Singular limit of Hele-Shaw flow and dispersive regularization of shock waves

Abstract

We study a family of solutions to the Saffman-Taylor problem with zero surface tension at a critical regime. In this regime, the interface develops a thin singular finger. The flow of an isolated finger is given by the Whitham equations for the KdV integrable hierarchy. We show that the flow describing bubble break-off is identical to the Gurevich-Pitaevsky solution for regularization of shock waves in dispersive media. The method provides a scheme for the continuation of the flow through singularites.

arXiv commentSome typos corrected, added journal reference

References in corpus (4)

• Normal random matrix ensemble as a growth problem
• Laplacian Growth and Whitham Equations of Soliton Theory
• Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
• Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy

Cited by in corpus (11)

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• Random Matrices in 2D, Laplacian Growth and Operator Theory
• Shocks and finite-time singularities in Hele-Shaw flow
• Novel Experimentally Observed Phenomena in Soft Matter
• The double scaling limit method in the Toda hierarchy
• Semiclassical expansions in the Toda hierarchy and the hermitian matrix model
• Critical behaviour for scalar nonlinear waves
• Toward the zero surface tension limit: The granular fingering instability
• Hamiltonian motions of plane curves and formation of singularities and bubbles