Abstract
The generalization of the Hamilton’s and Onsager’s variational principles for dissipative hydrodynamical systems is represented in terms of the mechanical and the heat displacement fields. A system of equations for these fields is derived from the extreme condition for action with a Lagrangian in the form of the difference between the kinetic and the free energies minus the time integral of the dissipation function. The generalized hydrodynamic equation system is then evaluated on the basis of the generalized variational principle. At low frequencies, this system corresponds to the traditional Navier–Stokes equation system, and in the high frequency limit it describes propagation of acoustical and heat modes with the finite propagation velocities.
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References
Biot, M.A.: Theory of propagation of elastic waves in fluid-saturated porous solid. I. Low-frequency range. J. Acoust. Soc. Am. 28, 168–178 (1956)
Biot, M.: Variational principles in theory of heat transfer. Mosc. Energy (1975)
Deresiewicz, H.: Plane wave in a thermoelastic solids. J. Acoust. Soc. Am. 29, 204–209 (1957)
Glensdorf, P., Prigogine, I.: Thermodynamic Theory of Structure, Stability and Fluctuations. New York (1971)
Gyarmati, I.: Non-Equilibrium Thermodynamics. Field Theory and Variational Principles. Springer, Berlin (1970)
Landau, L.D., Lifshitz, E.M.: Theoretical Physics. Statistical Physics, vol. 5. Nauka, Moscow (1964)
Landau, L.D., Lifshitz, E.M.: Theoretical Physics. Hydrodynamics, vol. 6. Nauka, Moscow (1986)
Landau, L.D., Lifshitz, E.M.: Theoretical Physics. Theory of Elasticity, vol. 7. Nauka, Moscow (1972)
Lykov, A.V.: Theory of Heat Conduction. Vysshaya Shkola, Moscow (1967)
Mandelshtam, L.I., Leontovich, M.A.: To the sound absorption theory in liquids. J. Exp. Theor. Phys. 7(3), 438–444 (1937). In Russian
Martynov, G.A.: Hydrodanamic theory of sound wave propagation. Theor. Math. Phys. 129, 1428–1438 (2001)
Maximov, G.A.: On the variational principle for dissipative hydrodynamics. Preprint 006-2006, Moscow, MEPhI (2006)
Maximov, G.A.: Generalized variational principle for dissipative hydrodynamics and its application to the Biot’s equations for multicomponent, multiphase media with temperature gradient. In: Weis, B.N. (ed.) New Research in Acoustics, pp. 21–61. Nova Science, New York (2008). ISNB:978-1-60456-403-7
Nettleton, R.E.: Relaxation theory of thermal conduction in liquids. Phys. Fluids 3, 216–223 (1960)
Onsager, L.: Reciprocal relations in irreversible process I. Phys. Rev. 37, 405–426 (1931)
Onsager, L.: Reciprocal relations in irreversible process II. Phys. Rev. 38, 2265–2279 (1931)
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Maximov, G.A. (2010). Generalized Variational Principle for Dissipative Continuum Mechanics. In: Maugin, G., Metrikine, A. (eds) Mechanics of Generalized Continua. Advances in Mechanics and Mathematics, vol 21. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5695-8_31
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DOI: https://doi.org/10.1007/978-1-4419-5695-8_31
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