# Faraday instability on a sphere: Floquet analysis

## TL;DR

This paper analyzes the Faraday instability on a spherical viscous liquid drop under radial oscillation, extending planar solutions to spherical geometry using Floquet analysis and spherical harmonics.

## Contribution

It introduces a spherical Floquet analysis of Faraday instability, incorporating viscosity effects and deriving new scaling laws for threshold behavior.

## Key findings

- Thresholds are similar to planar cases despite high curvature.
- Viscosity influences the instability threshold and scaling laws.
- A Floquet mode illustrating the instability dynamics is presented.

## 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 (1994) 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.