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 \(N \rightarrow \infty \) 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.