Frequency-dependent viscous flow in channels with fractal rough surfaces

The viscous dynamic permeability of some fractal-like channels is studied. For our particular class of geometries, the ratio of the pore surface area-to-volume tends to ∞ (but has a finite cutoff), and the universal scaling of the dynamic permeability, k(ω), needs modification. We performed accurate numerical computations of k(ω) for channels characterized by deterministic fractal wall surfaces, for a broad range of fractal dimensions. The pertinent scaling model for k(ω) introduces explicitly the fractal dimension of the wall surface for a range of frequencies across the transition between viscous and inertia dominated regimes. The new model provides excellent agreement with our numerical simulations.