Bubble pinch-off in viscous liquids

The effect of liquid viscosity on the final instants previous to pinch-off of an air bubble immersed in a stagnant viscous liquid is experimentally and theoretically investigated. Our experiments show that the use of a power-law to describe the collapse dynamics of the bubble is not appropriate in an intermediate range of liquid viscosities, for which a transition from an inviscid to a fully viscous pinch-off takes place. Instead, the instantaneous exponent $\alpha(\tau)$ varies during a single pinch-off event from the typical values of inviscid collapse, $\alpha\simeq 0.58$, to the value corresponding to a fully viscous dynamics, $\alpha\simeq 1$. However, we show that the pinch-off process can be accurately described by the use of a pair of Rayleigh-like differential equations for the time evolution of the minimum radius and the axial curvature evaluated at the minimum radius, $r_1$. This theoretical model is able to describe the smooth transition which takes place from inviscid to viscous-dominated pinch-off in liquids of intermediate viscosity, 10 $\leq\mu\leq$ 100 cP, and accounts for the fact that the axial curvature remains constant when the local Reynolds number becomes small enough, in agreement with our experimental measurements.