Derivation of Batchelor-Green formula for random suspensions

TL;DR

This paper rigorously derives the Batchelor-Green formula for the effective viscosity of dilute random suspensions, extending previous results to more realistic configurations with softer separation assumptions.

Contribution

It provides a rigorous derivation of the Batchelor-Green formula for random suspensions under softer separation and decorrelation assumptions, broadening its applicability.

Findings

Validates the Batchelor-Green formula for hardcore Poisson point processes.

Extends previous results to more realistic random configurations.

Provides explicit conditions for the applicability of the formula.

Abstract

This paper is dedicated to the effective viscosity of suspensions without inertia, at low solid volume fraction $\phi$. The goal is to derive rigorously a $o(\phi^2)$ formula for the effective viscosity. In previous works, such formula was given for rigid spheres satisfying the strong separation assumption $ d_{min} \ge c \phi^{-\frac13} r$, where $d_{min}$ 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 $O(\phi^2)$ coefficient. We consider here complementary (and certainly more realistic) random configurations, satisfying soft assumptions of separation and long range decorrelation. We justify in this setting the famous Batchelor-Green formula. Our result applies for instance to hardcore Poisson point process with almost minimal hardcore assumption $d_{min} > (2+\epsilon) r$, $\epsilon > 0$.