Continuum percolation in stochastic homogenization and the effective viscosity problem

TL;DR

This paper studies the homogenization of the steady Stokes system with randomly clustered particles, introducing a new approach based on cluster size bounds and percolation theory to handle physically realistic distributions.

Contribution

It develops a novel method using cluster size bounds and percolation techniques for homogenization in particle-laden fluids, accommodating clustering unlike previous moment-based assumptions.

Findings

Established homogenization results under cluster size bounds.

Proved bounds hold for various mixing particle distributions.

Extended analysis to elasticity problems with unbounded random stiffness.

Abstract

This contribution is concerned with the effective viscosity problem, that is, the homogenization of the steady Stokes system with a random array of rigid particles, for which the main difficulty is the treatment of close particles. Standard approaches in the literature have addressed this issue by making moment assumptions on interparticle distances. Such assumptions however prevent clustering of particles, which is not compatible with physically-relevant particle distributions. In this contribution, we take a different perspective and consider moment bounds on the size of clusters of close particles. On the one hand, assuming such bounds, we construct correctors and prove homogenization (using a variational formulation and $\Gamma$-convergence to avoid delicate pressure issues). On the other hand, based on subcritical percolation techniques, these bounds are shown to hold for various mixing particle distributions with nontrivial clustering. As a by-product of the analysis, we also obtain similar homogenization results for compressible and incompressible linear elasticity with unbounded random stiffness.