Purely pulsating flow of a viscoelastic fluid in a pipe revisited: The limit of large Womersley number

Abstract

The flow of a Maxwell fluid in a pipe generated by a pulsating pressure gradient is considered. The problem is governed by two non-dimensional parameters, the Womersley number α and a Deborah parameter γ. We obtain a simple approximate solution in the limit of large α (pipe diameter much larger than the viscous penetration depth) by solving the resulting boundary-layer problem through a two scales perturbation technique, which also helps to understand qualitatively the structure of the flow as γ increases. This limit is of interest, for instance, for the blood flow in arteries. The main qualitative differences between the Newtonian fluid flow (γ=0) and the highly elastic fluid with γ=O(1) or larger are easily discussed from the found solution in terms of just two algebraic functions of γ.