Derivation of the viscoelastic stress in Stokes flows induced by non-spherical Brownian rigid particles through homogenization

TL;DR

This paper rigorously derives a classical viscoelastic stress formula for suspensions of non-spherical Brownian particles in Stokes flow using homogenization, accounting for different Deborah number regimes.

Contribution

It provides a rigorous derivation of the viscoelastic stress in Doi models from microscopic particle dynamics, including the effects of particle shape and noise.

Findings

Derived the viscoelastic stress formula in the homogenization limit.

Analyzed the impact of Deborah number regimes on the stress.

Developed a PDE framework with Stratonovitch noise for particle orientations.

Abstract

We consider a microscopic model of $n$ identical axis-symmetric rigid Brownian particles suspended in a Stokes flow. We rigorously derive in the homogenization limit of many small particles a classical formula for the viscoelastic stress that appears in so-called Doi models which couple a Fokker-Planck equation to the Stokes equations. We consider both Deborah numbers of order $1$ and very small Deborah numbers. Our microscopic model contains several simplifications, most importantly, we neglect the time evolution of the particle centers as well as hydrodynamic interaction for the evolution of the particle orientations. The microscopic fluid velocity is modeled by the Stokes equations with given torques at the particles in terms of Stratonovitch noise. We give a meaning to this PDE in terms of an infinite dimensional Stratonovitch integral. This requires the analysis of the shape derivatives of the Stokes equations in perforated domains, which we accomplish by the method of reflections.