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Rayleigh-Taylor instability under curved substrates: An optimal transient growth analysis

Author(s): Balestra, G; Brun, PT; Gallaire, F

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Abstract: We investigate the stability of thin viscous films coated on the inside of a horizontal cylindrical substrate. In such a case, gravity acts both as a stabilizing force through the progressive drainage of the film and as a destabilizing force prone to form droplets via the Rayleigh-Taylor instability. The drainage solution, derived from lubrication equations, is found asymptotically stable with respect to infinitesimally small perturbations, although in reality, droplets often form. To resolve this paradox, we perform an optimal transient growth analysis for the first-order perturbations of the liquid’s interface, generalizing the results of Trinh et al. [Phys. Fluids 26, 051704 (2014)]. We find that the system displays a linear transient growth potential that gives rise to two different scenarios depending on the value of the Bond number (prescribing the relative importance of gravity and surface tension forces). At low Bond numbers, the optimal perturbation of the interface does not generate droplets. In contrast, for higher Bond numbers, perturbations on the upper hemicircle yield gains large enough to potentially form droplets. The gain increases exponentially with the Bond number. In particular, depending on the amplitude of the initial perturbation, we find a critical Bond number above which the short-time linear growth is sufficient to trigger the nonlinear effects required to form dripping droplets. We conclude that the transition to droplets detaching from the substrate is noise and perturbation dependent.
Publication Date: 7-Dec-2016
Citation: Balestra, G, Brun, PT, Gallaire, F. (2016). Rayleigh-Taylor instability under curved substrates: An optimal transient growth analysis. Physical Review Fluids, 1 (8), 10.1103/PhysRevFluids.1.083902
DOI: doi:10.1103/PhysRevFluids.1.083902
EISSN: 2469-990X
Type of Material: Journal Article
Journal/Proceeding Title: Physical Review Fluids
Version: Final published version. Article is made available in OAR by the publisher's permission or policy.

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