Extended thermodynamics of polymers and superfluids

J. Non-Newtonian Fluid Mech. 152 (2008) 36–44

Extended thermodynamics of polymers and superfluids J. Casas-Vazquez a , M. Criado-Sancho b , D. Jou a,d,∗ , M.S. Mongiovi c a

Departament de F´ısica, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Catalonia, Spain Departamento de Ciencias y T´ecnicas F´ısicoqu´ımicas, Facultad de Ciencias, UNED, 28040 Madrid, Spain c Dipartimento di Metodi e Modelli Matematici, Facolt` a di Ingegneria, Universit`a di Palermo, 90128 Palermo, Italy d Institut d’Estudis Catalans, 08001 Barcelona, Catalonia, Spain b

Received 29 November 2006; received in revised form 27 February 2007; accepted 12 March 2007

Abstract Polymer solutions and turbulent superfluids have in common the presence of a complex tangle of lines – macromolecules in the former, quantized vortex lines in the latter – which contribute to the internal friction and viscous pressure of the system and make them typical non-Newtonian fluids. Here we briefly review some recent studies on such tangles and their consequences on the dynamics and thermodynamics of the whole system, using the framework of extended irreversible thermodynamics. For polymer solutions, we deal with the coupling of diffusion and viscous pressure and its effects on the stability of the solution and shear-induced phase separation; for superfluids, we focus our attention on the coupling between heat flux and the vortex tangle, and its consequences on the second sound. Finally, we outline some of the features learned from the comparison of such diverse systems, concerning the contribution of the tangle’s geometry to the entropy of the system. © 2007 Elsevier B.V. All rights reserved. Keywords: Extended irreversible thermodynamics; Polymer solutions; Superfluids; Shear-induced phase transitions; Superfluid turbulence

1. Introduction The thermodynamics of flowing fluids is an active topic of research [1–9], because the influence of flow on thermodynamic potentials requires going beyond the local-equilibrium hypothesis and therefore it is a challenge for our current understanding of non-equilibrium macroscopic systems, especially those with long relaxation times. Here, we focus our attention on two different kinds of systems: polymer solutions and turbulent superfluids. In both situations, a dynamic set of filaments contributes to dissipation. In polymer solutions, these filaments are macromolecules [10–12], whereas in turbulent superfluids they are quantized vortex lines [13–16]. The common aim of the thermodynamic formalism is to describe the evolution of the tangle of lines and its influence on the dynamics of the system as a whole. Considering both kinds of systems may lead to fruitful analogies between them. As a unifying framework for such diverse systems we use extended irreversible thermodynamics, where the viscous pres∗

Corresponding author at: Departament de F´ısica, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Catalonia, Spain. E-mail address: [email protected] (D. Jou). 0377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2007.03.002

sure tensor and other fluxes, as heat or diffusion fluxes, is taken as independent variable [5,17–20]. In more general and precise terms, instead of the viscous pressure tensor one should refer to the extra stress tensor, namely, the part of the stress tensor that, in the context of classical hydrodynamics, corresponds to the viscous pressure, but which is not necessarily dissipative in the context of superfluid dynamics. The microscopic meaning of this tensor is different for polymer solutions and for turbulent superfluids, but several macroscopic aspects are common to them. One of the aims of thermodynamics is precisely to outline common perspectives to different kinds of systems. In contrast with usual rheological approaches, which focus on the constitutive equation for the viscous pressure tensor, we also pay attention to non-equilibrium equations of state. Modifications in the equations of state and the conditions of stability imply changes in the phase diagram of substances out of equilibrium, of practical importance in industrial and biological processes [1–6,21]. From the perspective of superfluids, turbulence is important in the refrigeration of systems by means of the flow of superfluids, because it confers to them a nonvanishing viscosity and increases the power necessary to drive the fluid in the refrigerating circuit, in comparison with laminar flows [14,15].

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A decisive step in the present thermodynamic formalism is the formulation of a free energy depending on the characteristics of the flow and to establish its role in transport phenomena. In Section 2, we present a microscopic description of the lines for both kinds of systems. Section 3 is devoted to the coupling of diffusion and viscous pressure and its influence on the phase diagram of polymer solutions. Section 4 deals with the coupling of heat flux and vortex tangle in turbulent superfluids and its influence on the dispersion relation of second sound. Section 5 analyzes the contribution of the geometry of the tangle to the entropy of the system, and it discusses some analogies and differences between both kinds of systems. 2. Line tangle, viscous pressure and internal friction In this section we briefly describe some characteristics of the macromolecular tangles in polymers and of vortex line tangles in turbulent superfluids, and their macroscopic effects through a viscous pressure tensor or a friction tensor. The general analogy between both systems based on a set of dynamic lines contributing to the internal friction should not hide several deep differences between the two systems. Amongst them, we must mention that the transversal radius of the vortex lines in the tangle is of the order of one atomic radius, instead of the size of a monomer – of the order of several atoms – the total length of the vortex tangle is not conserved, but depends on the value of the counterflow velocity – closely related to the heat flux across the system – the elementary lines are closed loops in turbulent superfluids and open chains in polymers; the vortex loops recombine, grow and break at a high rate in contrast with the slow polymerization rates in usual polymer solutions. 2.1. Polymer solutions We consider for dilute polymer solutions a usual bead-andspring model, where the viscous friction with the solvent is modelled by the beads and the internal energy of deformation is described by the elongation of the springs [9–12]. One usually describes the macromolecular configuration by means of the configuration tensor C  C = QQ = ψ(Q)QQ dQ, (2.1) with ψ being the configurational distribution function and Q is the end-to-end vector of the macromolecules. More detailed descriptions are possible in terms of Qi Qi , with Qi is the vector from bead i to bead i + 1, or the vector related with the ith normal mode in a Rouse–Zimm description or, in wormlike-chain models, one may use the local tangent vector to the filament. In equilibrium, the configurational distribution function has the form [11,12]  3(N−1)/2  H ψ(Q) = ψi (Qi ) = 2πkB T i   H  2 (2.2) × exp − Qi , 2kB T i

37

where H is an elastic constant characterizing the intramolecular interactions, T the absolute temperature and kB is the Boltzmann constant. Note that (2.2) does not depend on the orientation, i.e. all orientations are equally probable at equilibrium. As a simplifying hypothesis, it is common to consider a monodisperse solution, i.e. that all macro-molecules have the same number N of monomers, in other words, all of them are assumed to have the same average length. However, one may have polymers with different numbers N of monomers, and the distribution function f(N) should also be considered for polydisperse solutions. Another model of interest is the so-called wormlike chains, which describes the chains as continuous elastic lines. In this case the vectors Qi , of the several fragments constituting the chain, are substituted by the local tangent t(s) along the length of the line; the free energy of these chains does depend not only on the tangent vector but also on the curvature. When the intramolecular energy is Hookean and the macromolecules are advected by the solvent without slip, the configuration tensor (2.1) is directly related to the viscous pressure tensor (in more general conditions, the relation between the microscopic comfiguration tensor and the macroscopic pressure tensor is not one-to-one; in this case, a general connection between them may be obtained by following the Hamiltonian framework proposed in [42,43]). The contribution of the ith normal mode to the viscous pressure tensor, is given by [11,12] Pvi = −nHQi Qi  + nkB T U,

(2.3)

n being the number of polymer molecules per unit volume of the solution and U is the unit tensor. In equilibrium, expression (2.3) vanishes. The simplest description of the time evolution of (2.3) is the Maxwell model, where the viscous pressure tensor is described by dPv η 1 = − Pv − 2 (∇v)s , dt τ τ

(2.4)

with (∇v)s being the symmetric part of the velocity gradient, η the shear viscosity and τ is the relaxation time. Maxwell’s model captures the essential idea of viscoelastic models: the response to slow perturbations is that of a viscous fluid, whereas for fast perturbations it behaves as an elastic solid. However, the material time derivative in (2.4) is not fully satisfactory, and it must be replaced by a frame-indifferent derivative, as in the upperconvected Maxwell model for which the evolution equation for the viscous pressure tensor is [11,12] dPv 1 η − (∇v)T · Pv − Pv · (∇v) = − Pv − 2 (∇v)s . dt τ τ

(2.5)

˙ 0, 0) with γ˙ In a pure shear flow corresponding to v = (γy, v = Pv = the shear rate, Eq. (2.5) yields, in steady situations, P12 21 v 2 ˙ P11 = −2τηγ˙ and all the other components vanish. In −ηγ, contrast, if Eq. (2.4) is used, all the diagonal components in the viscous pressure vanish. Introduction of Pvi or of Qi Qi  as independent variables is essentially equivalent in the case of dilute polymer solutions. In non-equilibrium steady states, Pvi is directly controlled from the outside, whereas for the microscopic understanding of the

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processes, the use of Ci is more suitable though it cannot be directly controlled. 2.2. Superfluids The most outstanding peculiarities of helium II (namely, superfluid 4 He) are its ability to flow along narrow capillaries without resistance, the propagation of two kinds of longitudinal waves, namely first sound – a density or pressure wave – and second sound – a temperature wave – and the so-called fountain and thermomechanic effects, which are related to crossed effects between heat and mass transport. Helium II has a practically infinite thermal conductivity, and the heat flux has a practically infinite relaxation time. When the flow reaches a critical velocity, however, some resistance suddenly appears, and the attenuation of second sound increases. These effects can be explained by the formation of quantized vortex filaments, and their friction with the normal-fluid component. The most widely known model of superfluids is the twofluid model proposed initially by Tisza and Landau [13]. In it, it is assumed that the fluid is a mixture of two components: a superfluid one and a normal one. The superfluid component has zero viscosity and zero entropy, and is irrotational; the normal component has normal viscosity and nonvanishing entropy. Each component is characterized by a density and a velocity, namely, ρs , vs , ρn and vn , respectively; the total density ρ is ρ = ρs + ρn and the bulk velocity of the liquid v is ρv = ρs vs + ρn vn . The two-fluid model explains in an intuitive and efficient way the observations: the flow without resistance along capillaries corresponds to the flow of the superfluid component; the thermomechanical effects – the fact that the flow of heat is accompanied by a flow of momentum – is explained by the fact that the superfluid and the normal components carry both momentum and heat; the first sound corresponds to a wave where both normal and superfluid components oscillate in phase with each other, and the total density changes; in the second sound, the normal and superfluid have opposite oscillations, in such a way that density does not change but temperature does. The first surprises concerning vorticity in superfluids were experiments carried out in rotating cylindrical containers [14–16]. In principle, the superfluid, due to its nonviscous and irrotational character, should not participate in the rotation. However, it was realized that the free surface of the liquid in the rotating cylinder assumed a paraboidal shape of the same form that would result from the rigid rotation of both constituents, normal and superfluid. It was predicted that the superfluid could develop vorticity singularities corresponding to very thin vortices, with a radius of the order of the atomic radius, with quantized vorticity, the quantum of vorticity being κ = h/m, with h is the Planck’s constant and m is the mass of the helium atom. It was observed an array of straight quantized vortices, parallel to the rotation axis, whose areal density under an angular speed Ω (or the corresponding vortex line density per unit volume of the system, L) is LR = Ω/2κ. Furthermore, Hall and Vinen studied the second sound propagation in the rotating superfluid and realized that when its direction is perpendicular to the rotation axis it suffers an extra attenuation compared to the nonrotating con-

tainer, proportional to the angular velocity. In contrast, there was no extra attenuation when the second sound propagated along the rotation axis. It was then suggested that the extra attenuation should be due to a force between the normal fluid and the quantized vortices of the form [13–16] ρ s ρn κLR [Bt × (t × V ns ) − B t × V ns ], (2.6) F =− 2ρ with B and B are the Hall–Vinen coefficients, which depend on the temperature and express the longitudinal and transversal components of the friction force between the normal component and the vortices, t the tangent vector to the vortex line and Vns is the relative velocity of the normal component with respect to the superfluid component. We have taken into consideration that the vortices may be represented parametrically by a vectorial function s(ξ, t), ξ being the arc-length along the line, so that the derivative of s with respect to ξ, denoted as t, is the unit tangent vector. A further observation was carried out in the presence of a heat flux across the system when the flow of mass of superfluid component in one direction was compensated by an opposite flow of normal fluid. When the value of the heat flow is higher than some characteristic value, it was found a transition to a turbulent state, in which a disordered tangle of vortex lines appears, characterized by a given value of the vortex line density L. In fact, if the heat flow is further increased, another transition arises from the first turbulent state with a low value of L (TI turbulence) to a second turbulent state with a high value of L (TII turbulence), with an increase of the attenuation of second sound. It is possible to evaluate these effects by describing the frictional effects of the vortex array or the vortex tangle – i.e. their contribution to viscous dissipation in the superfluid – by means of a tensor Pω . This friction results from the collisions of the excitations constituting the normal fluid – mainly the rotons – with the core of the vortex lines. The collision cross section is maximum when the rotons travel perpendicular to this line, and it is zero when they move parallel to it. We are therefore led to take [13–15] Pω = 21 κLB(U − tt) + B W · t,

(2.7)

in such a way that the force which the tangle exerts on the normal fluid can be written as F = −ρs Pω ·Vns . The angular brackets denote an average over the different orientations of the vortices, and W is the third-order completely antisymmetric tensor. To have a complete description one needs to know L and tt. Usually, it is enough to know L, if the tangle is isotropic, but recently much interest in anisotropic tangles has arised [37]. In polymers, the total number of monomers is conserved, but the macromolecules are able of extracting some energy from the flow and storing it as elastic energy, in such a way that their length changes; in the Hookean elastic model, the square of such length variation is related to the stored energy. Thus, the length of the polymers depends to some extent on the velocity gradient. In superfluid vortex tangles, L is not conserved but it depends strongly on Vns , and the total energy is approximately proportional to the total length of the vortices. Futhermore, the tangle is formed by closed loops in continuous breaking and

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reconnection; these processes influence not only the topology of the tangle, but they also have some energetic consequences through Kelvin-wave excitation and energy cascading to smaller scales. Thus, an evolution equation is needed for L and another one for tt. In the simplest version it is assumed that the tangle is isotropic and that L is described by the so-called Vinen equation [14–16] dL = αV Vns L3/2 − βV κL2 , dt

(2.8)

where αV and βV are the dimensionless coefficients. In strict terms, the tangle is not purely isotropic, but the loops tend to develop on planes normal to the counterflow velocity [37,38], but here we will only deal with simple and illustrative situations and will assume isotropy, a reasonably satisfactory simplification when counterflow velocities are not very high. We have studied several generalizations of (2.8) to incorporate explicitly the effect of the walls and the presence of rotation [22–25]. For instance, the equation   dL L−1/2 V2 3/2 1− = αV Vns L + α1 ns L − βV κL2 , dt d κ (2.9) with  and α1 being dimensionless coefficients and d is the diameter of the tube that incorporates the effects of the walls. Its steady state solutions are 1/2

L1

1/2 L2,3

= 0;

  αV Vns 4βV κ α1 Vns  = 1± 1+ − . 2βV κ α V V αV κ d

(2.10)

The solutions L2,3 correspond to the + and − signs, respectively. These solutions are real only for values of Vns higher than Vc1 =

κ 4βV α κ ≈ c1 . d + 4βV α1 d

α2V

(2.11)

At the transition, when V = Vc1 , there is a discontinuity from L = 0, corresponding to the laminar regime, to L1/2 c1

2α2  1 . = 2 α + 4αβ d

(2.12)

Note from (2.11) that turbulence arises when Vns d/κ > c1 ; thus, this dimensionless quantity plays in counterflow superfluid turbulence a role analogous to the Reynolds number in classical turbulence. Instead of the kinematical viscosity, this combination has in the denominator the quantum of vorticity. These results are in qualitative agreement with experiments, which indicate that when the counterflow velocity increases beyond a first critical value Vc1 = c1 κ/d, suddenly appears a contribution to dissipation, and when it reaches a second critical value Vc2 = c2 κ/d, the dissipation has a second sudden increase [14,15]. The values of the numbers c1 and c2 depend on the temperature. In the first transition, a vortex tangle appears, but with a low value of the vortex line density. The discontinuity from L = 0 (laminar regime) to this first turbulent regime is of the order of 1/2 L1 d2 ≈ 2.5. Afterwards, L increases as L1 = γ1 V − (b1 /d),

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γ 1 and b1 being numerical parameters. A more detailed model explains also a second transition from a regime of low vortex-line density to a regime of higher values of L [22]. 3. Polymers: coupling of diffusion flux and viscous pressure Here, we consider a binary mixture of a polymer and a solvent under shear flow, with special attention to the coupling between diffusion and viscous pressure. We will introduce both quantities as independent variables in the entropy. The corresponding extended Gibbs equation is [5,17] ds = T −1 du − T −1 μ dcs −

vτ (1) vτ (2) v J · dJ − P : dPv , D0 T 2ηT (3.1)

with cs being the concentration (mass fraction) of the solute, μ ≡ μs − μ1 the difference between the specific chemical potentials of the solute and the solvent, D0 the related to the usual diffusion coefficient D as D = D0 (∂μ/∂cs ), and τ (1) and τ (2) are the relaxation times of J and Pv , respectively. Furthermore, we assume for the entropy flux the expression J s = T −1 q − μT −1 J + βPv · J.

(3.2)

The first two terms are classical; the latter one is characteristic of EIT, and β is a phenomenological coefficient coupling Pv and J given by β = −(RT2 )−1 , R being the molar gas constant. The energy and the mass balance equations are, respectively, du = −∇ · q − p(∇ · v) − Pv : (∇v)s , (3.3) dt dcs = −∇ · J. (3.4) ρ dt The corresponding evolution equations for the fluxes are [5,17] ρ

τ (1)

dJ = −(J + D0 ∇μ) + βD0 T ∇ · Pv , dt

(3.5)

and dPv = −(Pv + 2η(∇v)s ) + 2βTη(∇J)s , (3.6) dt where (J)s stands for the symmetric part of J. These equations exhibit couplings between diffusion and viscous stresses. The material time derivatives of J and Pv in these equations should be replaced, in general, by frame-invariant time derivatives, as in (2.5). When the coupling coefficient β and the relaxation times vanish, these equations reduce to the well-known Navier–Stokes and Fick equations, namely Pv = −2η(∇v)s , J = −D0 μ. It should be emphasised that the chemical potential appearing in (3.5) is not the local-equilibrium one, but is obtained by differentiation of the extended free energy related to the extended entropy (3.1) and therefore contains contributions from the fluxes. The chemical potential of component j is defined as   ∂G μj = . (3.7) ∂Nj T,p,Ni ,Pv τ (2)

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The Gibbs free energy in the presence of a viscous pressure is defined as G = U + pV − TS; in view of the non-equilibrium contributions to S, the Gibbs function G may be written as [5]  N−1  τ (2) 1 i v v G = Geq + Pi : (V Pi ) ≡ Geq + JPv : (V Pv ). 4ηi 4 i=1 (3.8) with Geq (T, p, ci ) the local-equilibrium Gibbs free energy. The steady state compliance J on the right-hand side of (3.8) is given by [11] N−1 (2) τ ηi J = i=1 i 2 , (3.9) N−1 η i i=0 where the sum in the denominator includes the solvent viscosity η0 and ηi and τ i are the viscosities and relaxation times of the several normal modes of the macromolecules. To be explicit, we use for the Gibbs free energy G in presence of Pv the expression (3.8). If we write N in terms of the polymer concentration n (moles per unit volume) as N = nV, the chemical potential μ of the solute following from (3.7) and (3.8) may be expressed as μ = μeq +

1 ∂ (1 − V  n) (JV )Pv : Pv , 4V ∂n

(3.10)

where V = ∂V/∂N is the partial molar volume of the polymer and μeq is the local-equilibrium chemical potential of the solute. We use for the diffusion flux the equation (3.5), which in the steady state is D0 J = −D0 ∇μ − ∇ · Pv . RT

(3.11)

In (3.11), the non-equilibrium chemical potential μ contains contributions of Pv , thus providing an additional coupling between viscous effects and diffusion, besides the term in ·Pv . Equations of this kind have been studied in the framework of extended irreversible thermodynamics for a number of years and have revealed their practical usefulness [40,41]. The generalised chemical potential (3.10) leads us to define an effective diffusion coefficient as Deff = D0 (∂μ/∂n) or, by writing D0 in terms of the classical diffusion coefficient D, ˙ where Ψ is defined as Deff = DΨ (n, γ), ˙ = Ψ (n, γ)

(∂μ/∂n) . (∂μeq /∂n)

(3.12)

Using (3.10), (3.12) takes the explicit form [1,2,5]   1 ∂ (1 − V  n) ∂ v v ˙ =1+ Ψ (n, γ) (JV )P : P . (∂μeq /∂n) ∂n 4V ∂n (3.13) When the contribution of the term in Pv :Pv is negative, it induces a flow of solute towards higher solute concentrations, i.e. contrary to the usual Fickian diffusion, and makes that the homogeneous solution becomes unstable [26,27]. Therefore, effective polymer diffusivity depends on the shear rate, molecular weight – through the dependence of the relax-

ation time – and concentration. When effective diffusivity is positive induced migration is expected to be very slow, the only thermodynamic force leading migration being the coupling of the second term in (3.13). However, when diffusivity becomes negative, the non-equilibrium contribution to the chemical potential considerably enhances polymer migration and makes it much faster. The definition of the chemical potential for non-equilibrium situations opens up the possibility of generalising the classical analysis of phase diagrams to non-equilibrium steady states, provided it is suitably complemented with dynamical considerations. Indeed, shear-induced phase segregation is well known in the literature. In references [1–6] a wide bibliography on the subject may be found. For instance, one may analyze the phase diagram of the solution in the plane T − φ (T being the absolute temperature and φ the polymeric volume fraction). In equilibrium situations, when (∂μ1 /∂φ) is positive, the homogeneous solution is stable; otherwise, it splits into two phases with different polymer concentrations. The line separating the stable and unstable regions is the so-called spinodal line, built from the condition (∂μ1 /∂φ) = 0. The equilibrium Gibbs function for a dilute polymer solution may be expressed according to the Flory–Huggins model. In [26–28] we have studied the shear-induced shift of the spinodal line for dilute and entangled polymer solutions and for polymer blends [1,2,5,26–28], with suitable expressions for the shear-state compliance. We will not discuss further on this point, which may be seen in the literature [26–28]. 4. Second sound and vortex tangle in turbulent superfluids In this section we present some applications of extended themodynamics to superfluids in the turbulent regime. The latter is characterized by an irregular tangle of quantized vortex filaments, whose dynamics is a topic of much interest not only in usual superfluids, but also in rotating Bose–Einstein condensates and in superconductors. As a basic theoretical framework one may choose the twofluid model, or a one-fluid model with a vectorial internal degree of freedom, which may be identified as the heat flux, or the relative velocity between normal and superfluid components. The heat flux q is related to the relative velocity V ns = vn − vs between normal and superfluid components by q = ρs Ts(vn − vs ) where ρs is the density of the superfluid component and s is the specific entropy. The idea of a one-fluid model with an internal degree of freedom is less intuitive than the two-fluid model, but it has also some physical appeal because, in actual fact, the two fluids cannot be directly separated. 4.1. Laminar regime In the one-fluid model of helium II, deduced from extended thermodynamics [22–25,29,30] the basic variables are the density ρ, velocity v, absolute temperature T, and heat flux q. The linearized set of evolution equations, neglecting viscous phe-

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In the rotating frame, the vortices are parallel to  and L = 2|Ω|κ; then, we may write

nomena, written in an inertial frame, is dρ + ρ∇ · v = 0, dt

(4.1)

dv + ∇p = 0, ρ dt

(4.2)

du + ∇ · q + p∇ · v = 0, ρ dt

(4.3)

dq + ξ∇T = 0, dt

(4.4)

(4.5)

We consider plane waves, with k = kr + iks , the complex wavenumber, ω the real frequency, and n is the unit vector in the direction of the wave propagation. This leads, for longitudinal waves, to the following dispersion relation (ω2 − k2 v21 )(ω2 − k2 v22 ) = 0, where v1 and v2 are the phase speeds, given by   ∂p ξ ; v22 = . v21 = ∂ρ T ρcv

(4.6)

(4.7)

where cv is the specific heat at constant volume. The first one corresponds to the normal sound wave, or first sound, and the second one is the temperature wave, or second sound.

The interaction between the vortices and the heat flux may be described by adding to the evolution equation for the counterflow velocity Vns a source term proportional to −Pvω · V ns , with Pvω is the friction tensor introduced in (2.7). To be specific, we generalize the evolution Eqs. (4.1)–(4.4) to a rotating framework in the presence of vortices. Eqs. (4.1) and (4.3) keep their form and (4.2) and (4.4) are modified as [23,25] dv + ∇p − i0 + 2ρΩ × v = 0. dt

ρ dV ns + s∇T + 2Ω × V ns = −Pω · V ns ≡ σω , dt ρn

Ω × Ω × q − B Ω × q, |Ω|

(4.10)

where B and B are the Hall–Vinen friction coefficients appearing in (2.7). In counterflow experiments, the vortices form an isotropic tangle. Then, U − tt = (2/3)U and (4.9) takes the form ρ dV ns B + s∇T + 2Ω × V ns = − κLV ns . dt ρn 3

(4.8) (4.9)

where B is the friction coefficient appearing in (2.7) and i0 − 2ρΩ × v denotes the inertial force. The source term σ ω on the right-hand side of (4.9) is able to describe in an unified way the effect of the vortices, both in the rotating system as in the counterflow situation.

(4.11)

Often this expression is written by assuming that L = γ H V2 , which is a good approximation for high values of Vns , and one writes the right hand side of (4.11) as σ ω = −B q2 q with B = (1/3)BκλH . Thus, the production term in (4.9) summarizes in a single expression the vortex contribution to dissipation, both in rotating cylinders and in counterflow turbulence, the difference between both cases being the geometrical distribution of vortex orientation. Propagation of low-amplitude second sound is a very useful experimental tool to obtain information on the density and distribution of vortex lines. We may study them by starting from Eqs. (4.1), (4.4)–(4.8), with (4.10) for rotating cylinders and (4.11) for thermal counterflow. For waves parallel to the rotation axis, up to first order terms in |Ω|, the dispersion relation is the same as for the fluid at rest, namely (4.5). Thus, neither the first nor the second sounds are influenced by the rotation. In contrast, for waves propagating orthogonally to the rotation axis, one obtains, up to first order in |Ω|, (ω2 − k2 v21 )(ω2 + iBω|Ω| − k2 v22 ) = 0,

4.2. Fixed quantized vortices: rotation experiments and counterflow experiments

ρ

ρ dV ns + s∇T + 2Ω × V ns dt ρn = −B

where u is the specific energy and p is the pressure. In liquid helium II both thermal conductivity and relaxation time of the heat flux are extremely high, but their ratio is finite and can be linked with the helium entropy by the relation [30] ξ = λ/τ = ρ(ρs /ρn )Ts2 , with τ is the relaxation time of the heat flux, and λ is the thermal conductivity. Using these relations, and in terms of Vns (4.4) may be rewritten as ρ dV ns + s∇T = 0. dt ρn

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(4.12)

with v1 and v2 defined in (4.7). The first sound is not influenced, whereas the second sound is attenuated, the imaginary part of (2) the wavevector being ks = (B/2v2 )|Ω|. Thus, measurement of the attenuation coefficient of these waves allows us to obtain the friction coefficient B. In counterflow turbulence, the dispersion relation, independent of the direction, is (ω2 − k2 v21 )(ω2 + 13 iωκBL − k2 v22 ) = 0.

(4.13)

As in the previous situation, the sound wave is not influenced by the vortex array, whereas the second sound experiences changes in phase speed and attenuation given by ξ v2 = ρcv



κ 2 B 2 L2 1− 72ω2

 ;

(2)

ks,2 =

κBL . 6v2

(4.14)

The measurements of ks and v2 allow us to obtain the vortex line density L. This has become a standard probe of the vortex density in turbulent superfluids.

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4.3. Coupling second sound and vortex waves In the precedent calculation it has been assumed that the vortex tangle is insensitive to the second sound, i.e. that it behaves like a rigid tangle which only contributes to friction in a passive way. In fact, however, the vortex tangle is flexible and it is deformed by the second sound; on its turn, this deformation produces a back-reaction on the second sound and contributes to the dispersion relation. A detailed study is necessary to relate the dispersion relation with the average value of L, taking into account this coupling between the tangle and the second sound. In [31] we have studied in detail a hydrodynamic model of inhomogeneous vortex tangles, which allows us to incorporate the deformation produced by the second sound on the tangle. The corresponding coupled equations for q and L are dV ns ρ 1 χ0 + s∇T + ∇L = − BκLV ns , dt ρn ρs Ts 3

(4.15)

dL + L∇ · v + v0 ∇ · q = −βV κL2 + αV Vns L3/2 , dt

(4.16)

where the coefficients ν0 , χ0 describe the mentioned couplings. The thermodynamic analysis imposes the restriction that their product must be positive. The resulting dispersion relation for the second sound is ω2 = [v22 (1 − ρεL ν0 ) + ν0 χ0 ]k2 + N1 N2 − iω(N1 + N2 )   k2 2 N3 N4 , (4.17) − i v2 N2 − ω ω − iN1 where v2 is the velocity of the second sound in the absence of vortex tangles, and the coefficients Ni are N1 = (1/3)κBL0 , N2 = 1/2 (0) (0) 2βV κL0 − (3/2)αV L0 Vns , N3 = (1/3)κBVns , and N4 = (0) 3/2 α2 Vns L0 . The progression from (4.6) to (4.13) to (4.17) shows explicitly the increasing level of complexity of the description of the interaction between second sound and vortex tangle when more realistic descriptions are carried out. 5. Contributions of the geometry of the lines to the entropy of polymers and turbulent superfluids In (4.11) we have assumed that L is an independent quantity and have taken for tt the constant value (1/3)U, because t is a unit vector. In more general situations, the tensor tt describing the geometry of the vortices will also be an independent variable, with an evolution equation of its own and a contribution to the entropy. For instance, in the simultaneous presence of rotation and counterflow, tt is no longer isotropic, as the rotation tends to align t parallely to it and the counterflow tends to randomize them [22,25]. This compromise between order and disorder is analogous to paramagnetic systems, where the magnetic field tends to orient the magnetic dipoles whereas the thermal motion tends to randomize them. In these situations one may learn from a parallelism between polymers and vortex tangles. According to EIT, the generalised Gibbs Eq. (3.1) for a polymer solution contains a contribution in Pv [5,17]. This non-equilibrium contribution may also be written

in terms of the configuration tensor C defined in (2.1) because of the relation (2.3) between this tensor and the viscous pressure tensor. From polymer physics it is known that the contribution to the entropy is [5,11,12]   (5.1) Sgeometry (n, C) = kB n 21 Tr(U − C) + ln|det C| . Analogously to polymer physics, the geometry of the tangle of vortex lines in superfluid turbulence may be described by the tensor tt, or by U − tt. The geometrical aspects of the tangle would also have a contribution to the entropy. In analogy with polymer physics, the configurational entropy St of the tangle will be given by   (5.2) St (L, tt) = kB L 21 TrU − tt + ln|dettt| . Indeed, expression (5.1) follows from general arguments based on the orientation distribution function, which are not exclusive of polymer lines. In fact, it follows from the usual expression −f ln f for the entropy combined with a Gaussian hypothesis for the dependence of the probability distribution function of the bead-to-bead vectors and the constraint on the trace of C (to avoid purely energetic rather than entropic contributions). In turbulent superfluids, a Gaussian hypothesis with respect to the tangent vector has been proposed and developed explicitly by Nemirovskii [44]; the restriction on the trace of the corresponding configuration tensor is related to the length of the vortex filament, which is proportional to the energy. In this way, the analogy between polymer physics and turbulent vortex tangles finds a useful aspect. The geometrical contribution to the entropy may be useful, for instance, for a detailed study of vortex diffusion. In this case, a further analogy between polymer physics and vortex tangles may be useful. Indeed, analogously to (3.11), the vortex diffusion in inhomogeneous tangles should be described by D ∇ · U − tt. (5.3) RT where D is the diffusion coefficient of vortex lines, which may be expressed in terms of the coefficients appearing in (4.15) and (4.16) by [31] D = 3ν0 χ0 /BL. This equation incorporates not only inhomogeneities in the vortex line density L, through the chemical potential of the vortex lines, given by μL ≈ (ρs κ2 /4π) ln(L/L* ), with L* is a reference vortex line density, but also in the configuration entropy. A further learning from polymer physics which may be useful in superfluid turbulence is the form of the evolution equation for the friction tensor Pω defined in (2.7) in the presence of inhomogeneous field of the counterflow velocity. It is expected that a relaxational equation for it should have a form analogous to (2.5). Indeed, the non-linear contribution related to the velocity gradient comes from rather general invariance arguments, which may be substantiated in Hamiltonian formulations, which pay a special attention to the general form of the convective contributions [7,17]. However, the fact that the vortex lines are continuously breaking and recombinating makes a relevant difference with usual polymers. An interesting perspective in this context could be learnt from wormlike-chain models and “living” polymers.

J L = −D∇μL −

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In this case, it is suggested an evolution equation of the form [32,33] dPω − (∇V ns )T · Pω − Pω · (∇V ns ) dt 1 2η = − Pω − (∇V ns )s − ζ[(∇V ns )s · Pω + Pω · (∇V ns )s ] τ τ (5.4) Note that this equation is not a direct translation of (2.5) but it has an extra term in ζ. This term, of the kind of the Gordon–Schowalter derivative, describes the nonafineness in the model as a consequence of the fact of the continuous breaking and reconnection of the filaments [33]. Whereas vortex diffusion and tangle entropy are meaningful both for polymers and superfuid vortex tangles, a relevant difference concerning the governing dynamical laws of both systems is the fact that in polymeric fluids the linear objects are material particles with a given mass, whereas the vortices have in fact zero mass, because the location of vortivity is a perfect vacuum. Therefore, the inertia plays a different role in both kinds of systems. It may be argued, however, that the vortices may exhibit inertial effects non on their own, but because of the rotating superfluid around them; thus, the vortices could be associated an inertia of the order of the superfluid mass density multiplied by the volume of a thin cylinder along the vortex lines, whose effective radius would be of the order of the average vortex separation. This would be the mass of the superfluid which must be stopped or set in (rotating) motion in a displacement of the vortex lines. This may be relevant at very high frequency, when vortex density may exhibit ondulatory behaviour. One could also ask on the role of the velocity gradient on the evolution equation for the vortex line density, and on the role of the latter on the effective viscosity of the superfluid, where reconnection induced wave cascades may increase dramatically the kinetic energy dissipation, thus contributing to an effective viscosity [39]. In (2.9) we have shown a generalization of the Vinen equation incorporating the effects of the walls, and in [25] we have studied a generalization incorporating the rotation. It is logical to assume that in the presence of a velocity gradient, the latter will also influence the dynamics of the vortex line density. For instance, one could assume, for a Couette flow with shear ˙ in analogy with a recent proposal by Lipniacki [34], that rate γ,  2  ∂L γ˙ Vns L3/2 + ∇ · (LvL ) = αV 1 − ∂t κL  2  γ˙ κL2 , − βV 1 − (5.5) κL with vL is the velocity of the vortex tangle. In fact, the dependence of γ˙ on this equation is far from being known, for the moment. Thus, the dynamics of the vortex lines is an open topic, with applications in refrigeration of systems by means of flowing superfluid helium, or in the dynamics of superfluid rotating neutron stars [35].

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6. Conclusions In this paper we have presented a brief review of some aspects of polymer solutions and superfluid turbulence in the context of extended irreversible thermodynamics. The connection between these two dissimilar kinds of systems is the presence of a dynamical tangle of lines which contribute to the internal friction of the system. In polymer solution, the fields used in the description are the classical fields (concentration, velocity, temperature) and the viscous pressure tensor Pv · In turbulent superfluids, the fundamental fields are, in addition to the classical fields, the heat flux q (or the counterflow velocity Vns ), the line density L and the tensor Pω describing the geometry of the tangle. Other theories [31,36] use as fundamental fields also the flux of vortex lines JL . In this case, turbulent superfluids can be considered as systems with three components with three densities ρs , ρn and L, and three velocities vs , vn and vL . Some deeper parallelisms between both systems could probably be obtained in a Hamiltonian framework, as the one proposed in [42,43], which yield a general connection between the extra stress tensor and the configurational tensor; we leave this topic for a future exploration. We think that the comparison of polymer solutions and turbulent superfluids may provide a stimulus for opening new questions in the latter, in analogy with some topics which have a long tradition in polymer physics. Here, we have found in polymer physics a way to evaluate the entropy related to the geometry of the tangle in Eq. (5.2) and the suggestion for an evolution Eq. (5.4) for the tensor describing the geometry of the tangle when the counterflow velocity is not uniform, and have seen how the breaking and reconnection of the tangles could influence the dynamical equation. On the other side, we have mentioned two open questions: how the vortex tangle affects the effective viscosity of turbulent superfluids, and how the shear rate of the barycentric velocity may affect the dynamical equation for the vortex line density. To study them in detail, the analogy between both kinds of systems may be a source of inspiration but, of course, each system requires, in the end, a deep understanding of its microscopic specificities. Acknowledgements The study received the financial support of the Spanish Ministry of Education under grants FIS 2006-12296-C02-01, and from the Direcci´o General de Recerca of the Generalitat of Catalonia (2005 SGR 00087). We also acknowledge the support of the Acci´on Integrada Espa˜na-Italia (Grant S2800082F-HI20040316 of the Spanish Ministry of Science and Technology and grant IT2253 of the Italian MIUR). MSM acknowledges the financial support from MIUR under grant PRIN 2005 17439-003 and by Fondi 60% of the University of Palermo. References [1] D. Jou, J. Casas-V´azquez, M. Criado-Sancho, Adv. Polym. Sci. 120 (1995) 207. [2] D. Jou, J. Casas-V´azquez, M. Criado-Sancho, Contrib. Sci. 2 (2003) (2002) 315.

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