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ОбложкаGray W.G. Introduction to the thermodynamically constrained averaging theory for porous medium systems / W.G.Gray, C.T.Miller. - Cham: Springer, 2014. - xxxiv, 582 p.: ill. - (Advances in geophysical and environmental mechanics and mathematics). - Bibliogr. a la fin des chapitres. - Ind.: p.573-582. - ISBN 978-3-319-04009-7; ISSN 1866-8348
Шифр: (И/З.3-G76) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
1    Elements of Thermodynamically Constrained Averaging
     Theory ..................................................... 1
1.1  Overview ................................................... 1
1.2  Identification of Scales for Modeling ...................... 4
     1.2.1  Length Scales ....................................... 4
     1.2.2  Time Scales ........................................ 11
1.3  Thermodynamically Constrained Averaging Theory Approach ... 13
     1.3.1  Overview ........................................... 13
     1.3.2  Microscale Conservation Principles ................. 13
     1.3.3  Microscale Thermodynamics .......................... 15
     1.3.4  Microscale Equilibrium Conditions .................. 16
     1.3.5  Averaging Theorems ................................. 18
     1.3.6  Larger-scale Conservation Principles ............... 19
     1.3.7  Larger-scale Thermodynamics ........................ 20
     1.3.8  Evolution Equations ................................ 22
     1.3.9  Constrained Entropy Inequality ..................... 22
     1.3.10 Simplified Entropy Inequality ...................... 24
     1.3.11 Closure Relations .................................. 25
     1.3.12 Closed Models ...................................... 26
     1.3.13 Subscale Modeling and Applications ................. 27
1.4  Summary ................................................... 28
     Exercises ................................................. 29
     References ................................................ 29

2    Microscale Conservation Principles ........................ 37
2.1  Overview .................................................. 37
2.2  General Conservation and Balance Principles ............... 39
     2.2.1  General Qualitative Formulation .................... 39
     2.2.2  General Quantitative Formulation ................... 40
     2.2.3  Species-based Quantitative Formulation ............. 42
2.3  Conservation and Balance Principles for a Phase ........... 44
     2.3.1  General Microscale Point Forms ..................... 44
     2.3.2  Specific Conservation and Balance Principles ....... 47
2.4  Conservation and Balance Principles for an Interface ...... 56
     2.4.1  General Microscale Point Form ...................... 57
     2.4.2  Specific Conservation and Balance Principles ....... 61
2.5  Conservation and Balance Principles for a Common Curve .... 68
     2.5.1  General Microscale Point Form ...................... 69
     2.5.2  Specific Conservation and Balance Principles ....... 73
2.6  General Multispecies Formulation for a Common Point ....... 78
     2.6.1  General Microscale Point Form ...................... 79
2.7  Summary ................................................... 81
     Exercises ................................................. 82
     References ................................................ 84

3    Microscale Thermodynamics ................................. 87
3.1  Overview .................................................. 87
3.2  Essence of Equilibrium Thermodynamics ..................... 90
3.3  Fluid-phase Equilibrium Thermodynamics .................... 91
     3.3.1  Fundamental and Differential Forms ................. 91
     3.3.2  Euler Equation for Internal Energy ................. 96
     3.3.3  Gibbs-Duhem Equation for a Fluid ................... 97
3.4  Normalized Internal Energy Formulation .................... 98
3.5  Other Thermodynamic Potentials ........................... 100
     3.5.1  Helmholtz Free Energy ............................. 101
     3.5.2  Enthalpy .......................................... 103
     3.5.3  Gibbs Free Energy ................................. 105
     3.5.4  Comments on Energy Potentials ..................... 107
3.6  Relation between fig.1P and fig.1V .............................. 108
3.7  Solid-phase Equilibrium Thermodynamics ................... 110
3.8  Interface and Common Curve Equilibrium Thermodynamics .... 114
     3.8.1  Interface Thermodynamics .......................... 115
     3.8.2  Common Curve Thermodynamics ....................... 117
3.9  Microscale Multiphase System Notation .................... 118
3.10 Partial Mass Quantities .................................. 120
     3.10.1 Fluid Phase ....................................... 120
     3.10.2 Solid Phase ....................................... 122
     3.10.3 Interface ......................................... 123
     3.10.4 Common Curve ...................................... 124
3.11 Classical Irreversible Thermodynamics (CIT) .............. 124
3.12 Other Thermodynamic Theories ............................. 128
     3.12.1 Rational Thermodynamics (RT) ...................... 128
     3.12.2 Extended Irreversible Thermodynamics (EIT) ........ 129
     3.12.3 Theory of Internal Variables (TIV) ................ 129
3.13 Summary .................................................. 130
     Exercises ................................................ 131
     References ............................................... 132

4    Microscale Equilibrium Conditions ........................ 135
4.1  Overview ................................................. 135
4.2  Components of Variational Analysis ....................... 136
4.3  Variation of Microscale Quantities ....................... 137
     4.3.1  Variation of Green's Strain Tensor ................ 137
4.4  Variation of Energy Integrals ............................ 139
     4.4.1  Analysis of the Integral of as:8(Cs/js) ........... 144
     4.4.2  General Condition of Equilibrium .................. 147
4.5  Single-fluid-phase Porous Medium System .................. 148
     4.5.1   Equilibrium Conditions ........................... 151
4.6  Two-fluid-phase Porous Medium System ..................... 152
     4.6.1  Identification of Index Sets ...................... 153
     4.6.2  Identification of Simpler Equilibrium Conditions .. 154
     4.6.3  Equilibrium Variational Analysis .................. 155
     4.6.4  Additional Equilibrium Conditions ................. 160
4.7  Summary .................................................. 163
     Exercises ................................................ 163
     References ............................................... 164

5    Microscale Closure for a Fluid Phase ..................... 167
5.1  Overview ................................................. 167
5.2  System Definition ........................................ 169
5.3  Conservation and Thermodynamic Equations ................. 170
     5.3.1  Entropy Inequality ................................ 171
     5.3.2  Conservation Equations ............................ 172
     5.3.3  Thermodynamic Relations ........................... 172
5.4  Constrained Entropy Inequality ........................... 173
     5.4.1  Introduction of Constraints ....................... 173
     5.4.2  Selection of Lagrange Multipliers ................. 175
     5.4.3  Reduction to the CEI .............................. 178
5.5  Simplified Entropy Inequality ............................ 181
     5.5.1  Introduction of Approximations .................... 182
     5.5.2  Consideration of Equilibrium Conditions ........... 183
5.6  Closure Relations ........................................ 184
     5.6.1  Count of Variables ................................ 184
     5.6.2  Diffusive Flux Rearrangement ...................... 186
5.7  Special Cases ............................................ 186
     5.7.1  Single-species Phase .............................. 187
     5.7.2  Single-species, Isothermal Phase .................. 187
5.8  Conjugate Force-flux Closure ............................. 187
     5.8.1  Stress Tensor ..................................... 188
     5.8.2  Diffusion Vector .................................. 191
     5.8.3  Non-advective Heat Flux ........................... 192
     5.8.4  Chemical Reaction ................................. 193
5.9  Cross-coupled Closure .................................... 193
5.10 Summary .................................................. 196
     Exercises ................................................ 197
     References ............................................... 198

6    Macroscale Conservation Principles ....................... 201
6.1  Overview ................................................. 201
6.2  Averaging Conventions and Notation ....................... 203
     6.2.1  Intrinsic Averages ................................ 205
     6.2.2  Mass Averages ..................................... 205
     6.2.3  Unique Averages ................................... 206
     6.2.4  Examples of Averaging Notation .................... 206
6.3  Averaging Theorems ....................................... 210
     6.3.1  Averaging Theorems for Phases ..................... 211
     6.3.2  Averaging Theorems for Interfaces ................. 214
     6.3.3  Averaging Theorems for Common Curves .............. 216
     6.3.4  Averaging Theorem for Common Points ............... 217
6.4  Application of Averaging Process ......................... 217
6.5  Macroscale Principles for a Phase ........................ 218
     6.5.1  Conservation of Mass .............................. 218
     6.5.2  Conservation of Momentum .......................... 225
     6.5.3  Conservation of Energy ............................ 230
     6.5.4  Balance of Entropy ................................ 235
     6.5.5  Body Force Potential .............................. 238
6.6  On the Forms of Macroscale Equations ..................... 240
6.7  Macroscale Principles for an Interface ................... 242
     6.7.1  Example: Conservation of Species Mass ............. 242
     6.7.2  Comment on Interface Equations .................... 245
6.8  Macroscale Principles for a Common Curve ................. 246
     6.8.1  Example: Conservation of Common Curve Momentum .... 247
     6.8.2  Comment on Common Curve Equations ................. 249
     6.9  Mixed Forms of Macroscale Equations ................. 250
     6.9.1  Species-based Equations ........................... 250
     6.9.2  Entity-based Energy and Entropy ................... 251
     6.9.3  Entity-based Momentum, Energy, and Entropy ........ 254
6.10 Internal Energy Equation ................................. 256
     6.10.1 Species-and Entity-based Equations ................ 256
     6.10.2 Mixed Formulation with Species Conservation ....... 257
6.11 Summary .................................................. 258
     Exercises ................................................ 260
     References ............................................... 260

7    Macroscale Thermodynamics ................................ 263
7.1  Overview ................................................. 263
7.2  Macroscale Euler Equations ............................... 264
     7.2.1  Fluid Phase ....................................... 264
     7.2.2  Solid Phase ....................................... 267
     7.2.3  Interface and Common Curve ........................ 268
7.3  Macroscale Energy Differentials .......................... 269
7.4  Fluid Energy Dynamics .................................... 271
     7.4.1  Fluid Species Energy .............................. 272
     7.4.2  Fluid Species Potential Energy .................... 275
     7.4.3  Fluid-phase Energy ................................ 276
     7.4.4  Fluid-phase Potential Energy ...................... 277
7.5  Solid-phase Energy Dynamics .............................. 278
     7.5.1  Solid Species Energy .............................. 278
     7.5.2  Solid Species Potential Energy .................... 284
     7.5.3  Solid-phase Energy ................................ 284
     7.5.4  Solid-phase Potential Energy ...................... 285
7.6  Interface Energy Dynamics ................................ 285
     7.6.1  Interface Species Energy .......................... 286
     7.6.2  Interface Species Potential Energy ................ 289
     7.6.3  Interface-entity Energy ........................... 289
     7.6.4  Interface-entity Potential Energy ................. 290
7.7  Common Curve Energy Dynamics ............................. 291
     7.7.1  Common Curve Species Energy ....................... 291
     7.7.2  Common Curve Species Potential Energy ............. 292
     7.7.3  Common Curve Entity Energy ........................ 292
     7.7.4  Common Curve Entity Potential Energy .............. 293
7.8  Equilibrium Conditions ................................... 293
     7.8.1  Two-phase Equilibrium Conditions .................. 294
     7.8.2  Three-phase Equilibrium Conditions ................ 296
7.9  Summary .................................................. 299
     Exercises ................................................ 299
     References ............................................... 300

8    Evolution Equations ...................................... 301
8.1  Overview ................................................. 301
8.2  Derivation of Evolution Equations ........................ 303
     8.2.1  General Expression ................................ 303
8.3  Single-fluid-phase Flow .................................. 305
     8.3.1  Phases ............................................ 305
     8.3.2  Interface ......................................... 306
8.4  Single-fluid-phase Flow Geometric Relations .............. 306
8.5  Two-fluid-phase Flow ..................................... 308
     8.5.1  Solid Phase ....................................... 309
     8.5.2  Fluid Phases ...................................... 310
     8.5.3  Fluid-solid Interfaces ............................ 311
     8.5.4  Fluid-fluid Interface ............................. 312
     8.5.5  Common Curve ...................................... 314
8.6  Two-fluid-phase Flow Geometric Relations ................. 316
     8.6.1  Solid Phase and Fluid-solid Interfaces ............ 316
     8.6.2  Fluid-fluid Interface Evolution ................... 317
     8.6.3  Common Curve Evolution ............................ 320
8.7  Average Normal Velocities ................................ 321
     8.7.1  Fluid-fluid Interface Velocity Approximation ...... 322
     8.7.2  Common Curve Velocity Approximation ............... 323
8.8  Summary .................................................. 324
     Exercises ................................................ 325
     References ............................................... 326

9    Single-fluid-phase Flow .................................. 327
9.1  Overview ................................................. 327
9.2  Single-phase-flow System Definition ...................... 329
9.3  Conservation and Thermodynamic Equations ................. 332
     9.3.1  Entropy Inequality ................................ 332
     9.3.2  Conservation Equations ............................ 333
     9.3.3  Thermodynamic Relations ........................... 334
9.4  Constrained Entropy Inequality ........................... 335
     9.4.1  Augmented Entropy Inequality ...................... 335
     9.4.2  Selection of Lagrange Multipliers ................. 336
     9.4.3  Elimination of Time Derivatives ................... 339
     9.4.4  Manipulation Insights ............................. 341
     9.4.5  Formulation of the CEI ............................ 344
9.5  Simplified Entropy Inequality ............................ 348
     9.5.1  The Need for Approximations ....................... 349
     9.5.2  Elimination of Terms .............................. 349
     9.5.3  Approximation of Averages ......................... 351
     9.5.4  General SEI ....................................... 354
9.6  Example Restricted Application ........................... 355
     9.6.1  Statement of Secondary Restriction ................ 356
     9.6.2  Count of Variables ................................ 357
     9.6.3  Reduction in Number of Variables .................. 358
     9.6.4  Conjugate Force-flux Closure ...................... 360
     9.6.5  Closed Conservation Equation Set .................. 362
9.7  Model of Fluid and Elastic Solid ......................... 363
     9.7.1  Compressible Elastic Solid with Small
            Deformation ....................................... 364
     9.7.2  Passive Solid Phase ............................... 367
9.8  Summary .................................................. 370
     Exercises ................................................ 370
     References ............................................... 371

10   Single-fluid-phase Species Transport ..................... 373
10.1 Overview ................................................. 373
10.2 System Definition by Primary Restrictions ................ 375
10.3 Constrained Entropy Inequality ........................... 377
     10.3.1 Augmented Entropy Inequality ...................... 377
     10.3.2 Determination of Lagrange Multipliers ............. 380
     10.3.3 Formulation of the CEI ............................ 384
10.4 Simplified Entropy Inequality ............................ 389
     10.4.1 Elimination of Small Terms ........................ 390
     10.4.2 Breaking of Averages .............................. 392
     10.4.3 General SEI ....................................... 394
10.5 SEI for Application to Non-isothermal Species Transport .. 396
     10.5.1 Imposition of Secondary Restrictions .............. 396
10.6 Isothermal Transport with No Interphase Mass Exchange .... 400
     10.6.1 Simplification of SEI ............................. 400
     10.6.2 Count of Equations and Variables .................. 401
10.7 Isothermal Transport with Interphase Mass Exchange ....... 402
     10.7.1 Simplification of the SEI ......................... 402
     10.7.2 Additional Variables and Constraints .............. 403
10.8 Unitemperature, Nonisothermal Transport .................. 404
     10.8.1 Simplification of the SEI for a Single
            Temperature ....................................... 404
     10.8.2 Accounting for Additional Variables ............... 405
10.9 Multi-temperature Species Transport ...................... 407
     10.9.1 SEI for the Multi-temperature Case ................ 407
     10.9.2 Treatment of Additional Variables ................. 408
10.10 Example of Conjugate Force-flux Closure ................. 409
     10.10.1 Closure Relations ................................ 410
     10.10.2 Closed Set of Conservation Equations ............. 414
10.11 Summary ................................................. 418
     Exercises ................................................ 419
     References ............................................... 420

11   Two-phase Flow ........................................... 421
11.1 Overview ................................................. 421
11.2 Primary Restrictions ..................................... 423
11.3 Constrained Entropy Inequality Statement ................. 424
     11.3.1 Entropy Inequality ................................ 424
     11.3.2 Augmented Entropy Inequality ...................... 424
     11.3.3 Selection of Lagrange Multipliers ................. 425
     11.3.4 Expanded CEI ...................................... 425
11.4 Simplified Entropy Inequality ............................ 429
     11.4.1 Required SEI Approximations ....................... 430
     11.4.2 Basic SEI Approximations .......................... 431
     11.4.3 Complex SEI Approximations ........................ 435
     11.4.4 Product Breaking SEI Approximations ............... 442
     11.4.5 General SEI ....................................... 445
11.5 Example Application ...................................... 448
     11.5.1 Selection of Secondary Restrictions ............... 448
     11.5.2 Identification of Variables ....................... 450
     11.5.3 Specification of Closure Conditions ............... 450
     11.5.4 Closed Set of Conservation Equations .............. 454

11.6 Simplified Momentum Equations for Example Model .......... 456
11.7 Immobile Solid ........................................... 459
11.8 Summary .................................................. 461
     Exercises ................................................ 462
     References ............................................... 463

12   Modeling Approach and Extensions ......................... 465
12.1 Overview ................................................. 465
12.2 Modeling Process ......................................... 466
     12.2.1 On Primary Restrictions ........................... 467
     12.2.2 On SEI Approximations ............................. 468
     12.2.3 On Secondary Restrictions and Closure ............. 469
12.3 Subscale Modeling ........................................ 470
     12.3.1 Capillarity Effects ............................... 471
     12.3.2 Testing of Approximations ......................... 473
12.4 Macroscale Modeling ...................................... 473
     12.4.1 Model Validation .................................. 474
     12.4.2 Model Verification ................................ 476
12.5 Extensions of TCAT Models ................................ 477
     12.5.1 Model Class Extensions: Equations ................. 477
     12.5.2 Mixed-scale Dimensionality: Averaging ............. 479
     12.5.3 Multiphysics: Linking of Larger-scale Systems ..... 480
     12.5.4 Alternative Thermodynamic Theories ................ 481
     12.5.5 Nonlinearities: Closure Relations ................. 482
     12.5.6 Applications: Dynamically Coupled Multiscale
            Systems ........................................... 483
     12.5.7 Subscale Modeling and Applications: Stochastic
            Systems ........................................... 483
12.6 Summary .................................................. 484
     Exercises ................................................ 485
     References ............................................... 485

A    Considerations on Calculus of Variations ................. 489
A.l  Fundamentals of Variational Approaches ................... 489
A.2  Classical Approach to Volume Integrals ................... 490
A.3  Indicator Functions ...................................... 493
     A.3.1  Universal Properties .............................. 494
     A.3.2  Integral over a Phase ............................. 495
     A.3.3  Integral over an Interface ........................ 495
     A.3.4  Integral over a Common Curve ...................... 496
A.4  Variation of a Volume Integral ........................... 497
A.5  Variation of a Surface Integral .......................... 498
A.6  Variation of an Integral over a Curve .................... 501
A.7  Summary .................................................. 505
     Exercises ................................................ 506
     References ............................................... 506

В    Derivations of Averaging Theorems ........................ 509
B.1  Overview ................................................. 509
     B.1.1  Naming Convention ................................. 510
B.2  Coordinate Systems ....................................... 510
B.3  Averaging Theorems for Volumes ........................... 512
     B.3.1  D[3,(3,0),0] ...................................... 514
     B.3.2  G[3,(3,0),0] ...................................... 514
     B.3.3  T[3,(3,0),0] ...................................... 515
B.4  Averaging Theorems for Surfaces .......................... 515
     B.4.1  D[2,(3,0),0] ...................................... 517
     B.4.2  G[2,(3,0),0] ...................................... 519
     B.4.3  T[2,(3,0),0] ...................................... 519
B.5  Averaging Theorems for Curves ............................ 521
     B.5.1  D[1,(3,0),0] ...................................... 522
     B.5.2  G[1,(3,0),0] ...................................... 524
     B.5.3  T[1,(3,0),0] ...................................... 524
B.6  Averaging Theorems for Points ............................ 526
     B.6.1  T[0,(3,0),0] ...................................... 526
     Exercises ................................................ 526
     References ............................................... 528

С    Constrained Entropy Inequality Derivations ............... 529
C.l  CEI for Single-fluid-phase Flow, Eq. (9.43) .............. 529
C.2  CEI for Single-fluid-phase Transport, Eq. (10.14) ........ 541
C.3  CEI for Two-fluid-phase Flow, Eq. (11.5) ................. 555

Index ......................................................... 573

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