New bounds on the permeability of a random array of spheres

Abstract

The recently derived variational principle of Rubinstein and Torquato (submitted to J. Fluid Mech.) is applied to obtain new rigorous two- and three-point upper bounds on the fluid permeability k for slow viscous flow around a random array of identical spheres which may penetrate one another in varying degrees. The n-point bounds involve up to n-point correlation function information. Both bounds are simplified and computed for the special case of mutually impenetrable spheres for a wide range of sphere volume fractions. The three-point bound is sharp and provides significant improvement over the two-point bound, especially at high sphere volume fractions (low porosities). It is the sharpest upper bound on k for a random array of impenetrable spheres developed to date and begins to approach the Kozeny–Carman empirical relation at low porosities.