Laminar flow in a porous channel with large wall suction and a weakly oscillatory pressure

The laminar oscillatory flow inside a rectangular channel with wall suction is considered here. The scope is limited to large suction imposed uniformly along the permeable walls. Inside the channel, the onset of small amplitude pressure disturbances produces an oscillatory field that we wish to investigate. Based on the normalized pressure-wave amplitude, the conservation equations are linearized and split into leading-order (steady) and first-order (time-dependent) equations. The first-order set is subdivided into an acoustic, pressure-driven, wave equation, and a vortical, boundary-driven, viscous equation. For longitudinal pressure oscillations, both equations are written to the order of the wall suction Mach number. The resulting equations are then solved in an exact fashion. The novelty lies in the vortical response that reduces to a Weber equation following a Liouville–Green transformation. The emerging rotational solution is expressible in terms of confluent hypergeometric functions of the suction Reynolds number, Strouhal number, and spatial coordinates. The total solution is then constructed and found to coincide with the numerical solution of the linearized momentum equation. The oscillatory velocity exhibits similar characteristics to the exact Stokes profile for oscillations inside a long channel with hard walls. In particular, a thin rotational layer is observed in addition to the small velocity overshoot near the wall. Both depth and overshoot are nowhere near their values obtained by switching from mass extraction to mass addition. In contrast to former studies involving injection, the so-called acoustic boundary layer is found to depreciate when suction is increased or when viscosity is reduced. This response is similar to that of the Stokes layer over hard walls. Overall, the effect of increasing frequency is that of compressing the rotational layer near the wall.