Transition to ballistic regime for heat transport in helium II

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, is 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 ℓ 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 Rℓ (ballistic regime). We consider two kinds of flows: along cylindrical pipes and along two infinite parallel plate

Effective thermal conductivity of superfluid helium: laminar, turbulent and ballistic regimes

“…Note that for channel with infinite plates we recover the same results obtained in [9] (see equation (4.9)). In the case of a very small distance between plates then the wall effects are emphasized and expression (22) can be written …”

“…Note that for a channel with infinite parallel plates the second term in the square brackets in the above expressions has to be removed, hence the heat flux acrossing the transversal section of width a and thickness b is [9] (14)Q = − b 3 a 12…”

“…The heat carriers in Helium II are phonons and rotons as pointed out by Landau [8], even though in the following we refer to phonons for the sake of simplicity, because at very low temperatures their contribution is higher than that of rotons. The aim of this paper is indeed to pay attention to the ballistic regime and continue the studies begun in [9,10], but without dismissing the other two regimes, i.e. in the framework of heat transfer equations able to deal with the three regimes.…”

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

“…In the more usual two-fluid model by Landau and Tisza [86][87][88][89], instead, it is considered to be composed of a normal viscous component and a superfluid one. Their densities, respectively, are ρ n and ρ s in such a way that ρ = ρ n + ρ s .…”

“…In steady states, the heat transfer in turbulent liquid helium [85,[89][90][91][92] may be described by an equation analogous to Eq. (6), taking the form…”