Non-existence of mean-field models for particle orientations in suspensions

TL;DR

This paper demonstrates that for a suspension of inertialess spherical particles in a Stokes flow, a mean-field model for particle orientations does not exist due to the singular perturbation effects, contrasting with common models like FENE and Doi.

Contribution

It proves the non-existence of mean-field models for particle orientations in certain suspensions, highlighting limitations of existing models in non-cubic geometries.

Findings

No valid mean-field limit for particle orientation behavior.

Single particle mobility problem shows coupling between angular velocity and strain.

Contrasts with established models like FENE and Doi.

Abstract

We consider a suspension of spherical inertialess particles in a Stokes flow on the torus $\mathbb T^3$. The particles perturb a linear extensional flow due to their rigidity constraint. Due to the singular nature of this perturbation, no mean-field limit for the behavior of the particle orientation can be valid. This contrasts with widely used models in the literature such as the FENE and Doi models and similar models for active suspensions. The proof of this result is based on the study of the mobility problem of a single particle in a non-cubic torus, which we prove to exhibit a nontrivial coupling between the angular velocity and a prescribed strain.