Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1897573 | Physica D: Nonlinear Phenomena | 2009 | 16 Pages |
Abstract
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.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
S.-Y. Lee, R. Teodorescu, P. Wiegmann,