# Faraday instability on a sphere: numerical simulation

## TL;DR

This paper investigates the Faraday instability on a spherical drop using numerical simulations, revealing pattern formations like Platonic solids and oscillations driven by parametric forcing.

## Contribution

It introduces a comprehensive 3D simulation approach to study spherical Faraday instability, exploring pattern formation and wave responses across various wavenumbers.

## Key findings

- Observation of spherical harmonic wave excitation from 1 to 6.
- Identification of patterns analogous to Platonic solids.
- Detection of both subharmonic and harmonic responses at low viscosity.

## Abstract

We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate-prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry.