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VOLUME 92 (2010) | ISSUE 2 | PAGE 95
Weak solution for the Hele-Shaw problem: viscous shocks and singularities
In Hele-Shaw flows a boundary of a viscous fluid develops unstable fingering patterns. At vanishing surface tension, fingers evolve to cusp-like singularities preventing a smooth flow. We show that the Hele-Shaw problem admits a weak solution where a singularity triggers viscous shocks. Shocks form a growing, branching tree of a line distribution of vorticity where pressure has a finite discontinuity. A condition that the flow remains curl-free at a macroscale uniquely determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2,3)-cusp singularity - an elementary branching event of a branching shock graph.