On the derivation of a Stokes-Brinkman problem from Stokes equations around a random array of moving spheres

TL;DR

This paper derives a Stokes-Brinkman model as an asymptotic limit of the Stokes equations around a large number of randomly distributed small moving spheres, providing insights into fluid behavior in complex particulate systems.

Contribution

It establishes the convergence of the Stokes system with many small moving spheres to a Stokes-Brinkman problem under random distribution assumptions.

Findings

Convergence of the fluid flow to a Stokes-Brinkman model as the number of spheres increases.

Validation of the model under the assumption of chaotic distribution of spheres.

Quantitative description of the asymptotic behavior of solutions.

Abstract

We consider the Stokes system in $\mathbb R^3,$ deprived of $N$ spheres of radius $1/N,$ completed by constant boundary conditions on the spheres. This problem models the instantaneous response of a viscous fluid to an immersed cloud of moving solid spheres. We assume that the centers of the spheres and the boundary conditions are given randomly and we compute the asymptotic behavior of solutions when the parameter $N$ diverges. Under the assumption that the distribution of spheres/centers is chaotic, we prove convergence in mean to the solution of a Stokes-Brinkman problem.