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Deduction of the Quasi-linear Transport Equations of Hydro-thermodynamics from the Gyarmati Principle.

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A N N A L E N D E R PHYSIK
~
~~
7. F O L G E
*
B A N D 27,
HEFT 3
Y
1971
Deduction of the Quasi-linear Transport Equations
of Hydro-thermodynamics from the GYARMATI
Principle
By GY. VINCZE
Abstract
From the universal form of GYARMATI’Svariational principle of thermodynamics the
differential equations governing the internal energy and impulse transport of one component
hydro-thermodynamic systems are derived. In our particular case GYARMATI’S“supplementary theorem” is confirmed, by which the validity of the universal form of GYARMATI’S
variational principle is guaranted also in non-linear cases. Finally some problems of the
GYARXATIprinciple and of non-linear thermodynamics are discussed.
1. Introdiiction
Recently GYARMATI
has reformulated his integral principle in a universal
form [I, 21 including the hitherto known variational principles of irreversible
thermodynamics particularly the different partial forms of ONSACER’S
principle of the least dissipation of energy-and which is applicable also for the
treatment of non-linear problems. GYARMATI’S
terminology - we may call it
“the governing principle of dissipative processes”. It is well-known that the
application of the partial forms of the variational principle in question for the
derivation of differential equations describing linear irreversible transport
processes have been demonstrated by GYARMATI and his coworkers. (See
[ 3 . . . 9 ] and particularly [lo].) Therefore now the aim of present author is to
derive from “the governing principle” the non-linear (more exactly quasilinear) differential equations together with the boundary conditions describing
the transports of internal energy, momentum and moment of momentum taking
place in isotropic, one-component, non-isothermal fluid systems, by making
[l, 21. Remarkable that for
use of the supplementary theorem of GYARMATI
rigorously linear cases, when the coefficients of the constitutive equations are
constants, the adequate linear forms of the above mentioned equations have been
already derived by BOROCZ
in different pictures from the previously used partial
form of GYARMATI’S
principle [l I].
Let us start from the following form of “the governing principle of dissipative processes”
sj
v
z d v = 0;
z=
- Y - Q,
(1)
where the integration is carried out over the volume V of the continuum. Here
3 is the LAGRANCian density which is identical t o the local OM (ONSAGER1.3 Bnn. Phssih.
7. Folge, Bd. 27
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MACHLUP)function, o is the entropy production (which is a bilinear expression
in terms of the thermodynamic forces X i ;
and fluxes Ji),!P and @ are the local
dissipation potentials defined by
where L,, and Ri,are the elements of the conductivity and resistance matrices
reciprocal relations hold [lo, 121. It should be
respectively, for which ONSAQER’S
noted, that in (2) scalar denotation was used for fluxes and forces, i.e. in case
of transport processes of different tensorial order the quantities X iand J i are
the scalars of the forces and current densities of the corresponding tensorial
quantities. It is also relevant that the coefficients L,, and Rikof the homogeneous
quadratic forms of (2) are constant quantities in the linear theory, whereas in
quasi-linear case they are functions of the state parameters (for instance temperature, velocity etc. [3]). By using the entropy balance
(where B is the substantial time derivative of the specific entropy and J , is the
entropy current density) the alternative form of the LAQRANGian above
L?
= eB
+ V . J , - !€’-
@
(4)
can be used formulating the variational principle (1).
Starting with the form (1)of the LAQRAiwiandensity the actual form of the
LAGRANGian is attained by the balance equations, whereas in the formulation
(4) by the direct application of the GIBss-relation. On the other hand the principle apart from the formulation in the “entropy picture” (1)can also be given
in different representations similarly to the partial formulations [9, 10, 111.
Considering however, that our calculations are the most simple ones in “energy
picture” and keeping in mind that this “picture” is most suitable for the formal
recuperation of the “inertial terms” of motions, in the following “energy picture’’ will be used [lo, 111. Evidently “the governing principle of the dissipative processes” will be in energy representation
6J2Y*dV = 0;
L F * s TO - !€’* - @*,
V
(5)
where T is the absolute temperature, To is the energy dissipation per unit
volume and unit time, finally !P* and @* are local dissipation potentials of
energy dimension, i.e. for instance the dissipation potential !P* is now determined not by the thermodynamic forces occuring in the expression o,but in the
expression To.
Owing to the fact that in case of transport processes such a set of the state
parameters ‘‘I‘,’’ can always be given that the thermodynamic forces X , can
i.e. X , = VPz thus the LAbe generated as the gradients of the parameters
GRANGian density, in general, is a function of the variables P,, PP, and J, i.e.
2*(I‘,,VTz, J , ) . If we vary independently with respect to I‘, and J , the first
set of the EULER-LAQRANGE
equations belonging to the (5) “governing prin-
r,;
227
GY. VINCZE: Deduction of the Quasi-linear Transport Equations
ciple"
are identic to the transport equations of dissipative processes, while the second
equations
set of the EULER-LACRANOE
-a-Y*
=o
aJ,
(7)
are identic to the kinematical constitutive equations [3, 101.
Our task will be in the following t o determine the actual form of the LACRANOian density (5) for the case of the system model of our investigation. For
this purpose let us consider the following expression of the energy dissipation
[lo, 121:
0
To =
-
0
-
J4 . Vln T - p?' V . v - Pvs: ( V V ) ~ Paa. ( V x
2,
- 2cu) 2 0.
(8)
I n our case the following balances are valid for the internal energy, momentum
and moment of momentum [lo, 121.
Balance of internal energy :
0
e l i + V ' . J q = - p V ~ v - P p " s : ( ~ ~ ) s - P v a ~ ( ~ X v - 2 2 0 ) (9)
.
Balance of momentum :
Q.ri
+ V . P = @F.
(10)
Balance of moment of momentum
:1)
Q@& = --pe a
(11)
in which expressions the usual decomposition of the pressure tensor
+
+
+
+
P =p d
P" = ( p 13') 6
P.'
F'5
(12)
has been taken into consideration [ l o , 121. I n the above equations u is the specific internal energy J4 the heat current density, p the hydrostatic pressure, Pv
is the viscous part of the pressure tensor P,pt' is the one third of the spur of the
viscous pressure tensor, Pvs is the symmetric part of P", Pvais the adequate axial
vector which may be formed from the antisymmetric part of Pv, F is the specific
external body force d the unit tensor and v is the velocity. I n (8) of the spatial
derivatives of the latest quantity the expressions
l) The more general form of the balance equation of the angular momentum has been
examined by BARANOWSKI
and ROMOTOWSKI
[14, 151.
l5*
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1971
must be understood [lo, 121. Finally o in (8) and (11)denotes the mean angular
velocity belonging to the “internal rotation”, of the continuum whereas 0 is
the macroscopic mean of the inertia momentum of corpuscules forming the
unit mass.
The relation between the thermodynamic forces and fluxes (occurring in
expression (8) of the energy dissipation) is given (linear approximation and in
“energy picture”) by the following constitutive equations [9, 10, 111:
J~ = - L:*
v In T,
pv = - L;v P . v,
where 2 is the heat conductivity coefficient, [,is the volume viscosity, [ the
shear viscosity and 5, the rotational viscosity coefficient. By using (14) and (8)
we have the actual forms of the dissipation potentials in “energy picture”:
and
which may be considered as & priori given quantities.
2. The actual form of the GYARMATIprinciple
The form of the GYARMATI
principle valid for hydrothermodynamic systems
can be attained by substituting the expressions (8), (15) and (16) into the variational principle ( 5 ) . Thus we have
GY. VINCZE: Deduction of the Quasi-linear Transport Equations
229
I n the following let us turn to those “internal variables” Tz,
from which the
forces can be generated as gradients. I n the LAGRANGian density there are, as it
is evident from (17) two “internal parameters”: u and In T of this kind.
In order to turn t o the “internal variables” let us take into consideration the
following vector-analytical identities :
where puis the transponed tensor of PI’. By using these indentities, further on by
taking into consideration that the quantities V . J, and V . P“ can be expressed
with the aid of the balances (9) and ( l o ) , the following actual form of the “governing principle” can be obtained
$ [ P ? ’ . P - + J,lnT].dQ=O
--6
R
while the volume integrals of divergencies occurring in the identities (1 8) has
been transformed into surface integrals by GAUSS’theorem.
It should be noted that instead of the substantial balances (9) and (10) also
their local forms can be used, because the source terms of the balances are invariant against the description choosed [lo]. Introducing the local forms of
the balances into the variational principle (17), instead of (19) the variational
task
- 6 $ [ ~ ” . v + ~ , l n ~ ] . d o~ d =
11
is obtained, as it can be demonstrated by simple manipulations.
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*
1971
3. The Deduction of the Transport Equations of Hydro-thermodynamics
I n order to derive from the variational principle (19) or (20) the transport
equations corresponding to the EULER-LACRANGE
equations given in (6) the
restriction
+
+
0
+
0
p P * 2,
P"': ( V V ) ~ Pva ( P x w - 2 w ) j = - In T SJ,
In T 6 [p'-i
_ - V.(lnTSJ,)+ ( P 1 n T ) . S J q w . 6 [ e & - e F + Pp]
=
- 2 , . S(P.P")
= -
+ ( P . v)6p" +
V . (v. SP")
+ (Pu): 6P" = - c7. (v . BP")
s b - (Pxw)
(21)
6p""
*
( h ) S :
stipulated by the balances (9) and (10) should be taken into account, where the
and the decomposition adequate to
commutability of the operators 6 and
(12) has been used. Let us primarily consider the linear case, when the conductivities and resistances are considered as constants and are not varied [9]. I n this
case with the following identities
v
2
6
(V In ~
1 =
2
F . [LI,*Q(V
In T ) 6 In
TI - P . (L:~P In T )s In
1 1 ,
*
&Y
2
6( c7 . w)2
-
=
P . [L,*,(P . v)6w] - c7 . (L,*,P . v)627,
V x [L("")'(VXW- 2 w ) l .6v - 2 L ( 4 * ( V X V- 'Lw) . d o ,
(22)
we can turn also in the dissipational potentials to the "internal" variables Ti.
By considering the restrictions (21) and the identities of ( 2 ) in the variational
principle (19) and by transforming with GAUSS' theorem the corresponding
parts into surface integrals then the expression
i{---[p'-i
V
+p
- [pv - p F
+
~v .
0
0
P~S:
+
( v v ) ~~
+ V p l . 6~ - 2w
- P . (L:" V . 27) 6v
-
(
. ~ f ' a
v .[L(tt)*
(Cw)S]
~- 2w)ld
x v ln T
~ 7 (. L : ~
VIn T)61n T
. 6w
- v x [L("")*
( P xw) am)] . 6v - 2LC"a)' (V x v - 2w) . 6w
+ R;, Jq. SJ, + R,*,pv Spv + R@t)*Pv8: SPUs+ Rcaa)*Pcn. 6Pc"
-
0
0
0
(V In T ) SJ, - (P . v) Spv - ( V V ) ~6Pv*
: - ( V x w ). SPrlL
+ 2w . Spa}d V + $ { -6 ( Pv . a ) + v P O + In T 6Jy
-
u
-
+
S(J, In T )
+ (L:,V In T ) 6 In T + (L:8 P . v) Sv
0
[L('l)*
(PV)S]
. 6v + [L("")*
(V x v
- 2w)l
x Sv} * dl2 = 0
(23)
231
GY. VINCZE:Deduction of the Quasi-linear Transport Equations
is attained. Let the variational restriction
-2P"a. 6 0 = QO & ' 6 0
(24)
following from the angular momentum balance (11)be considered, further on
by the expansion of the expressions 6(P" . v) and 6 ( J qIn T ) and by substituting
i t into (23) the final form of the variational principle is obtained:
Since 6 In T, 6 0 , 6v, SJ,, 6p7',SPcs, 6Pra adre arbitrary and independent
variations thus from the vanishing of the volume integral - and by considering
also (14) - the transport equations of the internal energy
+ p I 7 . v + Pcs: ( F v )+~ P L u .(4x27 - 2 0 ) + P . (AT V1n T )
O
plj.
D
=
o
(26)
of the momentuni
eg
+ F'P + ~7(rFF'.v)+17.[2i(c;2))s]+V X [ ~ ' , ( P X V - ~ C O ) ] - ~ F = O
(27)
and of the moment of momentum
+
@ O h 2g',(PXV - 2 0 ) = 0
finally t h e linear constitutive equations
1
--PO" - v. v
5,
= 0,
1 "
- P8. - ( V V ) S = 0 ,
25
3
- (Vxv- 2 0 ) = 0
- pzsa
5,
(28)
23’2
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7. Folge
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Band 27, Heft 3
*
1971
are obtained. It is evident that the vanishing of the surface integral in (25) leads
to the transversality conditions :
[J, - IT P I n TJ,&
= 0,
1P” - C,(P . V)l,l
0
=
0,
0
[P‘S
-
2[( vu)”,L= 0 ,
[ P a - C,(VXV -
20)],, = 0,
where n is the external normal of the boundary surface Q.
As a matter of course in order to give account of the evolution in space-time
of the investigated hydro-thermodynamic system beside of given initial and
boundary conditions, the transport equations (as), ( 2 7 ) and (28) must be
completed by the continuity equation
@
+ e r.2,= 0
(31)
further on by the equations of state [11, 121:
4. The Deduction of the Transport Equations of the Stationary Hydrothermodynamic System from the Governing Principle
I n order to derive the transport equations rel&tedto the stationary state it is
advisable t o start from the form (20) of the variational principle. For simplicity’s
sake in the following it is assumed that the pressure tensor is symmetric (i.e.
Pa = 0) and along the boundaries Q of the region we do not vary. I n case of
such conditions the variational task (20) is reduced as it follows :
Let us the case of “total” stationarity examine, i.e. when
ae-au -=
at
at
av
at = O
(34)
233
UY. VINCZE
: Deduction of the Quasi-linear Transport Equations
conditions are valid. By taking into consideration these conditions and the mass
balance (31), from (33) we get:
(35)
The EULER-LAQRANGE
equations (6) belonging t o this variational task will be
the following ones by taking into account the relations (14)
p . ~ 7 . 1 1 .+ p
$7
v . v + b: ( ?v)s + ~7 . (AT P In T )= 0 ,
( T ) - e ( v x r’v) + v p + V(C, v . v) + V ‘ [ 2 [ ( i v ) S ]
v 2
(36)
- QF = 0
(37)
which are the stationary forms of the transport equations (26) and (27). Of
course, by the EULER-LACRANQE
equations ( 7 ) the constitutive equations are
recuperated :
1
;ITJQ- V l n T = 0 ,
1
-pu
-
i v
1 P
25
OZ.9
F7.v
=
0,
0
-
(b)”
= 0.
The stationary transport equations (36) and (37) together with the form valid
for the stationary case of the continuity equation (31)
B . (ev) = 0
(39)
means five scalar equations for the unknowns. By considering them completed
with the state equations (32) then by the equations in question the stationary
states of the one-component, isotrop, non-isotherm fluid systems are described.
5. The Case of quasi-linear Coefficients and the GYARMATISupplementary
Theorem
I n the hitherto examinations it was assumed that the conductivities L:k
and resistances RZ are functions of the space coordinates. I n the majority of the
practical cases these coefficients, may also depend on the state parameters
(quasi-linear case) more over on their gradients, i.e. also on the thermodynamic
forces (rigorously non-linear case) [3].
234
Annalen der Physik
*
7. Folge
*
Band 27, Heft 3
*
1971
I n the following only the quasi-linear case will be dealt with when the linear
(14) constitutive equations are considered as such ones in which the conductivities L7k and resistances RZ are functions of the parameters Tidiscussed in detail
in the precedings. For this case the following theorem was formulated by GYARMATI [3] ; the variation with respect to the parameters Ti of the sum of the dissipational potentials vanishes, i.e.
S,(Y*
+ @*I
=
F -ap( py* -+ @*I
sri = 0.
(40)
This theorem which may be appropriately called in its relation to the variational principle as “supplementary theorem” - is mathematically based on the
symmetry properties of the total LEaENDRE-transformations of the dissipation
potentials [ 101. A very important corollarium of the “supplementary theorem”
is that GYARMATI’S“governing principle” remains valid also in the quasi-linear
case which fact cannot be said about the partial forms of the principle [1, 2,
10, 131.
Now the validity of GYARMATI’S
supplementary theorem will be demonstrated in case of our hydro-thermodynamic system model. Let us assume that the
coefficients L z and RTk are functions of the state parameters X and v.
Now by considering (40) first of all the part of the dissipational potential related
to heat conduction can be written:
It follows from ( 1 4 ) that R,*,L,*,= 1, from which the relations
and
result. On the other hand by making use of these relations and taking into
consideration the constitutive equations (14) the relations
are obtained.
Similarly to the precedings for the quantities related to the hydrodynamic
flow the relations
GY. VrNozE : Deduction of the Quasi-linear Transport Equations
235
can be obtained. From the comparison of (45), (46), (47) and (48) our statement,
i.e. the validity of the supplementary theorem in case of our actual instance is
demonstrated :
+
+
+
6,(Y*
@*) = 6*(Y* @*)
S,;(Y*-+@*) = 0 ,
(49)
where the lower index refers to the state parameter according to which the
variation has been carried out. It should be noted that for those state parameters
(for instance hydrostatic pressure) - forces generable from which by gradient
operator do not occur in the entropy production and in the energy dissipation
respectively - the validity of the “suplementary theorem” is trivial.
6. Discussion
It is evident that with respect to the continuity equation (31) expressing the
principle does not represent a genuine variaconservation of mass the GYARMATI
t,ional principle. However, it can be demonstrated that the continuity equation
(31) is obtainable from this principle only “formally”, since, the GYARMATI’S
“governing principle” is a genuine variational principle only for the “dissipative
terms” of the transport equations [Ill. Let us consider the group (6) of the
EULER-LAGRANGE
equations belonging to the variational principle (20) (in which
the local balance equations are included) representing the following transport
equations
$ + p . ( p v )+ p
+ Pus: (pw).$- p . (AT PIn T ) = 0,
0
p. w
0
(50)
apr:
at+ v . ( e l : ~ ) + Y ~ ; d ( r , ~ . n ) + ~ . [ 2 t ( ~ v ) ~ ] - - (51)
eF=O
in which for simplicity’s sake Pvawas taken as zero. I n order t o confirm our above
statement, i.e. that the equations (50) and (51) are not yet real transport equations let us carry out the following transformation :
236
Annalen der Physik
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7. Folge
*
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*
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It can be seen that the bracketed [...I parts of (52) and (53) are just the local
forms of the equations of mass continuity (31) multiplied by u and v. If in the
variational principle the local form of the balances are used, then the equations
obtained are also including the equations of the continuity of mass. This problem
is related to the fact that though the variational principle of GYARMATI
is
directly equivalent to the differential equations governing dissipative processes,
also the equations valid for the reversible motion are contained in it [ l o , 111.
The reason for this-according to our mind-is that in the course of the variafrom which the forces causing
tion we are going over to the state parameters
also
the energy dissipation can be generated as the gradients of Ti and thus in
such state variations are contained which are directly not related to the dissipative effects. We are meaning such cases even in each point of the space Ti
are varying with identical time functions. The GYARMATI
principle is invariant
against such “rigid like” motions and orthogonal transformations. These properties of invariance - as it will be demonstrated in detail in a paper to be published
in the near future - are meaning that GYARMATI’S
principle is satisfying the
“principle of objectivity” discovered by NOLL[16, 171.
r,,
r,
References
GYARMATI,
I., Z. phys. Chem. 239 (1968) 133.
GYARMATI,
I., Ann. Phys. 23 (1969) 353.
GYARMATI,
I., Zsurn. Fiz. Himii (Moszkva) T. 39, N. 6 (1965) 1489.
GYARMATI,
I., Acta Chim. Hung. 43 (1965) 353.
V E R H ~ SJ.,
, Zsurn. Fiz. Himii (Moszkva) T.40,N.6 (1966) 1213.
V E R ~ SJ.,
, Z. Phys. Chem. 234 (1967) 226.
V E R H ~ SJ.,
, Ann. Phys. 20 (1967) 90.
BOROCZ,Sz., Z. phys. Chem. 234 (1967) 26.
FARBAS,
H., Z. phys. Chem. 239 (1968) 124.
[lo] GYARMATI,
I., Non-equilibrium Thermodynamics, Field-Theory and Variational Principles, Engineering Science Library, Springer-Verlag Berlin-Heidelberg-New York
(1970). Originally Published as : Nemegyensulyi Termodinamika, Miiszaki Konyvkiad6,
Budapest 1967.
[ll]BOROCZ,Sz., Acta Chim. Hung. (in press).
[12] DE GROOT,S. R., and P. MAZUR,Non-equilibrium Thermodynamics, North-Holland
Publ. Co. Amsterdam (1962).
[13] VINCZE,GY,, Diplomarbeit, Budapest (1968).
[i4] BARANOWSKI,
B., and T. ROMOTOWSKI,
Bull. Acad. polon. Sci. Serie des sciences chimiques XII. 1. 71. (1964).
Bull. Acad. polon. Sci. Serie des sciences chir151 BARANOWSKI.
B.. and T. ROMOTOWSKI,
miques XII. 2. 127 (1964).
[16] NOLL,W., La mkcanique classique, bas6e sur un axiome d’objectivite. La Methode
Axiomatique dans les M6caniques Classiques e t Nouvelles. Colloque International,
Paris (1959). Paris: Gauthiers-Villar.
[17] TRUESDELL,
C., and W. NOLL,Non-linear Field Theories of Mechanics, Handbuch der
Physik, III/3. Springer-Verlag, Berlin-Heidelberg-New York (196.5).
[I]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
L
A
Godollb’ ( H u n g a r y ) , Institute for Physics, University of Agricultural Sciences.
Bei der Redaktion eingegangen am 2. Dezember 1970.
Anschr. d. Verf. : Dr. GY. VINCZE
Godoll8 (Hungary), Institute for Physics,
University of Agricultural Sciences
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