Acoustic-drift equation

Abstract

The aim of this paper is to derive a new equation (the \emph{acoustic-drift equation} (ADE)) describing the generation of a flow by an acoustic wave. We consider acoustic waves of perfect barotropic gas as the zero-order solution and derive the equation for the averaged flow of the first order. The used small parameter of our asymptotic study is dimensionless inverse frequency, and the leading term for a velocity field is chosen to be a purely oscillating acoustic field. The employed mathematical approach combines the two-timing method and the notion of a distinguished limit. The properties of commutators are used to simplify calculations. The derived averaged equation is similar to the original vorticity equation, where the Reynolds stresses has been transformed to an additional advection with the drift velocity. Hence ADE can be seen as a compressible version of the Craik-Leibovich equation. In particular, ADE shows that if the averaged vorticity is absent in the initial data, then it will always be equal to zero. This property confirms that the acoustic streaming is of viscous nature. At the same time ADE can be useful for the description of secondary motions which take place for `strong' acoustic streaming flow.