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Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics
002~72?.(183/MO297-17903.WiO 0 I983 Pergamon Press Ltd.
ht. J. Engng Sri. Vol. 21. No. 4. pp 297-313. 1983. Printed in Great Britain.
LAGRANGIAN FORMULATION OF THE MAXWELL-CATTAN~O HYDRODYNAMICS MIROSLAV GRMELA Department of Chemical En~neeri~, Ecole Polytechnique, Montrkal, Qutbec, Canada and Centre de Recherches de Math~matiques Appliqutes, Universite de Monlrial, Montr~ai, Qutbec, Canada and JXfii TEICHMANN D~partement de Physique, Universit6 de Montr6a1,Montrkal, Quebec, Canada Abstract-The Maxwell-Cattaneo modification of the Fourier constitutive relation allows to formulate hydrodynamics as dynamics of a material and a caloric point. The Maxwell-Cattaneo hydrodynamics is compatible with equilibrium thermodynamics, it represents an approximation of a kinetic equation, and it reduces to the standard hydrodynamics if the new transport coefficient (a rela~tion time) tends to zero. I. INTRODUCTION
A THERMO-DYNAMICAL process in a one component fluid is defined by five fields p(x, t), v(x, t) and e(x, t) denoting respectively the mass density, the vetocity and the internal energy. The time evolution of the five fields is governed by a set of the time evolution equations
lb
-
1
at
= VjdjVi+ f
ajHij
and a set of constitutive relations for fi and 4. We shall consider the NavierStokes-Fourier constitutive relations 4; = f
0
Aa,f
H, = PSij - V(a;Vj + 8,s) - Kd~V~S;~
(1.2) (1.3)
The scalars p, T, A, v and K remain indetermined functionals of p(x, t) and e(x, t), q represents the heat flux, fi is the pressure tensor. In eqns (l.l)-(1.3) we have used the notation ai = (a/ax,), x denotes the position vector, t is the time. We have used also the summation convection. The thermo-dynami~a1 process is called physically admissible (also compatible with equilibrium thermodynamics)[l, 21 if there exists a functional s(x, t) of p(x, t) and e(x, t) satisfying four properties (Hl)-(H4) (see Refs. I, 2 and Section 3). We shall show in Section 3 that s(x, t) satisfying (HI)-(H4) has the physical meaning of the non-equilibrium entropy. The time evolution eqns (1.1)-(1.3) represent a physically admissible thermo-dynamical process provided a compatibility condition, restricting the choice of s, p, T, A, v and K, is satisfied. This statement is proved in Section 3.2. The thermo-dynamical process (l.l)-( 1.3) can be reformulated in the Lagrangian form as a process defined by one deformation field x = X(%, t) and one scalar field[3]. In the particular case when one of the scalar fields e, Z’,s is considered invariant during the time evolution, the thermo-dynamical process will be called a dynamical process. Its time evoIution equations, in the Lagrangian formulation are equivalent to Newton’s equations governing the motion of a material point. This view of the time evolution has several important ES Vol.21,No.
4-A
297
298
advantages.
M. GRMELA and J. TEICHMANN
We mention here:
(i) A possibility of using a geometric point of view of the time evolution[41. (ii) A possibility of adopting concepts and results of theory of elasticity [3,5]. (iii) A possibility of using a well developed theory of linear Lagrangian stability [6,7]. (iv) The Lagrangian formulation is particularly well suited for the study of turbulence [8]. The presence of a scalar field, in addition to the material deformation field, in the Lagrangian description of the thermo-dynamical processes prevents the full use of the above advantages. Maxwell[9] and Cattaneo[lO] have suggested the following extension of the thermodynamical process (l.l)-(1.3): The extended thermo-dynamical process in a one component fluid is defined by eight fields p(x, t), e(x, t), v(x, t), q(x, t) denoting respectively the mass density, the internal energy, the velocity and the heat flux. The time evolution of the eight fields is governed by a set of eight time evolution equations. The first five time evolution equations are the same as (1.1). The remaining three time evolution equations are
-
$$=vjajqi + $
[ ai
(f) -t
Pqi].
(1.4)
The time evolution eqns (1.4) replace the Fourier constitutive relation (1.2). The Navier-Stokes constitutive relation (1.3) remains unchanged. The scalars p, T, A, V, K, 7 are the indetermined quantities entering the extended thermo-dynamical process. The aim of this paper is to show that the extended thermodynamical process satisfies the following properties: (i) The extended thermo-dynamical process allows a Lagrangian formulation in terms of two deformation fields, namely the material deformation field and another deformation field that we shall call a caloric deformation field. In other words, the equations governing the time evolution of the extended thermo-dynamical process become equivalent to equations governing the time evolution of a material and a caloric point. (ii) If ~+0 the thermo-dynamical process (l.l)-(1.3) is recovered. (iii) The extended thermo-dynamical process is physically admissible (i.e.) compatible with equilibrium thermodynamics-see Section 3). (iv) The extended thermo-dynamical process represents an intermediate dynamical theory of fluids, intermediate between the theory represented by eqns (l.l)-(1.3) and the kinetic theory. This point is briefly discussed in Section 3 but remains basically open. We also do not discuss situations and experiments relevant to the new information contained in the extended theory. It is interesting to note that the Lagrangian formulation of the Maxwell-Cattaneo extension of thermo-dynamical process has common features with caloric theoryt developed in 18th century[ll]. The extended thermo-dynamical process in a one component fluid can be regarded as a dynamical process in a two component fluid. We shall call the second component a caloric component. For the caloric component the total energy takes the role of the mass, its time evolution cannot be however obtained from Newton’s law. 2. LAGRANGIAN FORMULATION THERMO-DYNAMICAL
OF THE EXTENDED PROCESS
The time evolution equations of the-extended thermo-dynamical process (l.l), (1.4), (1.3) will be brought into a form suitable for the Lagrangian formulation. We introduce new variables P=P
E =i
wj
=
Vj +
PVjUj
+
pe
’E njkVk +i pqj.
tWe thank W. G. Laidlaw for pointing to one of us (M. G.) the historial references.
(2.1)
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics and
299
rewrite (1.1) (1.4) into
(2.2) -
dV
L
= VjajVi + i
at
djllij
and (2.3) aw. -2=VjdjWi+&di
-
at
-UiUjaj
+i
-
-
vi [
+1
(k)
$
p(
l+P
aj(puj) + g aj(Ewj) -
vj
E
a, VjajVl
$ aj(Ew,)]
ujdj uk +fa[ nk, 1
P&
li. vk
V;
ajnij
(;PikUk)
($Y)-Vjaj
kzPik#kaj (EWj)+ $
-2
E)
+ f
djnij
H
aj
vi - f aj nij
>I
>I .
We have used II, = PSij+ Pin The quantity E denotes the total energy. The first eqn in (2.3) represents the conservation of the total energy. We note that if e and q are known functionals of p and v then (2.3) is just another form of (2.2). The eqns (2.2) and (2.3) are written in the Eulerian form. Before we rewrite (2.2) and (2.3) into the Lagrangian form we review the Lagrangian formulation in the context of (2.2) (E and p entering (2.2) are assumed to be known functions of p). Let us assume that (2.2) are solved and thus the fields p(x, t) and v(x, t) are known. It follows from the physical meaning of p and v that trajectories of material points are obtained as solutions to the system of ordinary differential equations dx s = v(x, t)
(2.4)
By X(Q, t) we denote the solution of (2.4) that satisfies X(x0, to) = ~0. Thus
awx,,t) at
I= ,=t
v(x, t)
(2.5)
The first eqn in (2.2) is the Liouville [ lo] (also called continuity) equation corresponding to (2.4). Because the material points cannot be created or annihilated during the time evolution we have
dxo, to)d Vo= Ax, t) d v,,
(2.6)
300
M.GRMELAandJ.TElCHMANN
where d V, is an element of the volume at t = to, dV, is its image at t = t under the transformation X. Since d V, = IMI,, d V,, where M,,
=
f,
aXi(XO3 t,
I
dXOj
12.7)
x0
and
/Ml,,,= det (Mfj) we
have
Pk t) = PolMl$
(2.8)
p. denotes p(~, to). The impenetrability of matter implies jM/ $: 0 for all x and t. Thus we see that by knowing the trajectory mapping X (also called a material deformation field) we obtain p and v from (2.5) and (2.8). Conversely, by knowing the fields p, v one needs to solve (2.4) to obtain the mapping X. Let us now seek the mappings X such that p and v obtained (according to (2.5), (2.8)) from X satisfy (2.2). First we note that if p = 0, Y = 0 and K = 0 then
a’x(x(),f) at2
o
(2.9)
=
is the equation we look for. Indeed, we have from (2.5) a’Xi(Q, f) =-L&= d V.(X at2
f)
~jQi$avi(X, t) at (
The eqn (2.9) is just Newton’s equation for a free material point. If p+O then the equation determining the time evolution of X will have to include derivatives with respect to x0 and will become therefore a partial differential equation. We see easily that a... &d,lzM_,
T&- =
dXi 8x0,
Jr
a..._ m,1 LXj*X,9*- -1, axoj
(2.10)
where i, j, k means an even permutation of the indices (1, 2,3) and
1% axol [X, Xk,U] = det
ax,
ax,
axo2
axo3
ax, ax, ax, ax”1
ah2
aa
aa
1 \axol (we use the notation of Ref. 8). If p+ 0, Y= 0,
K =
axo2
ax,, aa
ax03
(2.11)
0 then (2.9) becomes
a2Xitxo,6 PO at2 = - [Xj, Xk, PI.
(2.12)
We can regard (2.12) also as Newton’s equation, the force is now however unusual. This reflects the fact that surface forces were introduced in the derivation of the second equation of (2.2) (the second equation of (2.2) is a local form of Newton’s law for a fluid confined in a finite
Lagrangian formulation
of the Maxwell-Cattaneo
hydrodynamics
301
volume). In the general case, p+ 0, v+ 0, K$ 0 thus we obtain
J2Xi(X”, 0=
pn
a2
-
[Xjf
xk,
PI
+ [X2,
x3t
(iho
v
+[x*,x3,(~~[x*,x3,~])] +[X3,X,,(~v[X3A~])] +[x,x*, (&v [x,Ay)]
(2.13)
We shall proceed now to the Lagrangian formulation of (2.3). We consider (2.3) as the time Similarly as in the study of the evolution equations of the second (caloric) component. Lagrangian formulation of (2.2) we introduce. X = Y(yn, r)
(2.14)
g = w(x, t)
(2.15)
yn = Y(Yn, GJ.
(2.16)
as solutions of
satisfying
the initial condition
The points x0 and y. are not independent;
we need to satisfy
X(x0, t) = VYO, 0
(2.17)
for all t since both (2.2) and (2.3) involve the same x and t. Note that (2.17) implies ~0 = y. at t = to, but for t+ to the points xg and y. are in general different. From (2.14) and (2.15) we have
t)
WYO,
at
,=,
= w(x, t).
(2.18)
We introduce c,,
= 1,
dyi(YO*
t,
and we assume ICI = det (C) =I 0 for all y. and t. The same arguments .Q.
(2.19)
aYOj
t) = E”lCl,:,
that led to (2.8) imply now (2.20)
302
M.GRMELAandJ.TEICHMANN
where En stands for E(y,, to). By using a ~=!$a--=~+=I[yi, ih; axi aYOj
yk,...) ] JYni
I%,,
we can rewrite the second eqn of (2.3) as differential equation simplicity we shall write the resulting equation only for
(2.21)
for Y(y,,, t). For the sake of
dW,
-=-~aiif)+~[~,-(l+~)ci] at
1+ ~
-f (
d;p - aj
(va;vj
+
VajU,)
(2.22)
I[
-
&(Kh)l.
that is obtained from the second eqn in (2.3) after neglecting the terms of the second and higher order in v and w. The Lagrangian form of (2.22) reads
-‘a(&ip~clyo)[Yjt Pn ICI,,
yk,P]
(2.23)
The main conceptual advantage of the Lagrangian formulation of fluid mechanics is that solutions to the Lagrangian equations represent trajectories. The solutions thus include a global information about the flow of a fluid. Such information is particularly pertinent in the study of secondary and turbulent flows[S]. The systems of eqns (2.13), (2.23) extends the Lagrangian formulation of a dynamical process to thermo-dynamical processes. The mathematical complexity of the Lagrangian equations (even in the case of dynamical processes) has not allowed so far to transform the conceptual advantages into practical advantages (some attempts are discussed in Ref. 8). A notable exception is the study of the Lagrangian linear stability of
303
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics
stationary plasma configurations that arises in the study of plasma confinement ]6,7]. The system (2.13), (2.23) allows now to include thermal processes while keeping the same mathematical approach and using the same general mathematical results as in Refs. [6,7]. For the purpose of the stability analysis, we shall now write the linearized system of evolution equations. We shall still consider only neutral fluids. We reformulate the Lagrangian form of evolution equations in case of small perturbations around the equilibrium state. Again, we consider the systems (2.2) and (2.3) separately. A small displacement of the fluid element from its equilibrium position is defined by a displacement field 5, (2.24)
x = x0 + &(x0,t) The displacement gradient matrix takes now a simplified form Mij = 6, +
& [i 01
and the corresponding Jacobian is, to first order in &. (2.25) where the gradient operator is taken at x = Q. Any function of x can be expanded in the vicinity of x0 in Taylor series in f either in form of expansion in Eulerian formulation with the first order term in 5 equal to
Sf(xo+ 5) or in form of expansion in Lagrangian formulation with the first order term in 5 of the form
Af (x0). Equating both quantities we obtain the relation between Eulerian and Lagrangian first order displacements
Af = sf + & &
(2.26)
f.
Let us now return to the system (2.2). Expressing the integral of the continuity equation in (2.2) with help of (2.8) we obtain the Eulerian displacement for density (2.27) The Eulerian displacements for velocity and pressure p = p(E, p) are similarly
6u, =
ii +
OOj
&01
&
-
5j
& vOi I
(2.28) where v. is the unperturbed velocity of stationary equilibrium velocity and 4 = (a&,, flat). The second system of evolution equations for the caloric fluid (2.5) is treated in a similar way. Here we introduce the displacement field 9 Y=%+ho,th
(2.29)
304
M.GRMELAandJ.TEICHMANN
The displacement gradient matrix is now
(2.30)
lcI=lt-&%.
It follows immediately from the system (2.5) that the Eulerian displacement for the total energy is
ljE=
-T&
CE077i)
01
(2.31)
and the velocity (2.32) The quantity w. means the unperturbed equilibrium velocity of the caloric fluid. In general, the velocities v. and w. do not coincide. To obtain the Euler-Lagrangian evolution equation to first order in 5 and q for the momentum transport we linearize eqn (1. l), introducing
P=Po+SP
with the Eulerian displacements defined by (2.27) and (2.28). To simplify the following discussion, we assume that we linearize around an equilibrium state, i.e. v. = w. = 0. The first order equation may be cast in the form A,~tB,j+C,~tD&E,~=O
(2.33)
The operators AM, BM, CM, DM and EM are defined as follows (AM&i=
PO&
The operator AM is self adjoint and positive definite. The remaining operators have, in general, self adjoint and skew adjoint parts. The transport equation for the caloric fluid is, to the first order in q and 6 Acrl+B,itC,~+D,lltE,5=0 Here, the new operators A,, B,, Cc, D, and E, take the form
(2.34)
305
Lagrangianformulation of the Maxwell-Cattaneo hydrodynamics (&ij)i
= PO%
(Cc&),= - (1 + 2) [djV(di[j + Jj(i) + ~jKa&Jijl
-$ 0 ai($ f 14 (Eoqj) (EcS)i= - & -k(z)
($f )aj(p&) + [i (1+f$)aj(PO&)
aj(Po&)]
@o-$-+4
(i+)
aj(Potj).
It is convenient to define a new Lagrangian displacement field
which allows to combine (2.33) and (2.34) to give (3.36)
Atf+Bb+D[=O
The operators A, B, D are defined as follows
The evolution equation (3.36) is particularly suitable for stability analysis. 3. COMPATIBILITY WITH KINETIC
OF THE EXTENDED THEORY
THERMO-DYNAMICAL
AND EQUILIBRIUM
PROCESS
THERMODYNAMIC
It has been demonstrated in the previous section that the extension of a thermo-dynamical process introduced in this paper is convenient because it allows an equivalent Lagrangian formulation describing the time evolution of material and caloric points, and because for ~+0 the extension reduces to the standard thermo-dynamical process. The purpose of this section is to demonstrate that the extension is also physically meaningful in the sense that (i) the extended
306
M.GRMELAandJ.TEICHMANN
theory can be regarded as a reduction of a kinetic equation, reduction that keeps more information than the reduction resulting in (l.l), (1.2) (1.3), (ii) the extended theory is compatible with equilibrium thermodynamics. We consider first the relation of extended theory to a kinetic equation. A reduction of a kinetic theory consists in extracting the qualitative information about solutions of the kinetic equation that is relevant to the macroscopic measurements. The qualitative information that is relevant to thermo-dynamical process is the long time behaviour of solutions to the kinetic equation[l3,14]. Indeed, since mass, energy and momentum are conserved in particle interactions the fields p(x, t), e(x, t) and v(x, t) are intuitively expected to possess the information about the long time behaviour one particle distribution function. This intuitive insight can be supported by rigorous arguments [ 13, 141. Let us suppose now that we want to keep more information about the long time behaviour of solutions to the kinetic equation than those that can be included in the five fields p, e, v. What additional fields shall we consider. Since a rigorous study of solutions to kinetic equations is very difficult, it is also very difficult to answer the above question. It has been suggested to consider higher moments of one particle distribution function[151, (p, e, v are the moments of the lowest order). It has not been shown however that higher is the order of the moment less important is the moment for the long time behaviour. Our extended eqns (1. I), (1.4), (I .3) can be clearly regarded as arising from thirteen moment approximation of a kinetic eqn [IS]-where the equations determining the time evolution of II, have been approximately solved to give (I .3), the equation determining the time evolution of the moment q has been kept and an appropriate closure with respect to higher moments has been chosen. We are not in position to argue in which sense this approximation of a kinetic equation is better (i.e. providing more faithful information about the long time behaviour of one particle distribution function) than the approximation that uses only the five fields p, e, v. We shall not therefore proceed in this discussion. Our attention will be focused on the compatibility of the extended theory with more macroscopic theory namely with equilibrium thermodynamics. 3.1 Compatibility of a dynamical theory with equilibrium thermo-dynamics. Entropy principle Detailed comparison of the Gibbs method [ 161 of deriving equilibrium thermodynamics from Boltzmann’s method of deriving equilibrium therthe classical Hamiltonian dynamics, modynamics from Boltzmann’s kinetic equation[l7] and studies of the compatibility of continuum theories with equilibrium thermodynamics [18-201 revealed a common structure [l, 21. Our fundamental assumption is that also other physically meaningful dynamical theories must be compatible with equilibrium thermodynamics in the sense of the abstract formulation that was extracted from classical mechanics, kinetic theory and continuum theories. The genera1 formulation has been already published and applied [l, 21. In order to make this paper selfcontained we shall review the general formulation, illustrate it on the example of the thermodynamical process and then apply it in the context of the extended thermo-dynamical process. Let the time evolution equation for f (f stands for state variables) take the form
$=W&f)
(3.1)
where %! is an operator and 11 denotes indetermined quantities through which individuality of systems are expressed. In order to analyze the compatibility of (3.1) with equilibrium thermodynamics, one has to recognize first the thermodynamic equilibrium states (i.e. the states considered in equilibrium thermodynamics) among all possible solutions of (3.1). This problem can be solved directly by finding all solutions to (3.1) that are approached as f--)30 for a class of boundary conditions (expressed in terms off) that is used in thermodynamic measurements. Only the specification of such boundary conditions is a very difficult problem by itself. We shall identify the thermodynamic equilibrium states indirectly. Among the state variables f we recognize a-variables (denoted (Y,,. . . , a,) and p-variables (denoted /3,, . . . , Pm). Thus f = (a,, . . 1a,, p,, . . . >Pm).
(3.2)
307
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics
We introduce
further an operator
.I such that
J(a,, . . . , %I, PI,. . . , Pm)= (a,, . . . , a,, - B,, . . . , - Pm) Note that J.1 = identity operator
(3.3)
(i.e. J is an involution).
The thermodynamic equilibrium states are defined as time independent solutions of (3.1) that are invariant with respect to J Equivalently, the thermodynamic equilibrium states are solutions of S!(f) = 0 such that all their /?-variables vanish. In concise form
~={flmLf)=O, where 8 is chosen to denote the equilibrium
(3.3)
Jf=fl,
states. The operator
2 can be split into two parts (3.4)
LWA,f) = B”(A, f) + %-(A, f), where
(3.5)
.%“(A, f) =; (%(A, f) 2 .M(Jj)).
The set of equilibrium
states (3.4) can be equivalently %‘={(fl%+(A,f)=O,
written as
%-(A,f)=O,
(3.6)
Jf=f}.
We say that the time evolution eqn (3.1) obeys the entropy functional H satisfying the four properties
principle
if there exists a
U-Ill
w;
61,
az, si) = - w-l + crlM(f) + a&(f) + u3
v,
(3.7)
where S denotes the nonequilibrium entropy. The functional S is sufficiently regular, -S is convex. The functional M denotes the total mass and the functional E denotes the total energy. The functionals S, M, E are assumed to satisfy
W-V-I= M(f)
(3.8)
E(Jf) = E(f). Since the system considered
is assumed to be isolated from its surroundings
we have further
y!$o, $=o. The last quantity V appearing system under consideration.
in eqn (3.7) denotes
the volume of the region occupied
(3.9) by the
0-U
dH
The equality in eqn (3.10) holds if and only if fE 8+, where
(3.10)
308
M. GRMELA and J. TEICHMANN 8, ={flB+(f)
= 0)
(3.11)
is the set that we shall call the set of dissipative equilibrium states. By comparing (3.11) and (3.6) we see that g C g, where g = {f/f E 8+, Jf = f}. (H3) The functional H reaches its minimum for equilibrium states. Thus, the set of equilibrium states may be characterized also by
iX=(f&o)
9
(3.12)
where the (SlSf) denotes the functional derivative. By using (3.7) we can define the equilibrium states as the states for which the nonequilibrium entropy reaches maximum under the constraint that M(fl and E(f) are kept constant. The parameter cr3is chosen in such a way that the minimal value of H, i.e. H evaluated at equilibrium states, vanishes
H
I
=0
(3.13)
r
(H4) The linear operators A, P+ and P- have the following properties: API is formally self adjoint and dissipative (3.14)
AP- is formally skew adjoint.
The operator A, P’ and Pm are introduced as follows. Let f0 represent a thermodynamic equilibrium state that is assumed to be independant of the position coordinates. We expand %(A, f) and H(f) around fO, Let f = f0 + cp.In the Taylor expansion of the functional H(f) the first and second terms vanish due to (3.13) and (3.17) respectively. The third term, quadratic in cp,is
(a APL
(3.15)
where A is a linear, formally self adjoint and positive definite operator (since H reaches its minimum at cp= 0). By writing the quadratic term in the form (3.15) we introduce the linear operator A as well as the inner product (. , .). Now we develop %?(A,f) around jO. Clearly S(A, fO)= 0 due to eqn (3.6). We thus have %(A, f) = Pq + O((P2),
(3.16)
PC+3 = P+cp+ P-q,
P+p resp. P-q are linear parts of %+(A,fl resp. %(A, f). The properties of self adjointness and skew adjointness remain formal since we avoid rather technical discussion of domains of definition of the linear operators P’, Pm and A. A linear operator B is called dissipative if (cp, BP) I 0 for all cpfor which (rp, &) is defined. An important consequence of the properties (Hl)-(H4) is that the functional H determines the thermodynamic equation of state (3.13) implied by the time evolution eqn (3.1). With help of (3.7) one can write eqn (3.13) as c3
=
g33(g,,
g22).
By differentiating (3.17) and by using (3.12) and (3.13) we obtain
(3.17)
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics
309
(3.18) *-
au2 - - + Hfo).
By comparison
of (3.18) with thermodynamic
relations,
we have
1 a3 =
a?=-, rth
where Tth, P,,, and PCh are thermodynamic potential respectively.
equilibrium
Pth r,h
temperature,
(3.19)
pressure
and chemical
3.2 Illustration of the general formulation on the example of the thermo-dynamical process The genera1 formulation of the preceding Section [ 1,2] is now applied to classical thermodynamical process (see Section I). The time evolution eqn (3.1) is now (1.1) the state variables (p, e) are the a-variables, u is the p-variable. The indetermined quantities are p, T, v, K, A, they are assumed to be local functionals of p and e. We require that the time evolution eqns (l.l)-(1.3) are compatible with equilibrium thermodynamics in the sense of Section 3.1. As a consequence we obtain restrictions on the freedom of choice of the indetermined quantities. It follows from the definition of%+(A, f) in (3.4) and (3.5) that
e, v) = -
K(p,
(3.20)
and 0
v(djVi+
aiVj)diVj
+
K(a,v,)* - aj
%+(p,e, v) = f i The functionals
aj(
M, E, S introduced
E(t) =
V( aiVj
+
djVi))
+
di(Ka,ul).
(3.21)
in (3.7) are now
d 3xp(x, t) i v*(x, t) + e(x, t) (3.22) S(t) = j- d 3x p(x, t)s(x, t)
where s is an unspecified local functional of p, e. The property (H2) will be considered as follows. We note that (dS/dt) = (dS/dt)l+ + (dS/dt)J-. Here (dS/dt)(+ denotes the change in time of S if only %!’ governs the time evolution, similarly (dS/dt)[_ denotes the change of S in time if only K govern the time evolution, i.e.
M.GRMELA andJ.'TEICHMANN
(3.23) dS = d 3X (s, Z+ I I t
5 V( 8iOj + ajO;)( 3iUj + ajs)
s,K(c?,v/)’ f Jj(S,)Adj
(3.24)
By s, and s,, we denote as/Je and &lap, respectively. In order to satisfy the condition (H2) we shall investigate first conditions that will guarantee (dS/dt) I+z 0 and conditions that make (dS[di) (_= 0 for all states. It follows from (3.24) that if
v>o,
s,=+o
h>o,
K>o,
(3.25)
then indeed (dS/dr) /+ 3 0, the equality holds if and only if l?iVj
=
0
(3.26) a;+=0 The eqn (3.26) determines the set %+of dissipative equilibrium states. The set S?+is thus defined by ai (i/T) ==O. One can easily verify that solutions of (3.26) are the only solutions to %‘+(p,e, v) = 0. The relation (3.25) is the first compatibility condition. We proceed now to consider (dS(dt) (_. We shall find conditions under which (dS/dt) I_= 0. The eqns (3.1), (3.20) imply then another conservation equation
I -a,$.
d(ps)= at
-
(3.27)
Here 6 is an unspecified local vector valued functional of p, e and v. It follows from (3.27) that (3.28)
By comparing (3.28) and (3.20) one obtains (3.29)
The integrability conditions !$,’ = sitar e.t.c. imply (3.30)
s,p = -p$
Relation (3.30) is the second compatibility condition. Let us note that the integrability conditions imply s,p + pzsP= const. We put the constant equal to zero in order to obtain $ = pSUi* The total entropy flux g is thus (see eqn 3.24)
Si= pSV;t A r
1
1
ai 7,
(3.31)
It is easy to verify that the compatibility conditions (3.25) and (3.30) guarantee also that the
Lagrangianformulationof the Maxwell-Cattaneohydrodynamics
311
remaining properties (H3) and (H4) are satisfied. Let us note that we have considered the property (H2) in a stronger form i.e. (dH/dt) /+ 2 0, (dH/dt) I_ = 0. These two relations imply clearly (dH/dt)l 5 0, the converse is not true. The compatibility conditions (3.20) and (3.30) imply that p and T introduced in constitutive relations (1.2) and (1.3) and s, introduced in (3.22) are not independent. It follows from the thermodynamic equation of state implied by (l.l)-(1.3) and from (3.25) and (3.30) that s depends on e(x, t) and (p(x, tj)-’ in the same way as the equilibrium entropy depends on thermodynamic energy and the thermodynamic volume. Thus p and T can be interpreted as the local pressure and the local temperature. Discussing the compatibility between the NavierStokes-Fourier hydrodynamics and the equilibrium thermodynamics we have thus obtained the local equilibrium viewpoint as a set of compatibility conditions. In many presentations of hydrodynamics[l6] the local equilibrium view-point is accepted a priori as a starting assumption. This method cannot be applied in the context of generalized hydrodynamic theories. The additional state variables introduced in more general fluid theory do not have any analog in equilibrium thermodynamics and therefore it is not clear how the entropy functional and constitutive relations should depend on the state variables. 3.3 Comments to other formulations of the entropy principle in Continuum theories Various formulations of compatibility of the continuum theories with equilibrium thermodynamics have been developed in the past (see e.g. Refs. [3, 18, 19, 201. The method developed in Ref. [I91 has been used to discuss the compatibility of the Maxwell-Cattaneo hydrodynamics with equilibrium thermodynamic[21]. As far as we know, no detailed relationship between the various formulations have been established. Let us mention briefly only obvious relations. In the section 3.2 we have shown that in a particular setting of classical thermodynamical process our formulation leads to the local equilibrium viewpoint. In this particular setting we obtain results compatible with Ref. [18]. Coleman and Noll[l8] in their formulation of entropy principle postulate a particular relation between the entropy flux !? and the heat flux q. Miiller[l9] considers s as a quantity for which a constitutive relation is required. Miiller has also observed that the postulate of Coleman and Noll does not generalize to more microscopic levels such as kinetic theory. Our formulation of the entropy principle agrees at these points with Miiller. However, our formulation contains a structure, namely the distinction between CY and 0 variables and a definition of an equilibrium state using the properties (H3) and (H4), that are absent in Miiller’s analysis. The additional structure helps to single out the entropy functional. Principal advantage of our formulation is its applicability to a whole hierarchy of levels that include kinetic theory and classical mechanics. 3.4 Compatibility of the extended thermo-dynamical process with equilibrium thermodynamics We shall follow closely section 3.2. The state variables of the extended theory are: p(x, t), e(x, t), v(x, t), q(x, t), the state variables p, e are u-variables, v, q are /%variables. We shall replace the functional S introduced in (3.22) by S(t) =
I
d ‘xp(x, t)
(
s(x, t) + i a(x, t)q; (x, t) qi (x, t , 1
(3.32)
where s, a remain indetermined local functionals of p and e. We shall prove now that the extended thermo-dynamical process (l.l), (1.4), (1.3) is compatible with equilibrium thermodynamics provided the following compatibility conditions are satisfied
0,
;>o
=f>O_ ps,
= -p?s,,
K
s,
>o,
Y >
a
;=P
and the terms of the third and higher order in v and q are neglected.
(3.33)
312
M. GRMELA and J. TEICHMANN
First we consider (dS/dt) I+. We find easily that SCk
V (JiUj
+
diV;)(
divj
+
8jVi)
+
S,K(
d/U!)2
+ 5
qiqi
.
(3.34)
Thus (dS/dt) /+3 0 provided (3.33) is satisfied. Now we prove that (dS/dt) )_ = 0. By following the arguments that led to (3.27), (3.28), (3.29) we obtain (the terms of the third and higher order in v and q are neglected)
0
L!?i, = pS,Ui +4 f
$k,= (P(P)p+
q1
(3.35)
c
SePlsij
One verifies easily that the integrability conditions implied by (3.35) are satisfied provided (3.33) is satisfied. The total entropy flux fi equals
f3i= PSUi
+
pS,lJi
(3.36)
Also the other properties (H2)-(H4) are satisfied provided (3.33) is satisfied. The proof (a direct verification) is omitted. 4. CONCLUSION
We have shown that the Maxwell-Cattaneo hydrodynamics is compatible with equilibrium thermodynamics and allows equivalent Lagrangian formulation in which the evolution of fluid can be interpreted as the time evolution of a material and caloric points. All conceptual and practical advantages associated with the Lagrangian description of a dynamical process can be applied now to processes that include a nontrivial evolution of the internal energy. In the study of the compatibility of the Maxwell-Cattaneo hydrodynamics with equilibrium thermodynamics we used the general formulation (reviewed in Section 3.1 and developed in Refs. [I, 21 that unifies the Gibbs prescription for the transition from the Hamiltonian mechanism to equilibrium thermodynamics, the Boltzmann’s H-theorem that relates the Boltzmann kinetic equation with equilibrium thermodynamics and the assumption of local equilibrium that relates the classical Navier-Stokes-Fourier hydrodynamics with equilibrium thermodynamics. REFERENCES [I] M. GRMELA, Lecture Notes in Mathematics In Global Analysis, (Edited by M. Grmela and J. E. Marsden), Vol. 755 (1979). [2] M. GRMELA and L. S. CARCIA-COLIN, Phys. Reo. A, Part I, 22, 1295 (1980). Part II, 22, 1304 (1980). [3] A. C. ERINGEN, In ConrinuumPhysics, (Edited by A. C. Eringen). Vol. 2, Academic Press, New York (1975). [4] V. 1. ARNOLD, Ann. Inst. Fourier XVI, 319 (1966); V. I. ARNOLD, Les Mtf’thodesMathimatiques de la Mticanique Classique, Appendice 2, Edition Mir, Moscou (1976); D. G. EBIN and J. MARDEN, Annals of Math. 92, 102 (1970). [S] A. S. LODGE, Body Tensor Fields in ContinuumMechanics. Academic Press, New York (1974). [6] E. M. BARSTON, J. Math. Phys. 8, 523, 1886, (1%7); 9, 2069 (1%8); 12, 1116 (1971). [7] J. TEICHMANN, Con. .l. Phys. 59.82 (1981); 60,640 (1982). [8] A. S. MONIN and A. M. YAGLOM, Statistical Fluid Mechanics. MIT Press, Cambridge, Massachusetts (1973). [9] J. C. MAXWELL, Phil. Trans. Roy. Sot. 157, 49 (1867). [IO] C. CATTANEO, Camp. Rend. Acad. Sci. Paris 247,431 (1958); 247, 3154 (1958). [II] S. G. BRUSH, Kinetic Theory, Vols. l-3. Pergamon Press, Oxford (1%6). [12] J. LIOUVILLE, J. de Math. Pures et Appliquies. 3, 342 (1838). [13] M. GRMELA, R. ROSEN and L. S. GARCIA-COLIN, I. Chem. Phys. 75.5474 (1981). [14] J. A. McLENNAN, Phys. Fluids 8, 1580 (1%5). [15] H. Grad, Hondbuch der Physik, Vol. XII. Springer-Verlag,Berlin (1958). [l6] W. GIBBS, Elementary Principles of Statistical Mechanics (Dover, New York l%O), 1st Edn. Yale University Press, New Haven (1902). [17] L. BOLTZMANN, In Wissenschofiichen Abhondlungen uon Ludwig Boltzmnnn, Vol. 2, Chelsea New York (1968).
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics
313
[18] S. R.$e GROOT and P. MAZUR, Non-Equilibrium Thermodynamics, McGraw-Hill, New York (1969). [19] I. MULLER, Z. fiir Physik 198, 329 (1%7), In Foundations of Continuum Thermodynamics, (Edited by J. J. J. Domingos, M. N. R. Mina and J. H. Whitelaw). Wiley, New York (1973). [ZO]C. TRUESDELL, Rational Thermodynamics. McGraw-Hill, New York (1%9). 1211G. LEBON, D. JOU and J. CASAS-VAZQUEZ, J. Phys. A. 13,275 (1980). (Received 1 July 1982)
ES Vol.
21,
No. 4-B
ht. J. Engng Sri. Vol. 21. No. 4. pp 297-313. 1983. Printed in Great Britain.
LAGRANGIAN FORMULATION OF THE MAXWELL-CATTAN~O HYDRODYNAMICS MIROSLAV GRMELA Department of Chemical En~neeri~, Ecole Polytechnique, Montrkal, Qutbec, Canada and Centre de Recherches de Math~matiques Appliqutes, Universite de Monlrial, Montr~ai, Qutbec, Canada and JXfii TEICHMANN D~partement de Physique, Universit6 de Montr6a1,Montrkal, Quebec, Canada Abstract-The Maxwell-Cattaneo modification of the Fourier constitutive relation allows to formulate hydrodynamics as dynamics of a material and a caloric point. The Maxwell-Cattaneo hydrodynamics is compatible with equilibrium thermodynamics, it represents an approximation of a kinetic equation, and it reduces to the standard hydrodynamics if the new transport coefficient (a rela~tion time) tends to zero. I. INTRODUCTION
A THERMO-DYNAMICAL process in a one component fluid is defined by five fields p(x, t), v(x, t) and e(x, t) denoting respectively the mass density, the vetocity and the internal energy. The time evolution of the five fields is governed by a set of the time evolution equations
lb
-
1
at
= VjdjVi+ f
ajHij
and a set of constitutive relations for fi and 4. We shall consider the NavierStokes-Fourier constitutive relations 4; = f
0
Aa,f
H, = PSij - V(a;Vj + 8,s) - Kd~V~S;~
(1.2) (1.3)
The scalars p, T, A, v and K remain indetermined functionals of p(x, t) and e(x, t), q represents the heat flux, fi is the pressure tensor. In eqns (l.l)-(1.3) we have used the notation ai = (a/ax,), x denotes the position vector, t is the time. We have used also the summation convection. The thermo-dynami~a1 process is called physically admissible (also compatible with equilibrium thermodynamics)[l, 21 if there exists a functional s(x, t) of p(x, t) and e(x, t) satisfying four properties (Hl)-(H4) (see Refs. I, 2 and Section 3). We shall show in Section 3 that s(x, t) satisfying (HI)-(H4) has the physical meaning of the non-equilibrium entropy. The time evolution eqns (1.1)-(1.3) represent a physically admissible thermo-dynamical process provided a compatibility condition, restricting the choice of s, p, T, A, v and K, is satisfied. This statement is proved in Section 3.2. The thermo-dynamical process (l.l)-( 1.3) can be reformulated in the Lagrangian form as a process defined by one deformation field x = X(%, t) and one scalar field[3]. In the particular case when one of the scalar fields e, Z’,s is considered invariant during the time evolution, the thermo-dynamical process will be called a dynamical process. Its time evoIution equations, in the Lagrangian formulation are equivalent to Newton’s equations governing the motion of a material point. This view of the time evolution has several important ES Vol.21,No.
4-A
297
298
advantages.
M. GRMELA and J. TEICHMANN
We mention here:
(i) A possibility of using a geometric point of view of the time evolution[41. (ii) A possibility of adopting concepts and results of theory of elasticity [3,5]. (iii) A possibility of using a well developed theory of linear Lagrangian stability [6,7]. (iv) The Lagrangian formulation is particularly well suited for the study of turbulence [8]. The presence of a scalar field, in addition to the material deformation field, in the Lagrangian description of the thermo-dynamical processes prevents the full use of the above advantages. Maxwell[9] and Cattaneo[lO] have suggested the following extension of the thermodynamical process (l.l)-(1.3): The extended thermo-dynamical process in a one component fluid is defined by eight fields p(x, t), e(x, t), v(x, t), q(x, t) denoting respectively the mass density, the internal energy, the velocity and the heat flux. The time evolution of the eight fields is governed by a set of eight time evolution equations. The first five time evolution equations are the same as (1.1). The remaining three time evolution equations are
-
$$=vjajqi + $
[ ai
(f) -t
Pqi].
(1.4)
The time evolution eqns (1.4) replace the Fourier constitutive relation (1.2). The Navier-Stokes constitutive relation (1.3) remains unchanged. The scalars p, T, A, V, K, 7 are the indetermined quantities entering the extended thermo-dynamical process. The aim of this paper is to show that the extended thermodynamical process satisfies the following properties: (i) The extended thermo-dynamical process allows a Lagrangian formulation in terms of two deformation fields, namely the material deformation field and another deformation field that we shall call a caloric deformation field. In other words, the equations governing the time evolution of the extended thermo-dynamical process become equivalent to equations governing the time evolution of a material and a caloric point. (ii) If ~+0 the thermo-dynamical process (l.l)-(1.3) is recovered. (iii) The extended thermo-dynamical process is physically admissible (i.e.) compatible with equilibrium thermodynamics-see Section 3). (iv) The extended thermo-dynamical process represents an intermediate dynamical theory of fluids, intermediate between the theory represented by eqns (l.l)-(1.3) and the kinetic theory. This point is briefly discussed in Section 3 but remains basically open. We also do not discuss situations and experiments relevant to the new information contained in the extended theory. It is interesting to note that the Lagrangian formulation of the Maxwell-Cattaneo extension of thermo-dynamical process has common features with caloric theoryt developed in 18th century[ll]. The extended thermo-dynamical process in a one component fluid can be regarded as a dynamical process in a two component fluid. We shall call the second component a caloric component. For the caloric component the total energy takes the role of the mass, its time evolution cannot be however obtained from Newton’s law. 2. LAGRANGIAN FORMULATION THERMO-DYNAMICAL
OF THE EXTENDED PROCESS
The time evolution equations of the-extended thermo-dynamical process (l.l), (1.4), (1.3) will be brought into a form suitable for the Lagrangian formulation. We introduce new variables P=P
E =i
wj
=
Vj +
PVjUj
+
pe
’E njkVk +i pqj.
tWe thank W. G. Laidlaw for pointing to one of us (M. G.) the historial references.
(2.1)
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics and
299
rewrite (1.1) (1.4) into
(2.2) -
dV
L
= VjajVi + i
at
djllij
and (2.3) aw. -2=VjdjWi+&di
-
at
-UiUjaj
+i
-
-
vi [
+1
(k)
$
p(
l+P
aj(puj) + g aj(Ewj) -
vj
E
a, VjajVl
$ aj(Ew,)]
ujdj uk +fa[ nk, 1
P&
li. vk
V;
ajnij
(;PikUk)
($Y)-Vjaj
kzPik#kaj (EWj)+ $
-2
E)
+ f
djnij
H
aj
vi - f aj nij
>I
>I .
We have used II, = PSij+ Pin The quantity E denotes the total energy. The first eqn in (2.3) represents the conservation of the total energy. We note that if e and q are known functionals of p and v then (2.3) is just another form of (2.2). The eqns (2.2) and (2.3) are written in the Eulerian form. Before we rewrite (2.2) and (2.3) into the Lagrangian form we review the Lagrangian formulation in the context of (2.2) (E and p entering (2.2) are assumed to be known functions of p). Let us assume that (2.2) are solved and thus the fields p(x, t) and v(x, t) are known. It follows from the physical meaning of p and v that trajectories of material points are obtained as solutions to the system of ordinary differential equations dx s = v(x, t)
(2.4)
By X(Q, t) we denote the solution of (2.4) that satisfies X(x0, to) = ~0. Thus
awx,,t) at
I= ,=t
v(x, t)
(2.5)
The first eqn in (2.2) is the Liouville [ lo] (also called continuity) equation corresponding to (2.4). Because the material points cannot be created or annihilated during the time evolution we have
dxo, to)d Vo= Ax, t) d v,,
(2.6)
300
M.GRMELAandJ.TElCHMANN
where d V, is an element of the volume at t = to, dV, is its image at t = t under the transformation X. Since d V, = IMI,, d V,, where M,,
=
f,
aXi(XO3 t,
I
dXOj
12.7)
x0
and
/Ml,,,= det (Mfj) we
have
Pk t) = PolMl$
(2.8)
p. denotes p(~, to). The impenetrability of matter implies jM/ $: 0 for all x and t. Thus we see that by knowing the trajectory mapping X (also called a material deformation field) we obtain p and v from (2.5) and (2.8). Conversely, by knowing the fields p, v one needs to solve (2.4) to obtain the mapping X. Let us now seek the mappings X such that p and v obtained (according to (2.5), (2.8)) from X satisfy (2.2). First we note that if p = 0, Y = 0 and K = 0 then
a’x(x(),f) at2
o
(2.9)
=
is the equation we look for. Indeed, we have from (2.5) a’Xi(Q, f) =-L&= d V.(X at2
f)
~jQi$avi(X, t) at (
The eqn (2.9) is just Newton’s equation for a free material point. If p+O then the equation determining the time evolution of X will have to include derivatives with respect to x0 and will become therefore a partial differential equation. We see easily that a... &d,lzM_,
T&- =
dXi 8x0,
Jr
a..._ m,1 LXj*X,9*- -1, axoj
(2.10)
where i, j, k means an even permutation of the indices (1, 2,3) and
1% axol [X, Xk,U] = det
ax,
ax,
axo2
axo3
ax, ax, ax, ax”1
ah2
aa
aa
1 \axol (we use the notation of Ref. 8). If p+ 0, Y= 0,
K =
axo2
ax,, aa
ax03
(2.11)
0 then (2.9) becomes
a2Xitxo,6 PO at2 = - [Xj, Xk, PI.
(2.12)
We can regard (2.12) also as Newton’s equation, the force is now however unusual. This reflects the fact that surface forces were introduced in the derivation of the second equation of (2.2) (the second equation of (2.2) is a local form of Newton’s law for a fluid confined in a finite
Lagrangian formulation
of the Maxwell-Cattaneo
hydrodynamics
301
volume). In the general case, p+ 0, v+ 0, K$ 0 thus we obtain
J2Xi(X”, 0=
pn
a2
-
[Xjf
xk,
PI
+ [X2,
x3t
(iho
v
+[x*,x3,(~~[x*,x3,~])] +[X3,X,,(~v[X3A~])] +[x,x*, (&v [x,Ay)]
(2.13)
We shall proceed now to the Lagrangian formulation of (2.3). We consider (2.3) as the time Similarly as in the study of the evolution equations of the second (caloric) component. Lagrangian formulation of (2.2) we introduce. X = Y(yn, r)
(2.14)
g = w(x, t)
(2.15)
yn = Y(Yn, GJ.
(2.16)
as solutions of
satisfying
the initial condition
The points x0 and y. are not independent;
we need to satisfy
X(x0, t) = VYO, 0
(2.17)
for all t since both (2.2) and (2.3) involve the same x and t. Note that (2.17) implies ~0 = y. at t = to, but for t+ to the points xg and y. are in general different. From (2.14) and (2.15) we have
t)
WYO,
at
,=,
= w(x, t).
(2.18)
We introduce c,,
= 1,
dyi(YO*
t,
and we assume ICI = det (C) =I 0 for all y. and t. The same arguments .Q.
(2.19)
aYOj
t) = E”lCl,:,
that led to (2.8) imply now (2.20)
302
M.GRMELAandJ.TEICHMANN
where En stands for E(y,, to). By using a ~=!$a--=~+=I[yi, ih; axi aYOj
yk,...) ] JYni
I%,,
we can rewrite the second eqn of (2.3) as differential equation simplicity we shall write the resulting equation only for
(2.21)
for Y(y,,, t). For the sake of
dW,
-=-~aiif)+~[~,-(l+~)ci] at
1+ ~
-f (
d;p - aj
(va;vj
+
VajU,)
(2.22)
I[
-
&(Kh)l.
that is obtained from the second eqn in (2.3) after neglecting the terms of the second and higher order in v and w. The Lagrangian form of (2.22) reads
-‘a(&ip~clyo)[Yjt Pn ICI,,
yk,P]
(2.23)
The main conceptual advantage of the Lagrangian formulation of fluid mechanics is that solutions to the Lagrangian equations represent trajectories. The solutions thus include a global information about the flow of a fluid. Such information is particularly pertinent in the study of secondary and turbulent flows[S]. The systems of eqns (2.13), (2.23) extends the Lagrangian formulation of a dynamical process to thermo-dynamical processes. The mathematical complexity of the Lagrangian equations (even in the case of dynamical processes) has not allowed so far to transform the conceptual advantages into practical advantages (some attempts are discussed in Ref. 8). A notable exception is the study of the Lagrangian linear stability of
303
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics
stationary plasma configurations that arises in the study of plasma confinement ]6,7]. The system (2.13), (2.23) allows now to include thermal processes while keeping the same mathematical approach and using the same general mathematical results as in Refs. [6,7]. For the purpose of the stability analysis, we shall now write the linearized system of evolution equations. We shall still consider only neutral fluids. We reformulate the Lagrangian form of evolution equations in case of small perturbations around the equilibrium state. Again, we consider the systems (2.2) and (2.3) separately. A small displacement of the fluid element from its equilibrium position is defined by a displacement field 5, (2.24)
x = x0 + &(x0,t) The displacement gradient matrix takes now a simplified form Mij = 6, +
& [i 01
and the corresponding Jacobian is, to first order in &. (2.25) where the gradient operator is taken at x = Q. Any function of x can be expanded in the vicinity of x0 in Taylor series in f either in form of expansion in Eulerian formulation with the first order term in 5 equal to
Sf(xo+ 5) or in form of expansion in Lagrangian formulation with the first order term in 5 of the form
Af (x0). Equating both quantities we obtain the relation between Eulerian and Lagrangian first order displacements
Af = sf + & &
(2.26)
f.
Let us now return to the system (2.2). Expressing the integral of the continuity equation in (2.2) with help of (2.8) we obtain the Eulerian displacement for density (2.27) The Eulerian displacements for velocity and pressure p = p(E, p) are similarly
6u, =
ii +
OOj
&01
&
-
5j
& vOi I
(2.28) where v. is the unperturbed velocity of stationary equilibrium velocity and 4 = (a&,, flat). The second system of evolution equations for the caloric fluid (2.5) is treated in a similar way. Here we introduce the displacement field 9 Y=%+ho,th
(2.29)
304
M.GRMELAandJ.TEICHMANN
The displacement gradient matrix is now
(2.30)
lcI=lt-&%.
It follows immediately from the system (2.5) that the Eulerian displacement for the total energy is
ljE=
-T&
CE077i)
01
(2.31)
and the velocity (2.32) The quantity w. means the unperturbed equilibrium velocity of the caloric fluid. In general, the velocities v. and w. do not coincide. To obtain the Euler-Lagrangian evolution equation to first order in 5 and q for the momentum transport we linearize eqn (1. l), introducing
P=Po+SP
with the Eulerian displacements defined by (2.27) and (2.28). To simplify the following discussion, we assume that we linearize around an equilibrium state, i.e. v. = w. = 0. The first order equation may be cast in the form A,~tB,j+C,~tD&E,~=O
(2.33)
The operators AM, BM, CM, DM and EM are defined as follows (AM&i=
PO&
The operator AM is self adjoint and positive definite. The remaining operators have, in general, self adjoint and skew adjoint parts. The transport equation for the caloric fluid is, to the first order in q and 6 Acrl+B,itC,~+D,lltE,5=0 Here, the new operators A,, B,, Cc, D, and E, take the form
(2.34)
305
Lagrangianformulation of the Maxwell-Cattaneo hydrodynamics (&ij)i
= PO%
(Cc&),= - (1 + 2) [djV(di[j + Jj(i) + ~jKa&Jijl
-$ 0 ai($ f 14 (Eoqj) (EcS)i= - & -k(z)
($f )aj(p&) + [i (1+f$)aj(PO&)
aj(Po&)]
@o-$-+4
(i+)
aj(Potj).
It is convenient to define a new Lagrangian displacement field
which allows to combine (2.33) and (2.34) to give (3.36)
Atf+Bb+D[=O
The operators A, B, D are defined as follows
The evolution equation (3.36) is particularly suitable for stability analysis. 3. COMPATIBILITY WITH KINETIC
OF THE EXTENDED THEORY
THERMO-DYNAMICAL
AND EQUILIBRIUM
PROCESS
THERMODYNAMIC
It has been demonstrated in the previous section that the extension of a thermo-dynamical process introduced in this paper is convenient because it allows an equivalent Lagrangian formulation describing the time evolution of material and caloric points, and because for ~+0 the extension reduces to the standard thermo-dynamical process. The purpose of this section is to demonstrate that the extension is also physically meaningful in the sense that (i) the extended
306
M.GRMELAandJ.TEICHMANN
theory can be regarded as a reduction of a kinetic equation, reduction that keeps more information than the reduction resulting in (l.l), (1.2) (1.3), (ii) the extended theory is compatible with equilibrium thermodynamics. We consider first the relation of extended theory to a kinetic equation. A reduction of a kinetic theory consists in extracting the qualitative information about solutions of the kinetic equation that is relevant to the macroscopic measurements. The qualitative information that is relevant to thermo-dynamical process is the long time behaviour of solutions to the kinetic equation[l3,14]. Indeed, since mass, energy and momentum are conserved in particle interactions the fields p(x, t), e(x, t) and v(x, t) are intuitively expected to possess the information about the long time behaviour one particle distribution function. This intuitive insight can be supported by rigorous arguments [ 13, 141. Let us suppose now that we want to keep more information about the long time behaviour of solutions to the kinetic equation than those that can be included in the five fields p, e, v. What additional fields shall we consider. Since a rigorous study of solutions to kinetic equations is very difficult, it is also very difficult to answer the above question. It has been suggested to consider higher moments of one particle distribution function[151, (p, e, v are the moments of the lowest order). It has not been shown however that higher is the order of the moment less important is the moment for the long time behaviour. Our extended eqns (1. I), (1.4), (I .3) can be clearly regarded as arising from thirteen moment approximation of a kinetic eqn [IS]-where the equations determining the time evolution of II, have been approximately solved to give (I .3), the equation determining the time evolution of the moment q has been kept and an appropriate closure with respect to higher moments has been chosen. We are not in position to argue in which sense this approximation of a kinetic equation is better (i.e. providing more faithful information about the long time behaviour of one particle distribution function) than the approximation that uses only the five fields p, e, v. We shall not therefore proceed in this discussion. Our attention will be focused on the compatibility of the extended theory with more macroscopic theory namely with equilibrium thermodynamics. 3.1 Compatibility of a dynamical theory with equilibrium thermo-dynamics. Entropy principle Detailed comparison of the Gibbs method [ 161 of deriving equilibrium thermodynamics from Boltzmann’s method of deriving equilibrium therthe classical Hamiltonian dynamics, modynamics from Boltzmann’s kinetic equation[l7] and studies of the compatibility of continuum theories with equilibrium thermodynamics [18-201 revealed a common structure [l, 21. Our fundamental assumption is that also other physically meaningful dynamical theories must be compatible with equilibrium thermodynamics in the sense of the abstract formulation that was extracted from classical mechanics, kinetic theory and continuum theories. The genera1 formulation has been already published and applied [l, 21. In order to make this paper selfcontained we shall review the general formulation, illustrate it on the example of the thermodynamical process and then apply it in the context of the extended thermo-dynamical process. Let the time evolution equation for f (f stands for state variables) take the form
$=W&f)
(3.1)
where %! is an operator and 11 denotes indetermined quantities through which individuality of systems are expressed. In order to analyze the compatibility of (3.1) with equilibrium thermodynamics, one has to recognize first the thermodynamic equilibrium states (i.e. the states considered in equilibrium thermodynamics) among all possible solutions of (3.1). This problem can be solved directly by finding all solutions to (3.1) that are approached as f--)30 for a class of boundary conditions (expressed in terms off) that is used in thermodynamic measurements. Only the specification of such boundary conditions is a very difficult problem by itself. We shall identify the thermodynamic equilibrium states indirectly. Among the state variables f we recognize a-variables (denoted (Y,,. . . , a,) and p-variables (denoted /3,, . . . , Pm). Thus f = (a,, . . 1a,, p,, . . . >Pm).
(3.2)
307
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics
We introduce
further an operator
.I such that
J(a,, . . . , %I, PI,. . . , Pm)= (a,, . . . , a,, - B,, . . . , - Pm) Note that J.1 = identity operator
(3.3)
(i.e. J is an involution).
The thermodynamic equilibrium states are defined as time independent solutions of (3.1) that are invariant with respect to J Equivalently, the thermodynamic equilibrium states are solutions of S!(f) = 0 such that all their /?-variables vanish. In concise form
~={flmLf)=O, where 8 is chosen to denote the equilibrium
(3.3)
Jf=fl,
states. The operator
2 can be split into two parts (3.4)
LWA,f) = B”(A, f) + %-(A, f), where
(3.5)
.%“(A, f) =; (%(A, f) 2 .M(Jj)).
The set of equilibrium
states (3.4) can be equivalently %‘={(fl%+(A,f)=O,
written as
%-(A,f)=O,
(3.6)
Jf=f}.
We say that the time evolution eqn (3.1) obeys the entropy functional H satisfying the four properties
principle
if there exists a
U-Ill
w;
61,
az, si) = - w-l + crlM(f) + a&(f) + u3
v,
(3.7)
where S denotes the nonequilibrium entropy. The functional S is sufficiently regular, -S is convex. The functional M denotes the total mass and the functional E denotes the total energy. The functionals S, M, E are assumed to satisfy
W-V-I= M(f)
(3.8)
E(Jf) = E(f). Since the system considered
is assumed to be isolated from its surroundings
we have further
y!$o, $=o. The last quantity V appearing system under consideration.
in eqn (3.7) denotes
the volume of the region occupied
(3.9) by the
0-U
dH
The equality in eqn (3.10) holds if and only if fE 8+, where
(3.10)
308
M. GRMELA and J. TEICHMANN 8, ={flB+(f)
= 0)
(3.11)
is the set that we shall call the set of dissipative equilibrium states. By comparing (3.11) and (3.6) we see that g C g, where g = {f/f E 8+, Jf = f}. (H3) The functional H reaches its minimum for equilibrium states. Thus, the set of equilibrium states may be characterized also by
iX=(f&o)
9
(3.12)
where the (SlSf) denotes the functional derivative. By using (3.7) we can define the equilibrium states as the states for which the nonequilibrium entropy reaches maximum under the constraint that M(fl and E(f) are kept constant. The parameter cr3is chosen in such a way that the minimal value of H, i.e. H evaluated at equilibrium states, vanishes
H
I
=0
(3.13)
r
(H4) The linear operators A, P+ and P- have the following properties: API is formally self adjoint and dissipative (3.14)
AP- is formally skew adjoint.
The operator A, P’ and Pm are introduced as follows. Let f0 represent a thermodynamic equilibrium state that is assumed to be independant of the position coordinates. We expand %(A, f) and H(f) around fO, Let f = f0 + cp.In the Taylor expansion of the functional H(f) the first and second terms vanish due to (3.13) and (3.17) respectively. The third term, quadratic in cp,is
(a APL
(3.15)
where A is a linear, formally self adjoint and positive definite operator (since H reaches its minimum at cp= 0). By writing the quadratic term in the form (3.15) we introduce the linear operator A as well as the inner product (. , .). Now we develop %?(A,f) around jO. Clearly S(A, fO)= 0 due to eqn (3.6). We thus have %(A, f) = Pq + O((P2),
(3.16)
PC+3 = P+cp+ P-q,
P+p resp. P-q are linear parts of %+(A,fl resp. %(A, f). The properties of self adjointness and skew adjointness remain formal since we avoid rather technical discussion of domains of definition of the linear operators P’, Pm and A. A linear operator B is called dissipative if (cp, BP) I 0 for all cpfor which (rp, &) is defined. An important consequence of the properties (Hl)-(H4) is that the functional H determines the thermodynamic equation of state (3.13) implied by the time evolution eqn (3.1). With help of (3.7) one can write eqn (3.13) as c3
=
g33(g,,
g22).
By differentiating (3.17) and by using (3.12) and (3.13) we obtain
(3.17)
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics
309
(3.18) *-
au2 - - + Hfo).
By comparison
of (3.18) with thermodynamic
relations,
we have
1 a3 =
a?=-, rth
where Tth, P,,, and PCh are thermodynamic potential respectively.
equilibrium
Pth r,h
temperature,
(3.19)
pressure
and chemical
3.2 Illustration of the general formulation on the example of the thermo-dynamical process The genera1 formulation of the preceding Section [ 1,2] is now applied to classical thermodynamical process (see Section I). The time evolution eqn (3.1) is now (1.1) the state variables (p, e) are the a-variables, u is the p-variable. The indetermined quantities are p, T, v, K, A, they are assumed to be local functionals of p and e. We require that the time evolution eqns (l.l)-(1.3) are compatible with equilibrium thermodynamics in the sense of Section 3.1. As a consequence we obtain restrictions on the freedom of choice of the indetermined quantities. It follows from the definition of%+(A, f) in (3.4) and (3.5) that
e, v) = -
K(p,
(3.20)
and 0
v(djVi+
aiVj)diVj
+
K(a,v,)* - aj
%+(p,e, v) = f i The functionals
aj(
M, E, S introduced
E(t) =
V( aiVj
+
djVi))
+
di(Ka,ul).
(3.21)
in (3.7) are now
d 3xp(x, t) i v*(x, t) + e(x, t) (3.22) S(t) = j- d 3x p(x, t)s(x, t)
where s is an unspecified local functional of p, e. The property (H2) will be considered as follows. We note that (dS/dt) = (dS/dt)l+ + (dS/dt)J-. Here (dS/dt)(+ denotes the change in time of S if only %!’ governs the time evolution, similarly (dS/dt)[_ denotes the change of S in time if only K govern the time evolution, i.e.
M.GRMELA andJ.'TEICHMANN
(3.23) dS = d 3X (s, Z+ I I t
5 V( 8iOj + ajO;)( 3iUj + ajs)
s,K(c?,v/)’ f Jj(S,)Adj
(3.24)
By s, and s,, we denote as/Je and &lap, respectively. In order to satisfy the condition (H2) we shall investigate first conditions that will guarantee (dS/dt) I+z 0 and conditions that make (dS[di) (_= 0 for all states. It follows from (3.24) that if
v>o,
s,=+o
h>o,
K>o,
(3.25)
then indeed (dS/dr) /+ 3 0, the equality holds if and only if l?iVj
=
0
(3.26) a;+=0 The eqn (3.26) determines the set %+of dissipative equilibrium states. The set S?+is thus defined by ai (i/T) ==O. One can easily verify that solutions of (3.26) are the only solutions to %‘+(p,e, v) = 0. The relation (3.25) is the first compatibility condition. We proceed now to consider (dS(dt) (_. We shall find conditions under which (dS/dt) I_= 0. The eqns (3.1), (3.20) imply then another conservation equation
I -a,$.
d(ps)= at
-
(3.27)
Here 6 is an unspecified local vector valued functional of p, e and v. It follows from (3.27) that (3.28)
By comparing (3.28) and (3.20) one obtains (3.29)
The integrability conditions !$,’ = sitar e.t.c. imply (3.30)
s,p = -p$
Relation (3.30) is the second compatibility condition. Let us note that the integrability conditions imply s,p + pzsP= const. We put the constant equal to zero in order to obtain $ = pSUi* The total entropy flux g is thus (see eqn 3.24)
Si= pSV;t A r
1
1
ai 7,
(3.31)
It is easy to verify that the compatibility conditions (3.25) and (3.30) guarantee also that the
Lagrangianformulationof the Maxwell-Cattaneohydrodynamics
311
remaining properties (H3) and (H4) are satisfied. Let us note that we have considered the property (H2) in a stronger form i.e. (dH/dt) /+ 2 0, (dH/dt) I_ = 0. These two relations imply clearly (dH/dt)l 5 0, the converse is not true. The compatibility conditions (3.20) and (3.30) imply that p and T introduced in constitutive relations (1.2) and (1.3) and s, introduced in (3.22) are not independent. It follows from the thermodynamic equation of state implied by (l.l)-(1.3) and from (3.25) and (3.30) that s depends on e(x, t) and (p(x, tj)-’ in the same way as the equilibrium entropy depends on thermodynamic energy and the thermodynamic volume. Thus p and T can be interpreted as the local pressure and the local temperature. Discussing the compatibility between the NavierStokes-Fourier hydrodynamics and the equilibrium thermodynamics we have thus obtained the local equilibrium viewpoint as a set of compatibility conditions. In many presentations of hydrodynamics[l6] the local equilibrium view-point is accepted a priori as a starting assumption. This method cannot be applied in the context of generalized hydrodynamic theories. The additional state variables introduced in more general fluid theory do not have any analog in equilibrium thermodynamics and therefore it is not clear how the entropy functional and constitutive relations should depend on the state variables. 3.3 Comments to other formulations of the entropy principle in Continuum theories Various formulations of compatibility of the continuum theories with equilibrium thermodynamics have been developed in the past (see e.g. Refs. [3, 18, 19, 201. The method developed in Ref. [I91 has been used to discuss the compatibility of the Maxwell-Cattaneo hydrodynamics with equilibrium thermodynamic[21]. As far as we know, no detailed relationship between the various formulations have been established. Let us mention briefly only obvious relations. In the section 3.2 we have shown that in a particular setting of classical thermodynamical process our formulation leads to the local equilibrium viewpoint. In this particular setting we obtain results compatible with Ref. [18]. Coleman and Noll[l8] in their formulation of entropy principle postulate a particular relation between the entropy flux !? and the heat flux q. Miiller[l9] considers s as a quantity for which a constitutive relation is required. Miiller has also observed that the postulate of Coleman and Noll does not generalize to more microscopic levels such as kinetic theory. Our formulation of the entropy principle agrees at these points with Miiller. However, our formulation contains a structure, namely the distinction between CY and 0 variables and a definition of an equilibrium state using the properties (H3) and (H4), that are absent in Miiller’s analysis. The additional structure helps to single out the entropy functional. Principal advantage of our formulation is its applicability to a whole hierarchy of levels that include kinetic theory and classical mechanics. 3.4 Compatibility of the extended thermo-dynamical process with equilibrium thermodynamics We shall follow closely section 3.2. The state variables of the extended theory are: p(x, t), e(x, t), v(x, t), q(x, t), the state variables p, e are u-variables, v, q are /%variables. We shall replace the functional S introduced in (3.22) by S(t) =
I
d ‘xp(x, t)
(
s(x, t) + i a(x, t)q; (x, t) qi (x, t , 1
(3.32)
where s, a remain indetermined local functionals of p and e. We shall prove now that the extended thermo-dynamical process (l.l), (1.4), (1.3) is compatible with equilibrium thermodynamics provided the following compatibility conditions are satisfied
0,
;>o
=f>O_ ps,
= -p?s,,
K
s,
>o,
Y >
a
;=P
and the terms of the third and higher order in v and q are neglected.
(3.33)
312
M. GRMELA and J. TEICHMANN
First we consider (dS/dt) I+. We find easily that SCk
V (JiUj
+
diV;)(
divj
+
8jVi)
+
S,K(
d/U!)2
+ 5
qiqi
.
(3.34)
Thus (dS/dt) /+3 0 provided (3.33) is satisfied. Now we prove that (dS/dt) )_ = 0. By following the arguments that led to (3.27), (3.28), (3.29) we obtain (the terms of the third and higher order in v and q are neglected)
0
L!?i, = pS,Ui +4 f
$k,= (P(P)p+
q1
(3.35)
c
SePlsij
One verifies easily that the integrability conditions implied by (3.35) are satisfied provided (3.33) is satisfied. The total entropy flux fi equals
f3i= PSUi
+
pS,lJi
(3.36)
Also the other properties (H2)-(H4) are satisfied provided (3.33) is satisfied. The proof (a direct verification) is omitted. 4. CONCLUSION
We have shown that the Maxwell-Cattaneo hydrodynamics is compatible with equilibrium thermodynamics and allows equivalent Lagrangian formulation in which the evolution of fluid can be interpreted as the time evolution of a material and caloric points. All conceptual and practical advantages associated with the Lagrangian description of a dynamical process can be applied now to processes that include a nontrivial evolution of the internal energy. In the study of the compatibility of the Maxwell-Cattaneo hydrodynamics with equilibrium thermodynamics we used the general formulation (reviewed in Section 3.1 and developed in Refs. [I, 21 that unifies the Gibbs prescription for the transition from the Hamiltonian mechanism to equilibrium thermodynamics, the Boltzmann’s H-theorem that relates the Boltzmann kinetic equation with equilibrium thermodynamics and the assumption of local equilibrium that relates the classical Navier-Stokes-Fourier hydrodynamics with equilibrium thermodynamics. REFERENCES [I] M. GRMELA, Lecture Notes in Mathematics In Global Analysis, (Edited by M. Grmela and J. E. Marsden), Vol. 755 (1979). [2] M. GRMELA and L. S. CARCIA-COLIN, Phys. Reo. A, Part I, 22, 1295 (1980). Part II, 22, 1304 (1980). [3] A. C. ERINGEN, In ConrinuumPhysics, (Edited by A. C. Eringen). Vol. 2, Academic Press, New York (1975). [4] V. 1. ARNOLD, Ann. Inst. Fourier XVI, 319 (1966); V. I. ARNOLD, Les Mtf’thodesMathimatiques de la Mticanique Classique, Appendice 2, Edition Mir, Moscou (1976); D. G. EBIN and J. MARDEN, Annals of Math. 92, 102 (1970). [S] A. S. LODGE, Body Tensor Fields in ContinuumMechanics. Academic Press, New York (1974). [6] E. M. BARSTON, J. Math. Phys. 8, 523, 1886, (1%7); 9, 2069 (1%8); 12, 1116 (1971). [7] J. TEICHMANN, Con. .l. Phys. 59.82 (1981); 60,640 (1982). [8] A. S. MONIN and A. M. YAGLOM, Statistical Fluid Mechanics. MIT Press, Cambridge, Massachusetts (1973). [9] J. C. MAXWELL, Phil. Trans. Roy. Sot. 157, 49 (1867). [IO] C. CATTANEO, Camp. Rend. Acad. Sci. Paris 247,431 (1958); 247, 3154 (1958). [II] S. G. BRUSH, Kinetic Theory, Vols. l-3. Pergamon Press, Oxford (1%6). [12] J. LIOUVILLE, J. de Math. Pures et Appliquies. 3, 342 (1838). [13] M. GRMELA, R. ROSEN and L. S. GARCIA-COLIN, I. Chem. Phys. 75.5474 (1981). [14] J. A. McLENNAN, Phys. Fluids 8, 1580 (1%5). [15] H. Grad, Hondbuch der Physik, Vol. XII. Springer-Verlag,Berlin (1958). [l6] W. GIBBS, Elementary Principles of Statistical Mechanics (Dover, New York l%O), 1st Edn. Yale University Press, New Haven (1902). [17] L. BOLTZMANN, In Wissenschofiichen Abhondlungen uon Ludwig Boltzmnnn, Vol. 2, Chelsea New York (1968).
Lagrangian formulation of the Maxwell-Cattaneo hydrodynamics
313
[18] S. R.$e GROOT and P. MAZUR, Non-Equilibrium Thermodynamics, McGraw-Hill, New York (1969). [19] I. MULLER, Z. fiir Physik 198, 329 (1%7), In Foundations of Continuum Thermodynamics, (Edited by J. J. J. Domingos, M. N. R. Mina and J. H. Whitelaw). Wiley, New York (1973). [ZO]C. TRUESDELL, Rational Thermodynamics. McGraw-Hill, New York (1%9). 1211G. LEBON, D. JOU and J. CASAS-VAZQUEZ, J. Phys. A. 13,275 (1980). (Received 1 July 1982)
ES Vol.
21,
No. 4-B