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Asymptotic analysis of heat propagation models

Moszyński K., Palczewski A.

Archives of Mechanics

2000

Vol. 52, nr 2

225-246

The subject of this paper is the analysis of different models of heat propagation. As is well known, one of essential disadvantages of the classical model proposed by Fourier is the infinite velocity at which heat propagates. To avoid that unphysical phenomenon, Cattaneo has proposed a hyperbolic model. An essential feature of that model is the introduction of a relaxation time for thermal processes. In recent years several new models have been proposed which retain the relaxation time phenomenon but are parabolic in their character. When the relaxation time is small, all these models lead to singularly perturbed equations. We analyze some of these models and prove that the solution of the classical heat equation (Fourier model) is a bulk approximation to exact solutions of these models. We show also that the behaviour of the Fourier model depends on the way in which it is applied. Finally, we present numerical comparison of exact solutions with the bulk solution for the test problem of heat propagation in thin metal films heated by a laser beam.