Spheroidal droplet deformation, oscillation and breakup in uniform outer flow

This work is focused on the development of an analytical model for the axisymmetric deformation of droplets with a given velocity lag in an outer flow. This leads either to oscillations around a deformed equilibrium shape or to strong deformations which may ultimately lead to their breakup when the equilibrium shape loses its stability. To obtain an evolution equation for the droplet shape evolution, it is first supposed that the droplet deforms like a spheroid. Then, a balance between kinetic energy of deformation, surface energy creation, external pressure work and viscous dissipation within the droplet is written. This is close to the approach developed in the classical Taylor analogy breakup or droplet deformation and breakup model. The main originality of the present modelling is that pressure work can be computed at any time, assuming an outer potential flow. Pressure is assumed to work on either the whole droplet or only on the forward half of the droplet due to flow separation. Results of this model are then compared with reported droplet oscillation frequencies. For suddenly accelerated droplets, oscillations are present in the liquid-liquid metal case and successfully compared with some new direct numerical simulation results. Comparison is also successful when compared with previous results on droplet oscillating at terminal falling velocity, whether in the liquid-liquid case or for falling raindrop. Lastly, comparisons are made with previously published secondary fragmentation experiments and direct numerical simulations whether in a shock tube or for droplets falling in a cross-flow.