J. Comp. & Math. Sci. Vol.2 (4), 595-604 (2011)
Unsteady effects of Thermal Radiation on MHD Free Convection Flow past a Vertical Porous Plate 1
SEETHAMAHALAKSHMI, 2G. V. RAMANA REDDY and 3B. D. C. N PRASAD
1,3
2
P.V.P.Siddartha Institute of Tecnology, Vijayawada - 521007, A. P. India Usha Rama College of Engineering and Technology, Telaprolu-521109, A. P. India. ABSTRACT The effects of thermal radiation on Magnetohydrodynamics (MHD) free convective heat and mass transfer flow past a vertical porous plate are studied. The governing time dependent boundary layer equations are transformed to ordinary differential equations containing radiation parameter, permeability parameter, Prandtl number, and Thermal diffusion parameter. These equations are solved by a closed analytical form method. The velocity profiles, temperature profiles, concentration profiles and the skin friction coefficient are computed and discussed in details for various values of the different parameters. Keywords: Heat and mass transfer; free convection; MHD; Porous medium; Vertical plate; Finite difference method; thermal diffusion.
1. INTRODUCTION The convection problem in a porous medium has important applications in geothermal reservoirs and geothermal extractions. The process of heat and mass transfer is encountered in aeronautics, fluid fuel nuclear reactor, chemical process industries and many engineering applications in which the fluid is the working medium. The wide range of technological and industrial applications has stimulated considerable amount of interest in the study of heat and mass transfer in convection flows. The radiation effect on
heat transfer over a stretching surface has been studied by Elbashbeshy1. Takhar et al.2 studied the radiation effects on MHD freeconvection flow of a gas past a semi-infinite vertical plate. Thermal radiation and buoyancy effects on MHD free convective heat generating flow over an accelerating permeable surface with temperature dependent viscosity has been studied by Seddeek3. Ghaly and Elbarbary4 have investigated the radiation effect on MHD free convection flow of a gas at a stretching surface with a uniform free stream. In all the above studies, only steady-state flows over semi infinite vertical plate have been
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
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Seethamahalakshmi, et al., J. Comp. & Math. Sci. Vol.2 (4), 595-604 (2011)
considered. The unsteady free-convection flows over a vertical plate have been studied Takhar et al.6. by Gokhale5, 7 Muthucumaraswamy and Ganesan . Chamka et al.8 have studied the radiation effects free convection flow past a semi-infinite vertical plate with mass transfer. Ganesan and Loganadhan9 studied the radiation and mass transfer effects on the flow of incompressible viscous fluid past a moving cylinder. Abd El-Naby et al.10 analyzed the radiation effects on MHD unsteady free convection flow over a vertical plate with variable surface temperature. Das et al.11 have studied the radiation effects on flow past an impulsively started infinite vertical plate. Thermal radiation on unsteady free convection and mass transfer boundary layer over a vertical moving plate were analyzed by Raptis and Perdikis12. Free convective flow past a vertical plate has been studied extensive by Ostrach13. Siegel14 investigated the transient free convection from a vertical flat plate. Cheng and Lau15 and Cheng and Teckchandani16 obtained numerical solutions for the convective flow in a porous medium bounded by two isothermal parallel plates in the presence of the withdrawal of the fluid. In all the above mentioned studies, the effect of porosity, permeability and the thermal resistance of the medium is ignored or treated as constant. However, porosity measurements by Benenati and Broselow17 show that porosity is not constant but varies from the surface of the plate to its interior to which as a result permeability also varies. In case of unsteady free convective flows, Soundalgekar18 studied the effects of viscous dissipation on the flow past an infinite vertical porous plate. The combined effect of buoyancy forces from thermal and mass
diffusion on forced convection was studied by Chen et al.19. The free convection on a horizontal plate in a saturated porous medium with prescribed heat transfer coefficient was studied by Ramanaiah and Malarvizhi20. Bejan and Khair21 have investigated the vertical free convective boundary layer flow embedded in a porous medium resulting from the combined heat and mass transfer. Lin and Wu22 were analyzed the problem of simultaneous heat and mass transfer with the entire range of buoyancy ratio for most practical and chemical species in dilute and aqueous solutions. Rushi Kumar and Nagarajan23 studied the mass transfer effects of MHD free convection flow of an incompressible viscous dissipative fluid past an infinite vertical plate. Mass transfer effects on free convection flow of an incompressible viscous dissipative fluid have been studied by Manohar and Nagarajan24. Recently, Sivaiah et al.25 have discussed on heat and mass transfer effects on MHD free convection flow past a vertical porous plate. Ramana Reddy et al.26 have investigated the effects of radiation and mass transfer on MHD free convective dissipative fluid in the presence of heat source/sink. The aim of this present paper is to study the effects of thermal radiation on MHD free convective heat and mass transfer flow of viscous incompressible fluid through a porous medium confined in a vertical porous plate. The velocity, temperature, concentration profiles and the skin friction have been analyzed for variations in the different parameters involved in the problem.
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Seethamahalakshmi, et al., J. Comp. & Math. Sci. Vol.2 (4), 595-604 (2011)
2. MATHEMATICAL ANALYSIS We study the two dimensional free convection and mass transfer flow of an incompressible and electrically conducting viscous Newtonian fluid past an infinite vertical porous plate, with thermal radiation. The plate temperature is oscillating with time, about a constant non-zero mean value. Viscous and Darcy’s resistance terms are taken into account with constant permeability of the medium. The suction velocity normal to the plate is a function of time and can be written as, v ∗ = −U 0 .A system of rectangular co-ordinates
(
∗
∗
O x ,y ,z
∗
) is taken, such that
∗
y = 0 on
the plate and z∗ axis is along its leading edge. All the fluid properties are considered constant except that the influence of the density variation with temperature is considered. The influence of the density variation in other terms of the momentum and the energy equation and the variation of the expansion coefficient with temperature is considered negligible. This is the well known Boussinesq approximation. We regard the porous medium as an assemblage of small identical and spherical particles of fixed space. Under these conditions, the problem is governed by the following system of equations: ∂v
*
∂y
*
=0
(1)
* * 2 * σ B2 ∂u ∂ u ν * ∂u +v =ν + g β (T ′−T∞′ )+ g β ∗ ( C′−C∞′ )− * u* − 0 u* *2 * * ρ ∂y ∂t ∂y K
(2)
∂T
*
∂t
*
+v
* ∂T
*
∂y
*
2 * * ∂ T 1 ∂qr − ρ C p ∂y *2 ρ C p ∂y* k
=
(3) ∂C
*
∂t
*
+v
* ∂C
*
∂y
*
=D
∂
2
∂y
C
*
∗2
+ D1
∂
2
∂y
T
*
∗2
(4) The boundary conditions for the velocity, temperature, and concentration fields are given as follows: u * = 0 , T * = T w* , C * = C w* u * → 0 , T * → T∞* ,
at y * = 0
C * → C ∞* , as y * → ∞
(5) Where x*, y* and t* are the dimensional distances along and perpendicular to the plate and dimensional time, respectively. u* and v* are the components of dimensional velocities along x*and y*directions.ρ is the fluid density, is the viscosity, Cp is the specific heat at constant pressure, σ is the fluid electrical conductivity, B0 is the magnetic induction, K * is the permeability of the porous medium, T* is the dimensional temperature, D is the mass diffusivity, D1 is the heat diffusivity, c* is the dimensional concentration, k is the thermal conductivity of the fluid, g is the acceleration due to gravity, and qr* is the radiative heat flux, By using the Rosseland approximation, the radiative heat flux vector qr can be written as: qr =
4σ ∗ ∂ T ' 4 3k ∗ ∂y '
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Seethamahalakshmi, et al., J. Comp. & Math. Sci. Vol.2 (4), 595-604 (2011)
It is assumed that the temperature differences within the flow are sufficiently small so that T 4 can be expanded in a Taylor series about the free stream temperature T ∞ so that after rejecting the higher order terms:
T '4 ≅ 4T∞'3T '−3T∞4
(7)
The energy equation after substitution of equations (5) and (6) can be written as:
∂u ∂ u ∂ 2 u 1 − = 2 + Gr (θ + NC ) − M + ∂ t ∂ y ∂y K
u
(10)
∂θ ∂θ 1 ∂ 2θ − = − Fθ ∂t ∂y Pr ∂y 2 ∂C ∂C 1 ∂ 2C ∂ 2θ − = + A ∂t ∂y Sc ∂y 2 ∂y 2
(12)
The corresponding boundary conditions are: ∂T ' ∂T ' k ∂ T ' 16σ T ' ∂ T ' + v' = + + Q (T '−T ' ∞ ) ∂t ' ∂y ' ρc p ∂y ' 2 3 ρc p k ∂y ' 2 *
2
3 ∞ *
2
(8) Let us introduce variables,
the
( (
non-dimensional
) )
u=
T * − T∞* t *U 02 y*U 0 u* ,t = ,y= ,θ = * , U0 ν ν Tw − T∞*
C=
(C (C
Sc =
*
− C∞*
* w
−C
ν D
* ∞
,F =
) ,K = U K ν ) 2 0
2
*
D1 (Tw − T∞ ) µC p σ B02ν , Pr = ,A= k ρU 02 ν ( Cw* − C∞* ) *
,M =
(
)
( (
ν g β Tw* − T∞* β * Cw* − C∞* 16σ ∗T∞3 , Gr = ,N = * 3 3k k U0 β Tw* − T∞*
)
*
)
(9) where Pr is the Prandtl number, Gr is the Grashof number, N is the buoyancy ratio, Sc is the Schmidt number, M is the magnetic parameter, D is the mass D diffusivity, 1 is the heat diffusivity, K is the permeability parameter, β is the thermal expansion coefficient, β is the concentration expansion coefficient and A is the thermal diffusion parameter. Other physical variables have their usual meanings. The governing equations reduce to, *
u = 0, θ = 1,
C =1
u = 0, T = 0, C = 0
at y = 0 as y →∞
(13)
3. METHOD OF SOLUTION In order to reduce the above system of partial differential equations to a system of ordinary differential equations in dimensionless form, we assume the trial solution for the velocity, temperature and concentration as:
u ( y , t ) = u0 ( y ) + ε u1 ( y )e iωt
(14)
θ ( y , t ) = θ 0 ( y ) + εθ1 ( y )eiω t
(15)
C ( y , t ) = C 0 ( y ) + ε C1 ( y )e iω t
(16)
Substituting equations (14)-(16) in equations (10)-(12) respectively, equating the harmonic and non-harmonic terms and neglecting the coefficients of ε 2 , we get zeroth order, u 0′′ + u 0′ − ( M +
1 ) u 0 = − Gr (θ 0 + NC 0 ) K
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Seethamahalakshmi, et al., J. Comp. & Math. Sci. Vol.2 (4), 595-604 (2011)
θ0′′ + Pr θ0′ − F Pr θ0 = 0
4. RESULTS AND DISCUSSION (18)
C0′′ + ScC0′ + ScAθ0′′ = 0
(19)
First order, u1′′ + u1′ − ( M +
1 + iω ) u1 = − G r (θ 1 + N C 1 ) K
θ1′′+ Pr θ1′ − ( F + iω) Pr θ1 = 0
(20)
(21)
C1′′+ ScC1′ − iωScC1 = − AScθ1′′
(22)
In view of the above, the corresponding boundary conditions can be re written as u0 = 0, u1 = 0, θ 0 = 1, θ1 = 0, C0 = 1, C1 = 0
at
y=0
u0 = 0, u1 = 0, θ 0 = 0, θ1 = 0, C0 = 0, C1 = 0
as
y→∞
(23) Solving the equations (17)-(22) under the boundary conditions (23), weget
u ( y, t ) = m5 e − m6 y + m3e − m1 y − m4 e − Scy
θ ( y, t ) = e − m y 1
C ( y, t ) = (1 − m2 )e− Scy + m2 e− m1 y Knowing the velocity field, the skin friction is evaluated in non-dimensional form using,
∂u = Scm4 − m3 m1 − m6 m5 . ∂y y =0
τ =
where m1 =
1 Pr + Pr 2 + 4 F Pr , 2
m3 = −
m2 = −
ASc , m1 − Sc
Gr + m2 GrN (1 − m2 , m4 = − 2 , m12 − m1 − ( M + 1/ K ) Sc1 − Sc1 − ( M + 1/ K )
m5 = m4 − m3 , m6 = −
(1 +
1 + 4( M + 1 / K ) 2
599
)
Numerical calculations have been carried out for dimensionless velocity, temperature and concentration distributions for different values of parameters and are displayed in Figures 1 to 15. Figures 1 to 8 represent the velocity profiles for different parameters. Figure 1 shows the variation of velocity u with magnetic parameter M. It is observed that the velocity decreases as M increases. From Figure 2, it is observed that the velocity increases as the Grashof number Gr increases. The variation of u with N is shown in Figure 3. It is noticed that increase in N leads to increase in velocity. Figure 4 shows that an increase in permeability parameter K causes an increase in velocity. From Figure 5 shows the variation of velocity u with Prandtl number Pr. It is observed that the velocity decreases as Pr increases. The velocity profile for Schmidt number Sc is shown in Figure 6. It is clear that velocity u decreases with increasing in Sc. In Figure 7, the velocity profile increases due to increasing thermal diffusion parameter A. From Figure 8 shows the variation of velocity u with radiation parameter F. It is observed that the velocity decreases as F increases. From Figure 9, it is observed that increase in Prandtl number Pr causes decrease in temperature. Figure 10 illustrates that the effect of radiation parameter F on the temperature profiles. It is observed that an increase of radiation parameter, the temperature decreases. In Figure 11, the concentration profile increases due to increasing thermal diffusion parameter A. Figure 12 illustrates
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
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Seethamahalakshmi, et al., J. Comp. & Math. Sci. Vol.2 (4), 595-604 (2011) 1.4
0.8 Pr = 0.71, Sc = 0.60, Gr = 1, A = 0.2, K=10, F=10, M=1
0.6
0.4
0.2
0
0
1
2
3
4
5 y
6
7
8
9
10
Fig.3. Effects of buoyancy ratio parameter on velocity profiles. 0.35 K K K K
0.3
= = = =
1 2 3 4
0.25
1 2 3 4
0.2 u
0.3
= = = =
1 2 3 4
1
0.35 M M M M
N= N= N= N=
1.2
u
that the effect of thermal radiation parameter F on the concentration field. It is noticed that as F increases the concentration field decreases. From Figure 13, it is noticed that an increase in Schmidt number Sc leads to decrease in concentration. Figure 14 illustrated that the effect of Schmidt number on the skin-friction. It is observed that for fixed values various parameters, as Schmidt number increases the skin friction decreases. Figure 15 illustrated that the effect of radiation parameter on the skin-friction. It is noticed that as radiation parameter increases the skin-friction decreases for a fixed values of various parameters.
0.25
0.15
0.2
Pr = 0.71, Sc = 0.60, Gr = 1, A = 0.2, M=1, F=10, N=1
u
0.1
0.15
Pr = 0.71, Sc = 0.60, Gr = 1, A = 0.2, K=10, F=10,N=1
0.05
0.1
0
0
1
2
3
4
0.05
0
0
1
2
3
4
5 y
6
7
8
9
10
Fig.1. Effects of magnetic parameter on velocity profiles.
5 y
6
7
8
9
10
Fig.4. Effects of permeability parameter on velocity profiles. 0.5
1.4 Gr = Gr = Gr = Gr =
1.2
1 2 3 4
Pr = Pr = Pr = Pr =
0.45 0.4
0.1 0.7 1.0 7.0
0.35
1
0.3 0.25
u
u
0.8
0.6
M = 1, Sc = 0.60, Gr = 1, A = 0.2, K=10, F=10, N=1
0.2
Pr = 0.71, Sc = 0.60, M = 1, A = 0.2, K=10, F=10,N=1
0.15
0.4
0.1 0.05
0.2
0 0
0
1
2
3
4
5 y
6
7
8
9
10
Fig.2. Effects of Grash of number on velocity profiles.
0
1
2
3
4
5 y
6
7
8
9
10
Fig.5. Effects of Prandtl number on velocity profiles.
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Seethamahalakshmi, et al., J. Comp. & Math. Sci. Vol.2 (4), 595-604 (2011) 1
0.5 Sc Sc Sc Sc
0.45
0.3 0.4 0.5 0.6
0.8
0.35
0.7
0.3
0.6
0.25 Pr = 0.71, Gr = 1, M = 1, A = 0.2, K=10, F=10,N=1
0.2
0.3 0.2
0.05
0.1
1
2
3
4
5 y
6
7
8
9
0
10
Fig.6. Effects of Schmidt number on velocity profiles. A A A A
= = = =
0
1
2
3
4
5 y
6
7
8
9
10
Fig.9. Effects of Prandtl number on temperature profiles.
0.4 0.35
0.71 3.00 7.00 10.00
Gr=1, Sc = 0.60, M = 1, A = 0.2, K=10, F=10, N=1
0.4
0.1
0
= = = =
0.5
0.15
0
Pr Pr Pr Pr
0.9
θ
u
0.4
= = = =
1
0.1 0.3 0.7 1.0
F F F F
0.9 0.8
0.3
= = = =
1 5 10 20
0.7
0.25
θ
u
0.6
0.2
0.5 Pr = 0.71, Sc = 0.60, M = 1, A = 0.2, K=10, Gr=1, N=1
0.4
0.15 Pr = 0.71, Sc = 0.60, M = 1, Gr = 1, K=10, F=10,N=1
0.1
0.3 0.2
0.05 0.1
0
0
1
2
3
4
5 y
6
7
8
9
0
10
Fig.7. Effects of thermal diffusion parameter on velocity profiles.
0
1
2
3
4
5 y
6
7
8
9
10
Fig.10. Effects of radiation parameter on temperature profiles.
0.45
1 F F F F
0.4 0.35
= = = =
1 5 10 20
F F F F
0.9 0.8
= = = =
1 5 10 20
0.7
0.3
0.6 C
u
0.25
0.5
0.2
Pr = 0.71, Gr=1, Sc = 0.60, M = 1, A = 0.2, K=10, N=1
0.4
Pr = 0.71, Sc = 0.60, M = 1, A = 0.2, K=10, Gr =1, N=1
0.15
0.3 0.1
0.2
0.05 0
0.1 0
1
2
3
4
5 y
6
7
8
9
10
Fig.8. Effects of radiation parameter on velocity profiles.
0
0
1
2
3
4
5 y
6
7
8
9
Fig.11. Effects of radiation parameter on concentration profiles.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 4, 31 August, 2011 Pages (581-692)
10
602
Seethamahalakshmi, et al., J. Comp. & Math. Sci. Vol.2 (4), 595-604 (2011) 1.4 A A A A
1.2
= = = =
1.7
0.2 0.5 1.0 1.5
1
Skin- Friction
C 0.6
Pr = 0.71, Sc = 0.60, Gr = 1, M =1, K=10, F=10,N=1
0.2
1.1
0
1
2
3
4
5 y
6
7
8
9
1
10
1 Sc Sc Sc Sc
0.9 0.8
= = = =
1.
0.5 Pr = 0.71, M = 1, Gr = 1, A = 0.2, K=10, F=10,N=1
2.
0.3 0.2 0.1
0
1
2
3
4
5 y
6
7
8
9
10
Fig.13. Effects of Schmidt number on concentration profiles.
3.
1.85 Pr = 0.71, A = 0.2, Gr = 1, M =1, K=10, F=10, N=1
1.8
Sc Sc Sc Sc
1.75
= = = =
0.3 0.4 0.5 0.6
1.7 1.65
4.
1.6 1.55 1.5 1.45 1.4 1.35
0
0.1
0.2
0.3
0.4
0.5 F
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5 A
0.6
0.7
0.8
0.9
1
REFERENCES
0,3 0,4 0.5 0,6
0.6
0.4
0
Fig.15. Effects of radiation parameter on skinfriction.
0.7
C
Pr = 0.71, Sc = 0.62, Gr = 1, M =1, K=10, N=1
1.3
1.2
Fig.12. Effects of thermal diffusion parameter on concentration profiles.
Skin-Friction
1 5 10 20
1.4
0.4
0
= = = =
1.5
0.8
0
F F F F
1.6
1
Fig.14. Effects of Schmidt number on skin-friction.
5.
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