Theoretical study on the interfacial instability of a spherical droplet subject to vertical vibration

TL;DR

This paper provides a linear theoretical analysis of interfacial instability in spherical liquid droplets under vertical vibration, revealing differences from radial vibration cases and supported by experimental validation.

Contribution

It introduces a novel linear analysis framework for spherical droplet instability under vertical vibration, highlighting differences from radial vibration and including experimental confirmation.

Findings

Unstable regions depend on spherical mode numbers and vibration parameters.

Latitudinal waves oscillate harmonically, not subharmonically, under vertical vibration.

Experimental results support the theoretical predictions.

Abstract

Interfacial instability would be aroused on a spherical liquid droplet when it is subject to external vertical vibration. In this paper, a linear analysis was conducted on this instability problem. The polar-angle dependent acceleration in the spherical coordinate is strongly coupled with the temporal and spatial component of the surface deformation displacement, which gives a recursion equation that implicitly expresses the dispersion relation between the growth rate and spherical mode numbers. The unstable regions (or unstable tongues) for the inviscid fluids considering latitudinal mode (longitudinal mode number m = 0) were derived and presented in the parameter plane. Compared with the solution of the spherical Faraday instability under radial vibration acceleration, the regions of harmonic unstable tongues for the mono-directional vibration case is much narrowed and the subharmonic unstable tongues almost become straight lines. The analysis shows that the latitudinal waves emerging on the spherical droplet surface ought to oscillate harmonically instead of subharmonically, which is opposite to the results for the case under radial vibration acceleration. A corresponding experiment of a liquid droplet lying on a vertically vibrating plate was conducted and the observations substantiate our theoretical predictions.