Second Sound Waves Research Articles

A novel approach for generalized Green-Naghdi-type electro-magneto-thermo-hyperelasticity wave propagation and reflection investigations in near-incompressible layers under shock loads

A novel nonlinear coupled finite-strain electro-magneto-thermo-hyperelasticity (EMTHE) model is developed for the first- and second-sound wave propagation and reflection investigations in media that are exposed to electromagnetic fields and mechanical shocks. In contrast to the previous studies which studied behaviors of the finite-strain elastic or incompressible materials with small deformations, the current research proposes a much more complicated but much more accurate novel practical model for wave propagation and reflection analyses in near-incompressible finite-strain materials. Furthermore, to evaluate the effects of the electro-magneto-thermomechanical coupling, the strain energy density function of the hyperelastic material is expanded in a new way. The governing equations are obtained according to a nonlinear version of the Helmholtz free energy. The energy equations comprise the first- and second-order time rates of the temperature to enable the modeling of the finite-speed heat transfer; i.e., the establishment of a second-sound model. A nonlinear iterative finite element solution algorithm is proposed and implemented for the resulting coupled time-dependent generalized electro-magneto-thermo-hyperelasticity equations. The results show significant differences between the predicted wave propagation and reflection characteristics and behaviors of the near-incompressible and incompressible finite-strain models.

Boiling and cavitation caused by transient heat transfer in superfluid helium-4

Superfluid helium-4 (He II) has been widely utilized as a coolant in various scientific and engineering applications due to its superior heat transfer capability. An important parameter required in the design of many He II based cooling systems is the peak heat flux ${q}^{*}$, which refers to the threshold heat flux above which boiling spontaneously occurs in He II. Past experimental and numerical studies showed that ${q}^{*}$ increases when the heating time ${t}_{h}$ is reduced, which leads to an intuitive expectation that very high ${q}^{*}$ may be achievable at sufficiently small ${t}_{h}$. Knowledge on how ${q}^{*}$ actually behaves at small ${t}_{h}$ is important for applications such as laser ablation in He II. Here we present a numerical study on the evolution of the thermodynamic state of the He II in front of a planar heater by solving the He II two-fluid equations of motion. For an applied heat flux, we determine the heating time beyond which the He II near the heater transits to the vapor phase. As such, a curve correlating ${q}^{*}$ and ${t}_{h}$ can be obtained, which nicely reproduces some relevant experimental data. Surprisingly, we find that there exists a critical peak heat flux ${q}_{c}^{*}$, above which boiling occurs nearly instantaneously regardless of ${t}_{h}$. We reveal that the boiling in this regime is essentially cavitation caused by the combined effects of the first-sound and the second-sound waves in He II. Based on this physical picture, an analytical model for ${q}_{c}^{*}$ is developed, which reproduces the simulated ${q}_{c}^{*}$ values at various He II bath temperatures and hydrostatic head pressures. This work represents a major progress in our understanding of transient heat transfer in He II.

Transient heat transfer of superfluid He4 in nonhomogeneous geometries: Second sound, rarefaction, and thermal layer

Transient heat transfer in superfluid $^{4}\mathrm{He}$ (He II) is a complex process that involves the interplay of the unique counterflow heat-transfer mode, the emission of second-sound waves, and the creation of quantized vortices. Many past researches focused on homogeneous heat transfer of He II in a uniform channel driven by a planar heater. In this paper, we report our systematic study of He II transient heat transfer in nonhomogeneous geometries that are pertinent to emergent applications. By solving the He II two-fluid equations of motion coupled with Vinen's equation for vortex-line density, we examine and compare the characteristics of transient heat transfer from planar, cylindrical, and spherical heaters in He II. Our results show that as the heater turns on, an outgoing second-sound pulse emerges, within which the vortex-line density grows rapidly. These vortices attenuate the second sound and result in a heated He II layer in front of the heater, i.e., the thermal layer. In the planar case where the vortices are created throughout the space, the second-sound pulse is continuously attenuated, leading to a thick thermal layer that diffusely spreads following the heat pulse. On the contrary, in the cylindrical and the spherical heater cases, vortices are created mainly in a thin thermal layer near the heater surface. As the heat pulse ends, a rarefaction tail develops following the second-sound pulse, in which the temperature drops. This rarefaction tail can promptly suppress the thermal layer and take away the deposited thermal energy. The effects of the heater size, heat flux, pulse duration, and temperature on the thermal-layer dynamics are discussed. We also show how the peak heat flux for the onset of boiling in He II can be studied in our model.

Non-equilibrium effective field theory and second sound

We investigate the phenomenon of second sound in various states of matter from the perspective of non-equilibrium effective field theory (EFT). In particular, for each state of matter considered, we find that at least two (though sometimes multiple) qualitatively different EFTs exist at finite temperature such that there is always at least one EFT with a propagating second-sound wave and at least one with no such second-sound wave. To aid in the construction of these EFTs, we use the method of cosets developed for non-equilibrium systems. It turns out that the difference between the EFTs with and without second-sound modes can be understood as arising from different choices of a new kind of inverse Higgs constraint. Finally, we demonstrate that it is possible to bypass the need for new inverse Higgs constraints by formulating EFTs on a new kind of manifold that is like the usual fluid worldvolume, but with reduced gauge symmetries.

Nonlinear thermoelastic waves in functionally graded materials: Application to nanowires

We propose a thermodynamic model to describe the thermoelastic behavior of Functionally Graded Materials. Specifically, the thermoelastic response of these materials to coupled thermal and elastic perturbations is studied in view of a nonlinear heat-transport equation. The necessary and sufficient conditions ensuring the compatibility of the model with second law of thermodynamics are carried out. Propagation of thermoelastic waves, both in equilibrium and nonequilibrium states, is analyzed. The influence of the stoichiometry on the speed of propagation of coupled first-and second-sound waves is pointed out.

Dissipation mechanisms of tuning fork vibrations in superfluid 3He–4He solutions

The amplitude–frequency characteristics of tuning forks immersed in superfluid 3He–4He solutions were measured in the temperature range of 0.1–2.5 K. The tuning fork resonance frequency and the resonance width were measured as functions of temperature for 5% and 15% concentrations of 3He and, for comparison, for pure 4He. The experimental results for the key dissipation mechanisms, that is, viscous friction and first and second-sound emission of a tuning fork, were analyzed. For separation and evaluation of the contributions of viscous damping and first and second-sound waves, “closed” (in a factory capsule) and “open” (without a capsule) tuning forks were used. The resonance width, which is a measure of dissipation of tuning fork vibrations, was found to be higher in solutions than in pure 4He, and to increase with increasing 3He concentration. It was shown that the existing analytical expression for the contribution of viscous friction provides a good description of the experimental data for 4He only in the hydrodynamic region. For the solutions, the viscosity contribution is consistent with the experiment only at high temperatures (above 1.4 K). For the “open” tuning fork, the contribution of the first-sound is consistent with the calculation results only for 4He, whereas for the solutions, the calculated values are underestimated compared to the experiment. The possible contribution of the second-sound to the dissipation of the tuning fork vibrations in solutions was estimated using experiments with the “closed” tuning fork. This contribution was found to vary non-monotonically with a maximum at temperatures of ≈ 0.6–0.8 K. The ratio of the tuning fork energy loss due to the thermal diffusion wave versus the loss due to the radiation flux of the second-sound wave in superfluid solutions was calculated using literature data with an accuracy of 10–3–10–4.

Non-stationary thermoelectric effect in He II and how it is affected by the thermal vibrations’ transition from laminar to the turbulent regime

The observation of the thermoelectric effect, which is the spontaneous electric polarization of a cell with liquid He II during the thermal excitation of standing second-sound waves, has been confirmed [Low Temperature Physics 30, 1321 (2004)]. The relationship of this effect with the thermal and hydrodynamic properties of He II is studied in detail in the temperature range of 1.4 K < T < 2 K. It is established that the dependence of the amplitude of electric potential oscillations on the excitation intensity changes significantly during the thermal vibrations’ transition from the laminar to the turbulent regime. The threshold value of the excitation power w = w0 (T) is recorded: in the region w < w0, the potential oscillations are regular and their amplitude increases in proportion to the power; at w > w0, the electric response becomes random in nature as the fluctuations increase and the amplitude decreases to zero, with a peculiar electromagnetic “noise” being observed. The experimental results are compared with the conclusions drawn from the theory of flexoelectric polarization of liquid helium. The polarization of liquid helium upon excitation of the first-sound waves, as well as pressure and temperature shock waves, has also been discussed.

Shock waves in narrow channels filled with superfluid helium

Formation of fourth-sound shock waves in narrow channels filled with superfluid helium is studied. Physical and mathematical conditions at the surface of discontinuity are established. These conditions differ somewhat from those for first- and second-sound waves. The velocity of discontinuity coincides with the speed of fourth sound. The jumps of temperature and the superfluid velocity are shown to be of the first order of the pressure jumps.

Probing the critical exponent of the superfluid fraction in a strongly interacting Fermi gas

We theoretically investigate the critical behavior of a second-sound mode in a harmonically trapped ultracold atomic Fermi gas with resonant interactions. Near the superfluid phase transition with critical temperature ${T}_{c}$, the frequency or the sound velocity of the second-sound mode crucially depends on the critical exponent $\ensuremath{\beta}$ of the superfluid fraction. In an isotropic harmonic trap, we predict that the mode frequency diverges like ${(1\ensuremath{-}T/{T}_{c})}^{\ensuremath{\beta}\ensuremath{-}1/2}$ when $\ensuremath{\beta}<1/2$. In a highly elongated trap, the speed of the second sound reduces by a factor of $1/\sqrt{2\ensuremath{\beta}+1}$ from that in a homogeneous three-dimensional superfluid. Our prediction could readily be tested by measurements of second-sound wave propagation in a setup, such as that exploited by Sidorenkov et al. [Nature (London) 498, 78 (2013)] for resonantly interacting lithium-6 atoms, once the experimental precision is improved.

On the possibility of entropy wind in superfluid helium

The possibility of quasistationary normal-component flow arising in superfluid helium in the presence of sufficiently intense first- and second-sound waves propagating in the helium is examined. The equations describing the flow are derived. The contributions of the energy dissipation at the shock front and the viscosity of the medium to the appearance of flows are analyzed for the case of second-sound waves. The possibility of observing the phenomenon experimentally is assessed.

Second sound waves in orthodeuterium, parahydrogen quantum crystals and neon crystals

The possibility of second sound wave propagation in isotopic pure ideal single crystals of orthodeuterium, parahydrogen, and neon is studied. The same collision frequencies as those used in Callaway’s model to describe experimental data on thermal conductivity are used to find the region of existence of second-sound waves, which lies in the region of the maximum value of the thermal conductivity. The concentrations of the isotopes H1, D2, and Ne22 below which weakly damped second sound waves can exist are determined. It is important to take account of phonon scattering by the boundaries of the sample.

Second sound in an anisotropic quasiparticle system of superfluidH4e

The phase and group velocities of second sound modes in superfluid helium are obtained for arbitrary values of the relative velocity of the normal and superfluid components. We show that the phase and group velocities of second sound, in general, depend on the angle between the wave vector and the relative velocity between the normal and superfluid components $\mathbf{w}$. We have found the relationship between the amplitudes of the oscillating variables that describe second sound. In the general case, the normal fluid not only has a velocity component parallel to the wave vector, but also a transverse velocity component. The general expressions for the velocities and amplitudes are analyzed when the normal fluid is only due to phonons. We find that there is a certain value of $\mathbf{w}$ which makes the second-sound wave stationary in the laboratory frame. We show that the amplitude of the temperature oscillation, in a second-sound wave in an anisotropic phonon system, can be zero under some conditions.

Numerical simulation of evolution of intensive waves of the second sound in superfluid helium turbulent flow

Results of simulation study of evolution of solitary intensive second-sound waves spreading in superfluid helium are presented. Quantitative description was carried out on the basis of equations of hydrodynamics of superfluid turbulence (HST). HST equations with second-order accuracy (relative parameter deviation from equilibrium) were written for the cases of planar, cylindrical, and spherical geometries. The system of equations was solved using the disruption decay technique. Calculations were carried out for the temperature of undisturbed helium T0 = 1.4 K. Simulation results were compared with experimental data.

Experiments on the rapid mechanical expansion of liquidHe4through its superfluid transition

Phenomena following a rapid mechanical quench of liquid 4He from its normal to its superfluid phase are reported and discussed. The mechanical expansion apparatus is an improved version of that described previously. It uses a double-cell geometry to effect a partial separation of the sample from the convolutions of the bellows that form the outer wall of the cell. Consistent with earlier work, no evidence is found for the production of quantized vortices via the Kibble-Zurek (KZ) mechanism. Although the expansion is complete within 15ms , the second-sound velocity and attenuation continue to increase for a further approximately 60ms ; correspondingly the temperature decreases. Subsequently, the temperature rises again toward its final value as the second-sound velocity and attenuation decrease. It is shown that this unexpected behavior is apparently associated with a large-amplitude second-sound oscillation produced by the expansion, and it is suggested that the observed vortices are created by the normal fluid-superfluid counterflow that constitutes the second-sound wave. If production of large-amplitude second sound is inherent to the mechanical expansion of liquid 4He through the superfluid transition, as appears to be the case for final temperatures more than 3mK from the lambda transition, the phenomenon sets a lower bound on the density of KZ vortices that can be detected in this type of experiment.

Formation of a Direct Kolmogorov-Like Cascade of Second-Sound Waves in He II

Based on measurements of nonlinear second-sound resonances in a high-quality resonator, we have observed a steady-state wave energy cascade in He II involving a flux of energy through the spectral range towards high frequencies. We show that the energy balance in the wave system is nonlocal in K space and that the frequency scales of energy pumping and dissipation are widely separated. The wave amplitude distribution follows a power law over a wide range of frequencies. Numerical computations yield results in agreement with the experimental observations. We suggest that second-sound cascades of this kind may be useful for model studies of acoustic turbulence.

Three dimensionality of pulsed second-sound waves in He II

Three dimensionality of three-dimensional (3D) pulsed second-sound waves in He II emitted from a finite size heater is experimentally investigated and theoretically studied based on the two-fluid model in this study. The detailed propagation of the 3D pulsed second-sound wave is presented and reasonable agreement between the experimental and theoretical results is obtained. Heater size has a big influence on the profile of the 3D second-sound wave. The counterflow between the superfluid and normal fluid components becomes inverse in the rarefaction of the 3D second-sound wave. The amplitude of rarefaction decreases due to the interaction between second-sound wave and quantized vortices, which explains the experimental results about the second-sound wave near ${T}_{\ensuremath{\lambda}}$ [L. C. Krysac, Phys. Rev. Lett. 73, 2480 (1994)]. The accumulation of dense quantized vortices in the vicinity of heater surface leads to the formation of a thermal boundary layer, and further increase of heating duration results in the occurrence of boiling phenomena.

On the description of electrical effects in the two-fluid model of superfluidity

A model description of the electrothermal effect in a superfluid liquid is proposed. It is hypothesized that the superfluid state is an ordered state with an isotropic quadrupole moment which does not manifest electrical properties in an equilibrium liquid. The polarization properties of the liquid are excited by an inhomogeneous distribution of the superfluid flow. The Landau equations of superfluid hydrodynamics are supplemented by an equation relating the electrical polarization of the medium with the density of the superfluid component and the inhomogeneity of the superfluid flow. The value of the expected oscillations of the electrical potential in a second-sound wave agrees with the measurements of A. S. Rybalko, Fiz. Nizk. Temp. 30, 1321 (2004) [Low Temp. Phys. 30, 994 (2004)].

Observation of the electric induction due to a second-sound wave in He II

It is shown experimentally that the relative motion of the superfluid and normal components of He II in a second-sound wave is accompanied by the appearance of electric induction. The process is reversible. It is shown that the amplitude ratio of the temperature and induction potential oscillations is a temperature-independent constant equal to 2.3×104 K/V.

Nonlinear second-sound waves in He4–3He superfluid mixtures

The Hamiltonian formalism in two-fluid hydrodynamics is generalized to the case of He4–3He superfluid mixtures. The resulting formalism is used to calculate the amplitude of the three-wave interaction for second sound and the nonlinearity coefficient of second sound in He II with a He3 impurity. It is shown that the temperature at which the nonlinearity coefficient of the second sound changes sign from negative to positive is lowered from 1.88 to 1.7 K when a 10% He3 impurity is added to the superfluid He4. Thus for weak He4–3He solutions there exists a rather wide interval of temperatures below the superfluid transition temperature Tλ in which the nonlinearity coefficient of the second sound is negative, and the shock front is formed at the trailing edge of the nonlinear second-sound heating wave.

Fiber bolometer and emitter with negligible reflection for second sound measurements near the λ-point

Superconducting Pb–Au thin film bolometers for second sound (SS) measurements near the λ-point have been made on 8 μm diameter glass fibers. The small diameter of the fibers provides negligible cross-section to SS waves as close as ∼10 −5 K to the λ-point (for f=10 kHz), thus eliminating reflections that may unduly complicate the wave distribution in the experimental cell. The bolometers have isotropic sensitivity to SS waves in the plane perpendicular to the fiber. The operating temperature of the bolometers can be adjusted by changing an external magnetic field and/or by varying the bias current, while still maintaining a figure of merit of ∼30. A SS emitter producing planar waves was made using the same fibers.