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 $o({\varphi }^2)$ formula for the effective viscosity. In [17], [19], such formula was given for rigid spheres satisfying the strong separation assumption $d_{min}\ge c{\varphi }^{-\frac{1}{3}}r$, 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 $O({\varphi }^2)$ 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 $d_{min}>(2+\epsilon )r$, $\epsilon >0$.

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$○({\phi }^2)$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$d_{米\mathrm{一世}n}\ge C{\phi }^{-\frac{1}{3}}r$， 在哪里 $d_{米\mathrm{一世}n}$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$哦({\phi }^2)$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$d_{米\mathrm{一世}n}>(2+\epsilon )r$, $\epsilon >0$.