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Generalization of the evolution criterion in extended irreversible thermodynamics
Physics Letters A 324 (2004) 262–271 www.elsevier.com/locate/pla
Generalization of the evolution criterion in extended irreversible thermodynamics S.I. Serdyukov Chemistry Department, Moscow State University, 119992 Moscow, Russia Received 6 November 2003; received in revised form 20 February 2004; accepted 24 February 2004 Communicated by C.R. Doering
Abstract A postulate of extended irreversible thermodynamics is considered, according to which the entropy density is a function of ordinary thermodynamic variables and their material time derivatives. Within the thermodynamic formalism proposed, an entropy balance equation is derived and it is shown that a system in a mechanical stationary state meets an extended evolution criterion, which is a generalization of the known Glansdorff–Prigogine criterion. 2004 Elsevier B.V. All rights reserved. PACS: 05.60.+w; 44.60.+k Keywords: Postulates of thermodynamics; Extended irreversible thermodynamics; Evolution criterion
1. Introduction One of the most general inequalities of irreversible thermodynamics is the Glansdorff–Prigogine evolution criterion, which is valid for any changes in a macroscopic system for time independent boundary conditions [1,2]. In particular, for a case of a mechanical stationary state, Glansdorff and Prigogine showed that the entropy production Ji Xi dV P= i
(Ji are the generalized fluxes, Xi are the thermodynamic forces, and V is the volume of the system) can be transformed to the form, where Ji and Xi are some other fluxes and thermodynamic forces, which satisfy the evolution criterion ∂X ∂X P ≡ Ji i dV 0. (1) ∂t ∂t i
The equality sign in criterion (1) takes place only in a stationary or equilibrium state. E-mail address: [email protected] (S.I. Serdyukov). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.02.068
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The quantity ∂X P /∂t is a part of the time derivative ∂P /∂t of the entropy production and ∂Ji ∂J P ≡ X dV ∂t ∂t i i
is the other part of this time derivative, i.e., ∂P /∂t = ∂J P /∂t + ∂X P /∂t. However, in the general case, both ∂J P /∂t, and ∂P /∂t have no definite sign. The evolution criterion (1) is a generalization of the famous theorem of minimum entropy production, and is reduced to this theorem if the relationship between the fluxes and thermodynamic forces is linear and the matrix of phenomenological coefficients is constant and symmetric [1,2]. The Glansdorff–Prigogine criterion was derived within classical irreversible thermodynamics, which is based on the local equilibrium hypothesis. According to this hypothesis, the entropy density s (per unit mass) is a function of ordinary thermodynamic variables: s = s(u, v, c1 , . . . , cK ),
(2)
where u is the internal energy density per unit mass, v = 1/ρ is the specific volume, ρ is the density of the medium, ck = ρk /ρ is the mass fraction of the component k, and ρk is the partial density of the component k (k = 1, . . . , K). However, when irreversible processes are very fast or very steep, the local equilibrium hypothesis is inapplicable and a more general postulate should be used. It is well known that a generalization of classical irreversible thermodynamics is extended irreversible thermodynamics (EIT) [3–7], for which the question of the existence of an evolution criterion is quite natural. EIT is based on the postulate that the entropy density s is a function of both ordinary thermodynamic variables and certain additional variables. In the conventional version of the extended theory [3–7], the additional variables are the heat flux q, the diffusion fluxes Jk , and the viscous pressure tensor Pv . When in the system, there are only dissipative processes (there is no convective transfer), we have s = s(u, c1 , . . . , cK ; q, J1, . . . , JK ).
(3)
There is experimental evidence that fluxes are the additional variables [8]. This evidence is based on the discovered effect of the fluxes on intensive thermodynamic quantities. However, the formulation of EIT that is based on the assumption that dissipative fluxes are the additional variables is not unique. For example, in [9], discussion was made of various versions of extended theory and the possibilities of their experimental confirmation. In pioneer works [10,11], temperature was regarded as an ordinary variable and time derivatives of temperature were considered as additional variables. Constitutive equations for dual- and triple-phase-lag heat transfer were obtained in [12,13], where the entropy was postulated to be a function of the internal energy and its first, second, and third time derivatives. Jeffreys-type constitutive equations can be obtained under the assumption that the additional variables are the material time derivatives of the internal energy and the specific volume [14]. In this Letter, a more general postulate is considered and it is assumed that the additional variables are the material time derivatives u, ˙ v, ˙ c˙1 , . . . , c˙K . In a case where there are only dissipative processes, which was studied earlier [15], we obtain s = s(u, c1 , . . . , cK ; ∂t u, ∂t c1 , . . . , ∂t cK ),
(4)
where ∂t ≡ ∂/∂t is the local time derivative. The main postulate of this Letter was chosen for the following reason. Inequality (1) shows that, in a stationary state, the evolution criterion degenerates (the left-hand side of inequality (1) becomes zero). Function (4) in this case also reduces to the classical expression (2). However, function (3) does not reduce. This suggests that the evolution criterion should be extended on the basis of postulate (4), rather than postulate (3).
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An extended evolution criterion was previously considered only for heat conduction in solid [12]. Within EIT, only particular variational principles [16,17] and the theorem of the minimum entropy production [18,19] were examined.1 In this Letter, on the basis of a more general postulate s = s(u, v, c1 , . . . , cK ; u, ˙ v, ˙ c˙1 , . . . , c˙K )
(5)
an extended evolution criterion is constructed, which is a direct generalization of the classical Glansdorff–Prigogine criterion. In the next section, the classical theory is described. In Section 3, an extended thermodynamic formalism is developed, on the basis of which an entropy balance equation (in Section 4) and an evolution criterion (in Section 5) are obtained.
2. Classical theory Let us consider a multicomponent system in the absence of external forces and chemical reactions. The classical evolution criterion is obtained using the following fundamental thermodynamic relations [2,20]. (1◦ ) The Gibbs fundamental equation T ds = du + p dv − µk dck ,
(6)
k
which can also be written as T d(ρs) = d(ρu) − µk dρk ,
(7)
k
where T is the temperature, p the pressure, and µk are the chemical potentials. (2◦ ) The Gibbs–Duhem equation written in the form u d T −1 + v d T −1 p − ck d T −1 µk = 0.
(8)
k
By introducing the enthalpy density h = u + pv, Eq. (8) is transformed to the form h d T −1 + vT −1 dp − ck d T −1 µk = 0.
(9)
k
The replacement of the differentials d(T −1 ), dp, and d(T −1 µk ) in Eq. (9) by ∇T −1 , ∇p, and ∇(T −1 µk ), respectively, allows one to obtain the Gibbs–Duhem equation in the form ρhv · ∇T −1 + T −1 v · ∇p − (10) ρk v · ∇ T −1 µk = 0, k
where v = k ck vk is the velocity of the multicomponent medium and vk is the velocity of the component k (k = 1, . . . , K). (3◦ ) The condition of the convexity of the entropy density ρs as a function of ρu, ρ1 , . . . , ρK : ∂T −1 ∂(ρu) ∂(T −1 µk ) ∂ρk − 0, ∂t ∂t ∂t ∂t k
1 See also [4, Section 5.3].
(11)
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which can be obtained directly from Eq. (7). Glansdorff and Prigogine [2] showed that the sign of inequality (11) is related to the conditions of thermodynamic stability of local equilibrium. Let us further consider the continuity equation and the internal energy balance equation in the form dv = −∇ · v, (12) dt du = −∇ · q − P : ∇v. ρ (13) dt Let us decompose the pressure tensor into two parts: P = pU + Pv , where U is the unit matrix and Pv is the viscous pressure tensor, which is assumed to be symmetric. Then, using Eq. (12), the balance equation (13) is transformed to the form dv du +p = −∇ · q − Pv : ∇v. ρ (14) dt dt ρ
Along with Eq. (14), let us consider the mass balance equation for each component of the system (in the absence of chemical reactions): dck (15) = −∇ · Jk , k = 1, . . . , K, dt where Jk = ρk (vk − v) is the conductive flux of the component k. The Gibbs fundamental equation (6) and the mass and energy balance equations (15) and (14) lead to the entropy balance equation ρ
ds = −∇ · Js + σ, dt from which the expressions for the entropy flux Js and the entropy production σ follow: Js = T −1 q − T −1 µk J k , ρ
k
σ = q · ∇T −1 − T −1 Pv : ∇v −
Jk · ∇ T −1 µk 0.
(16)
(17) (18)
k
According to the second law of thermodynamics, the entropy production is non-negative. The entropy production (18) is seen to be a bilinear form of the generalized fluxes and the thermodynamic forces. However, the expression for the entropy production as a bilinear form can be represented non-uniquely. Adding the Gibbs–Duhem equation (10) to expression (18) gives the bilinear form [2] ρk vk · ∇ T −1 µk 0. σ = (q + ρhv) · ∇T −1 − T −1 Pv : ∇v + T −1 v · ∇p − (19) k
A comparison between expressions (18) and (19) shows that expression (19) involves the new fluxes q + ρhv, v, and ρk vk and the new thermodynamic force T −1 ∇p. Let us now show that a part of the time derivative of the entropy production in the entire system, which is given by expression (19), is a form of a fixed sign. For this purpose, let us consider the balance equations for ρu and ρk , which can be derived from Eqs. (13) and (15): ∂(ρu) = −∇ · (q + ρhv) + v · ∇p − Pv : ∇v, ∂t ∂ρk = −∇ · (ρk vk ), k = 1, . . . , K. ∂t
(20) (21)
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Using the balance equations (20) and (21), and also the condition for a mechanical stationary state ∂t v = ∂t p = 0,
(22)
let us transform inequality (11) to the form −1 −1 ρk vk ∂t T µk + (q + ρhv) · ∂t ∇T −1 − Pv : ∂t T −1 ∇v + v · ∂t T −1 ∇p −∇ · (q + ρhv)∂t T − −
ρk vk · ∂t ∇ T
k
−1
µk 0.
k
Let us integrate the last inequality with respect to volume. Using Gauss’ theorem, one can obtain
ρk vk ∂t T −1 µk · n dΣ − (q + ρhv)∂t T −1 − k
ρk vk · ∂t ∇ T −1 µk dV 0, + (q + ρhv) · ∂t ∇T −1 − Pv : ∂t T −1 ∇v + v · ∂t T −1 ∇p − k
(23) where n is the unit vector directed outward along a normal to the surface and dΣ is a surface element. If the quantities T and µk at the boundary of the system are constant, the surface integral is zero. A comparison of the second integral in inequality (23) with expression (19) for the entropy production shows that inequality (23) is a bilinear form of form (1): ∂X P (q + ρhv) · ∂t ∇T −1 − Pv : ∂t T −1 ∇v + v · ∂t T −1 ∇p − = ρk vk · ∂t ∇ T −1 µk dV 0. ∂t k (24) The equality sign in (24) takes place only in an equilibrium or stationary state. Thus, a specific expression for the evolution criterion has been obtained. Below, it will be shown how one can generalize the classical theory, including inequality (24) obtained by Glansdorff and Prigogine.
3. Extended formalism Within extended thermodynamics, to postulate (5), the generalized Gibbs fundamental equation corresponds: T ds = du + p dv − (25) µk dck + Λ d u˙ + Ω d v˙ − Γk d c˙k , k
k
(∂s/∂u)−1
where T = is the extended temperature, p = T (∂s/∂v) is the extended pressure, µk = −T (∂s/∂ck ), k = 1, . . . , K is the extended chemical potential, Λ = T (∂s/∂ u) ˙ is an intensive quantity corresponding to the variable u, ˙ Ω = T (∂s/∂ v) ˙ is an intensive quantity corresponding to the variable v, ˙ and Γk = −T (∂s/∂ c˙k ), k = 1, . . . , K is an intensive quantity corresponding to the variable c˙k . However, for constructing an extended thermodynamic formalism, there is no need to consider the entropy density as a function of u˙ and v˙ in general form (5) and to introduce two intensive quantities, Λ and Ω (25). It is sufficient to analyze s as a function of u˙ and v˙ in the form of the sum u˙ + pv, ˙ i.e., ˙ c˙1 , . . . , c˙K ), s = s(u, v, c1 , . . . , cK ; u˙ + pv,
(26)
where p is the equilibrium pressure, which will be defined below. Here, u˙ + pv˙ is interpreted as an independent variable of the entropy density s.
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Based on postulate (26), let us write the extended Gibbs equation µk dck + Λ d(u˙ + pv) ˙ − Γk d c˙k , T ds = du + p dv − k
267
(27)
k
where T = T (u, v, c1 , . . . , cK ; u˙ + pv, ˙ c˙1 , . . . , c˙K ) =
∂s ∂u
−1
is the extended temperature, µk = µk (u, v, c1 , . . . , cK ; u˙ + pv, ˙ c˙1 , . . . , c˙K ) = −T
∂s , ∂ck
k = 1, . . . , K
is the extended chemical potential, Λ = Λ(u, v, c1 , . . . , cK ; u˙ + pv, ˙ c˙1 , . . . , c˙K ) = T
∂s ∂(u˙ + pv) ˙
is an intensive quantity corresponding to the variable u˙ + pv, ˙ and Γk = Γk (u, v, c1 , . . . , cK ; u˙ + pv, ˙ c˙1 , . . . , c˙K ) = −T
∂s , ∂ c˙k
k = 1, . . . , K
is an intensive quantity corresponding to the variable c˙k . According to the first law of thermodynamics, u˙ + pv˙ is equal to the rate q˙ of change in the amount of heat per unit mass, q˙ = u˙ + pv. ˙ The pressure p is introduced under the condition q˙ = u˙ + pv˙ = 0,
ck = const,
k = 1, . . . , K
(28)
(and, consequently, s = const, which is introduced according to the second law). Then, p = p(u, v, c1 , . . . , cK ) = −
∂u ; ∂v
i.e., in terms of the formalism based on postulate (26), the pressure is not an extended quantity, unlike T , Λ and Γk . In classical thermodynamics, the condition (28) means that, in the system, there is an isentropic process. In terms of postulate (26) and the generalized Gibbs equation (27), a process in which condition (28) is met is also isentropic. However, in terms of a more general postulate (5) and the generalized Gibbs equation (25), this statement is generally incorrect. Thus, let us construct a thermodynamic formalism using only additional intensive quantities Λ, Γ1 , . . . , ΓK according to the fundamental equation (27). The approach stated in this section allows one to consider the following thermodynamic relations. (1◦ ) The fundamental equation (27), which corresponds to postulate (26). For the entropy density ρs per unit volume, the corresponding postulate has the form ˙ ρ c˙1 , . . . , ρ c˙K ). ρs = s˜ (ρu, ρ1 , . . . , ρK ; ρ u˙ + pρ v,
(29)
Based on the postulate (29), let us write the extended Gibbs equation T d(ρs) = d(ρu) − µk dρk + Λ d(ρ u˙ + pρ v) ˙ − Γk d(ρ c˙k ). k
k
(30)
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(2◦ ) To the extended fundamental equation (27), the Gibbs–Duhem equation corresponds, which can be written in the form u d T −1 + v d T −1 p − (31) ˙ d T −1 Λ − ck d T −1 µk + (u˙ + pv) c˙k d T −1 µk = 0. k
k
As in the classical case, Eq. (31) can be transformed to the form ρhv · ∇T −1 + T −1 v · ∇p − ρk v · ∇ T −1 µk + ρ(u˙ + pv)v ˙ · ∇ T −1 Λ −
k
ρ c˙k v · ∇ T −1 Γk = 0.
(32)
k
(3◦ ) Based on Eq. (30), let us write the condition for the convexity of the entropy density ρs as a function of ρu, ˙ and ρ c˙k : ρk , ρ u˙ + pρ v, ∂(T −1 Γk ) ∂(ρ c˙k ) ˙ ∂T −1 ∂(ρu) ∂(T −1 µk ) ∂ρk ∂(T −1 Λ) ∂(ρ u˙ + pρ v) − + − 0. ∂t ∂t ∂t ∂t ∂t ∂t ∂t ∂t k
(33)
k
Thus, an almost complete analogy with the classical formalism is reached. Along with Eqs. (14) and (15), let us also consider the balance equations for the quantities u˙ + pv˙ and c˙k . For this purpose, Eq. (14) should be rewritten in the form u˙ + pv˙ = −v∇ · q − vPv : ∇v
(34)
and its left and right sides should be considered. Further, in the well-known equality [20] da ∂(ρa) (35) = + ∇ · (ρav), dt ∂t where a is a field quantity, let us make the change of variables a = u˙ + pv˙ on the left side and the change of variables a = −v∇ · q − vPv : ∇v on the right side. This gives the balance equation for the variable u˙ + pv: ˙ d(u˙ + pv) ˙ = −∇ · ∂t q + v∇ · q + vPv : ∇v − ∂t Pv : ∇v . ρ (36) dt Apparently, in the balance equation (36), ∂t q + v∇ · q + vPv : ∇v is the flux of u˙ + pv˙ and −∂t (Pv : ∇v) is the source of u˙ + pv. ˙ Let us perform similar transformations in Eq. (15). The change of variables a = c˙k on the left-hand side of Eq. (35) and the change of variables a = −v∇ · Jk on the right side lead to the balance equation ρ
d c˙k (37) = −∇ · (∂t Jk + v∇ · Jk ), k = 1, . . . , K, dt where ∂t Jk + v∇ · Jk is the flux of the quantity c˙k . Thus, in this section, two groups of quantities related to the balance equations (14), (15) and (36), (37) were considered. The first group comprises the internal energy density u, the specific volume v, the mass fractions ˙ v, ˙ and c˙1 , . . . , c˙K . The second c1 , . . . , cK , the pressure p(u, v, c1 , . . . , cK ), and the material time derivatives u, group includes the dissipative fluxes q, Pv , J1 , . . . , JK , the velocity v (the flux of the specific volume v), and their local time derivatives. The postulate was introduced that the quantities of the first group, namely, u, v, c1 , . . . , cK , and u˙ + pv, ˙ c˙1 , . . . , c˙K are independent variables of the extended quantities, specifically, the entropy density s, the temperature T , and the intensive quantities Λ, Γ1 , . . . , ΓK . In later sections, the entropy balance equation will be considered and an explicit expressions for the flux and source of entropy will be given. It will be seen that independent variables of the flux and source of entropy are dissipative fluxes and their local time derivatives, i.e., quantities of the second group. ρ
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4. Entropy balance equation Using the extended Gibbs equation (27), Eqs. (14), (15) and (36), (37), time derivative of the entropy density can be transformed to the form ds −1 −1 −1 v −1 = −∇ · T q − T µk Jk + T Λ ∂t q + v∇ · q + vP : ∇v − T Γk (∂t Jk + v∇ · Jk ) ρ dt k k Jk · ∇ T −1 µk + ∂t q + v∇ · q + vPv : ∇v · ∇ T −1 Λ + q · ∇T −1 − T −1 Pv : ∇v − −T
−1
k
Λ∂t P : ∇v − (∂t Jk + v∇ · Jk ) · ∇ T −1 Γk .
v
(38)
k
Comparing this equality with the entropy balance equation (16), one can obtain the expressions for the entropy flux: Js = T −1 q − T −1 µk Jk + T −1 Λ ∂t q + v∇ · q + vPv : V − T −1 Γk (∂t Jk + v∇ · Jk ), k
k
and the entropy production: σ = q · ∇T −1 − T −1 Pv : ∇v −
Jk · ∇ T −1 µk + ∂t q + v∇ · q + vPv : ∇v · ∇ T −1 Λ
k
(∂t Jk + v∇ · Jk ) · ∇ T −1 Γk 0. − T −1 Λ∂t Pv : ∇v −
(39)
k
As in the classical case, the non-negativity in expression (39) is determined by the second law of thermodynamics. From the last expression for the entropy production, one can obtain the extended fluxes and the thermodynamic forces, and also their relating phenomenological equations (see [14]). In this Letter, the relationship between the fluxes and forces is not considered, since the evolution criterion is independent of the form of phenomenological equations. Note only that the expressions for the entropy flux and the entropy production were obtained without additional assumptions or approximations, which are necessary in conventional extended thermodynamics (the assumption of the form of the entropy flux and the constancy of some coefficients2). Let us now obtain another expression for the entropy production. Adding Eq. (32) to expression (39) gives σ = (q + ρhv) · ∇T −1 − T −1 Pv : ∇v + T −1 v · ∇p − ρk vk · ∇ T −1 µk + (∂t q) · ∇ T −1 Λ −T
−1
Λ∂t P : ∇v − (∂t Jk ) · ∇ T −1 Γk 0.
v
k
(40)
k
Formula (40) is more compact than (39) and is here the main result of the extended theory. For a stationary state, formula (40) reduces to the classical expression. The expression for the entropy production in this form allows one to obtain an extended evolution criterion.
5. Extended evolution criterion Let us consider a system in a mechanical stationary state. Using condition (22), one can obtain ∂t (Pv : ∇v) = (∂t Pv ) : ∇v, which enables one to make a change of variable in formula (40) and write the final expression for the 2 See, e.g., Jou et al. [3, pp. 42–47].
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entropy production: σ = (q + ρhv) · ∇T −1 − T −1 Pv : ∇v + T −1 v · ∇p − −T
−1
Λ ∂t Pv : ∇v − (∂t Jk ) · ∇ T −1 Γk 0.
ρk vk · ∇ T −1 µk + (∂t q) · ∇ T −1 Λ
k
(41)
k
Expression (41) shows that, to the additional thermodynamic forces ∇(T −1 Λ), −T −1 Λ∇v, and −∇(T −1 Γk ), the flux rates ∂t q, ∂t Pv , and ∂t Jk correspond. Along with the balance equations (14) and (15), let us consider the local time derivatives of these equations: ∂(ρ u˙ + pρ v) ˙ = −∇ · (∂t q) − ∂t Pv : ∇v , ∂t ∂(ρ c˙k ) = −∇ · (∂t Jk ), k = 1, . . . , K. ∂t
(42) (43)
Using Eqs. (14), (15), (42), and (43), let us transform inequality (33) to the form −1 −1 −1 −1 −∇ · (q + ρhv)∂t T − ρk vk ∂t T µk + (∂t q)∂t T Λ − (∂t Jk )∂t T Γk k
+ (q + ρhv) · ∂t ∇T
−1
− ∂t T
k
−1
v P : ∇v + ∂t T −1 v · ∇p − ρk vk · ∂t ∇ T −1 µk k
∂t Jk · ∂t ∇ T −1 Γk 0. + (∂t q) · ∂t ∇ T −1 Λ − ∂t T −1 Λ ∂t Pv : ∇v −
(44)
k
Let us integrate the last inequality with respect to volume. Using Gauss’ theorem, the constancy of the quantities T , µk , Λ, and Γk at the boundary, and also the condition (22) for a mechanical stationary state, the extended evolution criterion is obtained: ∂X P ρk vk · ∂t ∇ T −1 µk = (q + ρhv) · ∂t ∇T −1 − Pv : ∂t T −1 ∇v + v · ∂t T −1 ∇p − ∂t k −1 v −1 −1 + (∂t q) · ∂t ∇ T Λ − ∂t P : ∂t T Λ∇v − (∂t Jk ) · ∂t ∇ T Γk dV 0. (45) k
One can see that the last inequality becomes the equality only in an equilibrium or stationary state. The evolution criterion (45) involves the classical and additional parts, which contain the time derivatives of the thermodynamic forces. The additional part also includes the flux rates ∂t q, ∂t Pv , ∂t J1 , . . . , ∂t JK and, therefore, in the vicinity of a stationary state, is of the second order of smallness. The evolution criterion (45) is essentially simplified for dissipative processes without convection: ∂X P = q · ∂t ∇T −1 − Jk · ∂t ∇ T −1 µk + (∂t q) · ∂t ∇ T −1 Λ ∂t k − (∂t Jk ) · ∂t ∇ T −1 Γk dV 0. (46) k
Apparently, in a particular case where the phenomenological equations are linear and the matrix of phenomenological coefficients is constant and symmetric, criterion (46) reduces to the extended theorem of minimum entropy production in a stationary state.
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6. Final remarks In this Letter, the thermodynamic formalism was considered, which is based on the postulate that the entropy density is a function of both ordinary thermodynamic variables and their material time derivatives. A feature of the formalism proposed is the fact that pressure, unlike temperature and chemical potentials, is introduced within the framework of the first law of thermodynamics and is not an extended quantity. The expressions for the entropy flux and the entropy production were obtained, the new fluxes and thermodynamic forces were introduced, and a particular case of a mechanical stationary state was considered. In this case, the entropy production is a function of the fluxes and the flux rates: σ = σ q + ρhv, Pv , v, ρ1 v1 , . . . , ρK vK ; ∂t q, ∂t Pv , ∂t J1 , . . . , ∂t JK , and there is an evolution criterion, which is a generalization of the classical Glansdorff–Prigogine criterion. The fact that the extended evolution criterion can be formulated suggests that the extended formalism considered in this work is a direct generalization of the formalism of classical irreversible thermodynamics.
Acknowledgement This work was supported by the Russian Fund of Base Researches (Grant No. 01-03-32662).
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P. Glansdorff, I. Prigogine, Physica 30 (1964) 351. P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability, and Fluctuations, Wiley–Interscience, New York, 1971. D. Jou, J. Casas-Vázquez, G. Lebon, Extended Irreversible Thermodynamics, second ed., Springer, Berlin, 1996. D. Jou, J. Casas-Vázquez, G. Lebon, Rep. Prog. Phys. 62 (1999) 1035. I. Müller, T. Ruggeri, Extended Thermodynamics, Springer, New York, 1993. L.S. Garsia-Cólin, F.J. Uribe, J. Non-Equilib. Thermodyn. 16 (1991) 89. R.E. Netletton, S.L. Sobolev, J. Non-Equilib. Thermodyn. 20 (1995) 205; R.E. Netletton, S.L. Sobolev, J. Non-Equilib. Thermodyn. 20 (1995) 297; R.E. Netletton, S.L. Sobolev, J. Non-Equilib. Thermodyn. 21 (1996) 1. R. Luzzi, A.R. Vasconcellos, J. Casas-Vázquez, D. Jou, Physica A 248 (1998) 111. A. Barletta, E. Zanchini, Int. J. Heat Mass Transfer 40 (1997) 1007. I. Müller, Arch. Rational Mech. Anal. 41 (1971) 319. R.C. Batra, Arch. Rational Mech. Anal. 53 (1974) 359. S.I. Serdyukov, Phys. Lett. A 281 (2001) 16. S.I. Serdyukov, N.M. Voskresenskii, V.K. Bel’nov, I.I. Karpov, J. Non-Equilib. Thermodyn. 28 (2003) 207. S.I. Serdyukov, Phys. Lett. A 316 (2003) 177. S.I. Serdyukov, Zh. Fiz. Khim. 71 (1997) 1572, Russ. J. Phys. Chem. 71 (1997) 1412. M. Chen, J. Math. Phys. 32 (1991) 744. F. Vázquez, J.A. del Río, Phys. Rev. E 47 (1993) 178. M. Ichiyanagi, Physica A 215 (1995) 123. G. Lebon, C. Dauby, Phys. Rev. A 42 (1990) 4710. S.R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam, 1962.
Generalization of the evolution criterion in extended irreversible thermodynamics S.I. Serdyukov Chemistry Department, Moscow State University, 119992 Moscow, Russia Received 6 November 2003; received in revised form 20 February 2004; accepted 24 February 2004 Communicated by C.R. Doering
Abstract A postulate of extended irreversible thermodynamics is considered, according to which the entropy density is a function of ordinary thermodynamic variables and their material time derivatives. Within the thermodynamic formalism proposed, an entropy balance equation is derived and it is shown that a system in a mechanical stationary state meets an extended evolution criterion, which is a generalization of the known Glansdorff–Prigogine criterion. 2004 Elsevier B.V. All rights reserved. PACS: 05.60.+w; 44.60.+k Keywords: Postulates of thermodynamics; Extended irreversible thermodynamics; Evolution criterion
1. Introduction One of the most general inequalities of irreversible thermodynamics is the Glansdorff–Prigogine evolution criterion, which is valid for any changes in a macroscopic system for time independent boundary conditions [1,2]. In particular, for a case of a mechanical stationary state, Glansdorff and Prigogine showed that the entropy production Ji Xi dV P= i
(Ji are the generalized fluxes, Xi are the thermodynamic forces, and V is the volume of the system) can be transformed to the form, where Ji and Xi are some other fluxes and thermodynamic forces, which satisfy the evolution criterion ∂X ∂X P ≡ Ji i dV 0. (1) ∂t ∂t i
The equality sign in criterion (1) takes place only in a stationary or equilibrium state. E-mail address: [email protected] (S.I. Serdyukov). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.02.068
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The quantity ∂X P /∂t is a part of the time derivative ∂P /∂t of the entropy production and ∂Ji ∂J P ≡ X dV ∂t ∂t i i
is the other part of this time derivative, i.e., ∂P /∂t = ∂J P /∂t + ∂X P /∂t. However, in the general case, both ∂J P /∂t, and ∂P /∂t have no definite sign. The evolution criterion (1) is a generalization of the famous theorem of minimum entropy production, and is reduced to this theorem if the relationship between the fluxes and thermodynamic forces is linear and the matrix of phenomenological coefficients is constant and symmetric [1,2]. The Glansdorff–Prigogine criterion was derived within classical irreversible thermodynamics, which is based on the local equilibrium hypothesis. According to this hypothesis, the entropy density s (per unit mass) is a function of ordinary thermodynamic variables: s = s(u, v, c1 , . . . , cK ),
(2)
where u is the internal energy density per unit mass, v = 1/ρ is the specific volume, ρ is the density of the medium, ck = ρk /ρ is the mass fraction of the component k, and ρk is the partial density of the component k (k = 1, . . . , K). However, when irreversible processes are very fast or very steep, the local equilibrium hypothesis is inapplicable and a more general postulate should be used. It is well known that a generalization of classical irreversible thermodynamics is extended irreversible thermodynamics (EIT) [3–7], for which the question of the existence of an evolution criterion is quite natural. EIT is based on the postulate that the entropy density s is a function of both ordinary thermodynamic variables and certain additional variables. In the conventional version of the extended theory [3–7], the additional variables are the heat flux q, the diffusion fluxes Jk , and the viscous pressure tensor Pv . When in the system, there are only dissipative processes (there is no convective transfer), we have s = s(u, c1 , . . . , cK ; q, J1, . . . , JK ).
(3)
There is experimental evidence that fluxes are the additional variables [8]. This evidence is based on the discovered effect of the fluxes on intensive thermodynamic quantities. However, the formulation of EIT that is based on the assumption that dissipative fluxes are the additional variables is not unique. For example, in [9], discussion was made of various versions of extended theory and the possibilities of their experimental confirmation. In pioneer works [10,11], temperature was regarded as an ordinary variable and time derivatives of temperature were considered as additional variables. Constitutive equations for dual- and triple-phase-lag heat transfer were obtained in [12,13], where the entropy was postulated to be a function of the internal energy and its first, second, and third time derivatives. Jeffreys-type constitutive equations can be obtained under the assumption that the additional variables are the material time derivatives of the internal energy and the specific volume [14]. In this Letter, a more general postulate is considered and it is assumed that the additional variables are the material time derivatives u, ˙ v, ˙ c˙1 , . . . , c˙K . In a case where there are only dissipative processes, which was studied earlier [15], we obtain s = s(u, c1 , . . . , cK ; ∂t u, ∂t c1 , . . . , ∂t cK ),
(4)
where ∂t ≡ ∂/∂t is the local time derivative. The main postulate of this Letter was chosen for the following reason. Inequality (1) shows that, in a stationary state, the evolution criterion degenerates (the left-hand side of inequality (1) becomes zero). Function (4) in this case also reduces to the classical expression (2). However, function (3) does not reduce. This suggests that the evolution criterion should be extended on the basis of postulate (4), rather than postulate (3).
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An extended evolution criterion was previously considered only for heat conduction in solid [12]. Within EIT, only particular variational principles [16,17] and the theorem of the minimum entropy production [18,19] were examined.1 In this Letter, on the basis of a more general postulate s = s(u, v, c1 , . . . , cK ; u, ˙ v, ˙ c˙1 , . . . , c˙K )
(5)
an extended evolution criterion is constructed, which is a direct generalization of the classical Glansdorff–Prigogine criterion. In the next section, the classical theory is described. In Section 3, an extended thermodynamic formalism is developed, on the basis of which an entropy balance equation (in Section 4) and an evolution criterion (in Section 5) are obtained.
2. Classical theory Let us consider a multicomponent system in the absence of external forces and chemical reactions. The classical evolution criterion is obtained using the following fundamental thermodynamic relations [2,20]. (1◦ ) The Gibbs fundamental equation T ds = du + p dv − µk dck ,
(6)
k
which can also be written as T d(ρs) = d(ρu) − µk dρk ,
(7)
k
where T is the temperature, p the pressure, and µk are the chemical potentials. (2◦ ) The Gibbs–Duhem equation written in the form u d T −1 + v d T −1 p − ck d T −1 µk = 0.
(8)
k
By introducing the enthalpy density h = u + pv, Eq. (8) is transformed to the form h d T −1 + vT −1 dp − ck d T −1 µk = 0.
(9)
k
The replacement of the differentials d(T −1 ), dp, and d(T −1 µk ) in Eq. (9) by ∇T −1 , ∇p, and ∇(T −1 µk ), respectively, allows one to obtain the Gibbs–Duhem equation in the form ρhv · ∇T −1 + T −1 v · ∇p − (10) ρk v · ∇ T −1 µk = 0, k
where v = k ck vk is the velocity of the multicomponent medium and vk is the velocity of the component k (k = 1, . . . , K). (3◦ ) The condition of the convexity of the entropy density ρs as a function of ρu, ρ1 , . . . , ρK : ∂T −1 ∂(ρu) ∂(T −1 µk ) ∂ρk − 0, ∂t ∂t ∂t ∂t k
1 See also [4, Section 5.3].
(11)
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which can be obtained directly from Eq. (7). Glansdorff and Prigogine [2] showed that the sign of inequality (11) is related to the conditions of thermodynamic stability of local equilibrium. Let us further consider the continuity equation and the internal energy balance equation in the form dv = −∇ · v, (12) dt du = −∇ · q − P : ∇v. ρ (13) dt Let us decompose the pressure tensor into two parts: P = pU + Pv , where U is the unit matrix and Pv is the viscous pressure tensor, which is assumed to be symmetric. Then, using Eq. (12), the balance equation (13) is transformed to the form dv du +p = −∇ · q − Pv : ∇v. ρ (14) dt dt ρ
Along with Eq. (14), let us consider the mass balance equation for each component of the system (in the absence of chemical reactions): dck (15) = −∇ · Jk , k = 1, . . . , K, dt where Jk = ρk (vk − v) is the conductive flux of the component k. The Gibbs fundamental equation (6) and the mass and energy balance equations (15) and (14) lead to the entropy balance equation ρ
ds = −∇ · Js + σ, dt from which the expressions for the entropy flux Js and the entropy production σ follow: Js = T −1 q − T −1 µk J k , ρ
k
σ = q · ∇T −1 − T −1 Pv : ∇v −
Jk · ∇ T −1 µk 0.
(16)
(17) (18)
k
According to the second law of thermodynamics, the entropy production is non-negative. The entropy production (18) is seen to be a bilinear form of the generalized fluxes and the thermodynamic forces. However, the expression for the entropy production as a bilinear form can be represented non-uniquely. Adding the Gibbs–Duhem equation (10) to expression (18) gives the bilinear form [2] ρk vk · ∇ T −1 µk 0. σ = (q + ρhv) · ∇T −1 − T −1 Pv : ∇v + T −1 v · ∇p − (19) k
A comparison between expressions (18) and (19) shows that expression (19) involves the new fluxes q + ρhv, v, and ρk vk and the new thermodynamic force T −1 ∇p. Let us now show that a part of the time derivative of the entropy production in the entire system, which is given by expression (19), is a form of a fixed sign. For this purpose, let us consider the balance equations for ρu and ρk , which can be derived from Eqs. (13) and (15): ∂(ρu) = −∇ · (q + ρhv) + v · ∇p − Pv : ∇v, ∂t ∂ρk = −∇ · (ρk vk ), k = 1, . . . , K. ∂t
(20) (21)
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Using the balance equations (20) and (21), and also the condition for a mechanical stationary state ∂t v = ∂t p = 0,
(22)
let us transform inequality (11) to the form −1 −1 ρk vk ∂t T µk + (q + ρhv) · ∂t ∇T −1 − Pv : ∂t T −1 ∇v + v · ∂t T −1 ∇p −∇ · (q + ρhv)∂t T − −
ρk vk · ∂t ∇ T
k
−1
µk 0.
k
Let us integrate the last inequality with respect to volume. Using Gauss’ theorem, one can obtain
ρk vk ∂t T −1 µk · n dΣ − (q + ρhv)∂t T −1 − k
ρk vk · ∂t ∇ T −1 µk dV 0, + (q + ρhv) · ∂t ∇T −1 − Pv : ∂t T −1 ∇v + v · ∂t T −1 ∇p − k
(23) where n is the unit vector directed outward along a normal to the surface and dΣ is a surface element. If the quantities T and µk at the boundary of the system are constant, the surface integral is zero. A comparison of the second integral in inequality (23) with expression (19) for the entropy production shows that inequality (23) is a bilinear form of form (1): ∂X P (q + ρhv) · ∂t ∇T −1 − Pv : ∂t T −1 ∇v + v · ∂t T −1 ∇p − = ρk vk · ∂t ∇ T −1 µk dV 0. ∂t k (24) The equality sign in (24) takes place only in an equilibrium or stationary state. Thus, a specific expression for the evolution criterion has been obtained. Below, it will be shown how one can generalize the classical theory, including inequality (24) obtained by Glansdorff and Prigogine.
3. Extended formalism Within extended thermodynamics, to postulate (5), the generalized Gibbs fundamental equation corresponds: T ds = du + p dv − (25) µk dck + Λ d u˙ + Ω d v˙ − Γk d c˙k , k
k
(∂s/∂u)−1
where T = is the extended temperature, p = T (∂s/∂v) is the extended pressure, µk = −T (∂s/∂ck ), k = 1, . . . , K is the extended chemical potential, Λ = T (∂s/∂ u) ˙ is an intensive quantity corresponding to the variable u, ˙ Ω = T (∂s/∂ v) ˙ is an intensive quantity corresponding to the variable v, ˙ and Γk = −T (∂s/∂ c˙k ), k = 1, . . . , K is an intensive quantity corresponding to the variable c˙k . However, for constructing an extended thermodynamic formalism, there is no need to consider the entropy density as a function of u˙ and v˙ in general form (5) and to introduce two intensive quantities, Λ and Ω (25). It is sufficient to analyze s as a function of u˙ and v˙ in the form of the sum u˙ + pv, ˙ i.e., ˙ c˙1 , . . . , c˙K ), s = s(u, v, c1 , . . . , cK ; u˙ + pv,
(26)
where p is the equilibrium pressure, which will be defined below. Here, u˙ + pv˙ is interpreted as an independent variable of the entropy density s.
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Based on postulate (26), let us write the extended Gibbs equation µk dck + Λ d(u˙ + pv) ˙ − Γk d c˙k , T ds = du + p dv − k
267
(27)
k
where T = T (u, v, c1 , . . . , cK ; u˙ + pv, ˙ c˙1 , . . . , c˙K ) =
∂s ∂u
−1
is the extended temperature, µk = µk (u, v, c1 , . . . , cK ; u˙ + pv, ˙ c˙1 , . . . , c˙K ) = −T
∂s , ∂ck
k = 1, . . . , K
is the extended chemical potential, Λ = Λ(u, v, c1 , . . . , cK ; u˙ + pv, ˙ c˙1 , . . . , c˙K ) = T
∂s ∂(u˙ + pv) ˙
is an intensive quantity corresponding to the variable u˙ + pv, ˙ and Γk = Γk (u, v, c1 , . . . , cK ; u˙ + pv, ˙ c˙1 , . . . , c˙K ) = −T
∂s , ∂ c˙k
k = 1, . . . , K
is an intensive quantity corresponding to the variable c˙k . According to the first law of thermodynamics, u˙ + pv˙ is equal to the rate q˙ of change in the amount of heat per unit mass, q˙ = u˙ + pv. ˙ The pressure p is introduced under the condition q˙ = u˙ + pv˙ = 0,
ck = const,
k = 1, . . . , K
(28)
(and, consequently, s = const, which is introduced according to the second law). Then, p = p(u, v, c1 , . . . , cK ) = −
∂u ; ∂v
i.e., in terms of the formalism based on postulate (26), the pressure is not an extended quantity, unlike T , Λ and Γk . In classical thermodynamics, the condition (28) means that, in the system, there is an isentropic process. In terms of postulate (26) and the generalized Gibbs equation (27), a process in which condition (28) is met is also isentropic. However, in terms of a more general postulate (5) and the generalized Gibbs equation (25), this statement is generally incorrect. Thus, let us construct a thermodynamic formalism using only additional intensive quantities Λ, Γ1 , . . . , ΓK according to the fundamental equation (27). The approach stated in this section allows one to consider the following thermodynamic relations. (1◦ ) The fundamental equation (27), which corresponds to postulate (26). For the entropy density ρs per unit volume, the corresponding postulate has the form ˙ ρ c˙1 , . . . , ρ c˙K ). ρs = s˜ (ρu, ρ1 , . . . , ρK ; ρ u˙ + pρ v,
(29)
Based on the postulate (29), let us write the extended Gibbs equation T d(ρs) = d(ρu) − µk dρk + Λ d(ρ u˙ + pρ v) ˙ − Γk d(ρ c˙k ). k
k
(30)
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(2◦ ) To the extended fundamental equation (27), the Gibbs–Duhem equation corresponds, which can be written in the form u d T −1 + v d T −1 p − (31) ˙ d T −1 Λ − ck d T −1 µk + (u˙ + pv) c˙k d T −1 µk = 0. k
k
As in the classical case, Eq. (31) can be transformed to the form ρhv · ∇T −1 + T −1 v · ∇p − ρk v · ∇ T −1 µk + ρ(u˙ + pv)v ˙ · ∇ T −1 Λ −
k
ρ c˙k v · ∇ T −1 Γk = 0.
(32)
k
(3◦ ) Based on Eq. (30), let us write the condition for the convexity of the entropy density ρs as a function of ρu, ˙ and ρ c˙k : ρk , ρ u˙ + pρ v, ∂(T −1 Γk ) ∂(ρ c˙k ) ˙ ∂T −1 ∂(ρu) ∂(T −1 µk ) ∂ρk ∂(T −1 Λ) ∂(ρ u˙ + pρ v) − + − 0. ∂t ∂t ∂t ∂t ∂t ∂t ∂t ∂t k
(33)
k
Thus, an almost complete analogy with the classical formalism is reached. Along with Eqs. (14) and (15), let us also consider the balance equations for the quantities u˙ + pv˙ and c˙k . For this purpose, Eq. (14) should be rewritten in the form u˙ + pv˙ = −v∇ · q − vPv : ∇v
(34)
and its left and right sides should be considered. Further, in the well-known equality [20] da ∂(ρa) (35) = + ∇ · (ρav), dt ∂t where a is a field quantity, let us make the change of variables a = u˙ + pv˙ on the left side and the change of variables a = −v∇ · q − vPv : ∇v on the right side. This gives the balance equation for the variable u˙ + pv: ˙ d(u˙ + pv) ˙ = −∇ · ∂t q + v∇ · q + vPv : ∇v − ∂t Pv : ∇v . ρ (36) dt Apparently, in the balance equation (36), ∂t q + v∇ · q + vPv : ∇v is the flux of u˙ + pv˙ and −∂t (Pv : ∇v) is the source of u˙ + pv. ˙ Let us perform similar transformations in Eq. (15). The change of variables a = c˙k on the left-hand side of Eq. (35) and the change of variables a = −v∇ · Jk on the right side lead to the balance equation ρ
d c˙k (37) = −∇ · (∂t Jk + v∇ · Jk ), k = 1, . . . , K, dt where ∂t Jk + v∇ · Jk is the flux of the quantity c˙k . Thus, in this section, two groups of quantities related to the balance equations (14), (15) and (36), (37) were considered. The first group comprises the internal energy density u, the specific volume v, the mass fractions ˙ v, ˙ and c˙1 , . . . , c˙K . The second c1 , . . . , cK , the pressure p(u, v, c1 , . . . , cK ), and the material time derivatives u, group includes the dissipative fluxes q, Pv , J1 , . . . , JK , the velocity v (the flux of the specific volume v), and their local time derivatives. The postulate was introduced that the quantities of the first group, namely, u, v, c1 , . . . , cK , and u˙ + pv, ˙ c˙1 , . . . , c˙K are independent variables of the extended quantities, specifically, the entropy density s, the temperature T , and the intensive quantities Λ, Γ1 , . . . , ΓK . In later sections, the entropy balance equation will be considered and an explicit expressions for the flux and source of entropy will be given. It will be seen that independent variables of the flux and source of entropy are dissipative fluxes and their local time derivatives, i.e., quantities of the second group. ρ
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4. Entropy balance equation Using the extended Gibbs equation (27), Eqs. (14), (15) and (36), (37), time derivative of the entropy density can be transformed to the form ds −1 −1 −1 v −1 = −∇ · T q − T µk Jk + T Λ ∂t q + v∇ · q + vP : ∇v − T Γk (∂t Jk + v∇ · Jk ) ρ dt k k Jk · ∇ T −1 µk + ∂t q + v∇ · q + vPv : ∇v · ∇ T −1 Λ + q · ∇T −1 − T −1 Pv : ∇v − −T
−1
k
Λ∂t P : ∇v − (∂t Jk + v∇ · Jk ) · ∇ T −1 Γk .
v
(38)
k
Comparing this equality with the entropy balance equation (16), one can obtain the expressions for the entropy flux: Js = T −1 q − T −1 µk Jk + T −1 Λ ∂t q + v∇ · q + vPv : V − T −1 Γk (∂t Jk + v∇ · Jk ), k
k
and the entropy production: σ = q · ∇T −1 − T −1 Pv : ∇v −
Jk · ∇ T −1 µk + ∂t q + v∇ · q + vPv : ∇v · ∇ T −1 Λ
k
(∂t Jk + v∇ · Jk ) · ∇ T −1 Γk 0. − T −1 Λ∂t Pv : ∇v −
(39)
k
As in the classical case, the non-negativity in expression (39) is determined by the second law of thermodynamics. From the last expression for the entropy production, one can obtain the extended fluxes and the thermodynamic forces, and also their relating phenomenological equations (see [14]). In this Letter, the relationship between the fluxes and forces is not considered, since the evolution criterion is independent of the form of phenomenological equations. Note only that the expressions for the entropy flux and the entropy production were obtained without additional assumptions or approximations, which are necessary in conventional extended thermodynamics (the assumption of the form of the entropy flux and the constancy of some coefficients2). Let us now obtain another expression for the entropy production. Adding Eq. (32) to expression (39) gives σ = (q + ρhv) · ∇T −1 − T −1 Pv : ∇v + T −1 v · ∇p − ρk vk · ∇ T −1 µk + (∂t q) · ∇ T −1 Λ −T
−1
Λ∂t P : ∇v − (∂t Jk ) · ∇ T −1 Γk 0.
v
k
(40)
k
Formula (40) is more compact than (39) and is here the main result of the extended theory. For a stationary state, formula (40) reduces to the classical expression. The expression for the entropy production in this form allows one to obtain an extended evolution criterion.
5. Extended evolution criterion Let us consider a system in a mechanical stationary state. Using condition (22), one can obtain ∂t (Pv : ∇v) = (∂t Pv ) : ∇v, which enables one to make a change of variable in formula (40) and write the final expression for the 2 See, e.g., Jou et al. [3, pp. 42–47].
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entropy production: σ = (q + ρhv) · ∇T −1 − T −1 Pv : ∇v + T −1 v · ∇p − −T
−1
Λ ∂t Pv : ∇v − (∂t Jk ) · ∇ T −1 Γk 0.
ρk vk · ∇ T −1 µk + (∂t q) · ∇ T −1 Λ
k
(41)
k
Expression (41) shows that, to the additional thermodynamic forces ∇(T −1 Λ), −T −1 Λ∇v, and −∇(T −1 Γk ), the flux rates ∂t q, ∂t Pv , and ∂t Jk correspond. Along with the balance equations (14) and (15), let us consider the local time derivatives of these equations: ∂(ρ u˙ + pρ v) ˙ = −∇ · (∂t q) − ∂t Pv : ∇v , ∂t ∂(ρ c˙k ) = −∇ · (∂t Jk ), k = 1, . . . , K. ∂t
(42) (43)
Using Eqs. (14), (15), (42), and (43), let us transform inequality (33) to the form −1 −1 −1 −1 −∇ · (q + ρhv)∂t T − ρk vk ∂t T µk + (∂t q)∂t T Λ − (∂t Jk )∂t T Γk k
+ (q + ρhv) · ∂t ∇T
−1
− ∂t T
k
−1
v P : ∇v + ∂t T −1 v · ∇p − ρk vk · ∂t ∇ T −1 µk k
∂t Jk · ∂t ∇ T −1 Γk 0. + (∂t q) · ∂t ∇ T −1 Λ − ∂t T −1 Λ ∂t Pv : ∇v −
(44)
k
Let us integrate the last inequality with respect to volume. Using Gauss’ theorem, the constancy of the quantities T , µk , Λ, and Γk at the boundary, and also the condition (22) for a mechanical stationary state, the extended evolution criterion is obtained: ∂X P ρk vk · ∂t ∇ T −1 µk = (q + ρhv) · ∂t ∇T −1 − Pv : ∂t T −1 ∇v + v · ∂t T −1 ∇p − ∂t k −1 v −1 −1 + (∂t q) · ∂t ∇ T Λ − ∂t P : ∂t T Λ∇v − (∂t Jk ) · ∂t ∇ T Γk dV 0. (45) k
One can see that the last inequality becomes the equality only in an equilibrium or stationary state. The evolution criterion (45) involves the classical and additional parts, which contain the time derivatives of the thermodynamic forces. The additional part also includes the flux rates ∂t q, ∂t Pv , ∂t J1 , . . . , ∂t JK and, therefore, in the vicinity of a stationary state, is of the second order of smallness. The evolution criterion (45) is essentially simplified for dissipative processes without convection: ∂X P = q · ∂t ∇T −1 − Jk · ∂t ∇ T −1 µk + (∂t q) · ∂t ∇ T −1 Λ ∂t k − (∂t Jk ) · ∂t ∇ T −1 Γk dV 0. (46) k
Apparently, in a particular case where the phenomenological equations are linear and the matrix of phenomenological coefficients is constant and symmetric, criterion (46) reduces to the extended theorem of minimum entropy production in a stationary state.
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6. Final remarks In this Letter, the thermodynamic formalism was considered, which is based on the postulate that the entropy density is a function of both ordinary thermodynamic variables and their material time derivatives. A feature of the formalism proposed is the fact that pressure, unlike temperature and chemical potentials, is introduced within the framework of the first law of thermodynamics and is not an extended quantity. The expressions for the entropy flux and the entropy production were obtained, the new fluxes and thermodynamic forces were introduced, and a particular case of a mechanical stationary state was considered. In this case, the entropy production is a function of the fluxes and the flux rates: σ = σ q + ρhv, Pv , v, ρ1 v1 , . . . , ρK vK ; ∂t q, ∂t Pv , ∂t J1 , . . . , ∂t JK , and there is an evolution criterion, which is a generalization of the classical Glansdorff–Prigogine criterion. The fact that the extended evolution criterion can be formulated suggests that the extended formalism considered in this work is a direct generalization of the formalism of classical irreversible thermodynamics.
Acknowledgement This work was supported by the Russian Fund of Base Researches (Grant No. 01-03-32662).
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