A Kinetic Model for the Sedimentation of Rod--Like Particles
KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Preprint Posting Date2015-10-12
Online Publication Date2017-03-23
Print Publication Date2017-01
AbstractWe 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 advectiondiffusion 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. For rectilinear flow we show numerical results which indicate that the quasi-dynamic provides accurate approximations. 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.
CitationHelzel C, Tzavaras AE (2017) A Kinetic Model for the Sedimentation of Rod--Like Particles. Multiscale Modeling & Simulation 15: 500–536. Available: http://dx.doi.org/10.1137/15M1023907.
AcknowledgementsThe authors express their thanks to Professor Felix Otto for his valuable comments and his involvement in several discussions concerning this work, and to Professor Jian-Guo Liu for helpful remarks.
PublisherSociety for Industrial & Applied Mathematics (SIAM)
JournalMultiscale Modeling & Simulation