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.

Some typos corrected, added journal reference

References in corpus (3)

• Normal random matrix ensemble as a growth problem
• 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 (7)

• Fingering versus stability in the limit of zero interfacial tension
• Bubble break-off in Hele-Shaw flows : Singularities and integrable structures
• Random Matrices in 2D, Laplacian Growth and Operator Theory
• Shocks and finite-time singularities in Hele-Shaw flow
• The double scaling limit method in the Toda hierarchy
• Semiclassical expansions in the Toda hierarchy and the hermitian matrix model
• Toward the zero surface tension limit: The granular fingering instability