On the correction to Einstein’s formula for the effective viscosity

Abstract

This paper is a follow-up to Gérard-Varet and Hillairet (2020) on the derivation of accurate effective models for viscous dilute suspensions. The goal is to identify an effective Stokes equation providing an $o({\lambda }^2)$ approximation of the exact fluid–particle system, with $\lambda$ 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 ${\mu }_{\mathrm{eff}}(x)=\mu +\frac{5}{2}\mu \rho (x)\lambda +{\mu }_2(x){\lambda }^2$. Under a separation assumption on the particles, we proved in the article above that if an $o({\lambda }^2)$ Stokes effective approximation exists, the correction ${\mu }_2$ is necessarily given by a mean field limit, which can then be studied and computed under further assumptions on the particle configurations. Roughly, we go here from the conditional result of the article above to an unconditional result: we show that such an $o({\lambda }^2)$ Stokes approximation indeed exists, as soon as the mean field limit exists. This includes the case of periodic and random stationary particle configurations.