Sedimentation of particles with very small inertia I: Convergence to the transport-Stokes equation

We consider the sedimentation of \(N\) spherical particles with identical radii \(R\) in a Stokes flow in \(\mathbb R^3\). The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particl...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-06
Main Authors: Höfer, Richard M, Schubert, Richard
Format: Article
Language:English
Subjects:
Boundary conditions
Coercivity
Fluid flow
Navier-Stokes equations
Newton Theory
Particle inertia
Representations
Sedimentation
Sedimentation & deposition
Stokes flow
Stokes law (fluid mechanics)
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We consider the sedimentation of \(N\) spherical particles with identical radii \(R\) in a Stokes flow in \(\mathbb R^3\). The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particle inertia as \(N\) tends to infinity and \(R\) to \(0\). In a mean-field scaling, we show that the particle evolution is well approximated by the transport-Stokes system which has been derived previously as the mean-field limit of inertialess particles. In particular this justifies to neglect the particle inertia in the microscopic system, which is a typical modelling assumption in this and related contexts. The proof is based on a relative energy argument that exploits the coercivity of the particle forces with respect to the particle velocities in a Stokes flow. We combine this with an adaptation of Hauray's method for mean-field limits to \(2\)-Wasserstein distances.Moreover, in order to control the minimal distance between particles, we prove a representation of the particle forces. This representation makes the heuristic \enquote{Stokes law} rigorous that the force on each particle is proportional to the difference of the velocity of the individual particle and the mean-field fluid velocity generated by the other particles.
ISSN:2331-8422