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

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 $N=N(\epsilon )\to \mathrm{\infty }$ as soon as $$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}[{\mathcal{S}}_{\epsilon }^i]\to 0as\epsilon \to 0,i=1,\cdots ,N(\epsilon )\mathrm{.}$$ 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.