VARIATIONAL FORMULATION OF GRADIENT IRREVERSIBLE THERMODYNAMICS

Abstract


This work proposes the elaboration of the variational principle of L.I. Sedov for modeling dissipative processes. The formulated variational principle makes it possible to propose the dissipative models using the known model of a reversible process (the known Lagrangian), adding the required number of dissipation channels. Dissipation channels are non-integrable variational forms that are linear in the variations of the ar-guments. The arguments of the dissipation channels are the generalized variables of the corresponding bilinear terms of the Lagrangian. Variational models of heat transfer pro-cesses are considered as examples. The paper introduces the thermal potential, which is taken as the main kinematic variable. The temperature and heat flux are determined from the expression of the possible work done on variations of the first derivatives of the thermal potential, by analogy with continuum mechanics, where internal forces do the possible work on the strain variations. The equations of heat conduction laws of the considered heat transfer models are obtained as compatibility equations by eliminating the thermal potential from the equations of constitutive relations for temperature and heat flow. It is shown that the proposed procedure for elaboration of the dissipative models makes it possible to obtain the laws of thermal conductivity of Fourier, Maxwell – Catta-neo, Gaer – Krumhaksl, Jeffrey and more general laws of thermal conductivity. For the simplest heat transfer model, a single dissipation channel was introduced, which made it possible to obtain a heat transfer equation containing the second and first time deriva-tives. This model takes into account the wave properties and dissipation by the diffusion mechanism. In a particular case, it is reduced to the classical model of heat conduction. For more general gradient models of heat transfer, additional dissipation channels are sequentially introduced. In accordance with the differential order of the heat balance equation, the variational method makes it possible to formulate a consistent spectrum of boundary conditions at each non-singular point of the surface. In addition, for a bounda-ry value problem in time, the variational principle determines pairs of alternative condi-tions at the initial and final times of the process under consideration.

About the authors

P. A. Belov

Instiutute of Applied Mechanics of RAS, Moscow, Russian Federation

S. A. Lurie

Instiutute of Applied Mechanics of RAS, Moscow, Russian Federation

References

  1. Седов Л.И. Об основных принципах механики сплошной среды. – М.: Изд-во МГУ, 1961. – 26 с.
  2. Седов Л.И. Об основных концепциях механики сплошной среды // Некоторые проблемы математики и механики. – 1961. – С. 227–235.
  3. Седов Л.И., Эглит М.Э. Построение неголономных моде-лей сплошных сред с учетом конечности деформаций и не-которых физико-химических эффектов // Докл. АН СССР. – 1962. – Т. 142, № 1. – С. 54–59.
  4. Бердичевский В.Л. Построение моделей сплошных сред при помощи вариационного принципа // ПММ. – 1966. – Т. 30. – Вып. 3. – С. 510–530.
  5. Бердичевский В.Л. Вариационные методы построения мо-делей сплошных сред с необратимыми процессами в спе-циальной теории относительности // ПММ. – 1966. – Т. 30. – Вып. 6. – С. 1081–1086.
  6. Лурье С.А., Белов П.А. Вариационная модель неголоном-ных сред // Механика композиционных материалов и кон-струкций. – 2001. – Т. 7, № 2. – С. 436–444.
  7. Белов П.А., Горшков А.Г., Лурье С.А. Вариационная мо-дель неголономных 4D-сред // Механика твердого тела. – 2006. – № 6. – С. 41–58.
  8. Lurie S.A., Belov P.A., Volkov-Bogorodskii D.B. Variational models of coupled gradient thermoelasticity and thermal con-ductivity // Mater. Phys. Mech. – 2019. – № 42. – Р. 564–581.
  9. Lurie S., Belov P. From generalized theories of media with fields of defects to closed variational models of the coupled gradient thermoelasticity and thermal conductivity // In Higher Gradient Materials and Related Generalized Continua / Eds.: Altenbach, H., Muller, W.H., Abali, B.E. – Springer: Cham, Switzerland. – 2019. – Vol. 11. – Р. 135–154.
  10. Lurie S.A., Belov P.A. On the nature of the relaxation time, the Maxwell-Cattaneo and Fourier law in the thermodynamics of a continuous medium, and the scale effects in thermal conductivi-ty // Continuum. Mech. Thermodyn. – 2020. – № 32. – Р. 709–728.
  11. Sellitto A., Cimmelli V.A., Jou D. Mesoscopic Theories of Heat Transport in Nanosystems. – S.: Springer International Publishing Switzerland – 2016. – 170 p.
  12. Zhukovsky K.V., Srivastava H.M. Analytical solutions for heat diffusion beyond Fourier law // Appl. Math. Comput. – 2017. – Vol. 293. – P. 423–437.
  13. Sobolev S.L. Nonlocal two-temperature model: application to heat transport in metals irradiated by ultrashort laser pulses // Int. J. Heat Mass Tran. – 2016. – Vol. 94. – P. 138–144.
  14. Maxwell J.C. On the Dynamical Theory of Gases // Philosoph-ical Transactions of the Royal Society of London. – 1867. – Vol. 157. – P. 49–88.
  15. Cattaneo C. Sulla Condizione Del Calore // Atti Del Semin. Matem. E Fis. Della Univ. Modena. – 1948. – Vol. 3. – P. 83–101.
  16. Vernotte M.P. La véritableéquation de chaleur // Comptes rendus hebdomadaires des séances de l'Académie des sciences. – 1958. – Vol. 247. – P. 2103–2105.
  17. Vernotte M.P. Les paradoxes de la théorie continue de léqua-tion de la chaleur // C.R. Hebd. Seances Acad. Sci. – 1958. – Vol. 246, no. 22. – P. 3154–3155.
  18. Joseph D.D. Preziosi L. Heat waves. Reviews of modern physics // Rev. Mod. Phys. – 1989. – Vol. 61. – P. 41–73.
  19. Sobolev S.L. Hyperbolic heat conduction, effective tempera-ture, and third law for nonequilibrium systems with heat flux // Physical review. – 2018. – Vol. E 97. – P. 022122.
  20. Kovács R., Fehér A., Sobolev S. On the two-temperature de-scription of heterogeneous materials // International Journal of Heat and Mass Transfer. – 2022. – Vol. 194. – P. 123021.

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