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 (1)

• Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy

Cited by in corpus (1)

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