Mixtures of Euler’s fluids and second sound propagation in superfluid helium (original) (raw)

Second sound and multiple shocks in superfluid helium

Zeitschrift für angewandte Mathematik und Physik, 2009

A temperature pulse propagating in superfluid helium is studied through the simple waves theory. Our aim is to determine the shape change of this pulse, initially represented by a gaussian profile, using a generalized non-linear Cattaneo model proposed, in the framework of Extended Thermodynamics, by Ruggeri and co-workers in the case of a rigid conductor. The theoretical basis of our arguments is given in a paper [1] where the differential system of a binary mixture of Euler's fluids is written as a system for a single heat conducting fluid. We prove that there exist three characteristic temperaturesθ, playing an essential role in the shape change of the propagating second sound wave; in particular, several families of multiple shocks (i.e. usual double shocks, double shocks only ahead or behind the wave profile, and very strange quadrishocks) can appear, depending on the relation among the unperturbed temperature of Helium II and the characteristic temperatures and, in some cases, on the wave's amplitude. Both the cases of a hot wave and a cold wave are discussed, proving that this last process is not symmetric with respect to the previous one. Finally, suitable choices of some parameters are suggested in order to better point out the changes of the wave profile and, in particular, the formation of multiple shocks.

Thermodynamics and Second Sound in a Two-Fluid Model of Helium II; Revisited

Journal of Non-Equilibrium Thermodynamics, 2003

Helium II is modelled as a mixture, within macroscopic hydrodynamics, with two partial pressures and a single temperature. The temperature-dependency of concentrations of the superfluid and the normal fluid is accounted for by regarding the mixture as reacting. Moreover, a force of interaction between the superfluid and the normal fluid which traces back to Landau is considered. Accounting for dissipative processes turns out to be a standard application of classical descriptions of viscosity and heat conduction. A detailed analysis of the thermodynamic restrictions is developed. The effciency of the resulting model is tested by revisiting the main phenomena in helium II and evaluating the expression of the wave speed of sound. The mass production has no effect on the wave speed, while the force of interaction affects the propagation of second sound. Also, a simple non-local model is given for the description of persistent currents.

Parametric Generation of Second Sound by First Sound in Superfluid Helium

Physical Review Letters, 1996

We report the first experimental observation of parametric generation of second sound (SS) by first sound (FS) in superfluid helium in a narrow temperature range in the vicinity of T λ . The temperature dependence of the threshold FS amplitude is found to be in a good quantitative agreement with the theory suggested long time ago [1] and corrected for a finite geometry. Strong amplitude fluctuations and two types of the SS spectra are observed above the bifurcation. The latter effect is quantitatively explained by the discreteness of the wave vector space and the strong temperature dependence of the SS dissipation length. 67.40.Nj, 67.40.Pm, 43.25.+y, 67.90.+z

Nonlinear acoustics of superfluid helium

Soviet Physics Uspekhi, 1990

The results of theoretical investigation of first and second sound in mathrmHmathrmemathrmImathrmI\mathrm{H}\mathrm{e}\mathrm{I}\mathrm{I}mathrmHmathrmemathrmImathrmI are reviewed. The variety of "standard" nonlinear phenomena are described such as nonlinear transformation of wave modes into one another, formation of shock fronts, nonlinear renormaJization of sound velocity, stability and parametric transformation of nonlinear waves etc. The effects of mathrmdmathrmamathrmmmathrmI)mathrmgmathrmimathrmn\mathrm{d}\mathrm{a}\mathrm{m}_{\mathrm{I}^{)}\mathrm{g}}\mathrm{i}\mathrm{n}mathrmdmathrmamathrmmmathrmI)mathrmgmathrmimathrmn and dispersion are studied. The possibility of self-focusing of the second sound in cubically nonlinear case as well as in the quadratically one is discussed. We also presented the investigation of stochastic wave fields and acoustic turbulence. In conclusion some of the open problems and the paths for further development touched upon are discussed.

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

In this work a one-fluid model of liquid helium II, based on extended irreversible thermodynamics, is further developed, in order to describe the action of quantized vortices on second sound propagation. The vortex array is described by means of a pressure tensor P 򬇐 V , which may be related to some parameters describing the vorticity. The influence of the vortices on the heat flux is taken to be given by the contraction of this pressure tensor and the heat flux. Two situations with different geometries of the vortices are examined: superfluid in a rotating cylinder and thermal counterflow. In both of them, previously known results are recovered from this more general perspective. Finally, a generalization of Vinen's equation for the evolution of the vortex line density including fractal effects is proposed on dimensional grounds, and its relation to recent experiments is briefly discussed.

Velocity of the fourth sound in liquid helium II via extended thermodynamics

Zeitschrift f�r Angewandte Mathematik und Physik (ZAMP), 2003

This work continues a study begun in previous works, where, using Extended Thermodynamics, a monofluid model of liquid helium II is formulated. The wave propagation in bulk liquid helium II is studied in the hypothesis that the thermal dilatation is not zero. The propagation of fourth sound, studied previously neglecting both the thermal dilatation and finite volume of the powder, is studied without these simplified hypotheses: a scattering correction n is introduced to take into account the porosity. The model is more general than the standard two-fluid model because it allows that a small amount of entropy is associated with helium when it flows through a very thin capillary or a porous medium. A comparison with experimental data is performed. From experimental values for velocities and attenuations of the two sounds in bulk liquid helium, the model provides the velocity of fourth sound in a porous medium. These values are determined at various temperatures and pressures and compared with fourth sound measurements in a packed powder.

Parametric generation of second sound in superfluid helium: Linear stability and nonlinear dynamics

Physical Review B, 2001

We report the experimental studies of a parametric excitation of a second sound (SS) by a first sound (FS) in a superfluid helium in a resonance cavity. The results on several topics in this system are presented: (i) The linear properties of the instability, namely, the threshold, its temperature and geometrical dependencies, and the spectra of SS just above the onset were measured. They were found to be in a good quantitative agreement with the theory. (ii) It was shown that the mechanism of SS amplitude saturation is due to the nonlinear attenuation of SS via three wave interactions between the SS waves. Strong low frequency amplitude fluctuations of SS above the threshold were observed. The spectra of these fluctuations had a universal shape with exponentially decaying tails. Furthermore, the spectral width grew continuously with the FS amplitude. The role of three and four wave interactions are discussed with respect to the nonlinear SS behavior. The first evidence of Gaussian statistics of the wave amplitudes for the parametrically generated wave ensemble was obtained. (iii) The experiments on simultaneous pumping of the FS and independent SS waves revealed new effects. Below the instability threshold, the SS phase conjugation as a result of three-wave

Resolving the puzzle of sound propagation in liquid helium at low temperatures

Low Temperature Physics, 2019

Experimental data suggests that, at temperatures below 1 K, the pressure in liquid helium has a cubic dependence on density. Thus the speed of sound scales as a cubic root of pressure. Near a critical pressure point, this speed approaches zero whereby the critical pressure is negative, thus indicating a cavitation instability regime. We demonstrate that to explain this dependence, one has to view liquid helium as a mixture of three quantum Bose liquids: dilute (Gross-Pitaevskii-type) Bose-Einstein condensate, Ginzburg-Sobyanin-type fluid, and logarithmic superfluid. Therefore, the dynamics of such a mixture is described by a quantum wave equation, which contains not only the polynomial (Gross-Pitaevskii and Ginzburg-Sobyanin) nonlinearities with respect to a condensate wavefunction, but also a non-polynomial logarithmic nonlinearity. We derive an equation of state and speed of sound in our model, and show their agreement with experiment.

Novel sound phenomena in superfluid helium in aerogel and other impure superfluids

Physica B-condensed Matter, 2003

During the last decade new techniques for producing impure superfluids with unique properties have been developed. This new class of systems includes superfluid helium confined to aerogel, HeII with different impurities (D 2 , N 2 , Ne, Kr), superfluids in Vycor glasses, and watergel. These systems exhibit very unusual properties including unexpected acoustic features. We discuss the sound properties of these systems and show that sound phenomena in impure superfluids are modified from those in pure superfluids. We calculate the coupling between temperature and pressure oscillations for impure superfluids and for superfluid He in aerogel. We show that the coupling between these two sound modes is governed either by c∂ρ/∂c or σρ a ρ s (for aerogel) rather than thermal expansion coefficient ∂ρ/∂T , which is enormously small in pure superfluids. This replacement plays a fundamental role in all sound phenomena in impure superfluids. It enhances the coupling between the two sound modes that leads to the existence of such phenomena as the slow mode and heat pulse propagation with the velocity of first sound observed in superfluids in aerogel. This means that it is possible to observe in impure superfluids such unusual sound phenomena as slow pressure (density) waves and fast temperature (entropy) waves. The enhancement of the coupling between the two sound modes decreases the threshold values for nonlinear processes as compared to pure superfluids. Sound conversion, which has been observed in pure superfluids only by shock waves should be observed at moderate sound amplitude in impure superfluids. Cerenkov emission of second sound by first sound (which never been observed in pure superfluids) could be observed in impure superfluids.