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

TL;DR

This paper provides a straightforward method to justify effective media models for PDEs in media with many dilute spherical inclusions, covering classical examples like the strange term, Brinkman term, and effective viscosity.

Contribution

It introduces a simple approach to derive effective models for PDEs in media with numerous dilute spheres, applicable to both periodic and random distributions.

Findings

Recovered effective models as the number of spheres tends to infinity.

Derived classical effective terms using simple arguments.

Applicable to both periodic and random sphere distributions.

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.