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.
Categories: math.AP