High-Reynolds-number viscous flow in collapsible tubes

Abstract

This study is concerned with steady laminar high-Reynolds-number flow in collapsible tubes, where the position of the tube wall is a function (the tube law) only of the pressure exerted by the fluid on the wall. The system is controlled by two main parameters: the Reynolds number of the incoming flow, and the ‘compliance’, which characterizes the response of the wall to a change in fluid pressure. Restrictions are placed on these parameters so that the streamwise lengthscale is large, and, to the order worked, the pressure is uniform across the tube. Attention is restricted to (axi)symmetric systems.Channels are considered in most detail, the results for axisymmetric pipes being largely similar.For a model tube law the flow in the converging section of a channel is investigated in detail. Solutions are presented for certain of the parameter values. For some channels a singularity is found in the solution such that the channel width tends to zero at a finite distance downstream. No way was found to integrate past this singularity.For particular channels and pipes, solutions are found only for flows in which the mean fluid velocity is less than the propagation speed of frictionless waves. This is consistent with experimental results.