On the effect of a large cloud of rigid particles on the motion of an incompressible non--Newtonian fluid

Abstract

We show that the collective effect of $N$ rigid bodies $({\mathcal{S}}_{n,N}{)}_{n=1}^N$ of diameters $(r_{n,N}{)}_{n=1}^N$ immersed in an incompressible non--Newtonian fluid is negligible in the asymptotic limit $N\to \infty$ as long as their total packing volume ${\sum }_{n=1}^N{r}_{n,N}^d$, $d=2,3$ tends to zero exponentially -- ${\sum }_{n=1}^N{r}_{n,N}^d\approx A^{-N}$ -- for a certain constant $A>1$. The result is rather surprising and in a sharp contrast with the associated homogenization problem, where the same number of obstacles can completely stop the fluid motion in the case of shear thickening viscosity. A large class of non--Newtonian fluids is included, for which the viscous stress is a subdifferential of a convex potential.