Bounds on the permeability of a random array of partially penetrable spheres

Abstract

A bound on the fluid permeability k for viscous flow through a random array of N identical spherical particles (distributed with arbitrary degree of impenetrability) due to Weissberg and Prager [Phys. Fluids 1 3, 2958 (1970)] is rederived by averaging flow quantities with respect to the ensemble of particle configurations. The ensemble‐averaging technique enables one to obtain series representations of certain n‐point distribution functions that arise in terms of probability density functions which characterize a particular configuration of n(<N) spheres. The Weissberg–Prager bound on k is computed exactly through second order in the sphere volume fraction for arbitrary λ (where λ is the impenetrability parameter, 0≤λ≤1) for two different interpenetrable‐sphere models. It is found that, at the same sphere volume fraction, the permeability of an assembly of partially overlapping spheres is greater than that of one characterized by a higher degree of impenetrability. The results of this study indicate that the Weissberg–Prager bound will not only yield the best available bound on k for an assemblage of totally impenetrable spheres (λ=1) but will provide a useful estimate of k for this model for a wide range of sphere volume fractions. It is also demonstrated that bounds which incorporate a certain level of statistical information on the medium are not always necessarily sharper than bounds which involve less information.