Drag forces of porous-medium acoustics

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Abstract
The drag forces controlling the amount of relative flow induced in a fluid-saturated porous material by a mechanical wave are modeled here from first principles. Specifically, analytical expressions for the drag are derived for material models that possess variable-width pores; i.e., pores that have widths that vary with distance along their axis. The dynamic (complex, frequency-dependent) permeability determined for such a variable-width pore model is compared to estimates made using the models of Johnson, Koplik, and Dashen (JKD) and of Biot. Both the JKD model and the Biot model underestimate the imaginary part of the dynamic permeability at low frequencies with the amount of discrepancy increasing with the severity of the convergent or divergent flow, i.e., increasing with the magnitude of the maximum pore-wall slope relative to the channel axis. It is shown how to modify the JKD model to obtain proper low-frequency behavior. It is also shown that a simple series sum of constant-width flow channels does a poor job in approximating the drag of a variable-width channel.