On the motion of a large number of small rigid bodies in a viscous incompressible fluid

Abstract

We consider the motion of $N$ rigid bodies -- compact sets $({\mathcal{S}}_{\epsilon }^1,\cdots ,{\mathcal{S}}_{\epsilon }^N{)}_{\epsilon >0}$ -- immersed in a viscous incompressible fluid contained in a domain in the Euclidean space ${\mathbb{R}}^d$, $d=2,3$. We show the fluid flow is not influenced by the presence of the infinitely many bodies in the asymptotic limit $\epsilon \to 0$ and $N=N(\epsilon )\to \infty$ as soon as \[ {\rm diam}[\mathcal{S}^i_\varepsilon ] \to 0 \ \mbox{as}\ \varepsilon \to 0 ,\ i=1,\cdots, N(\varepsilon). \] The result depends solely on the geometry of the bodies and is independent of their mass densities. Collisions are allowed and the initial data are arbitrary with finite energy.

Cited by in corpus (1)

• Interaction of liquid crystals with a rigid body