The Hamilton-Type Principle in Fluid Dynamics (eBook)

Angel Fierros Palacios was born in March 1933 in Pochutla, Oaxaca, México. He studied Physics at the Universidad Nacional Autonoma de México (UNAM), where he also obtained a Master degree and Ph. D. in Science. For many years he was Professor at the UNAM and a researcher at the Instituto Mexicano del Petróleo and at the Instituto Nacional de Investigaciones Nucleares. He also worked at different areas of the public administration such as the Instituto Mexicano del Seguro Social, the Secre-taría de Energía, as well as the Comisión Federal de Electricidad (CFE). He is the author of the project which originated the Instituto de Investiga-ciones Eléctricas (IIE) and also, he was General Director of the Museo Tecnológico de la CFE. Presently he is working at the IIE as Director de la División de Energías Alternas.

Contents 7
About the Author 14
Foreword 15
Acknowledgements 17
Notation 18
1 General Principles 23
§1. Introduction 23
§2. Kinematics 24
§3. Deformation 25
§4. Transformation of a volume element 26
§5. Rate of change of a volume element 27
§6. The hydrodynamics derivative 29
§7. Reynolds´ transport theorem 30
§8. Hamilton´s principle 31
Selected Topics 33
The velocity field 33
Jacobian variation 37
Euler´s relation 37
Hamilton´s principle in fluid dynamics 38
References 40
2 Mass Density 41
§9. The continuity equation 41
§10. Thesmall deformation strain tensor as a fundamental metric tensor 43
§11. The scalar equation for the mass density 47
§12. The dilute gas 50
Selected Topics 53
The continuity and motion equations† 53
Bossinesq´s approaches 55
The mass balance equation 56
Demonstration of a few formulae 58
The molecular chaos hypothesis 59
3 Analytical Mechanics 61
§13. Analytical treatment of mechanics 61
§14. The Hamilton-Type variational principle 62
§15. Temporary variations 63
§16. The field equation for the mass density 66
Selected Topics 69
The Hamilton-Type least action principle 69
Noether´s theorem 71
Description of classical fields 72
The calculus of variations 72
The fundamental processes of the calculus of variations 73
The commutative properties of the 75
Temporary variation of the action 77
The variational method 78
Green´s theorem 79
The mass balance 81
References 82
4 Ideal Fluids 83
§17. Field functions for an ideal fluid 83
§18. Hamilton-Type variational principle and field differential equations 84
§19. Generalized energy balance equation 87
§20. Euler´s motion equation 90
§21. Energy conservation law 91
Selected Topics 97
Euler-Lagrange´s equations 97
Isentropic flux 99
Equivalence between the isentropic flux and the energy balance equation 100
Bernoulli´s equation 101
Proper time 103
The action integral and the lagrangian for a free particle 105
The relativistic form of lagrangian density 107
The geometrical form of relativistic lagrangian density 109
The mass-energy relation 110
Momentum and energy 113
Momentum and energy in fluid dynamics 114
References 116
5 Potential Flow 117
§22. Wave equation 117
§23. Field differential equations 120
§24. Potential energy density 122
§25. The lagrangian density and the specific lagrangian 125
§26. The linearised continuity equation 128
§27. The energy conservation law of sound waves 132
§28. Pressure and density variations 133
§29. Deformation waves 136
§30. Pressure waves 138
Selected Topics 139
The sound velocity 139
Plane waves 141
The wave equation in terms of the change in the density 145
Variations in pressure and density 146
Geometrical acoustics 147
References 148
6 Viscous Fluids 149
§31. Dissipative processes in a viscous fluid 149
§32. The thermodynamics of deformation 150
§33. Field differential equations 154
§34. Generalized energy balance equation 158
§35 Cauchy´s and Navier-Stokes´ equations 161
§36. Energy conservation law 163
§37.The general equation of heat transfer 165
Selected Topics 167
The concept of viscosity 167
Viscosity stress tensor 168
The entropy 170
The thermodynamic entities in a real fluid 174
Fourier´s equation 176
References 177
7 Free Convection 179
§38. The start up mechanism of free convection 179
§39. The conditions so that the mechanical equilibrium be unstable 180
§40. The velocity field 183
§41. The general equation of heat transfer 187
§42. Politropic atmosphere 191
§43. Some numerical results 194
Selected Topics 195
The atmosphere 195
The mechanical equilibrium in the atmosphere 198
Air density changes with temperature and humidity 200
The control of atmospheric pollution 202
Wind generation 204
References 206
8 Magnetohydrodynamics 207
§44. Field equations in a mobile conducting continuous medium 207
§45 The change in the generalized specific internal energy 210
§46. Momentum balance equation 212
§47. Generalized energy balance equation 214
§48. The equation of motion 216
§49. Energy conservation law 218
§50. General equation of heat transfer 219
§51. Thermal equation of state 221
Selected Topics 223
Magnetohydrodynamics and plasma physics 223
Generalized Bernoulli´s equation 225
References 228
9 Potential Flow in a Magnetic Field 229
§52. The equation of motion 229
§53. Field differential equations 233
§54. The lagrangian functions 233
§55. The energy conservation law 237
Selected Topics 238
The drift of the lines of force 238
Magnetic diffusion 241
The method of separation of variables 242
References 244
10 Magnetohydrodynamic Waves 245
§56. Alfven´s waves 245
§57. Alfven´s wave equation 247
§58. Kinetic and potential energy densities 249
§59. Field differential equations 253
§60. Energy conservation law 256
Selected Topics 257
Transversal displacements 257
The pressure and the velocity of sound 258
References 260
11 The Sunspots 261
§61. The problem of the sunspots 261
§62. Dynamic equilibrium between regulatory and startup mechanisms 263
§63. The velocity of the fluid in the sunspots 267
§64. General equation of heat transfer 273
§65. Magnetic field of the sunspots 274
§66. Persistency of the sunspots 278
§67. Origin, permanency, disappearance, and properties of the sunspots 279
Selected Topics 283
The spontaneous magnetic field produced by a turbulent motion 283
Diffusion of magnetic field into a plasma 285
The thermal equation of the sunspots 287
References 288
12 The Hamilton Equations of Motion 289
§68. Legendre´s transformation 289
§69. Hamilton´s canonical equations 292
§70. The energy balance equation 297
§71. The field of the specific enthalpy 299
§72. Nature of interactions in fluid dynamics 303
Selected Topics 304
The specific hamiltonian 304
The canonical integral 305
Energy theorem 308
The variation of generalized momentum 310
The homogeneous form of the canonical equations 312
Cyclic variables 313
References 314
13 Thermodynamics 315
§73. The system of a single phase 315
§74. Total internal energy 316
§75. The change in the total internal energy 319
§76. The second law of thermodynamics 321
§77. Entropy and quantity of heat 324
§78. Temperature 326
§79. Hamilton´s formulation 330
§80. Generalized momenta and the hamiltonian density 331
§81. Helmholtz free energy 332
§82. The heat function 334
§83. Mechanical work and quantity of heat 335
§84. The thermodynamic identity 336
§85. Gibbs´ free energy 338
§86. Maxwell relations 340
Selected Topics 343
Adiabatic processes and mean values 343
Entropy and temperature 344
Temperature and energy 347
Pressure 348
The Helmholtz free energy as a mechanical potential 350
The generalized thermodynamic potentials 351
References 354
14 The Magnetic Field in the Stability of the Stars 355
§87. The self-generated magnetic field 355
§88. The internal structure and the stability of a gaseous star 357
§89. The magnetic field on the surface of a star 361
§90. The mass-luminosity relation and the coefficient of opacity 363
§91. Luminosity and opacity 365
§92. The central temperature 366
§93. The problem of variable stars of the cepheid type 367
§94. The magnetic field in the inner part of a gaseous star 371
Selected Topics 382
The scale of stellar magnitudes 382
Luminosity and stellar radius 383
Stellar distances 385
Time scale of stellar evolution 387
A simple model to estimate 390
and 390
Polytropic gas sphere 393
Gaseous stars 396
A numerical equation for 397
The magnetic field in homologous inner points 400
The mass and the luminosity 402
The effective temperature and the absolute magnitude 403
The absolute magnitude 411
The magnetic field self-generated by gaseous stars 414
The self-generated geomagnetic field 420
References 424
Index 425