PHYSICAL REVIEW E 80, 026317 共2009兲
Average temperature and Maxwellian iteration in multitemperature mixtures of fluids Tommaso Ruggeri* Department of Mathematics and Research Center of Applied Mathematics (C.I.R.A.M.), University of Bologna, Via Saragozza 8, 40123 Bologna, Italy
Srboljub Simić† Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, 21000 Novi Sad, Serbia 共Received 14 March 2009; revised manuscript received 9 June 2009; published 28 August 2009兲 This paper treats the nonequilibrium processes in mixtures of fluids under the assumption that each constituent is characterized by its own velocity and temperature field. First we discuss the concept of the average temperature of mixture based upon considerations that the internal energy of the mixture is the same as in the case of a single-temperature mixture. As a consequence, it is shown that the entropy of the mixture reaches a local maximum in equilibrium. An illustrative example of homogeneous mixtures is given to support the theoretical considerations. Through the procedure of Maxwellian iteration a new constitutive equation for nonequilibrium temperatures of constituents is obtained in a classical limit, together with the Fick’s law for the diffusion flux. These results obtained for n-species are in perfect agreement with a recent classical approach of thermodynamics of irreversible processes in multitemperature case due to Gouin and Ruggeri and generalize our previous papers concerning the case of a binary mixture. DOI: 10.1103/PhysRevE.80.026317
PACS number共s兲: 47.51.⫹a, 51.30.⫹i
I. INTRODUCTION
Modeling and analysis of mixtures is challenging and stimulating problem. In the case of gaseous mixtures, it can be successfully studied using the methods of kinetic theory of gases, but also by means of continuum theory of fluids. In either case, appropriate macroscopic equations can be derived. However, many questions are still open and are the fields of active research. This study tends to provide some results concerned with the modeling problem of a mixture of gases. It is not only about derivation of a set of governing equations, but also about more subtle questions such as the definition of an average temperature, the entropy of the mixture and the comparison between the extended thermodynamic approach and the classic limit of thermodynamics of irreversible processes 共TIP兲. These results can be regarded as a continuation of previous research in this field and a generalization of the results recently obtained by Ruggeri and Simić 关1–3兴, and Gouin and Ruggeri 关4兴. The paper is organized as follows. First, the multitemperature model of homogeneous mixtures, established within the framework of extended thermodynamics, is reviewed. It is followed by the first principal result—the definition of an average temperature based upon the definition of the internal energy of the mixture 关Eq. 共6兲兴. As its natural consequence, it appears that the total entropy of the mixture reaches a local maximum in equilibrium 关Eq. 共14兲兴. These observations are supported by an example of spatially homogeneous mixtures which emphasizes the dissipative character of governing equations and the importance of multitemperature assumption through the asymptotic behavior of the solution.
In the second part of the paper we consider, for a mixture of n species, the so-called Maxwellian iteration 关5兴, i.e., the limit equations when the relaxation times in the momentum equations and energy equations of each species are negligible. We obtain in this case the classic constitutive equation for mechanical diffusion 关Fick law, Eq. 共30兲兴, together with a new constitutive equation that permit to determine also the nonequilibrium temperatures of the constituents 关Eq. 共34兲兴. These results are in perfect agreement with a recent classical TIP approach of multitemperature mixture given by Gouin and Ruggeri 关4兴. II. MULTITEMPERATURE MODEL OF HOMOGENEOUS MIXTURES
The theory of homogeneous mixtures was developed, within the framework of rational thermodynamics by Truesdell 关6兴, under the assumption that each constituent obeys the same balance laws as a single fluid. A huge amount of literature appeared after that in the context of continuum approach, see e.g., 关7–13兴. The system of governing equations read,
*
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共1a兲
共 ␣v ␣兲 + div共␣v␣ 丢 v␣ − t␣兲 = m␣; t
共1b兲
冊
再冉
1 ␣v␣2 + ␣␣ 2 + div t = e␣ ,
†
1539-3755/2009/80共2兲/026317共9兲
冉
␣ + div共␣v␣兲 = ␣; t
冊
1 ␣v␣2 + ␣␣ v␣ − t␣v␣ + q␣ 2
冎
共1c兲 ©2009 The American Physical Society
TOMMASO RUGGERI AND SRBOLJUB SIMIĆ
PHYSICAL REVIEW E 80, 026317 共2009兲
where ␣ = 1 , . . . , n. On the left-hand side, ␣ is the density, v␣ is the velocity, ␣ is the internal energy, q␣ is the heat flux and t␣ is the stress tensor. The stress tensor t␣ can be decomposed into a pressure part −p␣I and a viscous part ␣ as
b + div共bvb兲 = b; t
共3d兲
共 bv b兲 + div共bvb 丢 vb − tb兲 = mb; t
共3e兲
t ␣ = − p ␣I + ␣ . Source terms ␣, m␣, and e␣, which describe the interactions between constituents, satisfy the following relations: n
n
兺 ␣ = 0; ␣兺=1 m␣ = 0; ␣兺=1 e␣ = 0.
The appropriate definitions of field variables for the whole mixture, n
兺 ␣ ;
␣=1
n
v =
冉
n
␣=1
=
兺 ␣v ␣ ;
u␣ = v␣ − v;
␣=1
冊
再冉
1 bv2b + bb 2 + div t
I =
b = ˆ b;
共2a兲
1 兺 ␣u␣2 ; 2 ␣=1
= I +
共2b兲
n
兺 共t␣ − ␣u␣ 丢 u␣兲;
共2c兲
␣=1
n
q=
兺
␣=1
再 冉 q␣ + ␣
冎
冊
1 ␣ + u␣2 u␣ − t␣u␣ , 2
冉
共3a兲
共v兲 + div共v 丢 v − t兲 = 0; t
共3b兲
冊
再冉
冊
冎
v2 ˆ b · v + eˆb , +m 2
ˆ b and eˆb are independent of v. Their form can be where ˆ b, m determined through application of the entropy inequality. In particular, this was done in 关1兴 for the mixture of Eulerian fluids, n−1
ˆ b = − 兺 bc c=1
+ div共v兲 = 0; t
1 2 v + 2 + div t
eb = ˆ b
共2d兲
permit to obtain, by summation of Eqs. 共1兲, the conservation laws of mass, momentum, and energy for the whole mixture in the same form as for a single fluid 共, v, u␣, I, , t, and q are, respectively, the total mass density, the mixture velocity, the diffusion velocity of the component ␣, the intrinsic internal energy, the total internal energy, the stress tensor and the flux of internal energy兲. In such a way the balance laws for one constituent, say n, may be replaced by the conservation laws, leading to the following system of governing equations:
共b = 1, . . . ,n − 1兲,
n
兺 ␣ ␣ ; ␣=1 t=
共3f兲
with b = 1 , . . . , n − 1. The source terms are determined in accordance with the general principles of extended thermodynamics—Galilean invariance and entropy principle. The Galilean invariance 关14兴 dictates the velocity dependence in the source terms 关1兴:
ˆ b; mb = ˆ bv + m n
冎
冊
1 bv2b + bb vb − tbvb + qb = eb , 2
冢
冣
1 1 c − u2c n − u2n 2 2 − ; Tc Tn
共4a兲
冉
uc un − ; Tc Tn
冊
共4b兲
eˆb = − 兺 bc −
1 1 , + Tc Tn
冊
共4c兲
n−1
ˆ b = − 兺 bc m c=1
n−1 c=1
冉
where T␣, ␣ = 1 , . . . , n, are the temperatures of the constituents and
␣ = ␣ − T ␣S ␣ +
p␣ ␣
共5兲
are their chemical potentials. Coefficients bc, bc, and bc, 共b , c = 1 , . . . , n − 1兲 are the elements of phenomenological symmetric positive definite matrices. In the sequel, our analysis will be restricted to a model of nonreacting mixtures, for which b = 0.
III. AVERAGE TEMPERATURE
1 2 v + v − tv + q = 0; 2 共3c兲
The theory of mixture of fluids in which each constituent has its own temperature seems to be more realistic than the models in which all the constituents have a common tem-
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perature. It is indispensable in some particular physical situations, such as plasma physics 关15兴 and some recent questions of astrophysics 关16兴. Nevertheless, from the practical point of view, the main problem regards the measurement of the temperature of each constituent. Modeling the mixtures of gases, in the way it is intended to be done here, calls for appropriate definition of an average temperature. It is not only the common temperature of all constituents in equilibrium, T␣ = T = const, ␣ = 1 , . . . , n, but also the temperature of the mixture expected to be identified during any thermodynamic process. In principle, T␣ ⫽ T during the process 共T ⫽ const兲, and this discrepancy can be described by nonequilibrium temperature difference ⌰␣ = T␣ − T. Therefore, a question of definition of an average temperature has to be posed. The average temperature is not thoroughly discussed in the literature. In fact, in continuum approach the authors usually use the single-temperature 共ST兲 theory, whereas multitemperature 共MT兲 theory is mainly considered in the kinetic theory context. In the latter approach the temperature of a single constituent is evaluated by the knowledge of the distribution functions. Some previous definitions of the average temperature are proposed for a two-component system at kinetic level in terms of the algebraic average of the mean square velocities for the two distributions 关17兴, or by the requirement that the total pressure of the mixture has the same form as in the ST theory, see, e.g., 关18,19兴 and also 关1兴. In this paper we reconsider the definition of an average temperature proposed recently by Ruggeri and Simić 关3兴 in the case of binary mixtures, and then implemented by Gouin and Ruggeri 关4兴 in the context of classical thermodynamics of irreversible processes. The main idea is to exploit the definition of internal energy to introduce the 共average兲 temperature T as a state variable for the mixture, and to do it in such a way that the intrinsic internal energy I 关see Eq. 共2b兲兴 of the multitemperature mixture resembles the structure of intrinsic internal energy of a single-temperature one. Therefore, the following implicit definition of an average temperature is adopted: n
I共1, . . . , n,T兲 =
兺 ␣␣共␣,T兲 =
␣=1
n
兺 ␣␣共␣,T␣兲.
case of mixture of ideal gases. Furthermore, the following relation holds for non-equilibrium temperature difference in linear approximation close to equilibrium 关4兴, ⌰n = −
The new definition of average temperature has several advantages. The first one is a consequence of definition 共6兲 that the conservation law for the energy of mixture 关Eq. 共3c兲兴 becomes an evolution equation for the average temperature T. The second advantage is related to the entropy of the whole mixture defined as n
S =
,
␣=1
n
共7兲
␣cV共␣兲 兺 ␣=1
S =
␣=1
n
S
␣ ␣ 共␣,T兲⌰␣ 兺 T ␣ ␣=1
2S ␣ 1 兺 ␣ 共␣,T兲⌰␣2 + O共⌰␣3 兲. 2 ␣=1 T␣2
共11兲
However, only n − 1 nonequilibrium variables ⌰␣ are independent because they are related through the definition of average temperature Eq. 共6兲. Therefore, by expanding 共6兲 in the neighborhood of T␣ = T the following relation is obtained: n
␣ 2 ␣ 1 ␣ 共␣,T兲⌰␣ = − 兺 ␣ 2 共␣,T兲⌰␣2 + O共⌰␣3 兲. 兺 2 ␣=1 T␣ T␣ ␣=1
共8兲
is specific heat at constant volume of constituent ␣. We observe that Eq. 共7兲 gives the exact average temperature in the
共10兲
n
n
␣共␣,T兲 T␣
兺 ␣S␣共␣,T兲 +
+
where cV共␣兲 =
兺 ␣S␣共␣,T␣兲,
␣=1
where S␣ are the entropy densities of the constituents. It appears that Eq. 共10兲 reaches its maximum value in equilibrium, i.e., when ⌰b = 0 for all b = 1 , . . . , n − 1, and it is a direct consequence of the definition 共6兲. To prove this statement it is necessary to expand Eq. 共10兲 in the neighborhood of T␣ = T,
n
T=
共9兲
A. Entropy of the mixture
By expanding this relation in the neighborhood of the average temperature we have:
兺 ␣cV共␣兲T␣
兺 bcV共b兲⌰b .
ncV共n兲 b=1
Equation 共6兲 uniquely determines the average temperature T, and there exist a one-to-one mapping of state variables T␣ ↔ 共T , ⌰b兲, 共␣ = 1 , . . . , n ; b = 1 , . . . , n − 1兲. Therefore, as a consequence of definition 共6兲 and of the aforementioned change of variables, the conservation law for the energy of the whole mixture 关Eq. 共3c兲兴 becomes an evolution equation for the average temperature T and the energy Eqs. 共3f兲 for the n − 1 species become the evolution equations for the ⌰b. The ⌰b that can be regarded as nonequilibrium variables because when ⌰b = 0 for all b = 1 , . . . , n − 1, all the source terms eˆb vanish 关see Eq. 共4c兲兴. The ⌰b’s are the thermal counterpart of diffusion velocity fluxes ub, and for this reason we call them diffusion temperature fluxes.
共6兲
␣=1 n
n−1
1
共12兲 From the Gibbs’ equations for each constituent,
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p␣ d␣ , ␣2
T␣dS␣ = d␣ −
共13兲
Two remarks have to be given about dynamical pressure. First, Eq. 共16兲 represents exact expression for dynamical pressure in the case of mixture of ideal gases, for which
we have T␣
S␣ ␣ = . T␣ T␣
p␣ =
Starting from this equation, necessary relations in equilibrium can be derived, 1 ␣ S␣ 共␣,T兲 = 共␣,T兲; T␣ T T␣
and all higher order derivatives 共greater than 1兲 of p␣ with respect to T␣ vanish. Therefore, for ideal gases, taking into account
1 ␣ 1 2 ␣ 2S ␣ 共 ,T兲 = − 共 ,T兲 + 共␣,T兲. ␣ ␣ T2 T␣ T T␣2 T␣2
␣ = cV共␣兲T␣,
Putting them into Eq. 共11兲 and exploiting the constraint 共12兲, it can be shown that entropy of the mixture reads n
S =
1
n
␣S␣共␣,T兲 − 2 兺 ␣cV共␣兲⌰␣2 + O共⌰␣3 兲. 兺 2T ␣=1 ␣=1
共14兲
Since cV共␣兲 ⬎ 0 there exist a neighborhood of equilibrium state ⌰␣ = 0 in which the nonequilibrium part of total entropy is a negative definite quadratic form of ⌰␣, and thus entropy of the mixture reaches its local maximum in equilibrium. This result, which is a consequence of Eq. 共6兲, is particularly interesting because such an estimate cannot be achieved in such a direct way with other definitions of average temperature. B. Dynamical pressure
k ␣T ␣ , m␣
cV共␣兲 =
k , m␣共␥␣ − 1兲
共k and m␣ and ␥␣ are, respectively, the Boltzmann constant, atomic mass, and ratio of specific heats of constituent ␣兲, we have n−1
p = p0 + = p0 + 兺 bcV共b兲共␥b − ␥n兲⌰b . b=1
Second, when two ideal gases, say bˆ and n, have the same ratio of specific heats, ␥bˆ = ␥n, then dynamical pressure does not depend on diffusion temperature flux ⌰bˆ, since rbˆ = 0. When all the constituents have the same ratio of specific heats, the total pressure of the mixture is equal to equilibrium one, regardless of possible diffusion temperature fluxes.
The total pressure of the mixture can be regarded as a sum of partial pressures, C. Alternative form of the differential system
n
p=
兺 p␣共␣,T␣兲.
␣=1
Nevertheless, it can be viewed also as a sum of total pressure evaluated in equilibrium and an additional term, called dynamical pressure, which appears as a nonequilibrium pressure due to diffusion temperature flux ⌰b,
In the context of multitemperature mixtures of nonreacting gases, the equilibrium field variables are , v, T, and b 共or the concentration variables cb = b / 兲. However, it is useful to introduce diffusion velocity flux J␣ related to diffusion velocity
n
p = p0 + 共,T,⌰b兲; 共 = 1,2, . . . ,n;
p0 =
兺
␣=1
冉兺 冊 n
p␣共␣,T兲;
J ␣ = ␣u ␣, 共15兲
b = 1,2, . . . ,n − 1兲.
In the first approximation dynamical pressure can be expressed as
␣=1
J␣ = 0 ,
so that Jb and ⌰b could be regarded as nonequilibrium variables. It is convenient for the following to rewrite the system 共2e兲 using the material derivatives
n−1
= 兺 r b⌰ b ,
共16兲
d = + v · , dt t
b=1
where 关4兴 rb =
1
ncV共n兲
再
ncV共n兲
冎
pb pn 共b,T兲 − bcV共b兲 共n,T兲 . Tb Tn
db = + vb · dt t
for b = 1 , . . . , n − 1, and taking into account the definition of the average temperature given by Eq. 共6兲, 026317-4
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冦
d + div v = 0, dt dv − div t = 0, dt I dT I n−1 I = 2 div v + 兺 b=1 div Jb + t grad v − div q, T dt cb
d b b + b div vb = 0, dt d bv b ˆ b, − div tb = m b dt d b b − tb · vb + div qb = eˆb . b dt
The differential equations govern the evolution of , v, T , b, Jb, and ⌰b, respectively, provided we assign the constitutive equations p␣, ␣ and for dissipative fluids also the heat fluxes q␣ and the viscous stress tensors ␣, 共␣ = 1 , 2 , . . . , n兲. Moreover we have also to prescribe the matrices bc and bc , 共b = 1 , ¯ n − 1兲 that appear in Eq. 共4兲.
are valid for Eulerian gases, still remain valid also for viscous and heat conducting fluids. Therefore, in the linearized case we obtain n−1
b
0 dvb = − 兺 bc 共vc − vn兲; dt c=1 T0
IV. EXAMPLE OF SPATIALLY HOMOGENOUS MIXTURE
In this section a simple example will be provided in order to support previous theoretical considerations, and to stress the main features of multitemperature approach. To do it in the simplest possible way, a nonreacting mixture of gases will be considered in the special case of spatially homogeneous fields, i.e., the case in which field variables depend solely on time. The governing Eqs. 共17兲 can be written in the following form: d = 0; dt db = 0; dt
b
dv = 0; dt dvb ˆ b; =m dt
dT = 0; dt
b
db = eˆb . dt
共18a兲
= const.;
v = const.;
b = const.;
T = const.;
b = 1, . . . ,n,
共19a兲 共19b兲
and due to Galilean invariance we may choose v = v0 = 0 without loss of generality. It is also remarkable that the average temperature of the mixture remains constant during the process: T共t兲 = T0. In the sequel we shall regard only small perturbations of equilibrium state, v␣ = v0 = 0, T␣ = T0, ␣ = 1 , . . . , n, and analyze their behavior. Therefore, the rhs of Eq. 共18b兲 could be linearized in the neighborhood of equilibrium. Taking into account that solution depends only on time, and not on space variables, production terms given by Eqs. 共4b兲 and 共4c兲 that
共20a兲
n−1
bcV共b兲
0 dTb = − 兺 bc2 共Tc − Tn兲, dt c=1 T0
共20b兲
0 0 where bc and bc are entries of positive definite matrices evaluated in equilibrium. Note that vb − vn = ub − un and Tb − T n = ⌰ b − ⌰ n. Although the system 共20兲 is linear, the explicit solution would not give a suggestive answer about behavior of nonequilibrium variables. Instead, the results of qualitative nature can be obtained due to a fact that source terms have been constructed in accordance with the entropy principle. First note that we have n−1
共18b兲
where now d / dt = / t. From Eq. 共18兲 it is easy to conclude
冧
共17兲
dv
dv
b n = − n , b 兺 dt dt b=1
as a consequence of definition of mixture velocity and solutions 共19a兲 and v = 0. If we do a scalar multiplication of Eq. 共20a兲 with vb − vn and then sum up all the equations, the following result will be obtained n
冉 冊
n−1
0 bc 1 d 2 = − v 兺 ␣ ␣ b,c=1 兺 T0 共vb − vn兲 · 共vc − vn兲 ⱕ 0, dt ␣=1 2
the last inequality being a consequence of positive definiteness of the matrix 0. It can be concluded that the kinetic energy of the system, viewed as a measure of mechanical nonequilibrium in this example, is a decreasing function which asymptotically tends to zero. On the other hand, as a consequence of Eqs. 共9兲 and 共19a兲 we have
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1
v1/v1(0)
0.20
v=0
0
0.15
-1
q2/T0
0.10
v2/v1(0)
0.05
-2
0.00
-3 -4
5
0
10
15
20
t/tv
25
FIG. 1. Dimensionless velocities of the constituents versus dimensionless time.
兺
=−
q1/T0
-0.10 0
5
10
15
20
t/tv
25
FIG. 2. Dimensionless diffusion temperature fluxes of the constituents versus dimensionless time.
n−1
bcV共b兲⌰b
-0.05
0 11 =
ncV共n兲⌰n .
3m1m2 kT2⌫⬘ ; 共m1 + m2兲2 0 12
0 11 =
2m1m2 ⬘, T0⌫12 m1 + m2
b=1
If we multiply Eq. 共20b兲 with ⌰b − ⌰n and then sum up all the equations, the following inequality is obtained, n
冉
冊
where ⌫12 ⬘ represents volumetric collision frequency, and the following estimate can be obtained: 2 共m1 + m2兲 T 2 ⬎ ⱖ 1, = v 3 1m2共␥2 − 1兲 + 2m1共␥1 − 1兲 3共␥max − 1兲
n−1
0 bc 1 d 共␣兲 2 c ⌰ = − 兺 ␣ V ␣ b,c=1 兺 T2 共⌰b − ⌰n兲共⌰c − ⌰n兲 ⱕ 0. dt ␣=1 2 0
The first term—a positive definite quadratic form of ⌰␣—can be regarded as a measure of thermal nonequilibrium. Its time derivative is a decreasing function due to definiteness of the matrix 0. Therefore, diffusion temperature fluxes asymptotically tend to zero. In the particular case of a binary mixture the explicit solution of Eqs. 共20兲 can be obtained and it reads v1共t兲 = v1共0兲e−t/v ;
T1共t兲 = T0 + 共T1共0兲 − T0兲e−t/T ,
共␥max = max兵␥1, ␥2其 ⱕ 5/3兲.
With the aim to compare the thermal diffusion, associated with the multitemperature assumption, with the mechanical diffusion, in Figs. 1–3 we present the graphs of normalized velocities, diffusion temperature fluxes, and dynamical pressure are shown, respectively, in terms of scaled time variable t / v, for the following choice of parameters of the mixture, m2 = 0.4; m1
where v and T represent relaxation times that for ideal gas assume the expression
v =
T =
1 2T 0 ; 0 11 k12T20
0 11 关1m2共␥2 − 1兲 + 2m1共␥1 − 1兲兴
5 ␥1 = ; 3 .
Starting from these solutions, other field variables can be obtained by means of defining equations
共21兲
1 = 0.8; 7 ␥2 = ; 5
2 = 0.2;
T = 3.18 v
共22兲
together with initial data: T1共0兲 = 300 K; T2共0兲 = 400 K. In such a way we have T0 = 327.3 K. It can be observed that, due to the inequality 共21兲, the 0.035
1v1 + 2v2 = v = 0;
0.030
1cV共1兲T1 + 2cV共2兲T2 = 共1cV共1兲 + 2cV共2兲兲T = 共1cV共1兲 + 2cV共2兲兲T0 .
0.025
It is obvious that, due to dissipative character of the system, all the nonequilibrium variables exponentially decay and converge to their equilibrium values including also the dynamical pressure 关Eq. 共16兲兴 which in this case reads
0.020
= 1cV共1兲共␥1 − ␥2兲⌰1 .
0.005
In order to compare the values of v and T for ideal gases, and also to compute the actual values of variables in numerical example, the relations from kinetic theory has to be recalled 关15兴,
pq/p0
0.015 0.010
0.000 0
5
10
15
20
t/tv 25
FIG. 3. Dimensionless dynamical pressure versus dimensionless time.
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AVERAGE TEMPERATURE AND MAXWELLIAN ITERATION…
mechanical diffusion vanishes more rapidly than the thermal one. This is in sharp contrast with widely adopted approach which ignores the multitemperature assumption. V. MAXWELLIAN ITERATION
The name and the procedure of the so-called Maxwellian iteration was first introduced in the paper by Ikenberry and Truesdell 关5兴 共see also the book of Truesdell and Muncaster 关20兴兲 and was successfully applied in many circumstances in continuum mechanics as a sort of macroscopic ChapmanEnskog procedure, i.e., an expansion in power series in appropriate relaxation times. In this manner it is possible to recover TIP approach as a limit case of extended theories. For example, the Navier-Stokes-Fourier constitutive relations emerge as a first-order Maxwellian iteration of the Grad’s 13 moments theory 关21兴 in the context of extended thermodynamics 关9兴. Another example is the Fick’s law obtained as a limiting case of momentum equations in ST theory 关8,9兴. Before proceeding to multitemperature mixture, the idea of Maxwellian iteration will be illustrated using the simplest system of 2 ⫻ 2 balance laws in one space dimension,
tu + x f共u, v兲 = 0;
共23a兲
1 tv + xg共u, v兲 = 关v − h共u兲兴,
共23b兲
TIP approach. We will see that, in the present case of mixture with multitemperature also, the Maxwellian iteration at the first order give the same result obtained earlier within the TIP framework of multitemperature mixture proposed by Gouin and Ruggeri 关4兴. As we are interested mainly in mechanical and thermal diffusion phenomena in mixtures, for the sake of simplicity we limit our calculation to the case of Euler gases for which Eqs. 共4b兲 and 共4c兲 hold. Nevertheless, our results remain confirmed also in the case of Navier-Stokes-Fourier fluids, as proved directly by extended TIP calculation given in 关4兴. Application of Maxwellian iterative procedure requires to put in the lhs of the system Eq. 共17兲5,6 the zeroth iterate, i.e., the equilibrium state and in the rhs the first iterate. Taking into account that in zeroth iteration v␣共0兲 = v and consequently 共0兲 共0兲 共0兲 共0兲 共0兲 d共0兲 b / dt = d / dt, Jb = ub = 0, and moreover T␣ = T, q = qb 共0兲 共0兲 共0兲 共0兲 = 0, t = −p I = −p0I, tb = −pb I, we obtain
冉 冊 再冉 冊 冉 冊 冉 冊 冉 冊 b
b
= eˆ共1兲 b ,
F共u兲 = f关u,h共u兲兴.
冉 冊 冉 冊 冉 冉 冊
共0兲
冎
+ p共0兲 b div v 共26兲
共0兲
dv dt
I dT T dt
= − grad p0
共0兲
db dt
= 2
共0兲
冊
I − p0 div v
= − b div v,
共27兲
and therefore inserting Eq. 共27兲 in Eq. 共26兲 we obtain −
b ˆ 共1兲 grad p0 + grad p共0兲 b =m b ,
⍀b div v = eˆ共1兲 b ,
共b = 1, . . . ,n − 1兲,
where
冦
⍀b = p共0兲 b + b − b
共25兲
共1兲
Putting v into the lhs of Eq. 共23b兲 one obtains in the same manner the second iterate v共2兲 from the rhs, etc. In such a way nonequilibrium variable is expanded in powers of relaxation time . The iterative procedure being stopped at the first order, the limit of the evolution equations implies that the nonequilibrium variables v共1兲 − h共u兲 are proportional to the gradients of equilibrium variables xu. In the physical cases where this procedure was applied it was verified that expression 共25兲 is equivalent to the constitutive equations of the corresponding
dT dt
共b = 1, ¯ n − 1兲.
v共1兲 = h共u兲 + 兵− h⬘共u兲兵f u关u,h共u兲兴 + h⬘共u兲f v关u,h共u兲兴
+ gu关u,h共u兲兴 + h⬘共u兲gv关u,h共u兲兴其其xu
共0兲
b Tb
+
1 tv共0兲 + xg关u, v共0兲兴 = 关v共1兲 − h共u兲兴. By a simple calculation, using Eq. 共23a兲, one obtains the following expression for v共1兲:
ˆ 共1兲 + grad p共0兲 b =m b ,
共0兲
db dt
共24兲
In this iterative procedure equilibrium value of v is treated as zeroth iterate, v共0兲 = vE = h共u兲, and should be put into lhs of Eq. 共23b兲. First iterate v共1兲 is then calculated form the rhs, i.e.,
共0兲
On the other hand form the zero order of Eq. 共17兲2,3,4 we have
where denotes small parameter—relaxation time. In Eq. 共23兲 u can be regarded as an equilibrium variable, whereas v is a nonequilibrium one, Eq. 共23b兲 being its governing equation. Relation vE = h共u兲 determines equilibrium manifold on which source term in Eq. 共23b兲 vanishes and the system is reduced to a conservation law,
tu + xF共u兲 = 0;
共0兲
b b
dv dt
冉 冊 冉 b b
共0兲
+ 2
I − p0
冊
共28兲
冉 冊
冧
b 共0兲 Tb . I T 共29兲
These relations have to be transformed further in order to recognize terms which represent thermodynamic forces in the sense of TIP. A. Diffusion velocity flux
Taking into account that p共0兲 = p0 = 兺␣n =1 p␣共␣ , T兲, Eq. 共28兲1 can be rewritten as
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PHYSICAL REVIEW E 80, 026317 共2009兲
TOMMASO RUGGERI AND SRBOLJUB SIMIĆ n−1
兺 a=1
冉
冊 冉
冉 冊
冊
b b
1 1 ˆ 共1兲 ␦abb − a b ⫻ grad p共0兲 grad p共0兲 =m a − n b a n
From the definition of chemical potential 共5兲 one obtains the following relations:
␣ 1 p␣ = ; ␣ ␣ ␣
冉 冊 再 冉 冊冎 冉 冊
= T grad −
T
T
p共0兲 b
+ T2 T
grad
b
T
n−1
1 兺 共0兲 T d=1 bd
冊 冉 冊
共0兲
冉
n−1
= 兺 Lab grad b=1
共0兲
共31兲
.
共0兲 b T
冉兺 冊
共0兲
S T
=T
共0兲
I
1 T
␣=1,2, ... ,n; b
and taking the map ␣ ↔ 共 , cb兲, = 1 , 2 , . . . , n − 1, we obtain
I T
=
共0兲
,
S p共0兲 2 +T
共0兲
共32兲
.
Inserting Eqs. 共31兲 and 共32兲 in Eq. 共29兲, we have ⍀b =
冉 冊冉 冊 冉 冊冉 冊冎 冉 冊再 Sb bT Tb S 共0兲 T
共0兲
S
共0兲
− b
S T
共0兲
Sb b
共0兲
.
n−1
FadJ共1兲 , a
a=1
冊
冉冊
b − n 1 + La grad , T T
共30兲
where phenomenological coefficients, evaluated in equilibrium, are
The first iterate of the source term eˆb is calculated through expansion of Eq. 共4c兲 in the neighborhood of T, and using the relation between diffusion temperature fluxes, n−1
eˆ共1兲 b =−
n−1
1 兺 共0兲 兺 共⌽cd⌰共1兲 d 兲. T2 c=1 bc d=1
共33兲
where ⌽cd = ␦cd + 共dcV共d兲兲 / 共ncV共n兲兲. By combining Eq. 共28兲2 with Eq. 共33兲, an expression for the first iterate of the diffusion temperature flux is obtained through a simple algebraic manipulation, ⌰共1兲 a = − ka div v.
共0兲 ; Lab = − T2关F−1−1F−1兴ab
共34兲
where
n−1
La = 兺 共− T 兲关F 3
−1
共0兲 −1F−1兴ab
n−1
再 冉 冊 冉 冊冎
共0兲 ⍀b . ka = T 兺 关⌽−1−1兴ab
b=1
⫻ T
Sb b
冉 冊 冉 冊 冉 冊 冉 冊
ˆ 共1兲 where Fad = ␦ad / a − 1 / n. Since m b is a linear combination 共1兲 of Ja , after some algebraic manipulations the explicit equation for the first iterate of diffusion flux reads J共1兲 a
冉 冊
n−1
To reach the goal of this section, the rhs of Eq. 共28兲1 has to be transformed in accordance with Eq. 共4b兲. Taking into account that u␣共1兲 = J␣共1兲 / ␣, and using the relation between difn−1 Ja, the following relation is obtained, fusion fluxes Jn = −兺a=1 ˆ 共1兲 m b =−
+T
2b
p0 d − 兺 共b − n兲共0兲dcb 2 b=1
TdS = dI −
1 pb pb 1 grad p共0兲 grad b + grad Tb b = b b b Tb
共0兲 b
=
From the Gibbs equation of the whole mixture in equilibrium,
␣ 1 p␣ = − S␣ . T␣ ␣ T␣
Using the first of them, the following transformation could be done:
冉
p共0兲 b
共0兲
b − n pb pn − − T T T b n
共0兲
b=1
,
and 共F−1兲ad = ␦add − 共ad兲 / . Thus, Eq. 共30兲 recovers the Fick’s law well-known from classical thermodynamics of irreversible processes and is in complete agreement with the result obtained trough the same procedure for singletemperature mixtures in extended thermodynamics 关9兴.
Equation 共34兲, obtained by means of Maxwellian iteration, gives the temperature of each species as a constitutive equation in the same manner as the Fick law give the velocities of each species. The present result which is in complete accordance with the classical TIP approach to multitemperature mixtures, proposed recently by Gouin and Ruggeri 关4兴. For a mixture of ideal gases
B. Diffusion temperature flux in the multitemperature model
n
Concerning the Eq. 共28兲2, we can put first ⍀b in a more convenient form taking into account that from the Gibbs equations for each species 关Eq. 共13兲兴 we can deduce the wellknown thermodynamical relations,
冉 冊 冉 冊 b Tb
共0兲
=T
Sb Tb
⍀b = bTcV共b兲
␣cV共␣兲共␥b − ␥␣兲 兺 ␣=1 n
.
兺 ␣cV共␣兲
␣=1
共0兲
When all the constituents have the same ratio of specific 026317-8
PHYSICAL REVIEW E 80, 026317 共2009兲
AVERAGE TEMPERATURE AND MAXWELLIAN ITERATION…
heats we have ⍀b = 0, and consequently ⌰共1兲 b = 0, a = 1 , . . . , n − 1. In this case the diffusion temperature flux cannot be observed in the first approximation. In the case of a binary mixture, these results reduce to the ones presented in 关2兴. VI. CONCLUSIONS
the case of ideal gas when all the constituents have the same ratio of specific heats 共the same degree of freedom兲, the diffusion temperature flux cannot be observed in the first approximation. The same occurs if the mixture is globally incompressible 共div v = 0兲 or in the static case 共v = 0兲. We observe that in the static case where the Maxwellian iteration does not allow to evaluate the temperature of constituents it is possible to use the recent strategy proposed by Ruggeri and Lou, in which by considering the nonequilibrium one-dimensional steady heat conduction between two walls maintained at two different temperatures, it was proved that the measure of the average temperature in 2共n − 1兲 points enable us to evaluate the temperature of constituents in all points 关22兴. The results of this study are supposed to be the starting point for future analysis of mixtures. The shock structure problem in multicomponent mixtures, as well as problems of combustion theory, like flame propagation and detonation waves, could be the possible fields of application of the developed theory.
In this paper we have reconsidered the definition of an average temperature in the context of multitemperature approach to the theory of mixtures of fluids. It was based upon the assumption that internal energy of the mixture should retain the same form as in a single-temperature approach. The supremacy of this definition is supported by a simple derivation of entropy maximization result in equilibrium as its consequence and the result that the average temperature remains constant for spatially homogenous mixture. Furthermore, by means of Maxwellian first iterative procedure we have derived constitutive equations for nonequilibrium variables, mechanical diffusion flux Ja, and diffusion temperature flux ⌰a, in the neighborhood of equilibrium. It was shown that the first iterate of diffusion flux J共1兲 a coincides with the classical generalized Fick’s laws which can be obtained in TIP. However, diffusion temperature flux ⌰共1兲 a is found to be proportional to div v—a new result which is in accordance with recent observations within classical TIP framework of multitemperature mixture 关4兴. The result is particularly important: so, at least in first approximation, the temperature of each species can be evaluated as a constitutive equation in the same manner as the Fick law gives the velocity of each species. Unfortunately, in
The authors thank an anonymous referee for several suggestions allowing to improve the final version of the paper. S.S. acknowledges support from GNFM and CIRAM– University of Bologna. It was supported in part 共T.R.兲 by the GNFM-INdAM and in part 共S.S.兲 by the Ministry of Science of the Republic of Serbia within the project “Contemporary problems of mechanics of deformable bodies” 共Project No. 144019兲.
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ACKNOWLEDGMENTS
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