shock Wave Structure 205
8.5.2 Type B: Asymmetric Shock Wave Structure 206
8.5.3 Type C: Shock Wave Structure Composed of Thin and Thick Layers 206
8.5.4 Critical Mach Numbers for the Transitions Between the Types A-B and B-C 207
8.5.5 Reexamination of the Bethe-Teller Theory 207
8.6 Comparison with Experimental Data 208
8.7 Conclusion 208
References 209
9 Light Scattering, Heat Conduction, and Fluctuation 210
9.1 Light Scattering 210
9.1.1 Introduction 210
9.1.2 Basic Equations 211
9.1.2.1 ET14 Theory 211
9.1.2.2 Navier-Stokes Fourier Theory 214
9.1.3 Comparison with Experimental Data for CO2 214
9.2 Heat Conduction 215
9.2.1 Basis of the Present Analysis 216
9.2.1.1 Basic System of Equations 216
9.2.1.2 Reduced Basic System of Equations 217
9.2.1.3 Navier-Stokes Fourier Theory 218
9.2.2 Boundary Conditions 218
9.2.3 Effect of the Dynamic Pressure 218
9.2.4 An Example: Polyatomic Effect in a Para-Hydrogen Gas 219
9.3 Fluctuating Hydrodynamics of ET14 220
9.3.1 Introduction 220
9.3.2 Theory of Fluctuating Hydrodynamics Based on ET 220
9.3.3 Two Subsystems of the Stochastic Field Equations 221
9.3.3.1 System-L 222
9.3.3.2 System-T 223
9.3.4 Relationship to the Landau-Lifshitz Theory 223
9.3.4.1 System-L 224
9.3.4.2 System-T 224
9.3.5 Conclusion 225
References 225
Part V Maximum Entropy Principle and Nesting Theories of Many Moments 228
10 Molecular ET Theory of Rarefied Polyatomic Gas 229
10.1 Introduction 229
10.2 MEP Closure for Rarefied Polyatomic Gases with Many Moments 230
10.2.1 Galilean Invariance 233
10.2.2 Closure of the (N,M)-System via the Maximum Entropy Principle 234
10.2.3 Closure of the (N,M)-System via the Entropy Principle 235
10.2.4 Closure and Symmetric Hyperbolic Form 237
10.3 Closure in the Neighborhood of a Local Equilibrium State and Principal Subsystems 238
10.3.1 14-Moment System and Its Principal Subsystems 239
10.3.2 Closure for Higher-Order Systems 240
10.3.2.1 17-Moment System (N=3-, M=1) 241
10.3.2.2 18-Moment System (N=3-, M=2-) 241
10.3.2.3 30-Moment System (N=3, M=2) 242
10.4 Characteristic Velocities of the (N,M)-System 243
10.4.1 Characteristic Velocities of the 14-, 11-, 6- and 5-Moment Systems 244
10.4.2 Systems with D-Independent Characteristic Velocities 246
10.5 Characteristic Velocities of the (N,N-1)-System and the Analysis of the Cases: D?3 and D?? 249
10.5.1 Limit Case: D?3 250
10.5.2 Limit Case: D?? 251
10.5.3 The Case: 3 < D <
10.6 Dependence of the Maximum Characteristic Velocity on the Order N 254
References 256
Part VI ET6: A Theory of Far-from-Equilibrium Thermodynamics 258
11 Non-linear ET6 and the Role of the Dynamic Pressure: Phenomenological Approach 259
11.1 Introduction 259
11.2 ET Theory with Six Fields 261
11.2.1 Galilean Invariance 261
11.2.2 Entropy Principle 262
11.2.3 Convexity Condition and Stability 264
11.2.4 Residual Inequality and Production Term 265
11.2.5 Main Field and Symmetric Form 266
11.2.6 Alternative Form of the Differential System 267
11.2.7 Euler Fluid as a Principal Subsystem of the ET6 System and Subcharacteristic Conditions 268
11.3 Comparison Between Nonlinear Systems of the ET6 Theory and of the Meixner Theory 269
11.4 ET6 Theory Near Equilibrium 273
11.4.1 Comparison Between the ET Theory Near Equilibrium and the Meixner Theory 274
11.5 Examples of an Ideal Polytropic Gas 276
11.5.1 Far-from-Equilibrium Case 276
11.5.2 Near-Equilibrium Case 279
11.5.3 Rarefied Monatomic-Gas Limit 279
11.6 Conclusion 280
References 281
12 Molecular Non-linear ET6 for Rarefied Polyatomic Gas 284
12.1 Introduction 284
12.2 Nonequilibrium Distribution Function 285
12.3 Closure and the Field Equations 288
12.3.1 Entropy Density 289
12.4 Main Field and Symmetric System 290
References 290
13 Application of ET6: Shock Wave and Sub-shock Formation 292
13.1 Introduction 292
13.2 Basis of the Present Analysis 292
13.2.1 Characteristic Velocities 292
13.2.2 Parameters 293
13.2.3 Dimensionless Form of the Balance Equations 294
13.2.4 Boundary Conditions 294
13.2.5 RH Conditions for a Sub-shock in Type C 295
13.2.6 Numerical Methods 295
13.2.7 Case 1: M0< ?max/c0
13.2.8 Case 2: M0> ?max/c0
13.3 Shock Wave Structure with and without a Sub-shock 296
13.3.1 Shock Wave Structure without a Sub-shock 296
13.3.2 Shock Wave Structure with a Sub-shock 297
13.3.3 Discussions 299
13.4 Strength and Stability of a Sub-shock 301
13.4.1 Mach Number Dependence of the Strength of a Sub-shock 301
13.4.2 Stability of a Sub-shock 302
13.5 Meixner's Temperature and the Temperature Overshoot 302
13.6 Conclusion 303
References 304
14 Acceleration Wave, K-Condition, and Global Existence in ET6 305
14.1 Characteristic Velocities and the K-Condition 305
14.2 Time-Evolution of the Amplitude and the Critical Time 307
14.3 Conclusion 310
References 310
15 Nonequilibrium Temperature and Chemical Potential 311
15.1 Generalized Gibbs Equation, Nonequilibrium Temperature and Chemical Potential 311
15.2 Nonequilibrium Temperature and Chemical Potential in ET with the Binary Hierarchy 313
15.3 Nonequilibrium Temperature and Chemical Potential in ET6 and ET14 314
15.4 Conclusion 316
References 316
Part VII Mixture of Gases with Multi-Temperature 318
16 Multi-Temperature Mixture of Fluids 319
16.1 Introduction 319
16.2 Mixtures in Rational Thermodynamics 320
16.2.1 Galilean Invariance of Field Equations 323
16.3 Coarse-Grained Theories: Single Temperature Model and Classical Mixture 325
16.4 Mixtures of Euler Fluids 327
16.4.1 Entropy Principle and Its Restrictions 328
16.4.2 Symmetric Hyperbolic Systemand Principal Subsystems 331
16.4.3 Characteristic Velocities and Their Upper Bound in the ST Model 332
16.4.4 Qualitative Analysis and K-Condition in the Mixture Theories 332
16.5 Average Temperature 333
16.5.1 Alternative Form of the Differential System 334
16.6 Examples of Spatially Homogeneous Mixture and Static Heat Conduction 335
16.6.1 Solution of a Spatially Homogenous Mixture 335
16.6.2 Solution of Static Heat Conduction 337
16.7 Maxwellian Iteration 340
16.8 A Classical Approach to Multi-Temperature Mixtures 342
References 345
17 Shock Structure and Temperature Overshoot in Macroscopic Model of Mixtures 348
17.1 Introduction 348
17.2 Binary Mixture of Euler Fluids 349
17.3 Shock Structure Problem 350
17.3.1 Dimensionless Shock Structure Equations 351
17.3.2 Boundary Conditions and Numerical Procedure 352
17.3.3 Profile of Shock Structure 354
17.4 Shock Structure and Temperature Overshoot 355
17.5 Shock Thickness and the Knudsen Number 357
References 357
Part VIII Maxwellian Iteration and Objectivity 359
18 Hyperbolic Parabolic Limit, Maxwellian Iterationand Objectivity 360
18.1 Different Constitutive Equations 360
18.2 Frame-Dependence of the Heat Flux 361
18.2.1 Maxwellian Iteration and the Parabolic Limit 362
18.3 Maxwellian Iteration and the Entropy Principle 363
18.4 Regularized System and Non-subshock Formation 365
18.5 Conclusion 367
References 367
19 Open Problems 369
19.1 Open Problems 369
Author Index 371
Subject Index 376