Generalized heat-transport equations: Parabolic and hyperbolic models

TL;DR

This paper derives two generalized heat-transport models, one hyperbolic and local, the other including nonlocal effects, both thermodynamically consistent and capable of describing finite-speed thermal wave propagation.

Contribution

It introduces two new generalized heat equations, one local and hyperbolic, the other nonlocal, expanding the modeling capabilities for heat conduction phenomena.

Findings

The first model encompasses classical heat equations and describes hyperbolic regimes.

The second model includes nonlocal effects and predicts finite-speed thermal wave propagation.

Both models are thermodynamically consistent through extended Liu and Coleman-Noll procedures.

Abstract

We derive two different generalized heat-transport equations: The most general one, of the first order in time and second order in space, encompasses some well known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman-Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed.