J. Comp. & Math. Sci. Vol.2 (2), 372-383 (2011)
Non-Darcy Convective Heat Transfer Flow in A Vertiocal Channel with Radiation Effect with Constant Heat Flux A. S. R. MURHY1 and D. R. V. PRASADA RAO2 1
Department of Mathematics. Mallareddy College of Engineering and Technology Hyderabad, A.P, India 2 Department of Mathematics. S. K. University, Anantapur-515055 Andhra Pradesh, India email: drv_atp@yahoo.in ABSTRACT In this paper we investigate the effect of radiation on convective heat transfer flow of a viscous fluid through a porous medium in a vertical channel which is maintained at constant heat flux. The governing equations of momentum and energy are solved by using a perturbation technique. The velocity, temperature, rate of heat transfer are analysed for different values of parameters. Keywords: Heat transfer, Non-darcy flow, porous medium.
1. INTRODUCTION Any substance with a temperature above zero transfers heat in the form of radiation. Thermal radiation always exits and can strongly interact with convection in many situations of engineering interest. The influence of radiation on natural or mixed convection is generally stronger than that on forced convection because of the inherent coupling between temperature and flow field2. Convection in a channel (or enclosed space) in the presence of thermal radiation continues to receive considerable attention because of its importance in many practical applications such as furnaces, combustion chambers, cooling towers, rocket engines and solar collectors. During the past several decades, a number of experiments and numerical computations have been presented
for describing the phenomenon of natural (or missed) convection in channel or enclosures. These studies aimed at clarifying the effect of mixed convection of flow and temperature regimes arising from variations in the shape of the channel (or enclosure), in fluid properties, in the transition to turbulence, etc. Chawla and Chan5 studied the effect of radiation heat transfer on thermally developing Poiseuille flow. The interaction of thermal radiation with conduction and convection in thermally developing, absorbing-emitting, non gray gas flow in a circular tube is investigated by Tabanfar and Modest18. Natural convectionradiation interaction is studied by Yucel et al.,22 for a square cavity, Lauriat10 for a vertical cavity. Akiyama and Chang1 numerically analysed the influence of gray surface radiation on the convection of
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nonparticipating fluid in a rectangular enclosed space. Non-Darcy effects on natural convection in porous media have received a great deal of attention in recent years because of the experiments conducted with several combinations of solids and fluids covering wide ranges of governing parameters which indicate that that the experimental data of systems other than glass water at low Rayleigh members do not agree with theoretical predictions based on the Darcy flow model. This divergence in the heat transfer results has been reviewed in detail Prasad et al.,12 among others. Extensive efforts are thus being made to include the inertia and viscous diffusion terms in the flow equations and to examine their effects in order to develop a reasonable accurate mathematical model for convective transport in porous media. The work of Vafai and Tien21 was once of the early attempts to account for the boundary and inertia effects in the momentum equations for a porous medium. They found that the momentum boundary layer thickness is of
k ε . Vafai and Thyagaraja20 order of presented analytical solutions for the velocity and temperature fields for the interface region using Brinkmann – Forchhimer – Extended Darcy equation. Detailed accounts of accounts of recent efforts on non-Darcy convection have been recently reported in Tien and Hong18a, Chang6, Prasad et al.,14 and Kalidas and Prasad9. Poulikakos and Bejan11 investigated the inertia effects through the inclusion of Fochhimer’s velocity squared terms and presented the boundary layer analysis for all cavities. They also obtained numerical
results for a few cases in order to verify the accuracy of their boundary layer solutions. Later Prasad and Tuntomo13 reported an extensive numerical work for a wide range of parameters and demonstrated that effects of Prandtl number remain almost unaltered while the dependence on the modified Grashof number G changes significantly with an increase in the Forchheimer number. They also reported a criterion for the Darcy flow limit. All the above studies are based on the hypothesis that the effect of dissipation is neglected. This is possible in case of ordinary fluid flow like air and water under gravitational force. But this effect is expected to be relevant for fluids with high values of the dynamic viscosity force. More over Gebhart and Mollendorf7 have shown that viscous dissipation heat in the natural convective flow is important when the fluid of extreme size or at extremely low temperature or in high gravitational field. On the other hand Barletta3 pointed out that relevant effects of viscous dissipations in the temperature profiles and in the Nusselt Number may occur in the fully developed convection in tubes .In view of this several authors notably, Soundalgekar17, Soundalgekar et al.19, Barletta3, El-Hain8 and Bulent Yasilats4 have studied the effect of viscous dissipation on the convective flows past an infinite vertical plates and through vertical channels and ducts. Ravindranath15 has analysed the dissipative effects on the convective heat transfer in channels under different conditions. In the paper we investigate the effect of radiation on the Non-Darcy convection heat transfer flow of a viscous dissipative fluid through a porous medium confined in a
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
A. S. R. Murhy, et al., J. Comp. & Math. Sci. Vol.2 (2), 372-383 (2011)
vertical channel. The non-liner, coupled equations governing the flow and heat transfer have been solved by employing a perturbation technique with δ the porous parameter as perturbation parameter. The velocity and temperature dissipation are analysed for different values of G, D-1, N, Ec and NT. Also the shear stress and the rate of heat transfer on the walls are numerically evaluated for different sets of parameters. Nomenclature u velocity component in the x-direction p pressure k porous permeability g acceleration due to gravity x,y axial and vertical coordinates T temperature CP Specific heat at constant pressure qr radiative heat flux Greek symbols µ dynamic viscosity ρ density of the fluid δ porosity of the medium λ coefficient of thermal conductivity
σ ∗ Stefan − Boltzman constant β R mea absorption coefficient
374
the plate at y=+L is maintained at a constant flux. The temperature gradient in the flow is sufficient to cause natural convection in the flow field. A constant axial pressure gradient is also imposed so that this resultant flow is a mixed convection flow. We also take dissipation into account in the energy equation. The porous matrix is assumed to be isotropic and homogeneous with constant porosity and the effective thermal diffusivity. The equations governing the flow and heat transfer are Momentum equation −
∂p µ ∂2u µ ρδ F 2 + ( 2 ) − ( )u − u −ρg =0 k ∂x δ ∂y k
2.1
Energy equation:
ρ0C pu
∂T ∂ 2T ∂(q ) ∂u = λ 2 − r + µ ( )2 ∂x ∂y ∂y ∂y
2.2
Equation of State:
ρ − ρ e = − β (T − Te )
2.3
Boundary conditions: u = 0 on y = ± L
2. MATHEMATICAL ANALYSIS We consider the flow of a viscous, incompressible fluid through a porous medium confined in a vertical channel in the presence of heat sources. A Cartesian coordinate system O(x,y,z) is used so that the boundaries are taken at y = ± L ,where 2L is the distance between the walls. Boussinesq approximation is used so that the density variation is taken only in the buoyancy force term. The plate sat y=-L is maintained at constant temperature T1and
T = T1
on
∂T = −q ∂y
y = −L
on
The axial gradient
y = +L
∂T is assumed to be a ∂x
constant, say A. Invoking Rosseland radiation flux we get
qr = −
2.4
approximation
4σ • ∂ (T ′4 ) β R ∂y
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for
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and linearising T ′4 about Te by using Taylor’s expansion and neglecting higher order terms we get T ′ 4 ≈ 4Te3T − 3Te4 where σ is the Stefan-Boltzman constant and β R is the mean absorbing coefficient. •
We introduce the following non-dimensional variables as ( x ′, y ′ ) = ( x , y ) / L , u ′ = u /(ν / L ), p ′ = pδ /( ρν 2 / L2 ) 2.5 θ =
T − T2 T1 − T2
Substituting (2.5) the equations in the nondimensional form are −
dp d 2u = − δ D −1u − δ Gθ dx dy 2
2.6
d 2θ du 4 d 2θ PN T u = 2 + PEc( ) 2 + 2.7 dy 3 N dy 2 dy u(±1) = 0, θ (−1) = 1,
dθ (+1) = −1 dy
2.8
3. SOLUTION OF THE PROBLEM Assuming the parameter δ to be small we assume the solutions as u ( y ) = uo ( y ) + δu1 ( y ) + δ 2u 2 ( y ) + ....... θ ( y ) = θ o ( y ) + δθ1 ( y ) + δ θ 2 ( y ) + .......... 3.1
d 2θ 0 du = ( P1 N T )u 0 + P1 E c ( 0 ) 2 2 dy dy to the first order are
d 2u1 − β12u1 = −Gθ 0 2 dy
d 2u 0 =π dy 2
3.2
3.4
du du d 2θ1 = ( P1 NT )u1 + 2 P1 Ec ( 0 )( 1 ) 2 dy dy dy 3.5 and to the second order are
d 2u 2 − β 12 u 2 = n(u 0 ) 2 − G θ 1 2 dy
3.6
d 2θ 2 = ( P1 N T ) dy2 du0 du2 +2( )( )) dy dy
3.7
u
2
+ P1 E c (
d u1 2 ) dy
Boundary Conditions
u0 (+1)=0
u0 (-1)= 0
;
dθ 0 ( +1) = −1 dy
;
u1 ( ± 1)=0
;
θ1 (-1)=0 :
2
Substituting (3.1) in equations (2.5)-(2.8) and equating the like powers of δ the equations to the zeroth order are
3.3
u2 ( ± 1)=0 θ 2 (-1)=0
;
θ 0 (-1)= 1 ;
dθ 1 ( +1) = 0 dy
3.8
3.9
;
dθ 2 ( +1) = 0 dy
3.10
Solving the equations(3.2)-(3.7) subject to the boundary conditions(3.8)-(3.10) we get.
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u0 = 0.5π ( y 2 − 1),θ 0 ( y ) = 0.5a1 y ( y + 1) + u 1 ( y ) = a 1 1 (1 −
376
a2 y ( y 3 − 1) − a5 ( y + 1) + ( y + 1) 12
C h(β1 y) Sh(β1 y) C h(β1 y) ) + a12 ( y − ) + a13 ( y 2 − ) + C h(β1) Sh(β1) C h(β1)
+ a14 ( y 4 −
C h(β1 y) ) C h(β1)
θ1 ( y ) = B1 ( ySh( β1 y ) − ySh( β1 ) − β1 yCh( β1 ) − 2Sh( β1 ) − β1Ch( β1 )) + + B2 ( yCh( β1 y ) + β1 ySh( β1 ) − yCh( β1 ) − β1Sh( β1 ) − 2Ch( β1 )) + + B3 (Ch( β1 y ) − β1 ySh( β1 ) − β1Sh( β1 ) − Ch( β1 )) + + B4 ( Sh( β1 y ) + β1 yCh( β1 ) + Sh( β1 )) + B5 ( y 3 − 3 y + 2) + + B6 ( y 4 − 4 y − 5) + B7 ( y 7 − 6 y − 7) + 3
u 2 ( y ) = B17 ( y 2 − 1) Sh( β1 y ) + B18 ( y 2 − 1)Ch( β1 y )) + + B19 ( ySh( β1 y ) −
Ch( β1 y ) Sh( β1 )) + Ch( β1 )
+ B20 ( yCh( β1 y ) −
Ch( β1 y ) Ch( β1 )) + Ch( β1 )
+ B21 (
Ch( β1 y ) Ch( β1 y ) − y 6 ) + B22 ( y 4 − )+ Ch( β1 ) Ch( β1 )
+ B23 ( y 2 − + B25 (1 −
Ch( β1 y ) Sh( β1 y ) ) + B24 ( y − )+ Ch( β1 ) Sh( β1 )
Ch( β1 y ) ) Ch( β1 )
where β12 = D −1 . In the absence of porous medium the results are in good agreement with Ruksana Begum15a 4. SHEAR STRESS AND NUSSELT NUMBER The shear stress on the walls is given by
τ = µ(
du ) y =± L dy
which in the non-dimensional form is
τ =
τ
µ (T1 − T2 ) / L
=(
du ) y = ±1 dy
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and the corresponding expressions are
(τ ) y = +1 = π − δ (a11 β 1Th( β 1 ) + a12 (1 − β 1Cth( β 1 ) + a13 (2 − β 1th( β 1 ) + + a14 (4 − β 1th( β 1 )) + δ 2 (2 B17 Sh( β 1 ) + 2 B18 Ch( β 1 ) + B19 ( Sh( β 1 ) + + β 1Ch( β 1 ) − β 1Th ( β 1 ) Sh( β 1 )) + B20 (Ch( β 1 ) + β 1 Sh( β 1 ) − − β 1Cth( β 1 )Ch( β 1 ) + B21 (6 − β 1Th( β 1 )) + B22 (4 − β 1Th( β 1 )) +
+ B23 (2 − β 1Th( β 1 )) + B24 (1 − β 1Cth ( β 1 )) − B25 β 1Th( β 1 ) (τ ) y =−1 = −π + δ (a11β1Th( β1 ) + a12 (1 − β1Cth( β1 ) − a13 (2 − β1Th( β1 ) + − a14 (4 − β1Th( β1 )) + δ 2 (2 B17 Sh( β1 ) − 2 B18Ch( β1 ) − B19 ( Sh( β1 ) + + β1Ch( β1 ) + β1Th( β1 ) Sh( β1 )) + B20 (Ch( β1 ) + β1Sh( β1 ) + + β1Cth( β1 )Ch( β1 )) − B21 (6 − β1Th( β1 )) − B22 (4 − β1Th( β1 )) + − B23 (2 − β1Th( β1 )) + B24 (1 − β1Cth( β1 )) + B25 β1Th( β1 ) The rate heat transfer (Nusselt number) on the walls is given by
q = −k (
dT ) y =± L dy
which in the non-dimensional form is
Nu = −
qL dθ = ( ) y = −1 (T1 − T2 ) k dy
and the corresponding expressions are
( Nu ) y =−1 = − B1δ β1Ch( β1 ) + Sh( β1 )) + B2δ (Ch( β1 ) − β1 Sh( β1 )) − − δ B3 β1 Sh( β1 ) + δ B4 (Ch( β1 )) + 3δ B5 − 4δ B6 − 6δ B7 + + a24 + a1 −
a2 + a3 3
5. RESULTS AND DISCUSSION In this analysis we discuss the effect of radiation on the non-Darcy convective heat transfer flow of viscous fluid through a porous medium in a vertical channel with constant heat flux by using a regular perturbation technique. The Governing
equations are solved to obtain the expressions for velocity, temperature, shear stress and the rate of heat transfer. Fig. 1-7 represent the variation of axial velocity with different values of G, D-1, α, M, NT and Ec. Fig. 1 represents the variation u with Grashof number G. We find that the velocity u is in the vertically
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downward direction. The magnitude of u depreciates everywhere in the flow region with increase in G>0 and enhances with |G| (<0). The variation of u with D-1 is shown in fig.3 . It is found that lesser the permeability porous medium lesser the magnitude of u in the entire flow region. From fig.2. We notice that an increase in the radiation parameter N leads to a depreciation in the magnitude of u. Also an increase in the temperature gradient NT depreciates |u| (fig.3). The variation u with Eckert number Ec is shown in fig.5. It is found that |u| experiences an enhancement in magnitude with increase in Ec, NT(fig.4) and depreciates with higher Ec > 0.5. Thus the viscous dissipative heat enhances |u| and for higher Ec, |u| experiences a depreciation in the entire flow region. The non-dimensional temperature (θ) is shown in figs 8-14 for different parametric values. The profile θ gradually reduces from prescribed value 1 on the left boundary y =-1 to attain the value ‘0’ on the right boundary y = 1.The variation of θ with Grashof number G is shown in fig. 6. It is found that the actual temperature experiences an enhancement with increase in G>0 and depreciates with G<0. Fig. 7 shows that lesser the permeability of the porous medium larger the actual temperature. From fig.8 we find that the actual temperature reduces with increase in the radiation parameter N. Also it depreciates with temperature gradient NT.With respect to the behaviour of θ with Ec, we find that θ exhibits an increasing tendency with Ec ≤ 0.1 and depreciates with higher Ec ≥ 0.5 Thus higher the dissipative heat larger the actual temperature and for still higher dissipative heat smaller the actual temperature in the entire flow region (fig. 10).
378
The shear stress (τ) at y = ±1 is exhibited in tables 1 to 8 for different values of G, D-1, N, NT and Ec. We find that the magnitude of the stress enhances with increase in G<0 and reduces with G>0 at both the walls. The variations of τ with D-1 shows that lesser the permeability porous medium the magnitude of τ at y = ±1 in the heating case and smaller |τ| in the cooling case. With respect to radiation parameter N we find that |τ| experiences a depreciation and enhances for G<0 at both the walls (tables 2 & 6).Also it reduces with increase in the temperature gradiant NT for G>0 and enhances for G<0 (tables 3 & 7) at y = ±1 The variation of τ with Eckert number Ec exhibits that at y = ±1, |τ| enhances with Ec ≤ 0.15 and depreciate with higher Ec ≥ 0.35 in the heating case while a reversed effect is observed in the cooling of channel walls at y = ±1. The Nusselt number (Nu) which measures the rate of heat transfer at the boundaries y = ±1 is shown in tables in 9-16 for different values of G, D-1, N, NT, and Ec. It is found the rate of heat transfer at y = +1 enhances with G>0 and depreciates with G<0 while it enhances with G at y = -1. The variation of Nu with D-1 shows that lesser the permeability of the porous medium smaller the rate of heat transfer for G>0 and larger |Nu| for G<0 at y = 1 while at y = -1 larger |Nu| in the heating case and smaller |Nu| in the cooling case. (Tables 9 and 3). The variation of Nu with radiation parameter N shows that |Nu| at y = 1 enhances in the heating case and reduces in the cooling case with increase in N while at y = -1 the rate of heat transfer experiences an enhancement with increase in N for all G. (tables 10 &
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14).An increase in the temperature gradient NT enhances |Nu| in the heating case and depreciates in the cooling case at y = 1 while at y = -11 it enhances with NT for all G. (Table 11 and 15). The variation of Nu with Eckert number Ec shows that the rate of heat transfer depreciates with Ec ≤ 0.15 and
enhances with higher Ec ≥ 0.35 in the heating case while in the cooling case a reversed effect is noticed. At y = -1, Nu depreciates with Ec ≤ 0.15 and enhances with Ec ≥ 0.35 for all G (tables 12 & 16).
Fig. 1 : Variation of u with G
G
I 102
II 3x102
III -102
IV -3x102
Fig. 2 : Variation of u with N
N
Fig. 3 : Variation of u with D-1 I D-1 102
II 2x102
III 3x102
IV 5x102
I 0.5
II 1.5
III 4.5
IV 10
Fig. 4 : Variation of u with NT NT
I 0.5
II 1.5
III 2.5
IV 3.5
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Fig. 5 : Variation of u with Ec I II III Ec 0.05 0.1 0.5
D-1
Fig. 7 : Variation of θ with D-1 I II III IV 102 2x102 3x102 5x102
NT
Fig. 9 : Variation of θ with NT I II III IV 0.5 1.5 2.5 3.5
IV 1.5
G
Fig. 6 : Variation of θ with G I II III IV 102 3x102 -102 -3x102
Ec
Fig. 8 : Variation of θ with Ec I II III IV 0.5 1.5 4.5 10
Ec
Fig. 10 : Variation of θ with Ec I II III IV 0.5 1.5 4.5 10
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380
381
A. S. R. Murhy, et al., J. Comp. & Math. Sci. Vol.2 (2), 372-383 (2011) Table – 1 Shear stress (ττ) at y = 1
G 103 3x103 -103 -3x103 D-1 N NT Ec
I 0.9889 0.9663 1.0115 1.0341 102 0.5 0.5 0.05
II 0.9919 0.9753 1.0084 1.0250 2x102 0.5 0.5 0.05
III 0.9933 0.9795 1.0070 1.0207 3x102 0.5 0.5 0.05
IV 0.9863 0.9584 0.0142 1.0421 102 1.5 0.5 0.05
V 0.9771 0.9306 1.0237 1.0702 102 10 0.5 0.05
VI 0.9806 0.9410 1.0201 1.0597 102 4 0.5 0.05
VII 0.9736 0.9198 1.0274 1.0812 102 0.5 1.5 0.05
VIII 0.9583 0.8733 1.0433 1.1283 102 0.5 2.5 0.05
IX 0.9820 0.9454 1.0186 1.0552 102 0.5 0.5 0.150
X 0.9682 0.9036 1.0328 1.0973 102 0.5 0.5 0.350
XI 0.9578 0.8722 1.0434 1.1290 102 0.5 0.5 0.5
VIII -0.9831 -0.9492 -1.0170 -1.0509 102 0.5 2.5 0.05
IX -0.9945 -0.9835 -1.0055 -1.0166 102 0.5 0.5 0.150
X -0.9915 -0.9743 -1.0086 -1.0257 102 0.5 0.5 0.350
XI -0.9892 -0.9675 -1.0109 -1.0326 102 0.5 0.5 0.5
Table – 2 Shear stress (ττ) at y = – 1 G 103 3x103 -103 -3x103 D-1 N NT Ec
I -0.9960 -0.98803 -1.0040 -1.0120 102 0.5 0.5 0.05
II -0.997 -0.9912 -1.0030 -1.0089 2x102 0.5 0.5 0.05
III -0.9976 -0.9926 -1.0025 -1.0074 3x102 0.5 0.5 0.05
IV -0.9951 -0.9852 -1.0049 -1.0148 102 1.5 0.5 0.05
V -0.9918 -0.9753 -1.0082 -1.0247 102 10 0.5 0.05
VI -0.9930 -0.9790 -1.0070 -1.0210 102 4 0.5 0.05
VII -0.9896 -0.9686 -1.0105 -1.0314 102 0.5 1.5 0.05
Table – 3 Nusselt Number Nu at y = 1 G 103 3x103 -103 -3x103 D-1 N NT Ec
I 0.8945 1.0007 0.9998 0.9993 102 0.5 0.5 0.05
II 1.0011 1.00003 0.9999 0.9996 2x102 0.5 0.5 0.05
III 1.0006 1.0002 0.9999 0.9998 3x102 0.5 0.5 0.05
IV 1.0635 1.0016 0.9995 0.9984 102 1.5 0.5 0.05
V 1.0422 1.0294 0.9899 0.9702 102 10 0.5 0.05
VI 1.0570 1.0085 0.9971 0.9914 102 4 0.5 0.05
VII 1.0539 1.0016 0.9994 0.9983 102 0.5 1.5 0.05
VIII 1.1657 1.0006 0.9991 0.9974 102 0.5 2.5 0.05
IX 0.7164 1.0011 0.9996 0.9989 102 0.5 0.5 0.150
X 1.0919 1.0020 0.9993 0.9980 102 0.5 0.5 0.350
XI 1.1093 1.0027 0.9991 0.9973 102 0.5 0.5 0.5
VIII 1.4181 1.2527 1.2539 1.2544 102 0.5 2.5 0.05
IX 0.7665 1.0501 1.0509 1.0513 102 0.5 0.5 0.150
X 1.1416 1.0525 1.0511 1.0519 102 0.5 0.5 0.350
XI 1.1586 1.0501 1.0513 1.0524 102 0.5 0.5 0.5
Table – 4 Nusselt Number Nu at y = – 1 G 103 3x103 -103 -3x103 D-1 N NT Ec
I -0.8441 1.0504 1.0508 1.0510 102 0.5 0.5 0.05
II 1.0507 1.0506 1.0504 1.0509 2x102 0.5 0.5 0.05
III 1.0509 1.0509 1.0502 1.0506 3x102 0.5 0.5 0.05
IV 1.1255 1.0620 1.0629 1.0633 102 1.5 0.5 0.05
V 1.1330 1.0923 1.1087 1.1168 102 10 0.5 0.05
VI 1.1418 1.1053 1.1200 1.1223 102 4 0.5 0.05
VII 1.2054 1.1516 1.1523 1.1527 102 0.5 1.5 0.05
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6. CONCLUSIONS
3.
1.
4.
2. 3.
4. 5.
6. 7.
8.
The velocity u depreciates with increase in G>0 and enhances with G<0. Lesser the permeability of the porous medium smaller the magnitude of u An increase in the radiation parameter N depreciates u in the entire flow region. An increase in Ec leads to an enhancement in |u| The actual temperature enhances with G>0 and reduces with G<0.Also an increase in D-1 enhances the actual temperature. The actual temperature depreciates with N and enhances with Ec. An increase in Ec through smaller values results in an enhancement in τ at y= ± 1 and reduces with higher Ec for G>0 and a reversed effect is noticed in τ. The rate of heat transfer enhances with N for G>0 and reduces for G<0.An increase in Ec<0.15 reduces Nu and decreases it with Ec>0.35 at y=1 in the heating case and a reversed effect is observed in the cooling case. At y=-1 it enhances with Ec<0.15 and reduces with higher Ec>0.35 for all G.
5.
6. 7. 8.
9.
2.
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Barletta, A : Int. J. Heat Mass Transfer V. 40, pp. 15-26 (1997). Bulent Yasilata : Int. J. Heat Mass Transfer (2002). Chawla TC, Chan SH. Combined radiation and convection in thermally developing Poiseuille flow with scattering. J. Heat Transfer ; 102 : 297-302 (1980). Cheng. P. Proc. ASME/JSME Heat Transfer Conf. pp. 297-303 (1987). Gebhart and Mollendrof : J. Fluid Mech. V. 38, pp. 97-107 (1969). A. Hakein, M. A. Int. Comm. Heat Mass Transfer V. 27, pp. 581-590 (2000). Kalida. N. and Prasad, V. Benard convection in porous media Effects of Darcy and Pransdtl Numbers. Int. Syms. Convection in porous media, Non-Darcy effects, Proc. 25th Nat. Heat Transfer Conf. V. 1, pp. 593-604 (1988).
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Laurati G. Combined radiationconvection in gray fluids enclosed in vertical cavities. J. Heat Transfer; 104: 609-15 (1982).
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Poulikakos D. and Bejan, A. The Departure from Darcy flow in Nat. Convection in a vertical porous layer, physics fluids V. 28, pp. 3477-3484 (1985).
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Prasad V. Kulacki, F.A. and Keyhani, M. Natural convection in a porous media, J. Fluid Mech. V. 150, pp. 89119 (1985). Prasad, V. and Tuntomo, A. Inertia Effects on Natural Convection in a vertical porous cavity, numerical Heat Transfer, V. 11, pp. 295-320 (1987).
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Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)