Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent

TL;DR

This paper links viscous shocks in Hele-Shaw flow at zero surface tension to Stokes phenomena of the Painleve I transcendent, revealing an integrable structure underlying flow singularities.

Contribution

It demonstrates that viscous shocks in Hele-Shaw flow correspond to Stokes lines of Painleve I, establishing a novel connection between fluid dynamics and integrable systems.

Findings

Viscous shocks are equivalent to Stokes level-lines of Painleve I.

Painleve I provides an integrable deformation of Hele-Shaw flow.

Flow passing through singularities is described by Painleve I solutions.

Abstract

In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers [1, 2] we have showed that singularities trigger viscous shocks propagating through the viscous fluid. Here we show that the weak solution of the Hele-Shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the Painleve I equation. We argue that the Painleve I equation provides an integrable deformation of the Hele-Shaw problem which describes flow passing through singularities. In this interpretation shocks appear as Stokes level-lines of the Painleve linear problem.