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

Article type: Research Article

Authors: Gérard-Varet, David^{; *}

Affiliations: IMJ-PRG, Université de Paris, Campus des Grands Moulins, 75013 Paris, France

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We present a gentle approach to the justification of effective media approximations, for PDE’s set outside the union of n≫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→+∞ by simple arguments.

Keywords: Homogenisation, fluid mechanics

DOI: 10.3233/ASY-211696

Journal: Asymptotic Analysis, vol. 128, no. 1, pp. 31-53, 2022