论文标题
爱因斯坦有效粘度对非常小的颗粒体积分数下沉积的影响
The Influence of Einstein's Effective Viscosity on Sedimentation at Very Small Particle Volume Fraction
论文作者
论文摘要
我们研究了在许多小颗粒极限的Stokes流体中相同惯性的球形颗粒的沉积。众所周知，颗粒的存在导致悬浮液的有效粘度增加。通过爱因斯坦的公式，这种效果是粒子体积分数$ ϕ $的顺序。导致粘度增加的流体流动的干扰非常单数（例如$ | x |^{ - 2} $）。然而，对于精心准备的初始配置和$ ϕ \至0 $，我们表明微观动力学通过宏观耦合的传输stokes系统近似于$ ϕ^2 | \ log ϕ | $，根据爱因斯坦的配方，具有有效的粘度。我们提供了定量估计，以$ p $ $ p $的$ p $ wasserstein距离和勒贝格空间中的流体速度收敛，以$ p $ -wasserstein的初始数据距离。我们的证明是基于通过反射方法和Hauray无限Wasserstein指标中平均场限制的经典结果的概括。
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 $|x|^{-2}$). Nevertheless, for well-prepared initial configurations and $ϕ\to 0$, we show that the microscopic dynamics is approximated to order $ϕ^2 |\log ϕ|$ 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.
