# Analysis of the viscosity of dilute suspensions beyond Einstein's formula

## TL;DR

This paper mathematically refines Einstein's formula for suspension viscosity by deriving the second-order correction involving particle interactions, applicable to large systems with specific spatial distributions.

## Contribution

It introduces a rigorous derivation of the second-order correction to Einstein's viscosity formula for dilute suspensions, including the limit as the number of particles grows large.

## Key findings

- Derived explicit formulas for the $O(\,\phi^2)$ correction to viscosity.
- Extended analysis to infinite particle systems with periodic or ergodic distributions.
- Validated the correction's applicability beyond classical Einstein approximation.

## Abstract

We provide a mathematical analysis of the effective viscosity of suspensions of spherical particles in a Stokes flow, at low solid volume fraction $\phi$. Our objective is to go beyond the Einstein's approximation $\mu_{eff}=(1+\frac{5}{2}\phi)\mu$. Assuming a lower bound on the minimal distance between the $N$ particles, we are able to identify the $O(\phi^2)$ correction to the effective viscosity, which involves pairwise particle interactions. Applying the methodology developped over the last years on Coulomb gases, we are able to tackle the limit $N \rightarrow +\infty$ of the $O(\phi^2)$-correction, and provide explicit formula for this limit when the particles centers can be described by either periodic or stationary ergodic point processes.