High-frequency permeability of porous media with thin constrictions. I. Wedge-shaped porous media

In this series of publications, the high-frequency behavior of the dynamic permeability of porous media with thin constriction is investigated. In Part I, the classical theory of Johnson et al. ["Theory of dynamic permeability and tortuosity in fluid saturated porous media," J. Fluid Mech. 176, 379 (1987)] for soft-curved pore geometries is recalled. For wedge-shaped pore geometries, numerical computations (by finite element method) and analysis by Cortis et al. ["Influence of pore roughness on high-frequency permeability," Phys. Fluids 15, 1766 (2003)] are revisited and confirmed, while leading to important new conclusions. Because the electric field is singular at the tip of wedges, the original model developed by Johnson et al., which links the viscous fluid flow problem to the electrical conduction problem, is inappropriate for describing the high-frequency behavior of the viscous fluid flow through wedge-shaped porous media. In particular, in the case of small wedge angles, we show that the real part of the dynamic permeability behaves in the high-frequency regime as R(k(omega)) alpha omega(-(3/2))(ln (omega) + constant), which differs from the predictions of the Johnson et al. model [R(k(omega)) alpha omega(-(3/2))]. However, our results show that the classical Johnson et al. high frequency limit can be a good approximation of the viscous fluid flow if the electrical conduction problem is solved over a fluid domain truncated by a boundary layer having a thickness comparable to the viscous skin depth. In Part II, we consider foam with perforated membranes involving different microstructural characteristic lengths: pore size, membrane aperture size, and membrane thickness. We assess the validity domain of the Johnson et al. approximation and test our modified high-frequency approximation for such porous materials.