Singularity Formation in Hele-Shaw Bubbles

We provide numerical and analytic evidence for the formation of a singularity driven only by surface tension in the mathematical model describing a two-dimensional Hele-Shaw cell with no air injection. Constantin and Pugh have proved that no such singularity is possible if the initial shape is close to a circle; thus we show that their result is not true in general. Our evidence takes the form of direct numerical simulation of the full problem, including a careful assessment of the effects of limited spatial resolution, and comparison of the full problem with the lubrication approximation. 1 Introduction The Hele-Shaw problem is one of the richest in fluid mechanics. As discussed in numerous review articles [1, 2], beyond its intrinsic interest it has served as a paradigm for pattern formation in many diverse fields such as dendritic solidification and porous media flow. The experimental geometry can take many forms: an open plane or a confined channel, with injection of air or withou..