Bubbles in the Hele-Shaw cell: Pattern selection and tip perturbations.

Abstract

We consider the motion of a symmetric finite bubble in a two-dimensional Hele-Shaw cell. In the absence of surface tension, the Saffman-Taylor solution contains two free parameters, U and \ensuremath{\lambda}, for a given bubble area J, where U is the dimensionless speed and \ensuremath{\lambda} is the dimensionless width of the bubble. It is shown that, in the presence of surface tension, a solution does not exist for Ug2 for any bubble area. We also derive the following scaling relations: (a) 2-U\ensuremath{\approxeq}${\ensuremath{\epsilon}}^{2/3}$ for large J; and (b) 2-U\ensuremath{\approxeq}${\ensuremath{\epsilon}}^{2}$ for small J, where \ensuremath{\epsilon} is a small parameter which is proportional to the surface tension. We show that by creating a cusp at the tip of the bubble, one can increase the speed of the bubble Ug2. We present predictions for the shape and speed of the symmetric bubble as a function of the external parameters in the presence of a cusp at the tip. This picture may explain the recent experimental results of Maxworthy, where substantially enhanced velocities were measured for the anomalous bubbles with a tiny bubble at the tip.