  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.
