Reiterated homogenization applied to heat conduction in heterogeneous media with multiple spatial scales and perfect thermal contact between the phases

Abstract

Several types of heterogeneous media with multiple spatial scales presently offer good potential to improve upon more traditional materials used in heat transfer and other engineering applications. An example might be a large scale, otherwise homogeneous medium filled with dispersed small-scale particles that form aggregate structures at an intermediate scale. In this paper, the strong-form Fourier heat conduction problem in such media is formulated using the method of reiterated homogenization. The constituent phases are assumed to have a perfect thermal contact at the interface. The ratio of two successive length scales of the medium is a constant small parameter \(\varepsilon\). The method is an up-scaling procedure that writes the temperature field as an asymptotic multiple-scale expansion in powers of the small parameter \(\varepsilon\). The technique leads to two pairs of local and homogenized problems, linked by effective coefficients. In this manner the phenomenon description at the smallest scale is seen to affect the medium macroscale, or effective, behavior, which is the main interest in engineering. To facilitate the physical understanding of the derived sub-problems, an analytical solution is obtained for the heat conduction problem in a laminated binary composite. The present formulation shall serve as a basis for future efforts to numerically compute effective properties of heterogeneous media with multiple spatial scales.