Gradient thermodynamics and heat equations

Quoc-Son Nguyen

doi:10.1016/j.crme.2010.07.010

Gradient thermodynamics and heat equations
[Modèle thermodynamique à gradients et l'équation thermique]

Quoc-Son Nguyen ¹

¹ Laboratoire de mécanique des solides, CNRS (umr7649), École polytechnique, 91128 Palaiseau, France

Comptes Rendus. Mécanique, Volume 338 (2010) no. 6, pp. 321-326.

Résumé
Abstract

On propose dans cette Note un cadre thermodynamique incluant le gradient de la température comme une variable d'état supplémentaire. Cette théorie est construite à partir d'une relation originale reliant les expresssions de l'énergie interne et l'énergie libre par transformation de Legendre étendue aux variables $(T, \nabla T)$ . La description proposée peut être justifiée par un passage micro–macro. En particulier, l'équation thermique résultante est discutée et comparée avec les résultats antérieurs de la littérature. On montre que cette équation enlève le paradoxe de propagation instantanée.

In this Note, a thermodynamic description is proposed to include the gradient of the temperature in the set of state variables. It is based upon an original expression of the entropy and of the internal energy taking account of the presence of the temperature gradient by Legendre transform with respect to the variable $(T, \nabla T)$ . The proposed description can be justified in a homogenization process. In particular, the associated heat equation is derived and discussed in comparison with the existing results of the literature. It is shown that this equation removes the paradox of instantaneous propagation.

Reçu le : 2010-04-19
Accepté le : 2010-07-16
Publié le : 2010-06-01

DOI : 10.1016/j.crme.2010.07.010

Keywords: Continuum mechanics, Continuum thermodynamics, State variable, Temperature gradient, Heat equation
Mots-clés : Milieux continus, Thermodynamique des milieux continus, Variable d'état, Gradient de température, Équation thermique

Affiliations des auteurs :

Quoc-Son Nguyen ¹

¹ Laboratoire de mécanique des solides, CNRS (umr7649), École polytechnique, 91128 Palaiseau, France

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Quoc-Son Nguyen. Gradient thermodynamics and heat equations. Comptes Rendus. Mécanique, Volume 338 (2010) no. 6, pp. 321-326. doi : 10.1016/j.crme.2010.07.010. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.07.010/

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