1 Introduction ............................................... 1
1.1 Heat waves in a rigid conductor ............................ 1
1.1.1 Second sound ........................................ 1
1.1.2 Notation, definitions ............................... 7
1.1.3 Overview ........................................... 10
1.2 Maxwell-Cattaneo theory ................................... 11
1.3 Guyer-Krumhansl theory .................................... 14
1.4 High order relaxation dynamics ............................ 16
1.5 Phase lag models .......................................... 17
1.6 Heat flux history models .................................. 19
1.6.1 Gurtin - Pipkin theory ............................. 20
1.6.2 Grain - Fabrizio theory ............................ 22
1.7 Two temperature model ..................................... 24
1.8 Green-Laws theory ......................................... 25
1.9 Temperature dependent conductivity ........................ 28
1.10 Type II rigid body ........................................ 29
1.11 Type III rigid body ....................................... 31
1.12 Microtemperatures ......................................... 33
1.13 Exercises ................................................. 35
2 Interaction with elasticity ............................... 38
2.1 Cattaneo theories ......................................... 38
2.1.1 Cattaneo-Lord-Shulman theory ....................... 38
2.1.2 Cattaneo-Fox theory ................................ 41
2.1.3 Hidden variables ................................... 43
2.2 Green-Lindsay theory ...................................... 48
2.3 Green-Naghdi type II theory ............................... 52
2.4 Green-Naghdi type III theory .............................. 56
2.5 Thermoelasticity with Voids ............................... 57
2.5.1 Basic theory of elastic materials with voids ....... 59
2.5.2 Thermodynamic restrictions ......................... 61
2.5.3 Voids and Green - Lindsay thermoelasticity ......... 62
2.5.4 Voids and type II thermoelasticity ................. 65
2.5.5 Voids and type III thermoelasticity ................ 67
2.5.6 Linear voids type III thermoelasticity ............. 69
2.6 Generalized thermoelasticity with microstructure .......... 72
2.6.1 Hetnarski-Ignaczak theory .......................... 72
2.6.2 Micropolar, dipolar, affine microstructure ......... 72
2.6.3 Piezoelectricity and thermoelasticity .............. 75
2.6.4 Other theories ..................................... 78
2.7 Exercises ................................................. 80
3 Interaction with fluids ................................... 82
3.1 Cattaneo theories ......................................... 82
3.1.1 Cattaneo-Fox theory ................................ 85
3.1.2 Cattaneo-Christov theory ........................... 86
3.1.3 Guyer-Krumhansl model .............................. 87
3.1.4 Alternative Guyer-Krumhansl model .................. 87
3.1.5 Further Cattaneo type fluid models ................. 87
3.2 Green-Laws theory ......................................... 89
3.3 Type II fluid ............................................. 91
3.4 Type III fluid ............................................ 94
3.4.1 Type III viscous fluid ............................. 96
3.4.2 Type III inviscid fluid ............................ 96
3.5 Green-Naghdi extended theory .............................. 97
4 Acceleration waves ....................................... 100
4.1 Maxwell-Cattaneo theory .................................. 100
4.1.1 Wave into equilibrium ............................. 103
4.1.2 Acceleration wave in three dimensions ............. 104
4.1.3 More general Maxwell - Cattaneo theory ............ 107
4.1.4 Dual phase lag theory ............................. 108
4.2 Type II rigid heat conductor ............................. 108
4.2.1 Acceleration waves in type II theory .............. 110
4.2.2 Region with no x variation ........................ 111
4.2.3 Amplitude solution ................................ 111
4.3 Acceleration waves with microtemperatures ................ 113
4.4 Type II thermoelasticity ................................. 116
4.4.1 Wavespeeds ........................................ 119
4.4.2 Amplitude behaviour ............................... 121
4.5 Type III thermoelasticity ................................ 123
4.5.1 Fundamental jump relations ........................ 124
4.6 Acceleration waves in a type II fluid .................... 126
4.7 Acceleration waves in a type III fluid ................... 130
4.8 Exercises ................................................ 134
5 Shock waves and numerical solutions ...................... 137
5.1 Shock development ........................................ 139
5.2 Type II and type III thermoelasticity .................... 149
5.3 Temperature dependent thermal conductivity ............... 152
5.4 Exercises ................................................ 153
6 Qualitative estimates .................................... 159
6.1 Decay in time ............................................ 159
6.1.1 Decay of temperature .............................. 160
6.1.2 Decay of heat flux ................................ 162
6.1.3 Decay with other effects .......................... 163
6.2 Uniqueness in type II thermoelasticity ................... 165
6.3 Growth in type II thermoelasticity ....................... 168
6.4 Uniqueness in type III thermoelasticity .................. 170
6.5 Uniqueness on an unbounded domain ........................ 173
6.5.1 The Graffi method ................................. 174
6.5.2 The weighted energy method ........................ 177
6.6 Non-standard problems in thermoelasticity ................ 178
6.6.1 Energy bounds, |α|, |β| > 1 ....................... 180
6.6.2 Energy bounds, |α|, |β| < 1 ....................... 185
6.6.3 Non-homogeneous boundary conditions ............... 186
6.7 Explosive instabilities in heat transfer ................. 187
6.7.1 Third order theory ................................ 187
6.7.2 Nonexistence of a solution ........................ 189
6.8 Qualitative results for fluids ........................... 192
6.8.1 Decay for a solution to (6.160)? .................. 193
6.9 Exercises ................................................ 195
7 Spatial decay ............................................ 202
7.1 Generalized Maxwell - Cattaneo theory .................... 203
7.1.1 Temperature spatial decay ......................... 204
7.1.2 Spatial decay of heat flux ........................ 205
7.1.3 Spatial decay with heat flux prescribed ........... 206
7.2 MC theory backward in time ............................... 206
7.3 Green-Lindsay thermoelasticity ........................... 208
7.3.1 Dirichlet boundary conditions ..................... 209
7.3.2 Neumann boundary conditions ....................... 210
7.4 Type III thermoelasticity ................................ 211
7.5 Strong ellipticity in thermoelasticity ................... 213
7.5.1 Monoclinic materials .............................. 215
7.5.2 Triclinic materials ............................... 219
8 Thermal convection in nanofluids ......................... 222
8.1 Heat transfer enhancement in nanofluids .................. 222
8.2 The Tzou model ........................................... 224
8.2.1 Coefficient dependence on nanoparticles ........... 227
8.3 Convection with Cattaneo theories ........................ 228
8.3.1 Cattaneo - Fox law ................................ 228
8.3.2 Cattaneo - Christov law ........................... 233
8.3.3 Cattaneo theories and porous materials ............ 238
8.4 Green - Naghdi model ..................................... 241
9 Other applications ....................................... 246
9.1 Applications in continuum mechanics ...................... 246
9.1.1 Nanoscale heat transport .......................... 246
9.1.2 Heat transport in nanowires ....................... 247
9.1.3 Heat transport in thin films ...................... 248
9.1.4 Reactor fuel rods ................................. 251
9.1.5 Phase changes ..................................... 252
9.2 Stellar, planetary heat propagation ...................... 255
9.2.1 Cryovolcanism on Enceladus ........................ 255
9.2.2 Thermohaline convection ........................... 256
9.3 Traffic flow ............................................. 258
9.4 Applications in biology .................................. 260
9.4.1 Population dynamics ............................... 260
9.4.2 Migration of a school of fish ..................... 261
9.4.3 Spread of the Hantavirus .......................... 262
9.4.4 Chemotaxis ........................................ 264
9.4.5 Radiofrequency heating ............................ 267
9.4.6 Skin burns ........................................ 269
9.5 Exercises ................................................ 271
References .................................................... 275
Index ......................................................... 315
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