Peristaltic pumping in water waves

In this paper we calculate the streaming induced by gravity waves passing over a thin fluid layer, one side of which is rigid while the other is a flexible, inextensible membrane. The problem is relevant to some recent laboratory experiments by Allison (1983) on the pumping action of water waves. On the assumption that the flow is laminar and that the lateral displacement b of the membrane is small compared with the thickness Δ of the fluid layer, we calculate the velocity profile of the streaming U within the layer. This depends on the ratio Δ/δ, where δ is the thickness of the Stokes layers at the upper and lower boundaries. When 0 < Δ/δ > 6 the boundary layers interact strongly and the velocity profile is unimodal. At large values of Δ/δ the profile of U exhibits thin ‘jets’ near the boundaries. The calculated drift velocities agree as regards order of magnitude with those observed. However, the pressure gradients observed were larger than those calculated, due possibly to turbulence, but probably also to finite-amplitude and end-effects. The theory given here can be considered as an extension of the theory of peristaltic pumping to flows at higher Reynolds number.