- Email: [email protected]
Nonlinear extended thermodynamics of a dilute nonviscous gas
MATHEMATICAL COMPUTER MODELLING PERGAMON
Mathematical
and Computer
Modelling
36 (2002) 951-962 www.elsevier.comJlocateJmcm
Nonlinear Extended Thermodynamics of a Dilute Nonviscous Gas M. S. MONGIOV~ Dipartimento di Matematica ed Applicazioni Facolta di Ingegneria, Viale delle Scienze University I-90128
of Palermo
Palermo,
It,aly
MongioviQunipa.it (Received
July 2002:
accepted
August
2002)
Abstract-This paper deals with further developments of a nonlinear theory for a nonviscous gas in the presence of heat flux, which has been proposed in previous papers, using extended thermodynamics. The fundamental fields used are the density, the velocity, the internal energy density, and the heat flux. Using the Liu procedure, the constitutive theory is built up without approximations and the consistence of the model is showed: it is shown that the model is determined by the choice of three scalar functions which must satisfy a system of partial differential equations, which always has solutions. Different changes of field variables are carned out, using different Legendre transformations, passing from variables which are convenient from a mathematical pomt of view to intermediate nonequilibrium variables and finally to equilibrium variables. In particular, two possible extensions of the temperature far from equilibrium are considered: the “nonequilibrium temperature” of extended irreversible thermodynamics [defined as the reciprocal of the derivative with respect to the energy of a generalized nonequilibrium entropy) and the “thermodynamic temperature” by Miiller (defined as the mverse of the coefficient linking the nonequilibrium entropy flux to the heat flux). The link between these two quantities is established, general expressions for these quantities, as functions of the density, of the absolute temperature and of the heat flux are obtained. @ 2002 Elsevier Science Ltd. All rights reserved.
Keywords-Extended
thermodynamics,
Kinetic
theory, Entropy
closure.
Hydrodynamic
models
1. INTRODUCTION The derivation of hydrodynamic equations from the microscopic kinetic, Boltzmann-type. equations is a fundamental research field of applied mathematics. The macroscopic description can be obtained by different mathematical approaches. For instance, the asymptotic theory for small Knudsen numbers related theory of gases means, as known [l], analyzing the macroscopic description
to models of the kinetic delivered by the kinetic
equations when the distance between particles tends to zero. Then one obtains a macroscopic description delivered by the microscopic one as an alternative to the purely phenomenological derivation. The method applies to various kinetic equations. Different scaling generate d&rent Partially supported by M.I.U.R. of Italy, under Grant and Stability in Models of Continuous Media”. 0895.7177/02/S - see front matter PII: SO895-7177(02)00249-Z
@ 2002 Elsevier
“Nonlinear
Mathematical
Problems
Science Ltd. All rights reserved.
in Wave Propagation
Typeset
by dJvIs-T)$C
952
M. S. M~NGIOV~
macroscopic
equations
as documented
in the formal
expansions
proposed
121. The literature
in
the field is documented in (11 starting to relatively more recent contributions On the other
hand,
is well understood Indeed, shown
at the formal level, by now, its full mathematical
the justification to be difficult
approaches
of the formal considering
to overcome
In prin
although
approximation
that
many
these difficulties
the same method mathematical
have emerged
can be applied
model
to the
a formal procedure in (111. The interest
in [lo], overcoming above
methods
Boltzmann
questions
remain
theory
Boltzmann
equation unsolved.
equations
of gases describing
has Two
by extended
[8,9], which
the evolution
of velocities.
some of the above mathematical is offered
is still missing.
recently.
to discretized
of the kinetic
justification
for the classical
basic regularity
allowed to move in the space with a finite number
as documented
An alternative
from the pioneer papers on the topic, e.g., Caflisch (31, up [4-71, which exploits suitable moment closures. the derivation of fluid dynamical equations from kinetic theory
of a
In this case. it is difficulties.
thermodynamics
which
offers
to obtain higher-order macroscopic models as documented, among others, in high-order models of fluids is documented in recent papers by Hutter et
ul. [12]. In detail, in this paper, a nonlinear extended theory for a nonviscous gas is developed without approximations. The fundamental fields used are the density p, the velocity v,, the internal energy density E, and the heat flux 4%. This first section provides an introduction The contents are organized into eight sections. Section 2 deals with a brief recall of extended to the aims and organization of the paper. thermodynamics of gases with 13 fields. as formulated by Liu and Muller [13]. In Section 3, the mathematical implications of the hypothesis of nonviscosity are studied.
Using
the Liu procedure [14], it is shown that the model is defined by the knowledge of some scalar satisfy a set of finite and constitutive functions of p, E, and q2, which are not independent, differential relations. In Section 4, the constitutive
theory
is analyzed
it is shown that the model is determined Lagrange multipliers, which must satisfy
in detail:
through
a Legendre
by the choice of three scalar a system of partial differential
transformation,
functions of the intrinsic equations, which always
has solutions. In Section 5, two possible
extensions of the temperature far from equilibrium are considered: 0 of extended irreversible thermodynamics by Jou, Casasthe “nonequilibrium temperature” Vazquez and Lebon [15-171 (defined as the reciprocal of the derivative with respect to the energy of a generalized nonequilibrium entropy) and the “thermodynamic temperature” Tth by Miiller et al. [18-211 (defined as the inverse of the coefficient linking the nonequilibrium entropy flux to the heat flux). The link between these two quantities is established and general expressions for these quantities are obtained. In Section 6, different changes of field variables are carried out, using different Legendre transformations, passing from variables which are convenient from a mathematical point of view to intermediate nonequilibrium variables and finally to equilibrium variables. In particular, for the quantity 0, defined as the reciprocal of the derivative with respect t,o the energy of a generalized nonequilibrium entropy the general expression
e = p5/2A where T is the thermostatic temperature tity Tth is linked to B by the relation
and A an arbitrary
1 -=_ Tth
1 e + ti&qz,
(1.1) function,
is obtained.
The quan-
(1.2)
where X, are the Lagrange multipliers associated to the heat flux and Q a coefficient linked to the stress tensor and to the flux of the heat flux. Further, the complete expressions furnished by the
Nonlinear
theory
for the energy
Extended
Thermodynamics
E and for the Lagrange
density
953
multipliers
X, of the heat flux, as functions
of p, 8, and q2, are determined. The
analysis
of Sections
7 and 8 aims to shed some light on the physical
the quantities 0 and Tth:’ it is shown that equilibrium the “nonequilibrium temperature” Miiller thermodynamic
temperature
in the extensions of the Gibbs appears, while in the extension
interpretation
of
equation far from of Fourier law the
appears.
2. A REMINDER OF EXTENDED THERMODYNAMICS OF GASES The derivation
of fluid dynamic
equations
from kinetic theory
has been in the past years the ob-
ject of many investigations. In this section, we present a brief recall of extended of gases with 13 fields, as formulated, by Liu and Miiller [13]. In the kinetic
theory
of gases, a dilute
gas is described
by the Boltzmann
thermodynamics
equation
where f = f(t,x, u) is the distribution function, t the time, x = (z,) the space, u = (ul) the velocity, f = (fZ) the external force, and S the collision term. Grad’s method is developed [22] considering
the moments
of various
order of the phase density
Pa]%*..2,,
and writing
the transport
equations
=
I
mu,,
G,
(2.2)
u,,, f du,
for these moments
g+gJ=p,
(2.3)
3
whereM = (PA) = (P,P~,P~~~~,. ,P~~~~...~,,), F, = (PAj), and P = (PA). This paper
deals with the transport
equations
for the first 13 moments,
therefore,
the notation
M = (P,pi, PZJ,~zjj) will be used. As it is known,
the above-mentioned
13 moments
correspond,
(2.4)
respectively,
to the mass density,
the momentum density, the momentum flux density, and the energy flux density. In this paper, the gas is supposed referred to an inertial frame, in the absence of external forces. In [13], Liu and Miiller first imposed general physical principles to this system, such as that of material objectivity and the entropy principle. Now, we report briefly this methodology, as reported in the book of Miiller and Ruggeri [23], which furnish an elegant and powerful method for obtain high-order models of gases. Restrictions on the constitutive relations for the nonfundamental the validity of the entropy principle: there are an entropy density density h, = hj(p~), such that the entropy thermodvnamic orocess
production
fields are obtained imposing h = h(pA) and an entropy flux
g is supposed
to be nonnegative
for every
(2.5) It must be taken into account that with the Boltzmann entropy of the 13 moments development, proposed Note that equations (2.3) can be the entropy principle a procedure,
h is a phenomenological macroscopic function with coincides kinetic theory of gases in the truncated form limited to the by Grad [21]. considered as constraints for the fields PA. In order to satisfy known as Liu method of Lagrange multipliers [14:23], will
M. S. MONGIOV~
954
he used; this method following
inequality
states
that,
introducing
must be satisfied
a Lagrange
for arbitrary
multiplier
AA for each equation,
values of the field M:
dh
u=x+ In the kinetic
theory
the
(2.6)
of gases, the central
PZiZZ.
.1,,
=
moments
&‘I = (CA), defined
s
by
c,,r,f dc,
mc,,c,,
(2.7)
where c, = v1 - w, (with v, = pz/p) is the peculiar velocity, are often chosen as independent fields instead of complete moments. An exponential matrix of the form X = eArvr, with A, constant matrices,
can be introduced
such that
the vector M can be written
as [23.24]
M=Xti.
(2.8)
where M is composed with objective tensors. In particular, the central part bz of the momentum pz is identically zero. Regards to the fluxes, in [24] it has been shown that, putting h, = hv, + $3 and Fj = MU, + G,, q53 is an intrinsic vector and G, can be expressed as Further P = XP and i = AX! where P and A are G, = XG:,, where G, is objective. intrinsic
[24].
In terms of &I. equations
(2.3) can be written
(23,241
(2.9) where $ denotes
the material
derivative.
This system can be easily put in normal form with respect to time derivatives solving the vector component of (2.9) with respect to % and substituting the result in the other equations. If momentum
is supposed
conserved,
the right side of the resulting
system
will be still the vector P.
In extended thermodynamics, system (2.9) instead of balance equations (2.3) is frequently employed. If one considers only the first 13 fields, the density p = 6, the velocity V, = pz/p, and the intrinsic parts & and bzjj of the other fields, are chosen as independent variables. The fields G:3 and @ are treated as dependent variables and are expressed in terms of the fundamental variables by means of constitutive equations, satisfying objectivity and entropy principles. In terms of the intrinsic dh -+i,s+$!-,i. dt .l It is plain that system (2.9).
quantities
inequality
. %+a%-+x3 inequality
(2.6) becomes
[23,24]
ae:, dz 3 + Ar
(2.10) can also be obtained
3. NONVISCOUS
directly,
by applying
the method
of Liu to
GASES
Sometimes, in the description of gases in nonequilibrium, the approximation of nonviscous fluid may be heuristically applied [25,26]. In a previous paper [27], it has shown that in order to formulate nonlinear extended theories for nonviscous gases and fluids, we can use as fundamental fields only the density p, the velocity uzr the internal energy density E = &,/2, and the heat flux (Iz = k/2.
955
Nonlinear Extended Thermodynamics
The evolutions with respect equation
equations
for these fields can be written
to the material
derivatives
of the stress deviator
of the central
Pczj,; one obtains
solving
the first 13 balance
moments,
and neglecting
equations
the evolution
[25,26]
(3.1)
The field I
is not considered
as an independent
variable,
but it must be expressed
through
a constitutive relation as function of the other fields. This implies that the generalized entropy density h introduced in (2.5) is independent of the deviatoric part /i(,,) of the stress tensor. therefore, being h an objective scalar quantity, we can write (3.2)
h = h (P, E, 9’) From the mathematical Miiller
[13], imposing
that
point of view, this theory
is obtained
the constitutive
must satisfy
relations A (2J)
=
by the 13 field theory the further
condition
of Liu and [27] (3.3)
‘1
where A(,,) is the intrinsic part of the Lagrange multipliers of the deviator of t,hc second monn~nt,. i.e., the derivative with respect to the stress deviator of the nonequilibrium entropy h. Constitutive eqUatiOnS for the flUXeS b(+), FzJkr and PtJjk are necessary t0 ClOSe system (3.1). Restrictions on these constitutive relations may be obtained imposing the validity of the entropy principle, applying the Liu procedure to system (3.1). This is equivalent to put constraint (3.3) in inequality (2.10). The following set of relations is obtained: (3.4) (3.5) (3.6) (3.7)
Recall that in the kinetic theory of dilute gases, the trace of the stress p is linked to the internal energy density E by the relation p = (2/3)E. Being the constitutive relations objective functions. they can be expressed in the form @k
=
4 (P,E, q2)qk,
(3.9)
fiti = ;E6,, +a (P,E, 4’) q(2q.4’
(3.10)
;4(,6,) +x (~7E, q2)ykwj) ihijk=P (P)El q2)&k + v (p,E, q2) i&k
=
i
q,+q,),
+‘:;’
= b (P, E, q2) 9%.
(3.11) (3.12) (3.13)
M. S. MONGIOV~
956
Substituting
now (3.9)-(3.13)
in (3.5)-(3.8),
A, = X,qi, we get
and putting
h - pii - ;E&
- 2Xqq2 = 0,
(3.14)
9 - 3xq2_ aIL+X,---- , 5
(3.15)
c#l- AE - x,q2 (v - UE) = 0,
(3.16)
dh = Adp + AEdE + Xqqidq,,
(3.17)
o
dc$ = A, {d+(dE+q2da+;dq2)+;q2dv+~dq2}.
(3.18)
X,bq’ 1 0,
(3.19)
CT =
where the following
position
has been made: (3.20)
Note that all the relations presented in this section are exact, because no approximation has been used for their determination. As can be seen, the constitutive theory is determined by the knowledge of the then scalar functions h, 4, a, x, ,6, v, b, A, AE, and X,, which are constrained by relations (3.14)-(3.19). In the following section, the relations will be analyzed in detail, in order to show the consistency of the proposed model and to establish which constitutive quantities can be chosen arbitrarily (and therefore, 4. Choose
be determined
ANALYSIS
as new variables
introduce defined
must
the scalar
experimentally)
OF
THE
the nonconvective
potential
and which are instead
CONSTITUTIVE parts
of the Lagrange
h’ = h’(& AE, i2) and the vector
fixed by the theory.
THEORY multipliers potential
(A, A,,
and i,),
Qi = @k(A, AE, i,),
by
h’ = -h + Ap +
AEE + Xtq,,
(4.1)
m;=-m,-n,q,+h*(i-?“q2) and rewrite
(3.5) and (3.6) in the alternative dh’ = pdit + Ed& da;
First,
= qkd&
(4.2)
-Jlc&(p&-E), form
+ ;qidiii,
+ (/%,
+
(4.3) d&i,.
u’?(,qk)
>
(4.4)
from (3.14) and (4.1), one can write 3 (4.5)
substituting
in (4.3), one obtains &’ = - -&
(h’ + itqt)
d&
-&
i,qi)
$
+ ;q,dh,.
(4.6)
Therefore. dh’ 7
ail.5
=
(h’+
,
= p,
s
= 4%
(4.7)
Nonlinear Extended Thermodynamics
it can be concluded
whose solution
that
h’ must satisfy
the following
PDE:
is
$,A , [ I
h’=X’H
where ‘H an arbitrary Writing
957
function,
and X the absolute
value of X,.
now @L = c++‘X,and qk = qxXk (q,j = l/X,),
&+’ 7
=
-
RA
(4.9)
from (4.4), we can write
2.
(pE - E)
14.10)
+ (p& - E) -
qx = -$
i3E
(4.11)
aii, ’
E
(4.12)
3~-2Et‘-30’=2X2~+2X2(pE-E)~,
(u - aE) q; = 2%
(4.13)
+ 2 (P& - E) g.
Now. regarding h’ as a known function of the Lagrange multipliers, satisfying from (4.3) we obtain the following expressions of the quantities p, E, and qx: dh’
equations
ah'
(4.14)
9x=23$
P=x’
(4.8),
From (4.12) and (4.13) one obtains &= from (4.10))(4.13)
(3-&?)
one can write u - u& + -q;dX’
dq’ = (E - p&)d& + qxdh, and d& =
-!.-
@’ - q/idAE-
E - p&
Integrability
(4.15)
+(3E-pE):
conditions
for the first equation
u - aE
-q;dX2 2
yields
(4.16)
I
(4.17)
(4.18) remembering
(4.14), one can write a2h'
- d2h’ = 0,
a2h'
aA,aA,
Z&
aliax2
(4.19)
which is a PDE in the unknown E = f(A, AE, X2). Substituting (4.10) and (4.11) in (4.18), one obtains “-E-$--g=
(g-E!$)
(E-qA)+s(&p&).
(4.20)
8hE
Now, compatibility
conditions
for (4.17) yields aE -Eg ah
Subtracting
(4.21) to (4.20), one obtains
-/$)
also the following
(E-Q)
+ g(E-pL).
(4.21)
PDE for 4’: (4.22)
Finally, a is computed by (3.20). x by (3.15), v and 0 by (4.12) and (4.13). Equations (4.1) and (4.15) reduce to identities. The main result of the analysis of this section can be summarized in the following proposition.
PROPOSITION. The constitutive theory is determined by knowledge of the three functions h’, E. and 4’ as function of the variables Aj AE, and i2, which must satisfy the following system of partial differential equations:
h’ +2x2$
+ ;I&
a2h’
ah’ aA,
ash’ ^ ahaX
a&ah,
=
o
(4.23) )
=o.
(4.24)
a& a$f a& a@'+zdh'!$=o. --_-_ aA, aA ahi, a2aA Note that system integrated which,
(4.23)-(4.25)
and its solution
always has solutions.
is given by (4.9); equation
one fixed h’, can be integrated
(4.23) can be immediately
(4.24) is a quasi-linear
first-order
of characteristics;
last equation
by the method
once h’ and E are known, is a linear first-order The compatibility of the constitutive theory
5. GENERALIZED
Indeed equation
(4.25)
equation (4.25),
equation. is such established.
TEMPERATURES
FAR
FROM
EQUILIBRIUM
The problem of t,he definition of the temperature far from equilibrium has received attention by several authors. An extensive discussion can be found in the book of Jou, Casas-Vazquez and Lebon [17]! which is a fundamental tool for people working in nonequilibrium thermodynamics. In classical thermodynamics of solids or fluids, the equilibrium temperature is equal to the inverse of the partial derivative of the entropy with respect to the energy, at constant, density and it is identical to the inverse of the coefficient linking the entropy flux and the heat flux. Some authors [15,16] h ave proposed the use of a generalized “nonequilibrium temperature“ B. defined as the reciprocal of the derivative with respect to the energy of a generalized nonequilibrium entropy h depending not only on the classical variables, but also on the dissipative fluxes, as, for instance
the heat flux q, %
1 _-ah H-aE [
1
i&q2
)
h = h (P, E, q2)
Using (3.17), we see that the nonequilibrium quantity Lagrange multiplier of the internal energy density d=
B can be identified
(5.1)
as the reciprocal
of the
1
AE(P, E, q2)
Other authors [18-211, recalling that both the entropy flux and the heat flux are continuous .. Tth far from equrhbrrum as the across an ideal wall, define the “thermodynamic temperature” inverse of the coefficient linking the nonequilibrium entropy flux to the heat flux. The theory developed here furnishes the complete nonequilibrium expression of the entropy flux @.l~:in fact, from (3.9), putting $ = v - a&, (5.3) one obtains (5.4) so that I
T~I, =
@(P, -L q2) = AE + &h2
(5.5)
Nonlrnear
When
phenomena
Tth is different
Extended
far from equilibrium
are considered,
from the nonequilibrium
959
Thermodynamics
temperature
the Miiller B of extended
thermodynamic irreversible
temperature
thermodynamics:
in fact 19= rftt, iff u = a&. In the following section, the mathematical expressions for 0 and Ttk, will be analyzed. In Sections 7 and 8 it will be shown that both these quantities play an important, role in extended thermodynamics
of a nonviscous
gas.
6. NONEQUILIBRIUM In this section,
the quantity
the heat flax q2, instead Observe
that
STATE
EQUATIONS
8 will be chosen as a fundamental
of the internal
energy
density
field, beside
t,he density
p and
E.
(3.17) can be written (6.1)
Integrability
conditions
for this equation
yield
(6.2) (6.3) (6.4) Consequently, the internal partial differential equations:
energy
E and the coefficient
density
;E = pdE+ +g aP
A, must
satisfy
the following
+ 4q2dE
(6.5)
aq2 ’
(6.6)
whose solutions
are
E (p, 0, q2) = JQ (PJd2)
Q5’2~(2, <) ,
(6.7) (6.8)
= p-"G(z,C).
where we have put es/2 z=----, P Obviously.
only one of the two functions
3 and C is arbitrary;
(6.9) in fact, from (6.4) we obtain
G<= f32, that
(6.10)
is (6.11)
We finally
recall that
the state equation
for an ideal monoatomic
gas, found by Liu and Miiller
in 1131, is . E=_&(p,T)=T5b'
(6.12)
960
M. S. MONGIOV~
where T is the thermostatic
temperature;
consequently,
the relation (6.13)
E (P, 0, q2) = fi (P, T) can be interpreted
as constitutive
relation
a0 E, - I$, -=Ea ’ a/, (6.14) in (6.5) and observing
Substituting
for 0. In particular, d0 _=dT that
_&
dr9
E,z
Eo ’
dQ2 -
Ee
(6.14)
is also (6.15)
the following
partial
differential
equation
in the unknown
+ 4q26,z,
+9 = ~0, + ;TH* whose solution
(6.16)
is
0 (P,T Observing
0 is obtained
that
near equilibrium
q2)= p2j3A
(6.17) A must satisfy
0 equals T, the function
the relation (6.18)
7. GIBBS
EQUATIONS
FOR A NONVISCOUS
GAS
In [28]. it has been observed that, in the study of nonequilibrium phenomena, different nonequilibrium variables can be used. In particular, in a theory in which the evolution time of the heat flux is not negligible either the heat flux q and the quantity J = Vq (being V = l/p the specific volume) can be chosen as nonequilibrium variables [25,29]. From a mathematical point of view: two different definitions for the pressure and chemical potential far from equilibrium can q and J, respectively, and be introduced: the nonequilibrium pressures 7rg and ?rJ at constant the nonequilibrium
In these
chemical
potentials
pu, and ~LJ at constant
q and JI respectively,
TiTq= p + 28&q,,
715 =P+~~&
pq = -s;i
PLJ = -/3 (A + x,J,)
expressions
= --E - B7j i- 3, P E = E/p
is the internal
specific
[28] (7.1)
energy
= --E - 87 + 7.
and 7 = h/p
(7.2)
the nonequilibrium
specific entropy. Using the definitions two alternative
of the nonequilibrium
pressures,
equation
(3.6) can be rewritten
Odq = de + r,dV
+ HVi,dq,,
(7.3)
Odq = de + -ir_,dV + &dJ,, while using the expressions
Therefore, equilibrium,
in the
ways [28]
of the nonequilibrium
chemical
(7.4) potentials,
these equations
become
c&q = Vdn, - VdQ - BVi,dq,,
(7.5)
dpJ = VdrJ
(7.6)
- r/d0 - ki,dJ,.
one sees that different generalizations of the Gibbs equation are possible far from which can be all useful in different physical situations. In all these generalizations,
the same quantity 6 appears. It is therefore, reasonable to call this quantity nonequilibrium temperature. Finally, observe that the potential h’ is linked to the nonequilibrium thermodynamic pressure at constant J, in fact. recalling that in this theory 3p = 2E, from (4.5) one obtains h’ = -;
(?? + ei,q,)
= _!$
(7.7)
Nonlinear
Extended
8. NONEQUILIBRIUM
Thermodynamics
STATIONARY
As we have seen, the nonequilibrium
temperature
HEAT
of the extended
ics [17] is the quantity
which must replace the thermostatic
of the Gibbs
far from equilibrium.
equation
961
temperature
On the contrary,
FLUX
irreversible
we will show that
modynamic temperature is the quantity which is effectively measured stationary situations. Specifically. we will treat the case of a stationary
thermodynam-
in the various
extensions
the Miiller ther-
in some nonequilibrium state with constant heat,
flux q and v = 0. To this purpose, relations
we write
(3.10)-(3.13)
the system
of field equations
in (3.1), the following
system
for a nonviscous
of field equations
gas.
Substituting
is obtained:
(8.1)
da
dt
7 dv, + s”“z
2 +
au,
iqkds
2 2.
3
+
+
dv,
g4’kz
8%
+ Xq(&qk)G
a[p&k+
??(d?k)]
_
5E&j + 3aq(,qj)
8 [2Efijk + 3”q(j%j]
9P
dxk
axk
= bq
z
Observe that the quantity -l/b appearing in the right side of equation (8.1) has the dimension of time and near equilibrium can be interpreted as relaxation time of the heat Aux 4%. We make, therefore, the following position: b=Consider now a nonequilibrium vu, = 0. Under these hypotheses,
’
T(P,Q,$)’
stationary state with constant equations (8.1) become
a
1
2
-&--
1
heat flux q = (91, 0:0) and with
p+jaq;
=o,
[
aql ---IO
(8.2)
1
&
Using
(3.18) the latter
equation
2 1
(D+$“i
1 =
-p
can be written 7 34 q1 =-X,&&
Remembering
the definition
of thermodynamic
temperature
r
(8.3) (8.3) can be written
aTth
ql=T:hX,a.T,
This equation shows that the Miiller thermodynamic temperature is the quantity effectively measured in the considered nonequilibrium stationary situations.
(8.4) which
is
REFERENCES 1. M. Lachowicz, Asymptotic analysis of nonlinear kinetic equations-The hydrodynamic limit. In Lecture Notes on the Mathematical Theory of the Boltzmann Equatzon, (Edited by N. Bellomo), World Sci London. (1995) 2. L. Bonilla and J. Soler, High field limit of the Vlasov-Poisson-Fokker-Plank system. A comparison of different perturbation methods, Math. Models n/leth. Appl. SC%.ll? 1457-1468, (2001).
962 3
M. S. MONGIOV‘I
22.
H. Cafhsh. The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 38, 651-666, (1980). A. DeMasi, R. Esposito and J.L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math. 42, 321-366, (1989). C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations: Convergence proofs for the Boltzmann equation, Comm. Pure. Appl. Math. 46, 321-366, (1993). C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equations, Math. Models Meth. Appl. Sci. 2, 235-257, (1991). PL. Lions and N. Masmoudi. From Boltzmann equation to the equation of mcompressible fluid mechanics, Arch. Rntv~nel. Mech. Anal. 158, 173-193, (2001). D Borsch. Generalized discrete models Math. Models Meth. Appl. Scz. 12. 49-75, (2002). 1-I. Babowski, .4 kinetic multiscale model, Math. Models Meth. Appl. Scz. 12, 309%331. (2002). A. Bellouquid. A diffusive limit for nonlinear discrete velocity models, Math. Models Meth. Appl. Ser. 13, (2003). T. Hayat, R.G. Aitapetyan and K. Hutter, Flow of a fourth grade fluid, Math. Models Meth. Appl. Sci. 12, 7955810, (2002). T. Hayat, Y. Wang, K. Hutter and A.M. Siddequi, Peristaltic transport of a third-order fluid in a circular cilindric tube, Math. Models Meth. Appl. Sci 12, (2002). 1 Liu and 1. Muller, Extended thermodynamics of classical and degenerate gases, Arch. Rat. Mech. Anal. 4 (X3). 285~332. (1983) I LIII. I\lethod of Lagrange multipliers for exploitation of the entropy principle, Arch. Rat. Mech. Anal. 46. 13lm l-18. (1972) D. Jou and J. Casas-Vazquez, Possible experiments to check the reality of a non-equilibrium temperature, Phys. Rev. E 45 (12), 45-47, (1992). J Casas-Vazquez and D. Jou, Nonequilibrium temperature versus local-equilibrium temperature, Phys. Rev. E 49 (2), 1040-1048, (1994). D. Jou. J. Casas-Vazquez and G. Lebon, Extended Irreversible Thermodynamics. Springer-Verlag, Berlin, (2001). I. Miiller, Die Kdltefunktion, eine universelle Funktion in der Thermodynamik viskoser warmeleitender Fliissigkeiten, Arch. Rational Mech. Anal. 40, l-36, (1971). I. Miiller. The coldness, a universal function in thermoelastic bodies, Arch. Ratzonal Mech. Anal. 41, 19-332, (1971) E. Barbera. I. Muller and M. Sugiyama, On the temperature of a rarefied gas in nonequilibrium, Meccanzca 34, 1033113, (1999). J. Au, I. Miiller and T. Ruggeri, Temperature jumps at the boundary of a rarefied gas, Contznuum Mech. il’hermodyn. 12, 19-29, (2000). der Physik XII, p. 205, Springer, Berlin, H Grad, Principles of the Kznetac Theory of Gases, Handbuch
23 24
(1958). 1. Miiller and T. Ruggeri. Ratronal Extended Thermodynamzcs, Springer-Verlag, New York, (1998.). T Ruggeri, Galilean invariance and entropy principle for systems of balance laws, Contznuum Mech.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13
II. 15 16. 17. 18. 19. 20 21.
25. 26. 27 28. 29,
Ther-
modyn. 1, 3-20, (1989). MS. Mongiovi. Thermodynamic pressure in nonlinear nonequilibrium thermodynamics of dilute nonviscous gases, Phys. Rev. E 63, (2001). M.S. \longiovi, Nonlinear nonviscous hydrodynamical models for charge transport in the framework of extended thermodynamic methods. Mathl. Comput. Modellzng 35 (7/8), 813-820, (2002) hl S. Mongiovi. Some consideration about nonlinear extended thermodynamic theories with different numbers of fields, J. Non-Equzlzb. Thermodyn. 24, 147-153, (1999). h1.S. Mongiovi, Gibbs equation in the nonlinear nonequilibrium thermodynamics of dilute nonviscous gases, Appl. Math. Lett. (to appear). R. Dominguez and D. Jou, Thermodynamic pressure in non equilibrium gases, Phys. Rev. E 51 (l), 158-163, (1995).
Mathematical
and Computer
Modelling
36 (2002) 951-962 www.elsevier.comJlocateJmcm
Nonlinear Extended Thermodynamics of a Dilute Nonviscous Gas M. S. MONGIOV~ Dipartimento di Matematica ed Applicazioni Facolta di Ingegneria, Viale delle Scienze University I-90128
of Palermo
Palermo,
It,aly
MongioviQunipa.it (Received
July 2002:
accepted
August
2002)
Abstract-This paper deals with further developments of a nonlinear theory for a nonviscous gas in the presence of heat flux, which has been proposed in previous papers, using extended thermodynamics. The fundamental fields used are the density, the velocity, the internal energy density, and the heat flux. Using the Liu procedure, the constitutive theory is built up without approximations and the consistence of the model is showed: it is shown that the model is determined by the choice of three scalar functions which must satisfy a system of partial differential equations, which always has solutions. Different changes of field variables are carned out, using different Legendre transformations, passing from variables which are convenient from a mathematical pomt of view to intermediate nonequilibrium variables and finally to equilibrium variables. In particular, two possible extensions of the temperature far from equilibrium are considered: the “nonequilibrium temperature” of extended irreversible thermodynamics [defined as the reciprocal of the derivative with respect to the energy of a generalized nonequilibrium entropy) and the “thermodynamic temperature” by Miiller (defined as the mverse of the coefficient linking the nonequilibrium entropy flux to the heat flux). The link between these two quantities is established, general expressions for these quantities, as functions of the density, of the absolute temperature and of the heat flux are obtained. @ 2002 Elsevier Science Ltd. All rights reserved.
Keywords-Extended
thermodynamics,
Kinetic
theory, Entropy
closure.
Hydrodynamic
models
1. INTRODUCTION The derivation of hydrodynamic equations from the microscopic kinetic, Boltzmann-type. equations is a fundamental research field of applied mathematics. The macroscopic description can be obtained by different mathematical approaches. For instance, the asymptotic theory for small Knudsen numbers related theory of gases means, as known [l], analyzing the macroscopic description
to models of the kinetic delivered by the kinetic
equations when the distance between particles tends to zero. Then one obtains a macroscopic description delivered by the microscopic one as an alternative to the purely phenomenological derivation. The method applies to various kinetic equations. Different scaling generate d&rent Partially supported by M.I.U.R. of Italy, under Grant and Stability in Models of Continuous Media”. 0895.7177/02/S - see front matter PII: SO895-7177(02)00249-Z
@ 2002 Elsevier
“Nonlinear
Mathematical
Problems
Science Ltd. All rights reserved.
in Wave Propagation
Typeset
by dJvIs-T)$C
952
M. S. M~NGIOV~
macroscopic
equations
as documented
in the formal
expansions
proposed
121. The literature
in
the field is documented in (11 starting to relatively more recent contributions On the other
hand,
is well understood Indeed, shown
at the formal level, by now, its full mathematical
the justification to be difficult
approaches
of the formal considering
to overcome
In prin
although
approximation
that
many
these difficulties
the same method mathematical
have emerged
can be applied
model
to the
a formal procedure in (111. The interest
in [lo], overcoming above
methods
Boltzmann
questions
remain
theory
Boltzmann
equation unsolved.
equations
of gases describing
has Two
by extended
[8,9], which
the evolution
of velocities.
some of the above mathematical is offered
is still missing.
recently.
to discretized
of the kinetic
justification
for the classical
basic regularity
allowed to move in the space with a finite number
as documented
An alternative
from the pioneer papers on the topic, e.g., Caflisch (31, up [4-71, which exploits suitable moment closures. the derivation of fluid dynamical equations from kinetic theory
of a
In this case. it is difficulties.
thermodynamics
which
offers
to obtain higher-order macroscopic models as documented, among others, in high-order models of fluids is documented in recent papers by Hutter et
ul. [12]. In detail, in this paper, a nonlinear extended theory for a nonviscous gas is developed without approximations. The fundamental fields used are the density p, the velocity v,, the internal energy density E, and the heat flux 4%. This first section provides an introduction The contents are organized into eight sections. Section 2 deals with a brief recall of extended to the aims and organization of the paper. thermodynamics of gases with 13 fields. as formulated by Liu and Muller [13]. In Section 3, the mathematical implications of the hypothesis of nonviscosity are studied.
Using
the Liu procedure [14], it is shown that the model is defined by the knowledge of some scalar satisfy a set of finite and constitutive functions of p, E, and q2, which are not independent, differential relations. In Section 4, the constitutive
theory
is analyzed
it is shown that the model is determined Lagrange multipliers, which must satisfy
in detail:
through
a Legendre
by the choice of three scalar a system of partial differential
transformation,
functions of the intrinsic equations, which always
has solutions. In Section 5, two possible
extensions of the temperature far from equilibrium are considered: 0 of extended irreversible thermodynamics by Jou, Casasthe “nonequilibrium temperature” Vazquez and Lebon [15-171 (defined as the reciprocal of the derivative with respect to the energy of a generalized nonequilibrium entropy) and the “thermodynamic temperature” Tth by Miiller et al. [18-211 (defined as the inverse of the coefficient linking the nonequilibrium entropy flux to the heat flux). The link between these two quantities is established and general expressions for these quantities are obtained. In Section 6, different changes of field variables are carried out, using different Legendre transformations, passing from variables which are convenient from a mathematical point of view to intermediate nonequilibrium variables and finally to equilibrium variables. In particular, for the quantity 0, defined as the reciprocal of the derivative with respect t,o the energy of a generalized nonequilibrium entropy the general expression
e = p5/2A where T is the thermostatic temperature tity Tth is linked to B by the relation
and A an arbitrary
1 -=_ Tth
1 e + ti&qz,
(1.1) function,
is obtained.
The quan-
(1.2)
where X, are the Lagrange multipliers associated to the heat flux and Q a coefficient linked to the stress tensor and to the flux of the heat flux. Further, the complete expressions furnished by the
Nonlinear
theory
for the energy
Extended
Thermodynamics
E and for the Lagrange
density
953
multipliers
X, of the heat flux, as functions
of p, 8, and q2, are determined. The
analysis
of Sections
7 and 8 aims to shed some light on the physical
the quantities 0 and Tth:’ it is shown that equilibrium the “nonequilibrium temperature” Miiller thermodynamic
temperature
in the extensions of the Gibbs appears, while in the extension
interpretation
of
equation far from of Fourier law the
appears.
2. A REMINDER OF EXTENDED THERMODYNAMICS OF GASES The derivation
of fluid dynamic
equations
from kinetic theory
has been in the past years the ob-
ject of many investigations. In this section, we present a brief recall of extended of gases with 13 fields, as formulated, by Liu and Miiller [13]. In the kinetic
theory
of gases, a dilute
gas is described
by the Boltzmann
thermodynamics
equation
where f = f(t,x, u) is the distribution function, t the time, x = (z,) the space, u = (ul) the velocity, f = (fZ) the external force, and S the collision term. Grad’s method is developed [22] considering
the moments
of various
order of the phase density
Pa]%*..2,,
and writing
the transport
equations
=
I
mu,,
G,
(2.2)
u,,, f du,
for these moments
g+gJ=p,
(2.3)
3
whereM = (PA) = (P,P~,P~~~~,. ,P~~~~...~,,), F, = (PAj), and P = (PA). This paper
deals with the transport
equations
for the first 13 moments,
therefore,
the notation
M = (P,pi, PZJ,~zjj) will be used. As it is known,
the above-mentioned
13 moments
correspond,
(2.4)
respectively,
to the mass density,
the momentum density, the momentum flux density, and the energy flux density. In this paper, the gas is supposed referred to an inertial frame, in the absence of external forces. In [13], Liu and Miiller first imposed general physical principles to this system, such as that of material objectivity and the entropy principle. Now, we report briefly this methodology, as reported in the book of Miiller and Ruggeri [23], which furnish an elegant and powerful method for obtain high-order models of gases. Restrictions on the constitutive relations for the nonfundamental the validity of the entropy principle: there are an entropy density density h, = hj(p~), such that the entropy thermodvnamic orocess
production
fields are obtained imposing h = h(pA) and an entropy flux
g is supposed
to be nonnegative
for every
(2.5) It must be taken into account that with the Boltzmann entropy of the 13 moments development, proposed Note that equations (2.3) can be the entropy principle a procedure,
h is a phenomenological macroscopic function with coincides kinetic theory of gases in the truncated form limited to the by Grad [21]. considered as constraints for the fields PA. In order to satisfy known as Liu method of Lagrange multipliers [14:23], will
M. S. MONGIOV~
954
he used; this method following
inequality
states
that,
introducing
must be satisfied
a Lagrange
for arbitrary
multiplier
AA for each equation,
values of the field M:
dh
u=x+ In the kinetic
theory
the
(2.6)
of gases, the central
PZiZZ.
.1,,
=
moments
&‘I = (CA), defined
s
by
c,,r,f dc,
mc,,c,,
(2.7)
where c, = v1 - w, (with v, = pz/p) is the peculiar velocity, are often chosen as independent fields instead of complete moments. An exponential matrix of the form X = eArvr, with A, constant matrices,
can be introduced
such that
the vector M can be written
as [23.24]
M=Xti.
(2.8)
where M is composed with objective tensors. In particular, the central part bz of the momentum pz is identically zero. Regards to the fluxes, in [24] it has been shown that, putting h, = hv, + $3 and Fj = MU, + G,, q53 is an intrinsic vector and G, can be expressed as Further P = XP and i = AX! where P and A are G, = XG:,, where G, is objective. intrinsic
[24].
In terms of &I. equations
(2.3) can be written
(23,241
(2.9) where $ denotes
the material
derivative.
This system can be easily put in normal form with respect to time derivatives solving the vector component of (2.9) with respect to % and substituting the result in the other equations. If momentum
is supposed
conserved,
the right side of the resulting
system
will be still the vector P.
In extended thermodynamics, system (2.9) instead of balance equations (2.3) is frequently employed. If one considers only the first 13 fields, the density p = 6, the velocity V, = pz/p, and the intrinsic parts & and bzjj of the other fields, are chosen as independent variables. The fields G:3 and @ are treated as dependent variables and are expressed in terms of the fundamental variables by means of constitutive equations, satisfying objectivity and entropy principles. In terms of the intrinsic dh -+i,s+$!-,i. dt .l It is plain that system (2.9).
quantities
inequality
. %+a%-+x3 inequality
(2.6) becomes
[23,24]
ae:, dz 3 + Ar
(2.10) can also be obtained
3. NONVISCOUS
directly,
by applying
the method
of Liu to
GASES
Sometimes, in the description of gases in nonequilibrium, the approximation of nonviscous fluid may be heuristically applied [25,26]. In a previous paper [27], it has shown that in order to formulate nonlinear extended theories for nonviscous gases and fluids, we can use as fundamental fields only the density p, the velocity uzr the internal energy density E = &,/2, and the heat flux (Iz = k/2.
955
Nonlinear Extended Thermodynamics
The evolutions with respect equation
equations
for these fields can be written
to the material
derivatives
of the stress deviator
of the central
Pczj,; one obtains
solving
the first 13 balance
moments,
and neglecting
equations
the evolution
[25,26]
(3.1)
The field I
is not considered
as an independent
variable,
but it must be expressed
through
a constitutive relation as function of the other fields. This implies that the generalized entropy density h introduced in (2.5) is independent of the deviatoric part /i(,,) of the stress tensor. therefore, being h an objective scalar quantity, we can write (3.2)
h = h (P, E, 9’) From the mathematical Miiller
[13], imposing
that
point of view, this theory
is obtained
the constitutive
must satisfy
relations A (2J)
=
by the 13 field theory the further
condition
of Liu and [27] (3.3)
‘1
where A(,,) is the intrinsic part of the Lagrange multipliers of the deviator of t,hc second monn~nt,. i.e., the derivative with respect to the stress deviator of the nonequilibrium entropy h. Constitutive eqUatiOnS for the flUXeS b(+), FzJkr and PtJjk are necessary t0 ClOSe system (3.1). Restrictions on these constitutive relations may be obtained imposing the validity of the entropy principle, applying the Liu procedure to system (3.1). This is equivalent to put constraint (3.3) in inequality (2.10). The following set of relations is obtained: (3.4) (3.5) (3.6) (3.7)
Recall that in the kinetic theory of dilute gases, the trace of the stress p is linked to the internal energy density E by the relation p = (2/3)E. Being the constitutive relations objective functions. they can be expressed in the form @k
=
4 (P,E, q2)qk,
(3.9)
fiti = ;E6,, +a (P,E, 4’) q(2q.4’
(3.10)
;4(,6,) +x (~7E, q2)ykwj) ihijk=P (P)El q2)&k + v (p,E, q2) i&k
=
i
q,+q,),
+‘:;’
= b (P, E, q2) 9%.
(3.11) (3.12) (3.13)
M. S. MONGIOV~
956
Substituting
now (3.9)-(3.13)
in (3.5)-(3.8),
A, = X,qi, we get
and putting
h - pii - ;E&
- 2Xqq2 = 0,
(3.14)
9 - 3xq2_ aIL+X,---- , 5
(3.15)
c#l- AE - x,q2 (v - UE) = 0,
(3.16)
dh = Adp + AEdE + Xqqidq,,
(3.17)
o
dc$ = A, {d+(dE+q2da+;dq2)+;q2dv+~dq2}.
(3.18)
X,bq’ 1 0,
(3.19)
CT =
where the following
position
has been made: (3.20)
Note that all the relations presented in this section are exact, because no approximation has been used for their determination. As can be seen, the constitutive theory is determined by the knowledge of the then scalar functions h, 4, a, x, ,6, v, b, A, AE, and X,, which are constrained by relations (3.14)-(3.19). In the following section, the relations will be analyzed in detail, in order to show the consistency of the proposed model and to establish which constitutive quantities can be chosen arbitrarily (and therefore, 4. Choose
be determined
ANALYSIS
as new variables
introduce defined
must
the scalar
experimentally)
OF
THE
the nonconvective
potential
and which are instead
CONSTITUTIVE parts
of the Lagrange
h’ = h’(& AE, i2) and the vector
fixed by the theory.
THEORY multipliers potential
(A, A,,
and i,),
Qi = @k(A, AE, i,),
by
h’ = -h + Ap +
AEE + Xtq,,
(4.1)
m;=-m,-n,q,+h*(i-?“q2) and rewrite
(3.5) and (3.6) in the alternative dh’ = pdit + Ed& da;
First,
= qkd&
(4.2)
-Jlc&(p&-E), form
+ ;qidiii,
+ (/%,
+
(4.3) d&i,.
u’?(,qk)
>
(4.4)
from (3.14) and (4.1), one can write 3 (4.5)
substituting
in (4.3), one obtains &’ = - -&
(h’ + itqt)
d&
-&
i,qi)
$
+ ;q,dh,.
(4.6)
Therefore. dh’ 7
ail.5
=
(h’+
,
= p,
s
= 4%
(4.7)
Nonlinear Extended Thermodynamics
it can be concluded
whose solution
that
h’ must satisfy
the following
PDE:
is
$,A , [ I
h’=X’H
where ‘H an arbitrary Writing
957
function,
and X the absolute
value of X,.
now @L = c++‘X,and qk = qxXk (q,j = l/X,),
&+’ 7
=
-
RA
(4.9)
from (4.4), we can write
2.
(pE - E)
14.10)
+ (p& - E) -
qx = -$
i3E
(4.11)
aii, ’
E
(4.12)
3~-2Et‘-30’=2X2~+2X2(pE-E)~,
(u - aE) q; = 2%
(4.13)
+ 2 (P& - E) g.
Now. regarding h’ as a known function of the Lagrange multipliers, satisfying from (4.3) we obtain the following expressions of the quantities p, E, and qx: dh’
equations
ah'
(4.14)
9x=23$
P=x’
(4.8),
From (4.12) and (4.13) one obtains &= from (4.10))(4.13)
(3-&?)
one can write u - u& + -q;dX’
dq’ = (E - p&)d& + qxdh, and d& =
-!.-
@’ - q/idAE-
E - p&
Integrability
(4.15)
+(3E-pE):
conditions
for the first equation
u - aE
-q;dX2 2
yields
(4.16)
I
(4.17)
(4.18) remembering
(4.14), one can write a2h'
- d2h’ = 0,
a2h'
aA,aA,
Z&
aliax2
(4.19)
which is a PDE in the unknown E = f(A, AE, X2). Substituting (4.10) and (4.11) in (4.18), one obtains “-E-$--g=
(g-E!$)
(E-qA)+s(&p&).
(4.20)
8hE
Now, compatibility
conditions
for (4.17) yields aE -Eg ah
Subtracting
(4.21) to (4.20), one obtains
-/$)
also the following
(E-Q)
+ g(E-pL).
(4.21)
PDE for 4’: (4.22)
Finally, a is computed by (3.20). x by (3.15), v and 0 by (4.12) and (4.13). Equations (4.1) and (4.15) reduce to identities. The main result of the analysis of this section can be summarized in the following proposition.
PROPOSITION. The constitutive theory is determined by knowledge of the three functions h’, E. and 4’ as function of the variables Aj AE, and i2, which must satisfy the following system of partial differential equations:
h’ +2x2$
+ ;I&
a2h’
ah’ aA,
ash’ ^ ahaX
a&ah,
=
o
(4.23) )
=o.
(4.24)
a& a$f a& a@'+zdh'!$=o. --_-_ aA, aA ahi, a2aA Note that system integrated which,
(4.23)-(4.25)
and its solution
always has solutions.
is given by (4.9); equation
one fixed h’, can be integrated
(4.23) can be immediately
(4.24) is a quasi-linear
first-order
of characteristics;
last equation
by the method
once h’ and E are known, is a linear first-order The compatibility of the constitutive theory
5. GENERALIZED
Indeed equation
(4.25)
equation (4.25),
equation. is such established.
TEMPERATURES
FAR
FROM
EQUILIBRIUM
The problem of t,he definition of the temperature far from equilibrium has received attention by several authors. An extensive discussion can be found in the book of Jou, Casas-Vazquez and Lebon [17]! which is a fundamental tool for people working in nonequilibrium thermodynamics. In classical thermodynamics of solids or fluids, the equilibrium temperature is equal to the inverse of the partial derivative of the entropy with respect to the energy, at constant, density and it is identical to the inverse of the coefficient linking the entropy flux and the heat flux. Some authors [15,16] h ave proposed the use of a generalized “nonequilibrium temperature“ B. defined as the reciprocal of the derivative with respect to the energy of a generalized nonequilibrium entropy h depending not only on the classical variables, but also on the dissipative fluxes, as, for instance
the heat flux q, %
1 _-ah H-aE [
1
i&q2
)
h = h (P, E, q2)
Using (3.17), we see that the nonequilibrium quantity Lagrange multiplier of the internal energy density d=
B can be identified
(5.1)
as the reciprocal
of the
1
AE(P, E, q2)
Other authors [18-211, recalling that both the entropy flux and the heat flux are continuous .. Tth far from equrhbrrum as the across an ideal wall, define the “thermodynamic temperature” inverse of the coefficient linking the nonequilibrium entropy flux to the heat flux. The theory developed here furnishes the complete nonequilibrium expression of the entropy flux @.l~:in fact, from (3.9), putting $ = v - a&, (5.3) one obtains (5.4) so that I
T~I, =
@(P, -L q2) = AE + &h2
(5.5)
Nonlrnear
When
phenomena
Tth is different
Extended
far from equilibrium
are considered,
from the nonequilibrium
959
Thermodynamics
temperature
the Miiller B of extended
thermodynamic irreversible
temperature
thermodynamics:
in fact 19= rftt, iff u = a&. In the following section, the mathematical expressions for 0 and Ttk, will be analyzed. In Sections 7 and 8 it will be shown that both these quantities play an important, role in extended thermodynamics
of a nonviscous
gas.
6. NONEQUILIBRIUM In this section,
the quantity
the heat flax q2, instead Observe
that
STATE
EQUATIONS
8 will be chosen as a fundamental
of the internal
energy
density
field, beside
t,he density
p and
E.
(3.17) can be written (6.1)
Integrability
conditions
for this equation
yield
(6.2) (6.3) (6.4) Consequently, the internal partial differential equations:
energy
E and the coefficient
density
;E = pdE+ +g aP
A, must
satisfy
the following
+ 4q2dE
(6.5)
aq2 ’
(6.6)
whose solutions
are
E (p, 0, q2) = JQ (PJd2)
Q5’2~(2, <) ,
(6.7) (6.8)
= p-"G(z,C).
where we have put es/2 z=----, P Obviously.
only one of the two functions
3 and C is arbitrary;
(6.9) in fact, from (6.4) we obtain
G<= f32, that
(6.10)
is (6.11)
We finally
recall that
the state equation
for an ideal monoatomic
gas, found by Liu and Miiller
in 1131, is . E=_&(p,T)=T5b'
(6.12)
960
M. S. MONGIOV~
where T is the thermostatic
temperature;
consequently,
the relation (6.13)
E (P, 0, q2) = fi (P, T) can be interpreted
as constitutive
relation
a0 E, - I$, -=Ea ’ a/, (6.14) in (6.5) and observing
Substituting
for 0. In particular, d0 _=dT that
_&
dr9
E,z
Eo ’
dQ2 -
Ee
(6.14)
is also (6.15)
the following
partial
differential
equation
in the unknown
+ 4q26,z,
+9 = ~0, + ;TH* whose solution
(6.16)
is
0 (P,T Observing
0 is obtained
that
near equilibrium
q2)= p2j3A
(6.17) A must satisfy
0 equals T, the function
the relation (6.18)
7. GIBBS
EQUATIONS
FOR A NONVISCOUS
GAS
In [28]. it has been observed that, in the study of nonequilibrium phenomena, different nonequilibrium variables can be used. In particular, in a theory in which the evolution time of the heat flux is not negligible either the heat flux q and the quantity J = Vq (being V = l/p the specific volume) can be chosen as nonequilibrium variables [25,29]. From a mathematical point of view: two different definitions for the pressure and chemical potential far from equilibrium can q and J, respectively, and be introduced: the nonequilibrium pressures 7rg and ?rJ at constant the nonequilibrium
In these
chemical
potentials
pu, and ~LJ at constant
q and JI respectively,
TiTq= p + 28&q,,
715 =P+~~&
pq = -s;i
PLJ = -/3 (A + x,J,)
expressions
= --E - B7j i- 3, P E = E/p
is the internal
specific
[28] (7.1)
energy
= --E - 87 + 7.
and 7 = h/p
(7.2)
the nonequilibrium
specific entropy. Using the definitions two alternative
of the nonequilibrium
pressures,
equation
(3.6) can be rewritten
Odq = de + r,dV
+ HVi,dq,,
(7.3)
Odq = de + -ir_,dV + &dJ,, while using the expressions
Therefore, equilibrium,
in the
ways [28]
of the nonequilibrium
chemical
(7.4) potentials,
these equations
become
c&q = Vdn, - VdQ - BVi,dq,,
(7.5)
dpJ = VdrJ
(7.6)
- r/d0 - ki,dJ,.
one sees that different generalizations of the Gibbs equation are possible far from which can be all useful in different physical situations. In all these generalizations,
the same quantity 6 appears. It is therefore, reasonable to call this quantity nonequilibrium temperature. Finally, observe that the potential h’ is linked to the nonequilibrium thermodynamic pressure at constant J, in fact. recalling that in this theory 3p = 2E, from (4.5) one obtains h’ = -;
(?? + ei,q,)
= _!$
(7.7)
Nonlinear
Extended
8. NONEQUILIBRIUM
Thermodynamics
STATIONARY
As we have seen, the nonequilibrium
temperature
HEAT
of the extended
ics [17] is the quantity
which must replace the thermostatic
of the Gibbs
far from equilibrium.
equation
961
temperature
On the contrary,
FLUX
irreversible
we will show that
modynamic temperature is the quantity which is effectively measured stationary situations. Specifically. we will treat the case of a stationary
thermodynam-
in the various
extensions
the Miiller ther-
in some nonequilibrium state with constant heat,
flux q and v = 0. To this purpose, relations
we write
(3.10)-(3.13)
the system
of field equations
in (3.1), the following
system
for a nonviscous
of field equations
gas.
Substituting
is obtained:
(8.1)
da
dt
7 dv, + s”“z
2 +
au,
iqkds
2 2.
3
+
+
dv,
g4’kz
8%
+ Xq(&qk)G
a[p&k+
??(d?k)]
_
5E&j + 3aq(,qj)
8 [2Efijk + 3”q(j%j]
9P
dxk
axk
= bq
z
Observe that the quantity -l/b appearing in the right side of equation (8.1) has the dimension of time and near equilibrium can be interpreted as relaxation time of the heat Aux 4%. We make, therefore, the following position: b=Consider now a nonequilibrium vu, = 0. Under these hypotheses,
’
T(P,Q,$)’
stationary state with constant equations (8.1) become
a
1
2
-&--
1
heat flux q = (91, 0:0) and with
p+jaq;
=o,
[
aql ---IO
(8.2)
1
&
Using
(3.18) the latter
equation
2 1
(D+$“i
1 =
-p
can be written 7 34 q1 =-X,&&
Remembering
the definition
of thermodynamic
temperature
r
(8.3) (8.3) can be written
aTth
ql=T:hX,a.T,
This equation shows that the Miiller thermodynamic temperature is the quantity effectively measured in the considered nonequilibrium stationary situations.
(8.4) which
is
REFERENCES 1. M. Lachowicz, Asymptotic analysis of nonlinear kinetic equations-The hydrodynamic limit. In Lecture Notes on the Mathematical Theory of the Boltzmann Equatzon, (Edited by N. Bellomo), World Sci London. (1995) 2. L. Bonilla and J. Soler, High field limit of the Vlasov-Poisson-Fokker-Plank system. A comparison of different perturbation methods, Math. Models n/leth. Appl. SC%.ll? 1457-1468, (2001).
962 3
M. S. MONGIOV‘I
22.
H. Cafhsh. The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math. 38, 651-666, (1980). A. DeMasi, R. Esposito and J.L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math. 42, 321-366, (1989). C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations: Convergence proofs for the Boltzmann equation, Comm. Pure. Appl. Math. 46, 321-366, (1993). C. Bardos and S. Ukai, The classical incompressible Navier-Stokes limit of the Boltzmann equations, Math. Models Meth. Appl. Sci. 2, 235-257, (1991). PL. Lions and N. Masmoudi. From Boltzmann equation to the equation of mcompressible fluid mechanics, Arch. Rntv~nel. Mech. Anal. 158, 173-193, (2001). D Borsch. Generalized discrete models Math. Models Meth. Appl. Scz. 12. 49-75, (2002). 1-I. Babowski, .4 kinetic multiscale model, Math. Models Meth. Appl. Scz. 12, 309%331. (2002). A. Bellouquid. A diffusive limit for nonlinear discrete velocity models, Math. Models Meth. Appl. Ser. 13, (2003). T. Hayat, R.G. Aitapetyan and K. Hutter, Flow of a fourth grade fluid, Math. Models Meth. Appl. Sci. 12, 7955810, (2002). T. Hayat, Y. Wang, K. Hutter and A.M. Siddequi, Peristaltic transport of a third-order fluid in a circular cilindric tube, Math. Models Meth. Appl. Sci 12, (2002). 1 Liu and 1. Muller, Extended thermodynamics of classical and degenerate gases, Arch. Rat. Mech. Anal. 4 (X3). 285~332. (1983) I LIII. I\lethod of Lagrange multipliers for exploitation of the entropy principle, Arch. Rat. Mech. Anal. 46. 13lm l-18. (1972) D. Jou and J. Casas-Vazquez, Possible experiments to check the reality of a non-equilibrium temperature, Phys. Rev. E 45 (12), 45-47, (1992). J Casas-Vazquez and D. Jou, Nonequilibrium temperature versus local-equilibrium temperature, Phys. Rev. E 49 (2), 1040-1048, (1994). D. Jou. J. Casas-Vazquez and G. Lebon, Extended Irreversible Thermodynamics. Springer-Verlag, Berlin, (2001). I. Miiller, Die Kdltefunktion, eine universelle Funktion in der Thermodynamik viskoser warmeleitender Fliissigkeiten, Arch. Rational Mech. Anal. 40, l-36, (1971). I. Miiller. The coldness, a universal function in thermoelastic bodies, Arch. Ratzonal Mech. Anal. 41, 19-332, (1971) E. Barbera. I. Muller and M. Sugiyama, On the temperature of a rarefied gas in nonequilibrium, Meccanzca 34, 1033113, (1999). J. Au, I. Miiller and T. Ruggeri, Temperature jumps at the boundary of a rarefied gas, Contznuum Mech. il’hermodyn. 12, 19-29, (2000). der Physik XII, p. 205, Springer, Berlin, H Grad, Principles of the Kznetac Theory of Gases, Handbuch
23 24
(1958). 1. Miiller and T. Ruggeri. Ratronal Extended Thermodynamzcs, Springer-Verlag, New York, (1998.). T Ruggeri, Galilean invariance and entropy principle for systems of balance laws, Contznuum Mech.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13
II. 15 16. 17. 18. 19. 20 21.
25. 26. 27 28. 29,
Ther-
modyn. 1, 3-20, (1989). MS. Mongiovi. Thermodynamic pressure in nonlinear nonequilibrium thermodynamics of dilute nonviscous gases, Phys. Rev. E 63, (2001). M.S. \longiovi, Nonlinear nonviscous hydrodynamical models for charge transport in the framework of extended thermodynamic methods. Mathl. Comput. Modellzng 35 (7/8), 813-820, (2002) hl S. Mongiovi. Some consideration about nonlinear extended thermodynamic theories with different numbers of fields, J. Non-Equzlzb. Thermodyn. 24, 147-153, (1999). h1.S. Mongiovi, Gibbs equation in the nonlinear nonequilibrium thermodynamics of dilute nonviscous gases, Appl. Math. Lett. (to appear). R. Dominguez and D. Jou, Thermodynamic pressure in non equilibrium gases, Phys. Rev. E 51 (l), 158-163, (1995).