Corrector equations in fluid mechanics: Effective viscosity of colloidal suspensions

TL;DR

This paper rigorously defines an effective viscosity for colloidal suspensions in steady Stokes flow using homogenization and corrector equations, extending Einstein's dilute regime approximation to more general particle distributions.

Contribution

It introduces a novel homogenization framework and corrector equations to define effective viscosity beyond dilute regimes in colloidal suspensions.

Findings

Established a homogenization result for random particle distributions.

Developed and analyzed corrector equations for effective viscosity.

Extended Einstein's approximation to non-dilute, ergodic particle configurations.

Abstract

Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations.