Semi-dilute rheology of particle suspensions: derivation of Doi-type models

TL;DR

This paper derives a semi-dilute mean-field model for the rheology of inertialess particle suspensions in Stokes flow, capturing non-Newtonian effects through a coupled macroscopic-microscopic framework.

Contribution

It introduces a Doi-type model derived from hydrodynamic principles, incorporating multibody interactions and non-Newtonian viscosity effects in dilute suspensions.

Findings

Derived a coupled Stokes-Vlasov model for particle suspensions.

Accounted for non-Newtonian viscosity via particle orientation distribution.

Analyzed multibody hydrodynamic interactions using cluster and multipole expansions.

Abstract

This work is devoted to the large-scale rheology of suspensions of non-Brownian inertialess rigid particles, possibly self-propelling, suspended in Stokes flow. Starting from a hydrodynamic model, we derive a semi-dilute mean-field description in form of a Doi-type model, which is given by a 'macroscopic' effective Stokes equation coupled with a 'microscopic' Vlasov equation for the statistical distribution of particle positions and orientations. This accounts for some non-Newtonian effects since the viscosity in the effective Stokes equation depends on the local distribution of particle orientations via Einstein's formula. The main difficulty is the detailed analysis of multibody hydrodynamic interactions between the particles, which we perform by means of a cluster expansion combined with a multipole expansion in a suitable dilute regime.