Heat conduction: hyperbolic self-similar shock-waves in solids

TL;DR

This paper derives analytic solutions for nonlinear hyperbolic heat conduction equations in solids, revealing self-similar shock-wave and continuous wave solutions with temperature-dependent properties.

Contribution

It generalizes the Fourier-Cattaneo law to include power-law temperature dependence and provides self-similar solutions for nonlinear hyperbolic heat conduction systems.

Findings

Existence of self-similar shock-wave solutions

Analytic forms for temperature and heat flux distributions

Application to materials with various temperature-dependent conductivities

Abstract

Analytic solutions for cylindrical thermal waves in solid medium is given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the relaxation time and heat propagation coefficient have a general power law temperature dependence. From such laws one cannot form a second order parabolic or telegraph-type equation. We consider the original non-linear hyperbolic system itself with the self-similar Ansatz for the temperature distribution and for the heat flux. As results continuous and shock-wave solutions are presented. For physical establishment numerous materials with various temperature dependent heat conduction coefficients are mentioned.