Lagrangian transport induced by peristaltic pumping in a tube

Abstract

This is an analytical study, based on equations of motion in Lagrangian form, for the steady Lagrangian fluid transport induced by long peristaltic waves (which can be progressive, purely or partially standing) of small amplitude traveling on the boundary of a flexible tube. The first-order oscillatory viscous flow and the higher-order time-mean Lagrangian drifts (or steady streaming) are obtained as functions of the wave properties. Two cases are considered. Firstly, the wave frequency is slow such that the steady-streaming Reynolds number (Res) is very small and the viscous diffusion is significant across the entire flow region. The time-mean flow can be found in the second-order problem. Secondly, high-frequency pumping is considered such that Res = O(1). Under this condition, the flow domain is divided into a thin Stokes boundary layer near the wall and the inviscid core region. The steady streaming in the core region is to be found in the fourth-order problem. Based on the Lagrangian coordinates, all the solutions are analytically expressed. Results are generated to illustrate the effects of wave properties on the Lagrangian transport. The phenomenon of reflux, a backward time-mean flow, is examined in particular.