Capillary pinch-off of inviscid fluids at varying density ratios: the bubble limit

The axisymmetric pinch-off of an inviscid blob of fluid of density $\rho_{1}$ in an ambient fluid of density $\rho_{2}$ is examined in the limit as the density ratio $D = \rho_{1}/\rho_ {2} \rightarrow 0$ using a boundary integral formulation. It has previously been shown (Leppinen \& Lister, {\em Phys. Fluids}, {\bf 15(2)}, 568-578, 2003) that pinch-off is a self-similar process in the droplet limit as $D \rightarrow$ with the radial and the axial length scales decreasing as $\tau^{2/3}$ where $\tau$ is the time to pinch-off. In the droplet limit, the similarity form is independent of the initial conditions. In the bubble limit, as $D \rightarrow 0$, it is seen that pinch-off is also a self-similar process, however, in this case the similarity form is dependent on initial conditions. In the bubble limit the radial length scale decreases as $\tau^{c_{1}}$ and the axial length scale descreases as $\tau^{c_{2}}$ with both $c_{1}$ and $c_{2}$ (and the associated prefactors) depending on the value of the density ratio $D$ and on the initial conditions. In the limit of $D=0$, $c_{1} \approx 0.55 \pm 0.01$ and $c_{2} \approx 0.48 \pm 0.05$ dependent on initial conditions.