Effective thermal conductivity of helium II: from Landau to Gorter–Mellink regimes

Abstract

The size-dependent and flux-dependent effective thermal conductivity of narrow channels filled with He II is analyzed. The classical Landau evaluation of the effective thermal conductivity of quiescent He II is extended to describe the transition to fully turbulent regime, where the heat flux is proportional to the cubic root of the temperature gradient (Gorter–Mellink regime). To do so, we use an expression for the quantum vortex line density L in terms of the heat flux considering the influence of the walls. From it, and taking into account the friction force of normal component against the vortices, we compute the effective thermal conductivity as a function of the heat flux, and we discuss in detail the corresponding size dependence

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

“…For it, a further equation has been necessary (equation (36)), which we have already proposed in [10], and we have used here for the sake of uniformity. Unfortunately, there are not enough papers dealing with parallelepiped counterflow channels.…”

“…[10] it was seen that it does not contribute to the full heat flow accross a transversal section of the channel in counterflow experiments.…”

“…In this section we assume the simplest expression for σ q , namely σ q = −q, while a more general assumption is required to take into account of the presence of vortices [10,26,27].…”

“…This expression may be compared with the more well-known thermal conductivity of superfluid helium in laminar flow in cylindrical tube of radius R, which is [10] (16)…”

“…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.…”