A comparison of macroscopic models describing the collective response of sedimenting rod-like particles in shear flows

TL;DR

This paper compares two macroscopic models derived from a kinetic model to describe sedimenting rod-like particles in shear flows, analyzing their stability and validating predictions through numerical simulations.

Contribution

It introduces and compares two new macroscopic models based on a simplified kinetic model for sedimenting rods in shear flow, including stability analysis and numerical validation.

Findings

Both macroscopic models predict cluster formation.

The diffusive and quasi-dynamic models show consistent results.

Numerical simulations confirm the models' accuracy in describing particle behavior.

Abstract

We consider a kinetic model, which describes the sedimentation of rod-like particles in dilute suspensions under the influence of gravity. This model has recently been derived by Helzel and Tzavaras in \cite{HT2015}. Here we restrict our considerations to shear flow and consider a simplified situation, where the particle orientation is restricted to the plane spanned by the direction of shear and the direction of gravity. For this simplified kinetic model we carry out a linear stability analysis and we derive two different macroscopic models which describe the formation of clusters of higher particle density. One of these macroscopic models is based on a diffusive scaling, the other one is based on a so-called quasi-dynamic approximation. Numerical computations, which compare the predictions of the macroscopic models with the kinetic model, complete our presentation.