Variational bounds on the diffusive and hydrodynamic permeabilities of randomly perforated sheets

Abstract

We define steady state permeabilities for a randomly perforated sheet of zero thickness with respect to both diffusive and hydrodynamic flow, and derive upper and lower variational bounds on them. For sparsely perforated sheets the bounds are sufficiently close to establish rigorously the existence of a logarithmic term in the expansion of the diffusive permeability in the perforation density; the hydrodynamic permeability shows no such behavior at low perforation densities. This contrast is one consequence of a quite general relationship between the diffusive permeability corresponding to a given perforation geometry and the hydrodynamic permeability for the complementary geometry generated by interchanging closed and open areas: logarithmic terms appear in the hydrodynamic permeability of a membrane constructed of randomly overlapping impermeable disks at the high porosity limit.

Variational bounds on the diffusive permeability can also be obtained if the perforation geometry is subject to temporal fluctuations, such as might be generated by allowing perforations to perform independent Brownian motions of their own in the plane of the membrane. Brownian motion of penetration sites can considerably enhance the diffusive permeability over what would have been obtained for stationary sites present at the same density.