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

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

A non-local model of thermal energy transport: The fractional temperature equation

Non-local models of thermal energy transport have been used in recent physics and engineering applications to describe several "small-scale" and/or high frequency thermodynamic processes as shown in several engineering and physics applications. The aim of this study is to extend a recently proposed fractional-order thermodynamics ([5]), where the thermal energy transfer is due to two phenomena: A short-range heat flux ruled by a local transport equation; a long-range thermal energy transfer that represents a ballistic effects among thermal energy propagators. Long-range thermal energy transfer accounts for small-scale effects that are assumed proportional to the product of the interacting masses, to a distance-decaying function, as well as to their relative temperature. In this paper the thermodynamic consistency of the model is investigated obtaining some restrictions on the functional class of the distance decaying function that rules the strength of the long-range thermal energy transfer. As the distance-decaying function is assumed in the form of a power-law decay a novel temperature equation involving multidimensional spatial Marchaud α-order derivatives (0 ≤ α ≤ 1) of the temperature field in the body is obtained. Some analytical and numerical solutions of the fractional-order temperature equation have been provided in the paper to show the capabilities of the proposed model and the influence of model parameter

Inhomogeneous vortex tangles in counterflow superfluid turbulence: flow in convergent channels

We investigate the evolution equation for the average vortex length per unit volume L of superfluid turbulence in inhomogeneous flows. Inhomogeneities in line density L and in counterflow velocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from a second order expansion of the Vinen equation itself, and those which are not related to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments

Hydrodynamics of superfluid 4He without dissipative effects

This review paper is the first of a series of papers focusing on the singular behavior of superfluids. Here, we will consider the laminar flow of superfluid $^4$He. It is shown that the properties of helium II can be explained both considering it as a two-fluid mixture or as a single fluid with extremely high thermal conductivity and extremely small viscosity. More specifically, in this paper is shown that the anomalous effects in helium II are, in a large measure, a consequence of entropy conservation. Indeed, it will be shown that these effects can be explained imposing entropy conservation in a two-fluid mixture as well as in a single fluid described by extended thermodynamics. Firstly, the rational theory of mixtures of ideal fluids is applied to describe the anomalous behaviour of superfluid helium II. Then, rational extended thermodynamics is applied to a fluid where entropy is conserved. The paper is not exclusively a review of already published works, but it contains several new results and proposals which are formulated in it for the first time

Thermodynamical derivation of a hydrodynamical model of unhomogeneous superfluid turbulence

In this paper, we build up a thermodynamical model of inhomogeneous superfluid turbulence to describe vortex diffusion in inhomogeneous turbulent tangles, and a coupling between second sound and vortex-density waves. The theory chooses as fundamental fields the density, the velocity, the energy density, the heat flux, and the averaged vortex line length per unit volume. The restrictions on the constitutive quantities are deduced from the entropy principle, using the Liu method of Lagrange multipliers. Field equations are written and the wave propagation is studied with the aim to describe the mutual interactions between the second sound and the vortex tangle

Transition to superfluidity in liquid 4He

In this work the transition from normal liquid helium I to superfluid liquid helium II, controlled by temperature and pressure, is studied in the simplified assumption of absence of viscosity. A macroscopic thermodynamical model is presented, which chooses as new independent fields the heat flux q and a phase field function f. For the heat flux a modification of Cattaneo equation is written, while for the function f a time dependent Ginzburg-Landau equation is proposed