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Curvature Singularity in the Asymmetric Breakup of an Underwater Air Bubble

The presence of slight azimuthal asymmetry in the initial shape of an underwater bubble entirely alters the final breakup dynamics. Here I examine the influence of initial asymmetry on the final breakup by simulating the bubble surface evolution as a Hamiltonian evolution corresponding to an inviscid, two-dimensional, planar implosion. I find two types of breakups: a previously reported coalescence mode in which distant regions along the air-water surface curve inwards and eventually collide with finite speed, and a hitherto unknown cusp-like mode in which the surface develops sharp tips whose radii of curvature are much smaller than the average neck radius. I present three sets of results that characterize the nature of this cusp mode. First, I show that the cusp mode corresponds to a saddle-node. In other words, an evolution towards a cross-section shape with sharp tips invariably later evolves away from it. In phase space, this saddle-node separates coalescence modes whose coalescence planes lie along different spatial orientations. Second, I show that the formation of the sharp tips can be interpreted as a weakly first-order transition which becomes second-order, corresponding to the formation of a finite-time curvature singularity, in the limit that the initial perturbation amplitude approaches zero. Third I show that, as the curvature singularity is approached, the maximum surface curvature diverges approximately as $(t_c - t)^{-0.8}$, where $t_c$ is the onset time of the singularity and the maximum velocity diverges approximately as $(t_c-t)^{-0.4}$. In practice, these divergences imply that viscous drag and compressibility of the gas flow, two effects not included in my analysis, become significant as the interface evolves towards the curvature singularity.