The generation of singular liquid jets in the axisymmetric bubble pinch-off at high Reynolds numbers

In this presentation we review the \emph{symmetric} and \emph{asymmetric} type of bubble pinch-off local geometries described in {\it PRL}, {\bf 95}, 194501, and provide with the different scalings for the minimum radius, $R_0$, as the singularity is approached. Moreover, in the case of gas inertia is not relevant in the description of the latest stages of bubble breakup (\emph{symmetric pinch-off}), local bubble shape is given by $F(z,t)/R_0(t)=1-[1/(6\,log(R_0))]\,(z/R_0)^2$. However, we also discuss that the asymptotic solutions for the \emph{symmetric} case are only reached for times so close to pinch off that they might be difficult to find and, therefore, bubble pinch-off strongly depends on initial conditions. Regarding the \emph{asymmetric} type of breakup, we provide new experimental evidence that support that, close to pinch-off, gas and liquid inertia are balanced. We will show that the velocity of the singular liquid jets formed within air and helium bubbles generated from a needle immersed in a coaxial co-flow strongly depends on gas density. More precisely, the ratio of the liquid jet velocity formed using air ($u_a$) to the liquid jet velocity formed using helium ($u_{h}$) is given, for the same operating conditions, by $u_a/u_h\simeq\,(\rho_{air}/\rho_{helium})^{1/n}$, with $2