Derivation of the Batchelor-Green formula for random suspensions

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 ( ϕ 2 ) formula for the effective viscosity. In [17] , [19] , such formula was given for rigid spheres satisfying the strong separation assumption d m i n ≥ c ϕ − 1 3 r , where d m i n 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 ( ϕ 2 ) 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 d m i n > ( 2 + e ) r , e > 0 .