In his PhD thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a random suspension of small rigid particles at low density. This formal derivation is based on two assumptions: (i) there is a scale separation between the size of particles and the observation scale, and (ii) particles do not interact with one another at first order. While the first assumption was addressed in a companion work in terms of homogenization theory, the second one is reputedly more subtle due to the long-range character of hydrodynamic interactions. In the present contribution, we provide a rigorous justification of Einsteins first-order expansion at low density in the most general setting. This is pursued to higher orders in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations, and we justify in particular a celebrated result by Batchelor and Green on the next-order correction. In addition, we address the summability of the cluster expansion in two specific settings (random deletion and geometric dilation of a fixed point set). Our approach relies on an intricate combination of combinatorial arguments, PDE analysis, and probability theory.
Download