Dimples, jets and self-similarity in nonlinear capillary waves

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 in our simulation develops from a smooth initial perturbation, chosen to be an eigenmode to the linearised ${O}(\epsilon )$ problem ( $\epsilon$ is the non-dimensional amplitude). We explain the physical mechanism of dimple formation and demonstrate that, for moderate $\epsilon$ , the sharp dimple seen in simulations is well captured by the weakly nonlinear ${O}(\epsilon ^3)$ theory developed here. For $\epsilon \gg 1$ 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, SIAM J. Appl. Maths, vol. 43, issue 2, 1983, pp. 268–277) 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 can form from radial inward focusing of capillary waves, and the formation of this may be described by a relatively simple model employing (nonlinear) inertial and capillary effects. Viscosity and gravity can, however, significantly influence the focusing process, either intensifying the singularity or weakening it (Walls et al., Phys. Rev. E, vol. 92, issue 2, 2015, 021002; Gordillo & Rodríguez-Rodríguez, J. Fluid Mech., vol. 867, 2019, pp. 556–571). This leads, in particular, to critical values of Ohnesorge and Bond numbers, which cannot be obtained from our minimal model.

Jet from a very large, axisymmetric, surface-gravity wave

“…Comparison of this analytical theory against numerical simulations demonstrated very good agreement (Kayal et al. 2022), additionally also shedding light into the physical mechanism at work driven by gradients of curvature. It was proven (Kayal et al.…”

“…Motivated partly by this observation, simulations of the incompressible, Euler's equation with only surface tension (no gravity) with the aforementioned initial condition of Farsoiya et al. (2017) have been reported recently by Kayal, Basak & Dasgupta (2022). It was demonstrated clearly in these simulations that surface tension alone at sufficiently small scales can produce a jet similar to what was observed at much larger capillary-gravity length scales by Farsoiya et al.…”

“…Concomitantly, the weakly nonlinear solution to the IVP for this initial condition and taking into account only surface tension has also been developed by Kayal et al. (2022). Comparison of this analytical theory against numerical simulations demonstrated very good agreement (Kayal et al.…”

“…It was proven (Kayal et al. 2022) that to describe the inception of the jet in the surface tension driven case, one needs to account for nonlinear effects of curvature (the gradient of which drives the flow). To capture this accurately, it was shown that the analytical theory needs to be at least third-order accurate, i.e.…”

Conical focusing: mechanism for singular jetting from collapsing drop-impact craters

“…2020). Kayal, Basak & Dasgupta (2022) recently studied a minimal model for converging capillary waves without gravity and viscosity. Their simulations have insufficient resolution to capture possible air sheets, but obtain normalized jetting velocities , which is significantly lower than our value of , using the maximum radius of our cavity.…”