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

TL;DR

This paper proves that particles with very small inertia in a sedimentation process can be accurately modeled by the transport-Stokes equation, justifying the neglect of inertia in such systems through a rigorous mean-field limit analysis.

Contribution

It provides a rigorous mathematical justification for neglecting particle inertia in sedimentation models by deriving the transport-Stokes system as a mean-field limit for particles with small inertia.

Findings

The particle dynamics are well approximated by the transport-Stokes system.

The force on each particle is proportional to the velocity difference with the mean fluid velocity.

The analysis confirms the validity of neglecting inertia in the microscopic model.

Abstract

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.