Vortex density waves and high-frequency second sound in superfluid turbulence hydrodynamics

In this paper we show that a recent hydrodynamical model of superfluid turbulence describes vortex density waves and their effects on the speed of high-frequency second sound. In this frequency regime, the vortex dynamics is not purely diffusive, as for low frequencies, but exhibits ondulatory features, whose influence on the second sound is here explored.

Description and evolution of anisotropy in superfluid vortex tangles with counterflow and rotation

We examine several vectorial and tensorial descriptions of the geometry of turbulent vortex tangles. We study the anisotropy in rotating counterflow experiments, in which the geometry of the tangle is especially interesting because of the opposite effects of rotation, which orients the vortices, and counterflow, which randomizes them. We propose to describe the anisotropy and the polarization of the vortex tangle through a tensor, which contains the first and second moments of the distribution of the unit vector ${\mathbf{s}}^{\ensuremath{'}}$ locally tangent to the vortex lines. We use an analogy with paramagnetism to estimate the anisotropy, the average polarization, the polarization fluc…

K-ϵ-L model in turbulent superfluid helium

Abstract We generalize the K − ϵ model of classical turbulence to superfluid helium. In a classical viscous fluid the phenomenological eddy viscosity characterizing the effects of turbulence depends on the turbulent kinetic energy K and the dissipation function ϵ , which are mainly related to the fluctuations of the velocity field and of its gradient. In superfluid helium, instead, we consider the necessary coefficients for describing the effects of classical and quantum turbulence, involving fluctuations of the velocity, the heat flux, and the vortex line density of the quantized vortex lines. By splitting the several fields into a time-average part and a fluctuating part, some expressions…

Spectral energy distribution and generalized Wien's law for photons and cosmic string loops

Physical objects with energy $u_w(l) \sim l^{-3w}$ with $l$ characteristic length and $w$ a dimensionless constant, lead to an equation of state $p=w\rho$, with $p$ the pressure and $\rho$ the energy density. Special entities with thisbproperty are, for instance, photons ($u = hc/l$, with $l$ the wavelength) with $w = 1/3$, and some models of cosmic string loops ($u =(c^4/aG)l$, with $l$ the length of the loop and $a$ a numerical constant), with $w = -1/3$. Here, we discuss some features of the spectral energy distribution of these systems and the corresponding generalization of Wien's law, which in terms of $l$ has the form $Tl_{mp}^{3w}=constant$, being $l_{mp}$ the most probable size of …

Hydrodynamic equations of anisotropic, polarized and inhomogeneous superfluid vortex tangles

We include the effects of anisotropy and polarization in the hydrodynamics of inhomogeneous vortex tangles, thus generalizing the well known Hall-Vinen-Bekarevich-Khalatnikov equations, which do not take them in consideration. These effects contribute to the mutual friction force ${\bf F}_{ns}$ between normal and superfluid components and to the vortex tension force $\rho_s{\bf T}$. These equations are complemented by an evolution equation for the vortex line density $L$, which takes into account these contributions. These equations are expected to be more suitable than the usual ones for rotating counterflows, or turbulence behind a cylinder, or turbulence produced by a grid of parallel th…

Relation Between the State of a System as Isolated and as Open

We discuss the differences in considering the description of a system as isolated or as a subsystem of a wider isolated system. In the latter case, a description in terms of a density operator directly arises without involving probability concepts, but as an orbit invariant. The non-probabilistic physical interpretation of the density operator is mathematically discussed.

A mathematical model of counterflow superfluid turbulence describing heat waves and vortex-density waves

The interaction between vortex density waves and high-frequency second sound in counterflow superfluid turbulence is examined, incorporating diffusive and elastic contributions of the vortex tangle. The analysis is based on a set of evolution equations for the energy density, the heat flux, the vortex line density, and the vortex flux, the latter being considered here as an independent variable, in contrast to previous works. The latter feature is crucial in the transition from diffusive to propagative behavior of vortex density perturbations, which is necessary to interpret the details of high-frequency second sound.

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.

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…

The Probability Law for Generic Density Operators

In this chapter, the probability law of the non-null eigenstates of a generic density operator—studied in the previous chapter—is determined, by showing that given the composite system and the subsystem being considered, a mapping arises which associates a universal probability distribution to the non-null eigenstates of the generic density operator. We thus recover the Born statistical interpretation without having assumed it as a postulate.

Energy and temperature of superfluid turbulent vortex tangles

We consider three aspects of turbulent vortex tangles in superfluids. First, we outline some contributions to the Vinen’s equation for the time evolution of the vortex line density, related to the presence of pinned vortices incorporating the effects of the walls. Afterwards, we analyze some aspects of the energy balance of the vortex tangle, related to frictional dissipation and to vortex formation and destruction. Finally, we explore the concept of an effective temperature for the vortex tangle, related to the average energy of the vortex loops and to the diffusion coefficient of vortex lines. The combination of these ideas suggests some formal similarities with other kinds of driven none…

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 as a function of the heat flux, and we disc…

Effective temperature and scaling laws of polarized quantum vortex bundles

Abstract An effective non-equilibrium temperature is defined for (locally) polarized and dense turbulent superfluid vortex bundles, related to the average energy of the excitations (Kelvin waves) of vortex lines. In the quadratic approximation of the excitation energy in terms of the wave amplitude A, a previously known scaling relation between amplitude and wavelength k of Kelvin waves in polarized bundles, namely A ∝ k − 1 / 2 , follows from the homogeneity of the effective temperature. This result is analogous to that of the well-known equipartition result in equilibrium systems.

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 …

Thermal duality and thermodynamics of micro black holes

Starting from a generalized black hole entropy with logarithmic area corrections, in this paper we obtain (for positive value of the coefficient of the correction term) a generalized equation of state for black holes with two dual branches. In one of them (the usual one for macro black holes) T ≃ 1/M, with T temperature and M mass. In the other one, for micro black holes, instead, T ≃ M. We compare the equilibrium and stability between macro black holes and electromagnetic radiation in a finite box with reflecting walls, with the dual situation corresponding to micro black holes and cosmic string loops, also in a finite box. In this model, the dual phenomenon of evaporation of unstable mac…

Superfluid turbulence in rotating containers: Phenomenological description of the influence of the wall

In this paper a previous equation for the evolution of vortex line density L in counterflow superfluid turbulence in rotating containers is generalized, in order to take into account the influence of the walls. This model incorporates the effects of counterflow velocity V and of angular velocity {omega} of the container, and introduces corrective terms depending on {delta}/d, {delta} being the intervortex spacing, of the order L{sup -1/2}, and d the diameter of the channel. The stability of the solutions for L, for several regimes of averaged counterflow velocity V and angular velocity {omega}, is analyzed. Our mathematical analysis reveals that qualitative consistency allows us to reduce t…

Transition to ballistic regime for heat transport in helium II

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 f…

Heat solitons and thermal transfer of information along thin wires

Abstract The aim of this paper is to consider soliton propagation of heat signals along a cylinder whose heat exchange with the environment is a non-linear function of the difference of temperatures of the cylinder and the environment and whose heat transfer along the system is described by the Maxwell–Cattaneo equation. To find the soliton solutions we use the auxiliary equation method. Our motivation is to obtain and compare the speed of propagation, the maximum rate of information transfer, and the energy necessary for the transfer of one bit of information for different solitons, by assuming that a localized soliton may carry a bit of information. It is shown that a given total power (e…

Entropy Flux in Non-Equilibrium Thermodynamics

Extended thermodynamics of polymers and superfluids

Abstract Polymer solutions and turbulent superfluids have in common the presence of a complex tangle of lines – macromolecules in the former, quantized vortex lines in the latter – which contribute to the internal friction and viscous pressure of the system and make them typical non-Newtonian fluids. Here we briefly review some recent studies on such tangles and their consequences on the dynamics and thermodynamics of the whole system, using the framework of extended irreversible thermodynamics. For polymer solutions, we deal with the coupling of diffusion and viscous pressure and its effects on the stability of the solution and shear-induced phase separation; for superfluids, we focus our …

Refrigeration of an array of cylindrical nanosystems by superfluid helium counterflow

Abstract Motivated by the challenge of computer refrigeration, we study the limits set by the transition to quantum turbulence on the cooling of an array of heat-producing cylindrical nanosystems by means of superfluid-helium counterflow. The effective thermal conductivity in laminar counterflow superfluid helium is obtained in channels with rectangular cross section, through arrays of mutually parallel cylinders and in the combined situation of arrays of orthogonal cylinders inside the rectangular channel. The maximum cooling capacity is analyzed on the condition that turbulence is avoided and that the highest temperature does not exceed the lambda temperature.

Fractal dimension of superfluid turbulence : A random-walk toy model

This paper deals with the fractal dimension of a superfluid vortex tangle. It extends a previous model [J. Phys. A: Math. Theor. {\bf 43}, 205501 (2010)] (which was proposed for very low temperature), and it proposes an alternative random walk toy model, which is valid also for finite temperature. This random walk model combines a recent Nemirovskii's proposal, and a simple modelization of a self-similar structure of vortex loops (mimicking the geometry of the loops of several sizes which compose the tangle). The fractal dimension of the vortex tangle is then related to the exponents describing how the vortex energy per unit length changes with the length scales, for which we take recent pr…

The State of a Quantum System as a Subsystem of a Composite System

The notion of state in quantum systems is analized, a non-probabilistic definition of state provided, the Zurek’s concept of envariance is mathematically formulated, and the characterization of a state through its properties is discussed.

A duality-invariant Einstein-Planck relation and its consequences on micro-black holes.

We discuss the consequences of a duality-invariant Einstein–Planck (DIEP) relation on the equation of state of micro black holes. The results are analogous to those obtained from the "world-crystal" model, but with some significative differences, as for instance a limiting vanishing value for temperature for very small black holes. The model leads to a total evaporation of micro black holes but with the final stage being very slow.

Duality-invariant Einstein-Planck relation and the speed of light at very short wavelengths

We propose a generalized Einstein-Planck relation for photons which is invariant under the change $\ensuremath{\lambda}/a{l}_{P}$ to $a{l}_{P}/\ensuremath{\lambda}$, $\ensuremath{\lambda}$ being the photon wavelength, ${l}_{P}$ Planck's length, and $a$ a numerical constant. This yields a wavelength-dependent speed of light $v(\ensuremath{\lambda})=c/(1+{a}^{2}({l}_{P}/\ensuremath{\lambda}{)}^{2})$, with $c$ the usual speed of light in vacuo, indicating that the speed of light should decrease for sufficiently short wavelengths. We discuss the conceptual differences with the previous proposals related to a possible decrease of the speed of light for very short wavelengths based on quantum flu…

Duality relation between radiation thermodynamics and cosmic string loop thermodynamics

We discuss thermodynamics of electromagnetic radiation, with p=(1/3){rho} and S{proportional_to}T{sup 3}V, and of cosmic string loops, with p=-(1/3){rho} and S{proportional_to}T{sup -3}V, where p stands for pressure, T temperature, {rho} energy density, S entropy, and V volume. We write the thermodynamic formalisms under a common framework that illustrates their formal relationship and allows us to go from one to the other through a smooth transformation. From a microscopic perspective, these relations arise from the energy relations u({lambda})=hc/{lambda} for the photons of electromagnetic radiation, and u(l)=(c{sup 4}/a{sup 2}G)l for cosmic string loops, a being a numerical (dimensionles…

Generalization of Vinen’s equation including transition to superfluid turbulence

A phenomenological generalization of the well known Vinen equation for the evolution of vortex line density in superfluid counterflow turbulence is proposed. This generalization includes nonlinear production terms in the counterflow velocity and corrections depending on the diameter of the tube. The equation provides a unified framework for the various phenomena (stationary states and transitions) present in counterflow superfluid turbulence: in fact, it is able to describe the laminar regime, the first-order transition from laminar to turbulent TI state, the two turbulent states, the transition from TI to TII turbulent states, and it yields a slower decay of the counterflow turbulence than…

Second sound, superfluid turbulence, and intermittent effects in liquid helium II

Thermal solitons along wires with flux-limited lateral exchange

We obtain some exact solutions in the context of solitons, for heat conduction with inertia along a cylinder whose heat exchange with the environment is a non-linear function of the difference of temperatures of the cylinder and the environment, due to a flux-limiter behavior of the exchange. We study the consequences of heat transfer and information transfer along the wire, and we compare the situation with analogous solitons found in nonlinear lateral radiative exchange studied in some previous papers. We also find further exact solutions in terms of Weierstrass elliptic functions for the sake of completeness.

Refrigeration of an Array of Cylindrical Nanosystems by Flowing Superfluid Helium

We consider the refrigeration of an array of heat-dissipating cylindrical nanosystems as a simplified model of computer refrigeration. We explore the use of He II as cooling fluid, taking into account forced convection and heat conduction. The main conceptual and practical difficulties arise in the calculation of the effective thermal conductivity. Since He II does not follow Fourier’s law, the effective geometry-dependent conductivity must be extracted from a more general equation for heat transfer. Furthermore, we impose the restrictions that the maximum temperature along the array should be less than (Formula presented.) transition temperature and that quantum turbulence is avoided, in o…

Generalization of Vinen's equation including transition to superfluid turbulence

Non-equilibrium temperature of well-developed quantum turbulence

Abstract A non-equilibrium effective temperature of quantum vortex tangles is defined as the average energy of closed vortex loops. The resulting thermodynamic expressions for the entropy and the energy in terms of the temperature of the tangle are confirmed by a microscopic analysis based on a potential distribution function for the length of vortex loops. Furthermore, these expressions for the entropy and energy in terms of temperature are analogous to those of black holes: this may be of interest for establishing further connections between topological defects in superfluids and cosmology.

A thermodynamical model of inhomogeneous superfluid turbulence

In this paper we perform a thermodynamical derivation of a nonlinear hydrodynamical model of inhomogeneous superfluid turbulence. 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 derived from the entropy principle, using the Liu method of Lagrange multipliers. The mathematical and physical consequences deduced by the theory are analyzed both in the linear and in the nonlinear regime. 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 t…

Three Duality Symmetries between Photons and Cosmic String Loops, and Macro and Micro Black Holes

We present a review of two thermal duality symmetries between two different kinds of systems: photons and cosmic string loops, and macro black holes and micro black holes, respectively. It also follows a third joint duality symmetry amongst them through thermal equilibrium and stability between macro black holes and photon gas, and micro black holes and string loop gas, respectively. The possible cosmological consequences of these symmetries are discussed.

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 …

Phenomenological description of sedimentation in turbulent vortex tangles

The aim of this Brief Report is to provide a simple intuitive derivation of the results for sedimentation velocity of a small spherical particle in a counterflow vortex tangle in turbulent superfluid. When the velocity of the tangle vortex lines is small as compared to that of the particle, our results reduce to those obtained previously by other authors through more complex arguments, except for a logarithmic dependence of one of the coefficients on the vortex line density. Comparison of both derivations may be useful to clarify the range of validity of the expressions for the forces between the particle and the tangle.

Vortex length, vortex energy and fractal dimension of superfluid turbulence at very low temperature

By assuming a self-similar structure for Kelvin waves along vortex loops with successive smaller scale features, we model the fractal dimension of a superfluid vortex tangle in the zero temperature limit. Our model assumes that at each step the total energy of the vortices is conserved, but the total length can change. We obtain a relation between the fractal dimension and the exponent describing how the vortex energy per unit length changes with the length scale. This relation does not depend on the specific model, and shows that if smaller length scales make a decreasing relative contribution to the energy per unit length of vortex lines, the fractal dimension will be higher than unity. F…

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…

Non-equilibrium thermodynamics, heat transport and thermal waves in laminar and turbulent superfluid helium

This review paper puts together some results concerning non equilibrium thermodynamics and heat transport properties of superfluid He II. A one-fluid extended model of superfluid helium, which considers heat flux as an additional independent variable, is presented, its microscopic bases are analyzed, and compared with the well known two-fluid model. In laminar situations, the fundamental fields are density, velocity, absolute temperature, and heat flux. Such a theory is able to describe the thermomechanical phenomena, the propagation of two sounds in liquid helium, and of fourth sound in superleak. It also leads in a natural way to a two-fluid model on purely macroscopical grounds and allow…

Nonlocal effects in superfluid turbulence: Application to the low-density- to high-density-state transition and to vortex decay

We discuss a phenomenological equation for the evolution of vortex tangle in counterflow superfluid turbulence, which takes into account the influence of the nonlocal effects, introducing into the original equation of Vinen two simple corrective terms dependent on a nonvanishing ratio between the average separation between vortex lines and the diameter of the channel. The equation allows one to describe, in relatively good agreement with experimental results, the two turbulent regimes present in counterflow superfluid turbulence and the transition between them. The decay rate of the vortex line density L, when the heat flux is suddenly turned off, is also investigated; due to the simplicity…

Entropy flux in non-equilibrium thermodynamics

Abstract An important problem in thermodynamics is the link between the entropy flux and the heat flux, for phenomena far from equilibrium. As an illustration we consider here the case of a rigid heat conductor subject to heating. The expression of the entropy flux is determined by the expressions of the evolution equations of the basic variables. It is shown that the coefficient relating entropy and heat fluxes differs far from equilibrium from the inverse of the non-equilibrium temperature θ . The particular case in which these two quantities are identical is examined in detail. A simple but intuitive physical illustration of the results is proposed. A comparison with information theory i…

Energy of string loops and thermodynamics of dark energy

We discuss the thermodynamic aspects of a simple model of cosmic string loops, whose energy is nonlinearly related to their lengths. We obtain in a direct way an equation of state having the form p=-(1+{alpha}){rho}/3, with {rho} the energy density and 1+{alpha} the exponent which relates the energy u{sub l} of a loop with its length l as u{sub l}{approx}l{sup 1+{alpha}}. In the linear situation ({alpha}=0) one has p=-{rho}/3, in the quadratic one ({alpha}=1) p=-2{rho}/3, and in the cubic case ({alpha}=2) p=-{rho}. For all values of {alpha} the entropy goes as S{approx}(2-{alpha})L{sup 3/2} (L being the string length density). The expression of S is useful to explore the behavior of such st…

Hydrodynamic Equations of Anisotropic, Polarized, Turbulent Superfluids

Thermodynamical derivation of a hydrodynamical model of inhomogeneous 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.

Vortex dynamics in rotating counterflow and plane Couette and Poiseuille turbulence in superfluid Helium

An equation previously proposed to describe the evolution of vortex line density in rotating counterflow turbulent tangles in superfluid helium is generalized to incorporate nonvanishing barycentric velocity and velocity gradients. Our generalization is compared with an analogous approach proposed by Lipniacki, and with experimental results by Swanson et al. in rotating counterflow, and it is used to evaluate the vortex density in plane Couette and Poiseuille flows of superfluid helium.

HYDRODYNAMICAL MODELS OF SUPERFLUID TURBULENCE

This review paper puts together some of our results concerning the application of non equilibrium Thermodynamics to superfluid liquid helium. Two of the most important situations of this quantum fluid are rotating superfluid and superfluid turbulence, both characterized by the presence of quantized vortices (vortex lines whose core is about 1 Angstrom and the quantum of circulation is $h/m$, $h$ being the Plank's constant and $m$ the mass of helium atom). In the first part of the work a non-standard model of superfluid helium, which considers heat flux as independent variable, is briefly recalled, and compared with the well known two-fluid model, in absence of vortices, proposed by Tisza an…

Phenomenological description of counterflow superfluid turbulence in rotating containers

In this paper a simple equation for the vortex line density describing some of the most relevant observed effects in counterflow superfluid turbulence in ${}^{4}\mathrm{He}$ in the presence of rotation is proposed. This model is based on a generalization of Vinen's equation which incorporates as additional quantity the angular velocity \ensuremath{\Omega}.

Non-Equilibrium Thermodynamics of Unsteady Superfluid Turbulence in Counterflow and Rotating Situations

The methods of nonequilibrium thermodynamics are used in this paper to relate an evolution equation for the vortex line density $L$, describing superfluid turbulence in the simultaneous presence of counterflow and rotation, to an evolution equation for the superfluid velocity ${\mathbf{v}}_{s}$, in order to be able to describe the full evolution of ${\mathbf{v}}_{s}$ and $L$, instead of only $L$. Two alternative possibilities are analyzed, related to two possible alternative interpretations of a term coupling the effects of the counterflow and rotation on the vortex tangle, and which imply some differences between situations where counterflow and rotation vectors are parallel or orthogonal …

Alternative Vinen equation and its extension to rotating counterflow superfluid turbulence

Two alternative Vinen's evolution equations for the vortex line density L in counterflow superfluid turbulence, are physically admissible and lead to analogous results in steady states. In Phys. Rev. B, 69, 094513 (2004) the most used of them was generalized to counterflow superfluid turbulence in rotating containers. Here, the analogous generalization for the alternative Vinen's equation is proposed. Both generalized Vinen's equations are compared with the experimental results, not only in steady-states but also in some unsteady situations. From this analysis follows that the solutions of the alternative Vinen's equation tend significantly faster to the corresponding final steady state val…

Nonequilibrium effective temperature of superfluid vortex tangle

An effective nonequilibrium temperature in counterflow superfluid turbulence is proposed, as a parameter characterizing a canonical probability distribution function of vortex orientation, and relating the diffusion coefficient of vortex lines to the vortex friction through an Einstein relation.

Thermodynamics of computation and linear stability limits of superfluid refrigeration of a model computing array

We analyze the stability of the temperature profile of an array of computing nanodevices refrigerated by flowing superfluid helium, under variations in temperature, computing rate, and barycentric velocity of helium. It turns out that if the variation in dissipated energy per bit with respect to temperature variations is higher than some critical values, proportional to the effective thermal conductivity of the array, then the steady-state temperature profiles become unstable and refrigeration efficiency is lost. Furthermore, a restriction on the maximum rate of variation in the local computation rate is found.