# A local version of Einstein's formula for the effective viscosity of suspensions

## TL;DR

This paper rigorously proves a local version of Einstein's effective viscosity formula for dilute suspensions at the level of the Stokes equation, considering a large number of particles with vanishing size and volume fraction.

## Contribution

It provides the first proof of Einstein's formula at the Stokes equation level for dilute suspensions with many particles, using a dipole approximation and homogenization techniques.

## Key findings

- Established $L^$ and $L^p$ estimates for microscopic and homogenized solutions.
- Validated Einstein's formula in a local setting for suspensions with many particles.
- Demonstrated the effectiveness of the dipole approximation and reflection method.

## Abstract

We prove a local variant of Einstein's formula for the effective viscosity of dilute suspensions, that is $\mu^\prime=\mu (1+\frac 5 2\phi+o(\phi))$, where $\phi$ is the volume fraction of the suspended particles. Up to now rigorous justifications have only been obtained for dissipation functionals of the flow field. We prove that the formula holds on the level of the Stokes equation (with variable viscosity). We consider a regime where the number $N$ of particles suspended in the fluid goes to infinity while their size $R$ and the volume fraction $\phi=NR^3$ approach zero. We establish $L^\infty$ and $L^p$ estimates for the difference of the microscopic solution to the solution of the homogenized equation. Here we assume that the particles are contained in a bounded region and are well separated in the sense that the minimal distance is comparable to the average one. The main tools for the proof are a dipole approximation of the flow field of the suspension together with the so-called method of reflections and a coarse graining of the volume density.