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Divergence-driven oscillations in a flexible-channel flow with fixed upstream flux

Published online by Cambridge University Press:  16 April 2013

Feng Xu
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
John Billingham
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Oliver E. Jensen*
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
Email address for correspondence: oliver.jensen@manchester.ac.uk

Abstract

We consider flow in a finite-length channel, one wall of which contains a membrane under longitudinal tension. The upstream flux and downstream pressure are prescribed and an external linear pressure distribution is applied to the membrane such that the system admits uniform Poiseuille flow as a steady solution. The system is described using a one-dimensional model that accounts for viscous and inertial effects. A linear stability analysis reveals that the uniform state is unstable to static (or divergent) and oscillatory instabilities. Asymptotic analysis in the neighbourhood of a Takens–Bogdanov bifurcation point shows how, when the downstream rigid section of the channel is not substantially longer than the membrane, an oscillatory mode arises through an interaction between two static eigenmodes. Perturbations to the uniform state exhibit the dynamics of a weakly dissipative Hamiltonian system for which low-frequency self-excited oscillations are forced by the divergent instability of two nearby steady solutions, before ultimately growing to large amplitudes. Simulations show that the subsequent dynamics can involve slamming motion in which the membrane briefly comes into near-contact with the opposite rigid wall over short length scales.

Type
Papers
Copyright
©2013 Cambridge University Press 

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Xu et al. supplementary movie

This movie shows (in blue) the evolution of the channel width h(x,t) along the length of the channel, corresponding to the solution shown in Figure 8 of the paper. The red curve shows the steady solution that exists for the same parameter values.

Download Xu et al. supplementary movie(Video)
Video 3.5 MB