Mild assumptions for the derivation of Einstein’s effective viscosity formula

Abstract

We provide a rigorous derivation of Einstein’s formula for the effective viscosity of dilute suspensions of n rigid balls, $n\gg 1,$ set in a volume of size 1. So far, most justifications were carried under a strong assumption on the minimal distance between the balls: $d_{\mathit{min}}\ge c{n}^{-\frac{1}{3}},$ c > 0. We relax this assumption into a set of two much weaker conditions: one expresses essentially that the balls do not overlap, while the other one gives a control of the number of balls that are close to one another. In particular, our analysis covers the case of suspensions modeled by standard Poisson processes with almost minimal hardcore condition.

Keywords:

• Effective viscosity
• homogenization
• Stokes equations
• suspensions

Notes

1 In detail: let ${\mathcal{E}}_{\eta }$ be the operator that erases all points without a neighboring point closer than η, and let T_x denote a translation by x. Now, let μ be the measure for the original process ${\Phi }^{\delta }.$ Then the measure for ${\Phi }_{\eta }^{\delta }$ is given by ${\mu }_{\eta }=\mu °{\mathcal{E}}_{\eta }^{-1}.$ Since ${\mathcal{E}}_{\eta }{T}_x=T_x{\mathcal{E}}_{\eta }$ (for all x, in particular for $T_{-x}=T_x^{-1}$), we have for any measurable set A that $T_x{\mathcal{E}}_{\eta }^{-1}A={\mathcal{E}}_{\eta }^{-1}{T}_{x}A.$ This immediately implies that the new process adopts stationarity and ergodicity.

Additional information

Funding

The first author acknowledges the support of the Institut Universitaire de France, and of the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR). The second author has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research FOUNDATION) through the collaborative research center “The Mathematics of Emerging Effects” (CRC 1060, Projekt-ID 211504053) and the Hausdorff Center for Mathematics (GZ 2047/1, Projekt-ID 390685813).