Acta Mech 194, 123–140 (2007) DOI 10.1007/s00707-007-0459-3 Printed in The Netherlands
Acta Mechanica
Mixed convection with viscous dissipation in a vertical channel filled with a porous medium A. Barletta, E. Magyari, Bologna, Italy, I. Pop, Cluj, Romania, L. Storesletten, Kristiansand, Norway Received December 21, 2006; revised January 23, 2007 Published online: April 19, 2007 Ó Springer-Verlag 2007
Summary. The fully developed laminar mixed convection flow in a vertical plane parallel channel filled with a porous medium and subject to isoflux isothermal wall conditions is investigated assuming that (i) the Darcy law and the Boussinesq approximation hold, (ii) the effect of viscous dissipation is significant, and (iii) the average flow velocity Um (as an experimentally accessible quantity) is prescribed. It is shown that under these conditions both upward (Um > 0) and downward (Um < 0) laminar flow solutions may exist as long as Um does not exceed a maximum value Um;max . The velocity field can either be unidirectional or bidirectional. Moreover, bidirectional flow configurations are possible also for Um ¼ 0. A remarkable feature of the problem is that for Um < Um; max even two solution branches (dual solutions) exist, which merge when Um approaches its maximum value Um; max . The mechanical and thermal characteristics of the flow configurations associated with the dual solutions are investigated in the paper analytically and numerically in detail.
1 Introduction Although one of the oldest chapters of the engineering sciences, due to its numerous industrial and environmental applications the fluid flow and heat transfer in porous media is still a topic of current research interest. Several comprehensive works published recently in this field (see, e.g., [1]–[4]) give a convincing evidence of this development. Special attention has been paid to the internal flows in ducts and channels filled with porous media, with a broad application in building physics, mechanical, electrical, chemical, energy and environmental engineering. The thermally developing forced convection flow in a parallel-plate channel or circular tube filled by a saturated porous medium with walls at uniform temperature or uniform heat flux, with axial conduction and viscous dissipation, has been investigated by an extended Graetz method in a series of papers by Nield et al. [5]–[8] and Kuznetsov et al. [9]. An exact analytical solution of the Graetz problem for these basic duct geometries when the axial conductivity is significant has very recently been reported by Minkowycz and Haji-Sheikh [10]. The heat transfer in the thermal entrance region of a rectangular passage has been studied by Haji-Sheikh et al. [11] with the aid of the Green’s function method. The computed heat transfer coefficients show that the thermally fully developed regime may not be attained in practical applications for very narrow passages with prescribed wall heat flux. Heterogeneity and variable viscosity effects in ducts filled with porous materials have been considered by Nield and Kuznetsov [12] and by Narashima and Lage [13], respectively.
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In spite of some similarities compared to the viscous flow of clear fluids, the internal (forced, free and mixed) convection in saturated porous media shows also essential differences. Nevertheless, the vast literature accumulated along the decades in the latter field (see, e.g., the comprehensive overviews by Shah and London [14] and Kakac and Yener [15]) offers a solid orientation and help in the investigation of analogous problems in porous media. Thus, Storesletten and Pop [16] have extended the problem of buoyancy–driven viscous flow in a vertical parallel plane channel posed by Banks and Zaturska [17] to the case of a vertical porous layer with non-uniform wall temperature. The effect of viscous dissipation has been included in the study of the combined free and forced convection in a porous medium between two vertical walls by Ingham et al. [18]. Al-Hadhrami et al. [19] proposed a new model for viscous dissipation in porous media across a range of permeability values, while Umavathi et al. [20] presented a numerical and analytical study of the mixed convection in a vertical porous channel using the Brinkman-Forchheimer model with various combinations of boundary conditions and with viscous dissipation effects included. More recent contributions to the effect of viscous dissipation in addition to the buoyancy effects have been published by Nield [21], [22] and by Magyari et al. [23]. The problem we consider in the present paper is the porous-medium analogue of the problems for a clear viscous fluid investigated comprehensively by Barletta [24]–[27], Zanchini [28], Barletta and Zanchini [29] and Barletta et al. [30]. Namely, analytical series solutions are given for the steady fully developed mixed convection flow with viscous dissipation in a vertical parallel plane channel of width L, filled with a fluid-saturated porous medium so that the left wall is subjected to a uniform heat flux qw ð0Þ, while the right wall of the channel is kept at the constant temperature Tw ð LÞ (isoflux isothermal wall conditions). It is assumed that the Darcy law and the Boussinesq approximation hold, and that the average flow velocity Um (as an experimentally accessible quantity) is prescribed. One of the aims of the paper is to put in evidence the quantitative and qualitative changes brought about by the presence of viscous dissipation. The main result of the paper is that both upward (Um > 0) and downward (Um < 0 ) laminar flow solutions may exist as long as Um does not exceed a maximum value Um;max . For Um < Um;max two solution branches (dual solutions) occur, which merge when Um approaches its maximum value Um;max . The mechanical and thermal characteristics of the flow configurations associated with the dual solutions are investigated in the paper analytically and numerically in some detail. These solutions are new for the internal mixed convection flows in porous media. In the case of external mixed convection boundary layer flow over a vertical surface embedded in a fluid-saturated porous medium, however, the existence of dual solutions has already been reported by Merkin [31] nearly three decades ago.
2 Governing equations 2.1 Problem formulation We consider laminar fully developed mixed convection flow in a vertical parallel plane channel of width L, filled with a porous medium. The vertical X-axis is opposite to the acceleration due to the gravity, and the Y -axis is perpendicular to the channel walls which are assumed to be w ð0Þ, impermeable. As shown in Fig. 1 the left wall (at Y ¼ 0) is subjected to uniform heat flux q and the right one (at Y ¼ L ) is kept at the constant temperature Tw ð LÞ. We further assume that the Darcy law and the Boussinesq approximation hold, and that the heat generation by viscous
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Mixed convection with viscous dissipation
X
g
Porous Medium qw(0)
Tw(L)
L
0
Y
Fig. 1. Coordinate system and isoflux isothermal boundary conditions at Y ¼ 0 and Y ¼ L, respectively
friction is non-negligible. Similarly to the case of clear fluids (see [25]), under these assumptions the only non-vanishing component of the seepage velocity field is its longitudinal component U (Xcomponent). Both U and the temperature T of the fluid (which is in thermal equilibrium with the porous matrix) depend only on the transversal coordinate Y . The continuity equation is satisfied identically, the transversal pressure gradient @P=@Y is vanishing, the longitudinal one, @P=@X is constant, and the corresponding Darcy and energy balance equations are l dP U¼ þ q g bðT Tm Þ ; K dX
ð1Þ
d2 T m U 2 ¼ 0: þ 2 dY K cp a
ð2Þ
In the above equations, K is the permeability of the porous medium, g is the gravitational acceleration, b is the coefficient of thermal expansion, cp is the specific heat at constant pressure, a ¼ k=ðqcp Þ is the thermal diffusivity and k the thermal conductivity of the porous medium, q is the fluid density, l is the dynamic viscosity, m ¼ l=q is the kinematic viscosity, P ¼ p þ q g X is the hydrodynamic pressure, and as reference temperature of the Boussinesq approximation the average fluid temperature Tm in a cross section of the channel has been chosen, 1 Tm ¼ L
ZL TdY :
ð3Þ
0
The desirable effect of this choice of the reference temperature is that it maximizes the accuracy of the Boussinesq approximation, by minimizing the averaged square deviation from the local temperature of the fluid (see [27]). The assumed thermal boundary conditions read dT w ð0Þ; TjY ¼L ¼ Tw ð LÞ: k ¼ q ð4Þ dY Y ¼0
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As an immediate consequence of the choice of the average temperature (3) as reference temperature of the Boussinesq approximation, between the axial pressure gradient dP=d X and the average fluid velocity, 1 Um ¼ L
ZL UdY ;
ð5Þ
0
the simple relationship
dP l ¼ Um dX K
ð6Þ
holds (which has been obtained by integrating Eq. (1) with respect to Y from 0 to L and noticing that due to Eq. (3) the integral contribution of the buoyancy term vanishes). In addition to the velocity and temperature field, in the present investigation the following quantities are of physical and engineering interest: The surface temperature Tw ð0Þ of the left wall of the channel, where the incoming heat flux w ð0Þ has been prescribed, q w ð LÞ, where the surface The outgoing heat flux through the right wall of the channel, q temperature Tw ð LÞ has been prescribed, The heat flux resulting from the volumetric heat generation by viscous friction, ZL l frictional ¼ q U 2 dY ; ð7Þ K 0
The bulk temperature, ZL 1 UT dY ; Tb ¼ Um L
ð8Þ
0
The Nusselt number Nu for the isothermal wall (at Y ¼ L), Nu ¼
w ð LÞdH q ; kðTw Tb Þ
ð9Þ
where dH ¼ 2L is the hydraulic diameter of the channel.
2.2 General solution scheme We first notice that the boundary value problem specified by Eqs. (1), (2), (4) is in fact a mathematically underdefined problem. The reason is that, on the one hand, the general solution of the second order differential equation (2) involves two integration constants and, on the other hand, the (constant) value of the pressure gradient dP=dX is not a priori known. Accordingly, the two conditions (4) are not sufficient to close a problem with three unknown quantities. On this reason, similarly to the duct flow studies of clear fluids (see e.g. Barletta et al. [30]), we prescribe in addition to the conditions (4) also the value Um of the average flow velocity (5) which, as being the volumetric flow rate through the transversal section of the channel, is an experimentally well accessible quantity. Accordingly, the unknown value of the pressure gradient is obtained from Eq. (6) in terms of Um immediately, and our solution procedure goes on as follows.
Mixed convection with viscous dissipation
The temperature T ðY Þ, on account on Eqs. (1) and (6), is m T ðY Þ ¼ Tm þ ½U ðY Þ Um : gbK
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ð10Þ
Substituting Eq. (10) in Eq. (2) we obtain for the velocity U the non linear ordinary differential equation of the second order d2 U gb 2 U ¼ 0: þ 2 dY cp a
ð11Þ
The two conditions necessary for the determination of the two integration constants involved in the general solution of Eq. (11) are dU gbK w ð0Þ; q ¼ dY Y¼0 km
1 L
ZL
UðY ÞdY ¼ Um ;
ð12Þ
0
where the former equation has been obtained from the first boundary condition (4) and w ð0Þ; Um Þ of the (well posed) problem (11), (12) is Eq. (10). Now, once the solution U ¼ U ðY; q known, Eq. (10) and the second boundary condition (4) give the average temperature in terms of the prescribed values Tw ð LÞ and Um , m Tm ¼ Tw ð LÞ þ ½Um U ð LÞ: ð13Þ gbK Thus, the temperature field results from Eqs. (10) and (13) in terms of the velocity solution w ð0Þ; Um Þ and the prescribed wall temperature Tw ð LÞ in the form U ¼ U ðY ; q m ½U ðY Þ U ð LÞ: ð14Þ T ðY Þ ¼ Tw ð LÞ þ gbK Accordingly, the surface temperature Tw ð0Þ of the left wall of the channel is given by m ½U ð0Þ U ð LÞ; Tw ð0Þ ¼ Tw ð LÞ þ gbK and the outgoing heat flux through the right wall of the channel by km dU w ð LÞ ¼ q : gbK dY
ð15Þ
ð16Þ
Y¼L
Furthermore, the heat flux corresponding to the volumetric heat generation by viscous friction results from Eqs. (7) and (11) as km dU dU frictional ¼ w ð0Þ: w ð LÞ q q ð17Þ ¼q gbK dY dY Y ¼L
Y¼0
w ð LÞ and incoming frictional equals the difference between the outgoing heat flux q As expected, q one qw ð0Þ. The bulk temperature as obtained from Eqs. (8), (14) and (7) reads Tb ¼ Tw ð LÞ
frictional mU ð LÞ q : þ gbK qgbLUm
ð18Þ
2.3 Nondimensionalization For the subsequent calculations it is convenient to introduce the velocity and temperature scales a cp mU ma cp ¼ U ¼ ; T ¼ ð19Þ gbL2 gbK g2 b2 L2 K
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as well as the dimensionless quantities y¼
Y U ðY Þ Um T ðY Þ Tw ð LÞ Tm Tw ð LÞ L w : ; u¼ q ; um ¼ ; hðyÞ ¼ ; hm ¼ ; qw ¼ L U T T kT U
ð20Þ
It is worth mentioning that the choice (19) of the velocity and temperature scales U and T implies that between the Peclet number Pe ¼ U L=a , the Darcy-Rayleigh number Ra ¼ gbKLT =ðamÞ and the Gebhart number Ge ¼ gbL=cp the simple relationship Pe ¼ Ra ¼ 1=Ge holds. In terms of the dimensionless variables (20), Eqs. (11) and (12) become u00 þ u2 ¼ 0
ð21Þ
and u0 ð0Þ ¼ qw ð0Þ k;
Z1
udy ¼ um ;
ð22Þ
0
respectively, where the primes denote differentiation with respect to y and the notation qw ð0Þ ¼ k for the dimensionless form of the prescribed incoming wall heat flux has been introduced for simplicity. The solution of the problem (21), (22) yields the dimensionless velocity field u ¼ uðy; k; um Þ, where the quantities k and um are prescribed. Then, in terms of this solution u ¼ uðyÞ, we obtain from Eq. (14) the dimensionless temperature field hðyÞ ¼ uðyÞ uð1Þ:
ð23Þ
The dimensionless form of the average temperature (3) and of the boundary conditions (4) is hm ¼ um uð1Þ
ð24Þ
and h0 ð0Þ ¼ k;
hð1Þ ¼ 0;
ð25Þ
respectively. Furthermore, the dimensionless surface temperature hð0Þ of the left wall of the channel is given by hð0Þ ¼ uð0Þ uð1Þ;
ð26Þ
and the dimensionless heat flux through the right wall of the channel by qw ð1Þ ¼ u0 ð1Þ:
ð27Þ
The dimensionless counterparts of the heat flux (17) corresponding to the volumetric heat generation by viscous friction, of the bulk temperature (18) and of the Nusselt number (9) are qfrictional hb ¼
frictional q ¼ qw ð1Þ qw ð0Þ ¼ u0 ð1Þ k; ðkT =LÞ
qfrictional uð1Þ um
ð28Þ ð29Þ
and Nu ¼
2qw ð1Þ ; hb
respectively.
ð30Þ
Mixed convection with viscous dissipation
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In the present problem, we are basically interested in the dependence of all the above physical quantities on the incoming heat flux qw ð0Þ ¼ k, for prescribed values of the (dimensionless) volumetric flow rate um . This will be done with the aid of the analytic solution u ¼ uðy; k; um Þ of the velocity problem (21), (22). When between the prescribed quantities k and um a special relationship holds, the analytical solution can be given in a closed elementary form (see Sect. 3.1.). In the general case, the analytical solution will be given in the form of a Taylor series with respect to y in Sect. 3.2.
3 Analytical solutions 3.1 An elementary closed form solution It is easy to see that Eq. (21) admits the elementary solution uðyÞ ¼
6 ðy þ y0 Þ2
ð31Þ
;
where y0 is a constant. The first boundary condition (22) determines the value of y0 as y0 ¼
1=3 12 ; k
ð32Þ
and the integral condition (22) implies that the special solution (31) is valid only when between the prescribed quantities k and um the relationship um ¼ ð12=kÞ
1=3
h
6 1 ð12=kÞ1=3
i
ð33Þ
holds. Having in mind that y 2 ½0; 1, the solution (31) is non-singular only when k < 12. This restricting inequality, as well as the relationship (33) between k and um confer to the solution (31) its special character. In the range k < 12, Eq. (31) describes a downward (um < 0) unidirectional flow (u is everywhere negative) except for k ¼ 0 where (31) reduces to the trivial solution u 0. The other quantities of physical and engineering interest (23)–(30) can also be calculated in this case easily.
3.2 Taylor series solution To obtain the general solution of Eq. (21), we first formally expand the dimensionless velocity u ¼ uðyÞ in a Taylor series to powers of y, uðyÞ ¼
1 X
An yn :
ð34Þ
n¼0
Thus, the first term of the series (34) represents the value of the seepage velocity at the left wall, A0 ¼ uð0Þ, while the second coefficient, A1 ¼ u0 ð0Þ, coincides according to Eq. (22) with the negative value of the incoming wall heat flux k, i.e. A0 ¼ uð0Þ c;
A1 ¼ u0 ð0Þ ¼ k:
ð35Þ
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Here c is a short notation for A0 and represents the velocity of the flow at the wall y ¼ 0. Substituting now Eq. (34) in Eq. (21) and identifying the coefficients of the same powers of y, we obtain the recurrence equations Anþ2 ¼
n X 1 Aj Anj ; ðn þ 1Þðn þ 2Þ j¼0
n ¼ 0; 1; 2; . . .
ð36Þ
which allows us to determine every one of the higher coefficients A2 ; A3 ; A4 ; ::: in terms of the parameters c and K . The first coefficients obtained from Eqs. (36) and (35) are 1 1 1 3 1 c k2 ; A5 ¼ c2 k; A2 ¼ c2 ; A3 ¼ ck; A4 ¼ 2 3 12 12 1 4 1 3 A6 ¼ c 2ck2 ; A7 ¼ k 4c k2 ; ::: 72 252
ð37Þ
In this way, the only remaining task is to determine the value(s) of c associated with the prescribed quantities ðk; um Þ of the problem (21), (22). This requires the solution of the second condition (22) for the unknown parameter c, when the values ðk; um Þ are specified. It is immediately seen that in terms of the coefficients of the series solution (34) the integral condition (22) reads 1 X An ¼ um : n þ1 n¼0
ð38Þ
The features of Eq. (38) will be discussed in Sect. 5 in detail. Once the solution c of Eq. (38) is known, the dimensionless velocity field u ¼ uðy; k; um Þ is also known explicitly, and the further quantities of physical and engineering interest are obtained as follows. The respective Eqs. (23) and (24) yield the temperature field h and the average temperature hm , 1 X An ð1 yn Þ; hðyÞ ¼
ð39Þ
n¼1
hm ¼ um
1 X
ð40Þ
An :
n¼0
Equation (26) gives the temperature hð0Þ of the isoflux surface (left wall), and Eq. (27) yields the outgoing heat flux qw ð1Þ through the isothermal surface (right wall) of the channel, hð0Þ ¼
1 X
ð41Þ
An ;
n¼1 1 X qw ð1Þ ¼ ðn þ 1ÞAnþ1 :
ð42Þ
n¼0
Furthermore the frictional heat flux qfrictional , the bulk temperature hb and the Nusselt number are obtained from the respective Eqs. (28)–(30) as follows: qfrictional ¼ k
1 X
ðn þ 1ÞAnþ1 ;
ð43Þ
n¼0
! 1 X 1 kþ ½um An þ ðn þ 1ÞAnþ1 ; hb ¼ um n¼0
ð44Þ
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Mixed convection with viscous dissipation
2um
1 P
ðn þ 1ÞAnþ1
n¼0
Nu ¼ kþ
1 P
ð45Þ
: ½um An þ ðn þ 1ÞAnþ1
n¼0
It is worth noticing here that the closed form solution reported in Sect. 3.1 can serve as a benchmark for the general series solution (34). Indeed, Eqs. (31), (32) and the first equation (35) show that the case of the exact solution (31) is that special case of the general series solution (34) in which between the parameters c and k the relationship 1=3 c ¼ 3k2 =2
ð46Þ
holds. For c given by Eq. (46) the series (34) until the fourth degree term is 1=3 2=3 4=3 3 3 k k5=3 5k2 4 2=3 u¼ y2 1=3 2=3 y3 y ... k ky 2 2 2 24 2 3
ð47Þ
and the corresponding series (38) of um reads um ¼
1=3 3 k k4=3 k5=3 k2 k2=3 5=3 1=3 7=3 2=3 . . . 2 2 2 3 24 2 3
ð48Þ
Now, it is easy to show (by using standard software for symbolic calculations) that Eq. (47) coincides exactly with the power series expansion of the elementary solution (31) around y ¼ 0, and Eq. (48) with the power series expansion of the average velocity (33) around k ¼ 0. This result means that u and um given by Eqs. (31) and (33) are the resurgent functions of the series (47) and (48), respectively.
4 Discussion 4.1 Convergence and numerical validation A shortcoming of power series solutions of differential equations consists often in their slow convergence, which requires that in their practical use a large number of terms must be considered. In the case of the Taylor series solution presented in Sect. 3 this problem may also arise in certain ranges of the parameters involved. Let us consider in this respect e.g. the case k ¼ 0 which corresponds to the adiabatic insulation of the left wall of the channel. In this case the first non-vanishing coefficients of the series (34) are A0 ¼ c;
A2 ¼
c2 ; 2
A4 ¼
c3 ; 12
A6 ¼
c4 ; 72
A8 ¼
c5 ;::: 504
ð49Þ
One immediately sees that for large values of jcj the convergence of all the series expressions of Sect. 3.2 becomes critical. Indeed, the radius of convergence of the series (38) of um in this case is jcj ¼ 8:84752. The convergence can sometimes be accelerated with the aid of the Euler-Knopp series transformation, which applied, e.g., to Eqs. (34) and (38) gives ! 1 n X n! X Aj yj uðyÞ ¼ ; ð50Þ 2nþ1 j¼0 ðn jÞ!j! n¼0
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1 X n! um ¼ nþ1 2 n¼0
n X j¼0
! Aj : ðn jÞ!ðj þ 1Þ!
ð51Þ
All the other equations (39)–(48) of Sect. 3 can be transformed similarly. In order to check the convergence of the initial and of the transformed series for specified values of the parameters involved (which in the present case are c and k), it is useful to plot their respective terms in the increasing order of the summation indices. This procedure provides at the same time information about the number of terms which has to be considered for a required accuracy of the results. As an illustration, in Figs. 2a and b the coefficients of the initial and of the transformed series (38) and (51) of um are plotted, respectively. In this case the convergence-accelerating effect of the Euler-Knopp series transformation is substantial.
6 4 2 0 –2 –4 –6 –8 0
10
20
30
a
40
50
n
3
2
1
0
–1 2
b
4
6
8
10
12
n
Fig. 2. a Plot of the first 50 terms An =ðn þ 1Þ, n ¼ 0; 1; 2; . . . ; 49 of the series (38) of um for the values c ¼ 7 and k ¼ 3 of the parameters involved. The value of the first term is 7 (itP coincides with c) and that of the 112th term amounts to 7:53 106 , b Plot of the first 11 terms n!=2nþ1 nj¼0 Aj =½ðn jÞ!ðj þ 1Þ! , n ¼ 0; 1; 2; . . . ; 49 of the transformed series (51) of um for the same values c ¼ 7 and k ¼ 3 of the parameters involved. The value of the first term is 3:5 and that of the 45th term is 7:18 106
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Mixed convection with viscous dissipation
A further check for the accuracy of the series results obtained for prescribed values of um is their direct validation by using the numerical solution of the initial value problem u00 þ u2 ¼ 0; uð0Þ ¼ c;
ð52Þ
u0 ð0Þ ¼ k;
where c is obtained either as a root of Eq. (38), or of its Euler-Knopp transformation (51). In this sense, the evaluation of the integral condition (22) with the aid of the numerical solution u ¼ uðyÞ of the problem (52), and the subsequent comparison of the result with the prescribed value of um is especially insightful. Obviously, one also can compare the velocity and temperature profiles and other characteristics of the flow obtained by the series method, on the one hand, and with the numerical solution of the initial value problem (52), on the other hand. In this respect, the exact elementary solution (31) furnishes useful benchmarks again.
4.2 General features of the velocity and temperature fields As we have shown in Sect. 3.2, for prescribed values of k and um the velocity field and all the other quantities of interest depend on the parameter c which has to be determined from Eq. (38). Accordingly, the features of the roots c of this equation are of basic importance for the present investigation. The main features of the function um ¼ um ðc; kÞ become manifest by a simple inspection of Fig. 3 which shows the following: Solutions only exist when, for a given k, the prescribed value of um does not exceed a maximum value um;max ðkÞ, i.e., the domain of existence of the solution is um um;max ðkÞ;
ð53Þ
where um;max ðkÞ is reached at a value c cmax of the parameter c which depends on k, cmax ¼ cmax ðkÞ.
6 4
l = –10 –5
2
0 4 6.38 l = 10
um
0 –2 –4 –6 –8 –5
0
g
5
10
15
Fig. 3. Plot of the average velocity as a function of the parameter c according to Eq. (38), for six different values of k, k ¼ 10; 5; 0; 4; 6:38484; þ10. With increasing values of k, the maximum value um;max of um (marked by dot) decreases and, at the same time, moves to the right (direction South-East)
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The existence domain um um;max is not bounded from below; all negative values of um being allowed. For any prescribed value of um in the range um < um;max ðkÞ, the mixed convection problem admits two solutions (‘‘dual solutions’’) corresponding to two different values c1 ðum Þ and c2 ðum Þ which become coincident for um ¼ um;max ðkÞ, i.e., c1 um;max ¼ c2 um;max ¼ cmax ðkÞ:
ð54Þ
With increasing values of k, the maximum value um;max decreases and, at the same time, moves to the right (direction South-East) i.e. the value of cmax ðkÞ also increases with increasing k. cmax ðkÞ is negative for k < 33:08, zero for k ¼ 33:08 (not shown in Fig. 3) and becomes positive for k > 33:08. The maximum value um;max ðkÞ of um is positive for k < 6:38484, zero for k ¼ 6:38484 (the second curve from below) and becomes negative for k > 6:38484. The vanishing value um;max ð6:38484Þ ¼ 0 of um;max ðkÞ is reached at cmax ð6:38484Þ ¼ 5:87. Accordingly, for k < 6:38484, both upward (um > 0) and downward (um < 0) flow regimes, as well as zero average velocity nontrivial flows can exist, while for k > 6:38484 only downward flows (um < 0) are possible. As a first illustration, in Fig. 4a and b the dual velocity and temperature profiles corresponding to the prescription ðk; um Þ ¼ ð5; 1:91606Þ are shown where the value um ¼ 1:91606 has been obtained from Eq. (33) for k ¼ 5. Thus, we expect that one of the dual velocity solutions uðyÞ coincides with the elementary solution given by Eqs. (31), and the corresponding dual temperature solution, according to Eq. (23), with hðyÞ ¼ uðyÞ uð1Þ, where uðyÞ is given by Eqs. (31). Indeed, we find that for ðk; um Þ ¼ ð5; 1:91606Þ the two corresponding values of the parameter c ¼ uð0Þ (i.e., the roots of Eq. (38)) are c1 ¼ 3:34716 and c2 ¼ 13:3860, respectively. Hence, the seepage velocity at the left wall given by solution ‘‘1’’, uð0Þ ¼ 3:34716, coincides with that resulting from the elementary solution (31). The other characteristics of the solution ‘‘1’’ also coincide with those given by Eqs. (31) and hðyÞ ¼ uðyÞ uð1Þ (see Table 1). We may therefore conclude that the velocity and temperature profiles uðy; c1 Þ and hðy; c1 Þ of Fig. 4a coincide exactly with those of the elementary solution reported in Sect. 3.1. While the solution ‘‘1’’ describes a unidirectional downward flow of which temperature hðy; c1 Þ is everywhere negative, its dual counterpart ‘‘2’’ shown in Fig. 4b describes a bidirectional flow which temperature hðy; c2 Þ is everywhere positive. However, the velocity uðy; c2 Þ changes from positive to negative values at y y0 ¼ 0:495881. In the above example, in addition to the prescribed value k ¼ 5 of the incoming heat flux, the value um ¼ 1:91606 of the average velocity has been chosen. The latter value lies far below the maximum value um;max ðkÞ ¼ 3:4002 which is reached at cmax ðkÞ ¼ 4:27747 for k ¼ 5 (see Fig. 3). As um approaches its maximum value, in the present case the value um;max ð5Þ ¼ 3:4002, the c-values c1 ðum Þ and c2 ðum Þ of the corresponding dual solutions also approach each other and become coincident for um ¼ um;max (see Eq. (54)). In Fig. 5, these coincident velocity and temperature profiles are shown for k ¼ 5, um ¼ um;max ð5Þ ¼ 3:4002 and c ¼ cmax ð5Þ ¼ 4:27747. In this case the seepage velocity at the right wall is uð1; cmax Þ ¼ 0:0585431. This small value of uð1; cmax Þ is the reason why in Fig. 5 the temperature profile hðy; cmax Þ, in agreement with Eq. (23), lies close below the velocity profile uðy; cmax Þ. The values of the other quantities of interest in this case are:
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Mixed convection with viscous dissipation
0 –0.5 q ( y;g1) –1 u (y;g1)
–1.5 –2 –2.5
l = –5 um = –1.91606
–3 0
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–20
0
b
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0.6 y
Fig. 4. Plot of the dual velocity and temperature profiles ‘‘1’’ (a) and ‘‘2’’ (b) for ðk; um Þ ¼ ð5; 1:91606Þ. The two corresponding values of the parameter c ¼ uð0Þ are c1 ¼ 3:34716 and c2 ¼ 13:3860, respectively. The velocity and temperature profiles uðy; c1 Þ and hðy; c1 Þ shown in Fig. 4a coincide exactly with those of the elementary solution (31)
Table 1. Characteristics of the dual velocity and temperature fields described by the dual solutions corresponding to the same prescribed values ðk; um Þ ¼ ð5; 1:91606Þ of the incoming heat flux k and the average velocity um
c ¼ uð0Þ uð1; cÞ hm ðcÞ hð0; cÞ qfrictional ðcÞ qw ð1; cÞ hb ðcÞ NuðcÞ
Solution ‘‘1’’
Solution ‘‘2’’
c1 ¼ 3:34716 uð1; c1 Þ ¼ 1:09683 hm ðc1 Þ ¼ 0:819226 hð0; c1 Þ ¼ 2:25033 qfrictional ðc1 Þ ¼ 4:06208 qw ð1; c1 Þ ¼ 0:937919 hb ðc1 Þ ¼ 1:02319 Nuðc1 Þ ¼ 1:83333
c2 ¼ 13:3860 uð1; c2 Þ ¼ 31:7807 hm ðc2 Þ ¼ 29:8646 hð0; c2 Þ ¼ 45:1666 qfrictional ðc2 Þ ¼ 156:734 qw ð1; c2 Þ ¼ 151:734 hb ðc2 Þ ¼ 50:0196 Nuðc2 Þ ¼ 6:06699
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4
u ( y;gmax) q ( y;gmax)
3
2 l = –5 um = um,max = 3.4002
1
g = g max = 4.27747
0 0
0.2
0.4
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0.8
1
y Fig. 5. Plot of the coincident dual velocity and temperature profiles for ðk; um ; cÞ ¼ k; um;max ; cmax ¼ ð5; 3:4002; 4:27747Þ. The maximum value of uðy; cmax Þ is 4:87369 and it is reached at y ¼ 0:228769
hm ðcmax Þ ¼ 3:34166, hð0; cmax Þ ¼ 4:21893, qw ð1; cmax Þ ¼ 8:78498, qfrictional ðcmax Þ ¼ 13:78498, hb ðcmax Þ ¼ 3:99562 and Nuðcmax Þ ¼ 4:39731. It is worth emphasizing here that the results obtained for the present isoflux isothermal wall conditions are also valid for the case of a symmetrical channel Y 2 ½L; L subject to symmetrical isothermalisothermal boundary conditions. Indeed, having in mind that in the midplane of this channel the symmetry condition dT=dY jY ¼0 ¼ 0 holds, the solution can immediately be obtained by substituting in the present half-channel equations k ¼ 0, and by extending the variation range of y simply to ½1; 1. In other words, the special case of the present half-channel problem with adiabatical left wall, k ¼ qw ð0Þ ¼ 0 provides the results for a symmetrical channel subject to symmetrical isothermal isothermal boundary conditions. In the case k ¼ 0 the maximum value of um is um;max ¼ 2:11512, and it is reached at c cmax ¼ 4:98332. The two zeros of um are located at c ¼ 0 and c c0 ¼ 10:5388 (see Fig. 3). In order to illustrate the extension procedure described above, let us prescribe for the half-channel the values ðk; um Þ ¼ ð0; 4Þ. For the corresponding roots of Eq. (38) we find the values c1 ¼ 2:53843 and c2 ¼ 13:9514, respectively. The symmetric dual velocity and temperature profiles associated with these values are plotted in Fig. 6. The left-half of the profiles shown in Fig. 6 have been obtained by a simple extension of the variation range of y from ½0; 1 to ½1; 1.
4.3 Heat transfer characteristics The main heat transfer characteristic of the flow, the dimensionless outgoing heat flux qw ð1Þ through the isothermal right wall of the channel, is plotted in Fig. 7 as a function of the parameter c for three different values of the incoming heat flux k ¼ qw ð0Þ, according to Eq. (42). Figure 7 shows that for non-negative values of the incoming heat flux (in Fig. 7, for k ¼ 0 and k ¼ 5 ) the outgoing heat flux qw ð1Þ is, as expected, positive for all c 6¼ 0. One also sees that for negative values of the incoming heat flux (in Fig. 7, for k ¼ 5 ) there exists a c-range
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q (y;g2)
40
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u (y;g2) q (y;g1)
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l=0 um = –4
–40 –1
–0.5
0
0.5
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y Fig. 6. The dual velocity and temperature profiles in a symmetrical channel with symmetrical isothermal isothermal boundary conditions and average flow velocity um ¼ 4 have been obtained by a straightforward extension of the present half-channel results corresponding to the case of an adiabatic (k ¼ 0) left wall
20
15
qw (1)
l = +5
10
l=0
5 l = –5
0
–4
–2
0
g
2
4
6
Fig. 7. Plot of the outgoing heat flux qw ð1Þ as a function of the parameter c for the values k ¼ 5; 0 and þ 5 of the incoming heat flux k ¼ qw ð0Þ
ðc1 ; c2 Þ in which the outgoing heat flux qw ð1Þ is also negative, i.e., the heat generated by viscous friction is not sufficient to compensate the heat extracted from the flow through the left wall of the channel. At the ends c1 and c2 one has qw ð1Þ ¼ 0. According to the flux balance equation (28), qfrictional ¼ qw ð1Þ qw ð0Þ, this means that for c ¼ c1 and c ¼ c2 the heat produced by viscous friction compensates the heat extracted through the left wall of the channel exactly. Out of the cinterval ðc1 ; c2 Þ, the heat flux through the right wall is positive, i.e., in these c-ranges the viscous friction generates more heat than that extracted from the flow through the left wall of the channel. In the case k ¼ 5 we have c1 ¼ 3:47294, c2 ¼ 0:30598, and the minimum value of qw ð1Þ, qw;min ð1Þ ¼ 3:56152 is reached at c cmin ¼ 2:37906. The c-values corresponding
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to the zeros and to the minimum value of qw ð1Þ are associated with different values of the average flow velocity um ¼ um ðk; cÞ. In the case k ¼ 5 we obtain um ð5; c1 Þ ¼ 2:17448, um ð5; cmin Þ ¼ 0:26087 and um ð5; c2 Þ ¼ þ1:9227, respectively.
5 Summary and conclusions The fully developed laminar Darcy mixed convection flow with viscous dissipation has been investigated in a vertical parallel plane channel filled with a porous medium. In addition to the isofluxisothermal wall conditions, the average flow velocity Um (as a quantity which can operatively be tuned in a real laboratory experiment) has been prescribed. The validity of Boussinesq approximation has been assumed. The governing Darcy and energy balance equations have been solved by an analytical series expansion method. The main results of the paper can be summarized as follows: In the range k < 6:38484 of the incoming surface heat flux qw ð0Þ k, both upward (Um > 0) and downward (Um < 0) laminar flow solutions may exist as long as Um does not exceed a maximum value Um;max ðkÞ, while for k > 6:38484 only downward flows are possible. The existence domain Um Um;max ðkÞ is not bounded from below; all negative values of Um being allowed. The velocity field can either be unidirectional or bidirectional. For every Um < Um;max two solution branches (dual solutions) exist. In the range k 6:38484 of the incoming heat flux, also (bidirectional) flow configurations with vanishing average velocity are possible. The mechanical and thermal characteristics of the flows described by the dual solutions are quite different but they approach each other continuously as the prescribed Um approaches its maximum allowed value Um;max . The origin of the flow features described in the paper resides in a subtle interplay of the driving external- and buoyancy forces on the one hand, and the nonlinear effect of the heat generation by viscous friction on the other hand. Accordingly, one is faced here with a quite general phenomenon encountered both in external [31] and internal mixed convection flows and for other geometries. No doubt, the ultimate answer on the feasibility of the dual solution can only be gained from a detailed stability analysis of the full unsteady Darcy and energy equations which, however, is beyond the scope of the present paper. In particular, the stability analysis could ascertain whether the dual solutions are the consequence of a bistable behavior of the flow system or, on the contrary, only one of these two solution branches is stable. In this respect, it appears to be decisive the transient process that drives the system to the steady parallel flow configuration. One can expect that by starting the fluid from the rest to some average flow rate Um 6¼ 0, the unidirectional flow configuration will first be reached. Imposing now a sudden change of the wall temperature or of the wall heat flux, and re-establishing subsequently the initial wall conditions, the transient buoyancy force induced in the vicinity of the wall can produce an inverted flow in this region and thus a possible transition to the bidirectional velocity profile. In this way, sufficiently intense cyclic perturbations of either thermal or dynamic origin can produce repeated transitions from one solution to its dual counterpart, especially in the range near to the maximum average velocity. Accordingly the stability analysis is a research opportunity which could yield a deeper insight in the nature of the dual solutions.
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Acknowledgement One of the authors (E.M.) expresses his gratitude to the Ministero Italiano dell’Universita` e della Ricerca for supporting his collaboration in a Progetto di Ricerca di Rilevante Interesse Nazionale concerning the convection heat transfer in internal flows.
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[22] Nield, D. A.: Comments on ‘A new model for viscous dissipation in porous media across a range of permeability values’. Transport Porous Media 55, 253–254 (2004). [23] Magyari, E., Rees, D. A. S., Keller, B.: Effect of viscous dissipation on the flow in fluid saturated porous media. In: Handbook of porous media (Vafai, K., ed.), 2nd ed., pp. 373–406. New York: Taylor and Francis 2005. [24] Barletta, A.: Laminar mixed convection with viscous dissipation in a vertical channel. Int. J. Heat Mass Transf. 41, 3501–3513 (1998). [25] Barletta, A.: Analysis of combined forced and free flow in a vertical channel with viscous dissipation and isothermal-isoflux boundary conditions. ASME J. Heat Transf. 121, 349–356 (1999). [26] Barletta, A.: Combined forced and free convection with viscous dissipation in a vertical circular duct. Int. J. Heat Mass Transf. 42, 2243–2253 (1999). [27] Barletta, A.: Heat transfer by fully developed flow and viscous heating in a vertical channel with prescribed wall heat fluxes. Int. J. Heat Mass Transf. 42, 3873–3885 (1999). [28] Zanchini, E.: Effect of viscous dissipation on mixed convection in a vertical channel with boundary conditions of the third kind. Int. J. Heat Mass Transf. 41, 3949–3959 (1998). [29] Barletta, A., Zanchini, E.: On the choice of the reference temperature for fully-developed mixed convection in a vertical channel. Int. J. Heat Mass Transf. 42, 3169–3181 (1999). [30] Barletta, A., Magyari, E., Keller, B.: Dual mixed convection flows in a vertical channel. Int. J. Heat Mass Transf. 48, 4835–4845 (2005). [31] Merkin, J. H.: Mixed convection boundary layer flow on a vertical surface in a saturated porous medium. J. Engng. Math. 14, 301–313 (1980). Authors’ addresses: A. Barletta and E. Magyari, Dipartimento di Ingegneria Energetica, Nucleare e del Controllo Ambientale (DIENCA), Universita` di Bologna, Via dei Colli 16, I-40136 Bologna, Italy (E-mail:
[email protected]); I. Pop, Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania; L. Storesletten, Department of Mathematics, Agder University College, Serviceboks 422, 4604 Kristiansand, Norway
