Global stability analysis of flexible channel flow with a hyperelastic wall

Miguel A Herrada, Sergio Blanco-Trejo, Jens G Eggers, Peter Stewart*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

Abstract

We consider the stability of flux-driven flow through a long planar rigid channel, where a segment of one wall is replaced by a pre-tensioned hyperelastic (neo-Hookean) solid of finite thickness and subject to a uniform external pressure. We construct the steady configuration of the nonlinear system using Newton’s method with spectral collocation and high-order finite differences. In agreement with previous studies, which use an asymptotically thin wall, we show that the thick-walled system always has at least one stable steady configuration, while for large Reynolds numbers the system exhibits three co-existing steady states for a range of external pressures. Two of these steady configurations are stable to non-oscillatory perturbations, one where the flexible wall is inflated (the upper branch) and one where the flexible wall is collapsed (the lower branch), connected by an unstable intermediate branch. We test the stability of these steady configurations to oscillatory perturbations using both a global eigensolver (constructed based on an analytical domain mapping technique) and also fully nonlinear simulations. We find that both the lower and upper branches of steady solutions can become unstable to self-excited oscillations, where the oscillating wall profile has two extrema. In the absence of wall inertia, increasing wall thickness partially stabilises the onset of oscillations, but the effect remains weak until the wall thickness becomes comparable to the width of the undeformed channel. However, with finite wall inertia and a relatively thick wall, higher-frequency modes of oscillation dominate the primary global instability for large Reynolds numbers.
Original languageEnglish
Article number2101131
Pages (from-to)1-32
Number of pages32
JournalJournal of Fluid Mechanics
Volume934
Early online date18 Feb 2022
DOIs
Publication statusPublished - 10 Mar 2022

Bibliographical note

Funding Information:
M.A.H. and S.B.T. acknowledge funding from the Spanish Ministry of Economy, Industry and Competitiveness under Grants DPI2016-78887 and PID2019-108278RB and from the Junta de Andalucía under Grant P18-FR-3623. P.S.S. acknowledges funding from Engineering and Physical Sciences Research Council (UK) grants EP/P024270/1, EP/N014642/1 and EP/S030875/1.

Funding Information:
Helpful discussions with Professor J.H. Snoeijer (University of Twente) are very gratefully acknowledged. We are very grateful to Professor M. Heil (University of Manchester) for providing the data used to plot figure 3(b).

Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press.

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