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Derivation of the Batchelor-Green formula for random suspensions
Journal de Mathématiques Pures et Appliquées ( IF 2.1 ) Pub Date : 2021-05-31 , DOI: 10.1016/j.matpur.2021.05.002
David Gérard-Varet

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 dmincϕ13r, 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.



中文翻译:

用于随机悬浮的 Batchelor-Green 公式的推导

本文致力于在低固体体积分数ϕ下无惯性悬浮液的有效粘度。目标是严格地推导出一个(φ2)有效粘度公式。在[17]、[19]中,对于满足强分离假设的刚性球体,给出了这样的公式d一世nCφ-13r, 在哪里 d一世n是球体与其半径r之间的最小距离。然后将其应用于具有分离的周期性和随机配置,以产生明确的值(φ2)系数。我们在这里考虑互补的(当然更现实的)随机配置,满足更软的分离假设和远程去相关。我们在这种情况下证明了著名的 Batchelor-Green 公式 [3]。我们的结果适用于例如具有几乎最小的硬核假设的硬核泊松点过程d一世n>(2+ε)r, ε>0.

更新日期:2021-07-01
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