Derivation of the Batchelor-Green formula for random suspensions

Abstract

This paper is dedicated to the effective viscosity of suspensions without inertia, at low solid volume fraction ϕ. The goal is to derive rigorously a o(ϕ2) formula for the effective viscosity. In [17,19], such formula was given for rigid spheres satisfying the strong separation assumption dmin≥cϕ−13r, where dmin is the minimal distance between the spheres and r their radius. It was then applied to both periodic and random configurations with separation, to yield explicit values for the O(ϕ2) coefficient. We consider here complementary (and certainly more realistic) random configurations, satisfying softer assumptions of separation, and long range decorrelation. We justify in this setting the famous Batchelor-Green formula [3]. Our result applies for instance to hardcore Poisson point process with almost minimal hardcore assumption dmin>(2+ε)r, ε>0.