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Steady streaming between two vibrating planes at high Reynolds numbers

We consider incompressible flows between two transversely vibrating solid walls and construct an asymptotic expansion of solutions of the Navier-Stokes equations in the limit when both the amplitude of vibrations and the thickness of the Stokes layer are small and have the same order of magnitude. Our asymptotic expansion is valid up to the flow boundary. In particular, we derive equations and boundary conditions, for the averaged flow. In the leading order, the averaged flow is described by the stationary Navier-Stokes equations with an additional term which contains the leading-order Stokes drift velocity. In a slightly different context (for a flow induced by an oscillating conservative body force), the same equations had been derived earlier by Riley (2001). The general theory is applied to two particular examples of steady streaming induced by transverse vibrations of the walls in the form of standing and travelling plane waves. In particular, in the case of waves travelling in the same direction, the induced flow is plane-parallel and the Lagrangian velocity profile can be computed analytically. This example may be viewed as an extension of the theory of peristaltic pumping to the case of high Reynolds numbers.