J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011)
MHD Effects on Moving Vertical Porous Plate with Homogeneous Chemical Reaction K.V. NAGESWARA REDDY1, B. RAMA BHUPAL REDDY2, and S. RAMAKRISHNA3 1 2
Assistant Professor, Dept. of Mathematics, K.S.R.M.C.E., Kadapa, India Associate Professor, Dept. of Mathematics, K.S.R.M.C.E., Kadapa, India 3 Professor, Dept. of Computer Science, S.V. University, Tirupati, India 2 reddybrb@gmail.com ABSTRACT The MHD flow of a viscous incompressible conducting fluid past on impulsively started infinite vertical porous plate with variable temperature in the presence of homogeneous chemical reaction is studied. The governing equations are solved by the Laplace transform technique. The effects of various emerging parameters on the velocity field are discussed. Keywords: Homogenous Chemical Reaction, Heat and Mass Transfer, Vertical Porous Plate, MHD.
INTRODUCTION Natural convection in a fluidsaturated porous medium is of fundamental importance in many industrial and natural problems. Few examples of the heat transfer by natural convection can be found in geophysics and energy related engineering problems such as natural circulation in geothermal reservoirs, acquifers, porous insulations, solar power collectors, spreading of pollutants etc. Natural convection occurs due to the spatial variations in density, which is caused by the non-uniform distribution of temperature or/and concentration of a dissolved substance. Kandaswamy et al.9 were presented to investigate the effects of thermophoresis and
variable viscosity on MHD mixed convective heat and mass transfer of viscous, incompressible and electrically conducting fluid past a porous wedge in the presence of chemical reaction. Also Anjalidevi and Kandaswamy3 studied an approximate solution for the steady laminar flow along a semi- infinite horizontal plate in the presence of species concentration and chemical reaction. The effects of thermal radiation on unsteady free convective flow over a moving vertical plate with mass transfer in the presence of homogeneous first order chemical reaction analyzed by Muthucumaraswamy et al.11. MHD effects on moving vertical plate with homogeneous chemical reaction studied by Muthucumaraswmy and Chandrakala10.
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There are many studies Nield and Bejan12, Cheng and Minkowyez5, Prasad and Kulacki13 and Angirasa and Peterson1 in which natural convection caused by immersing a hot surface in a fluid-saturated porous medium at constant ambient temperature has been considered. Also a few studies are found when the porous medium is thermally stratified, i.e. the ambient temperature is not uniform and it varies as a linear function of stream wise direction. This phenomenon has its applications in hot dike complexes in volcanic region for heating of ground water, development of advanced technologies for nuclear waste management, separation process in chemical engineering etc. Rees and Lage16, Takhar and Pop22 and Tewari and Singh23 analytically analyzed free convection from a vertical plate immersed in a thermally stratified porous medium under boundary layer assumptions. On the other hand Angirasa and Peterson2, Rathish Kumar et al.15 and Rathis Kumar and Singh14 have numerically investigated the natural convection process in a thermally stratified porous medium. Stewartson21 presented analytic solution to the viscous flow past an impulsively started semi-infinite horizontal plate whereas Hall8 solved the problem of Stewartson21 by finite-difference method. Soundalgekar20 first presented an exact solution to the flow of a viscous incompressible fluid past a impulsively started infinite vertical plate by the Laplacetransform technique. The fluid considered in this study was pure air or water. However, in nature, availability of pure air or water is very difficult. It is usually a very complicated phenomenon, however, by introducing suitable assumptions, the
governing equations can be simplified. These simplified equations were derived by Gebhart6 by assuming the concentration level to be very low. This enabled us to neglect Soret-Dufur effects. The solution to this problem governed by coupled linear differential equations was derived by the Laplace-transform technique. Freeconvection flow with mass-transfer past a semi-infinite vertical plate was presented by Gebhart and Pera7. Similarity solutions presented by Gebhart6. In all these studies the concentration level at the plate was assumed to be constant and at low level, which is true in some cases. Many times, mass is supplied at the plate at constant rate in the presence of species concentration and such a situation has not been studied in case of an impulsively started infinite vertical isothermal plate. Such a study will be found useful in chemical, aerospace and other engineering applications. Chambre and Young4 have analyzed a first order chemical reaction in the neighbourhood of a horizontal plate. Boundary layer flow on moving horizontal surfaces was studied by Sakiadis17. The effects of transversely applied magnetic field, on the flow of an electrically conducting fluid past an impulsively started infinite isothermal vertical plate was studied by Soundalgekar et al.18. MHD effects on impulsively started vertical infinite plate with variable temperature in the presence of transverse magnetic field studied by Soundalgekar et al.19. NOMENCLATURE
B0 C′
External magnetic field Species concentration in the fluid
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
K. V. Nageswara Reddy, et al., J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011)
Cw′ C∞′ C Cp
g Gr Gc k
Kl M Pr Sc T′
Tw′ T∞′ t′ t u′
u0 u y′ y
Concentration of the plate Concentration in the fluid far away from the plate Dimensionless concentration Specific heat at constant pressure Acceleration due to gravity Thermal Grashof number Mass Grashof number Thermal conductivity of the fluid Chemical reaction parameter Magnetic field parameter Prandtl number Schmidt number Temperature of the fluid near the plate Temperature of the plate Temperature of the fluid far away from the plate Time Dimensionless time Velocity of the fluid in the x’ direction Velocity of the plate Dimensionless velocity Coordinate axis normal to the plate Dimensionless coordinate axis normal to the plate
Greek symbols
α β β*
Thermal conductivity Volumetric coefficient of thermal expansion Volumetric coefficient of expansion with Concentration
µ ν ρ σ τ′ τ θ η erfc
276
Coefficient of viscosity Kinematic viscosity Density of the fluid Electric conductivity Skin-friction Dimensionless skin-friction Dimensionless temperature Similarity parameter Complementary error function
MATHEMATICAL ANALYSIS MHD flow of a viscous incompressible fluid past an impulsively started infinite vertical plate with variable temperature and uniform mass diffusion in the presence of homogeneous chemical reaction is studied. Here the x ' -axis is taken along the plate in vertically upward direction and the y ' -axis is taken normal to the plate. Initially, the plate and fluid are at the same temperature and concentration. At time t ' > 0 . The plate is given an impulse motion in the vertical direction against gravitational field with constant velocity u0 . The plate temperature is raised linearly. With time and the concentration level near the plate is also raised to cw . A transverse magnetic field of uniform strength Bo is assumed to be applied normal to the plate. The induced magnetic field and viscous dissipation is assumed to be negligible. It is also assumed that there exists a homogeneous first order chemical reaction between the fluid and species concentration. The governing equations are
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K. V. Nageswara Reddy, et al., J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011)
∂u ' ∂ 2 u ' σβ 02 ν * = g β (T ' − T∞ ' ) + g β (C ' − C ' ∞ ) + ν 2 − u '− u ' ∂t ' K ρ ∂t
∂T ' ∂ 2T ' ρC p =k ∂t ' ∂y '2 ∂C ' ∂ 2C ' =k − KlC ' ∂t ' ∂y '2
(1) (2) (3)
with the following initial and boundary conditions
t ' ≤ 0 : u ' = 0, T ' = T∞'
C ' = C∞' for all y '
t ' > 0 : u ' = u0 , T ' = T∞' + (Tw' − T∞' ) At '
C ' = Cw' at y ' = 0
u ' = 0, T ' → T∞'
C ' → C∞' as y ' → ∞
where A =
u02
ν
(4)
.
On introducing the following non-dimensional quantities
u' u= , u0
t ' u02 t= , v
T '− T ' θ = ' ∞' , Tw − T∞
Gr =
(C − C ) , C= (C − C ) Pr =
'
' ∞
' w
' ∞
µC p
,
k σB 2ν 1 N = 02 + Kl ρu 0
y=
Gc = Sc =
ν k
,
y ' u0 v
g β v (Tw' − T∞' ) u03
vg β * ( Cw' − C∞' ) u03 K=
(5)
vK l u02
By using non-dimensional variables the governing equations (1) to (4) leads to
∂u ∂ 2u = Grθ + GcC + 2 − Nu ∂t ∂y Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
(6)
K. V. Nageswara Reddy, et al., J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011)
∂θ 1 ∂ 2θ = ∂t Pr ∂y 2 1 ∂ 2C ∂C = − KC ∂t Sc ∂y 2
278 (7) (8)
The last term in (8) represents homogeneous first order chemical reaction, K being the dimensionless reaction rate constant. The initial and boundary condition in dimensionless form are u = 0,θ = 0, C = 0 for all y , t ≤ 0 at y = 0 t > 0 : u = 1,θ = t , C = 1 as y ' → ∞ u = 0,θ → 0, C → 0
(9)
SOLUTION The equations (6) to (8), subject to the boundary conditions (9), are solved by the usual Laplace transform technique and the solutions are derived as follows:
(
exp ( −η 2 Pr ) π
)
θ = t (1 + 2η 2 Pr ) erfc η Pr − 2η 1 Gr (1 + at ) Gc u = 1 + 2 + 2 a (1 − Pr ) b(1 − Sc)
(
) (
)
Pr
(
(10)
) ( )
)
exp 2η Nt erfc η + Nt + exp −2η Nt erfc η − Nt Grη t exp −2η Nt erfc η − Nt − 2a(1 − Pr ) N Gr − exp +2η Nt er c η + Nt − 2 erfc η Pr a (1 − Pr ) Gr exp( at ) exp 2η ( N + a)t erfc η + ( N + a)t 2a 2 (1 − Pr ) + exp −2η ( N + a )t erfc η − ( N + a )t Gc exp(bt ) − exp 2η ( N + b)t erfc η + ( N + b)t 2b(1 − Sc)
(
(
) ( (
(
) (
)
(
) (
) (
(
)
)
)
) (
)
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
279
K. V. Nageswara Reddy, et al., J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011)
(
) (
(
( )
(
(
(
2
where a =
(
)
(
C=
)
+ exp −2η ( N + b)t erfc η − ( N + b)t Grt Pr 2 + exp −η 2 Pr 1 + 2η Pr erfc η Pr − 2η π a (1 − Pr ) Gr exp(at ) + 2 exp 2η aPr t erf c η Pr + at 2a (1 − Pr ) + exp −2η aPr t erf c η Pr − at Gc exp 2η Kt Sc erfc η Sc + Kt − 2b(1 − Sc) + exp −2η Kt Sc erfc η Sc − Kt Gc exp(bt ) + exp 2η Sc( K + b)t erfc η Sc + ( K + b)t 2b(1 − Sc) + exp −2η Sc ( K + b)t erfc η Sc − ( K + b)t 1 exp(2η KtSc ) erfc (η Sc + Kt ) +
)
)
(
(
)
)
(
)
) (
) ( ( ) (
)
)
) (
)
)
exp(-2 η KtSc ) erfc (η Sc − Kt )
(11)
(12)
y N N − KSc , and η = . ,b = Pr − 1 Sc − 1 2 t
Also our results coincide with those results obtained by Muthucumaraswamy and Chandrakala11 in the limit of K → ∞ . RESULTS AND DISCUSSION The velocity profiles for the different values of Da with Pr = 7, Gr = 2, Gc = 5, t = 0.2, K = 0.2, M = 2 and Sc = 2.01 is shown in Figure 1. It is observed that the velocity increases with increasing Da . Figure 2 depicts the variation of velocity profiles with magnetic field parameter M with
Pr = 7, Gr = 2, Gc = 5, t = 0.2, K = 0.2 , Da = 0.1 and Sc = 2.01. It is observed that velocity decreases with increasing M . The effect of different values of dimensionless chemical reaction para meter K on the veloc ity profiles for Pr = 7, Gr = 2, Gc = 5, t = 0.2, Da = 0.1, M = 2 and Sc = 2.01 is
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
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K. V. Nageswara Reddy, et al., J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011)
presented in Figure 3. It is observed that the velocity increases with increasing K . Figure 4 depicts the velocity profiles for different values of Schmidt number Sc with
Pr = 7, Gr = 2, Gc = 5, t = 0.2, K = 0.2, M = 2 and Da = 0.1 . It is
observed that the velocity decreases with an increase in Sc . In order to study the effect of thermal Grashof number Gr on the velocity profiles for
Pr = 7, Da = 0.1, Gc = 5, t = 0.2, K = 0.2, M = 2 and Sc = 2.01 , we have
plotted Figure 5. It is noted that the velocity
increases with an increase in Gr . Figure 6 shows the effect of mass Grashof number Gc on the velocity profiles for
Pr = 7, Gr = 2, Da = 0.1, t = 0.2, K = 0.2, M = 2 and Sc = 2.01 . It is
observed that the velocity increases with increasing Gc . The effect of Prandtl number Pr on the velocity profiles for on the velocity profiles for
Gc = 5, Gr = 2, Da = 0.1, t = 0.2, M = 2 , K = 0.2 and Sc = 2.01
is depicted in Figure 7. It is observed that the velocity increases with increasing Pr .
2.5
Da → ∞
2
Da = 10
1.5
Da = 0.1
1
Da = 0.01
0.5 0
u -0.5 -1 -1.5 -2 -2.5
0
0.4
0.8
η
1.2
1.6
Figure 1 Velocity profile for different values of Da with Pr = 7, Gr = 2, Gc = 5, t = 0.2, K = 0.2, M = 2 and Sc = 2.01 . Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
2
281
K. V. Nageswara Reddy, et al., J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011) 0.8
M =0 0.6
u
M =2
M =5
0.4
0.2
M = 10 0
0
0.4
0.8
Ρ
1.2
Figure 2 Velocity profile for different values of
1.6
M
2
with
Pr = 7, Gr = 2, Gc = 5, t = 0.2, K = 0.2, Da = 0.1 and Sc = 2.01 . 0.9
K =2
0.6
K = 0.2
u 0.3
0
0
0.4
0.8
Ρ
1.2
Figure 3 Velocity profile for different values of
1.6
K
2
with
Pr = 7, Gr = 2, Gc = 5, t = 0.2, Da = 0.1, M = 2 and Sc = 2.01 . Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
K. V. Nageswara Reddy, et al., J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011) 0.8
0.6
Sc = 1.1
0.4
u
Sc = 2 0.2
Sc = 3 0
0
0.4
0.8
η
1.2
1.6
2
Figure 4 Velocity profile for different values of Sc with Pr = 7, Gr = 2, Gc = 5, t = 0.2, K = 0.2, M = 2 and Da = 0.1 . 0.8
0.6
Gr = 3, 2,1 0.4
u 0.2
0
0
0.4
0.8
η
1.2
1.6
2
Figure 5 Velocity profile for different values of Gr with Pr = 7, Da = 0.1, Gc = 5, t = 0.2, K = 0.2, M = 2 and Sc = 2.01 . Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
282
283
K. V. Nageswara Reddy, et al., J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011) 0.8
0.6
Gc = 5 0.4
Gc = 2
u
Gc = 0.2
0.2
0
0
0.4
0.8
η
1.2
1.6
2
Figure 6 Velocity profile for different values of Gc with Pr = 7, Gr = 2, Da = 0.1, t = 0.2, K = 0.2, M = 2 and Sc = 2.01 . 0.8
0.6
Pr = 7,3, 2
0.4
u 0.2
0
0
0.4
η
0.8
1.2
1.6
2
Figure 7 Velocity profile for different values of Pr with Gc = 5, Gr = 2, Da = 0.1, t = 0.2, K = 0.2, M = 2 and Sc = 2.01 . Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
K. V. Nageswara Reddy, et al., J. Comp. & Math. Sci. Vol.2 (2), 274-285 (2011)
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