Standing waves and jets on a sessile, incompressible bubble

Abstract

We show numerically that large amplitude, shape deformations imposed on a spherical-cap, incompressible, sessile gas bubble pinned on 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, linearized perturbations has been studied recently in D. Ding and J. Bostwick [“Oscillations of a partially wetting bubble,” J. Fluid Mech. 945, A24 (2022)]. in the potential flow limit. We reformulate this as an initial-value problem analogous in spirit to classical derivations in the inviscid limit by Kelvin [“Oscillations of a liquid sphere,” Math. Phys. Papers 3, 384–386 (1890)], Rayleigh [“On the instability of jets,” Proc. London Math. Soc. s1-10, 4–13 (1878)] or by Prosperetti [“Viscous effects on small-amplitude surface waves,” Phys. Fluids 19, 195–203 (1976)] and Prosperetti [“Motion of two superposed viscous fluids,” Phys. Fluids 24, 1217–1223 (1981)] for the viscous case. The first test of linear theory is reported here by distorting the shape of the pinned, spherical cap employing eigenmodes obtained from linearized theory. These are employed as the initial shape distortion of the bubble in numerical simulations. It is seen that linearized predictions show good agreement with nonlinear simulations at small distortion amplitude producing standing waves, which oscillate at the predicted frequency. Beyond the linear regime as the shape distortions are made sufficiently large, we observe the formation of a dimple followed by a slender, wall-directed jet, analogous to similar jets observed in other geometries from collapsing wave troughs [Farsoiya et al., “Axisymmetric viscous interfacial oscillations–theory and simulations,” J. Fluid Mech. 826, 797–818 (2017) and Kayal et al., “Dimples, jets and self-similarity in nonlinear capillary waves,” J. Fluid Mech. 951, A26 (2022).] This jet can eject with an instantaneous velocity exceeding nearly 20 times that predicted by linear theory. By projecting the shape of the bubble surface around the time instant of jet ejection, into the eigenspectrum we show that the jet ejection coincides with the nonlinear spreading of energy into a large number of eigenmodes. We further demonstrate that the velocity-field associated with the dimple plays a crucial role in evolving it into a jet and without which, the jet does not form. It is also shown that evolving the bubble shape containing a dimple but zero initial velocity-field everywhere, via linear theory, does not produce the jet. These conclusions accompanied by first principles analysis provide insight into the experimental observations of Prabowo and Ohl [“Surface oscillation and jetting from surface attached acoustic driven bubbles,” Ultrason. Sonochem. 18, 431–435 (2011)], where similar jets were reported earlier, albeit via acoustic forcing. Our inferences also complement well-known results of Naude and Ellis [“On the mechanism of cavitation damage by nonhemispherical cavities collapsing in contact with a solid boundary,” J. Fluids Eng. 83, 648–656 (1961)] and Plesset and Chapman [“Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary,” J. Fluid Mech. 47, 283–290 (1971)] demonstrating that wall-directed jets can be generated from volume preserving, shape deformations of a pinned bubble.