Droplet sizes from impulsive capillary jets: an exponential distribution with no lower cut-off

Abstract

A collapsing cavity throws up a ligament, which fragments into a train of droplets whose sizes are commonly given a lower bound at a fixed multiple of the viscocapillary length $\ell_\mu=\mu^2/\rho\sigma$. We show that no such bound exists, for a reason more general than the fate of one parameter: an impulsive jet possesses no length of its own. The similarity solutions that govern each pinch contain none, the cascade of stretching and iterated satellites introduces none, and $\ell_\mu$ belongs to the singularity rather than to the droplets. A size distribution built on such a process can inherit a scale from one place only, the cavity that launched the jet. Computations of bubble bursting, confined and unconfined, bear this out. The floor they display is the one the mesh imposes, not a property of the fluid, and wherever the mesh looks below $\ell_\mu$ it finds droplets. The census of unique emitted droplets is exponential, with a single scale set by the cavity radius, indifferent both to how vigorous the event is and to how much liquid surrounds it. An exponential is the least structured census compatible with a prescribed mean, so the fragmentation keeps the size of the cavity and no other memory of its parent. Sustained stretching does keep such a memory, imprinting the corrugation of the parent thread and yielding the peaked, gamma-like distributions reported for ligament-mediated fragmentation. The manner of loading, and not the fluid alone, selects the shape of a spray.