Faraday instability on a sphere: Floquet analysis

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...

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Bibliographic Details
Published inJournal of fluid mechanics Vol. 805; pp. 591 - 610
Main Authors Ebo Adou, Ali-higo, Tuckerman, Laurette S.
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 25.10.2016
Subjects
Bessel functions
Boundary conditions
Dependence
Drops (liquids)
Fluids
Instability
Investigations
Kinematics
Laboratories
Mathematical analysis
Mathematical models
Scaling
Scaling laws
Spherical harmonics
Stability analysis
Standing waves
Thresholds
Viscosity
Faraday waves
capillary waves
parametric instability
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Summary: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.
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2016.542