| The presence of slight azimuthal asymmetry in the initial shape of an underwater bubble entirely alters the final break-up dynamics. Vibrations in the cross-section shape of the bubble develop, grow relative to the average size of the bubble neck and bring about a coalescence mode of breakup in which distant regions along the air-water surface curve inwards and eventually collide with finite speed. Here we present boundary integral simulation results showing that these coalescence modes of breakups are interspersed with dynamics that give rise to sharp tips along the bubble surface. Our numerics show that when the initial condition is tuned towards some threshold value, the surface appears to evolve into a finite-time curvature singularity by developing sharp tips with infinite curvatures. However, starting with initial conditions at the threshold values, the surface evolution towards the curvature singularity is pre-empted by coalescence. We also show that the dynamics around the curvature singularity corresponds to a saddle-node evolution. In other words, an evolution towards a cross-section shape with sharp tips invariably later evolves away from it. The maximum curvature attained when the interface evolves towards the curvature singularity increases as the amplitude of the initial perturbation decreases. Taken together, the results suggest that the curvature singularity appears to be attained only in the limit that the initial perturbation amplitude approaches 0. For a phase space trajectory close to the curvature singularity, as the singularity is approached, the curvature of the sharp tip diverges approximately as (R-R c)-0.8, where R describes the average size of the horizontal cross-section of the bubble neck minimum and Rc corresponds to the onset of the singularity, and the velocity of the tip diverges approximately as (R-Rc) -0.4. In practice, these divergences imply that viscous drag and compressibility of the gas flow, two effects not included in our analysis, become significant as the interface evolves towards the curvature singularity. |
