Variational Theory of Heat Conduction

  • G. Lebon
  • P. Perzyna
Part of the International Centre for Mechanical Sciences book series (CISM, volume 262)


This chapter is devoted to the problem of heat conduction in rigid bodies. A representative collection of variational principles is examined. According to the nature of the problem to be handled, these criteria will be classical or not.


Heat Conduction Variational Principle Variational Theory Heat Conduction Equation Trial Function 
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  1. 1.
    Biot, M., Variational principles in heat transfer, Oxford Univ. Press, Oxford, 1970.MATHGoogle Scholar
  2. 2.
    Vujanovic, B., An approach to linear and non-linear heat transfer problems using a Lagrangian, Journal, 9, 131, 1971.ADSGoogle Scholar
  3. 3.
    Glansdorff, P. and Prigogine, I., On a general evolution criterion in macroscopic physics, Physica, 30, 351, 1964.MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Glansdorff, P. and Prigogine, I., Structure, Stability and fluctuations, J. Wiley, New-York, 1971.MATHGoogle Scholar
  5. 5.
    Lebon, G. and Lambermont, J., Some variational principles pertaining to non-stationary heat conduction and coupled thermoelasticity, Acta Mech., 15, 121, 1972.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Lebon, G., A new variational principle for the non-linear unsteady heat conduction problem, Quart. J. Mech. Appl. Math., 29, 499, 1976.MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Gurtin, M., Variational principles for linear initial value problems, Quart. Appl. Math, 22, 252, 1964.MathSciNetMATHGoogle Scholar
  8. 8.
    Tonti, E., On the variational formulation for linear initial value problems, Annali di Math. pura ed appl., 95, 331, 1973.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Reddy, J., A note on mixed variational principles for initial-value problems, Quart. J. Mech. Appl. Math., 28, 123, 1975.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Maxwell, J., On the dynamical theory of gases, Phil. Trans. Roy. Soc. London, 157, 49, 1867.CrossRefGoogle Scholar
  11. 11.
    Cattaneo, C., Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, C.R.Acad. Sc. Paris, 247, 431, 1958.MathSciNetGoogle Scholar
  12. 12.
    Vernotte, P., Les paradoxes de la théorie continue de l’équation de la chaleur, C.R.Acad. Sc. Paris, 247, 3154, 1958.Google Scholar
  13. 13.
    Onsager, L., Reciprocal relations in irreversible processes, Phys. Rev., 37, 405, 1931.ADSCrossRefGoogle Scholar
  14. 14.
    Gyarmati, I., Non-equilibrium thermodynamics, Springer-Verlag, Berlin, 1971.Google Scholar
  15. 15.
    Rosen, P., Variational approach to magnetohydrodynamics, Phys. Fluids, 1, 251, 1958.ADSCrossRefGoogle Scholar
  16. 16.
    Djukic, D. and Vujanovic, B., On a new variational principle of Hamiltonian type for classical field theory, Z.A.M.M., 21, 611, 1971.CrossRefGoogle Scholar
  17. 17.
    Djukic, D., Vujanovic, B., Tatic, N. and Strauss, A., On two variational methods for obtaining solutions to transport problems, Chem. Eng. J., 5, 145, 1973.CrossRefGoogle Scholar
  18. 18.
    Djukic, D., Hiemenz magnetic flow of power-law fluids, J. Appl. Mech., September 1974, 822, 1974.CrossRefGoogle Scholar
  19. 19.
    Prigogine, I., Etude thermodynamique des phénomènes irréversibles, Desoer, Liège, 1947.Google Scholar
  20. 20.
    Lebon, G. and Lambermont, J., A rather general variational principle for purely dissipative processes, Annalen Phys., 7, 15, 1972.ADSGoogle Scholar
  21. 21.
    Lebon, G. and Lambermont, J., Generalization of Hamilton’s principle to continuous dissipative systems, J. Chem. Phys., 59, 2929, 1973.MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Lebon, G., Distribution non-stationnaire de la température dans les milieux dont la conductibilité thermique dépend de la température, Ann. Soc. Sc. Bruxelles, 84, 304, 1970.Google Scholar
  23. 23.
    Lardner, T.J., Biot’s variational principle in heat conduction, J. A. I. A. A. 1, 1, 1963.Google Scholar
  24. 24.
    Richardson, P.D., Unsteady heat conduction with a non-linear boundary condition, J. Heat Transfer, 86, 298, 1964.CrossRefGoogle Scholar
  25. 25.
    Finlayson, B.A. and Scriven, L., On the search for variational principles, Int. J. Heat Mass Transfer, 10, 799, 1967.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1980

Authors and Affiliations

  • G. Lebon
    • 1
  • P. Perzyna
    • 2
  1. 1.University of LiegeBelgium
  2. 2.Polish Academy of SciencesPoland

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