1 Introduction ................................................. 1
1.1 Porous media ............................................ 1
1.1.1 Applications, examples ........................... 1
1.1.2 Notation, definitions ............................ 6
1.1.3 Overview ......................................... 9
1.2 The Darcy model ........................................ 10
1.2.1 The Porous Medium Equation ...................... 11
1.3 The Forchheimer model .................................. 12
1.4 The Brinkman model ..................................... 12
1.5 Anisotropic Darcy model ................................ 13
1.6 Equations for other fields ............................. 14
1.6.1 Temperature ..................................... 14
1.6.2 Salt field ...................................... 15
1.7 Boundary conditions .................................... 15
1.8 Elastic materials with voids ........................... 16
1.8.1 Nunziato-Cowin theory ........................... 16
1.8.2 Microstretch theory ............................. 17
1.9 Mixture theories ....................................... 18
1.9.1 Eringen's theory ................................ 18
1.9.2 Bowen's theory .................................. 22
2 Structural Stability ........................................ 27
2.1 Structural stability, Darcy model ...................... 27
2.1.1 Newton's law of cooling ......................... 28
2.1.2 A priori bound for T ............................ 30
2.2 Structural stability, Forchheimer model ................ 31
2.2.1 Continuous dependence on b ...................... 32
2.2.2 Continuous dependence on с ...................... 34
2.2.3 Energy bounds ................................... 35
2.2.4 Brinkman-Forchheimer model ...................... 37
2.3 Forchheimer model, non-zero boundary conditions ........ 37
2.3.1 A maximum principle for с ....................... 39
2.3.2 Continuous dependence on the viscosity .......... 39
2.4 Brinkman model, non-zero boundary conditions ........... 42
2.5 Convergence, non-zero boundary conditions .............. 43
2.6 Continuous dependence, Vadasz coefficient .............. 44
2.6.1 A maximum principle for T ....................... 45
2.6.2 Continuous dependence on α ...................... 46
2.7 Continuous dependence, Krishnamurti coefficient ........ 48
2.7.1 An a priori bound for T ......................... 49
2.7.2 Continuous dependence ........................... 53
2.8 Continuous dependence, Dufour coefficient .............. 55
2.8.1 Continuous dependence on γ ...................... 57
2.9 Initial - final value problems ......................... 69
2.10 The interface problem ............................. 72
2.11 Lower bounds on the blow-up time .................. 76
2.12 Uniqueness in compressible porous flows ........... 82
3 Spatial Decay ............................................... 95
3.1 Spatial decay for the Darcy equations .................. 95
3.1.1 Nonlinear temperature dependent density ......... 96
3.1.2 An appropriate "energy" function ................ 98
3.1.3 A data bound for E(0, t) ....................... 104
3.2 Spatial decay for the Brinkman equations .............. 1ll
3.2.1 An estimate for grad T ......................... 112
3.2.2 An estimate for grad u ......................... 114
3.3 Spatial decay for the Forchheimer equations ........... 120
3.3.1 An estimate for grad T ......................... 125
3.3.2 An estimate for E(0, t) ........................ 127
3.3.3 An estimate for uiui ........................... 129
3.3.4 Bounding Øi .................................... 131
3.4 Spatial decay for a Krishnamurti model ................ 132
3.4.1 Estimates for T,iT,i and C,iC,i .................. 134
3.4.2 An estimate for the uiui term .................. 136
3.4.3 Integration of the H inequality ................ 138
3.4.4 A bound for H(0) ............................... 138
3.4.5 Bound for uiui at z = 0 ........................ 141
3.5 Spatial decay for a fluid-porous model ................ 142
4 Convection in Porous Media ................................. 147
4.1 Equations for thermal convection in a porous medium ... 148
4.1.1 The Darcy equations ............................ 148
4.1.2 The Forchheimer equations ...................... 148
4.1.3 Darcy equations with anisotropic
permeability ................................... 149
4.1.4 The Brinkman equations ......................... 150
4.2 Stability of thermal convection ....................... 150
4.2.1 The Benard problem for the Darcy equations ..... 151
4.2.2 Linear instability ............................. 152
4.2.3 Nonlinear stability ............................ 154
4.2.4 Variational solution to (4.28) ................. 155
4.2.5 Benard problem for the Forchheimer equations ... 158
4.2.6 Darcy equations with anisotropic
permeability ................................... 159
4.2.7 Benard problem for the Brinkman equations ...... 163
4.3 Stability and symmetry ................................ 166
4.3.1 Symmetric operators ............................ 166
4.3.2 Heated and salted below ........................ 168
4.3.3 Symmetrization ................................. 170
4.3.4 Pointwise constraint ........................... 171
4.4 Thermal non-equilibrium ............................... 172
4.4.1 Thermal non-equilibrium model .................. 172
4.4.2 Stability analysis ............................. 174
4.5 Resonant penetrative convection ....................... 177
4.5.1 Nonlinear density, heat source model ........... 177
4.5.2 Basic equations ................................ 178
4.5.3 Linear instability analysis .................... 180
4.5.4 Nonlinear stability analysis ................... 181
4.5.5 Behaviour observed ............................. 182
4.6 Throughflow ........................................... 183
4.6.1 Penetrative convection with throughflow ........ 183
4.6.2 Forchheimer model with throughflow ............. 184
4.6.3 Global nonlinear stability analysis ............ 186
5 Stability of Other Porous Flows ............................ 193
5.1 Convection and flow with micro effects ................ 193
5.1.1 Biological processes ........................... 193
5.1.2 Glia aggregation in the brain .................. 194
5.1.3 Micropolar thermal convection .................. 196
5.2 Porous flows with viscoelastic effects ........... 198
5.2.1 Viscoelastic porous convection ................. 198
5.2.2 Second grade fluids ............................ 200
5.2.3 Generalized second grade fluids ................ 201
5.3 Storage of gases ...................................... 202
5.3.1 Carbon dioxide storage ......................... 202
5.3.2 Hydrogen storage ............................... 204
5.4 Energy growth ......................................... 205
5.4.1 Soil salinization .............................. 205
5.4.2 Other salinization theories .................... 208
5.4.3 Time growth of parallel flows .................. 210
5.4.4 Stability analysis for salinization ............ 218
5.4.5 Transient growth in salinization ............... 220
5.5 Turbulent convection .................................. 222
5.5.1 Turbulence in porous media ..................... 222
5.5.2 The background method .......................... 223
5.5.3 Selecting τ .................................... 225
5.6 Multiphase flow .................................. 227
5.6.1 Water-steam motion ............................. 227
5.6.2 Foodstuffs, emulsions .......................... 230
5.7 Unsaturated porous medium ............................. 231
5.7.1 Model equations ................................ 231
5.7.2 Stability of flow .............................. 232
5.7.3 Transient growth ............................... 233
5.8 Parallel flows ........................................ 234
5.8.1 Poiseuille flow ................................ 234
5.8.2 Flow in a permeable conduit .................... 236
6 Fluid - Porous Interface Problems .......................... 239
6.1 Models for thermal convection ......................... 239
6.1.1 Extended Navier-Stokes model ................... 240
6.1.2 Nield (Darcy) model ............................ 241
6.1.3 Forchheimer model .............................. 243
6.1.4 Brinkman model ................................. 244
6.1.5 Nonlinear equation of state .................... 244
6.1.6 Reacting layers ................................ 246
6.2 Surface tension ....................................... 246
6.2.1 Basic solution ................................. 246
6.2.2 Perturbation equations ......................... 248
6.2.3 Perturbation boundary conditions ............... 249
6.2.4 Numerical results .............................. 251
6.3 Porosity effects ...................................... 253
6.3.1 Porosity variation ............................. 253
6.3.2 Numerical results .............................. 255
6.4 Melting ice, global warming ........................... 258
6.4.1 Three layer model .............................. 258
6.4.2 Under ice melt ponds ........................... 260
6.5 Crystal growth ........................................ 262
6.6 Heat pipes ............................................ 265
6.7 Poiseuille flow ....................................... 267
6.7.1 Darcy model .................................... 267
6.7.2 Linearized perturbation equations .............. 269
6.7.3 (Chang et al., 2006) results ................... 271
6.7.4 Brinkman - Darcy model ......................... 272
6.7.5 Steady solution ................................ 273
6.7.6 Linearized perturbation equations .............. 274
6.7.7 Numerical results .............................. 276
6.7.8 Forchheimer - Darcy model ...................... 276
6.7.9 Brinkman - Forchheimer / Darcy model ........... 284
6.8 Acoustic waves, ocean bed ............................. 289
6.8.1 Basic equations ................................ 290
6.8.2 Linear waves in the Bowen theory ............... 291
6.8.3 Boundary conditions ............................ 293
6.8.4 Amplitude behaviour ............................ 294
7 Elastic Materials with Voids ............................... 297
7.1 Acceleration waves in elastic materials ............... 297
7.1.1 Bodies and their configurations ................ 297
7.1.2 The deformation gradient tensor ................ 298
7.1.3 Conservation of mass ........................... 298
7.1.4 The equations of nonlinear elasticity .......... 298
7.1.5 Acceleration waves in one-dimension ............ 300
7.1.6 Given strain energy and deformation ............ 303
7.1.7 Acceleration waves in three dimensions ......... 305
7.2 Acceleration waves, inclusion of voids ................ 307
7.2.1 Porous media, voids, applications .............. 307
7.2.2 Basic theory of elastic materials with voids ... 308
7.2.3 Thermodynamic restrictions ..................... 310
7.2.4 Acceleration waves in the isothermal case ...... 312
7.3 Temperature rate effects .............................. 314
7.3.1 Voids and second sound ......................... 314
7.3.2 Thermodynamics and voids ....................... 316
7.3.3 Void-temperature acceleration waves ............ 318
7.3.4 Amplitude behaviour ............................ 320
7.4 Temperature displacement effects ...................... 325
7.4.1 Voids and thermodynamics ....................... 325
7.4.2 De Cicco - Diaco theory ........................ 325
7.4.3 Acceleration waves ............................. 327
7.5 Voids and type III thermoelasticity ................... 329
7.5.1 Thermodynamic theory ........................... 329
7.5.2 Linear theory .................................. 331
7.6 Acceleration waves, microstretch theory ............... 332
8 Poroacoustic Waves ......................................... 337
8.1 Poroacoustic acceleration waves ....................... 337
8.1.1 Equivalent fluid theory ........................ 337
8.1.2 Jordan - Darcy theory .......................... 339
8.1.3 Acceleration waves ............................. 340
8.1.4 Amplitude equation derivation .................. 341
8.2 Temperature effects ................................... 344
8.2.1 Jordan-Darcy temperature model ................. 344
8.2.2 Wavespeeds ..................................... 345
8.2.3 Amplitude equation ............................. 346
8.3 Heat flux delay ....................................... 349
8.3.1 Cattaneo poroacoustic theory ................... 349
8.3.2 Thermodynamic justification .................... 351
8.3.3 Acceleration waves ............................. 353
8.3.4 Amplitude derivation ........................... 356
8.3.5 Dual phase lag theory .......................... 358
8.4 Temperature rate effects .............................. 360
8.4.1 Green-Laws theory .............................. 360
8.4.2 Wavespeeds ..................................... 362
8.4.3 Amplitude behaviour ............................ 364
8.5 Temperature displacement effects ...................... 366
8.5.1 Green-Naghdi thermodynamics .................... 366
8.5.2 Acceleration waves ............................. 369
8.5.3 Wave amplitudes ................................ 371
8.6 Magnetic field effects ................................ 373
9 Numerical Solution of Eigenvalue Problems .................. 375
9.1 The compound matrix method ............................ 375
9.1.1 The shooting method ............................ 375
9.1.2 A fourth order equation ........................ 376
9.1.3 The compound matrix method ..................... 377
9.1.4 Penetrative convection in a porous medium ...... 379
9.2 The Chebyshev tau method .............................. 381
9.2.1 The D2 Chebyshev tau method .................... 381
9.2.2 Penetrative convection ......................... 384
9.2.3 Fluid overlying a porous layer ................. 385
9.2.4 The D Chebyshev tau method ..................... 389
9.2.5 Natural variables .............................. 390
9.3 Legendre-Galerkin method .............................. 391
9.3.1 Fourth order system ............................ 391
9.3.2 Penetrative convection ......................... 395
9.3.3 Extension of the method ........................ 397
References .................................................... 399
Index ......................................................... 433
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