Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published June 15, 2011 | public
Journal Article

Viscous shocks in Hele–Shaw flow and Stokes phenomena of the Painlevé I transcendent

Abstract

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) 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 Painlevé I equation. We argue that the Painlevé 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 Painlevélinear problem.

Additional Information

© 2011 Elsevier B.V. Received 24 May 2010; revised 29 September 2010; accepted 30 September 2010. Communicated by M. Vergassola. Available online 16 March 2011. S.-Y.L. was supported by CRM-ISM postdoctoral fellowship. P.W. was supported by NSF DMR-0540811/FAS 5-2783, NSF DMR- 0906427, MRSEC under DMR-0820054 and the FASI of the Russian Federation under contract 02.740.11.5029. P.W. and R.T. also acknowledge the USF College of Engineering Interdisciplinary Scholarship Program and the support of USF College of Arts and Sciences. We thank A. Its and A. Kapaev for helpful discussions and are especially grateful to I. Krichever.

Additional details

Created:
August 22, 2023
Modified:
October 24, 2023