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

We provide a rigorous derivation of Einstein's formula for the effective viscosity of dilute suspensions of n rigid balls, n >> 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(min) >= cn(-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.

Automated summary

This study rigorously derives Einstein's formula for the effective viscosity of dilute suspensions of n rigid balls. By relaxing the assumption on minimal distance between the balls, it provides a more comprehensive analysis of different scenarios.