Hele–Shaw flow with a point sink: generic solution breakdown

Abstract

Recent numerical evidence [8, 28, 33] suggests that in the Hele–Shaw suction problem with vanishingly small surface tension $\gamma$, the free boundary generically approaches the sink in a wedge-like configuration, blow-up occurring when the wedge apex reaches the sink. Sometimes two or more such wedges approach the sink simultaneously [33]. We construct a family of solutions to the zero-surface tension (ZST) problem in which fluid is injected at the (coincident) apices of an arbitrary number $N$ of identical infinite wedges, of arbitrary angle. The time reversed suction problem then models what is observed numerically with non-zero surface tension. We conjecture that (for a given value of $N$) a particular member of this family of ZST solutions, with special complex plane singularity structure, is selected in the limit $\gamma\,{\to}\,0$.