We provide a mathematical analysis of the effective viscosity of suspensions of spherical particles in a Stokes flow, at low solid volume fraction $\phi$. Our objective is to go beyond the Einstein's approximation $\mu_{eff}=(1+\frac{5}{2}\phi)\mu$. Assuming a lower bound on the minimal distance between the $N$ particles, we are able to identify the $O(\phi^2)$ correction to the effective viscosity, which involves pairwise particle interactions. Applying the methodology developped over the last years on Coulomb gases, we are able to tackle the limit $N \rightarrow +\infty$ of the $O(\phi^2)$-correction, and provide explicit formula for this limit when the particles centers can be described by either periodic or stationary ergodic point processes.

中文翻译：

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超越爱因斯坦公式的稀悬浮液粘度分析

我们提供了斯托克斯流中球形颗粒悬浮液在低固体体积分数 $\phi$ 中的有效粘度的数学分析。我们的目标是超越爱因斯坦的近似值 $\mu_{eff}=(1+\frac{5}{2}\phi)\mu$。假设 $N$ 粒子之间的最小距离有一个下限，我们能够确定对有效粘度的 $O(\phi^2)$ 修正，这涉及成对粒子相互作用。应用过去几年在库仑气体上开发的方法，我们能够解决 $O(\phi^2)$ 修正的极限 $N \rightarrow +\infty$，并为该极限提供明确的公式，当粒子中心可以通过周期性或平稳遍历点过程来描述。