Steady streaming in channels with a porous interior

Hall, in his Ph.D. thesis (University of London, 1973), demonstrated that the viscous flow arising in a slender channel of slowly varying cross section subject to a purely oscillatory pressure difference between its open ends contains a steady streaming component. This study extends the analysis to channels with a porous interior. Using a homogenized flow model that incorporates a Darcy resistance term proportional to the local velocity, a closed-form solution for the streaming motion is derived in the asymptotic limit of small stroke-to-channel length ratios <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn></mml:math> . Consistent with Hall's findings, a net flow rate is seen to arise only when the channel ends have unequal widths. The presence of a porous medium significantly attenuates the streaming-flow magnitude, with the differences between porous and nonporous channels becoming more pronounced for large Womersley numbers, a limiting case considered separately. In nonporous channels with unequal end widths the streaming motion exhibits large velocities that are comparable in magnitude to those of the oscillatory flow, whereas in porous channels the streaming velocities remain a factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math> smaller. Channels with equal end widths feature streaming recirculating vortices within near-wall Stokes boundary layers. Nonporous channels display additional recirculating vortices in the central core region, which are absent in porous configurations. These results provide insights into transport processes in biomedical and technological applications involving oscillatory flow in wall-bounded porous media.