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

We show that the collective effect of N rigid bodies (Sn,N)ₙ₌₁N of diameters (rn,N)ₙ₌₁N immersed in an incompressible non--Newtonian fluid is negligible in the asymptotic limit N → ∞ as long as their total packing volume ∑ₙ₌₁N rn,Nᵈ, $d=2,3$ tends to zero exponentially -- ∑ₙ₌₁N rn,Nᵈ ≈ 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.