Self-similar Features in Secondary Breakup of a Droplet and Ligament Mediated Fragmentation under Extreme Conditions

Abstract

Droplet formation is relevant in many applications spanning natural and artificial settings. A physical understanding of aerobreakup or air-assisted secondary atomization and predicting size distributions in these applications is non-trivial. We show that extreme airflow speeds induce catastrophic breakup, which, although chaotic and seemingly obscure, is not hopelessly unstructured. In the present study, through shockwave-induced breakups, we investigate the associated intermediate processes at smaller spatiotemporal scales at very high Weber numbers ($We \sim 10^3-10^4$). We demonstrate that microscale protrusions decorate the disintegrating droplet interface and eventually fragment, resulting in the generation of daughter droplets. We discover these undulations to follow breakup patterns (sub-secondary breakup) that resemble a scaled-down version of secondary atomization. The consistent topology across a vast range of scales $(10^{-6}m-10^{-2}m)$ suggests a self-similar mechanism bridged by local Weber number. The normalized size distribution of the resultant droplets exhibits universality and $We$ invariance at all extreme conditions, including transient statistics for subsequent time periods. This conforms to a universal modified gamma distribution characterized by ligament shape factors, which tend toward the limiting behavior associated with the maximum corrugations physically possible. Scaling laws based on the $We$ are derived for the averaged diameter as $\sim We^{-1/3}$, using a high-energy aerodynamic breakup mechanism and subsequently used to derive a time-integrated distribution. These observations reinforce the idea of a self-similar mechanism for the catastrophic droplet breakups, encompassing multiscale deformation cascades, self-similar sub-secondary breakups, maximally corrugated ligaments, and universal droplet size distributions.