Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy

Abstract

We show that unstable fingering patterns of two dimensional flows of viscous fluids with open boundary are described by a dispersionless limit of the KdV hierarchy. In this framework, the fingering instability is linked to a known instability leading to regularized shock solutions for nonlinear waves, in dispersive media. The integrable structure of the flow suggests a dispersive regularization of the finite-time singularities.

Published version

References in corpus (2)

• Normal random matrix ensemble as a growth problem
• Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy

Cited by in corpus (11)

• Bubble break-off in Hele-Shaw flows : Singularities and integrable structures
• Singular limit of Hele-Shaw flow and dispersive regularization of shock waves
• Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent
• Shocks and finite-time singularities in Hele-Shaw flow
• Matrix Models, Complex Geometry and Integrable Systems. I
• The double scaling limit method in the Toda hierarchy
• Semiclassical expansions in the Toda hierarchy and the hermitian matrix model
• Matrix Models, Complex Geometry and Integrable Systems. II
• Integrability-preserving regularizations of Laplacian Growth
• Toward the zero surface tension limit: The granular fingering instability
• Classical and Quantum Integrability in Laplacian Growth