Exact solutions for nonlinear development of Kelvin-Helmholtz instability for counterflow of superfluid and normal components of Helium II

TL;DR

This paper derives exact nonlinear solutions for Kelvin-Helmholtz instability at the free surface of Helium II, revealing cusp formation and finite-time singularities unique to superfluid-normal fluid interfaces.

Contribution

It provides the first exact solutions for the nonlinear development of KHI in Helium II, highlighting the special dynamics due to superfluidity and normal fluid interaction.

Findings

Exact solutions show cusp formation at finite time

Dynamics reduce to Laplace growth equation

Unique behavior compared to classical KHI

Abstract

A relative motion of the normal and superfluid components of Helium II results in Kelvin-Helmholtz instability (KHI) at their common free surface. We found the exact solutions for the nonlinear stage of the development of that instability. Contrary to the usual KHI of the interface between two fluids, the dynamics of Helium II free surface allows decoupling of the governing equations with their reduction to the Laplace growth equation which has the infinite number of exact solutions including the formation of sharp cusps at free surface in a finite time.