Heat equation with temperature-dependent conductivity coefficients and macroscopic properties of microheterogeneous media

Abstract

The aim of this paper is a study of the quasi-linear transport equation, for instance, the stationary heat equation. For periodically microheterogeneous media, asymptotic homogenization has been performed with the local problem formulated as a minimization problem. The Hashin-Shtrikman type bounds and Golden-Papanicolaou integral representation theorem have been extended. In the case of layered composites, exact analytical formulae for the effective coefficients have been derived. The possibility of applying Padé approximants has been shown. Specific cases and examples have also been examined.