A simple justification of effective models for conducting or fluid media with dilute spherical inclusions

Abstract

We present a gentle approach to the justification of effective media approximations, for PDE's set outside the union of $n \gg 1$ spheres with low volume fraction. To illustrate our approach, we consider three classical examples: the derivation of the so-called strange term, made popular by Cioranescu and Murat, the derivation of the Brinkman term in the Stokes equation, and a scalar analogue of the effective viscosity problem. Under some separation assumption on the spheres, valid for periodic and random distributions of the centers, we recover effective models as $n$ goes to infinity by simple arguments.