Stability and Wave Motion in Porous Media / Edition 1

Stability and Wave Motion in Porous Media / Edition 1

by brian straughan

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ISBN-10: 0387765417

ISBN-13: 9780387765419

Pub. Date: 08/12/2008

Publisher: Springer New York

This book describes several tractable theories for fluid flow in porous media while the important mathematical questions about structural stability and spatial decay are addressed. Thermal convection and stability of other flows in porous media are covered and a chapter is devoted to the problem of stability of flow in a fluid overlying a porous layer.



This book describes several tractable theories for fluid flow in porous media while the important mathematical questions about structural stability and spatial decay are addressed. Thermal convection and stability of other flows in porous media are covered and a chapter is devoted to the problem of stability of flow in a fluid overlying a porous layer.

Nonlinear wave motion in porous media is analysed, and waves in an elastic body with voids are investigated. Acoustic waves in porous media are also analysed in some detail.

A chapter is included on efficient numerical methods for solving eigenvalue problems which occur in stability problems for flows in porous media.

About the Author:
Brian Straughan is a professor at the Department of Mathematical Sciences at Durham University, United Kingdom

Product Details

Springer New York
Publication date:
Applied Mathematical Sciences Series, #165
Edition description:
Product dimensions:
6.20(w) x 9.30(h) x 1.00(d)

Table of Contents

Preface     vii
Introduction     1
Porous media     1
Applications, examples     1
Notation, definitions     6
Overview     9
The Darcy model     10
The Porous Medium Equation     11
The Forchheimer model     12
The Brinkman model     12
Anisotropic Darcy model     13
Equations for other fields     14
Temperature     14
Salt field     15
Boundary conditions     15
Elastic materials with voids     16
Nunziato-Cowin theory     16
Microstretch theory     17
Mixture theories     18
Eringen's theory     18
Bowen's theory     22
Structural Stability     27
Structural stability, Darcy model     27
Newton's law of cooling     28
A priori bound for T     30
Structural stability, Forchheimer model     31
Continuous dependence on b     32
Continuous dependence on c     34
Energy bounds     35
Brinkman-Forchheimer model     37
Forchheimer model, non-zero boundaryconditions     37
A maximum principle for c     39
Continuous dependence on the viscosity     39
Brinkman model, non-zero boundary conditions     42
Convergence, non-zero boundary conditions     43
Continuous dependence, Vadasz coefficient     44
A maximum principle for T     45
Continuous dependence on [alpha]     46
Continuous dependence, Krishnamurti coefficient     48
An a priori bound for T     49
Continuous dependence     53
Continuous dependence, Dufour coefficient     55
Continuous dependence on [gamma]     57
Initial - final value problems     69
The interface problem     72
Lower bounds on the blow-up time     76
Uniqueness in compressible porous flows     82
Spatial Decay     95
Spatial decay for the Darcy equations     95
Nonlinear temperature dependent density     96
An appropriate "energy" function     98
A data bound for E(0, t)     104
Spatial decay for the Brinkman equations     111
An estimate for grad T     112
An estimate for grad u     114
Spatial decay for the Forchheimer equations      120
An estimate for grad T     125
An estimate for E(0, t)     127
An estimate for u[subscript i]u[subscript i]     129
Bounding [Phi subscript i]     131
Spatial decay for a Krishnamurti model     132
Estimates for T,[subscript i]T,[subscript i] and C,[subscript i]C,[subscript i]     134
An estimate for the u[subscript i]u[subscript i] term     136
Integration of the H inequality     138
A bound for H(0)     138
Bound for u[subscript i]u[subscript i] at z = 0     141
Spatial decay for a fluid-porous model     142
Convection in Porous Media     147
Equations for thermal convection in a porous medium     148
The Darcy equations     148
The Forchheimer equations     148
Darcy equations with anisotropic permeability     149
The Brinkman equations     150
Stability of thermal convection     150
The Benard problem for the Darcy equations     151
Linear instability     152
Nonlinear stability     154
Variational solution to (4.28)     155
Benard problem for the Forchheimer equations     158
Darcy equations with anisotropic permeability      159
Benard problem for the Brinkman equations     163
Stability and symmetry     166
Symmetric operators     166
Heated and salted below     168
Symmetrization     170
Pointwise constraint     171
Thermal non-equilibrium     172
Thermal non-equilibrium model     172
Stability analysis     174
Resonant penetrative convection     177
Nonlinear density, heat source model     177
Basic equations     178
Linear instability analysis     180
Nonlinear stability analysis     181
Behaviour observed     182
Throughflow     183
Penetrative convection with throughflow     183
Forchheimer model with throughflow     184
Global nonlinear stability analysis     186
Stability of Other Porous Flows     193
Convection and flow with micro effects     193
Biological processes     193
Glia aggregation in the brain     194
Micropolar thermal convection     196
Porous flows with viscoelastic effects     198
Viscoelastic porous convection     198
Second grade fluids      200
Generalized second grade fluids     201
Storage of gases     202
Carbon dioxide storage     202
Hydrogen storage     204
Energy growth     205
Soil salinization     205
Other salinization theories     208
Time growth of parallel flows     210
Stability analysis for salinization     218
Transient growth in salinization     220
Turbulent convection     222
Turbulence in porous media     222
The background method     223
Selecting [tau]     225
Multiphase flow     227
Water-steam motion     227
Foodstuffs, emulsions     230
Unsaturated porous medium     231
Model equations     231
Stability of flow     232
Transient growth     233
Parallel flows     234
Poiseuille flow     234
Flow in a permeable conduit     236
Fluid - Porous Interface Problems     239
Models for thermal convection     239
Extended Navier-Stokes model     240
Nield (Darcy) model     241
Forchheimer model     243
Brinkman model     244
Nonlinear equation of state     244
Reacting layers     246
Surface tension     246
Basic solution     246
Perturbation equations     248
Perturbation boundary conditions     249
Numerical results     251
Porosity effects     253
Porosity variation     253
Numerical results     255
Melting ice, global warming     258
Three layer model     258
Under ice melt ponds     260
Crystal growth     262
Heat pipes     265
Poiseuille flow     267
Darcy model     267
Linearized perturbation equations     269
(Chang et al., 2006) results     271
Brinkman - Darcy model     272
Steady solution     273
Linearized perturbation equations     274
Numerical results     276
Forchheimer - Darcy model     276
Brinkman - Forchheimer / Darcy model     284
Acoustic waves, ocean bed     289
Basic equations     290
Linear waves in the Bowen theory     291
Boundary conditions      293
Amplitude behaviour     294
Elastic Materials with Voids     297
Acceleration waves in elastic materials     297
Bodies and their configurations     297
The deformation gradient tensor     298
Conservation of mass     298
The equations of nonlinear elasticity     298
Acceleration waves in one-dimension     300
Given strain energy and deformation     303
Acceleration waves in three dimensions     305
Acceleration waves, inclusion of voids     307
Porous media, voids, applications     307
Basic theory of elastic materials with voids     308
Thermodynamic restrictions     310
Acceleration waves in the isothermal case     312
Temperature rate effects     314
Voids and second sound     314
Thermodynamics and voids     316
Void-temperature acceleration waves     318
Amplitude behaviour     320
Temperature displacement effects     325
Voids and thermodynamics     325
De Cicco - Diaco theory     325
Acceleration waves     327
Voids and type III thermoelasticity     329
Thermodynamic theory      329
Linear theory     331
Acceleration waves, microstretch theory     332
Poroacoustic Waves     337
Poroacoustic acceleration waves     337
Equivalent fluid theory     337
Jordan - Darcy theory     339
Acceleration waves     340
Amplitude equation derivation     341
Temperature effects     344
Jordan-Darcy temperature model     344
Wavespeeds     345
Amplitude equation     346
Heat flux delay     349
Cattaneo poroacoustic theory     349
Thermodynamic justification     351
Acceleration waves     353
Amplitude derivation     356
Dual phase lag theory     358
Temperature rate effects     360
Green-Laws theory     360
Wavespeeds     362
Amplitude behaviour     364
Temperature displacement effects     366
Green-Naghdi thermodynamics     366
Acceleration waves     369
Wave amplitudes     371
Magnetic field effects     373
Numerical Solution of Eigenvalue Problems     375
The compound matrix method      375
The shooting method     375
A fourth order equation     376
The compound matrix method     377
Penetrative convection in a porous medium     379
The Chebyshev tau method     381
The D[superscript 2] Chebyshev tau method     381
Penetrative convection     384
Fluid overlying a porous layer     385
The D Chebyshev tau method     389
Natural variables     390
Legendre-Galerkin method     391
Fourth order system     391
Penetrative convection     395
Extension of the method     397
References     399
Index     433

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