Nonlocal Internal Variable and Superfluid State in Liquid Helium II

Abstract

We present a model of superfluidity based on the internal variable theory. We consider a two-component fluid endowed with a scalar internal variable whose gradient is the counterflow velocity. The restrictions imposed by the second law of thermodynamics are obtained by applying a generalized Coleman–Noll procedure. A set of constitutive equations of the Landau type, with entropy, entropy flux and stress tensor depending on the counterflow velocity, is obtained. The propagation of acceleration waves is investigated as well. It is shown that the first-and-second sound waves may propagate along the system with speeds depending on the physical parameters of the two fluids. First sound waves may propagate in the same direction or in the opposite direction of the counterflow velocity, depending on the concentration of normal and superfluid components. The speeds of second sound waves have the same mathematical form of those propagating in dielectric crystals.

1. Introduction

Among all known fluids, helium II exhibits a very peculiar behavior due to quantum effects which manifest themselves at the macroscopic level [1,2]. Its properties have several practical applications in cryogenic problems. The boiling point of liquid helium is $4.2$ K. In the range of temperature from this boiling point to the temperature $T_{\lambda }=2.172$ K at vapor pressure, liquid helium behaves as a classical fluid. However, at the temperature $T_{\lambda }$ it undergoes a phase transition and transforms into a completely different fluid, called helium II (He II), which exhibits new and unusual thermodynamical properties. Quantum fluids, such as superfluid helium, neutron stars, or atomic Bose–Einstein condensates, are characterized by quantized vorticity, superfluidity and, at finite temperatures below the critical one, two-fluid behavior [3,4,5]. Such peculiar behavior cannot be observed in classical fluids [6,7]. From the macroscopic point of view, quantum vortices need a mathematical description which is different to that used for the description of classical vorticity. In the superfluid state, the fluid flows without viscosity and results in vortex filaments whose circulation is restricted by a quantic constraint. Such filaments interact, reconnect and exchange energy at different scales [3]. In spite of this complex scenario, the two-fluid behavior can be reasonably captured by considering gradient effects. This allows the modelization of stable flows at low speed and provides a simple thermodynamic framework for analyzing these specific phenomena.

The superfluid state is characterized by an extremely high thermal conductivity and the complete absence of viscosity. The possibility of flowing without friction can be observed by connecting a narrow capillary, called a superleak, and two vessels, both containing helium II. Heating one of them, a part of the fluid contained in it flows into the second vessel trough the superleak. Due to the incoming heated helium, the temperature of the fluid in the second vessel should also increase, but, surprisingly, a cooling of it is observed. Such an unusual phenomenon can be explained by observing that superfluid helium is capable of flowing through the superleak without carrying entropy. Thus, the fluid in the second vessel has the same entropy distributed over more mass, and hence it is cooled.

In liquid helium II, the heat conduction is not diffusive but propagative due to its ability of transmitting temperature waves, known as second sound, with very small attenuation. Besides temperature waves, pressure waves, known as first sound, can propagate too [8]. Such waves can also propagate along thin films or narrow capillaries, and in such cases they are called third and fourth sound, respectively [1,2]. Moreover, in helium II the fluid motion produces a temperature gradient, and a temperature gradient produces fluid motion. One of the most striking phenomena is that, despite the basic tenets of classical thermodynamics, the fluid flow opposes the heat flow instead of being in the same direction of it. Further non-classical behaviors of liquid helium II are heat transport even in the absence of a net matter flow and the absence of a boiling point as a consequence of the extremely high (infinite, for practical purposes) thermal conductivity [1,2].

Several phenomenological theories describing the behavior of liquid helium below the lambda point can be found in the literature [9]. The most popular one, from Landau [10], is known as the two-fluid model and assumes that the overall fluid behaves as a nonconventional mixture of two components, the normal and superfluid ones [3,11,12]. The first of them is a normal viscous fluid, while the other, called the superfluid component, is able to flow without dissipation of energy. The superfluid component disappears above the lambda point, while the normal one is absent at absolute zero. In this model, a fundamental role is played by the so-called counterflow velocity, defined as the difference in the velocities of the normal and superfluid components, respectively. In spite of the classical Fourier and Navier–Stokes theories, such a velocity determines the heat flux and the stress tensor [11]. However, one should note that this mixture is not of the conventional kind, since distinct constituents cannot exist independently of each other. An alternative macroscopic approach to describe the anomalous behavior of liquid helium II is the extended one-fluid model [9,13,14].

The aim of this paper is to develop a model of superfluidity based on the internal variable theory. The internal variables of state allow us to model nonequilibrium processes involving complex thermodynamical systems in which classical observable quantities, such as pressure and temperature, are insufficient to describe nonequilibrium phenomena. The nature of these additional nonequilibrium parameters depends on the phenomenon to be modeled, and their evolution is ruled either by ordinary differential equations (local behavior) or by partial differential equations (weakly nonlocal behavior) [15,16]. Thus, following the way already paved in [17], we consider a two-component fluid endowed with a scalar internal variable $\gamma$, whose gradient $\mathbf{w}$ is the counterflow velocity. In this way, $\gamma$ is related to an internal kinematic constraint of the fluid, expressing its possibility of flowing as two fluids which necessarily must move together. We follow the approach developed in [16] and suppose that both $\gamma$ and $\mathbf{w}$ enter the state space. The restrictions imposed by the second law of thermodynamics are derived by applying a generalized Coleman–Noll procedure [18], suitable for the thermodynamic analysis of processes in which the gradients of the unknown fields enter the state space. The basic idea of this methodology is to consider the spatial differential consequences (gradients) of the balance laws as additional equations to be substituted into the entropy inequality, up to the order of the gradients entering the state space [19]. A set of constitutive equations of the Landau type [10], with entropy, entropy flux and stress tensor depending on the counterflow velocity, is obtained. The system of balance laws is derived and, in the one-dimensional case, the propagation of acceleration waves is investigated. It is shown that the model allows first-and-second sound waves to propagate along the system. The speeds of propagation are calculated as well.

The paper has the following layout.

In Section 2, we write the balance equations for partial concentrations, linear momentum, energy, heat flux and internal variables, together with the local balance of entropy (second law of thermodynamics).

In Section 3, by applying a generalized Coleman–Noll procedure, we prove that the system of equations postulated in Section 2 is compatible with the second law of thermodynamics under the validity of a suitable set of thermodynamic restrictions on the constitutive equations for the stress tensor, the specific entropy and the entropy flux.

In Section 4, we obtain the system of jumps across a discontinuity surface and calculate the possible wave speeds by requiring that such a system admits nontrivial solutions.

In Section 5, we present an overview of the results obtained and some possible applications of them.

2. The Two-Fluid Model

The two-fluid model was for the first time proposed by Landau in 1941 [10] in order to describe the main properties of liquid helium below the $\lambda$ point. Such a model assumes that the overall fluid behaves as a nonconventional mixture of two components, the normal and superfluid ones, having different properties [11]. The balance equations for the mixture are postulated in the form

$$\varrho c_{n,t}+\varrho c_{n,k}{v}_k+{\left[{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{w}_k\right]}_{,k}=m_n,$$

(1)

$$\varrho c_{s,t}+\varrho c_{s,k}{v}_k-{\left[{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{w}_k\right]}_{,k}=m_s,$$

(2)

$$\varrho v_{i,t}+\varrho v_{i,k}{v}_k-T_{ij,j}=0,$$

(3)

$$\varrho {\epsilon }_{,t}+\varrho {\epsilon }_{,k}{v}_k+q_{i,i}-T_{ij}{D}_{ij}=0,$$

(4)

$$q_{i,t}+q_{i,k}{v}_k+{\displaystyle \frac{q_i}{\tau }}+{\displaystyle \frac{\overline{\kappa }}{\tau }}{\epsilon }_{,i}-{\displaystyle \frac{\phi (\Sigma )}{\tau }}{w}_i=0,$$

(5)

$${\gamma }_{,t}+{\gamma }_{,k}{v}_k-H(\Sigma )=0,$$

(6)

wherein

• The indices $i,j,k,$ range from 1 to 3;
• ${\varrho }_n$ is the mass density of the normal component;
• ${\varrho }_s$ is the mass density of the superfluid component;
• $\varrho ={\varrho }_n+{\varrho }_s$ is the total mass density;
• $c_n={\varrho }_n/\varrho$ and $c_s={\varrho }_s/\varrho$, are the partial mass concentrations, with $c_n+c_s=1$;
• $w_k=v_{nk}-v_{sk}$, with $v_{nk}$ and $v_{sk}$ as the components of the velocities of the two constituents, are the components of the counterflow velocity;
• $m_n$ and $m_s$ are the rates of mass production per unit volume of the normal and superfluid components, respectively;
• ${\displaystyle v_i=\frac{{\varrho }_n{v}_{ni}+{\varrho }_s{v}_{si}}{\varrho }}$ are the components of the local barycentric velocity in the actual configuration;
• $T_{ij}$ are the components of the Cauchy stress tensor;
• $\epsilon$ is the internal energy per unit mass;
• $q_i$ are the components of the local heat flux;
• ${\displaystyle D_{ij}=\frac{v_{i,j}+v_{j,i}}{2}}$ are the components of the symmetric part of the gradient of velocity;
• c is the specific heat, $\kappa$ is the thermal conductivity, and $\overline{\kappa }\equiv \kappa /c$;
• $\tau$ is a material parameter, depending on $\epsilon$, which represents the total relaxation time of the heat flux (i.e., the time elapsed between the application of a difference of temperature and the appearance of a heat flux);
• $\Sigma$ is the state space, which will be specified better below, and $\phi (\Sigma )$ a scalar function defined on $\Sigma$;
• $\gamma$ is a scalar internal variable such that ${\gamma }_{,k}=w_k$, and H a scalar function defined on the state space.

Some considerations of the system of Equations (1)–(6) are in order. From the mathematical point of view, it represents a mixture of fluids with 2 mass densities, ${\varrho }_n$ and ${\varrho }_s$, two velocities, ${\mathbf{v}}_n$ and ${\mathbf{v}}_s$, and single internal energy $\epsilon$ (or, equivalently, single temperature $\vartheta$).

Indeed, mixtures can be modeled at different degrees of detail, depending on the properties which is necessary to describe, namely the following:

• Models with different concentrations and the same velocity and temperature of the single constituents [20];
• Models with different concentrations and temperatures and the same velocity of the single constituents [21];
• Models with different concentrations and velocities and the same temperature of the single constituents [22];
• Models with different concentrations, velocities and temperatures of the single constituents [23].

In the present paper, we apply the approach mentioned in the item 3 because there is no physical evidence that the normal and superfluid components have different temperatures. Thus, we start from the classical model of the mixture (see [20]) in which the partial balances of mass are postulated in the form

$$\varrho c_{n,t}+\varrho c_{n,k}{v}_k+{\left[{\varrho }_n(v_{nk}-v_k)\right]}_{,k}=m_n,$$

(7)

$$\varrho c_{s,t}+\varrho c_{s,k}{v}_k+{\left[{\varrho }_s(v_{sk}-v_k)\right]}_{,k}=m_s.$$

(8)

On the other hand, by simple algebra we can express the fluxes of normal and superfluid mass in the form ${\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }{w}_k}$ and ${\displaystyle -\frac{{\varrho }_n{\varrho }_s}{\varrho }{w}_k}$, respectively, obtaining the balance Equations (1) and (2) for the partial concentrations. In what follows, we use the semilinear approximation ${\displaystyle {\left[\frac{{\varrho }_n{\varrho }_s}{\varrho }{w}_k\right]}_{,k}\simeq \frac{{\varrho }_n{\varrho }_s}{\varrho }{w}_{k,k}}$, i.e., we consider the products of the spatial derivatives of ${\displaystyle \left[\frac{{\varrho }_n{\varrho }_s}{\varrho }\right]}$ with the spatial derivatives of $\gamma$ as second-order quantities. It is worth observing that, in the case under study, we have only the transition of matter from the normal to the superfluid phase or vice versa, i.e., the net production of mass is zero, so that $m_n+m_s=0$. The gradient of Equation (6) yields

$${\gamma }_{,ht}+{\gamma }_{,hk}{v}_k+{\gamma }_{,k}{v}_{k,h}-H_{,k}=0,$$

(9)

or, equivalently, since ${\gamma }_{,h}=w_h$,

$$w_{h,t}+w_{h,k}{v}_k+w_k{v}_{k,h}-H_{,h}=0,$$

(10)

which is an evolution equation for the counterflow velocity. Thus, the motion of the mixture is described by the barycentric velocity $\mathbf{v}$, ruled by the balance of linear momentum (3), and by the counterflow velocity $\mathbf{w}$, ruled by Equation (10). In this sense, the model proposed here represents a mixture with two concentrations and two velocities (item 3).

As far as the heat flux is concerned, we go beyond Fourier’s law [24] by postulating a generalized Maxwell–Cattaneo equation [25], with a production term depending on the counterflow velocity (Equation (5)). In fact, mechanical and thermal waves can propagate in superfluid helium and interact each other [26]. Then, in order to describe such a phenomenology, an equation for the heat flux with relaxation terms is necessary. Equation (5) goes in that direction and incorporates also Landau’s result of the heat flux proportional to the counterflow velocity [10,11]. When the relaxation effects are negligible $(\tau \simeq 0)$, Equation (5) reduces to $q_i=-\overline{\kappa }{\epsilon }_{,i}-\phi (\Sigma )w_i$, which describes the property that heat flow is provoked by the presence of both the gradient of temperature and the difference between normal and superfluid velocity. In particular, close to the thermal equilibrium $({\epsilon }_{,i}\simeq 0)$, $q_i$ reduces to the classical Landau’s heat flux if $\phi (\Sigma )={\varrho }_s\vartheta s$, with s as the specific entropy [10,11]. It is worth noting that, in our approach, the heat flux is no longer a constitutive quantity, as in Rational Thermodynamics [18,20], but an independent thermodynamic variable ruled by its own balance equation according to the tenets of Extended Irreversible Thermodynamics [27,28].

In order to close the system (1)–(6), we need a set of constitutive equations for the functions $m_n$, $m_s$$T_{ij}$, $\phi$ and H to be assigned on a suitable state space $\Sigma$, which we choose as follows:

$$\Sigma =\{c_n,c_s,\epsilon ,q_k,\gamma ,{\gamma }_{,k}\}.$$

(11)

It is worth noticing that the gradient of velocity $v_{k,h}$ has not been included in $\Sigma$ because in this paper we aim at describing only the phase below the lambda point, in which the viscosity effects are negligibly small.

Note also that the model is local with respect to the standard variables $c_n$, $c_s$, $\epsilon$ and $q_k$ and weakly nonlocal with respect to the internal variable, since ${\gamma }_{,k}$ enters the state space.

The constitutive equations must be assigned in such a way that the unilateral differential constraint (second law of thermodynamics)

$$\varrho s_{,t}+\varrho s_{,k}{v}_k+J_{i,i}\ge =0,$$

(12)

with s as the specific entropy and $J_i$ as the entropy flux, is fulfilled in arbitrary thermodynamic processes [18]. Note that the entropy flux $J_i$ is more general with respect to the expression $q_i/\vartheta$ postulated in [20] and may include the extra flux of entropy due to the diffusion of matter inside the mixture. In [20], instead, an extra flux of energy is introduced in the local balance of energy.

Since the state space is weakly nonlocal, the compatibility of the assigned constitutive quantities will be explored by applying a generalized Coleman–Noll procedure, proposed recently by Cimmelli and co-workers [19]. Such a study will be carried out in Section 3.

3. Thermodynamics of Helium II

In view of the observations above on the vector $\mathbf{w}$, we rewrite the state space as follows:

$$\Sigma =\{c_n,c_s,\epsilon ,q_k,\gamma ,w_k\}.$$

(13)

Then, the entropy inequality (12) on the state space yields

$$\begin{array}{c}\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_n}}{c}_{n,t}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_s}}{c}_{s,t}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}{\epsilon }_{,t}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{q}_k}}{q}_{k,t}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\gamma }}{\gamma }_{,t}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_k}}{w}_{k,t}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_n}}{c}_{n,k}{v}_k+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_s}}{c}_{s,k}{v}_k\\ \varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}{\epsilon }_{,k}{v}_k+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{q}_h}}{q}_{h,k}{v}_k+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\gamma }}{\gamma }_{,k}{v}_k+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_h}}{w}_{h,k}{v}_k+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{c}_n}}{c}_{n,k}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{c}_s}}{c}_{s,k}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }\epsilon }}{\epsilon }_{,k}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{q}_h}}{q}_{h,k}\\ +{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }\gamma }}{\gamma }_{,k}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{w}_h}}{w}_{h,k}\ge 0.\end{array}$$

(14)

A further step is to substitute into the inequality above Equations (1), (2), (4)–(6) and (10).

Remark 1.

We note that it is not necessary to substitute Equation (3) in the entropy inequality, as such an equation is used to derive the local balance of energy (4) so that the information carried by Equation (3) is already included in Equation (4). In this way, it is enough to substitute in Equation (14) only Equation (4), which contains also information on the constraint (3).

On the state space Equation (10) reads

$$w_{h,t}+w_{h,k}{v}_k+w_k{v}_{k,h}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_n}}{c}_{n,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_s}}{c}_{s,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }\epsilon }}{\epsilon }_{,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{q}_h}}{q}_{h,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }\gamma }}{\gamma }_{,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{w}_h}}{w}_{h,k}=0.$$

(15)

Remark 2.

We note also that, as Equation (15) is the gradient of Equation (6), according to the extended Coleman–Noll procedure, it represents an additional constraint to be substituted into the entropy inequality [19].

In this way we obtain

$$\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_n}}\left(-\varrho c_{n,k}{v}_k-{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{w}_{k,k}-m_n\right)+{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_s}}\left(-\varrho c_{s,k}{v}_k+{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{w}_{k,k}-m_s\right)+\\ {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}\left(-\varrho {\epsilon }_{,k}{v}_k-q_{i,i}+T_{hk}{v}_{h,k}\right)+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{q}_k}}\left(-q_{i,k}{v}_k-{\displaystyle \frac{q_i}{\tau }}-{\displaystyle \frac{\overline{\kappa }}{\tau }}{\epsilon }_{,i}+{\displaystyle \frac{\phi (\Sigma )}{\tau }}{w}_i\right)+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\gamma }}H\\ +\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_k}}\left(-w_{h,k}{v}_k-w_k{v}_{k,h}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_n}}{c}_{n,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_s}}{c}_{s,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }\epsilon }}{\epsilon }_{,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{q}_h}}{q}_{h,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }\gamma }}{\gamma }_{,k}-{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{w}_h}}{w}_{h,k}\right)\\ +\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_n}}{c}_{n,k}{v}_k+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_s}}{c}_{s,k}{v}_k\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}{\epsilon }_{,k}{v}_k+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{q}_h}}{q}_{h,k}{v}_k+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\gamma }}{\gamma }_{,k}{v}_k+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_h}}{w}_{h,k}{v}_k\\ +{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{c}_n}}{c}_{n,k}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{c}_s}}{c}_{s,k}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }\epsilon }}{\epsilon }_{,k}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{q}_h}}{q}_{h,k}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }\gamma }}{\gamma }_{,k}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{w}_h}}{w}_{h,k}\ge 0.\end{array}$$

(16)

With some algebra, the inequality above can be rearranged as follows:

$$\begin{array}{c}\left(\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_k}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_n}}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{c}_n}}\right)c_{n,k}+\left(\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_k}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_s}}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{c}_s}}\right)c_{s,k}+\\ \left(\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_k}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }\epsilon }}-\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{q}_k}}{\displaystyle \frac{\overline{\kappa }}{\tau }}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }\epsilon }}\right){\epsilon }_{,k}+\left({\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}{T}_{hk}-\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_h}}{w}_k\right)v_{h,k}\\ 〈-{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_n}}{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{\delta }_{hk}+{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_s}}{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{\delta }_{hk}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_h}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{w}_k}}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{w}_h}}〉w_{h,k}\\ 〈-{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}{\delta }_{hk}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_h}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{q}_k}}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{q}_h}}〉q_{h,k}+g(\Sigma )\ge 0,\end{array}$$

(17)

wherein $g(\Sigma )$ is a suitable scalar function defined on the state space, whose expression is too long to be reported here, and the symbol $〈a_{hk}〉$ denotes the symmetric part of the generic tensor $a_{hk}$. The inequality above is linear in the derivatives $\{c_{n,k},c_{s,k},v_{h,k},w_{h,k},q_{h,k}\}$, which are not included in $\Sigma$, as they are independent of their coefficients, which, instead, are defined on the state space. Such derivatives can assume arbitrary values [29], and hence their coefficients must vanish; otherwise the inequality could be easily violated. Then, the following theorem ensues.

Theorem 1.

The entropy inequality (17) is satisfied along arbitrary thermodynamic processes if and only if the following restrictions on the constitutive equations hold:

$$\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_k}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_n}}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{c}_n}}=0,$$

(18)

$$\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_k}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_s}}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{c}_s}}=0,$$

(19)

$$\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_k}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }\epsilon }}-\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{q}_k}}{\displaystyle \frac{\overline{\kappa }}{\tau }}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }\epsilon }}=0,$$

(20)

$${\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}{T}_{hk}-\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_h}}{w}_k=0,$$

(21)

$$〈-{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_n}}{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{\delta }_{hk}+{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_s}}{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{\delta }_{hk}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_h}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{w}_k}}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{w}_h}}〉=0,$$

(22)

$$〈-{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}{\delta }_{hk}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{w}_h}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{q}_k}}+{\displaystyle \frac{\mathsf{\partial }{J}_k}{\mathsf{\partial }{q}_h}}〉=0,$$

(23)

$$g(\Sigma )\ge 0.$$

(24)

Remark 3.

The function $g(\Sigma )$ represents locally the net entropy production once the restrictions (18)–(23) have been satisfied.

In order to make the results above more explicit, in the following we limit ourselves to the one-dimensional case. We consider a system of reference $(O,x,y,z)$ and a one-dimensional continuous system placed on the axis x. Indeed, in some cases as, for instance, during its motion in the superleak, superfluid helium can be considered as a one-dimensional system. Moreover, for one-dimensional systems it is easier to calculate particular solutions of the set of thermodynamic restrictions, which allows us to point out its physical properties.

We denote with x the position of the points of the system in the actual configuration. The stress tensor, the heat flux and the entropy flux will be denoted by T, q and J, respectively, while the symbol $f_{,x}$ will denote the partial derivative of function f with respect to the spatial coordinate x in the actual configuration. The barycentric and the counterflow velocity fields will be denoted by v and w, respectively. Then, we can reformulate Theorem 1 as follows.

Theorem 2.

For a one-dimensional superfluid system the entropy inequality (17) is satisfied along arbitrary thermodynamic processes if and only if the following restrictions on the constitutive equations hold:

$$\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }w}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_n}}+{\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }{c}_n}}=0,$$

(25)

$$\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }w}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }{c}_s}}+{\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }{c}_s}}=0,$$

(26)

$$\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }w}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }\epsilon }}-\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }q}}{\displaystyle \frac{\overline{\kappa }}{\tau }}+{\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }\epsilon }}=0,$$

(27)

$${\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}T-\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }w}}w=0,$$

(28)

$$-{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_n}}{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}+{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }{c}_s}}{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }w}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }w}}+{\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }w}}=0,$$

(29)

$$-{\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}+\varrho {\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }w}}{\displaystyle \frac{\mathsf{\partial }H}{\mathsf{\partial }q}}+{\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }q}}=0,$$

(30)

$$g(\Sigma )\ge 0.$$

(31)

In order to test the restrictions above with a practical application, let us pursue our analysis under the following hypotheses:

Postulate 1.

$$s=s_0\left(\epsilon \right)-{\displaystyle \frac{1}{2}}{\displaystyle \frac{s_1{c}_n{c}_s}{\vartheta }}{w}^2,$$

(32)

$$H=H_0\left(\epsilon \right)-{\displaystyle \frac{1}{2}}{c}_n{c}_s{w}^2,$$

(33)

wherein

• $s_0\left(\epsilon \right)$ is the specific entropy above the lambda point;
• $s_1$ is a constant coefficient such that the second term of s has the physical dimension of a specific entropy;
• $H_0$ is a suitable function whose physical dimension is $m^2/s^2$.

Then, from Theorem 2, we have the following.

Theorem 3.

Under the hypotheses (32) and (33), the restrictions (25)–(30) take the form

$${\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }{c}_n}}={\displaystyle \frac{1}{2}}\varrho s_1{c}_s^2{c}_n{w}^3,$$

(34)

$${\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }{c}_s}}={\displaystyle \frac{1}{2}}\varrho s_1{c}_n^2{c}_s{w}^3,$$

(35)

$${\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }\epsilon }}=\varrho {\displaystyle \frac{\mathsf{\partial }{H}_0}{\mathsf{\partial }\epsilon }}{s}_1{c}_n{c}_{s}w,$$

(36)

$$T=-\varrho s_1{c}_n{c}_s{w}^2,$$

(37)

$${\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }w}}=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{s}_1{c}_s{w}^2-{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{s}_1{c}_n{w}^2-\varrho s_1{c}_n^2{c}_s^2{w}^2,$$

(38)

$${\displaystyle \frac{\mathsf{\partial }J}{\mathsf{\partial }q}}={\displaystyle \frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}.$$

(39)

In view of the thermodynamic relation ${\displaystyle \frac{1}{\vartheta }=\frac{\mathsf{\partial }s}{\mathsf{\partial }\epsilon }}$, Equation (39) shows that the constitutive equation for the entropy flux encompasses the classical equation $J=q/\vartheta$ postulated in Rational Thermodynamics [18]. More generally, the restrictions above allow an entropy flux of Landau’s type, depending on the counterflow velocity too. Finally, the stress tensor is quadratic in w, reflecting the property that in the three-dimensional case its components are obtained by the diadic product of the counterflow velocity with itself. This result is consistent with that obtained by Atkin and Fox in [11].

Remark 4.

It is worth observing that in the present paper we are modeling only the non-dissipative behavior of helium II and, as a consequence of the state space defined in Equation (13) and of the second law of thermodynamics, we obtained the nonviscous stress in Equation (37) (the Landau stress). In fact, in the system of Equations (1)–(6) the spatial derivatives of $v_{nk}$, which could take into account the viscosity effects, are not present. A dissipative model accounting for the viscosity of the normal component of helium II would require a state space of the type

$$\Sigma =\{c_n,c_s,\epsilon ,q_k,v_{nk,i}\},$$

(40)

while the system of equations should change as follows

$$\varrho c_{n,t}+\varrho c_{n,k}{v}_k+{\left[{\varrho }_n(v_{nk}-v_k)\right]}_{,k}=m_n,$$

(41)

$$\varrho c_{s,t}+\varrho c_{s,k}{v}_k+{\left[{\varrho }_s(v_{sk}-v_k)\right]}_{,k}=m_s,$$

(42)

$${\varrho }_n{v}_{ni,t}+{\varrho }_n{v}_{ni,k}{v}_k-T_{nij,j}=0,$$

(43)

$${\varrho }_s{v}_{si,t}+{\varrho }_s{v}_{si,k}{v}_k-T_{sij,j}=0,$$

(44)

$$\varrho {\epsilon }_{,t}+\varrho {\epsilon }_{,k}{v}_k+q_{i,i}-T_{ij}{D}_{ij}=0,$$

(45)

$$q_{i,t}+q_{i,k}{v}_k+{\displaystyle \frac{q_i}{\tau }}+{\displaystyle \frac{\overline{\kappa }}{\tau }}{\epsilon }_{,i}-{\displaystyle \frac{\phi (\Sigma )}{\tau }}{w}_i=0,$$

(46)

wherein $T_{nij}$ and $T_{sij}$ represent the stress of the normal and superfluid component, respectively. Note that in the system above the new unknown velocities are $v_{ni}$ and $v_{si}$, while the counterflow and the barycentric velocities can be derived by the constitutive equations $w_k=v_{nk}-v_{sk}$ and ${\displaystyle v_i=\frac{{\varrho }_n{v}_{ni}+{\varrho }_s{v}_{si}}{\varrho }}$. Hence, the introduction of an internal variable is not necessary. An alternative approach to describe the dissipative state would be to postulate the system of equations

$$\varrho c_{n,t}+\varrho c_{n,k}{v}_k+{\left[{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{w}_k\right]}_{,k}=m_n,$$

(47)

$$\varrho c_{s,t}+\varrho c_{s,k}{v}_k-{\left[{\displaystyle \frac{{\varrho }_n{\varrho }_s}{\varrho }}{w}_k\right]}_{,k}=m_s,$$

(48)

$${\varrho }_n{v}_{ni,t}+{\varrho }_n{v}_{ni,k}{v}_k-T_{nij,j}=0,$$

(49)

$$\varrho {\epsilon }_{,t}+\varrho {\epsilon }_{,k}{v}_k+q_{i,i}-T_{ij}{D}_{ij}=0,$$

(50)

$$q_{i,t}+q_{i,k}{v}_k+{\displaystyle \frac{q_i}{\tau }}+{\displaystyle \frac{\overline{\kappa }}{\tau }}{\epsilon }_{,i}-{\displaystyle \frac{\phi (\Sigma )}{\tau }}{w}_i=0,$$

(51)

$$w_{h,t}+w_{h,k}{v}_k+w_k{v}_{k,h}-H_{,h}=0,$$

(52)

where the unknown velocities are $v_{ni}$ and $w_h$, while the superfluid and barycentric velocities can be derived by the constitutive equations $v_{sk}=w_k-v_{nk}$ and ${\displaystyle v_i=\frac{{\varrho }_n{v}_{ni}+{\varrho }_s{v}_{si}}{\varrho }}$. The state space would be now

$$\Sigma =\{c_n,c_s,\epsilon ,q_k,\gamma ,w_k,v_{nk,i}\}.$$

(53)

Also, for the two models above, the state space is weakly nonlocal, so the compatibility of the assigned constitutive quantities must be explored by applying a generalized thermodynamic framework [19]. Although such approaches are worthy of consideration, since classical studies on superfluids underscore the importance of viscosity in governing the entropy production and the threshold for the occurrence of second sound, they are outside of the focus of the present research, which is aimed at investigating the nondissipative flow of helium II, together with the propagation of first-and-second sound waves. The dissipative state will be considered in future investigations.

4. First-and-Second Sound Waves

According to the point of view expressed Section 2, here we consider a two-fluid mixture with different concentrations and velocities for each constituent and the same temperature for both the constituents. Then, the evolution in space and time of the mixture is ruled by the balance equations for partial masses, linear momentum, energy and heat flux (9 fields), together with the evolution equation for the counterflow velocity (3 fields). In the one-dimensional case, the fields involved are only six, namely, $\{c_n,c_s,v,\epsilon ,q,w\}.$ Taking into account the constitutive Equations (33) and (37), under the hypothesis that the products of the spatial derivatives of $\varrho$ with the term $s_1{c}_n{c}_s{w}^2$ are second-order quantities, we can rewrite the balance Equations (1)–(5) and the differential consequence (10) as follows:

$$c_{n,t}+c_{n,x}v+c_{n,x}{c}_{s}w+c_{s,x}{c}_{n}w+c_n{c}_s{w}_{,x}-m_n=0,$$

(54)

$$c_{s,t}+c_{s,x}v-c_{n,x}{c}_{s}w-c_{s,x}{c}_{n}w-c_n{c}_s{w}_{,x}-m_s=0,$$

(55)

$$\varrho v_{,t}+\varrho v_{,x}v-\varrho s_1{c}_{s,x}{c}_n{w}^2-\varrho s_1{c}_{n,x}{c}_s{w}^2-2\varrho s_1{c}_s{c}_{n}w{w}_{,x}=0,$$

(56)

$$\varrho {\epsilon }_{,t}+\varrho {\epsilon }_{,x}v+q_{,x}-\varrho s_1{c}_s{c}_n{w}^2{v}_{,x}=0,$$

(57)

$$q_{,t}+q_{,x}v+{\displaystyle \frac{q}{\tau }}+{\displaystyle \frac{\overline{\kappa }}{\tau }}{\epsilon }_{,x}-{\displaystyle \frac{\phi (\Sigma )}{\tau }}w=0,$$

(58)

$$w_{,t}+w_{,x}v+w{v}_{,x}-{\displaystyle \frac{\mathsf{\partial }{H}_0}{\mathsf{\partial }\epsilon }}{\epsilon }_{,x}-{\displaystyle \frac{1}{2}}{c}_{s,x}{c}_n{w}^2-{\displaystyle \frac{1}{2}}{c}_{n,x}{c}_s{w}^2-c_s{c}_{n}w{w}_{,x}=0.$$

(59)

The nonlinear system of equations above allows the existence of nonregular solutions.

Definition 1.

An acceleration wave is a traveling surface $\mathcal{S}$ across which a solution $\left\{c_n,c_s,v,\epsilon ,q,w\right\}$ of Equations (54)–(59) is continuous, but its first and higher-order derivatives suffer jump discontinuities [30,31].

Remark 5.

Equations (54)–(59) hold in the points of the two regions behind and ahead $\mathcal{S}$, while on $\mathcal{S}$ those equations must be written in terms of the jumps of the discontinuous fields across the wavefront. Such jumps are given by the differences

$$\begin{array}{cccc}\Delta c_{n,x}=c_{n,x}^{-}-c_{n,x}^{+},\hfill & \Delta c_{s,x}=c_{s,x}^{-}-c_{s,x}^{+},\hfill & \Delta v_{,x}=v_{,x}^{-}-v_{,x}^{+},\hfill & \\ \Delta {\epsilon }_{,x}={\epsilon }_{,x}^{-}-{\epsilon }_{,x}^{+},\hfill & \Delta q_{,x}=q_{,x}^{-}-q_{,x}^{+},\hfill & \Delta w_{,x}=w_{,x}^{-}-w_{,x}^{+},\hfill \end{array}$$

(60)

where the superscript + denotes the value of the corresponding fields in the (equilibrium) region which $\mathcal{S}$ is about to enter, and the superscript − denotes the same value in the (nonequilibrium) region which $\mathcal{S}$ is about to leave.

We introduce the following definitions.

Definition 2.

$$\Delta {\displaystyle \frac{\mathsf{\partial }{c}_n}{\mathsf{\partial }x}}\equiv \delta c_n,\Delta {\displaystyle \frac{\mathsf{\partial }{c}_s}{\mathsf{\partial }x}}\equiv \delta c_s,\Delta {\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }x}}\equiv \delta v,\Delta {\displaystyle \frac{\mathsf{\partial }\epsilon }{\mathsf{\partial }x}}\equiv \delta \epsilon ,\Delta {\displaystyle \frac{\mathsf{\partial }q}{\mathsf{\partial }x}}\equiv \delta q,\Delta {\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }x}}\equiv \delta w,$$

(61)

with δ as the jump of the discontinuous derivatives in the one-dimensional case.

We suppose that across $\mathcal{S}$, the time derivatives of the fields $\left\{c_n,c_s,v,\epsilon ,q,w\right\}$ are discontinuous too. The following corollary is straightforward.

Corollary 1.

The definitions above imply

$$\Delta {\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}=-U\Delta {\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}=-U\delta u,$$

(62)

wherein U is the speed of propagation of thermomechanical disturbances along the x direction, and u denotes one of the fields $\left\{c_n,c_s,v,\epsilon ,q,w\right\}$.

Proof. 

Corollary 1 immediately follows by the continuity across $\mathcal{S}$ of the functions $\left\{c_n,c_s,v,\epsilon ,q,w\right\}$ and by the classical Hadamard identities (see, for instance, Ref. [31] and Equation (55) therein). □

By Definition 2 and Corollary 1, the following result ensues.

Corollary 2.

The jumps of the unknown fields across $\mathcal{S}$ satisfy the system of equations

$$\left(v-U+c_{s}w\right)\delta c_n+c_{n}w\delta c_s+c_n{c}_s\delta w=0,$$

(63)

$$\left(v-U-c_{n}w\right)\delta c_s-c_{s}w\delta c_n-c_n{c}_s\delta w=0,$$

(64)

$$\left(\varrho v-\varrho U\right)\delta v-\varrho s_1{c}_s{w}^2\delta c_n-\varrho s_1{c}_n{w}^2\delta c_s-2\varrho s_1{c}_n{c}_{s}w\delta w=0,$$

(65)

$$\left(\varrho v-\varrho U\right)\delta \epsilon +\delta q-\varrho s_1{c}_s{c}_n{w}^2\delta v=0,$$

(66)

$$\left(v-U\right)\delta q+{\displaystyle \frac{\overline{\kappa }}{\tau }}\delta \epsilon =0,$$

(67)

$${\displaystyle \left(v-U\right)\delta w+w\delta v-\frac{\mathsf{\partial }{H}_0}{\mathsf{\partial }\epsilon }\delta \epsilon -\frac{1}{2}{H}_0{c}_n{w}^2\delta c_s-\frac{1}{2}{H}_0{c}_s{w}^2\delta c_n-c_s{c}_{n}w\delta w=0,}$$

(68)

wherein it must be understood that the coefficients of the unknown jumps are evaluated in the equilibrium region which $\mathcal{S}$ is about to enter.

The system above can be rewritten as

$$\left[\begin{array}{cccccc}\left(v-U+c_{s}w\right)& c_{n}w& 0& 0& 0& c_n{c}_s\\ -c_{s}w& \left(v-U-c_{n}w\right)& 0& 0& 0& -c_n{c}_s\\ -\varrho s_1{c}_s{w}^2& -\varrho s_1{c}_n{w}^2& \left(\varrho v-\varrho U\right)& 0& 0& -2\varrho s_1{c}_n{c}_{s}w\\ 0& 0& -\varrho s_1{c}_s{c}_n{w}^2& \left(\varrho v-\varrho U\right)& +1& 0\\ 0& 0& 0& \overline{\kappa }/\tau & \left(v-U\right)& 0\\ -{\displaystyle \frac{1}{2}}{H}_0{c}_s{w}^2& -{\displaystyle \frac{1}{2}}{H}_0{c}_n{w}^2& w& {\displaystyle -\frac{\mathsf{\partial }{H}_0}{\mathsf{\partial }\epsilon }}& 0& \left(v-U-c_s{c}_{n}w\right)\end{array}\right]\left[\begin{array}{c}\delta c_n\\ \delta c_s\\ \delta v\\ \delta \epsilon \\ \delta q\\ \delta w\end{array}\right]=0.$$

(69)

Thus, from linear algebra, the following theorem ensues.

Theorem 4.

The system (69) admits nontrivial solutions if and only if the following compatibility condition is fulfilled.

$$det\left[\begin{array}{cccccc}\left(v-U+c_{s}w\right)& c_{n}w& 0& 0& 0& c_n{c}_s\\ -c_{s}w& \left(v-U-c_{n}w\right)& 0& 0& 0& -c_n{c}_s\\ -\varrho s_1{c}_s{w}^2& -\varrho s_1{c}_n{w}^2& \left(\varrho v-\varrho U\right)& 0& 0& -2\varrho s_1{c}_n{c}_{s}w\\ 0& 0& -\varrho s_1{c}_s{c}_n{w}^2& \left(\varrho v-\varrho U\right)& +1& 0\\ 0& 0& 0& \overline{\kappa }/\tau & \left(v-U\right)& 0\\ -{\displaystyle \frac{1}{2}}{H}_0{c}_s{w}^2& -{\displaystyle \frac{1}{2}}{H}_0{c}_n{w}^2& w& {\displaystyle -\frac{\mathsf{\partial }{H}_0}{\mathsf{\partial }\epsilon }}& 0& \left(v-U-c_s{c}_{n}w\right)\end{array}\right]=0.$$

(70)

The compatibility condition (70) leads to a sixth-degree equation for the unknown U, whose solutions are not necessarily real, whatever the value of the physical parameters is. The real solutions, however, represent the speeds of propagation of thermomechanical disturbances when both phases, normal and superfluid, coexist.

In order to illustrate the general result in Theorem 4 by practical examples, let us consider some particular cases of the general situation analyzed above.

Postulate 2.

The barycentric velocity v is zero while internal energy, heat flux and counterflow velocity remain constant.

Such an assumption means that the two-fluid composition of the system is evident by the variation in the partial densities only. The system (54)–(59) reduces to

$$c_{n,t}+c_{n,x}{c}_{s}w+c_{s,x}{c}_{n}w-m_n=0,$$

(71)

$$c_{s,t}-c_{n,x}{c}_{s}w-c_{s,x}{c}_{n}w-m_s=0.$$

(72)

Corollary 3.

For the system above, the compatibility condition (70) reduces to

$$det\left[\begin{array}{cc}\left(-U+c_{s}w\right)& c_{n}w\\ -c_{s}w& \left(-U-c_{n}w\right)\end{array}\right]=0.$$

(73)

By simple algebra we see that such a compatibility condition leads to the propagation of a mechanical wave (first sound) with speed $U=(c_s-c_n)w$. Thus, the wave propagates in the same verse as the counterflow velocity if $c_s>c_n$ and in the opposite verse if $c_s<c_n$. This implies that freezing the mixture from the lambda point toward absolute zero, where only the superfluid phase exists, one should observe a mechanical wave which first proceeds in a given verse, then disappears when $c_n$ becomes equal to $c_s$, and finally inverts its verse of propagation in the range of temperature where $c_n<c_s$. The detection of a wave with this characteristics would (i) validate the physical consistency of the model; (ii) give information on the verse of propagation of the counterflow velocity

Postulate 3.

The velocity field v vanishes, while $c_n$ and $c_s$ remain constant.

In such a case, the two-fluid composition of the system becomes apparent through the presence of the counterflow velocity. The system (54)–(59) yields now

$$\varrho {\epsilon }_{,t}+q_{,x}=0,$$

(74)

$$q_{,t}+{\displaystyle \frac{q}{\tau }}+{\displaystyle \frac{\overline{\kappa }}{\tau }}{\epsilon }_{,x}-{\displaystyle \frac{\phi (\Sigma )}{\tau }}w=0,$$

(75)

$$w_{,t}-{\displaystyle \frac{\mathsf{\partial }{H}_0}{\mathsf{\partial }\epsilon }}{\epsilon }_{,x}-c_s{c}_{n}w{w}_{,x}=0.$$

(76)

Corollary 4.

For the system above, the compatibility condition (70) reduces to

$$det\left[\begin{array}{ccc}-\varrho U& 1& 0\\ \overline{\kappa }/\tau & -U& 0\\ {\displaystyle -\frac{\mathsf{\partial }{H}_0}{\mathsf{\partial }\epsilon }}& 0& \left(-U-c_s{c}_{n}w\right)\end{array}\right]=0,$$

(77)

namely, to the the algebraic equation

$$\left(-U-c_s{c}_{n}w\right)\left(\varrho U^2-{\displaystyle \frac{\overline{\kappa }}{\tau }}\right)=0.$$

(78)

Thus, a mechanical wave of speed $U=-c_s{c}_{n}w$ can propagate inside the mixture together with a thermal wave [25,32,33] of speed $U=\pm \sqrt{{\displaystyle \frac{\overline{\kappa }}{\varrho \tau }}}$.

Postulate 4.

The velocity field v is different from zero, while $c_n$, $c_s$ and w remain constant.

Thus, the two-fluid composition of the system manifests itself only through a modified stress tensor, which depends on the two-fluid variables $c_n$, $c_s$ and w, and a modified evolution equation for the heat flux. The system (54) yields

$$\varrho v_{,t}+\varrho v_{,x}v=0,$$

(79)

$$\varrho {\epsilon }_{,t}+\varrho {\epsilon }_{,x}v+q_{,x}-\varrho s_1{c}_s{c}_n{w}^2{v}_{,x}=0,$$

(80)

$$q_{,t}+q_{,x}v+{\displaystyle \frac{q}{\tau }}+{\displaystyle \frac{\overline{\kappa }}{\tau }}{\epsilon }_{,x}-{\displaystyle \frac{\phi (\Sigma )}{\tau }}w=0.$$

(81)

Corollary 5.

For the system above, the compatibility condition (70) reduces to

$$det\left[\begin{array}{ccc}\left(\varrho v-\varrho U\right)& 0& 0\\ -\varrho s_1{c}_s{c}_n{w}^2& \left(\varrho v-\varrho U\right)& +1\\ 0& \overline{\kappa }/\tau & \left(v-U\right)\end{array}\right]=0.$$

(82)

Equation (83) leads to the propagation of a mechanical wave with speed $U=v$, and a thermomechanical wave with speed

$$U=v\pm \sqrt{{\displaystyle \frac{\overline{\kappa }}{\varrho \tau }}}.$$

(83)

We observe that, if the fluid is at rest, the first sound disappears and Equation (83) yields the second sound waves $U=\pm \sqrt{{\displaystyle \frac{\overline{\kappa }}{\varrho \tau }}}$. This in accordance with the theoretical results in [34], wherein it is shown that Čerenkov second sound emission processes, for which the threshold for their nonlinear excitation is lower than the decay threshold, become possible. The experimental evidence of the transition from first sound propagation to second sound propagation would also constitute a test for the validity of the present model.

Postulate 5.

The internal energy, heat flux and counterflow velocity remain constant.

We get the system

$$c_{n,t}+c_{n,x}v+c_{n,x}{c}_{s}w+c_{s,x}{c}_{n}w-m_n=0,$$

(84)

$$c_{s,t}+c_{s,x}v-c_{n,x}{c}_{s}w-c_{s,x}{c}_{n}w-m_s=0,$$

(85)

$$\varrho v_{,t}+\varrho v_{,x}v-\varrho s_1{c}_{s,x}{c}_n{w}^2-\varrho s_1{c}_{n,x}{c}_s{w}^2=0.$$

(86)

Corollary 6.

For the system above, the compatibility condition (70) reduces to

$$det\left[\begin{array}{ccc}\left(v-U+c_{s}w\right)& c_{n}w& 0\\ -c_{s}w& \left(v-U-c_{n}w\right)& 0\\ -\varrho s_1{c}_s{w}^2& -\varrho s_1{c}_n{w}^2& \left(\varrho v-\varrho U\right)\end{array}\right]=0.$$

(87)

The solutions of Equation (87) satisfy the third-grade algebraic equation

$$\varrho (v-U)\left[U^2+(c_{n}w-c_{s}w-2v)U+(c_s-c_n)wv+v^2\right]=0,$$

(88)

which, beside a wave propagating with the fluid $(U=v)$, admits two additional first sound waves of speed

$$U={\displaystyle \frac{-\left(c_{n}w-c_{s}w-2v\right)\pm \sqrt{{\left(c_{n}w-c_{s}w-2v\right)}^2-4\left[(c_s-c_n)wv+v^2\right]}}{2}},$$

(89)

provided

$${(c_{n}w-c_{s}w-2v)}^2-4\left[\left(c_s-c_n\right)wv+v^2\right]\ge 0.$$

(90)

Remark 6.

Let us notice that, in order to construct the figures representing first and second sound speeds, we started to collect the data on the physical parameters of the two-fluid system. However, the experiments do not have an easy interpretation since, as we said before, the two fluids cannot exist independently, and hence it is very difficult to point out the properties of the single components which are necessary to calculate the values of the mechanical and thermal waves [3,9]. The present investigation, where a model allowing the propagation of such waves is developed, can be considered a first step toward a more deep physical description of liquid helium, which encompasses unstable behaviors.

5. Conclusions

In this paper, we developped a model of superfluidity based on the internal variable theory. Following the way already paved in [17], we considered a two-component fluid endowed with a scalar internal variable $\gamma$, whose gradient $\mathbf{w}$ is the counterflow velocity. In this way, $\gamma$ is related to an internal kinematic constraint of the fluid, expressing its possibility of flowing as two fluids which necessarily must move together. The restrictions imposed by the second law of thermodynamics have been derived by applying a generalized Coleman–Noll procedure [18], suitable for the thermodynamic analysis when the gradients of the unknown fields enter the state space. A set of constitutive equations of the Landau type [10], with entropy, entropy flux and stress tensor depending on the counterflow velocity, has been obtained. In the one-dimensional case, the propagation of acceleration waves was investigated. It has been shown that first-and-second sound waves may propagate along the system with speeds depending on the physical parameters of the mixture. In particular, some first sound waves may propagate in the same direction or in the opposite direction of the counterflow velocity, depending on the concentration of normal and superfluid components, respectively. Thermal waves, instead, have the same expression of the ones calculated and detected in dielectric crystals at low temperatures [32,33].

It is worth observing that the system (1)–(6) does not contain terms describing either the interaction of normal and superfluid components or the transition from laminar to turbulent regime. We are aware that this simplification of the interaction between the normal and superfluid components and the neglect of quantum effects omits important quantum mechanical aspects that differentiate superfluid helium from classical fluids. Quantum fluids, such as superfluid helium, neutron stars, or atomic Bose–Einstein condensates, are characterized by quantized vorticity, superfluidity and, at finite temperatures below the critical one, by two-fluid behavior [3,4,5]. The transition from laminar to the turbulent motion, ruled by quantum physics, is very complex [3,4]. The underlying physics is the condensation of atoms according to Bose–Einstein statistics. At nonzero temperatures, quantum turbulence manifests itself in the inviscid superfluid, whose vorticity is confined to vortex lines of atomic thickness and quantized circulation (also referred to as quantized vortices). In this state, the fluid flows without viscosity and the quantization of the circulation results in vortex filaments that interact, reconnect and transfer energy across different scales [3,4]. These quantized vortices, with atomic-scale cores, form a disordered tangle whose dynamics shows the existence of energy transfer mechanisms distinct from those in conventional fluids. The interplay between vortex reconnections and the emission of helical distortions propagating along vortex filaments, also known as Kelvin waves, underpins much of the turbulent behavior observed in these systems. Recent experimental and computational works have yielded further insights into the complex behavior of quantum turbulence. For instance, experiments utilizing controlled rotation have uncovered novel regimes in which inertial waves mediate the energy cascade through a series of interacting dynamical processes bypassing the traditional mechanism. Furthermore, advanced particle tracking methods have allowed for the direct visualization of quantized vortices, revealing the resonant interactions between active particles and vortex filaments. These studies have not only confirmed the presence of distinct dynamical regimes in turbulent superfluids but have also illuminated the scaling behaviors that govern energy transfer at microscopic scales. Quantum vortices have a deep influence on the transport properties of superfluid helium, as they influence very much its thermal resistance. Thus, the quantum vortices influence the heat flux which, in turn, modifies the quantum vortices. The description of such a complex behavior is far from the aims of the present investigation, focused on sound propagation and thermal conductivity phenomena that can be reasonably captured by considering gradient effects. This simplification aligns with the paper’s goal of modeling stable behaviors at low speed and provides a simple thermodynamic framework for analyzing these specific phenomena. The lacking of stability [35] in helium flow at low temperature can be considered in future works once the model has been validated by more concrete experiments testing the wave propagation results described in Section 4.

In the present investigation, we supposed that nonlocal effects are limited to the internal variable only. Indeed, especially when the helium II flows in a narrow capillary (superleak), nonlocal effects can be evident for the whole set of thermodynamic variables. In such a case, the state space should contain also the gradients of all the unknown variables, resulting in a more general model encompassing the two-fluid one. Such an investigation will be carried out in future research.