Journal
PHYSICA D-NONLINEAR PHENOMENA
Volume 238, Issue 14, Pages 1113-1128Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.physd.2009.03.016
Keywords
Singular dynamics; Hydrodynamic instabilities; Stochastic growth
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Hele-Shaw flow at vanishing surface tension is ill-defined. in finite time, the flow develops cusp-like singularities. We show that this ill-defined problem admits a weak dispersive solution when singularities give rise to a graph of shock waves propagating into the Viscous fluid. The graph of shocks grows and branches. Velocity and pressure have finite discontinuities across the shock. We formulate a few simple physical principles which single out the dispersive solution and interpret shocks as lines of decompressed fluid. We also formulate the dispersive weak solution in algebro-geometrical terms as an evolution of the Krichever-Boutroux complex curve. We study in detail the most generic (2, 3)-cusp singularity, which gives rise to an elementary branching event. This solution is self-similar and expressed in terms of elliptic functions. Published by Elsevier B.V.
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