Derivation of a homogenized two-temperature model from the heat equation

Abstract

This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat, Collège de France Seminar, vol. II. Paris 1979-1980; vol. 60 of Res. Notes Math. Pitman, Boston, London (1982) 98-138].