Shocks and finite-time singularities in Hele-Shaw flow

PHYSICA D-NONLINEAR PHENOMENA

卷:238

Article

Lee, S. -Y.²  Teodorescu, R.¹  Wiegmann, P.^3,4 

[1] Los Alamos Natl Lab, Ctr Nonlinear Studies & T4, Los Alamos, NM 87545 USA

[2] Univ Montreal, Ctr Rech Math, Montreal, PQ H3C 3J7, Canada

[3] Univ Chicago, James Franck Inst, Chicago, IL 60637 USA

[4] Univ Chicago, Enrico Fermi Inst, Chicago, IL 60637 USA

关键词: Singular dynamics;    Hydrodynamic instabilities;    Stochastic growth;   

DOI  :  10.1016/j.physd.2009.03.016

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

Free