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Stability of falling liquid films on flexible substrates

Published online by Cambridge University Press:  13 August 2020

J. Paul Alexander Open the ORCID record for J. Paul Alexander [Opens in a new window] ,
Toby L. Kirk Open the ORCID record for Toby L. Kirk [Opens in a new window]  and
Demetrios T. Papageorgiou Open the ORCID record for Demetrios T. Papageorgiou [Opens in a new window]
J. Paul Alexander
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
Toby L. Kirk
Affiliation:
Mathematical Institute, Oxford University, Woodstock Road, OxfordOX2 6GG, UK
Demetrios T. Papageorgiou*
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

The linear stability of a liquid film falling down an inclined flexible plane under the influence of gravity is investigated using analytical and computational techniques. A general model for the flexible substrate is used leading to a modified Orr–Sommerfeld problem addressed numerically using a Chebyshev tau decomposition. Asymptotic limits of long waves and small Reynolds numbers are addressed analytically and linked to the computations. For long waves, the flexibility has a destabilising effect, where the critical Reynolds number decreases with decreasing stiffness, even destabilising Stokes flow for sufficiently small stiffness. To pursue this further, a Stokes flow approximation was considered, which confirmed the long-wave results, but also revealed a short wave instability not captured by the long-wave expansions. Increasing the surface tension has little effect on these instabilities and so they were characterised as wall modes. Wider exploration revealed mode switching in the dispersion relation, with the wall and surface mode swapping characteristics for higher wavenumbers. The zero-Reynolds-number results demonstrate that the long-wave limit is not sufficient to determine instabilities so the numerical solution for arbitrary wavenumbers was sought. A Chebyshev tau spectral method was implemented and verified against analytical solutions. Short wave wall instabilities persist at larger Reynolds numbers and destabilisation of all Reynolds numbers is achievable by increasing the wall flexibility, however increasing the stiffness reverts back to the rigid wall limit. An energy decomposition analysis is presented and used to identify the salient instability mechanisms and link them to their physical origin.

JFM classification

Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A40
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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