Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent

PHYSICA D-NONLINEAR PHENOMENA

卷:240

Article

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

[1] Univ S Florida, Dept Math, Tampa, FL 33620 USA

[2] CALTECH, Pasadena, CA 91125 USA

[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.2010.09.017

来源: Elsevier

【 摘 要 】

In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers Lee et al. (2009, 2008) [8,9] we have showed that singularities trigger viscous shocks propagating through the viscous fluid. Here we show that the weak solution of the Hele-Shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the Painleve I equation. We argue that the Painleve I equation provides an integrable deformation of the Hele-Shaw problem which describes flow passing through singularities. In this interpretation shocks appear as Stokes level-lines of the Painlevelinear problem. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

Free