Diffusion through a composite structure with a high contrasting diffusivity

Abstract

We consider the stationary diffusion equation in a periodic composite medium made of two components M_ε and B_ε having very different diffusivity, the ratio between the coefficients of the diffusion in that structure being 1/α_ε², where ε is the size of the period and α_ε which represents the amplitude of the diffusion in the inclusions B_ε is a decreasing sequence towards zero. We show that the inclusions B_ε work on the macroscopic diffusion as holes. In particular for scalings 0<α_ε�ε corresponding to very weak diffusion in B_ε, the volume fraction of the material at the limit is the well known one in the case of holes.