Turbulent Superfluid Profiles and Vortex Density Waves in a Counterflow Channel

In this paper we study the two-dimensional profiles of the superfluid component velocity and the quantized vortex-points density in a counterflow channel where the influence of the walls cannot be neglected. The numerical results obtained show the presence of vortex density waves in the channel, as shown in a recent paper by means of the one-fluid model.

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.

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 …

Non-classical Velocity Statistics in Counterflow Quantum Turbulence

In this work we analyse the statistical distribution of turbulent superfluid velocity components in a He II counterflow channel, via two-dimensional numerical simulations pre- sented in past studies. The Probability Density Functions (PDFs) of the superfluid velocity components are investigated at lengthscales smaller than the average intervortex spacing, for varying vortex densities and different wall-normal distances. The results obtained con- firm the non-classical signature of quantum turbulence already observed in past numerical studies.

Large-scale normal fluid circulation in helium superflows

We perform fully-coupled numerical simulations of helium II pure superflows in a channel, with vortex- line density typical of experiments. Peculiar to our model is the computation of the back-reaction of the superfluid vortex motion on the normal fluid and the presence of solid boundaries. We recover the uniform vortex-line density experimentally measured employing second sound resonators and we show that pure superflow in helium II is associated with a large-scale circulation of the normal fluid which can be detected using existing particle-tracking visualization techniques.

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…

Vortex line density in plane Couette flow in superfluid helium

Alternative Vinen's equation and its extension to counterflow rotational superfluid turbulence

A Mathematical Model of Counterflow Superfluid Turbulence in Presence of Combined Rotation and Counterflow

Onde di densità di vortici nella turbolenza superfluida

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.

Exact solutions of the Zakharov equations

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…

Two relaxation times and thermal nonlinear waves along wires with lateral heat exchange

Abstract We propose a model for studying several nonlinear waves for heat transport along a cylindrical system with lateral non-linear heat transfer to the environment. We consider relaxational equations, each with its own relaxation time, for longitudinal heat transfer and for lateral heat transfer across the wall. We consider two kinds of nonlinear lateral heat transport: radiative heat transport, and flux-limited heat transport. This work generalizes our previous studies in which the relaxation time for the lateral heat transfer was considered equal to that of the longitudinal heat flux. We explore the influence of both relaxation times on the propagation speed of linear and nonlinear wa…

Travelling wave solutions of nonlinear equations using the Auxiliary Equation Method

In this paper we obtain travelling wave solutions of nonlinear partial differential equations starting from a different reducible hyperelliptic equation as an auxiliary equation which does not appear in any other paper. We point out that all the cases, to our knowledge, considered in the literature are included in this paper, so our work exhausts all the reducible cases of the hyperelliptic equation to the genus one.

Onde di calore e onde di densità di vortici nella turbolenza superfluida

Painlev\'{e} analysis for a generalized nonlinear Schr\"{o}dinger equation

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…

A mathematical description of glitches in neutron stars

In a pulsar, there are gaps and difficulties in our knowledge of glitches, mainly because of the absence of information about the physics of the matter of the star. This has motivated several authors to suggest dynamical models that interpret most of the astronomical data. Many predictions are based on the assumption that the inner part is analogous to the structure of matter of superfluids. Here, we illustrate a new mathematical model, partially inspired by the dynamics of superfluid helium. We obtain two evolution equations for the angular velocities (of the crust and of superfluid), which are supported by another evolution equation for the average vortex line length per unit volume. This…

Nonlinear Kelvin waves on a quantized vortex line in superfluid helium

In this paper we show an exact solution (Kelvin wave) of an approximated dynamical equation for a quantized vortex line in helium superfluid at finite temperature. It is shown that the applied heat flux interacts with the vortex line, and the amplitude of the Kelvin wave can grow (the so-called Donnelly instability) or decrease according with the mutual direction between heat flux and wave vector.

Waves on a vortex filament: exact solutions of dynamical equations

In this paper we take into account the dynamical equations of a vortex filament in superfluid helium at finite temperature (1 K < T < 2.17 K) and at very low temperature, which is called Biot-Savart law. The last equation is also valid for a vortex tube in a frictionless, unbounded and incompressible fluid. Both the equations are approximated by the Local Induction Approximation (LIA) and Fukumoto's approximation. The obtained equations are then considered in the extrinsic frame of reference, where exact solutions (Kelvin waves) are shown. These waves are then compared one to each other in terms of their dispersion relations in the frictionless case. The same equations are then investigated…

A direct method to find solutions of some type of coupled Korteweg-de Vries equations using hyperelliptic functions of genus two

Abstract We suggest how one can obtain exact solutions of some type of coupled Korteweg–de Vries equations by means of hyperelliptic functions of genus two.

Matter-wave dark solitons in boxlike traps

Motivated by the experimental development of quasihomogeneous Bose-Einstein condensates confined in boxlike traps, we study numerically the dynamics of dark solitons in such traps at zero temperature. We consider the cases where the side walls of the box potential rise either as a power law or a Gaussian. While the soliton propagates through the homogeneous interior of the box without dissipation, it typically dissipates energy during a reflection from a wall through the emission of sound waves, causing a slight increase in the soliton's speed. We characterize this energy loss as a function of the wall parameters. Moreover, over multiple oscillations and reflections in the boxlike trap, the…

Study of the anisotropy in turbulent superfluids

In this review we are interested on the anisotropy and polarity of superfluid turbulence in helium II, a still open problem which needs more details. Though some of the results presented here have already been published in different papers, this short review aims to put the main results together and to extend them when necessary. From the mesoscopic viewpoint, an evolution equation for the vortex line density was proposed in rotating counterflow (heat flux without mass flux) by means of dimensional analysis. Then, starting from the microscopic viewpoint this evolution equation was further extended to include situations where turbulence is not homogeneously distributed. Indeed, microscopical…

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…

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.

MATHEMATICAL MODEL FOR GLITCHES IN PULSARS

Effective thermal conductivity of superuid helium: Laminar, turbulent and ballistic regimes

Abstract In this paper we extend previous results on the effective thermal conductivity of liquid helium II in cylindrical channels to rectangular channels with high aspect ratio. The aim is to compare the results in the laminar regime, the turbulent regime and the ballistic regime, all of them obtained within a single mesoscopic formalism of heat transport, with heat flux as an independent variable.

Solutions of some coupled Korteweg-de Vries equations in terms of hyperelliptic functions of genus two

Coupled normal fluid and superfluid profiles of turbulent helium II in channels

We perform fully coupled two--dimensional numerical simulations of plane channel helium II counterflows with vortex--line density typical of experiments. The main features of our approach are the inclusion of the back reaction of the superfluid vortices on the normal fluid and the presence of solid boundaries. Despite the reduced dimensionality, our model is realistic enough to reproduce vortex density distributions across the channel recently calculated in three--dimensions. We focus on the coarse--grained superfluid and normal fluid velocity profiles, recovering the normal fluid profile recently observed employing a technique based on laser--induced fluorescence of metastable helium molec…

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…

Copper(ii) and zinc(ii) dependent effects on Aβ42 aggregation: a CD, Th-T and SFM study

A? aggregation is a central event in Alzheimer's disease (AD). In vitro evidence indicates that A? aggregation and fibrillogenesis are significantly influenced by the employed experimental conditions. Indeed, although it is widely established that metal ions, such as copper and zinc, have significant effects on the A? aggregation process, their actual role in A? fibrillogenesis is still debated. In this work the effects of a molar excess of zinc(ii) and/or copper(ii) ions on the A?42 aggregation process and the morphology of the resultant aggregates have been compared in samples exhibiting different initial conformations. CD spectroscopy, Th-T-induced fluorescence and Scanning Force Microsc…

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.

Studies in thermal and dynamical duality

Generalized heat equation and transitions between different heat-transport regimes in narrow stripes

Abstract In the framework of weakly nonlocal thermodynamic theory, in this paper we derive a nonlocal and nonlinear heat-transport equation beyond the Fourier law by means of thermodynamic considerations in agreement with the second law. The obtained equation describes the transitions among different heat-transport regimes. The stability of the solution of that equation is also analyzed in a special case.

Closure to “Explicit Equations for Uniform Flow Depth” by Vito Ferro and Michele Sciacca

Explicit equations for uniform flow depth

Conventional approach in uniform open channel flow is to express the resistance coefficient in the Manning, Darcy-Weisbach or Chezy form. However, for practical cross-sections, including rectangular and trapezoidal ones, the governing equation is implicit in the uniform water depth. For these sections the water depth, corresponding to known values of the flow discharge, slope channel and resistance coefficient, is presently obtained by trial and error procedure. In this paper exact analytical solutions of uniform flow depth for rectangular and trapezoidal section have been obtained in the form of fast converging power series.

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…

The saturation of decaying counterflow turbulence in helium

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…

Phenomenological description of the vortex density in rotating BEC superfluids

The close analogy of a purely magnetic excitation scheme, used in the experiments on Bose-Einstein condensate by Hodby et al., and the rotating bucket experiments with liquid helium 4 has suggested us to apply a phenomenological equation for the vortex line density, previously proposed for rotating superfluid helium 4, to describe the vortex density in a rotating Bose-Einstein condensate as a function of the angular speed. In both systems, the phenomenological equation provides a reasonable description of the observed data.

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.

Waves Propagation in Turbulent Superfluid Helium in Presence of Combined Rotation and Counterflow

A complete study of the propagation of waves (namely longitudinal density and temperature waves, longitudinal and transversal velocity waves and heat waves) in turbulent superfluid helium is made in three situations: a rotating frame, a thermal counterflow, and the simultaneous combination of thermal counterflow and rotation. Our analysis aims to obtain as much as possible information on the tangle of quantized vortices from the wave speed and attenuation factor of these different waves, depending on their relative direction of propagation with respect to the rotation vector.

Soliton solutions for an higher order nonlinear Schroedinger equation in optical fiber

The new improvements to increase the bit rate in optical fiber require the propagation of pulse whose temporal width is always lesser. This causes the presence of further terms, linear and nonlinear, in the evolution equation of the pulse. The analysis on the complete integrability of the evolution equation, in a fiber optics with local properties and achieved in a previous paper, is improved dealing with the normal dispersion case, which allows the dark soliton propagation. In the last section special efforts are made to propose some interesting soliton solutions both bright and dark.

Waves Propagation in Superfluid Helium in Presence of Combined Rotation and Counterflow

Using the linear macroscopic mono-fluid model of liquid helium II, in which the fundamental fields are the density ?, the velocity v, the temperature T and heat flux q and taking into account the expression of an additional pressure tensor P(w), introduced to describe phenomena linked to vortices, a complete study of wave propagation is made in the complex situation involving thermal counterflow in a rotating cylinder.

Turbulent Superfluid Profiles in a Counterflow Channel

We have developed a two-dimensional model of quantised vortices in helium II moving under the influence of applied normal fluid and superfluid in a counterflow channel. We predict superfluid and vortex-line density profiles which could be experimentally tested using recently developed visualization techniques.

Statistical mechanics and thermodynamics of turbulent quantum vortex tangles

In this paper we present some phenomenological ideas about the thermodynamics of quantized vortex loops arising in superfluid turbulence. The system of vortex loops may be seen as a dissipative structure, not existing on its own but only under the influence of an external heat flux. Starting from a simple definition of the temperature of the vortex tangle and from the relation between energy and vortex length, we obtain the entropy of the system, as well as the caloric and thermal equations of state, relating internal energy and pressure to temperature and volume. We discuss the connection between our macroscopic results and microscopic results on vortex length distribution function having …

Classical and quantum vortex leapfrogging in two-dimensional channels

The leapfrogging of coaxial vortex rings is a famous effect which has been noticed since the times of Helmholtz. Recent advances in ultra-cold atomic gases show that the effect can now be studied in quantum fluids. The strong confinement which characterizes these systems motivates the study of leapfrogging of vortices within narrow channels. Using the two-dimensional point vortex model, we show that in the constrained geometry of a two-dimensional channel the dynamics is richer than in an unbounded domain: alongsize the known regimes of standard leapfrogging and the absence of it, we identify new regimes of backward leapfrogging and periodic orbits. Moreover, by solving the Gross-Pitaevskii…

Stability in the plane Couette flow of superfluid helium

An hydrodynamical model previously proposed to describe the presence of vortices in counterflow superfluid turbulence and in rotating containers is used to discuss plane Couette flow and the stability of the stationary solution.

Non-Equilibrium Thermodinamic analysis of rotating counterflow superfluid turbulence

Two alternative evolution equations for the vortex line density L in counterflow superfluid turbulence in 4He were proposed by Vinen in 1958. These equations was recently generalized to counterflow superfluid turbulence in rotating containers. Here, according with the formalism of Non-Equilibrium Thermodynamics, the compatibility between the alternative Vinen equation as evolution equation for the vortex line density in rotating counterflow turbulence and the velocity of the superfluid component is studied. From the compatibility request a new term dependent on the anisotropy of the tangle arises.

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…

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…

Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics

Abstract Taking into account many developments in fiber optics communications, we propose a higher nonlinear Schrodinger equation (HNLS) with variable coefficients, more general than that in [R. Essiambre, G.P. Agrawal, Opt. Commun. 131 (1996) 274], which governs the propagation of ultrashort pulses in a fiber optics with generic variable dispersion. The study of this equation is performed using the Painleve test and the zero-curvature method. Also, we prove the equivalence between this equation and its anomalous integrable counterpart (the so-called Sasa–Satsuma equation). Finally, in view of its physical relevance, we present a soliton solution which represents the propagation of ultrasho…

Non-equilibrium thermodynamics analysis of rotating counterflow superfluid turbulence

In two previous papers two evolution equations for the vortex line density $L$, proposed by Vinen, were generalized to rotating superfluid turbulence and compared with each other. Here, the already generalized alternative Vinen equation is extended to the case in which counterflow and rotation are not collinear. Then, the obtained equation is considered from the viewpoint of non-equilibrium thermodynamics. According with this formalism, the compatibility between this evolution equation for $L$ and that one for the velocity of the superfluid component is studied. The compatibility condition requires the presence of a new term dependent on the anisotropy of the tangle, which indicates how the…

The saturation of decaying counterflow turbulence in helium II

We are concerned with the problem of the decay of a tangle of quantized vortices in He II generated by a heat current. Direct application of Vinen's equation yields the temporal scaling of vortex line density $L \sim t^{-1}$. Schwarz and Rozen [Phys. Rev. Lett. {\bf 66}, 1898 (1991); Phys. Rev. B {\bf 44}, 7563 (1991)] observed a faster decay followed by a slower decay. More recently, Skrbek and collaborators [Phys. Rev. E {\bf 67}, 047302 (2003)] found an initial transient followed by the same classical $t^{-3/2}$ scaling observed in the decay of grid-generated turbulence. We present a simple theoretical model which, we argue, contains the essential physical ingredients, and accounts for t…

Equazione non lineare di Schroedinger del terzo ordine in fibre ottiche con caratteristiche di non omogeneità

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

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.

Quantum Reynolds number for superfluid counterflow turbulence

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…

Some exact solutions of the two dimensional Bussinesq equation

Integrability of an inhomogeneous nonlinear Schrödinger equation in Bose–Einstein condensates and fiber optics

In this paper, we investigate the integrability of an inhomogeneous nonlinear Schrödinger equation, which has several applications in many branches of physics, as in Bose-Einstein condensates and fiber optics. The main issue deals with Painlevé property (PP) and Liouville integrability for a nonlinear Schrödinger-type equation. Solutions of the integrable equation are obtained by means of the Darboux transformation. Finally, some applications on fiber optics and Bose-Einstein condensates are proposed (including Bose-Einstein condensates in three-dimensional in cylindrical symmetry).

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…

Generalization of the Alternative Vinen's Equation Describing the Superfluid Turbulence in Rotating Container

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.

Exact Travelling Wave Solutions of Nonlinear Equations Using the Auxiliary Equation Method