Curvature singularity in the asymmetric breakup of an underwater air bubble

Abstract

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.PACS numbers: 47.55.df, 47.20.Cq, 47.20.Ma

Pinch-off of underwater air bubbles with up--down asymmetry

“…The outcome is a side-to-side contact before the average radius reaches 0, cutting off the singularity. These vibrations were confirmed and studied in detail in later theory (Schmidt et al, 2009), experiments (Keim, 2011;Enríquez et al, 2011), and simulations (Turitsyn et al, 2009;Lai, 2012).…”

Multitude of dimple shapes can produce singular jets during the collapse of immiscible drop-impact craters

“…2006; Enriquez et al. 2012; Lai 2012), but they can also imprint a large variety of axial shapes on the free surface of the dimple, thereby modifying its singular collapse.…”

“…The above dimple shapes show that not only can the boundary conditions break the axisymmetry of the collapse of a pinching air cylinder (Burton et al 2005;Keim et al 2006;Enriquez et al 2012;Lai 2012), but they can also imprint a large variety of axial shapes on the free surface of the dimple, thereby modifying its singular collapse.…”

“…This memory of the boundaries has been best demonstrated for the pinch-off of a bubble from an elliptic nozzle (Schmidt et al. 2009; Lai 2012). Air cavities formed by the impact of circular plates with periodic edges produce kindred non-axisymmetric shapes, which can even split into multiple necks (Enriquez et al.…”

“…For the purely inertial scaling, the surface tension becomes irrelevant near the final pinch-off and there is a strong dependence on the initial or boundary conditions. This memory of the boundaries has been best demonstrated for the pinch-off of a bubble from an elliptic nozzle (Schmidt et al 2009;Lai 2012). Air cavities formed by the impact of circular plates with periodic edges produce kindred non-axisymmetric shapes, which can even split into multiple necks (Enriquez et al 2012).…”