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On the modelling of momentum and energy transfer in incompressible mixtures
In, J Non-Lmeo, Mechanrcu.Vol. 30, No. 4, pp. 419-431, 1995 Copyright 0 1995 Else&r Science Ltd Printed in Great Britain. All rights reserved 0020-7462/95 $9.50 + 0.00
Pergamon
0020-7462(95)00016-X
ON THE MODELLING OF MOMENTUM AND ENERGY TRANSFER IN INCOMPRESSIBLE MIXTURES Herald0 Department
of Mechanical
Maria Laboratkio
Laura
S. Costa
Mattos
Engineering, Universidade Federal Fluminense, 156, 24210-240, Niter&, RJ. Brazil
Martins-Costa
National
and Rogkrio
M. Saldanha
de ComputaGZo Cientifica, Rua Lauro Rio de Janeiro, Brazil
(Received 6 October
1994; in revisedfirm
Rua Passo da PBtria,
da Gama
Miiller, 455; 22290-160,
1 February
1995)
Abstract-Although many papers concerning the continuum theory ofmixtures have been produced in the last years, the adequate form for the first and second laws of thermodynamics is still a controversial subject. In this paper, the basic principles that govern the evolution of a continuous mixture are postulated and used to develop a systematic procedure to obtain thermodynamically admissible objective constitutive equations. The prescription of five thermodynamic potentials is sufficient to define a complete set of constitutive relations. Two examples concerning the thermal analysis of a binary mixture of rigid solids and fluid flow in a porous channel bounded by two isothermal impermeable plates are presented. They illustrate the possibilities of effective practical use of the proposed theory.
INTRODUCTION
Some papers concerning the continuum theory of mixtures have been produced in the last years. In these papers a mixture is regarded as a superposition of kinematically independent continuous constituents [l, 21. Nevertheless the adequate form of the basic governing principles (mass balance, linear and angular momentum balances, first and second laws of thermodynamics) in a continuum theory of mixtures context is still a controversial subject. In some of these papers the first and second laws of thermodynamics do not take into account the power of the interaction forces and they do not exclude the possibility of unusual behaviours such as a heat supply to a constituent arising from the interaction with another constituent at a lower temperature and negative work in a cyclic isothermal process. To set up a general theory for momentum and energy transfer in continuous mixtures it is necessary to consider aspects of both the first and second laws of thermodynamics, since dissipative phenomena must be taken into account. Such phenomena are important, for instance, in the simulations of processes like the manufacturing of some composites or petroleum flow through a porous matrix. In the present work, the basic equations that describe the dynamics of a continuous mixture are obtained from the virtual power principle. This leads to a natural interpretation of the interaction force among the constituents and the antisymmetrical part of the partial stress tensor. The use of the continuum mixture approach gives rise to parameters (absent in a classical continuum mechanics approach) to account for the thermomechanical interaction among the constituents. Restrictions, arising from the second law of thermodynamics, for these constitutive quantities are derived in this paper. It is shown that the local form of the first and second laws of thermodynamics must take into account the power of the interaction forces between constituents in order to describe adequately the thermomechanical couplings. A systematic procedure is also proposed to obtain a complete set of constitutive equations that always satisfy the second law of thermodynamics and the principle of objectivity, regardless of the geometry, the external actions on the mixture, the boundary 419
420
H. S. Costa Mattos et al.
and initial conditions. These equations are proposed aiming to offer a minimum of complexity in order to render favourable experimental confirmation. This study can be regarded as a sequence of the work of Costa Mattes [3] and Costa Mattes et al. [4] and this procedure is used to show that the equations proposed by Martins-Costa et al. [S-S] for the energy transfer in a binary mixture composed by a rigid solid matrix and an Ostwald-deWaele incompressible fluid are thermodynamically admissible. It is important to remark that we do not claim the generality of the second law version proposed in this paper and that it is rather restrictive. This work is basically concerned with a mathematically correct and physically realistic modelling of mixtures and this version is conceived to exclude physically unusual material behaviours.
BASIC THEORY
In this section, the basic governing principles for a continuous mixture are postulated. The basic assumption used in the theory is that the mixture may be viewed as a superposition of n kinematically independent continua [l, 2,9]. For the sake of simplicity the presentation is limited to processes such that in every instant t, the whole of some region of the Euclidean space E is simultaneously occupied by all the constituents. Besides, it is also assumed that there is no interchange of mass among the constituents. Similarly as in Green and Naghdi [9] a motion of a constituent g= is defined by a sufficiently smooth vector function xa which assigns position x, = x8(X,, t) to each material point X,. In the principles presented in this section for each constituent %a a material part P, occupying the whole of a region R, = R c E at the current configuration is considered, the boundary of R, being denoted by aR= G dR. Mass conservation
principle
Defining p. as the mass density of P,, the mass of the constituent occupying the region R at a given instant t can be written as: M=(P=, t) =
s R.
p=dV.
(1)
The densities of the constituents must be such that: ” dM” (P,, t) = 0. =:, dt
(2)
In the absence of mass interchange between constituents, the following identities hold: T
(P,, t) = 0 3 2
+ p,div v, = 0
VP=
where D”( )/Dt denotes material time derivative holding X, fixed. Virtual power principle
The virtual power principle is a convenient framework in which to define the dynamics of a continuous mixture. Compared to the traditional approach - conservation of linear and angular momentum - [l-lo] the principle of virtual power is commonly considered more abstract and hence more difficult to apply. However, in the case of mixtures, the great advantage of this alternative approach is that it allows a simple interpretation of the additional internal forces that arise in the theory. In order to determine the forces acting on the part P= that occupy the region R, G R at the current configuration it is necessary to introduce the space VIR(R=) of all possible velocity fields which take place at a given instant. VIR(R=) is called the space of virtual velocities and, for the sake of simplicity, throughout this work it is defined as the space of all vector fields v : R, --* E which are sufficiently regular to ensure the existence of its gradient. VIR(R=) = {v: R, --* ElVv is defined}.
(4)
421
Modelling of momentum and energy transfer
any constituent %?=the power of external forces PgXT and the power of inertial forces P& for a given virtual field of velocity w, E V,,(R,) are defined as: For
PbEXT(Pa, 44 =
s
b,.w,dV+
R.
Ph(P,, 4 w,) =
f,.w,dA s aR.
s
p,a,-w,dV
(6)
R.
where b, is the external volume force acting on P,, f, is the surface force acting on the boundary aR, and a, is the actual acceleration of the constituent. In the mechanics of a single continuum it is usual to consider a first order gradient theory in which the power of the internal forces is supposed to be a function of the velocity and its gradient [l 1,121. Extending this to a continuous mixture and, on the assumption that, for a fixed instant t and for any constituent V,, the power of the internal forces can be expressed as a linear functional on V,R(R,), it may be shown that PyNT(Pn,t, w,) must have the following form:
fYNT(Pm4 w,) =
s
[T,“.(V”w,) + T:.(V”w,) - m,.w,]dV
R.
(7)
where VuW, and V’w, are, respectively, the antisymmetrical and symmetrical parts of VW,; with T,” being an antisymmetric tensor, Ti a symmetric tensor and m, a vector. T,“, Ti and m, will be called the generalized internal “forces”. Making an analogy with the theory presented in Atkin and Craine [2], T,” and T,” may also be called the antisymmetrical and symmetrical parts of the partial stress tensor and m, the interaction force. Using the previous definitions, the virtual power principle may be stated as follows: (Pl) For the material parts P, c %?awhich occupy an arbitrary region R, = R at the present configuration, the antisymmetrical and symmetrical parts of the partial stress tensor, T,” and Tl”, the interaction force m,, the body force b,, the surface force& and the actual acceleration a, of Wgmust be such that: i
P;N(Po,t,~a)
a=1
+ i
P;,&Pa,
4wg) = i
P&(Pn,t,W,)
VW, E OR.
(8)
a=1
a=1
(P2) The term Cl= 1P&(P., t, w,) is zero for a virtual rigid body motion, that is when all the constituents have the same rigid body velocity: n c PTNT(Pa, t, w,) = 0 when w,(x,) = A(x, - x,,) + co (9) n=l
where A is an antisymmetrical tensor and co is the velocity of a reference point x0 E E. Under suitable regularity assumptions, it can be proved that (PI) implies the following local expressions: paa, = div (Ti + Ti) + m, + b,
in R
and (T: + T,“)n =h It
in i?R, VP,.
(10)
can also be proved that (P2) implies the following local expressions: n
1 m, = 0 a=1
The role understood undergoing deformation is necessary NLM 30-4-B
and
=$I T,” = 0.
(11)
of the additional internal forces T.” and m, in the internal power can be from simple examples. In the first example each constituent is supposed to be a different translation (i.e. u,(x,, t) = c,, c1 # c2 # . . . # c,). Although the of each constituent is zero, it is reasonable to develop a mixture theory where it to apply some external action in order to impose such kind of motion. From the
422
H. S. Costa Mattos et al
definition (7) of the power of the internal forces, it is possible to verify that: j1 PMP.9
t, %) = -
[m, Fc,] d V. i a=1 s R.
(12)
If the internal forces m, were not taken into account in definition (7), the power of the internal forces for such kind of motion would be zero. Since the power of the inertial forces is zero (because the acceleration of each constituent is zero), from the principle (Pl) it comes that the power of the external forces (and hence the external forces) should also be zero, which is not physically expected. Hence, in this case, the theory has to consider the (generalized) internal forces m, in order to allow a non zero internal power, which indicates that it is necessary to apply some external action in order to impose such kind of motion. In the second example each constituent is suposed to be undergoing a different rotation (i.e. u,(x,) = A,Jx, - x,,) = A1 # A2 # . . . # A,,). From the equations (7) and (1 l)i it may be concluded that:
s
CT,“- W,] dV
a=1
(13)
R.
Similarly to the first case, although the deformation of each constituent is zero, it is reasonable to expect a mixture theory where it is necessary to apply some external action in order to impose such kind of motion. If the internal forces T,” were not taken into account in the definition (7), the power of the internal forces for such kind of motion would be zero. Since the power of the inertial forces is zero (because the acceleration of each constituent is zero), from the principle (Pl) it comes that the power of the external forces (and hence the external forces) should also be zero, what is not physically expected. Hence, in this case, the theory has to consider the internal forces T,” in order to allow a non zero internal power, which indicates that it is necessary to apply some external action in order to impose such kind of motion. These simple examples show that the additional internai forces T,” and m, must be introduced in the continuum theory of mixtures in order to take into account the work done when the constituents are undergoing independent rigid body motion. First law of thermodynamics
order to postulate the energy balance for a mixture it is necessary to define the internal energy U’(R) t), the kinetic energy K”(R, t) and the nonmechanical power P&(R, t): In
U”(P=, t) =
K”(P=, t) = ;
PUP.3
t) = -
s R,
s R.
P=e=dV
(15)
p=v=.v=dV
q,.ndA
s dR.
(14)
+ s R.
(16)
para d V
where e, is the internal energy per unit mass, u, is the actual velocity, qa is the heat flux vector and r= is a heat supply or source per unit mass and time. The first law of thermodynamics can then be stated as: $
;
U”(P=,t) + =$, $ K”(P=,t) = =cl PbT(P=,
t>u=) + i
PaTH(P=,t).
(17)
a=1
Using the definition of the power of internal forces (7) and the mass conservation principle it is possible to prove that, under the appropriate continuity assumptions, (d/dt)K,(P,, t) = P;,(P=, t, v,) and the first law of thermodynamics may be written as: g
i
U,(P,, t) = i
Q=l
Phv(Par t,tl) + i a=1
KfPmt).
(18)
423
Modelling of momentum and energy transfer
The local form of the principle stated above is given by: (19) with D, = V’u, and W, = V%,. The version (19) of first law of thermodynamics is stated for the mixture as a whole. Nevertheless it is easy to verify that (19) is equivalent to the following identities: p=&e=-
T,“.W=-
VP, with i
T,.DD,+m,.v,+divq,-p,r,=IC/,
$a = 0
(20)
a=1
The scalar tja represents the thermal energy supply to the constituent %?,arising from its interaction with the other constituents of the mixture. Second law of thermodynamics
If the first law of thermodynamics states the possibility of conversion of mechanical energy into heat and vice-versa, the second law of thermodynamics provides a distintion between possible and impossible processes. In this section a version of the second law for continuous mixtures is postulated. It is important to remark that there are different opinions about the validity of some definitions. We do not claim the generality of our version. In order to state an entropy production inequality, it is assigned to each constituent a temperature 8,, which is assumed to be positive, and an entropy per unity of mass s,. The total entropy S, of the constituent %?aoccupying the region R at a given instant t is given by:
S,(R,t) =
s
E~,sald V.
(21)
R.
The entropy s, and the temperature
s
0, are postulated to obey:
!!!_?!dA+
aR.
ea
The entropy inequalities proposed above do not restrict some unusual behaviours since they do not take into account the dissipation arising from the thermomechanical interactions among the constituents. Thus, an additional restriction that takes into account the mixture as a whole is necessary: (23) Using (20) to eliminate r. on (22) and introducing the free energy A, of each constituent, A, = e, - O=s=,it is possible to obtain the following local forms of the inequalities (22) and (23): d== T,.D=-p, di=
D”A,
D”8,
Dt
+ ‘= Dt ) ---
q=.grade= 0,
,. -
vp =
i
(24)
a=1
in which d, is the dissipation of each constituent seen as a single continuum and di is the dissipation that arises from the interaction among the constituents. The inequality (24), implies the following inequality which is exactly the version of the second law of thermodynamics proposed by Atkin and Craine [2]:
C-D, - P=
q=wade= 8,
1
, o -*
(25)
Obviously, if the inequalities (24) are verified, then (25) is also verified. The inequalities (24) are an adequate local version of the second law restriction. The role of the additional inequality (24), can be easily understood by considering (11) and (20)2. If
424
H. S. Costa Mattos et al.
these identities are assumed to hold, it can be shown that the dissipation di is zero when all the constituents have the same temperature and the same velocity (or, in other words, when there is no interaction among the constituents).
CONSTITUTIVE
EQUATIONS
The balance equations (3), (lo), (1 l), (20), and the second law restrictions (24) are valid for any kind of continuous mixture. A complete modelling requires additional informations in order to characterize the behaviour of each kind of mixture. A systematic procedure to obtain a set of thermodynamically admissible constitutive equations that verify automatically the principle of objectivity, may be stated by the steps below. In order to simplify the presentation, the study is restricted to incompressible constituents. (Hl) Constituents will be considered for which the free energy A, is a differentiable function of the temperature 19,: A= = i=(e=)
a= 1,n.
(26)
1, n.
(27)
(H2) The following state law holds:
d-4
s== -de,
a=
A consequence of the hypotheses 1 and 2 is that the dissipation d=(see equation (24),) will always be equal to zero in equilibrium, that is when u, = 0 and grad 8, = 0. (H3) The symmetrical part of the partial stress tensor Ti is such that: T, = -p=l
+ TiR
a=
1,n
(28)
where the pressure p= is a Lagrange multiplier related to the incompressibility constraint and T,‘” represents the irreversible parcel of Ti. In order to characterize completely the behaviour of the constituent, additional information about the variables TiR, q,, T,“, ma and $= must be given. In this theory, the additional information is obtained from potentials, called dissipation potentials. The following compact notation will be adopted: 8=_, = 8, - O,, u=_~ = u, - vg, Warns = W, - W,, g. = (l/f?=)grad 8,. (H4) Associated to each constituent %?=there exist two objective and differentiable potentials 0’ and @r, such that: (29) (HS) For the mixture as a whole, there exist two objective and differentiable potentials Yr and Y”, such that: T,” = i
S,,;
8=1
s,,
=
ayz a wm+
-.
mar = f:
h,,;
/3=1
h,, = _ ay’. a0,_8’
*, = i
Las
J9=1
& = - $. QB
(30)
The terms S,,, h, and <=# can be interpreted as the contribution arising from each constituent V, to the antisymmetrical partial stress T,“, the interaction force m, and the energy supply @=of the constituent V,. The definition of the potentials A,, a”, @‘, Yz, Y’T is motivated by experimental observation in simple tests. Therefore, the constitutive equations (27)-(30) may eventually allow some thermodynamically impossible processes. Besides, the identities (11) and (20), must always hold. In order to assure that the local version (24) of the second law and that the identities (11) and (20), will always be satisfied, independently of the geometry, of the external actions and
Modelling of momentum
425
and energy transfer
of the boundary conditions, it is sufficient to restrict the choice of a’, QT, ‘I”’ and Y’Tin the following way;
(i) The
potentials
@’ and
@’ = i
QT
have the following particular form:
&,f(D,),
QT = i
Q(O) = Q(O) = 0
W&),
V%=
(31)
a=1
a=1
where for a given a(1 I a I n)@(D.) and &,$(g=) are non-negative convex differentiable objective functions. (ii) The potentials ‘-I” and YT have the following particular form: Y’=
i f: [Yy.ws-Y$] a=1 p=ol
YQ
iYfB 12=1 p=a
(32)
in which: Y&4 = %&-&,
Y$ = %#u-,),
Y’B,@ = %s(@z-s)
(33)
where, for a given pair (a, p>, (1 I a I n, 1 I B I; n), Y&, Ys and Y$ are non-negative convex differentiable objective functions such that: ‘y; E Y,w,, 9,w,(K-,)
‘u:, = Y;;,,
= - q,,“,cI+,,, %&?-,)
Y$ = Y&
%&,-,I
= - ~‘&s-=)
= - 9’f&-,)
9$(O) = 9$(O) = q:@(o) = 0.
(34)
From equations (34) and (30), it may be concluded that:
Thus, using (35) and the definition of &+, h,, and S,, in (30) it is easy to verify that the identities (11) and (20)2 will always be satisfied if (H6) holds. The demonstration that the inequalities (24) are verified if hypothesis (H6) holds may be done by using the following classical result of the Convex Analysis: Let X and Y be elements of a vector space V with an internal product (X. Y). If 0 : V + [0, + co) it is a non-negative convex and differentiable function such that O(O) = 0, then (X. Y) 2 O(X) 2 0 if Y = d@/dX. Taking X = (D,), Y = (TiR) and (X * Y) = D, - T,‘” it is easy to verify that D,sT,‘~
2 0
(36)
provided that (29), and item (i) of (H6) hold. Similarly, taking X = (g.), Y = ( -qJ and (X . Y) = - 9.. q, it is easy to verify that - (I,.& 2 0
(37)
provided that (29), and item (i) of (H6) hold. Hence, from (36) and (37) it comes that the inequality (24), always holds if (H4) and item (i) of (H6) hold. Using the same kind of reasoning it is possible to verify that: s,,. W.-s 2 0;
- haS.vm_8 2 0;
- 5,aen-s
2 0
provided that (H5) and item (ii) of (H6) hold. Hence, since the absolute temperature always positive, it may be concluded that:
(38) is
426
H. S. Costa Mattos et al.
and (39)
or:
and ”
n
=L &‘7=1
a=1
(40) provided that hypothesis (H6) holds. Using the definition of &, h,, and S,, in (30), it finally comes from (40) that the inequality (24), will always be satisfied if hypothesis (H6) holds. It is important to remark that this demonstration can be easily extended if other variables are taken as independent parameters in the potentials, allowing a more realistic model of the coupling among different mechanisms. For instance: Qi = &,h(D,; Di, BayOi)
(41)
@aT= g)$(g=; si, L
(42)
6)
I+-j,Bi-j,Vi-j)
Y,W,= q,Wa(W,_p;e,_p,V,-8,
Wa-p, Vi-j, Oi-j, Wi-j)
Yip = q;s(Va_a;ea_8, Y$ = ‘k:,(e,_,;
(43)
Wa_p,V,-~,ei-j
Wi-j,Ui-j).
(44) (45)
equation (41)-(42) i = 1, n; i # cc;and in equation (43)-(45) i = 1, n; i # a; j = i, n;j # fl. If A,, O”, W, Y”, YT are known, equations (27)-(30) form a complete set of objective and thermodynamically admissible constitutive equations, provided the hypothesis (H6) is verified. It is also interesting to remark that the relation (38), In
5,s(e, - 0,) 2 0 assures that the heat supply (aa from the constituent WPto the constituent Ws,,arising from the interaction between them must be positive if 8, < 8, and negative if 8, > 8,. From inequalities (38),,, it can also be shown that, for an isothermal process in which the constituents are undergoing independent rigid body motion, the work in a cycle must be non-negative. ANALYSIS
OF THE THERMOMECHANICAL
COUPLINGS
An alternative local form of the first law of thermodynamics (the energy balance equation) can be obtained by introducing the constitutive relations proposed in the previous section in equation (20). Initially, using the definition! of free energy A, and the state law (27) it can be concluded that: Dole, -=G’
D”B,
(46)
Dt
cm =- dZA,e de,’ a 1.
where c, is defined as:
i
(47)
Then, inserting (46) in the local form (20) of the first law of thermodynamics, it comes that: D”B,
divq, + P,C, --par,= Dt
T,“.D,+
T:.W,--m;v,+$,.
(48)
427
Modelling of momentum and energy transfer
Finally, using the constitutive relations (28)-(30), the following alternative expression is obtained: D”8
div q= + pat, $
n aw n a\yT 1 p-u 8=1av,-pa +c--8=1 a,-;
aV ,=,a~-,
$,.+
f
01
- para =
--Ww,+
(49)
The terms in the right side of equation (48) are responsible for the thermomechanical couplings: the temperature evolution on the constituents is coupled due to the term I+Q, and the mechanical problem is coupled to the thermal problem due to the term Ti. D, + Tt - W, -m, * II,. The parcel T,S- D, is due to the constituent deformation and the parcel T; - W, - m, - u, is due to the mechanical interaction among the constituents. If mechanical equilibrium is considered, the only non-zero term in the right side of (48) is ea, if each constituent is undergoing a particular rigid body motion the right side of (48) is reduced to Tl. W, - m, * u, + $a and, finally, if no restriction on the deformation of the constituents is considered, the whole right side of (48) must be taken into account.
RIGID
SOLID
MATRIX
AND OSTWALD-DE
WAELE
FLUID
The systematic procedure proposed in the previous section may be used to verify if a given set of constitutive equations is thermodynamically admissible. For instance, the constitutive equations proposed by Martins-Costa et al. [S--7] and Costa Mattos et al. [4] for a binary mixture composed by a rigid solid constituent and an Ostwald-de Waele incompressible fluid constituent. In this case, the free energy, the intrinsic and thermal dissipation potentials, for the solid and the fluid constituents represented respectively by the indices 1 and 2, are defined as:
s 0.
A, =
a,log(
dl - k b,(8, - 0,“)’
u= 1,2
(50)
0:
Y”,2 = -
&(U1_2.Vl_2)(2~+2):2 ‘yy2 =&w,_,.
Yy”12 = - RiZ(l + Slo,,,,)~
wl_2)‘+’
(53)
where a,, b,, a2, b2, m and c are positive constants of the mixture and n is a constant. k,, a = 1,2 is a tensorial function of 0, and 11 vlm2 /I = (u~._~~u~_2)1i2 with k,(0,, /Iv1-2 11)being a second order symmetric and positive definite tensor for all (e,, 11 u1_2 11)./? and E are positive functions of e2 and d, RI2 and 6 are positive-valued parameters which depend on both constituents thermal properties and on the internal structure of the mixture [4,3] and e,” is a reference temperature. Under these hypotheses and using equations (26)-(33) the constitutive equations for the mixture are: S, = a,[loge,
- l0ge,O]
Tl = p,l + 2fiW,-,
p = &(W2_1’ W2_i)C q. = - k,grad8,
a =
~1= 1,2
+ b,(e, - e,“) T, = - p,l
+
P = PP2
.D2Jn
(54)
2~02 + 2/iW,_,
1,2
(55) (56)
m, = -d~~vI_2~~2mu1._2 = -m2
(57)
$1 = - R12(1
(58)
+ ~~~~l-2i~)(el-2) = - *2.
428
H. S. Costa Mattos et al
Considering /?(0,) = p. and ~(0,) = PO, the fluid constituent is called thermically insensitive and p = pO(DZ.DZ)n, E = po( IV, - W,)‘. If n = 0 the fluid is called Newtonian. If II = - 1 the fluid has a plastic behaviour. It is important to remark that the material constants al, bi, a2, bZ, po, y, f3: and IZmust be obtained from experiments using a theory of mixtures viewpoint. Hence, although the constitutive equation for the symmetrical part of the partial stress in the fluid constituent Ti is similar to the constitutive equation for the stress in the fluid (regarded as a continuum), the material constants are not necessarily the same. The antisymmetrical part T; of the partial stress tensor T2 in the fluid constituent is zero in an irrotational flow. From now on the discussion with will be restricted to such kind of flow. If the mentioned constants, for each constituent of the mixture, are not available it is possible to relate some of them with the constants of the solid matrix and of the fluid, regarded as continua. An analogy with the partial stress tensor proposed by Williams [13] for a Newtonian fluid allows to express the coefficient po, related to the fluid constituent viscosity as: PO = A(P2Vo
(59)
where q. is the viscosity parameter related to the Ostwald-de Waele fluid, whose value is found in the literature, ,? is a positive-valued factor, accounting for the solid matrix structure and cpis the fluid fraction, coincident to the porosity in saturated flows. The pressure acting on the fluid constituent, pF, may be related to the pressure acting on the mixture as a whole, p, by the following relation: P2 = (PP.
(60)
Under the previously mentioned hypotheses, the fluid constituent partial stress tensor, for an incompressible and thermically insensitive Ostwald-de Waele fluid, may be expressed as: T, = - cppl + 21(p2qo(D2 *&)“&.
(61)
It is also important to remark that the thermal conductivities of the constituents in the mixture are not necessarily equal to the thermal conductivities of the constituents materials, regarded as continua. The partial heat flux for both constituents, according to equations (56), may be given by the equations used by Martins-Costa et al. [S]:
q2 = - Ak,cp grad o2
q1 = - Aks(l - rp)grad0,
(62)
where kF and ks are the fluid and the matrix thermal conductivities, scalar factors for isotropic media, A is an always positive scalar parameter, which may depend on both the internal structure and the kinematics of the mixture. According to Saldanha da Gama and Sampaio 1141, the interaction force m2 may be written as:
where K is the porous matrix permeability, provided that the stress tensor acting on the fluid (in a classical continuum mechanics description), T, is given by: T = - pl + 2qo(D. D)nD
(64)
where p is the pressure acting on the fluid and D is the symmetric part of the velocity gradient. Finally, considering the basic balance equations shown in the basic theory section and the constitutive equations (58) and (61)-(63) discussed in this section, the governing equations to describe the flow of an Ostwald-de Waele fluid in a rigid solid matrix and the heat transfer process are automatically obtained. If initial and boundary conditions are given, the solution of the resulting problem will always satisfy the principle of objectivity and the second law of thermodynamics.
Modelling of momentum and energy transfer
429
TWO EXAMPLES
In order to illustrate the theory, two distinct examples are considered. In the first one, the fluid flow in a porous channel bounded by two isothermal impermeable flat plates, as shown in Fig. 1, is simulated, Assuming one-dimensional steady-state flow, and the following material constants: a, = 0.46 x lo3 J/kg K, a2 = 4.2 x lo3 J/kg K, bi = b2 = 0, n = 0, ylo= 10-j kg/m.s, RI2 = 100 W.m/K, 6 = 0, cp = 0.5, A = 1, A = 1, kF = 0.6 W/m.K, ks = 60 W/m. K and K = 10e5rne2, the fluid constituent velocity profile can be obtained analytically [S], since each constituent’s velocity was prescribed on the boundaries. It should be noticed that the velocity profile is similar to those obtained by Johnson et al. [15,16]. Although the physical problem is different (isothermal mixture of solid particles and fluid) such results lend support to the present work. Two remarks are important at this point. First that unless the velocities (or displacements) were known, some restrictions on the partial stresses should be imposed on the boundaries. Rajagopal et al. [ 171 have proposed a physically meaningful criterion to obtain appropriate partial tractions on the boundaries of saturated mixtures of a fluid and a non-linear elastic solid. The second one is that the fluid constituent inlet temperature is the only boundary condition prescribed in the flow direction (x). This requires an iterative procedure to simulate the problem [ 181. Figure 2 shows both constituents centerline temperatures vs the x-variable and Fig. 3 plots both constituents’ temperatures for a section near the channel exit. In Fig. 2 a significative temperature difference can be observed at the channel entrance (where the fluid constituent temperature is prescribed). This difference decreases as the channel exit is reached. In Fig. 3 the behaviour of a section is shown. The difference is more
IMPERMEABLE
Y
t
AND ISOTHERMAL
A
j.:.:.:
T,=
0
”
Fig. 1. Scheme for Example 1.
SOLID
Fig. 2. Centerline temperature YSx. NlM 30-4-C
SURFACES
430
H. S. Costa Mattos et al.
Fig. 3. Temperature vs y - section x = 110.
Fig. 4. Scheme for Example 2.
acute at the central point (y = 0.5) and decreases as both the superior and inferior impermeable surfaces are reached (where both constituents temperatures have the same prescribed value). In the second example the heat conduction in a binary mixture of rigid solids [19] as shown in Fig. 4 is analitically described. The constitutive hypotheses (56) and (58) are reduced to: qi = - Akicpigrad8i i = 1,2 (65) rki = - Rij(ei - ej)
i,j = 1,2; j # i.
056)
The energy equation, supposing an infinite plate (x + 00) with height L,results in the following system: d2& Aklql- dyl + RIZ(& - 01)
krp,
(67)
d2&
-dy2 +
R,I@I
-
w-9
02)
subject to prescribed temperatures at the impermeable interfaces (y = 0 and y = L):
b(O)= 40
e,(o) = e20
(69)
e,(L) = OIL
e,(L) = ezL..
(70)
Assuming RI2= R2, constant the following temperature tained:
profiles are analytically ob-
e,(y) = e,(Y) =
$1
& 1
(Ae-“Y + Beay)+
Cy + D
(71)
(AemaY + Be”‘)+
Cy +
D
(72)
2
2
where: E 1
J&I R 12
i 2 = Adz R 12
1 a2=-+Ii,
1 i2
(73)
431
Modelling of momentum and energy transfer
where ki is a constituent thermal conductivity (regarded as a continuum), cpiits porosity and A a positive-valued factor to account for the porous matrix microstructure. Considering the boundary conditions stated in (69) and (70) the constants may be defined as: A
=
ezL- elL - e-9e20- elo) QL -
B=
e2L+ elL + dL(e20 -- ho)
-
e
OL -
{
(e2L+ elL) - (e2o+ eld +
(75)
e-aL
^
c =&
(74)
e-L
1
~c(e,, - ho) 1 2
I$ D
=
f
(ho
+
e2d
FINAL
-
$20
-
8,d
(e2L- elL)i
(76) I
-ii,
___ ii, +
(77) f,’
REMARKS
To set up a general constitutive theory for binary mixtures, which, of course, must take into account dissipative phenomena, it is necessary to consider aspects of the second law of thermodynamics. The systematic procedure to obtain constitutive relations that verify automatically the principle of objectivity and the second law of thermodynamics presented in this paper is a promising tool in the modelling of rigid solid-fluid mixtures. Interesting works can be made by analysing the dependence of the constitutive equations on both constituents temperatures and the resulting thermomechanical couplings.
REFERENCES 1. C. Truesdell, Sulle basi della termomeccanica. Rend. Accad. Lincei 22, 158-166 (1957). 2. R. J. Atkin and R. E. Craine, Continuum theories of mixtures. Basic theory and historical development. Quart. J. Mech. A&. Math. 29, 209-244 (1976). 3. H. Costa Mattos, Constitutive relations for fluids: an internal variable theory. In Proc. I1 th Braz. Congress Mech. Engng, 169-172 (1991). 4. H. Costa Mattos, M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, A thermodynamically consistent constitutive theory for a rigid solid-Stokesian fluid mixture. Mech. Research Comm. 20, 243-249 (1993). 5. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, Modelling and simulation of energy transfer in a saturated flow through a porous medium. Appi. Math. Model/ins 16, 589-597 (1992). 6. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, Modelling and simulation of natural convection flow in a porous cavity saturated by a non-Newtonian fluid. In Proc. 4th Braz. Thermal SC. Meeting 569-572 (1992). 7. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, Modelling and simulation of natural convection flow in a dilatant fluid-saturated porous cavity. In Proc. 13th Iberic-Latin-American Conaress Comp. Meth. Engng 2, 314-323 (1992). 8. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, Modelling and simulation of natural convection flow in a saturated porous cavity. Meccanica 29, 1-13 (1994). 9. A. E. Green and P. M. Naghdi, On the thermodynamics and nature of the second law for mixtures of interacting continua. Quart. J. Appl. Math. 31, 265-293 (1978). 10. A. Bedford and D. S. Drumheller, Theories of immiscible and structured mixtures. Int. J. Engng Sci. 21, 863-960 (1983). 11. P. Germain, Micanique des Milieu Continus. Masson, Paris (1973). 12. P. Germain, La mtthode des puissances virtuelles en mecanique des millieux continus l&e partie. Theorie du second gradient. J. Mecaniclue 12. 235-274 (1973). 13. W. 0. Williams, Constitutive equations fo; a flbw of an incompressible viscous fluid through a porous medium. Quart. Appl. Math. 36, 255-267 (1978). 14. R. M. Saldanha da Gama and R. Sampaio, Constitutive relations for the flow of non-Newtonian fluids through saturated Dorous media. J. Braz. Sot. Met. Sci. 7. 267-280 (1985). 15. G. Johnson, M. Massoudi and K. R. Rajagopal, Flbw of a flu&solid mixture between flat plates. Chem. Engng Sci. 46, 1713-1723 (1991). 16. G. Johnson, M. Massoudi and K. R. Rajagopal, Flow of a fluid infused with solid particles through a pipe. Int. J. Engng Sci. 29, 649-661 (1991). 17. K. R. Rajagopal, A. S. Winerman and M. Gandhi, On boundary conditions for a certain class of problems in mixture theory. Int. J. Engng Sci. 24, 1453-1463 (1986). 18. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, An algorithm for simulating the energy transfer process in a moving solid-fluid mixture. J. Braz. Sot. Met. Sci. 13, 337-359 (1991). 19. R. M. Saldanha da Gama, A model for heat transfer in rigid continuous mixtures. Int. Comm. Heat Mass Transjk 16, 123-132 (1989).
Pergamon
0020-7462(95)00016-X
ON THE MODELLING OF MOMENTUM AND ENERGY TRANSFER IN INCOMPRESSIBLE MIXTURES Herald0 Department
of Mechanical
Maria Laboratkio
Laura
S. Costa
Mattos
Engineering, Universidade Federal Fluminense, 156, 24210-240, Niter&, RJ. Brazil
Martins-Costa
National
and Rogkrio
M. Saldanha
de ComputaGZo Cientifica, Rua Lauro Rio de Janeiro, Brazil
(Received 6 October
1994; in revisedfirm
Rua Passo da PBtria,
da Gama
Miiller, 455; 22290-160,
1 February
1995)
Abstract-Although many papers concerning the continuum theory ofmixtures have been produced in the last years, the adequate form for the first and second laws of thermodynamics is still a controversial subject. In this paper, the basic principles that govern the evolution of a continuous mixture are postulated and used to develop a systematic procedure to obtain thermodynamically admissible objective constitutive equations. The prescription of five thermodynamic potentials is sufficient to define a complete set of constitutive relations. Two examples concerning the thermal analysis of a binary mixture of rigid solids and fluid flow in a porous channel bounded by two isothermal impermeable plates are presented. They illustrate the possibilities of effective practical use of the proposed theory.
INTRODUCTION
Some papers concerning the continuum theory of mixtures have been produced in the last years. In these papers a mixture is regarded as a superposition of kinematically independent continuous constituents [l, 21. Nevertheless the adequate form of the basic governing principles (mass balance, linear and angular momentum balances, first and second laws of thermodynamics) in a continuum theory of mixtures context is still a controversial subject. In some of these papers the first and second laws of thermodynamics do not take into account the power of the interaction forces and they do not exclude the possibility of unusual behaviours such as a heat supply to a constituent arising from the interaction with another constituent at a lower temperature and negative work in a cyclic isothermal process. To set up a general theory for momentum and energy transfer in continuous mixtures it is necessary to consider aspects of both the first and second laws of thermodynamics, since dissipative phenomena must be taken into account. Such phenomena are important, for instance, in the simulations of processes like the manufacturing of some composites or petroleum flow through a porous matrix. In the present work, the basic equations that describe the dynamics of a continuous mixture are obtained from the virtual power principle. This leads to a natural interpretation of the interaction force among the constituents and the antisymmetrical part of the partial stress tensor. The use of the continuum mixture approach gives rise to parameters (absent in a classical continuum mechanics approach) to account for the thermomechanical interaction among the constituents. Restrictions, arising from the second law of thermodynamics, for these constitutive quantities are derived in this paper. It is shown that the local form of the first and second laws of thermodynamics must take into account the power of the interaction forces between constituents in order to describe adequately the thermomechanical couplings. A systematic procedure is also proposed to obtain a complete set of constitutive equations that always satisfy the second law of thermodynamics and the principle of objectivity, regardless of the geometry, the external actions on the mixture, the boundary 419
420
H. S. Costa Mattos et al.
and initial conditions. These equations are proposed aiming to offer a minimum of complexity in order to render favourable experimental confirmation. This study can be regarded as a sequence of the work of Costa Mattes [3] and Costa Mattes et al. [4] and this procedure is used to show that the equations proposed by Martins-Costa et al. [S-S] for the energy transfer in a binary mixture composed by a rigid solid matrix and an Ostwald-deWaele incompressible fluid are thermodynamically admissible. It is important to remark that we do not claim the generality of the second law version proposed in this paper and that it is rather restrictive. This work is basically concerned with a mathematically correct and physically realistic modelling of mixtures and this version is conceived to exclude physically unusual material behaviours.
BASIC THEORY
In this section, the basic governing principles for a continuous mixture are postulated. The basic assumption used in the theory is that the mixture may be viewed as a superposition of n kinematically independent continua [l, 2,9]. For the sake of simplicity the presentation is limited to processes such that in every instant t, the whole of some region of the Euclidean space E is simultaneously occupied by all the constituents. Besides, it is also assumed that there is no interchange of mass among the constituents. Similarly as in Green and Naghdi [9] a motion of a constituent g= is defined by a sufficiently smooth vector function xa which assigns position x, = x8(X,, t) to each material point X,. In the principles presented in this section for each constituent %a a material part P, occupying the whole of a region R, = R c E at the current configuration is considered, the boundary of R, being denoted by aR= G dR. Mass conservation
principle
Defining p. as the mass density of P,, the mass of the constituent occupying the region R at a given instant t can be written as: M=(P=, t) =
s R.
p=dV.
(1)
The densities of the constituents must be such that: ” dM” (P,, t) = 0. =:, dt
(2)
In the absence of mass interchange between constituents, the following identities hold: T
(P,, t) = 0 3 2
+ p,div v, = 0
VP=
where D”( )/Dt denotes material time derivative holding X, fixed. Virtual power principle
The virtual power principle is a convenient framework in which to define the dynamics of a continuous mixture. Compared to the traditional approach - conservation of linear and angular momentum - [l-lo] the principle of virtual power is commonly considered more abstract and hence more difficult to apply. However, in the case of mixtures, the great advantage of this alternative approach is that it allows a simple interpretation of the additional internal forces that arise in the theory. In order to determine the forces acting on the part P= that occupy the region R, G R at the current configuration it is necessary to introduce the space VIR(R=) of all possible velocity fields which take place at a given instant. VIR(R=) is called the space of virtual velocities and, for the sake of simplicity, throughout this work it is defined as the space of all vector fields v : R, --* E which are sufficiently regular to ensure the existence of its gradient. VIR(R=) = {v: R, --* ElVv is defined}.
(4)
421
Modelling of momentum and energy transfer
any constituent %?=the power of external forces PgXT and the power of inertial forces P& for a given virtual field of velocity w, E V,,(R,) are defined as: For
PbEXT(Pa, 44 =
s
b,.w,dV+
R.
Ph(P,, 4 w,) =
f,.w,dA s aR.
s
p,a,-w,dV
(6)
R.
where b, is the external volume force acting on P,, f, is the surface force acting on the boundary aR, and a, is the actual acceleration of the constituent. In the mechanics of a single continuum it is usual to consider a first order gradient theory in which the power of the internal forces is supposed to be a function of the velocity and its gradient [l 1,121. Extending this to a continuous mixture and, on the assumption that, for a fixed instant t and for any constituent V,, the power of the internal forces can be expressed as a linear functional on V,R(R,), it may be shown that PyNT(Pn,t, w,) must have the following form:
fYNT(Pm4 w,) =
s
[T,“.(V”w,) + T:.(V”w,) - m,.w,]dV
R.
(7)
where VuW, and V’w, are, respectively, the antisymmetrical and symmetrical parts of VW,; with T,” being an antisymmetric tensor, Ti a symmetric tensor and m, a vector. T,“, Ti and m, will be called the generalized internal “forces”. Making an analogy with the theory presented in Atkin and Craine [2], T,” and T,” may also be called the antisymmetrical and symmetrical parts of the partial stress tensor and m, the interaction force. Using the previous definitions, the virtual power principle may be stated as follows: (Pl) For the material parts P, c %?awhich occupy an arbitrary region R, = R at the present configuration, the antisymmetrical and symmetrical parts of the partial stress tensor, T,” and Tl”, the interaction force m,, the body force b,, the surface force& and the actual acceleration a, of Wgmust be such that: i
P;N(Po,t,~a)
a=1
+ i
P;,&Pa,
4wg) = i
P&(Pn,t,W,)
VW, E OR.
(8)
a=1
a=1
(P2) The term Cl= 1P&(P., t, w,) is zero for a virtual rigid body motion, that is when all the constituents have the same rigid body velocity: n c PTNT(Pa, t, w,) = 0 when w,(x,) = A(x, - x,,) + co (9) n=l
where A is an antisymmetrical tensor and co is the velocity of a reference point x0 E E. Under suitable regularity assumptions, it can be proved that (PI) implies the following local expressions: paa, = div (Ti + Ti) + m, + b,
in R
and (T: + T,“)n =h It
in i?R, VP,.
(10)
can also be proved that (P2) implies the following local expressions: n
1 m, = 0 a=1
The role understood undergoing deformation is necessary NLM 30-4-B
and
=$I T,” = 0.
(11)
of the additional internal forces T.” and m, in the internal power can be from simple examples. In the first example each constituent is supposed to be a different translation (i.e. u,(x,, t) = c,, c1 # c2 # . . . # c,). Although the of each constituent is zero, it is reasonable to develop a mixture theory where it to apply some external action in order to impose such kind of motion. From the
422
H. S. Costa Mattos et al
definition (7) of the power of the internal forces, it is possible to verify that: j1 PMP.9
t, %) = -
[m, Fc,] d V. i a=1 s R.
(12)
If the internal forces m, were not taken into account in definition (7), the power of the internal forces for such kind of motion would be zero. Since the power of the inertial forces is zero (because the acceleration of each constituent is zero), from the principle (Pl) it comes that the power of the external forces (and hence the external forces) should also be zero, which is not physically expected. Hence, in this case, the theory has to consider the (generalized) internal forces m, in order to allow a non zero internal power, which indicates that it is necessary to apply some external action in order to impose such kind of motion. In the second example each constituent is suposed to be undergoing a different rotation (i.e. u,(x,) = A,Jx, - x,,) = A1 # A2 # . . . # A,,). From the equations (7) and (1 l)i it may be concluded that:
s
CT,“- W,] dV
a=1
(13)
R.
Similarly to the first case, although the deformation of each constituent is zero, it is reasonable to expect a mixture theory where it is necessary to apply some external action in order to impose such kind of motion. If the internal forces T,” were not taken into account in the definition (7), the power of the internal forces for such kind of motion would be zero. Since the power of the inertial forces is zero (because the acceleration of each constituent is zero), from the principle (Pl) it comes that the power of the external forces (and hence the external forces) should also be zero, what is not physically expected. Hence, in this case, the theory has to consider the internal forces T,” in order to allow a non zero internal power, which indicates that it is necessary to apply some external action in order to impose such kind of motion. These simple examples show that the additional internai forces T,” and m, must be introduced in the continuum theory of mixtures in order to take into account the work done when the constituents are undergoing independent rigid body motion. First law of thermodynamics
order to postulate the energy balance for a mixture it is necessary to define the internal energy U’(R) t), the kinetic energy K”(R, t) and the nonmechanical power P&(R, t): In
U”(P=, t) =
K”(P=, t) = ;
PUP.3
t) = -
s R,
s R.
P=e=dV
(15)
p=v=.v=dV
q,.ndA
s dR.
(14)
+ s R.
(16)
para d V
where e, is the internal energy per unit mass, u, is the actual velocity, qa is the heat flux vector and r= is a heat supply or source per unit mass and time. The first law of thermodynamics can then be stated as: $
;
U”(P=,t) + =$, $ K”(P=,t) = =cl PbT(P=,
t>u=) + i
PaTH(P=,t).
(17)
a=1
Using the definition of the power of internal forces (7) and the mass conservation principle it is possible to prove that, under the appropriate continuity assumptions, (d/dt)K,(P,, t) = P;,(P=, t, v,) and the first law of thermodynamics may be written as: g
i
U,(P,, t) = i
Q=l
Phv(Par t,tl) + i a=1
KfPmt).
(18)
423
Modelling of momentum and energy transfer
The local form of the principle stated above is given by: (19) with D, = V’u, and W, = V%,. The version (19) of first law of thermodynamics is stated for the mixture as a whole. Nevertheless it is easy to verify that (19) is equivalent to the following identities: p=&e=-
T,“.W=-
VP, with i
T,.DD,+m,.v,+divq,-p,r,=IC/,
$a = 0
(20)
a=1
The scalar tja represents the thermal energy supply to the constituent %?,arising from its interaction with the other constituents of the mixture. Second law of thermodynamics
If the first law of thermodynamics states the possibility of conversion of mechanical energy into heat and vice-versa, the second law of thermodynamics provides a distintion between possible and impossible processes. In this section a version of the second law for continuous mixtures is postulated. It is important to remark that there are different opinions about the validity of some definitions. We do not claim the generality of our version. In order to state an entropy production inequality, it is assigned to each constituent a temperature 8,, which is assumed to be positive, and an entropy per unity of mass s,. The total entropy S, of the constituent %?aoccupying the region R at a given instant t is given by:
S,(R,t) =
s
E~,sald V.
(21)
R.
The entropy s, and the temperature
s
0, are postulated to obey:
!!!_?!dA+
aR.
ea
The entropy inequalities proposed above do not restrict some unusual behaviours since they do not take into account the dissipation arising from the thermomechanical interactions among the constituents. Thus, an additional restriction that takes into account the mixture as a whole is necessary: (23) Using (20) to eliminate r. on (22) and introducing the free energy A, of each constituent, A, = e, - O=s=,it is possible to obtain the following local forms of the inequalities (22) and (23): d== T,.D=-p, di=
D”A,
D”8,
Dt
+ ‘= Dt ) ---
q=.grade= 0,
,. -
vp =
i
(24)
a=1
in which d, is the dissipation of each constituent seen as a single continuum and di is the dissipation that arises from the interaction among the constituents. The inequality (24), implies the following inequality which is exactly the version of the second law of thermodynamics proposed by Atkin and Craine [2]:
C-D, - P=
q=wade= 8,
1
, o -*
(25)
Obviously, if the inequalities (24) are verified, then (25) is also verified. The inequalities (24) are an adequate local version of the second law restriction. The role of the additional inequality (24), can be easily understood by considering (11) and (20)2. If
424
H. S. Costa Mattos et al.
these identities are assumed to hold, it can be shown that the dissipation di is zero when all the constituents have the same temperature and the same velocity (or, in other words, when there is no interaction among the constituents).
CONSTITUTIVE
EQUATIONS
The balance equations (3), (lo), (1 l), (20), and the second law restrictions (24) are valid for any kind of continuous mixture. A complete modelling requires additional informations in order to characterize the behaviour of each kind of mixture. A systematic procedure to obtain a set of thermodynamically admissible constitutive equations that verify automatically the principle of objectivity, may be stated by the steps below. In order to simplify the presentation, the study is restricted to incompressible constituents. (Hl) Constituents will be considered for which the free energy A, is a differentiable function of the temperature 19,: A= = i=(e=)
a= 1,n.
(26)
1, n.
(27)
(H2) The following state law holds:
d-4
s== -de,
a=
A consequence of the hypotheses 1 and 2 is that the dissipation d=(see equation (24),) will always be equal to zero in equilibrium, that is when u, = 0 and grad 8, = 0. (H3) The symmetrical part of the partial stress tensor Ti is such that: T, = -p=l
+ TiR
a=
1,n
(28)
where the pressure p= is a Lagrange multiplier related to the incompressibility constraint and T,‘” represents the irreversible parcel of Ti. In order to characterize completely the behaviour of the constituent, additional information about the variables TiR, q,, T,“, ma and $= must be given. In this theory, the additional information is obtained from potentials, called dissipation potentials. The following compact notation will be adopted: 8=_, = 8, - O,, u=_~ = u, - vg, Warns = W, - W,, g. = (l/f?=)grad 8,. (H4) Associated to each constituent %?=there exist two objective and differentiable potentials 0’ and @r, such that: (29) (HS) For the mixture as a whole, there exist two objective and differentiable potentials Yr and Y”, such that: T,” = i
S,,;
8=1
s,,
=
ayz a wm+
-.
mar = f:
h,,;
/3=1
h,, = _ ay’. a0,_8’
*, = i
Las
J9=1
& = - $. QB
(30)
The terms S,,, h, and <=# can be interpreted as the contribution arising from each constituent V, to the antisymmetrical partial stress T,“, the interaction force m, and the energy supply @=of the constituent V,. The definition of the potentials A,, a”, @‘, Yz, Y’T is motivated by experimental observation in simple tests. Therefore, the constitutive equations (27)-(30) may eventually allow some thermodynamically impossible processes. Besides, the identities (11) and (20), must always hold. In order to assure that the local version (24) of the second law and that the identities (11) and (20), will always be satisfied, independently of the geometry, of the external actions and
Modelling of momentum
425
and energy transfer
of the boundary conditions, it is sufficient to restrict the choice of a’, QT, ‘I”’ and Y’Tin the following way;
(i) The
potentials
@’ and
@’ = i
QT
have the following particular form:
&,f(D,),
QT = i
Q(O) = Q(O) = 0
W&),
V%=
(31)
a=1
a=1
where for a given a(1 I a I n)@(D.) and &,$(g=) are non-negative convex differentiable objective functions. (ii) The potentials ‘-I” and YT have the following particular form: Y’=
i f: [Yy.ws-Y$] a=1 p=ol
YQ
iYfB 12=1 p=a
(32)
in which: Y&4 = %&-&,
Y$ = %#u-,),
Y’B,@ = %s(@z-s)
(33)
where, for a given pair (a, p>, (1 I a I n, 1 I B I; n), Y&, Ys and Y$ are non-negative convex differentiable objective functions such that: ‘y; E Y,w,, 9,w,(K-,)
‘u:, = Y;;,,
= - q,,“,cI+,,, %&?-,)
Y$ = Y&
%&,-,I
= - ~‘&s-=)
= - 9’f&-,)
9$(O) = 9$(O) = q:@(o) = 0.
(34)
From equations (34) and (30), it may be concluded that:
Thus, using (35) and the definition of &+, h,, and S,, in (30) it is easy to verify that the identities (11) and (20)2 will always be satisfied if (H6) holds. The demonstration that the inequalities (24) are verified if hypothesis (H6) holds may be done by using the following classical result of the Convex Analysis: Let X and Y be elements of a vector space V with an internal product (X. Y). If 0 : V + [0, + co) it is a non-negative convex and differentiable function such that O(O) = 0, then (X. Y) 2 O(X) 2 0 if Y = d@/dX. Taking X = (D,), Y = (TiR) and (X * Y) = D, - T,‘” it is easy to verify that D,sT,‘~
2 0
(36)
provided that (29), and item (i) of (H6) hold. Similarly, taking X = (g.), Y = ( -qJ and (X . Y) = - 9.. q, it is easy to verify that - (I,.& 2 0
(37)
provided that (29), and item (i) of (H6) hold. Hence, from (36) and (37) it comes that the inequality (24), always holds if (H4) and item (i) of (H6) hold. Using the same kind of reasoning it is possible to verify that: s,,. W.-s 2 0;
- haS.vm_8 2 0;
- 5,aen-s
2 0
provided that (H5) and item (ii) of (H6) hold. Hence, since the absolute temperature always positive, it may be concluded that:
(38) is
426
H. S. Costa Mattos et al.
and (39)
or:
and ”
n
=L &‘7=1
a=1
(40) provided that hypothesis (H6) holds. Using the definition of &, h,, and S,, in (30), it finally comes from (40) that the inequality (24), will always be satisfied if hypothesis (H6) holds. It is important to remark that this demonstration can be easily extended if other variables are taken as independent parameters in the potentials, allowing a more realistic model of the coupling among different mechanisms. For instance: Qi = &,h(D,; Di, BayOi)
(41)
@aT= g)$(g=; si, L
(42)
6)
I+-j,Bi-j,Vi-j)
Y,W,= q,Wa(W,_p;e,_p,V,-8,
Wa-p, Vi-j, Oi-j, Wi-j)
Yip = q;s(Va_a;ea_8, Y$ = ‘k:,(e,_,;
(43)
Wa_p,V,-~,ei-j
Wi-j,Ui-j).
(44) (45)
equation (41)-(42) i = 1, n; i # cc;and in equation (43)-(45) i = 1, n; i # a; j = i, n;j # fl. If A,, O”, W, Y”, YT are known, equations (27)-(30) form a complete set of objective and thermodynamically admissible constitutive equations, provided the hypothesis (H6) is verified. It is also interesting to remark that the relation (38), In
5,s(e, - 0,) 2 0 assures that the heat supply (aa from the constituent WPto the constituent Ws,,arising from the interaction between them must be positive if 8, < 8, and negative if 8, > 8,. From inequalities (38),,, it can also be shown that, for an isothermal process in which the constituents are undergoing independent rigid body motion, the work in a cycle must be non-negative. ANALYSIS
OF THE THERMOMECHANICAL
COUPLINGS
An alternative local form of the first law of thermodynamics (the energy balance equation) can be obtained by introducing the constitutive relations proposed in the previous section in equation (20). Initially, using the definition! of free energy A, and the state law (27) it can be concluded that: Dole, -=G’
D”B,
(46)
Dt
cm =- dZA,e de,’ a 1.
where c, is defined as:
i
(47)
Then, inserting (46) in the local form (20) of the first law of thermodynamics, it comes that: D”B,
divq, + P,C, --par,= Dt
T,“.D,+
T:.W,--m;v,+$,.
(48)
427
Modelling of momentum and energy transfer
Finally, using the constitutive relations (28)-(30), the following alternative expression is obtained: D”8
div q= + pat, $
n aw n a\yT 1 p-u 8=1av,-pa +c--8=1 a,-;
aV ,=,a~-,
$,.+
f
01
- para =
--Ww,+
(49)
The terms in the right side of equation (48) are responsible for the thermomechanical couplings: the temperature evolution on the constituents is coupled due to the term I+Q, and the mechanical problem is coupled to the thermal problem due to the term Ti. D, + Tt - W, -m, * II,. The parcel T,S- D, is due to the constituent deformation and the parcel T; - W, - m, - u, is due to the mechanical interaction among the constituents. If mechanical equilibrium is considered, the only non-zero term in the right side of (48) is ea, if each constituent is undergoing a particular rigid body motion the right side of (48) is reduced to Tl. W, - m, * u, + $a and, finally, if no restriction on the deformation of the constituents is considered, the whole right side of (48) must be taken into account.
RIGID
SOLID
MATRIX
AND OSTWALD-DE
WAELE
FLUID
The systematic procedure proposed in the previous section may be used to verify if a given set of constitutive equations is thermodynamically admissible. For instance, the constitutive equations proposed by Martins-Costa et al. [S--7] and Costa Mattos et al. [4] for a binary mixture composed by a rigid solid constituent and an Ostwald-de Waele incompressible fluid constituent. In this case, the free energy, the intrinsic and thermal dissipation potentials, for the solid and the fluid constituents represented respectively by the indices 1 and 2, are defined as:
s 0.
A, =
a,log(
dl - k b,(8, - 0,“)’
u= 1,2
(50)
0:
Y”,2 = -
&(U1_2.Vl_2)(2~+2):2 ‘yy2 =&w,_,.
Yy”12 = - RiZ(l + Slo,,,,)~
wl_2)‘+’
(53)
where a,, b,, a2, b2, m and c are positive constants of the mixture and n is a constant. k,, a = 1,2 is a tensorial function of 0, and 11 vlm2 /I = (u~._~~u~_2)1i2 with k,(0,, /Iv1-2 11)being a second order symmetric and positive definite tensor for all (e,, 11 u1_2 11)./? and E are positive functions of e2 and d, RI2 and 6 are positive-valued parameters which depend on both constituents thermal properties and on the internal structure of the mixture [4,3] and e,” is a reference temperature. Under these hypotheses and using equations (26)-(33) the constitutive equations for the mixture are: S, = a,[loge,
- l0ge,O]
Tl = p,l + 2fiW,-,
p = &(W2_1’ W2_i)C q. = - k,grad8,
a =
~1= 1,2
+ b,(e, - e,“) T, = - p,l
+
P = PP2
.D2Jn
(54)
2~02 + 2/iW,_,
1,2
(55) (56)
m, = -d~~vI_2~~2mu1._2 = -m2
(57)
$1 = - R12(1
(58)
+ ~~~~l-2i~)(el-2) = - *2.
428
H. S. Costa Mattos et al
Considering /?(0,) = p. and ~(0,) = PO, the fluid constituent is called thermically insensitive and p = pO(DZ.DZ)n, E = po( IV, - W,)‘. If n = 0 the fluid is called Newtonian. If II = - 1 the fluid has a plastic behaviour. It is important to remark that the material constants al, bi, a2, bZ, po, y, f3: and IZmust be obtained from experiments using a theory of mixtures viewpoint. Hence, although the constitutive equation for the symmetrical part of the partial stress in the fluid constituent Ti is similar to the constitutive equation for the stress in the fluid (regarded as a continuum), the material constants are not necessarily the same. The antisymmetrical part T; of the partial stress tensor T2 in the fluid constituent is zero in an irrotational flow. From now on the discussion with will be restricted to such kind of flow. If the mentioned constants, for each constituent of the mixture, are not available it is possible to relate some of them with the constants of the solid matrix and of the fluid, regarded as continua. An analogy with the partial stress tensor proposed by Williams [13] for a Newtonian fluid allows to express the coefficient po, related to the fluid constituent viscosity as: PO = A(P2Vo
(59)
where q. is the viscosity parameter related to the Ostwald-de Waele fluid, whose value is found in the literature, ,? is a positive-valued factor, accounting for the solid matrix structure and cpis the fluid fraction, coincident to the porosity in saturated flows. The pressure acting on the fluid constituent, pF, may be related to the pressure acting on the mixture as a whole, p, by the following relation: P2 = (PP.
(60)
Under the previously mentioned hypotheses, the fluid constituent partial stress tensor, for an incompressible and thermically insensitive Ostwald-de Waele fluid, may be expressed as: T, = - cppl + 21(p2qo(D2 *&)“&.
(61)
It is also important to remark that the thermal conductivities of the constituents in the mixture are not necessarily equal to the thermal conductivities of the constituents materials, regarded as continua. The partial heat flux for both constituents, according to equations (56), may be given by the equations used by Martins-Costa et al. [S]:
q2 = - Ak,cp grad o2
q1 = - Aks(l - rp)grad0,
(62)
where kF and ks are the fluid and the matrix thermal conductivities, scalar factors for isotropic media, A is an always positive scalar parameter, which may depend on both the internal structure and the kinematics of the mixture. According to Saldanha da Gama and Sampaio 1141, the interaction force m2 may be written as:
where K is the porous matrix permeability, provided that the stress tensor acting on the fluid (in a classical continuum mechanics description), T, is given by: T = - pl + 2qo(D. D)nD
(64)
where p is the pressure acting on the fluid and D is the symmetric part of the velocity gradient. Finally, considering the basic balance equations shown in the basic theory section and the constitutive equations (58) and (61)-(63) discussed in this section, the governing equations to describe the flow of an Ostwald-de Waele fluid in a rigid solid matrix and the heat transfer process are automatically obtained. If initial and boundary conditions are given, the solution of the resulting problem will always satisfy the principle of objectivity and the second law of thermodynamics.
Modelling of momentum and energy transfer
429
TWO EXAMPLES
In order to illustrate the theory, two distinct examples are considered. In the first one, the fluid flow in a porous channel bounded by two isothermal impermeable flat plates, as shown in Fig. 1, is simulated, Assuming one-dimensional steady-state flow, and the following material constants: a, = 0.46 x lo3 J/kg K, a2 = 4.2 x lo3 J/kg K, bi = b2 = 0, n = 0, ylo= 10-j kg/m.s, RI2 = 100 W.m/K, 6 = 0, cp = 0.5, A = 1, A = 1, kF = 0.6 W/m.K, ks = 60 W/m. K and K = 10e5rne2, the fluid constituent velocity profile can be obtained analytically [S], since each constituent’s velocity was prescribed on the boundaries. It should be noticed that the velocity profile is similar to those obtained by Johnson et al. [15,16]. Although the physical problem is different (isothermal mixture of solid particles and fluid) such results lend support to the present work. Two remarks are important at this point. First that unless the velocities (or displacements) were known, some restrictions on the partial stresses should be imposed on the boundaries. Rajagopal et al. [ 171 have proposed a physically meaningful criterion to obtain appropriate partial tractions on the boundaries of saturated mixtures of a fluid and a non-linear elastic solid. The second one is that the fluid constituent inlet temperature is the only boundary condition prescribed in the flow direction (x). This requires an iterative procedure to simulate the problem [ 181. Figure 2 shows both constituents centerline temperatures vs the x-variable and Fig. 3 plots both constituents’ temperatures for a section near the channel exit. In Fig. 2 a significative temperature difference can be observed at the channel entrance (where the fluid constituent temperature is prescribed). This difference decreases as the channel exit is reached. In Fig. 3 the behaviour of a section is shown. The difference is more
IMPERMEABLE
Y
t
AND ISOTHERMAL
A
j.:.:.:
T,=
0
”
Fig. 1. Scheme for Example 1.
SOLID
Fig. 2. Centerline temperature YSx. NlM 30-4-C
SURFACES
430
H. S. Costa Mattos et al.
Fig. 3. Temperature vs y - section x = 110.
Fig. 4. Scheme for Example 2.
acute at the central point (y = 0.5) and decreases as both the superior and inferior impermeable surfaces are reached (where both constituents temperatures have the same prescribed value). In the second example the heat conduction in a binary mixture of rigid solids [19] as shown in Fig. 4 is analitically described. The constitutive hypotheses (56) and (58) are reduced to: qi = - Akicpigrad8i i = 1,2 (65) rki = - Rij(ei - ej)
i,j = 1,2; j # i.
056)
The energy equation, supposing an infinite plate (x + 00) with height L,results in the following system: d2& Aklql- dyl + RIZ(& - 01)
krp,
(67)
d2&
-dy2 +
R,I@I
-
w-9
02)
subject to prescribed temperatures at the impermeable interfaces (y = 0 and y = L):
b(O)= 40
e,(o) = e20
(69)
e,(L) = OIL
e,(L) = ezL..
(70)
Assuming RI2= R2, constant the following temperature tained:
profiles are analytically ob-
e,(y) = e,(Y) =
$1
& 1
(Ae-“Y + Beay)+
Cy + D
(71)
(AemaY + Be”‘)+
Cy +
D
(72)
2
2
where: E 1
J&I R 12
i 2 = Adz R 12
1 a2=-+Ii,
1 i2
(73)
431
Modelling of momentum and energy transfer
where ki is a constituent thermal conductivity (regarded as a continuum), cpiits porosity and A a positive-valued factor to account for the porous matrix microstructure. Considering the boundary conditions stated in (69) and (70) the constants may be defined as: A
=
ezL- elL - e-9e20- elo) QL -
B=
e2L+ elL + dL(e20 -- ho)
-
e
OL -
{
(e2L+ elL) - (e2o+ eld +
(75)
e-aL
^
c =&
(74)
e-L
1
~c(e,, - ho) 1 2
I$ D
=
f
(ho
+
e2d
FINAL
-
$20
-
8,d
(e2L- elL)i
(76) I
-ii,
___ ii, +
(77) f,’
REMARKS
To set up a general constitutive theory for binary mixtures, which, of course, must take into account dissipative phenomena, it is necessary to consider aspects of the second law of thermodynamics. The systematic procedure to obtain constitutive relations that verify automatically the principle of objectivity and the second law of thermodynamics presented in this paper is a promising tool in the modelling of rigid solid-fluid mixtures. Interesting works can be made by analysing the dependence of the constitutive equations on both constituents temperatures and the resulting thermomechanical couplings.
REFERENCES 1. C. Truesdell, Sulle basi della termomeccanica. Rend. Accad. Lincei 22, 158-166 (1957). 2. R. J. Atkin and R. E. Craine, Continuum theories of mixtures. Basic theory and historical development. Quart. J. Mech. A&. Math. 29, 209-244 (1976). 3. H. Costa Mattos, Constitutive relations for fluids: an internal variable theory. In Proc. I1 th Braz. Congress Mech. Engng, 169-172 (1991). 4. H. Costa Mattos, M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, A thermodynamically consistent constitutive theory for a rigid solid-Stokesian fluid mixture. Mech. Research Comm. 20, 243-249 (1993). 5. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, Modelling and simulation of energy transfer in a saturated flow through a porous medium. Appi. Math. Model/ins 16, 589-597 (1992). 6. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, Modelling and simulation of natural convection flow in a porous cavity saturated by a non-Newtonian fluid. In Proc. 4th Braz. Thermal SC. Meeting 569-572 (1992). 7. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, Modelling and simulation of natural convection flow in a dilatant fluid-saturated porous cavity. In Proc. 13th Iberic-Latin-American Conaress Comp. Meth. Engng 2, 314-323 (1992). 8. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, Modelling and simulation of natural convection flow in a saturated porous cavity. Meccanica 29, 1-13 (1994). 9. A. E. Green and P. M. Naghdi, On the thermodynamics and nature of the second law for mixtures of interacting continua. Quart. J. Appl. Math. 31, 265-293 (1978). 10. A. Bedford and D. S. Drumheller, Theories of immiscible and structured mixtures. Int. J. Engng Sci. 21, 863-960 (1983). 11. P. Germain, Micanique des Milieu Continus. Masson, Paris (1973). 12. P. Germain, La mtthode des puissances virtuelles en mecanique des millieux continus l&e partie. Theorie du second gradient. J. Mecaniclue 12. 235-274 (1973). 13. W. 0. Williams, Constitutive equations fo; a flbw of an incompressible viscous fluid through a porous medium. Quart. Appl. Math. 36, 255-267 (1978). 14. R. M. Saldanha da Gama and R. Sampaio, Constitutive relations for the flow of non-Newtonian fluids through saturated Dorous media. J. Braz. Sot. Met. Sci. 7. 267-280 (1985). 15. G. Johnson, M. Massoudi and K. R. Rajagopal, Flbw of a flu&solid mixture between flat plates. Chem. Engng Sci. 46, 1713-1723 (1991). 16. G. Johnson, M. Massoudi and K. R. Rajagopal, Flow of a fluid infused with solid particles through a pipe. Int. J. Engng Sci. 29, 649-661 (1991). 17. K. R. Rajagopal, A. S. Winerman and M. Gandhi, On boundary conditions for a certain class of problems in mixture theory. Int. J. Engng Sci. 24, 1453-1463 (1986). 18. M. L. Martins-Costa, R. Sampaio and R. M. Saldanha da Gama, An algorithm for simulating the energy transfer process in a moving solid-fluid mixture. J. Braz. Sot. Met. Sci. 13, 337-359 (1991). 19. R. M. Saldanha da Gama, A model for heat transfer in rigid continuous mixtures. Int. Comm. Heat Mass Transjk 16, 123-132 (1989).