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We consider incompressible flows between two transversely vibrating solidwalls and construct an asymptotic expansion of solutions of the Navier-Stokesequations in the limit when both the amplitude of vibrations and the thicknessof the Stokes layer are small and have the same order of magnitude. Ourasymptotic expansion is valid up to the flow boundary. In particular, we deriveequations and boundary conditions, for the averaged flow. In the leading order,the averaged flow is described by the stationary Navier-Stokes equations withan additional term which contains the leading-order Stokes drift velocity. In aslightly different context (for a flow induced by an oscillating conservativebody force), the same equations had been derived earlier by Riley (2001). Thegeneral theory is applied to two particular examples of steady streaminginduced by transverse vibrations of the walls in the form of standing andtravelling plane waves. In particular, in the case of waves travelling in thesame direction, the induced flow is plane-parallel and the Lagrangian velocityprofile can be computed analytically. This example may be viewed as anextension of the theory of peristaltic pumping to the case of high Reynoldsnumbers.

Steady streaming between two vibrating planes at high Reynolds numbers