Transition to ballistic regime for heat transport in helium II

TL;DR

This paper investigates how heat transport in superfluid helium transitions from classical to ballistic regimes in narrow channels, extending existing models to account for non-local effects and boundary slip.

Contribution

It introduces an extended heat flux equation with non-local terms to describe the transition to ballistic heat transport in superfluid helium within narrow channels.

Findings

Effective thermal conductivity scales with R^2 in Landau regime.

Effective thermal conductivity scales with R*ell in ballistic regime.

Transition depends on channel size relative to phonon mean-free path.

Abstract

The size-dependent and flux-dependent effective thermal conductivity of narrow capillaries filled with superfluid helium is analyzed from a thermodynamic continuum perspective. The classical Landau evaluation of the effective thermal conductivity of quiescent superfluid, or the Gorter-Mellinck regime of turbulent superfluids, are extended to describe the transition to ballistic regime in narrow channels wherein the radius $R$ is comparable to (or smaller than) the phonon mean-free path $\ell$ in superfluid helium. To do so we start from an extended equation for the heat flux incorporating non-local terms, and take into consideration a heat slip flow along the walls of the tube. This leads from an effective thermal conductivity proportional to $R^2$ (Landau regime) to another one proportional to $R\ell$ (ballistic regime). We consider two kinds of flows: along cylindrical pipes and along two infinite parallel plates.