European Journal of Applied Mathematics

Table of Contents - June 2012 - Volume 23, Issue 03  

Papers

The decay of a swirling perturbation of pipe Poiseuille flow

S. A. SHEPHERDa1*

a1 School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK email: P.Hydon@surrey.ac.uk

Abstract

Secondary flows consisting of two pairs of vortices arise when two fluid streams meet at a confluence, such as in the airways of the human lung during expiration or at the vertebrobasilar junction in the circulatory system, where the left and right vertebral arteries converge. In this paper the decay of these secondary flows is studied by considering a four-vortex perturbation from Poiseuille flow in a straight, three-dimensional pipe. A polynomial eigenvalue problem is formulated and the exact solution for the zero Reynolds number R is derived analytically. This solution is then extended by perturbation analysis to produce an approximation to the eigenvalues for R ≪ 1. The problem is also solved numerically for 0 ≤ R ≤ 2,000 by a spectral method, and the stability of the computed eigenvalues is analysed using pseudospectra. For all Reynolds numbers, the decay rate of the swirling perturbation is found to be governed by complex eigenvalues, with the secondary flows decaying more slowly as R increases. A comparison with results from an existing computational study of merging flows shows that the two models give rise to similar secondary flow decay rates.

(Received May 16 2010)

(Revised August 17 2011)

(Accepted November 28 2011)

(Online publication January 12 2012)

Key words:

  • Pipe flow;
  • perturbations;
  • numerical eigenvalue problems

Footnotes

* Sadly, Sarah Shepherd died in September 2011. In her short life, she accomplished much, including innovative contributions to the mathematics community through her magazine iSQUARED and her podcasts. This paper is the outcome of research that Sarah carried out as an undergraduate.