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

TL;DR

This paper links unstable fingering patterns in Hele-Shaw flows to a dispersionless KdV hierarchy, revealing a connection to shock solutions and suggesting a dispersive regularization of singularities.

Contribution

It introduces a novel integrable framework for understanding fingering instabilities in viscous flows using the dispersionless KdV hierarchy.

Findings

Fingering patterns are described by a dispersionless KdV hierarchy.

The instability relates to shock solutions in nonlinear waves.

Dispersive regularization may prevent finite-time singularities.

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.