Dimple, jets and self-similarity in nonlinear capillary waves

TL;DR

This paper develops a minimal nonlinear model to explain dimple and jet formation in collapsing capillary waves, revealing self-similar jet evolution and emphasizing inertial and capillary forces without viscosity or gravity.

Contribution

It introduces a simplified model capturing dimple and jet formation from capillary waves, including a novel self-similar jet regime and physical mechanism explanations.

Findings

Weakly nonlinear theory captures moderate steepness dimples.

Steep regimes produce sharply rising jets.

Self-similar jet evolution follows inviscid capillary wave scales.

Abstract

Numerical studies of dimple and jet formation from a collapsing cavity often model the initial cavity shape as a truncated sphere, mimicking a bursting bubble. In this study, we present a minimal model containing only nonlinear inertial and capillary forces, which produces dimples and jets from a collapsing, capillary wave trough. The trough develops from an initial perturbation, chosen to be an eigen-mode to the linearised problem.We explain the physical mechanism of dimple formation and demonstrate that, for moderate steepness, the sharp dimple seen in simulations is well captured by the weakly nonlinear theory developed here. For steepness >> 1 the regime is strongly nonlinear spreading surface energy into many modes and the precursor dimple now develops into a sharply rising jet. Here, simulations reveal a novel localised window (in space and time) where the jet evolves self-similarly following inviscid Keller & Miksis (1983) scales. We develop an analogy of this regime to a self-similar solution of the first kind, for linearised, capillary waves. Our first principles study demonstrates that at sufficiently small scales, dimples and jets form due to radial focussing of capillary waves requiring (nonlinear) inertial and capillary contributions, sans viscous or gravitational interventions.