Published July 2010
| public
Journal Article
Weak solution for the Hele-Shaw problem: Viscous shocks and singularities
- Creators
- Lee, S.-Y.
- Teodorescu, R.
- Wiegmann, P.
Abstract
In Hele-Shaw flows, a boundary of a viscous fluid develops unstable fingering patterns. At vanishing surface tension, fingers evolve to cusp-like singularities preventing a smooth flow. We show that the Hele-Shaw problem admits a weak solution where a singularity triggers viscous shocks. Shocks form a growing, branching tree of a line distribution of vorticity where pressure has a finite discontinuity. A condition that the flow remains curl-free at a macroscale uniquely determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2, 3)-cusp singularity—an elementary branching event of a branching shock graph.
Additional Information
© 2010 Pleiades Publishing, Inc. Received May 25, 2010. The article is published in the original. The work of S.-Y.L. was supported by CRM-ISM postdoctoral fellowship. The work of P.W. was supported by the US National Science Foundation (grant nos. DMR-0540811/FAS 5-2783, DMR-0906427), by the Materials Research Science and Engineering Centers (grant no. DMR-0820054), and by the Federal Agency for Science and Innovation, Ministry of Education and Science of the Russian Federation (state contract no. 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
- Eprint ID
- 20479
- DOI
- 10.1134/S0021364010140043
- Resolver ID
- CaltechAUTHORS:20101022-091940086
- CRM-ISM
- DMR-0540811/FAS 5-2783
- NSF
- DMR-0906427
- NSF
- DMR-0820054
- Materials Research Science and Engineering Centers
- 02.740.11.5029
- Federal Agency for Science and Innovation, Ministry of Education and Science of the Russian Federation
- USF College of Engineering Interdisciplinary Scholarship Program
- USF College of Arts and Sciences
- Created
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2010-11-30Created from EPrint's datestamp field
- Updated
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2021-11-09Created from EPrint's last_modified field