Rim thickness evolution in unsteady sheet fragmentation

Upon impact on a finite solid surface, a drop first expands into a sheet in the air, surrounded by a rim that destabilizes into ligaments that, in turn, shed secondary drops. The rim thickness is a key part of this system, set by the balance between incoming fluid from the expanding sheet and the fluid feeding the ligaments. This thickness is inherently unsteady, constantly changing with time. Linear stability analysis can explain the growth of an initial perturbation on a rim’s steady base-state, as long as the perturbation remains small. However, once the perturbation is large, the predictive power of such approach breaks down and the rim thickness is not known. We combine theory with advanced image analysis to study the unsteady rim destabilization and thickness evolution. We show that, at all times, the rim thickness is governed by a local instantaneous Bond number equal to unity, defined with the instantaneous, local, unsteady rim acceleration. We found that this criterion is robust and universal for a family of unsteady inviscid fluid sheet fragmentation phenomena, from impacts of drops on various surface geometries to impacts on films. We further discuss under which viscous and viscoelastic conditions the criterion continues to govern the unsteady rim thickness.