Boundary streaming with Navier boundary condition

Abstract

In microfluidic applications involving high-frequency acoustic waves over a solid boundary, the Stokes boundary-layer thickness $\delta$ is so small that some non-negligible slip may occur at the fluid-solid interface. This paper assesses the impact of the slip by revisiting the classical problem of steady acoustic streaming over a flat boundary with the Navier boundary condition $u|_{y=0} = L_\mathrm{s} \partial_y u|_{y=0}$, where $u$ is the velocity tangent to the boundary $y=0$, and the parameter $L_\mathrm{s}$ is the slip length. The limit outside the boundary layer provide an effective slip velocity. A general expression is obtained for the streaming velocity outside the boundary layer as a function of the dimensionless parameter $L_\mathrm{s}/\delta$. Particularising to travelling and standing waves shows that the boundary slip respectively increases and decreases the streaming velocity.