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Three-dimensional Rayleigh–Taylor instability under a unidirectional curved substrate

Published online by Cambridge University Press:  19 December 2017

Gioele Balestra ,
Nicolas Kofman ,
P.-T. Brun ,
Benoit Scheid  and
François Gallaire Open the ORCID record for François Gallaire [Opens in a new window]
Gioele Balestra*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, EPFL, CH 1015 Lausanne, Switzerland
Nicolas Kofman
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, EPFL, CH 1015 Lausanne, Switzerland
P.-T. Brun
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA
Benoit Scheid
Affiliation:
TIPs Laboratory, Université Libre de Bruxelles, C.P. 165/67, Avenue Franklin Roosevelt 50, 1050 Bruxelles, Belgium
François Gallaire
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, EPFL, CH 1015 Lausanne, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the Rayleigh–Taylor instability of a thin liquid film coated on the inside of a cylinder whose axis is orthogonal to gravity. We are interested in the effects of geometry on the instability, and contrast our results with the classical case of a thin film coated under a flat substrate. In our problem, gravity is the destabilizing force at the origin of the instability, but also yields the progressive drainage and stretching of the coating along the cylinder’s wall. We find that this flow stabilizes the film, which is asymptotically stable to infinitesimal perturbations. However, the short-time algebraic growth that these perturbations can achieve promotes the formation of different patterns, whose nature depends on the Bond number that prescribes the relative magnitude of gravity and capillary forces. Our experiments indicate that a transverse instability arises and persists over time for moderate Bond numbers. The liquid accumulates in equally spaced rivulets whose dominant wavelength corresponds to the most amplified mode of the classical Rayleigh–Taylor instability. The formation of rivulets allows for a faster drainage of the liquid from top to bottom when compared to a uniform drainage. For higher Bond numbers, a two-dimensional stretched lattice of droplets is found to form on the top part of the cylinder. Rivulets and the lattice of droplets are inherently three-dimensional phenomena and therefore require a careful three-dimensional analysis. We found that the transition between the two types of pattern may be rationalized by a linear optimal transient growth analysis and nonlinear numerical simulations.

JFM classification

Instability: Instability Nonlinear Dynamical Systems: Pattern formation Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 19 - 47
Copyright
© 2017 Cambridge University Press 

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Balestra et al. supplementary movie 1

Dynamics of the dripping droplets

Download Balestra et al. supplementary movie 1(Video)
Video 41.2 MB

Balestra et al. supplementary movie 2

Dynamics of the rivulets

Download Balestra et al. supplementary movie 2(Video)
Video 28.9 MB

Balestra et al. supplementary movie 3

Dynamics of the mixed regime

Download Balestra et al. supplementary movie 3(Video)
Video 28.6 MB

Balestra et al. supplementary movie 4

Dynamics of the dripping rivulets

Download Balestra et al. supplementary movie 4(Video)
Video 44.1 MB

Balestra et al. supplementary movie 5

Dynamics of rivulets in the cylinder experiment

Download Balestra et al. supplementary movie 5(Video)
Video 18.9 MB

Balestra et al. supplementary movie 6

Independent repetition of movie 5 for the same experimental conditions

Download Balestra et al. supplementary movie 6(Video)
Video 17.8 MB