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Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets

Published online by Cambridge University Press:  12 October 2020

Pier Giuseppe Ledda Open the ORCID record for Pier Giuseppe Ledda [Opens in a new window] ,
Gaétan Lerisson ,
Gioele Balestra Open the ORCID record for Gioele Balestra [Opens in a new window]  and
François Gallaire Open the ORCID record for François Gallaire [Opens in a new window]
Pier Giuseppe Ledda*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
Gaétan Lerisson
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
Gioele Balestra
Affiliation:
iPrint Institute, University of Applied Sciences and Arts of Western Switzerland, FribourgCH-1700, Switzerland
François Gallaire
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
*
†Email address for correspondence: pier.ledda@epfl.ch
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Abstract

We study the pattern formation of a thin film flowing under an inclined planar substrate. The phenomenon is studied in the context of the Rayleigh–Taylor instability using the lubrication equation. Inspired by experimental observations, we numerically study the thin film response to a streamwise-invariant sinusoidal initial condition. The numerical response shows the emergence of predominant streamwise-aligned structures, modulated along the direction perpendicular to the flow, called rivulets. Oscillations of the thickness profile along the streamwise direction do not grow significantly when the inclination is very large or the liquid layer very thin. However, for small inclinations or thick films, streamwise perturbations grow on rivulets. A secondary stability analysis of one-dimensional and steady rivulets reveals a strong stabilization mechanism for large inclinations or very thin films. The theoretical results are compared with experimental measurements of the streamwise oscillations of the rivulet profile, showing a good agreement. The emergence of rivulets is investigated by studying the impulse response. Both the experimental observation and the numerical simulation show a marked anisotropy favouring streamwise-aligned structures. A weakly nonlinear model is proposed to rationalize the levelling of all but streamwise-aligned structures.

JFM classification

Instability: Absolute/convective instability Instability: Nonlinear instability Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 904 , 10 December 2020 , A23
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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