The Hamilton-Type Principle in Fluid Dynamics : Fundamentals and Applications to Magnetohydrodynamics, Thermodynamics, and Astrophysics

The contents of this book in its English edition summarizes the basic re-search work accomplished by the author on the field of Fluid Dynamics, Magnetohydrodynaics (MHD), Classical Thermodynamics, and Astrophy-sics. Besides, it contains a section attached to each chapter with the title of Selected Topics. In these sections fine theoretical points are cleared up and a large number of illustrative problems are solved. This way the idea is to have readers who use this book as a text book as well as a research monography addressed to all those who study Theoretical Physics and who are interested in the analytical treatment of perfect and real fluids, as much as in the dynamic state of fluids electrically charged which flow in regions where there is a magnetic field, and also fluids at rest proper of Classical Thermodynamics. With this material we wish to facilitate the access to fundamental aspects of Fluid Dynamics, MHD, and Thermodynamics, and the application of these disciplines to Astrophysics. It is assumed that the reader is aqcuainted with Maxwell?s equations of Classical Electrodyna-mics, those of Hydrodynamics balance equations, and the fundamental postulates of Thermodynamics. A reasonable handling of the calculus of variations and the tensorial notation is required, as well as knowledge of the principles and methodology of Lagrange?s Analytical Mechanics. In general terms and from the view point of the required mathematics, the book is self-contained in the sense in which everything that is needed is clearly stated within the text. Contents......Page 6 About the Author......Page 13 Foreword......Page 14 Acknowledgements......Page 16 Notation......Page 17 §1. Introduction......Page 22 §2. Kinematics......Page 23 §3. Deformation......Page 24 §4. Transformation of a volume element......Page 25 §5. Rate of change of a volume element......Page 26 §6. The hydrodynamics derivative......Page 28 §7. Reynolds ́ transport theorem......Page 29 §8. Hamilton ́s principle......Page 30 The velocity field......Page 32 Euler ́s relation......Page 36 Hamilton ́s principle in fluid dynamics......Page 37 References......Page 39 §9. The continuity equation......Page 40 tensor......Page 42 §11. The scalar equation for the mass density......Page 46 §12. The dilute gas......Page 49 The continuity and motion equations†......Page 52 Bossinesq ́s approaches......Page 54 The mass balance equation......Page 55 Demonstration of a few formulae......Page 57 The molecular chaos hypothesis......Page 58 References......Page 59 §13. Analytical treatment of mechanics......Page 60 §14. The Hamilton-Type variational principle......Page 61 §15. Temporary variations......Page 62 §16. The field equation for the mass density......Page 65 The Hamilton-Type least action principle......Page 68 Noether ́s theorem......Page 70 The calculus of variations......Page 71 The fundamental processes of the calculus of variations......Page 72 The commutative properties of the -process......Page 74 Temporary variation of the action......Page 76 The variational method......Page 77 Green ́s theorem......Page 78 The mass balance......Page 80 References......Page 81 §17. Field functions for an ideal fluid......Page 82 §18. Hamilton-Type variational principle and field differential equations......Page 83 §19. Generalized energy balance equation......Page 86 §20. Euler ́s motion equation......Page 89 §21. Energy conservation law......Page 90 Euler-Lagrange ́s equations......Page 96 Isentropic flux......Page 98 Equivalence between the isentropic flux and the energy balance equation......Page 99 Bernoulli ́s equation......Page 100 Proper time......Page 102 The action integral and the lagrangian for a free particle......Page 104 The relativistic form of lagrangian density......Page 106 The geometrical form of relativistic lagrangian density......Page 108 The mass-energy relation......Page 109 Momentum and energy......Page 112 Momentum and energy in fluid dynamics......Page 113 References......Page 115 §22. Wave equation......Page 116 §23. Field differential equations......Page 119 §24. Potential energy density......Page 121 §25. The lagrangian density and the specific lagrangian......Page 124 §26. The linearised continuity equation......Page 127 §27. The energy conservation law of sound waves......Page 131 §28. Pressure and density variations......Page 132 §29. Deformation waves......Page 135 §30. Pressure waves......Page 137 The sound velocity......Page 138 Plane waves......Page 140 The wave equation in terms of the change in the density......Page 144 Variations in pressure and density......Page 145 Geometrical acoustics......Page 146 References......Page 147 §31. Dissipative processes in a viscous fluid......Page 148 §32. The thermodynamics of deformation......Page 149 §33. Field differential equations......Page 153 §34. Generalized energy balance equation......Page 157 §35 Cauchy ́s and Navier-Stokes ́ equations......Page 160 §36. Energy conservation law......Page 162 §37.The general equation of heat transfer......Page 164 The concept of viscosity......Page 166 Viscosity stress tensor......Page 167 The entropy......Page 169 The thermodynamic entities in a real fluid......Page 173 Fourier ́s equation......Page 175 References......Page 176 §38. The start up mechanism of free convection......Page 178 §39. The conditions so that the mechanical equilibrium be unstable......Page 179 §40. The velocity field......Page 182 §41. The general equation of heat transfer......Page 186 §42. Politropic atmosphere......Page 190 §43. Some numerical results......Page 193 The atmosphere......Page 194 The mechanical equilibrium in the atmosphere......Page 197 Air density changes with temperature and humidity......Page 199 The control of atmospheric pollution......Page 201 Wind generation......Page 203 References......Page 205 §44. Field equations in a mobile conducting continuous medium......Page 206 §45 The change in the generalized specific internal energy......Page 209 §46. Momentum balance equation......Page 211 §47. Generalized energy balance equation......Page 213 §48. The equation of motion......Page 215 §49. Energy conservation law......Page 217 §50. General equation of heat transfer......Page 218 §51. Thermal equation of state......Page 220 Magnetohydrodynamics and plasma physics......Page 222 Generalized Bernoulli ́s equation......Page 224 References......Page 227 §52. The equation of motion......Page 228 §54. The lagrangian functions......Page 232 §55. The energy conservation law......Page 236 The drift of the lines of force......Page 237 Magnetic diffusion......Page 240 The method of separation of variables......Page 241 References......Page 243 §56. Alfven ́s waves......Page 244 §57. Alfven ́s wave equation......Page 246 §58. Kinetic and potential energy densities......Page 248 §59. Field differential equations......Page 252 §60. Energy conservation law......Page 255 Transversal displacements......Page 256 The pressure and the velocity of sound......Page 257 References......Page 259 §61. The problem of the sunspots......Page 260 §62. Dynamic equilibrium between regulatory and startup mechanisms......Page 262 §63. The velocity of the fluid in the sunspots......Page 266 §64. General equation of heat transfer......Page 272 §65. Magnetic field of the sunspots......Page 273 §66. Persistency of the sunspots......Page 277 §67. Origin, permanency, disappearance, and properties of the sunspots......Page 278 The spontaneous magnetic field produced by a turbulent motion......Page 282 Diffusion of magnetic field into a plasma......Page 284 The thermal equation of the sunspots......Page 286 References......Page 287 §68. Legendre ́s transformation......Page 288 §69. Hamilton ́s canonical equations......Page 291 §70. The energy balance equation......Page 296 §71. The field of the specific enthalpy......Page 298 §72. Nature of interactions in fluid dynamics......Page 302 The specific hamiltonian......Page 303 The canonical integral......Page 304 Energy theorem......Page 307 The variation of generalized momentum......Page 309 The homogeneous form of the canonical equations......Page 311 Cyclic variables......Page 312 References......Page 313 §73. The system of a single phase......Page 314 §74. Total internal energy......Page 315 §75. The change in the total internal energy......Page 318 §76. The second law of thermodynamics......Page 320 §77. Entropy and quantity of heat......Page 323 §78. Temperature......Page 325 §79. Hamilton ́s formulation......Page 329 §80. Generalized momenta and the hamiltonian density......Page 330 §81. Helmholtz free energy......Page 331 §82. The heat function......Page 333 §83. Mechanical work and quantity of heat......Page 334 §84. The thermodynamic identity......Page 335 §85. Gibbs ́ free energy......Page 337 §86. Maxwell relations......Page 339 Adiabatic processes and mean values......Page 342 Entropy and temperature......Page 343 Temperature and energy......Page 346 Pressure......Page 347 The Helmholtz free energy as a mechanical potential......Page 349 The generalized thermodynamic potentials......Page 350 References......Page 353 §87. The self-generated magnetic field......Page 354 §88. The internal structure and the stability of a gaseous star......Page 356 §89. The magnetic field on the surface of a star......Page 360 §90. The mass-luminosity relation and the coefficient of opacity......Page 362 §91. Luminosity and opacity......Page 364 §92. The central temperature......Page 365 §93. The problem of variable stars of the cepheid type......Page 366 §94. The magnetic field in the inner part of a gaseous star......Page 370 The scale of stellar magnitudes......Page 381 Luminosity and stellar radius......Page 382 Stellar distances......Page 384 Time scale of stellar evolution......Page 386 A simple model to estimate pc , Tc , and Hc......Page 389 Polytropic gas sphere......Page 392 Gaseous stars......Page 395 A numerical equation for......Page 396 The magnetic field in homologous inner points......Page 399 The mass and the luminosity......Page 401 The effective temperature and the absolute magnitude......Page 402 The absolute magnitude......Page 410 The magnetic field self-generated by gaseous stars......Page 413 The self-generated geomagnetic field......Page 419 References......Page 423 Index......Page 424 The objective of this book is to contribute to specialized literature with the most significant results obtained by the author in Continuous Mechanics and Astrophysics. The nature of the book is largely determined by the fact that it describes Fluid Dynamics,Magnetohydrodynamics,and Classical Thermodynamics as branches of Lagrange's Analytical Mechanics; and in that sense, the approach presented in it is markedly different from the treatment given to them in traditional text books. In order to reach that goal, a Hamilton-Type Variational Principle, as the proper mathermatical technique for the theoretical description of the dy- mic state of any fluid is formulated. The scheme is completed proposing a new group of variations regarding the evolution parameter which is time; and with the demonstration of a theorem concerning the invariance of the action integral under continuous and infinitesimal temporary transfor- tions. With all that has been mentioned before and taking into account the methodsof thecalculusof variationsandtheadequateboundaryconditions, a general methodology for the mathematical treatment of fluid flows characteristic of Fluid Dynamics, Magnetohydrodynamics, and also fluids at rest proper of Classical Thermodynamics is presented. The Book Summarizes The Basic Research Work Accomplished By The Author On The Field Of Fluid Dynamics, Magnetohydrodynamics (mhd), Classical Thermodynamics, And Astrophysics. The Nature Of The Book Is Largely Determined By The Fact That It Describes Fluid Dynamics, Magnetohydrodynamics, And Classical Thermodynamics As Branches Of Lagrange's Analytical Mechanics; And In That Sense, The Approach Presented In It Is Markedly Different From The Treatment Given To Them In Traditional Text Books. In Order To Reach That Goal, A Hamilton-type Variational Principle, As The Proper Mathematical Technique For The Theoretical Description Of The Dynamic State Of Any Fluid Is Formulated.--jacket. I. General Principles -- Ii. Mass Density -- Iii. Analytical Mechanics -- Iv. Ideal Fluids -- V. Potential Flow -- Vi. Viscous Fluids -- Vii. Free Convection -- Viii. Magnetohydrodynamics -- Ix. Potential Flow In A Magnetic Field -- X. Magnetohydrodynamic Waves -- Xi. The Sunspots -- Xii. The Hamilton Equations Of Motion -- Xiii. Thermodynamics -- Xiv. The Magnetic Field In The Stability Of The Stars. Angel Fierros Palacios. Includes Bibliographical References And Index Describes Fluid Dynamics, Magnetohydrodynamics, and Classical Thermodynamics as branches of Lagrange's Analytical Mechanics. This book formulates a Hamilton-Type Variational Principle as the proper mathematical technique for the theoretical description of the dynamic state of any fluid