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arXiv 2002.048462020-02math.AP

Mild assumptions for the derivation of Einstein's effective viscosity formula

David Gérard-Varet, Richard M. Höfer

备注：: V3: In this new version we added references of results that have appeared after the first submission, and we refined the discussion on the assumptions (B1)-(B2) in Sections 5 and 6. To appear in Commun. Partial. Differ. Equ. V2: Assumption (B2) considerably weakened

英文摘要：

We provide a rigorous derivation of Einstein's formula for the effective viscosity of dilute suspensions of $n$ rigid balls, $n \gg 1$, set in a volume of size $1$. So far, most justifications were carried under a strong assumption on the minimal distance between the balls: $d_{min} \ge c n^{-\frac{1}{3}}$, $c > 0$. We relax this assumption into a set of two much weaker conditions: one expresses essentially that the balls do not overlap, while the other one gives a control of the number of balls that are close to one another. In particular, our analysis covers the case of suspensions modelled by standard Poisson processes with almost minimal hardcore condition.

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