Optimal Bounds on the Trapping Constant and Permeability of Porous Media

We derive exact expressions for so-called ``void'' bounds on the trapping constant $\gamma$ and fluid permeability $k$ for coated-spheres and coated-cylinders models of porous media. We find that in some cases the bounds are optimal, i.e., the void bounds coincide with the corresponding exact solutions of $\gamma$ and $k$ for these coated-inclusions models. In these instances, exact expressions are obtained for the relevant length scale that arises in the void bounds, which depends on a two-point correlation function that characterizes the porous medium. In contrast to bounds on the effective conductivity and elastic moduli of composite media, this is the first time that model microstructures have been found that exactly realize bounds on either the trapping constant or fluid permeability.