Waves and jets on a sessile bubble -- Part 1

We show numerically that large amplitude, \textit{shape deformations}, imposed on a spherical-cap, sessile gas bubble pinned on to a rigid wall can produce a sharp, wall-directed jet. For such a bubble filled with a permanent gas, the temporal spectrum for surface tension driven, linearised perturbations has been studied recently in \cite{ding2022oscillations} in the potential flow limit. We test these predictions against numerical simulations of the inviscid, incompressible, Euler's equation with surface tension \citep{popinet2014basilisk}. The bubble surface is deformed initially as a shape eigenmode and this is evolved in time. At small modal amplitude, the resulting standing waves show very good agreement with theory, both for axisymmetric and three-dimensional modes. In contrast, when the bubble is evolved temporally from a large amplitude initial distortion, we observe the formation of a sharp dimple at the symmetry axis. This is the precursor to a slender, wall-directed jet, as the distortion amplitude is increased further. We find dimples which eject with instantaneous velocities exceeding fifteen times that predicted from linear theory. These form via capillary wave focussing, analogous to their free-surface counterparts \citep{duchemin2002jet}. At sufficiently large distortion amplitude and in a small time window after the birth of the dimple, the temporal evolution of the jet occurs self-similarly following \cite{keller1983surface} scales. Our results complement \cite{naude1961mechanism,plesset1971collapse} demonstrating that wall-directed jets may be generated from volume conserving, shape deformations of a pinned, sessile bubble.