A kinetic model for the sedimentation of rod-like particles

TL;DR

This paper develops a kinetic model for sedimenting rod-like particles, analyzing cluster formation through a coupled kinetic-macroscopic system, deriving effective equations, and validating them with numerical simulations and stability analysis.

Contribution

It introduces a nonlinear moment closure model and an effective advection-diffusion equation for sedimenting rods, linking microscopic alignment to macroscopic clustering.

Findings

Cluster formation is driven by buoyancy-induced velocity gradients.

The effective equation accurately predicts cluster behavior in shear flows.

Linear stability analysis reveals a wavelength selection mechanism for clusters.

Abstract

We consider a coupled system consisting of a kinetic equation coupled to a macroscopic Stokes (or Navier-Stokes) equation and describing the motion of a suspension of rigid rods in gravity. A reciprocal coupling leads to the formation of clusters: The buoyancy force creates a macroscopic velocity gradient that causes the microscopic particles to align so that their sedimentation reinforces the formation of clusters of higher particle density. We provide a quantitative analysis of cluster formation. We derive a nonlinear moment closure model, which consists of evolution equations for the density and second order moments and that uses the structure of spherical harmonics to suggest a closure strategy. For a rectilinear flow we employ the moment closure together with a quasi-dynamic approximation to derive an effective equation, The effective equation is an advection-diffusion equation with nonisotropic diffusion coupled to a Poisson equation, and belongs to the class of the so-called flux-limited Keller-Segel models. For shear flows, we provide an argument for the validity of the effective equation and perform numerical comparisons that indicate good agreement between the original system and the effective theory. Finally, a linear stability analysis on the moment system shows that linear theory predicts a wavelength selection mechanism for the cluster width, provided that the Reynolds number is larger than zero.