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Faraday instability on a sphere: Floquet analysis

Published online by Cambridge University Press:  23 September 2016

Ali-higo Ebo Adou  and
Laurette S. Tuckerman
Ali-higo Ebo Adou
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), UMR CNRS 7636; PSL-ESPCI; Sorbonne Univ.- UPMC, Univ. Paris 6; Sorbonne Paris Cité-UDD, Univ. Paris 7, France LIMSI, CNRS, Université Paris-Saclay, 91405 Orsay, France
Laurette S. Tuckerman*
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), UMR CNRS 7636; PSL-ESPCI; Sorbonne Univ.- UPMC, Univ. Paris 6; Sorbonne Paris Cité-UDD, Univ. Paris 7, France
*
†Email address for correspondence: [email protected]
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Abstract

Standing waves appear at the surface of a spherical viscous liquid drop subjected to radial parametric oscillation. This is the spherical analogue of the Faraday instability. Modifying the Kumar & Tuckerman (J. Fluid Mech., vol. 279, 1994, pp. 49–68) planar solution to a spherical interface, we linearize the governing equations about the state of rest and solve the resulting equations by using a spherical harmonic decomposition for the angular dependence, spherical Bessel functions for the radial dependence and a Floquet form for the temporal dependence. Although the inviscid problem can, like the planar case, be mapped exactly onto the Mathieu equation, the spherical geometry introduces additional terms into the analysis. The dependence of the threshold on viscosity is studied and scaling laws are found. It is shown that the spherical thresholds are similar to the planar infinite-depth thresholds, even for small wavenumbers for which the curvature is high. A representative time-dependent Floquet mode is displayed.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Faraday waves Instability: Parametric instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 805 , 25 October 2016 , pp. 591 - 610
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Copyright
© 2016 Cambridge University Press 

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