Mathematical modeling of heat conduction in continuum thermodynamics

What is it about?

In this work we propose a new approach to heat conduction theories that is inherently thermodynamic. Local constitutive equations derive directly from the Clausius-Duhem inequality by applying a representation formula and specifying by constitutive laws the Helmholtz free energy and the non-negative entropy production. The set of independent variables includes only the macroscopically observable fields and their temporal and spatial derivatives, without making any recourse to internal variables or ambiguous state variables (such as thermal displacement). In particular, no constitutive prescription on the heat influx is made, rather it is treated as an independent variable. This strategy is compatible with both rigid and deformable bodies. We consider both local and (weakly) nonlocal models of heat conduction.

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Why is it important?

The entropy production (rate) in all models derived in this paper is explicitly represented as a positive-definite quadratic function of non-equilibrium quantities. As stated by Prigogine ``To extend thermodynamics to non-equilibrium processes we need an explicit expressions for the entropy production" (in I. Prigogine, Time, structure and fluctuations, Nobel lecture, 1977). As to nonlocal models, the local entropy supply is a quadratic function of first and second order gradients of the absolute temperature and the heat flux vector. According to the Prigogine's theorem, the entropy production (rate) is proved to decrease when the system approaches the steady-state (constant deformation, uniform temperature distribution and zero heat flux) and vanish as the steady-state is reached.

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