Exactly realizable bounds on the trapping constant and permeability of porous media

Sandstone, granular media, bone, wood, and cell membranes are just a few examples of porous media that abound in Nature and in synthetic situations. Two important effective properties of fluid-saturated porous media that have been extensively studied are the trapping constant γ and scalar fluid permeability k. Exact expressions for the "void" bounds on γ and k for coated-spheres and coated-cylinders models of porous media are derived. In certain instances, the bounds are shown to be optimal, i.e., the void bounds coincide with the corresponding exact solutions of γ and k for these coated-inclusions models. In the optimal cases, we obtain exact expressions for the relevant length scale that arises in the void bounds, which depends on a two-point correlation function that characterizes the porous medium. By contrast, optimal bounds on the effective conductivity and elastic moduli of composite media have long been known.