Causal Non-Stationary Thermodynamics of Non-Viscous Heat Conducting Fluids with Internal Variables

General Relativity and Gravitation, Vol. 33, No. 9, 2001

0001-7701/ 01/ 0900-1427$19.50/ 0 2001 Plenum Publishing Corporation

1427

Causal Non-Stationary Thermodynamics ofNon-Viscous Heat Conducting Fluids with InternalVariables

Vito Antonio Cimmelli* and Mauro Francaviglia**

Received January 28, 2000; revised version March 16, 2001

Relativistic non-viscous heat conducting fluids with a vectorial internal variable aremodeled according to the dissipation and causality principles. A set of constitutiveequations, ensuring the causal nature of the model, is postulated. The second lawof thermodynamics is exploited by analyzing a suitable covariant form of theClausius–Duhem inequality. A modification of the classical theory of heat conduction,allowing a finite speed of propagation of thermal disturbances, is considered.

KEY WORDS: Relativistic thermodynamics, causality principle.

1. INTRODUCTION

Thermodynamics with internal variables has obtained a wide success in contin-uum mechanics in the last years. Its fields of application deal with a large classof phenomena such as viscoplasticity, electric and magnetic relaxation, complexflows, superfluidity, superconductivity in deformable solids, second sound prop-agation at low temperatures, (see Maugin and Muschik, [1, 2], for an extensivediscussion). The basic idea of this approach is to assume that, besides the con-ventional variables, the space of thermodynamic states may be determined bysome additional internal variables related to certain peculiar properties of thesystem at hand. The time evolution of these variables is characterized by suit-able evolution equations, the form of which may be determined on the basis of

*Department of Mathematics, University of Basilicata, 85100, Potenza, Italy. e-mail: [email protected]

**Department of Mathematics, University of Torino, 10123, Torino, Italy. e-mail: [email protected]

Cimmelli and Francaviglia1428

phenomenological considerations or with the help of statistical mechanics, [3].In any case, internal variables must be measurable through measurements ofthe observable quantities which are related to them. The applications mentionedabove have been developed in the framework of Newtonian mechanics while,as far as we know, very few results are related to general relativity, [4], [5], [6].However, relativistic continuum mechanics enters many fields of modern physicswhere thermodynamics with internal state variables could fruitfully be applied.Meaningful examples are represented, for instance, by the dissipative effects dueto the interaction between radiation and matter or by the heat conduction in agas of neutrinos during a stellar collapse. The aim of this paper is to presenta thermodynamic model of a non viscous heat conducting fluid endowed withan internal state variable. Our point of view will be complementary, but not incontrast, with the set of theories known as Extended Irreversible Thermodynam-ics, whose starting point is represented by the Israel–Stewart formalism and itsvariants, [7], [8], [9], [10]. In the sequel we will be guided by the followinggeneral principles:

1) the second law of thermodynamics;2) the relativity principle;3) the causality principle.

The second law of thermodynamics, in its classical formulation, states that theentropy production, in all thermodynamic processes is non-negative. On theother hand, the relativity principle requires that both the field equations andthe entropy inequality should be invariant under general space-time diffeomor-phisms. Finally, in order to satisfy the causality requirement, it is necessary thatfield equations be of the hyperbolic type, thus allowing thermo-mechanical per-turbations to propagate with a finite speed. As a further requirement we seek formathematical properties of these equations which allow us to prove well-posed-ness of Cauchy’s problem under not too much restrictive hypotheses.1

In Section 2 we illustrate the physics of internal variables through someexamples involving Newtonian systems. Then, after describing the general geo-metric framework in which our theory is developed, in Section 3 we derive thefield equations for the fluid at hand, in a generally covariant form. Special atten-tion is paid to the definition of the system of reference respect to which our equa-tions are obtained. In Section 4, a new model of non viscous fluid with inter-nal variable is proposed and, moreover, the second law of thermodynamics isexploited in order to obtain thermodynamic restrictions on the constitutive equa-tions. To this end we propose a suitable relativistic form of the Clausius–Duhem

1 Some authors believe and claim that all field equations of both continuum mechanics and thermo-dynamics should be cast under a symmetric hyperbolic form, [11]. Although this property auto-matically guarantees well-posedness, [12], we consider it much too restrictive.

Causal Non-Stationary Thermodynamics with Internal Variables 1429

inequality which, at our knowledge, has never been presented in the literature.Section 5 is devoted to investigate the physical conditions under which the pro-posed theory is causally correct, i.e. hyperbolic. Starting from the general modeldeveloped in Section 4, a causal constitutive equation is proposed for the heatflux. Moreover, comparison is made with various forms of heat flux alreadypresent in literature, especially with those models allowing finite speeds of ther-momechanical disturbances. A critical discussion on the mathematical procedureapplied in the present paper is developed in Section 6.

2. INTERNAL VARIABLES IN NEWTONIAN IRREVERSIBLETHERMODYNAMICS

In this section we illustrate the physics of internal variables through someexamples involving Newtonian systems.

Second sound i.e. thermal wave propagation is a typical low temperaturephenomenon which is observed in solid He3 and He4 and in dielectric crys-tals such as sodium fluoride (NaF) and bismuth (Bi), [13], [14]. The classicalFourier’s theory of heat conduction cannot be applied in this case since it sim-ply leads to a parabolic evolution equation and, consequently, to infinite ther-mal wave speeds. Cimmelli and Kosinski [15], proposed a hyperbolic theory ofheat conduction which rests upon non-equilibrium thermodynamics with internalstate variables. These authors introduced an additional scalar variable, the semi-empirical temperature, whose gradient is related to the heat flux by a Fourier’stype heat conduction law. The basic equations for a rigid conductor are:

1) The balance of energy

re + divh c 0 (2.1)

where r is the mass density, e is the specific internal energy and h the heat fluxvector;

2) The Clausius–Duhem inequality

r w + rSv +hv

. ∇v ≤ 0, (2.2)

where v is the absolute temperature, S is the specific entropy and

w c e − vS (2.3)

the Helmholtz free energy;3) The non-equilibrium semi-empirical temperature b, defined through the

differential equation


b c f (v , b), (2.4)

with function f to be determined on the basis of phenomenological considera-tions;

4) The constitutive assumptions

h c −k (v)∇b, (2.5)

e c e(v), (2.6)

w c w(v , ∇b), (2.7)

where k (v) means the thermal conductivity.Compatibility of (2.5)–(2.7) with (2.2) forces w and f to have the form

w c w1(v) +12

w2(v)(∇b)2, (2.8)

f (v , b) c f 1(v) + f 2(b). (2.9)

Let us restrict ourselves to the linear case. Then the specific heat cv c ∂e/ ∂v andthe heat conductivity k are constant while the evolution equation for b takes theform

tb + b c v , (2.10)

where t is a constant relaxation time. Under the hypotheses above the balanceof energy (2.1) yields

rcvtb + rcvb − kDb c 0. (2.11)

This equation is of the hyperbolic type and allows the propagation of thermalwaves with finite speed

U ch

k

rtcv. (2.12)

A similar behaviour has been found for nonlinear material functions and fordeformable heat conductors, [19], [20].

A second example deals with helium II. This fluid remains liquid at tem-peratures close to zero degrees and exhibits some other fascinating propertiesdue to quantum effects which become apparent at the macroscopic scale. For


instance, it is able to flow through a narrow capillary without measurable resis-tance (fountain effect), [16]. According to an original Landau’s idea, [17], heliumII is regarded as a binary mixture of a normal (i.e. Newtonian) fluid with massdensity rn, velocity vn and normal viscosity, and a superfluid one, having den-sity rs, velocity vs and vanishing viscosity. The following relations are assumedto hold

r c rn + rs, (2.13)

rv c rnvn + rsvs, (2.14)

where r and v stand for the mass density and the velocity of the mixture regardedas a whole. Kosinski and Cimmelli, [18], introduced into the constitutive equa-tions a vectorial variable w, related to the so called counterflow velocity V cvn − vs, which is responsible of the main properties of liquid helium II, [16].The model is based on:

1) The balance equations of mass, linear momentum and energy

r + r div v c 0, (2.15)

rv c div T + rb, (2.16)

rddt

(e + 0, 5v . v) − div (vT) + div h − rb . v − rr c 0, (2.17)

with T Cauchy stress tensor, b the body force, e the specific internal energy, hheat flux vector. Moreover div means the divergence operator calculated withrespect to Eulerian coordinates and

f c :df

dtc ∂f/ ∂t + v . grad f. (2.18)

2) The Clausius–Duhem inequality

−r w − rSv + T . L − div k − hc

. grad v ≥ 0, (2.19)

where, w is the Helmholtz free energy, S is the specific entropy, L c grad v andk is an entropy extraflux due to the diffusion of matter inside the body, [21].

3) The balance of mass of the single constituents


rcs + div [rs(vs − v)] c ms, (2.20)

rcn + div [rn(vn − v)] c mn. (2.21)

In Eqs. (2.20) and (2.21) cs ≡ rs/ r and cn ≡ rn/ r mean the mass concentrationsof the superfluid and normal components, respectively, while ms and mn are themass productions of the same components.

4) The scalar internal variable g , defined by the evolution equation

g c H(.), (2.22)

where H(.) is a suitable smooth function to be specified in a similar way toother constitutive functions appearing in the model and w ≡ grad g. Finally, thecounterflow velocity V is related to the gradient w by a suitable constitutive law.

5) The constitutive assumption

F c F*(r, c, cs, g , w), (2.23)

where F is an element of the set (w , e, H, T, h, k, V). Hypotheses above lead toa very general model of mixture with internal variable. However, as a particularcase one can find, [18],

w c w1(r, c, cs) + (1/ 2)cncs |w | 2, (2.24)

T c −pI − (rsrn/ r)w ⊗ w, (2.25)

p c r2∂w1/ ∂r − 0.5(r2s rn/ r2) |w | 2, (2.26)

h c rscncSw. (2.27)

These relations represent just Landau’s two fluid model, derived so in a fullymacroscopic framework.

3. BALANCE EQUATIONS IN A RELATIVISTIC SPACE-TIME

According to the principles of general relativity we consider a relativisticspace-time (V4, dj 2) where V4 is a C ∞ differentiable manifold of dimension4 and dj 2 is a pseudo-Riemannian metric with Lorentzian signature + − − − ;the geometry of space-time is determined by the metric Riemannian connectionassociated with dj 2. We call dj 2 normal hyperbolic metric. If (xa), a c 0, 1, 2,3, is a system of local coordinates on V4, we express dj 2 in the usual coordinateform


dj 2 c gabdxadxb, (3.1)

and call the metric tensor gab the fundamental gravitational tensor, [22], [23],[24], [25]. It satisfies the system of Einstein equations,

Gab c Tab, (3.2)

with

Gab c Rab − 12

Rgab, (3.3)

where Rab is the Ricci curvature tensor, R c gabRab is the scalar curvature andTab the energy-momentum tensor which satisfies the conservation laws

∇aTab c 0. (3.4)

Here and in the sequel the units are such that the velocity of light is unitary, ∇a

is the covariant (metric) derivative and the indices are raised or lowered usingthe metric.

Furthermore, according to Landau viewpoint, [26], we derive our balanceequations in a system of reference drifting in the direction of the heat flux, whosevelocity is such that the mass-energy counterflow of particles relative to thisframe cancels exactly the heat flux, (energy frame). As proved in [26], if bulkand shear viscosity are ignored, the impulse energy tensor takes the form

Tab c r0 f uaub − pgab, (3.5)

p being the pressure and f, given by

f c 1 + e +pr0

, (3.6)

with e specific internal energy, is the specific enthalpy of the fluid. Finally, r0

is the proper material mass density and ua c dxa/ dj is the four-velocity vector,representing the fluid motion in space-time. Let us remark that the relativisticequivalence between mass and energy implies that to any variation of the heatflux ha of a continuous system there corresponds a variation of mass. As a con-sequence the conservation equation (3.4) must be supplemented by the additionalrelation, [26],


∇a(r0ua + qa) c 0, (3.7)

with

qa c 1f

ha. (3.8)

For notational convenience in the sequel we call qa heat flux.We assume that ua is time-like and unitary and qa is space-like and orthog-

onal to ua, i.e.

uaua c 1, qaua c 0. (3.9)

Let us notice that other authors (see e.g. [7], [8], [9]) derive the balance equa-tions with respect to a local observer which is at rest with respect to the averagemotion of particles, (particle system). Hence, to account for the energy con-tent of the heat flux, they add additional terms (like e.g. uaqb + qaub) to theenergy-momentum tensor Tab. In such a system the mass-energy conservationis represented by the equation

∇a(r0ua) c 0. (3.10)

Here and in the sequel we will use the energy frame of reference. Owing to (3.5)and (3.9), Eqn. (3.4) yields immediately

∇a(r0 f ua) − ua∂ap c 0. (3.11)

Moreover, taking (3.7) into account we get from (3.11) the following relation

r0ua �∂ae − p

r20

∂ar0� c f ∇aqa. (3.12)

Finally, by virtue of (3.11), we may rewrite (3.7) in the form

r0 f ua∇aub − gab∂ap c 0 (3.13)

where


gab c gab − uaub. (3.14)

is the space projector.

4. THERMODYNAMICS OF NON VISCOUS FLUIDS WITHINTERNAL VARIABLE

We model a non viscous fluid with internal variable through a suitable set ofconstitutive equations and investigate the consequences on them of the secondlaw of thermodynamics. To this purpose we need a relativistic expression ofthe Clausius–Duhem inequality which represents the local form of the secondlaw. Let us notice that Ehlers first, [28], and then Dixon [29], argued that forirreversible processes, up to first order in departures from equilibrium (linearapproximation), a Clausius–Duhem inequality of the form

∇aSa ≥ 0, (4.1)

where the entropy flux four-vector Sa depends linearly on the independent ther-modynamic variables, seems to be acceptable. Ambiguities do arise if one wishesto go beyond the linear approximation since, in such a case, the classical defini-tions of entropy and temperature, following from thermostatics, seem to be ques-tionable. However, as shown by Alts and Muller [30] and Israel, [7], in rationalrelativistic thermodynamics the necessity of going beyond the linear approxima-tion for Sa comes from the necessity of reducing the speed of propagation ofthermal effects to a finite value. In fact, the infinite value which results fromthe linearized theory is in contrast with the causality requirements of relativisticphysics. We do not proceed further in the analysis of these fundamental and deepproblems, which, to our knowledge, remain still open. In what follows we limitourselves to consider only non-equilibrium systems and processes for which anunambiguous definition of a complete set of independent state variables togetherwith an unambiguous definition of suitable state functions on the space of thestate variables, can be given.2 We propose a Clausius–Duhem inequality whichpresents strong analogies with that of Newtonian fluids and is sufficiently gen-eral to allow the analysis of both dissipative and conservative processes involv-ing those systems whose energy-momentum tensor may be given in the form(3.7). To our knowledge, such an inequality has never been derived in a trulyrelativistic framework. According to the general principles quoted in Section 1,we are interested to obtain it in a fully covariant form. To this purpose let us

2 Let us notice that this point of view is in accordance with the postulate of local equilibrium [32]of irreversible thermodynamics.


first introduce the objective time derivative of a vector field va defined, following[31], by:

Dva c ul∇lva +12

vb(gla∇lub − glb∇lua). (4.2)

For a scalar field f the objective derivative reduces to

Df c ul∇lf. (4.3)

As a consequence, Eqn. (3.12) may be cast in the form

r0De − pr0

Dr0 c f ∇aqa, (4.4)

which, formally, is similar to the classical balance of energy except for the factorf appearing in front of ∇aqa. This will be taken into account when choosing thecorrect form of the entropy inequality. We further observe that if again S meansthe specific entropy, a suitable form for the entropy rate seems to be given byDS. As far as the internal state variables are concerned we postulate that, besidesthe conventional thermodynamic variables r0 and T (T absolute temperature), avector state variable aa enters the constitutive equations and, moreover, it maycontribute to the total entroy production with an additional power term which,taking into account the isotropy of the fluid, may be cast in the form ADa2, wherea2 c −aaaa is the norm of aa 3 while, according with the general principles ofirreversible thermodynamics, the scalar function A, called affinity [1], can beinterpreted as the generalized force conjugated to a2. Finally, we propose thefollowing relativistic Clausius–Duhem inequality

r0DS + ADa2 − f ∇aFa ≥ 0, (4.5)

where Fa is the space-like component of the entropy flux four-vector dividedby the enthalpy f. Eqn. (4.5) presents a strong analogy with the classical one.The factor f in front of ∇aFa has been introduced in order to take (3.12) intoaccount while the negative sign reflects the signature of the hyperbolic metric.Furthermore, it is easily proved that (4.5) can be arranged in the form (4.1). Letus notice that the additional term ADa2 does not appear in the classical treat-

3 The negative sign is assumed for notational convenience. Notice, however, that our later hypotheseswill eventually imply that also aa is space-like.


ments of internal variables (see Section 2). However, in this case, only dissipativephenomena may be considered, [1]. It is proved in [33] that its presence may bejustified on the basis of the general properties of internal variables. Again dueto the isotropy, we are allowed to write constitutive equations of the followingtype

F c F*(r0, T , a2), (4.6)

for each scalar element F of the set of response functions ( p, e, S, A, Fa). Asfar as the vector Fa is concerned, it must be an isotropic vector function of r0,T and aa; we will consider the problem of its representation in Section 4. Herewe simply accept for Fa the classical decomposition

Fa c qa

T+

ka

T, (4.7)

where the extra-flux ka/ T accounts for dissipation not directly related to the heatconduction. An additional entropy flux has already been proposed for classicalcontinua by several authors (see for instance Maugin, [34], and Muller, [21]).

For the extra variable a2 the covariant evolution equation

t∂aa2 c ∂aT + l∂ar0 + jaa, (4.8)

will be postulated. The functions t, l, j are material parameters depending onr0 and T but not depending on a2. Introducing then the Helmholtz free energy w

by the Legendre’s transformation (2.3) and using (4.4) and (4.8), the inequality(4.5) may be rearranged as follows

− r0Dw − r0SDT + TADa2 +pr0

Dr0 +fT

qa∂aT

+fT

ka∂aT − f ∇aka ≥ 0. (4.9)

On the other hand, due to the constitutive equations (4.6) and the kinetic equation(4.7), we may also put the previous inequality in the more explicit form


�−r0Sua − r0∂w

∂Tua − f

∂ka

∂T � ∂aT

+ �−TAua + r0∂w

∂a2ua + (qa + ka)t − f

∂ka

∂a2 � ∂aa2

+ � pr0

ua − r0∂w

∂r0ua − f

T(qa + ka)l − f

∂ka

∂r0� ∂ar0

− fT

(qa + ka)jaa − f ka∂a(lnf−g) ≥ 0. (4.10)

Let us remark now that the three derivatives ∂aT , ∂ar0 and ∂aa2 are indepen-dent and, moreover, their coefficients do not depend on them. Then a classicalprocedure developed by Coleman and Gurtin [35], (see also [36]), which is notnecessary to repeat here, easily shows that (4.10) is satisfied whatever is thevalue of ∂aT , ∂ar0 and ∂aa2 if and only if their three coefficients vanish and,moreover, the following inequality holds

− fT

(qa + ka)jaa − f ka∂a(lnf−g) ≥ 0. (4.11)

This result leads hence us to impose the following thermodynamic restrictionson the constitutive equations

S c − ∂w

∂T− f

r0

∂ka

∂Tua, (4.12)

p c r20

∂w

∂r0+ f r0

∂ka

∂r0ua, (4.13)

A c r0

T∂w

∂a2+

fT

∂ka

∂a2ua, (4.14)

j (qa + ka)aa + f ka∂a(lnf−g) ≤ 0. (4.15)

Let us draw the attention of the reader on the circumstance that all constitutivequantities S, p, A contain an additional term due to the extra entropy flux. When-ever ka vanishes the classical expressions of S, p and A are then recovered (see[1]). Furthermore by (4.14) and (4.15) we may argue that dissipation of energy


is due to the heat flux qa and the energy extra flux ka and also to the internalvariable aa, while the generalized force conjugated to a2 is conservative.

5. CAUSALITY PRINCIPLE AND HYPERBOLICITY

In the present section we face with the causality principle, which requiresthat any physical perturbation propagates with a finite speed. This is not a pri-ori guaranteed when the system of field equations is not hyperbolic. The mostcelebrated example concerns the propagation of thermal waves. As it was firstnoticed by Cattaneo [37] and then reconsidered by Vernotte [38], the classicalFourier theory of heat conduction is, in a sense, paradoxical, allowing an infi-nite speed for thermal perturbations. Cattaneo proposed a rate type constitutiveequation for the heat flux vector h having the form

th + h c −k∇v , (5.1)

where t is a suitable relaxation time and k is the heat conductivity.Fichera [39], proved that in a certain range of temperatures the paradox is

physically irrelevant since all the perturbations having infinite speed are stronglydamped. However, this result does not apply near absolute zero, where thermalwaves have instead been observed. Moreover it is intolerable in any consistentrelativistic theory because it contradicts the causality principle. In dealing withnon-equilibrium thermodynamics heat conduction is a basic phenomenon, andthis motivates the wide literature on the subject. We refer the reader to reviewpapers [40], [41], [42], [9] and to the book [10]. Here we consider only thosepapers which are more close to the point of view expressed in Section 1. Hencewe recall Landau-Lifchitz’s book [26], where the heat flux is assumed to havethe form

qa c −kgab(m∂bT − T∂bm). (5.2)

The ensuing theory is parabolic since, using the constitutive equation (5.2), theanalysis of the relevant Cauchy problem leads to infinite speeds of propaga-tion, [23]. Later on Galipo and Giambo, [43], did generalize Landau’s model inthe framework of the extended irreversible thermodynamics (see also [44]), byproposing the heat flux qa in the form

qa c −kgab(m∂bT − T∂bm + r0TaDqb), (5.3)

with


a c −2T∂S∂q2

, (5.4)

and q2 c −qaqa. One of the purposes of the present paper is to compare theextended thermodynamics approach with the internal variable approach. Hencewe repeat the procedure used by these authors to obtain (5.3). The main assump-tion of extended irreversible thermodynamics is that the generalized fluxes enterthe constitutive equations. For a non viscous heat conducting fluid the basic pos-tulates are thence:

1) The constitutive equation

F c F*(r0, T , q2); (5.5)

2) The generalized Gibbs equation

TDS c r0De − p

r20

Dr0 + aqlDql , (5.6)

where a is that defined by Eqn. (5.4).The balance of impulse-energy (3.11) and equation (5.6) yield

TDS c f ∇lql + r0aqlDql , (5.7)

which may be regarded as a balance law for entropy. Our aim is to put (5.7) inthe general form

∇l(r0Sul + Jl) c j (S ), (5.8)

where Jl is a space-like vector, representing the entropy flux, and j (S ) is theentropy production. Owing to the isotropy of the fluid, Jl may be representedin the form

Jl c b(e, r0, q2)ql . (5.9)

A comparison of (5.7) and (5.8) yields

b c − m

T, (5.10)


where

m c f − TS (5.11)

is the relativistic chemical potential of matter. Consequently, j (S ) may beexpressed in the form

j (S ) c 1T 2

qb(m∂bT − T∂bm + r0TaDqb). (5.12)

Due to second law of thermodynamics, j (S ) must be positive semi-definite.Moreover, a direct inspection shows that (5.12) is in the standard form of aproduct of the flux qb and the generalized force

Ab c 1T 2

(m∂bT − T∂bm + r0TaDqb). (5.13)

According to the general principles of Onsager’s irreversible thermodynamics,[3], we may express the heat flux in the form

qa c −kgab(m∂bT − T∂bm + r0TaDqb), (5.14)

where function k represents the heat conductivity and may depend on the basicfields r0, T, q2. As we pointed out in Section 4, this procedure becomes ques-tionable beyond the approximation of the fluxes as linear functions of the forces,(for a critical analysis of Onsager’s approach to non-equilibrium thermodynam-ics we refer the reader to Truesdell’s book, [45]). Thence, we follow a differentprocedure, which starts by considering the restrictions due to the reduced entropyinequality (4.15) and then imposes that the requirements of relativity and causal-ity are fulfilled. For the sake of simplicity, since we are interested into the prob-lem of heat conduction only, in this section we assume ka c 0. As observed inSection 3, qa must be an isotropic vector function of r0, T and aa. There are rep-resentations theorems for such functions in E3, [46], which have been extendedby Pennisi to the present case, [47]. According to the results mentioned above,qa admits the following representation

qa c q(r0, T , a2)aa, (5.15)

which also shows the vector aa is space-like. It is convenient to write Eqn. (5.15)in the equivalent form


qa c q(r0, T , a2)gabab, (5.16)

making explicit use of the space projector defined by (3.11). Then Eqn. (5.16)is compatible with the causality principle, with second law of thermodynamicsand with the theory of representation of isotropic vector functions. However, itis still too general and some simplifications seem to be necessary. Accordingto the general principles of internal variables theory, [48], we observe that aa

should be measurable through a direct measurement of the heat flux qa and ofthe scalar function q, which should be related to some given physical propertiesof the fluid. Hence, we take q not depending on a2, i.e.

qa c −x(r0, T )gabab. (5.17)

Clearly, the properties of the heat flux and the subsequent characteristics of ourtheory depend now on the evolution equation for aa. If we substitute (4.8) in(5.17) the constitutive equation for qa becomes of the Cattaneo’s type, with t

playing the role of relaxation time. It reduces exactly to the Cattaneo’s equationif x is constant and l is zero. Hence we may argue that (5.17) leads to finitethermal wave speeds. This property is proved by a rigorous procedure in [49].

In order to compare our model with that quoted above let us rewrite (5.17)in a different form, by introducing the chemical potential m, which is a functionof r0, T and a2, and expressing the partial derivative ∂ar0 as a linear combinationof ∂aT , ∂am and ∂aa2. This is allowed because of the standard hypothesis

∂m

∂r0c 1

r0

∂p∂r0

> 0, (5.18)

whose physical meaning is evident. Finally, after some algebra we get the fol-lowing evolution equation

t*∂ba2 + ab c 1L1

m∂bT +1

L2T∂bm, (5.19)

where the material functions L1, L2 and t* depend on r0, T and a2 only. Then,the constitutive equation (5.17) for qa yields

qa c xgab �t*∂ba2 − 1L1

m∂bT − 1L2

T∂bm� . (5.20)


According to (5.20) we call x dynamical heat conductivity. It may reduce infact to the Fourier heat conductivity when t* vanishes. Landau’s model may berecovered by setting

t* c 0, L1 c x

k, L2 c −L1, (5.21)

while the constitutive equation proposed by Galipo’ and Giambo’ may beobtained if the material functions are such that

L1t*ubDa2 c −r0TaDqb, L1 c x

k, L2 c −L1. (5.22)

Let us draw the attention of the reader on the circumstance that this constitutesonly one of the possible applications of the general model of fluid proposed inSection 4, which can be applied to describe a wide class of non-equilibriumphenomena. We close this section by observing that at the thermal equilibrium,i.e. when qa c 0, the vector functions Daa, ∂aT and ∂am must vanish. This is adifferent situation from the classical one, in which only ∂jT , ( j c 1, 2, 3), vanish;this new fact is due both to relativistic effects and to the hyperbolic modificationof the theory, following from the evolution equation (5.20) for the heat flux. It isour opinion that allowing the extra heat flux ka different from zero, will result ina more general theory, encompassing the previous one but allowing aa to havecomponents non parallel to qa. This subject is currently under investigation.

6. CONCLUDING REMARKS

We have developed a model of a relativistic perfect fluid with a vecto-rial internal state variable aa, in the framework of general relativity. A suitablecovariant form of the Clausius–Duhem inequality has been proposed in a formwhich, to our knowledge, appears for the first time in literature. Moreover, thisinequality was exploited to obtain thermodynamic restrictions on constitutiveequations for a perfect fluid with the scalar internal variable a2 c −aaaa. Amodification of the Fourier’s theory of heat conduction, which allows a finitespeed of propagation of thermal disturbances, has been derived by using the rel-ativity and causality principles together with the restrictions due to second lawof thermodynamics.

In order to prevent some critical remarks which could be moved by thereader, let us conclude with some observations on the mathematical procedurewe have used in the previous sections. First of all one could observe that thetransport equation (4.8) may be substituted into (4.9) in different ways, giving


rise to different thermodynamic restrictions. This situation is typical of non-equi-librium thermodynamics if an additional balance law is used. For instance, theOnsager’s procedure exploits second law by analyzing the bilinear form of theentropy production. The latter quantity is evaluated by:

1) substituting the total time derivative of the entropy S (generalized Gibbsequation) into the first law of thermodynamics;

2) comparing the obtained equation with the balance law of entropy in theform (5.8).

However, step 1) may be performed in several ways, leading to differentexpressions of the entropy production and, as a consequence, to different ther-modynamic restrictions. As far as the present case is concerned, let us recall thatour exploitation of second law through Colemann-Gurtin procedure, [35], yieldsa set of thermodynamic restrictions which is equivalent to the Clausius–Duheminequality. Hence, a different use of (4.8) would imply a second set of relationswhich, being equivalent to Clausius–Duhem inequality, is also equivalent to thefirst one.

A second critical remark could concern the causality property of the trans-port equation (4.8), whose rigorous proof has been deferred to a separate paper(reference [49]). The wave analysis showing the causality of a physical modelis usually performed by writing the system of governing equations in the first-order, quasi-linear form and then proving that a suitable matrix, related to thecoefficients of the system, admits only real eigenvalues and a complete set ofeigenvectors spanning the state space. In the relativistic case the eigenvalues,which may be interpreted as the speeds of propagation of thermo-mechanicaldisturbances, must not exceed the velocity of light. This procedure may resultin some lengthy and cumbersome calculations which justify a paper in its ownweight. The method we mentioned above may be found in [50], [51] and [11].In the last reference the general scheme describing the propagation of thermo-acceleration waves in relativistic fluids is pointed out while the complete analysisis deferred to the papers [52] and [53]. Considerations above suggested us a dif-ferent approach. Indeed, as noticed in Section 5, our theory yields as a particularcase a Cattaneo’s type model. So, we may argue that the theory allows for finitewave speeds. The same point of view has been applied in references [5], [54]and [9]. The rigorous analysis is carried on in ref. [49] which will be sent forpublication in the near future.

Finally, it is our opinion that the thermodynamic formalism developed herecould be put in an even more meaningful and compact form by relying on a suit-able modification of the geometric approach to thermodynamic spaces developedby one of us (M.F.) and his coworkers in a series of papers, [55], [56].


ACKNOWLEDGMENTS

We thank the referees for their helpful criticism and comments which con-tributed to improve the first version of this paper.

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Causal Non-Stationary Thermodynamics of Non-Viscous Heat Conducting Fluids with Internal Variables

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