Exact Solutions for Nonlinear Development of a Kelvin-Helmholtz Instability for the Counterflow of Superfluid and Normal Components of Helium II

Relative motion of the normal and superfluid components of helium II results in the quantum Kelvin-Helmholtz instability (KHI) at their common free surface. We found the integrability and exact growing solutions for the nonlinear stage of the development of that instability. Contrary to the usual KHI of the interface between two classical fluids, the dynamics of a helium II free surface allows reduction to the Laplace growth equation, which has an infinite number of exact solutions, including the generic formation of sharp cusps at the free surface in a finite time.

Figure 1

A schematic of counterflow in superfluid ${}^4\mathrm{He}$. Heating results in the flux of heat $\mathbf{Q}$, which is carried by the normal fluid component with velocity ${\mathbf{v}}_n$, while the superfluid component moves in the opposite direction with the velocity ${\mathbf{v}}_s$. Both components coexist in the same volume of fluid and share the same free surface.

Figure 2

Evolution of an initial periodic perturbation of the free surface $y(x)$ for Eq. (22) with $kA(0)\approx 0.15$. The surface shape is shown over one spatial period for the times $ckt=0$, 0.8, 1.2, 1.4 until the cusp singularity is formed. The dashed line shows the unperturbed free surface, $y\equiv 0$.