Mode selection on breakup of a droplet falling into a miscible solution

When a droplet with a relatively high density falls into a miscible solution with a relatively low density, the droplet breaks up spontaneously. We investigated the number m of breakup in experiments with several density differences Delta rho between two solutions, viscosities mu, and droplet radii r. The mode number m has a distribution even under the same experimental conditions. We propose a simple model of mode selection based on the linear Rayleigh-Taylor instability and the growing radius of a vortex ring deformed from the droplet. The model provides the probability distribution P(m) and a relationship between the non-dimensional parameter G proportional to Delta rho gr(3) / mu(2) and the average value of m, which are consistent with experimental results.