On the Steady Streaming Induced by Vibrating Walls

Abstract

We consider incompressible viscous 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 the vibration and the thickness of the oscillatory boundary layers (the Stokes layers) 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. At 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. The same equations had been derived by Craik and Leibovich in 1976 when they proposed their model of Langmuir circulations in the ocean [A. D. D. Craik and S. Leibovich, J. Fluid Mech., 73 (1976), pp. 401--426]. In the context of flows induced by an oscillating conservative body force, these equations had also been derived by Riley [Ann. Rev. Fluid Mech., 33 (2001), pp. 43--65]. 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 traveling plane waves. In particular, in the case of waves traveling 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.