Longitudinal counterflow in turbulent liquid helium: velocity profile of the normal component

In this paper, the velocity profile of the normal component in the stationary flow of turbulent superfluid helium inside a cylindrical channel is determined, making use of a one-fluid model with internal variables derived from Extended Thermodynamics. In the hypothesis of null barycentric velocity of the fluid (the so-called counterflow situation) it is seen that, in the presence of a sufficiently high vortex length density, the velocity profile of the normal component becomes very flat in the central region of the channel. Thus, a central flat profile of the normal fluid does not necessarily imply that the flow of the normal component is turbulent.

Stationary heat flux profile in turbulent helium II in a semi-infinite cylindrical channel

In this paper we determine a set of solutions for a system of partial dif- ferential equations describing stationary heat flux in a semi-infinite cylindrical channel filled with turbulent superfluid helium. This study uses a continuous model for liquid helium II, derived from Extended Thermodynamics, in which the heat flux q is a fundamental variable. The influence of the vortex line den- sity on the radial distribution of the heat flux is especially discussed.

Heat rectification in He II counterflow in radial geometries

Abstract We consider heat rectification in radial flows of turbulent helium II, where heat flux is not described by Fourier's law, but by a more general law. This is different from previous analyses of heat rectification, based on such law. In our simplified analysis we show that the coupling between heat flux and the gradient of vortex line density plays a decisive role in such rectification. Such rectification will be low at low and high values of the heat rate, but it may exhibit a very high value at an intermediate value of the heat rate. In particular, for a given range of values for the incoming heat ow, the outgoing heat flow corresponding to the exchange of internal and external tem…

Effects of heat flux on lambda transition in liquid 4He,

This paper is concerned with the derivation of a phase field model for λ-transition in 4He, when the liquid is subject to pressure and heat flux. As parameter that controls the transition, a field f that is the geometrical mean between the density of the fluid and that of the superfluid is used. The resulting model, that is a generalization of previous papers on the same subject, chooses as field variables the density, the velocity, the temperature and the heat flux, in addition to this field f. The restrictions on the constitutive quantities are obtained by using the Liu method of Lagrange multipliers. New results with respect to previous models are the presence of non-local terms to descr…

Contribution of the normal component to the thermal resistance of turbulent liquid helium

Previous results for the velocity profile of the normal component of helium II in counterflow are used to evaluate the viscous contribution to the effective thermal resistance. It turns out that such a contribution becomes considerably higher than the usual Landau estimate, because in the presence of vortices, the velocity profile is appreciably different from the Poiseuille parabolic profile. Thus, a marked increase in the contribution of the normal component to the thermal resistance with respect to the viscous Landau estimate does not necessarily imply that the normal component is turbulent. Furthermore, we examine the influence of a possible slip flow along the walls when the radius of …

HEAT FLUX IN SUPERFLUID TRANSITION AND IN TURBULENT HELIUM COUNTERFLOW

Nonlocal model of Superfluid Turbulence: Constitutive Theory

In this paper, the constitutive restrictions for the fluxes in a nonlocal model of superfluid turbulence are deduced from the entropy principle, using the Liu method of Lagrange multipliers. The proposed model chooses as fundamental fields the density, the velocity, the energy density, the heat flux, and the averaged vortex line length per unit volume. The onstitutive quantities are assumed to depend on the fundamental fields and on their first derivative.

Non-equilibrium Thermodynamical Description of Superfluid Transition in Liquid Helium

In previous papers a phase field model for λ-transition in 4He was proposed, which is able to describe the influence of the heat flux on the temperature transition. The model presented here generalizes previous results taking into account of a homogeneous presence of quantized vortices below the λ-transition. As parameter that controls the transition, a dimensionless field f linked to the modulus of the condensate wave function is used. In addition to the field f , the resulting model chooses the following field variables: Density, velocity, temperature and heat flux. Nonlocal terms to describe inhomogeneities in the field variables and dissipative effects of mechanical and thermal origin…

Vortex diffusion and vortex-line hysteresis in radial quantum turbulence

Abstract We study the influence of vortex diffusion on the evolution of inhomogeneous quantized vortex tangles. A simple hydrodynamical model to describe inhomogeneous counterflow superfluid turbulence is used. As an illustration, we obtain solutions for these effects in radial counterflow of helium II between two concentric cylinders at different temperatures. The vortex diffusion from the inner hotter cylinder to the outer colder cylinder increases the vortex length density everywhere as compared with the non-diffusive situation. The possibility of hysteresis in the vortex line density under cyclical variations of the heat flow is explored.

Second sound near lambda transition in presence of quantum vortices

In this paper, temperature waves (also known as second sound) are consid- ered, with their respective coupling with waves in the order parameter describing the transition from normal phase to superfluid phase, and with waves in the vortex length density. We analyze the coupling between these three kinds of waves and explore its relevance in situations not far from the lambda transition. In particular, the expres- sions for the second sound speed and second sound attenuation are explicitly obtained within some approximations, showing the influence of the order parameter and the vortex length density, which is decisive close to the transition.

Thermodynamic approach to vortex production and diffusion in inhomogeneous superfluid turbulence

In this paper, we use a non-equilibrium thermodynamic framework to generalize a previous nonlocal model of counterflow superfluid turbulence to incorporate some new coupled terms which may be relevant in the evolution of inhomogeneous vortex tangles. The theory chooses as fundamental fields the energy density, the heat flux, and the averaged vortex line length per unit volume. The constitutive quantities are assumed to depend on the fundamental fields and on their first spatial derivatives, allowing us to describe thermal dissipation, vortex diffusion and a new contribution to vortex formation. The restrictions on the constitutive relations are deduced from the entropy principle, using the …

Coupling of heat flux and vortex polarization in superfluid helium

We consider a macroscopic description of the mutual influence between heat flux and vortex polarization in superfluid helium, in which the vortices produce a lateral deviation of the heat flux, and the heat flux produces a lateral drift of vortices. This coupling is a consequence of a microscopic Magnus force and mutual friction force between the vortices and the flow of excitations carrying the heat. We keep track of these effects with simplified macroscopic equations, and we apply them to second sound propagation between rotating concentric cylinders and to spatial distribution of polarization across a rectangular channel with vortices polarized orthogonally to the channel in the presence…

Effective thermal conductivity of superfluid helium in short channels

The aim of this paper is to explore how the effective thermal conductivity of small channels filled with superfluid helium II in the laminar regime separates from the classical Landau expression as the channel becomes shorter. The Landau expression is valid for fully developed Poiseuille flow for the normal component, and therefore is suitable for long channels. By taking into account entrance effects, we show a transition from a heat flux proportional to ∆T /l (Landau regime) for long channels, to a heat flux proportional to l^(1/3) (∆T /l)^(2/3) for short channels.

Propagation of plane and cylindrical waves in turbulent superfluid helium

In this paper, the equations that govern the propagation of plane and cylindrical waves in turbulent superfluid solutions in some simplified cases are determined.

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

Abstract 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 andincounterflowvelocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from asecondorder expansionofthe Vinenequationitself, andthose whichare notrelated to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments.