Paper detail

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

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.