Free Convection Flow of a Jeffrey Fluid between Vertical Plates Partially Filled with Porous Medium

International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882Volume 4, Issue 5, May 2015 Free Convection Flow of a Jeffrey Fluid between Vertical Plates Partially Filled with Porous Medium S. Dhananjaya1, P.V. Arunachalam1, S. Sreenadh1, A. Parandhama2 Department of Mathematics, Sri Venkateswara University, Tirupati- 517502, India. 2 Department of Mathematics, Sree Vidyanikethan Engineering College, Tirupati-517102, India. 1 ABSTRACT The laminar fully developed free-convection flow of a Jeffrey fluid in a channel bounded by two vertical plates having interface vertically, partially filled with porous matrix and partially with a clear fluid, has been investigated. The momentum transfer in porous medium has been described by the Brinkman-extended Darcy model. The effects of Darcy number and Jeffrey parameter on flow velocity and temperature has been discussed with the help of graphs. Applying perturbation method, the expressions for velocity and temperature are obtained. It is found that the velocity and temperature increase with increasing non-Newtonian Jeffrey parameter. Further increment in Jeffrey parameter gives rise to an increase in skin-friction and rate of heat transfer at both the walls of the vertical channel. Keywords -Convection, Darcy number, Jeffrey fluid, Porous medium. I. INTRODUCTION Viscous flow through and past porous media has important applications in various branches of science and engineering. In particular the study of porous flow associated with heat transfer has numerous applications in oil extraction processes and tapping ground water from underground. Flow mechanism at the fluid/porous interface, was first studied by Beavers & Joseph [1] and it was observed that the velocity gradient at the interface is proportional to the slip velocity. Taylor [2] and Richardson [3] continued the investigation in which they modeled fluid flow by Darcy law. Free convection flows in vertical slots were discussed by Aung et al. [4]. Rudraiah and Nagaraj [5] studied the fully developed free-convection flow of a viscous fluid through a porous medium bounded by two heated vertical plates. Extensive research work has been carried out in recent years to study the effects of various solid matrix and fluid flow parameters on the free convection in channels partially filled by porous materials. Beckermann et al. [6] examined free convection flow between a fluid and a porous layer in a rectangular enclosure. Singh [7] has studied transient free convective flow between two vertical walls for asymmetric heating when one of the walls is moving with constant velocity. Kim et al. [8] studied the effect of wall conduction on free convection between asymmetrically heated vertical plates with uniform wall heat flux. Using Darcy model for momentum transfer, mixed convection on a vertical cylinder embedded in a saturated porous medium is presented by Ramanaiah et al. [9] while momentum transfer based on Brinkman model in a circular cylinder is studied by Pop et al. [10]. Vajravelu et al. [11] studied combined free and forced convection in an inclined channel with permeable boundaries. Sacheti et al. [12] extended the Rudraiah and Nagaraj [5] problem in a rotating system. Convection effects are investigated in an inclined channel with porous substrates at thebounding rigid walls by Chauhan and Soni[13]. Vajravelu et al. [14] studied unsteady flow of two immiscible conducting fluids between two permeable beds. Mixed convection was studied by Chang and Chang [15] in a vertical channel partially filled with highly permeable porous media. Kuznetsov [16] presented an analytical solution of the fluid flow in the interface region between a porous medium and a clear fluid in channels partially filled with a porous medium. Paul et al. [17] has studied freeconvection effects in a vertical channel partially filled with porous medium. He has found that effect of Brinkman term is in entire porous domain for large values of Darcy number while its effect is confined nearer to interface and wall for small values of Darcy number. AlNimr and Hadded [18] examined the effects of free convection in a fully developed flow through open-ended vertical channels that are filled partially by porous medium. Al-Nimr and Khadrawi [19] investigated transient free convection flow in channels partially filled by porous substrates in various configurations. Buhler [20] studied on special solutions of the Boussinesqequations for free convection flows in a vertical gap. Elshehawey et al. [21] studied peristaltic transport in an asymmetric channel through a porous medium. The movement of bio-fluids in a physiological system has to be investigated thoroughly in order to solve diagnostic problems that arise in a living body. When we consider the flow of blood through an artery. We observe that there is an endothelium layer present in the arterial www.ijsret.org 503 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882 Volume 4, Issue 5, May 2015 wall and this layer is modeled as a porous layer Dulal pal et al. [22]. Similarly cartilages are also modeled as porous layers. In this way various biological tissues in physiological systems are modeled as porous layers. It has been observed from experimental investigations that blood, being a suspension of cells, behaves like a nonNewtonian fluid at low shear rates in smaller arteries. Since there is no universal model to describe all nonNewtonian fluids in physiological systems, several models are proposed to explain the behavior of these biofluids. Among them Jeffrey fluid model is a significant generalization of Newtonian fluid model as the later one can be deduced as a special case of the former. Several researchers have studied Jeffrey fluid flows under different conditions. The effects of mild stenosis on the resistive impedance and the wall shear stress in an artery have been put forward by Shukla et al. [23] who considered the flowing blood to be non-Newtonian characterized by the Power law, Jeffrey and Cason’s fluid models. Hayat and Ali [24] investigated the peristaltic motion of a Jeffrey fluid under the effect of a magnetic field. Vajravelu et al. [25] discussed the influence of heat transfer on peristaltic transport of a Jeffrey fluid in a vertical porous stratum. Mahmoud et al. [26] studied the effect of porous medium and magnetic field on the peristaltic transport of a Jeffrey fluid in an asymmetric channel. Misra et al. [27] has investigated a mathematical modeling of blood flow in porous vessel having double stenosis in the presence of an external magnetic field. Umesh Gupta et al. [28] studied free convection flow between vertical plates moving in opposite direction and partially filled with porous medium. Srivastava and Singh [29] studied free convection between coaxial vertical cylinders partially filled with a porous material having constant heat flux on inner cylinder. Turkyilmazoglu and Pop [30] have investigated the flow and heat transfer of a Jeffrey fluid near the stagnation point on a stretching/shrinking sheet with a parallel external flow. Devaki et al. [31] have considered the pulsatile flow of a Jeffrey fluid in a circular tube lined internally with porous material. Akbar et al. [32] has studied the Jeffrey fluid model for the peristaltic flow of chyme in the small intestine with magnetic field. Santhosh and Radhakrishnamacharya [33] studied a two-fluid model for the flow of Jeffrey fluid in tubes of small diameters. Recently Eldabe et al. [34] studied the peristaltic motion of non-Newtonian fluid with heat and mass transfer through a porous medium in the channel under the effect of magnetic field. Albalawi et al. [35] has investigated the peristaltic flow of a Jeffrey fluid in an asymmetric channel. In this chapter, flow and heat transfer of a Jeffrey fluid in a vertical channel partially filled with porous medium is investigated. The velocity field, the temperature distribution and skin friction at the walls are determined and discussed. II. MATHEMATICAL FORMULATION Let us consider the fully developed steady laminar free convective flow of a Jeffrey fluid between two vertical parallel walls partially filled with porous medium and partially filled with a clear fluid when one wall is heated and other is cooled as shown in Fig.1. The x -axis is taken along one of the walls and y -axis normal to it. The flow in the porous medium is described by Brink men extended Darcy law whereas the flow in the free fluid region is described by Jeffrey model, where U f and U p are the velocities of the fluid in the free fluid region and porous region respectively. The temperature of the walls y  0 and y  H is considered as Tf  Tc  A(Th  Tc ) Tp  Tc  B(Th  Tc ) respectively. Under usual and Boussinesq approximation, the flow in the fluid and porous regions is governed by the following equations of motion and energy: Free fluid region: 2  d Uf   g  T f  Tc   0 1  1 dy 2 2  d (Tf  Tc )   dU f K0   2 (1  1 ) (1  1 )  dy dy (1) 2    0 (2)  Porous region: 2  d Up  Up    g  (Tp  Tc )  0 (3) 2 1  1 dy 1  1 k K0 d 2 (Tp ) dy 2   dU p   U p    0 (4)    1  1  dy  1  1 k 2 2 The corresponding boundary conditions are www.ijsret.org 504 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882 Volume 4, Issue 5, May 2015 T f  Tc  A Th  Tc    y  H ; U p  0 Tp  Tc  B Th  Tc     (5) dU f dU p  y  d; Uf Up   dy dy  dT f dTp  y  d ; T f  Tp   dy dy  We introduce the following non-dimensional quantities: U f k y d y  0; U f  0, Da  Up  H2 H ;d  U p g  2 H ;U f  ; f  T  T  H T  T  ;  g H 2 N ; y 2 h c T T f h 4 h c g  H 2 Th  Tc   Tc   Tc  ; p  H T T p h ;  Tc   Tc  where, the matching conditions for velocity are due to continuity of velocity and shear stress at the interface. ; III. SOLUTION OF THE PROBLEM ;  g  H 2 Th  Tc  (6) In view of the above non-dimensional quantities, the basic equations (1) to (4) and the boundary conditions equation (5) can be expressed in non-dimensional form, dropping bars, as K Free Fluid region: 2 1 d Uf f  0 1  1 dy 2 d 2 f (7) 2 N  dU f   0 2 dy 1  1   dy  d 2 p The governing momentum and energy equations (7) to (10) are coupled partial differential equations that cannot be solved in closed form. It can be observed that problem is non-linear due to viscous and Darcy dissipation terms. This problem can be tackled by using a perturbation method as N is small in most of the practical problems. Accordingly we assume, for small N, the expansions: U f  U 0 f  N U1 f  0(N 2 )   U p  U 0 p  N U1 p  0(N 2 )    f   0 f  N 1 f  0(N 2 )    p   0 p  N 1 p  0(N 2 )  (12) Substituting (12) in the equations (7) to (10) give the U 0 f , U 0 P ,  0 f and  0 p which are the quantities (8) solutions for N equal to zero i.e., when the viscous and Darcy dissipations are neglected whereas U 1 f , U 1 p , 1 f , Porous region: 2 1 d Up 1 Up  p  0 2 1  1 dy 1  1 Da The boundary and matching conditions (5) in dimensionless form are y  0 : Uf  0 , f  A   dU f dU p  y  d: Uf Up ,  ; dy dy   (11) y 1: Up  0 ,  p  B;  d f d p   y  d :  f  p ,  dy dy  1 p are perturbed quantities relative to U 0 f , U 0 P ,  0 f (9) and  0 p respectively when the viscous and Darcy 2 2 N  dU p  N  U p   0 (10)     2 dy 1  1   dy  Da 1  1  dissipations are taken into account. In equation (9), the momentum transfer in porous domain is described based on Brinkman extended Darcy model, where Da is the Darcy number, d the distance of interface from the plate, g the acceleration due to gravity, H the distance between vertical plates, k the permeability of the porous matrix, K the thermal conductivity, N the buoyancy parameter,  the coefficient of thermal expansion,  the dynamic viscosity,  the kinematic viscosity,  the density,  the shear stress and  is the temperature. The subscripts frepresent fluid layer, pthe porous layer, h hot plate and c the cold plate. 2 1 d U0 f  0 f  0 1  1 dy 2 d 2 0 f dy 2 (13) 0 (14) 2 1 d U1 f  1 f  0 1  1 dy 2 d 21 f dy 2 www.ijsret.org (15) 2 1  dU 0 f    0 (16) 1  1   dy  505 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882 Volume 4, Issue 5, May 2015 2 U0 p 1 d U0 p    0 p  0 (17) 2 1  1 dy Da 1  1  U1 f  c7 y  c8  f1 y 8  f 2 y 7  f 3 y 6  f 4 y 5 d 20 p U1 p  a7 eb2 y  a8 eb2 y  e36 e2b2 y  e37 e 2b2 y  e38 y 2 eb2 y dy 2 0  f5 y 4  f 6 y 3 (18)  e39 y 2 eb2 y  e40 yeb2 y  e41 yeb2 y  e42 eb2 y 2 1 d U1 p 1  U1 p  1 p  0 2 1  1 dy Da 1  1  d 21 p  e43 eb2 y  e44 y 4  e45 y 3  e46 y 2  e47 y  e48 (19) 2 2 1  dU 0 p  1   U  0 (20)     0 p dy 2 1  1   dy  Da 1  1  IV. SKIN-FRICTION du f  du p  1   and  2     dy  y  0 dy  y  0 The corresponding boundary conditions are U0 f  0  0 f  A   at y  0 U1 f  0 1 f  0   U 0 p  0; 0 p  B    at y  1 U1 p  0; 1 p  0   dU 0 f dU 0 p  U0 f  U0 p   dy dy   at y  d (21) dU1 f dU1 p  U1 f  U1 p  dy dy   0 f  0 p 1 f  1 p d 0 f Its non-dimensional form, dropping bars d 0 p   dy dy   at y  d d1 f d1 p   dy dy   Solving the equations (13) to (20) subject to the boundary conditions (21), we obtain the following velocity and temperature components:  0 f  c1 y  c2  0 p  a1 y  a2 (22) e  e  1 f  c5 y  c6   3  y 6   4  y 5  20   30  e  e  e    5  y 4   6  y3   7  y 2  12   6   2  1 p  e27 e2b2 y  e28 e2b2 y  e29 yeb2 y  e30 ye b2 y e31eb2 y  e32 eb2 y  e33 y 4  e34 y 3  e35 y 2  a5 y  a6 du f   c3  Nc7  dy  y  0 2  du p  b2  b2   a3b2 e  a4 b2 e  b3  dy  y  1 (23)  a7 b2 b5  a8 b2 b6  2e36 b2 e 2b2  2e37 b2 e 2b2     e38 b5  b2  2   e39 b6  2  b2   e40 b5  b2  1  N   e41b6 1  b2   e42 b2 b5  e43 b2 b6  4e44   3e45  2e46  e47    c y3 c y 2  U 0 f  c3 y  c4  (1  1 )  1  2  2   6 U0 p  a3eb2 y  a4 eb2 y  b3 y  b4 1  V. RATE OF HEAT TRANSFER Apart from the velocity and temperature distribution in the channel, it is important to determine rate of heat transfer between the plates and the fluid. The rate of heat transfer through the channel wall to the fluid is given by  dT  Qk   dy  y  0, y  H The rate of heat transfer in dimensional form can be written as q d  dy  y  0, y 1 Based on the analytical solutions reported above, the rate of heat transfer at the walls y  0 and y  1 is given by qf  www.ijsret.org d f   c1  N c5  dy  y  0 (24) 506 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882 Volume 4, Issue 5, May 2015 qp  d p   dy  y  1  2b2 e27 e 2b2     2 b2  2b2 e28 e  e29 b5  b2  1     a1  N  e30 b6 1  b2   b2 e31b5   b2 e32 b6  4e33  3e34     2e35  a5  follows that increase in the magnitude of the nonNewtonian Jeffrey parameter or permeability of the porous layer enhances the velocity in the channel. We also note that the reduction of width of porous layer increases the velocity in the channel with the walls temperature A=1, B=0 and A=0, B=1 respectively. From Figs. 10, 11 and 12, we observe that the temperature 1 f or 1 p increases with increasing Jeffrey parameter   1 , Darcy number Da and width of free fluid region d . It VI. RESULTS AND DISCUSSION In this paper, the fully developed steady laminar free convective flow of a Jeffrey fluid between two vertical parallel walls partially filled with porous medium and partially filled with a clear fluid when one wall is heated and other is cooled as shown in Figure 1.The governing equations having non- linear nature have been solved by analytical method. The velocity and temperature distributions are analyzed for two primary regions (free region and porous region), for the cases when A=1, B =0 (plate y=0 is heated and plate y=1 is cooled) and A=0, B=1 (plate y=0 is cooled and y=1 is heated). The effect of Darcy number on the flow has been discussed. The variation of velocity and temperature with y is calculated, from equations (22), for different parameters Da and 1 for different cases, and is shown in Figures (2)-(13) and table-1. From Figs. 2, 3 and 4, 5 we observe that the velocity increases with increasing Jeffrey U 0 f orU 0 p   parameter 1 , Darcy number Da and width of free fluid region d . It follows that increase in the magnitude of the non-Newtonian Jeffrey parameter or permeability of the porous layer enhances the velocity in the channel. We also note that the reduction of width of porous layer increases the velocity in the channel with the walls temperature A=1, B=0 and A=0, B=1 respectively. From Figs. 6, 7 and 8, 9 we observe that the velocity U1 f orU1 p increases with increasing Jeffrey parameter   1 , Darcy number Da and width of free fluid region d . It follows that increase in the magnitude of the nonNewtonian Jeffrey parameter or permeability of the porous layer enhances the temperature in the channel. We also note that the reduction of width of porous layer increases the temperature in the channel with the walls temperature A=1, B=0 and A=0, B=1 respectively. Further it is clear that maximum temperature occurs in the porous region. From Fig. 13, it is found that the temperature 1 f or 1P increases with the increasing in Darcy   number for a given Da . Also for fixed Da the temperature decreases with the increment in the value of d . ie the effect of increasing width of the free flow region is to decrease the temperature in the composite channel. The values of skin-friction and rate of heat transfer on the walls y=0 and y=1are given in TABLE.1. It is clear that skin-friction increases on the both walls with increasing non-Newtonian Jeffrey parameter, width d and Darcy number. The interchange of temperatures at the walls also expresses the same behavior with respect to skin-friction on both the walls. Further it is clear that rate of heat transfer increases on both the walls with increasing non-Newtonian Jeffrey parameter, Darcy number and width d . We infer that the increase in permeability, width of the porous layer and Darcy number enhances rate of heat transfer in the channel. The interchange of temperatures at the walls expresses the same behavior with respect to rate of heat transfer on both the walls. www.ijsret.org 507 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882 Volume 4, Issue 5, May 2015 q TABLE.1: Values of skin-friction  1 ,  2  and rate of heat transfer 1 , q2  for different values of non- Newtonian Jeffrey parameter 1 and Darcy parameter Da with effect of width of the channel d and inter change of walls temperature  A  1, B  0 and A  0, B  1 𝜆1 Da 0.1 0.01 0.3 0.02 0.5 0.03 1 0.1 A=1 B=0 𝜏1 𝜏2 𝑞1 𝑞2 0.2732 0.503 0.8093 0.4331 0.7444 1.1211 0.637 1.0206 1.4468 2.302 2.5648 2.4181 0.0261 0.0487 0.132 0.0881 0.1566 0.3519 0.1897 0.3164 0.5981 0.9172 1.507 1.535 2.8687 6.5303 10.671 5.0187 9.0745 13.562 7.2687 11.53 16.247 23.953 22.24 23.54 2.3916 4.1125 6.9687 3.578 5.7498 8.8666 4.7888 7.3971 10.556 5.8275 12.693 14.583 d 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7 0.08 0.07 B=1 𝜏1 𝜏2 𝑞1 𝑞2 0.0671 0.1711 0.4314 0.1562 0.3413 0.7346 0.2885 0.5718 1.1049 1.3079 2.0695 3.0271 1.1146 0.1238 0.1978 0.2222 0.2586 0.4713 0.3657 0.4469 0.8248 1.3857 1.8445 3.0224 2.001 2.9519 5.5836 3.0947 4.4691 7.7498 4.314 6.1045 9.8896 10.976 14.176 18.28 0.9075 1.5651 4.7377 2.8188 3.8542 8.1808 4.8271 6.2043 11.501 13.738 16.268 24.764 0.07 1= 0.1 Da = 0.1 Da = 0.01 1= 0.3 d = 0. 7 A=0 0.06 1= 0.5 Da = 0.001 0.06 d = 0. 7 U0f or U0p 0.05 U0f or U0p 0.05 0.04 0.04 d = 0. 5 d = 0. 5 0.03 0.03 0.02 0.02 0.01 0.01 d = 0. 3 d = 0. 3 0 0 0.1 0.2 0.3 0.4  Y 0.5 0.6 0.7 0.8 0.9 0 1  0 0.1 0.2 0.3 0.4  Y 0.5 0.6 0.7 0.8 0.9 1  for different values of Fig.2: Velocity profiles U 0 f or U 0 P for different values of Fig.3: Velocity profiles U 0 f or U 0 P 1 , with the difference of width of the flow region, at walls Da , with the difference of width of the flow region, at walls temperature A=1 and B=0. temperature A=1 and B=0. www.ijsret.org 508 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882 Volume 4, Issue 5, May 2015 0.06  1 =0.1 d = 0.7 1.6 x 10 -4  1 = 0.1  1 =0.3  1 = 0.3 1.4  1 =0.5 0.05  1 = 0.5 d = 0. 7 1.2 d = 0.5 0.03 1 U1f or U1P U0f or U0p 0.04 0.8 d = 0. 5 0.6 0.02 d = 0. 3 0.4 d = 0.3 0.01 0.2 0 0 0.1 0.2 0.3 0.4 0.5 Y 0.6 0.7 0.8 0.9 0 1 0 0.1 0.2 0.3 0.4 Y 0.5 0.6 0.7 0.8 0.9 1 Fig.4: Velocity profiles U 0 f or U 0 P  for different values of 1 , Fig.8: Velocity profiles U1 f or U1P for different values of 1 , with the difference of width of the flow region, at walls temperature A=0 and B=1. with the difference of width of the flow region, at walls temperature A=0 and B=1. 0.07 Da = 0.1 Da = 0.01 Da = 0.001 d = 0.7 0.06 x 10   -4 Da = 0.1 Da = 0.01 Da = 0.001 2.5 0.05 U0f or U0p d = 0. 5 d = 0. 3 2 U1f or U1P d = 0. 5 0.04 1.5 0.03 0.02 1 0.01 0.5 0 0 0.1 0.2 0.3 0.4 Y 0.5 0.6 0.7 0.8 0.9 0 1 d = 0. 7 d = 0. 3 0 0.1 0.2 0.3  0.4 Y 0.5 0.6  0.7 0.8 0.9 1 Fig.5: Velocity profiles U 0 f or U 0 P  for different values of Da Fig.9: Velocity profiles U1 f or U1P for different values of , with the difference of width of the flow region, at walls temperature A=0 and B=1. Da , with the difference of width of the flow region, at walls temperature A=0 and B=1. 1.6 x 10 -4 2.5 Da = 0.1 Da = 0.01 Da = 0.001 1.4 x 10 -3  1 = 0.1  1 = 0.3  1 = 0.5 2 d = 0. 7 or  1P d = 0. 5 1 d = 0.7 1.5 1f U1f or U1P 1.2  0.8 d = 0. 3 0.6 1 0.4 0.5 0.2 d = 0.3 0 0 0.1 0.2 0.3 0.4 Fig.7: Velocity profiles Y U 0.5 1f 0.6 0.7 0.8 0.9 1 or U1P  for different values of Da , with the difference of width of the flow region, at walls temperature A=1 and B=0. 0 0 0.1 0.2 0.3 d = 0.5 0.4 Y 0.5 0.6 0.7 0.8 0.9 1 Fig.10: Temperature profiles 1 f or 1P  for different values of 1 , with the difference of width of the flow region, at walls temperature A=1 and B=0. www.ijsret.org 509 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882 Volume 4, Issue 5, May 2015 3 x 10 -3 VII. APPENDIX Da = 0.1 Da = 0.01 Da = 0.001 or  1P 2.5 2  1  c2  a2  A, c1  a1  B  A, b1    , b2  b1 ,  Da  L L L1  (1  1 ); b3  1 c1 , b4  1 c2 , c4  0, b5  eb2 , b1 b1 1f d = 0.5 d = 0.7  b6  e  b2 , b7  (b3  b 4 ), b8  b3 d  b4  1.5 c1 L1 d 2  c2 L1 d , 2 b12  b9 (1  db2 ), b13  b10 (1  db2 ), b14  b8  db11 , b9  eb2 d , b10  e b2d , b11  b3  1 0.5 0 c1 L1 d 3 c2 L1 d 2  , 6 2 d = 0.3 b15  b6 b12  b5 b13 , b16  b7 b12  b5 b14 , b17  b16 , b15 Fig.11: Temperature profiles 1 f or 1P  for different values of b18  Da , with the difference of width of the flow region, at walls temperature A=1 and B=0. b7  b17 b6 , b19  b2  b18 b9  b17 b10   b11 b5 e1  c1 L1 e2 2e e , e2  c 2 L1 , e3   1 , e4   1 2 , 2 L1 L1 e5  (2 e1 c3  e22 ) 2e c c2 a 2b2 , e6 2 3 , e7   3 ; e8   3 2 ; L1 L1 L1 L1 0 0.1 1.8 x 10 0.2 0.3 0.4 Y 0.5 0.6 0.7 0.8 0.9 1 -3  1 = 0.1 1.6  1 = 0.3 d = 0.7  1 = 0.5 or  1P 1.4 e8   d = 0.5 1 e12   1f 1.2 0.8 0.6 2a3 a4 b22  b32 a2 ; L11  Da (1  L1 ); e13   3 ; L1 L11 e14   2a b 2a b 2a b a42 ; e15   3 3 ; e16   4 3 ; e17   3 4 ; L11 L11 L11 L11 e18   b2 2b b 2a a  b42 2a4 b4 ; e19   3 ; e 20   3 4 ; e 21   3 4 L11 L11 L11 L11 0.4 0.2 d = 0.3 0 0 0.1 0.2 0.3 0.4 Y 0.5 0.6 0.7 0.8 0.9 1 Fig.12: Temperature profiles 1 f or 1P  for different values of 1 , with the difference of width of the flow region, at maintained walls temperature A=0 and B=1. 3 x 10 Da = 0.1 Da = 0.01 Da = 0.001 d = 0. 7 2 d = 0. 5  1.5 d = 0. 3 e e e e22 ; e28  232 ; e29  152 ; e30  162 ; 2 4b2 4b2 b2 b2 2e15 e e e e24  ; e32  25 ; e33  19 ; e34  20 ; 12 6 b22 4b23 b22 b25  b21  b24 ; b26  1 e3 d 5 e d3 e d2 e d4  4  5  6  e7 d 5 4 3 2  e27 b22  e28 b23  e29 db9  e30 db10  e31b9  b24   ; 4 3 2   e32 b10  e33 d  e34 d  e35 d  b 28  b26  b27 ; 0.5 0 -0.5 e26  e12  e21 ; e27   e27 e 2b2  e28 e 2b2  e29 b5  e26   e33  ; c6  0; b20    e30 b6  e31b5  e32 b6   ; 2 e  e  e  34 35  33  e3 d 6 e4 d 5 e5 d 4 e6 d 3 e7 d 2 2 b2 d 2b2 d b22  e ; b23  e ; b21      ; 30 20 12 6 2 1f or  1P 2.5 e22  e8  e13 ; e23  e9  e14 ; e24  e10  e17 ; e25  e11  e18 e31  -3 a32 b22 2a b b 2a b b ; e10   3 2 3 ; e11   4 2 3 ; L1 L1 L1 0 0.1 0.2 0.3 0.4 Y 0.5 0.6 0.7 0.8 0.9 1 Fig.13: Temperature profiles 1 f or 1P  for different values of Da , with the difference of width of the flow region, at maintained walls temperature A=0 and B=1. b27  e22 b22 e23b23 e15 db9 e15 b9 e24 b9   2   2b  2b  b  b b 2 2 2 2 2    e16 b10 e16 b10 e25 b10 e19 d 3 e20 d 2   2     e26 d   b b 3 2 b 2 2  2  www.ijsret.org 510 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882 Volume 4, Issue 5, May 2015 b29  b20  b28 d  b25 ; b30  b29  b28 ; b31  b20  b29 ; Le Le Le f1   1 3 ; f 2   1 4 ; f 3   1 5 ; 1680 840 360 L1e6 L1e7 L1c5 f4   ; f5   ; f6   ; c8  0; 120 24 6  24e33 2e35 a  e48  L1    6  ; b35  b33  b34 ; 3 b1  b12  b1 2 b2 2 b2  e27 e   e28 e  b5  e29  e31   b32  L1    e30  e32  b6  e33  e34  e35  a5  a6    e36   L1e27 Le Le Le ; e37   1 28 ; e38   1 29 ; e39  1 30 ; 3b1 3b1 4b2 4b2 e e e  e  Le e40  L1  29  31  ; e41  L1  30  32  ; e42   1 29 ; 8b1b2  4b1 2b2   4b1 2b2  12e Le Le Le e  e43  1 30 ; e44  1 33 ; e45  1 34 ; e46  L1  233  35  ; 8b2 b1 b1 b1 b1  b  1  6e a  e47  L1  234  5  ; b1   b1  e d8 e d6 e d5 e d4 c d3  e d7 b33   L1  3  4  5  6  7  5 ; 840 360 1120 24 6  1680 e d7 e d 5 e6 d 4 e d 3 c5 d 2  e d6 b36   L1  3  4  5   7   120 60 24 6 2   210 e36 b22  e37 b23  e38 d 2 b9  e39 d 2 b10  e40 db9  e41db10  b34    4 3 2  e42 b9  e43b10  e44 d  e45 d  e46 d  e47 d  e48   2b2 e36 b22  2b2 e37 b23  e38 b9  d 2 b2  2d      b37  e39 b10  2d  d 2 b2   e40 b9  db2  1  e41b10 1  db2      b2 e42 b9  e43b2 b10  4e44 d 3  3e45 d 2  2e46 d  e47    b38  b36  b37 ; b39  b9  b2 d  1 ; b40  b10  b2 d  1 ; b41  dn38  b35 ; b42  b40 b32  b41b6 ; b43  b40 b5  b39 b6 b44  b  b44 b5 b42 ; a7  b44 ; b45  32 ; a8  b45 ; b43 b6 b46  b44 b2 b9  b45 b2 b10  b38 ; c7  b46 ; REFERENCES [1] Beavers, G. 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