Linear analysis on the interfacial instability of a spherical liquid droplet subject to a radial vibration

Precursory surface standing waves for liquid atomization occur on a spherical droplet subjected to a radial time-periodic force. In this paper, we carried out a linear stability analysis on the spherical Faraday instability. With the Floquet analysis, a derived difference equation gives the dispersion relation between the Floquet exponent and the spherical modes. For inviscid instability, the problem can also be reduced to the standard Mathieu equation as the same as its planar counterpart, but the parameters in the equation correspond to different quantities due to the spherical configuration. The analysis shows that increasing the density ratio of the ambient fluid to the droplet narrows the range of possibly excited spherical modes under the same forcing condition. For viscous instability, an additional parameter corresponding to the viscous effects was introduced into the difference equation. With increasing the droplet viscosity, the surface waves with large mode numbers are stabilized and hence a larger forcing amplitude is required to cause instability. Furthermore, the most-unstable spherical mode of the largest growth rate excited in the experimental condition is determined and discussed for its physical interpretation for droplet atomization caused by Faraday instability.