论文标题
关于爱因斯坦公式的有效粘度公式的校正
On the correction to Einstein's formula for the effective viscosity
论文作者
论文摘要
本文是对粘性稀释悬浮液的准确有效模型的推导的条款[6]的随访。目的是确定一个有效的Stokes方程，该方程提供了确切的流体粒子系统的$ O（λ^2）$近似，并具有$λ$的粒子固体体积分数。这意味着我们正在寻找以$μ_{eff}（x）=μ+ \ frac {5} {2} {2}μρ（x）λ+μ_2（x）λ^2 $的形式改善爱因斯坦公式的有效粘度。在颗粒上的分离假设下，我们在[6]中证明，如果存在$ O（λ）^2 $ stokes有效近似，则校正$μ_2$必须由平均场限制给出，然后可以在粒子配置的进一步假设下研究并计算出来。粗略地，我们从[6]的条件结果到无条件的结果：我们表明，一旦存在平均场限制，就确实存在这样的$ O（λ^2）$ stokes $。这包括周期性和随机固定粒子配置的情况。
This paper is a follow-up of article [6], on the derivation of accurate effective models for viscous dilute suspensions. The goal is to identify an effective Stokes equation providing a $o(λ^2)$ approximation of the exact fluid-particle system, with $λ$ the solid volume fraction of the particles. This means that we look for an improvement of Einstein's formula for the effective viscosity in the form $μ_{eff}(x) = μ+ \frac{5}{2} μρ(x) λ+ μ_2(x) λ^2$. Under a separation assumption on the particles, we proved in [6] that if a $o(λ)^2$ Stokes effective approximation exists, the correction $μ_2$ is necessarily given by a mean field limit, that can then be studied and computed under further assumptions on the particle configurations. Roughly, we go here from the conditional result of [6] to an unconditional result: we show that such a $o(λ^2)$ Stokes approximation indeed exists, as soon as the mean field limit exists. This includes the case of periodic and random stationary particle configurations.
