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arXiv · 2008.06324

Derivation of Batchelor-Green formula for random suspensions

David Gerard-Varet
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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 $o(ϕ^2)$ formula for the effective viscosity. In previous works, such formula was given for rigid spheres satisfying the strong separation assumption $ d_{min} \ge c ϕ^{-\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 $O(ϕ^2)$ 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 $d_{min} > (2+ε) r$, $ε> 0$.

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David Gerard-Varet. 2021-03-14. Derivation of Batchelor-Green formula for random suspensions. https://arxiv.org/abs/2008.06324

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