We investigate the sedimentation of identical inertialess spherical particles in a Stokes fluid in the limit of many small particles. It is known that the presence of the particles leads to an increase of the effective viscosity of the suspension. By Einstein's formula this effect is of the order of the particle volume fraction ϕ. The disturbance of the fluid flow responsible for this increase of viscosity is very singular (like ${\left|x\right|}^{-2}$). Nevertheless, for well-prepared initial configurations and $\varphi \to 0$, we show that the microscopic dynamics is approximated to order ${\varphi }^2|\log \varphi |$ by a macroscopic coupled transport-Stokes system with an effective viscosity according to Einstein's formula. We provide quantitative estimates both for convergence of the densities in the p-Wasserstein distance for all p and for the fluid velocity in Lebesgue spaces in terms of the p-Wasserstein distance of the initial data. Our proof is based on approximations through the method of reflections and on a generalization of a classical result on convergence to mean-field limits in the infinite Wasserstein metric by Hauray.

我们研究了斯托克斯流体中相同的无惯性球形粒子在许多小粒子的限制下的沉降。已知颗粒的存在导致悬浮液的有效粘度增加。根据爱因斯坦的公式，这种效应是粒子体积分数ϕ 的数量级。造成这种粘度增加的流体流动的扰动是非常奇异的（如${\left|X\right|}^{-2}$）。然而，对于精心准备的初始配置和$\phi \to 0$，我们表明微观动力学近似于阶 ${\phi }^2|\mathrm{日志}\phi |$通过具有根据爱因斯坦公式的有效粘度的宏观耦合输运-斯托克斯系统。我们为所有p的p- Wasserstein 距离中的密度收敛和 Lebesgue 空间中根据初始数据的p- Wasserstein 距离的流体速度提供了定量估计。我们的证明是基于通过反射方法的近似和 Hauray 的无限瓦瑟斯坦度量收敛到平均场极限的经典结果的推广。