# Effective viscosity of a polydispersed suspension

## TL;DR

This paper derives the first order correction to the effective viscosity of a suspension with arbitrarily shaped particles, using homogenization and an extended reflection method, applicable even as particle number increases.

## Contribution

It introduces a novel homogenization approach and extends the method of reflections to handle suspensions with many particles of arbitrary shapes, providing convergence results.

## Key findings

- Derived the first order correction to effective viscosity.
- Extended the method of reflections for multiple particles.
- Proved convergence for small volume fractions regardless of particle count.

## Abstract

We compute the first order correction of the effective viscosity for a suspension containing solid particles with arbitrary shapes. We rewrite the computation as an homogenization problem for the Stokes equations in a perforated domain. Then, we extend the method of reflections to approximate the solution to the Stokes problem with a fixed number of particles. By obtaining sharp estimates, we are able to prove that this method converges for small volume fraction of the solid phase whatever the number of particles. This allows to address the limit when the number of particles diverges while their radius tends to 0. We obtain a system of PDEs similar to the Stokes system with a supplementary term in the viscosity proportional to the volume fraction of the solid phase in the mixture.