J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011)
Non-Darcy Hydromagnetic Convective Heat Transfer Flow of A Viscous Fluid in a Vertical Channel with Radiation and Dissipative Effects. V. NAGARADHIKA1 and D.R.V. PRASADA RAO2 1
Associate Prof, Department of Basic Sciences, Moula Ali College of Engg & Tech., Anantapur. India 2 Professor in Department of Mathematics, S. K. University, Anantapur. India ABSTRACT We consider the dissipative and radiation effects on non-darcy convective flow of a viscous fluid in a vertical channel. By employing a regular perturbation scheme the governing equations of flow and heat are solved. The effects of Ec and N on the flow characteristics are investigated in detail. Keywords: Non-Darcy model, radiation effect, dissipation, mixed convection.
1. INTRODUCTION 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 in Cheng5 and Prasad et al.,13 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 Tien22 was once of the early attempts to account for the boundary and inertia effects in the momentum equations for a porous medium. Vafai and Thyagaraga21 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 Hong19, Chang6, Prasad et al.,15 and Kalidas and Prasad10. Poulikakos and Bejan12
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investigated the inertia effects through the inclusion of Fochhimer’s velocity squared terms and presented the boundary layer analysis for all cavities. Prasad and Tuntomo14 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. The Brinkmann-Extended Darcy model was considered in Tong and Subramamian20 and Laurait and Prasad11 to examine the boundary effects on free convection in a vertical cavity. A numerical study based on the Forchheimer-BrinkmannExtended Darcy equation of motion has also been reported recently by Beckermann et al.,3. 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. Moreover Gebhart7, Gebhart and 8 Mollendorf have shown that viscous dissipation heat in the natural convective flow is important when the fluid of 1 extreme size or at extremely low temperature or in high gravitational field. On the other hand Barletta2 and Zanchini23,24 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, Soundalgekar and Pop18, Ramana Murthy
and Soundalgekar29, Soundalgekar et al.22, Barletta2,4, El-Hain9, Bulent Yasilats4 and Anwar Hossain1 have studied the effect of viscous dissipation on the convective flows past an infinite vertical plates and through vertical channels and ducts. Ravindranath15a 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 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 distribution are analyzed for different values of G,D-1,α and N. Also the shear stress and the rate of heat transfer on the walls are numerically evaluated for different sets of parameters. 2. FORMULATION OF THE PROBLEM We consider the flow of a viscous, incompressible, electrically conducting fluid through a porous medium confined in a vertical channel .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. A uniform magnetic field of strength Ho is applied normal to the walls. Assuming the magnetic Reynolds number Re to be small we neglect the induced magnetic field. The plate sat y=-L is maintained at constant temperature T1 and
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
V. Nagaradhika, et al., J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011)
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 ∂p µ ∂ 2 u µ ρδ F 2 + ( ) − ( )u − u ∂x δ ∂y 2 k k
−ρg−
(σµ e2 H o2 ) u
∂T = −q ∂y
on
y = +L
qr = −
4σ • ∂ (T ′ 4 ) β R ∂y
(2.2)
2
(2.3)
Boundary conditions : y = ±L
constant, say A. Invoking Rosseland approximation (3a) for radiation flux we get
T ′ 4 ≈ 4Te3T − 3Te4
( x ′, y ′) = ( x, y ) / L , u ′ = u /(ν / L), p ′
Equation of State : ρ − ρ e = − β (T − Te )
u = 0 on
∂T is assumed to be a ∂x
Where σ • is the Stefan-Boltzman constant and β R is the mean absorbing coefficient. We introducing the non-dimensional variables
=0
∂u ∂T ∂ 2T ∂(qr ) =λ 2 − + µ ∂x ∂ y ∂y ∂y
The axial gradient
(2.1)
Energy equation : ρ 0C p u
that depends on the Reynolds number and microstructure of porous medium .σ is the electrically conductivity of the fluid, µ is the magnetic permeability of the medium and Ho is the strength of the magnetic field.
and linearising T ′ 4 about Te by using Taylor’s expansion and neglecting higher order terms we get
Momentum equation: −
473
T = T1
= pδ /( ρν 2 / L2 )θ =
T − T2 T1 − T2
(2.5)
the equations reduces to on
y = −L
(2.4)
where T is the temperature of the fluid , ρ is the density of the fluid, C p is the specific heat at constant pressure, k is the permeability of the porous medium, µ is the coefficient of viscosity of the fluid, δ is the porosity of the medium, β is the coefficient of thermal expansion, λ is the coefficient of thermal Conductivity, F is the function
−
dp d 2 u = − δ D −1u − δ M 2 u − δ Gθ dx dy 2
PN I u =
d 2θ
4 d 2θ + Ec du 2 2 dy 3N dy dy u (±1) = 0, θ (−1) = 1,
2
+
dθ (+1) = −1 dy
Where G=
(2.6)
βgL3 (T1 − T2 ) (Grashof Number), ν2
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(2.7)
(2.8)
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V. Nagaradhika, et al., J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011)
σµ e 2 H 0 2 L2 (Hartman Number) ν2 L2 ( porousparameter), Ec (Eckert = K
number) =
to the first order are
d 2 u1 − β 12 u1 = −Gθ 0 dy 2
U2 C p .(T1 − T0 )
d 2θ1 dy 2
In the absence of radiation (N=0) the equations (2.6) & (2.7) reduces to Ruksana Begum25a
Assuming the parameter δ to be small we assume the solutions as
(3.1)
Substituting (3.1) in equations (2.5)-(2.8) and equating the like powers of δ the equations to the zeroth order are 2
d u0 =π dy 2 d θ0 2
dy 2
d 2u 2 2
(3.5)
d 2θ 2 dy 2
− β12 u2 = n(u0 ) 2 − G θ1
du = ( P1 N T )u2 + Ec 2 dy
u0 (+1)=0
;
u0 (-1)= 0
dθ 0 ( +1) = −1 ; θ 0 (-1)= 1 dy
u1 ( ± 1)=0 ; θ1 (-1)=0 :
(3.2)
du = ( P1 N T )u0 + Ec 0 dy
2
(3.6)
2
(3.7)
Boundary Conditions
u ( y ) = u o ( y ) + δu1 ( y ) + δ 2 u 2 ( y ) + .....
θ ( y ) = θ o ( y ) + δθ1 ( y ) + δ 2θ 2 ( y ) + .......
du + Ec 1 dy
and to the second order are
dy
3. SOLUTION OF THE PROBLEM
= ( P1 N T )u1
(3.4)
u 2 ( ± 1)=0 ; θ 2 (-1)=0
2
; ;
dθ1 ( +1) = 0 dy
dθ 2 ( +1) = 0 dy
(3.8)
(3.9)
(3.10)
(3.3)
Solving the equations (3.2)-(3.7) subject to the boundary conditions (3.8)-(3.10) we get. π u o = ( y 2 − 1) 2 a2 y ( y 3 − 1) − a5 ( y + 1) + ( y + 2) 12 Ch ( β 1 y ) Sh ( β 1 y ) Ch ( β 1 y ) u1 ( y ) = a11 (1 − ) + a12 ( y − ) + a13 ( y 2 − )+ Ch ( β 1 ) Sh ( β 1 ) Ch ( β 1 ) Ch ( β 1 y ) + a14 ( y 4 − ) Ch ( β 1 )
θ 0 ( y ) = 0.5a1 y ( y + 1) +
Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
V. Nagaradhika, et al., J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011)
θ1 ( y ) = B1 ( ySh( β1 y ) − ySh( β1 ) − β1 yCh( β1 ) − 2Sh( β1 ) − β1Ch( β1 )) + + B2 ( yCh( β1 y ) + β1 ySh( β1 ) − yCh( β1 ) − β1 Sh( 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 )
4. SHEAR STRESS, NUSSELT NUMBER AND SHERWOOD 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 du = ( ) y = ±1 µ (T1 − T2 ) / L dy
and the corresponding expressions are (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 ) Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)
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V. Nagaradhika, et al., J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011)
(τ ) y = −1 = −π + δ (a11 β1Th ( β1 ) + a12 (1 − β1Cth ( β1 ) − a13 (2 − β1Th ( β1 ) + − a14 (4 − β1Th ( β1 )) + δ 2 (2 B17 Sh( β1 ) − 2 B18 Ch ( 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 dy (T1 − T2 ) k
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 1) As we move from the left boundary to right boundary the velocity is negative for all values of G,D-1 and M. u enhances with increase in G ( 0) fig(1). 2) Lesser the permeability of the porous medium larger the magnitude of u everywhere in the flow region fig(2). 3) Higher the Lorenz force larger u in the entire flow region with maximum occurring at y=0. 4) Radiation parameter N, we find that for all values of N. u is negative for all N. The behavior of u with N is different in the left and right regions. The enhancement in u in the left region is uniform whose in the right region there is a steep rise in u in the region and depreciation in the vicinity of y=0.5 and
5). 6).
7).
8)
again enhances in the remaining region fig(4). It is find that u depreciates in every where in the flow region. The temperature gradually enhances from value ‘1’ on the left boundary to attain its value on the right boundary y = 1. It is found that the temperature enhances with increase in G > 0 fig(6). The behavior of θ with D-1 shows that lesser the permeability of the porous medium smaller the temperature in the mid region and with a dip at y=0.2. The variation of θ in the left region is marginal and there is appreciable change in θ in the right region with increase in D-1. θ attains maximum at y=0.8 fig(7). u experiences depreciation with increase in M in the right half. It is to be
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V. Nagaradhika, et al., J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011)
noticed that depreciation in θ in the left half is negligible and is more predominate in the right half of the channel fig(8). 9) An increase in N reduces θ marginally in the left half and appreciably in the right half fig(9). 10) The inclusion viscous dissipation reduces the temperature in the flow region fig(10). 11) The variation of τ with G shows that τ at y=±1 depreciates with G>0 and enhances with G. 12) Lesser the permeability to the porous medium larger τ for G>0 and smaller τ for G<0 at both the walls. Also higher the Lorenz force smaller the magnitude of τ at y=±1 in both heating and cooling cases τ experiences a depreciation with increase in the radiation parameter N, increase the Eckert number Ec reduces τ for G>0 and enhances it for G<0 and both the walls (table 1 & 2). 13) The rate of heat transfer enhances at
y=+1and depreciates y=-1 with increasing G>0 while it reduces at y=1 and enhances y=-1 forG<0. Lesser the permeability porous medium smaller Nu at y=1 and larger Nu at y=-1 at reverse effect is observed for G<0. 14) Also higher the Lorenz force larger the rate of heat transfer at y=1 and at y=-1 larger Nu for G>0 and smallerNu for G<0. Nu depreciates at y=1 and enhances at y=-1 for G<0 while for G>0 it exhibits an increasing tendency y=±1 at G=3x103 while at G=103 it fluxgates with N. 15) It is found that the rate of heat transfer reduces at y=1 and enhances at y=-1 with Ec while for G<0 it exhibits an increasing tendency at y=1 and depreciates at y=-1 at G<3x103 while at G≥3x103 it reduces with Ec≤0.15 and enhance at higher value Ec=0.35 and for still higher values of Ec it again reduces at the both the walls. In general we notice that the rate of heat transfer at y=1 is less than that y=-1(table 3 &4).
Fig. 2 u with D-1
Fig. 1 velocity u with G 3
D = 10 , M = 2, I G 103
N = 1 EC = 0.5 II III 3x103 -103
477
3
IV -3x103
G = 3x10 , N = 1 EC = 0.5, I II D-1 103 2x103
M=2 III 3x103
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IV 5x103
478
V. Nagaradhika, et al., J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011)
Fig. 3 u with M
Fig. 4 u with N
G = 3x103, D-1 = 103, EC = 0.05 I II III M 2 4 6
G = 3x103, D-1 = 103, M = 2, EC = 0.05 I II III IV V N 0.3 0.5 1.5 1.5 5
Fig. 6 θ with G
Fig. 5 u with EC D-1 = 103, M = 2, N = 1 I II III Ec 0.05 0.1 0.3
IV 0.5
G
G = 3x103, N = 1, EC = 0.05 I II III 103 2x103 3x103
IV 5x103
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V. Nagaradhika, et al., J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011 1)
Fig 7 θ with D-1
Fig. 8 θ with M
G = 3x103, M = 2, EC = 0.05 I II III D-1 103 2x103 3x103
G = 3x103, D-1 = 103, N = 1, EC = 0.05 I II III M 2 4 6
IV 5x103
Fig. 9 θ with N
N
G = 3x103, D-1 = 103, M = 2 I II III IV 0.3 0.5 1 1.5
Fig. 10 θ with EC V 5
VI 10
EC
G = 3x103, N = 1, M = 2 I II III IV 0.05 0.1 0.3 0.5
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479
480
V. Nagaradhika, et al., J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011)
Table. 1 Shear stress τ at y=1 G 103 3X103 -103 -3X103
I 0.9891 0.9669 1.0113 1.0335
II 0.9919 0.9755 1.0084 1.0248
D-1 M Ec N
102 2 0.05 1
III 0.9933 0.9796 1.0069 1.0206
2x102 2 0.05 1
IV 0.9994 0.97 1.0108 1.0319
3x102 2 0.05 1
102 4 0.05 1
V 0.9903 0.9706 1.01 1.0542
102 6 0.05 1
VI 0.9823 0.9463 1.0322 1.0956
102 2 0.15 1
VII 0.9687 0.9053 1.0322 1.956
102 2 0.35 1
102 2 0.05 5
VIII 0.9809 0.942 1.0198 1.0587
IX 0.9775 0.9318 1.0233 1.069
102 2 0.05 10
Table. 2 Shear stress τ at y=-1 G 103 3X103 -103 -3X103
I -0.9961 -0.9882 -1.0039 -1.0118
II -0.9971 -0.9912 -1.0029 -1.0088
D-1 M Ec N
102 2 0.05 1
III -0.9976 -0.9927 -1.0024 -1.0073
2x102 2 0.05 1
IV -0.9998 -0.98 -1.0038 -1.0113
3x102 2 0.05 1
102 4 0.05 1
V -0.9965 -0.9895 -1.0035 -1.0105
102 6 0.05 1
VI -0.9946 -0.9837 -1.0054 -1.0163
102 2 0.15 1
102 2 0.35 1
VII -0.9916 -0.9748 -1.0084 -1.0252
102 2 0.05 5
VIII -0.9931 -0.9794 -1.0069 -1.0252
IX -0.9919 -0.9757 -1.0081 -1.0243
102 2 0.05 10
Table. 3 Nusselt number(Nu)at y= 1 G 103 3X103 -103 -3X103
I -0.8945 1.0007 0.9998 0.9993 D-1 M Ec N
II 1.0001 1.0003 0.9999 0.9997 102 2 0.05 1
III 0.9997 1.0001 1.0002 0.9999
2x102 2 0.05 1
3x102 2 0.05 1
IV 0.9009 1.0001 1.0989 0.9994 102 4 0.05 1
V 1.0002 1.0005 0.9998 0.9995 102 6 0.05 1
VI 1.0919 1.0019 0.9993 0.998 102 2 0.15 1
102 2 0.35 1
VII 1.0693 1.0026 0.9991 0.9974 102 2 0.05 5
VIII 1.0569 1.0082 0.9972 0.9917
IX 1.0418 1.0283 0.9903 0.9713
102 2 0.05 10
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V. Nagaradhika, et al., J. Comp. & Math. Sci. Vol.2 (3), 471-482 (2011)
481
Table. 4 Nusselt number(Nu)at y= -1 G 103 3X103 -103 -3X103
I -0.8441 1.0504 1.0508 1.051 D-1 M Ec N
II 1.0507 1.0506 1.0506 1.0509 102 2 0.05 1
III 1.0509 1.0508 1.0505 1.0508
2x102 2 0.05 1
3x102 2 0.05 1
IV 0.9041 1.0503 1.1499 1.0508 102 4 0.05 1
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