Boundary Layers, Displacement Flow, and the High-Frequency Response of Porous Media: Beyond the Viscous and Thermal Characteristic Lengths, Order by Order
The Johnson-Koplik-Dashen viscous characteristic length $Λ$ and its thermal analogue $Λ'$ introduced by Champoux and Allard govern the leading $O(ω^{-1/2})$ boundary-layer behaviour of fluid-saturated porous media, but omit higher microgeometric effects associated with interface curvature and field gradients. We develop the asymptotic high-frequency expansions of dynamic tortuosity and dynamic compressibility beyond the leading boundary-layer terms. By resolving the oscillatory Stokes and heat conduction problems in the thin boundary-layer limit, we establish a recursive hierarchy, which can be continued to any order, defining successive microgeometric parameters, given by wall integrals. Each order requires transport equations of WKB type across the boundary layer, closed by a non-secularity condition, together with one Neumann problem for the potential displacement flow that the boundary layer induces. Applied to the values fitted on highly absorbing foams, these expressions show that the negative surface parameter is consistent with strong in-parallel disorder of the pore sizes, whereas the large positive third-order term points to strong constrictions or enlargements along part of the flow paths. The thermal parameters reduce to geometrical quantities: the integral mean curvature of the pore walls, a Willmore-type measure of their departure from umbilicity and, at the next order, this measure weighted by the mean curvature. Using the Avellaneda-Torquato relaxation-mode representation, we show that all these parameters are spectral data of the distribution of relaxation times. For isolated sharp edges across which the fluid flows, the singular terms of the formal expansion cancel, and the description persists with apparent, logarithmically varying values of the parameters.