Journal of Engineering Physics and Thermophysics, Vol. 86, No. 6, November, 2013

MHD FLOW OF A ROTATING FLUID PAST A VERTICAL POROUS FLAT PLATE IN THE PRESENCE OF CHEMICAL REACTION AND RADIATION S. Sivaiah

UDC 536.25

The MHD flow of a rotating fluid past a vertical porous flat plate in the presence of chemical reaction and radiation is studied. The chemical reaction is assumed to be of the first order. The dimensionless governing equations are solved numerically using the finite element method. The effects of various physical parameters on the complex velocity, temperature, and concentration fields across the boundary layer are investigated. It is observed that the concentration decreases with increase in the chemical reaction parameter and the temperature increases with the radiation parameter. Keywords: MHD, chemical reaction, radiation, finite element method. Introduction. Many practical diffusion operations involve molecular diffusion of a species in the presence of chemical reaction within the boundary or on it. There are two types of reactions: homogeneous and heterogeneous. A homogeneous reaction is one that occurs uniformly throughout a given phase. The species generation in a homogeneous reaction is analogous to an internal source of heat generation. In contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a phase. It can therefore be treated as a boundary condition similar to the constant heat flux condition. The study of heat and mass transfer with chemical reaction is of great practical importance for engineers and scientists because of its almost universal occurrence in many branches of science and engineering. The flow of an incompressible Boussinesq fluid in the presence of rotation is encountered in space science and engineering fluid dynamics. The Hall current effect on hydromagnetic free convection with mass transfer in a rotating fluid was studied by Agarwal, Ram, and Singh [1]. Bestman and Adjepong [2] investigated an unsteady hydromagnetic free convective flow with radiative heat transfer in a rotating fluid. The radiation effects on the MHD mixed free-fixed convective flow past a semi-infinite moving vertical plate for high temperature differences was studied by Azzam [3]. Bestman [4] considered free convective heat transfer to steady radiating non-Newtonian MHD flow past a vertical porous plate. Chamkha [5] studied the thermal radiation and buoyancy effects on hydromagnetic flow over an accelerating permeable surface with a heat source or sink. Chamkha [6] investigated unsteady convective heat and mass transfer past a semi-infinite permeable moving plate with heat absorption and found that an increase in the solutal Grashoff number enhanced the concentration buoyancy effects, leading to an increase in the velocity. Cookey, Ogulu, and Omubo-Pepple [7] studied the influence of viscous dissipation and radiation on unsteady MHD free convective flow past an infinite heated vertical plate in a porous medium with time-dependent suction. Elbarbary and Elgazery [8] considered the effect of variable viscosity on magnetic micropolar fluid flow with radiation by the finite difference method. Helmy [9] focused attention upon the MHD flow in a micropolar fluid. Ibrahim, Hassanein, and Bakr [10] investigated unsteady magnetohydrodynamic micropolar fluid flow and heat transfer in a porous medium over a vertical porous plate in the presence of thermal and mass diffusion with a constant heat source. Jha [11] considered MHD free convection and mass transfer through a porous medium but did not take into account the radiation effect, which is essential for astrophysical and cosmic studies. Kandasamy, Periasamy, and Prabhu [12] investigated the effects of chemical reaction, as well as heat and mass transfer along a wedge with a heat source and convection in the presence of suction or injection. Kim [13] studied unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction. Ogulu and Cookey [14] considered MHD free convection and mass transfer with radiative heat transfer. Heat transfer to unsteady magnetohydrodynamic flow past an infinite moving vertical plate with variable suction was the subject of the work by Ogulu and Prakash [15] who showed that an increase in the plate velocity increases the flow velocity. Department of Engineering Mathematics, GITAM University, Hyderabad Campus, Hyderabad — 502 329, A.P., India; email: [email protected]. Original article submitted July 18, 2012. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 86, No. 6, pp. 1249–1256, November–December, 2013. Original article submitted April 15, 2013. 1328

0062-0125/13/8606-1328 ©2013 Springer Science+Business Media New York

Prakash, Ogulu, and Zhandire [16] studied MHD free convection and mass transfer of a micropolar thermally radiating and reacting fluid with time-dependent suction. The non-Darcian forced convective heat transfer over a flat plate in a porous medium with variable viscosity and Prandtl number was studied by Pantokratoras [17]. Shivaiah and Anand Rao [18] investigated the chemical reaction effect on an unsteady MHD free convective flow past an infinite vertical porous plate with constant suction. Anand Rao and Shivaiah [19] considered the chemical reaction effects on unsteady MHD flow past a semi-infinite vertical porous plate with viscous dissipation. The main objective of the present investigation is to study the MHD flow of a rotating fluid past a vertical porous flat plate in the presence of chemical reaction and radiation. The equations of linear momentum, energy, and diffusion governing the flow field are solved numerically using the Galerkin finite element method, which is more economical from the computational viewpoint. The behavior of the velocity, temperature, concentration, and skin friction with changes in the governing parameters is discussed. Mathematical Formulation. We consider unsteady free convective flow of an incompressible electrically conducting viscous fluid near a moving infinite flat plate in a rotating porous medium with the velocity U 0 and angular velocity Ω (as in [2]). We assume that a uniform magnetic field B 0 is applied in the direction of the flow fixed relative to the plate. We also assume that the induced magnetic field is negligible in comparison with the applied field. We consider the absence of electric field and viscous dissipation heating in the energy equation. With these assumptions and those usually associated with the Boussinesq approximation, the proposed governing equations become

∂u′ ∂ 2u′ σB0 2u′ υu′ ′), − 2Ω′v′ = υ − − + g β (T − T∞ ) + g β* (C ′ − C∞ 2 ′ K ∂t ′ ρ ∂y ′

(1)

∂v′ ∂ 2v′ σB02v′ υv′ , − 2Ω′u′ = υ − − K′ ∂t ′ ρ ∂y ′2

(2)

∂T k = ∂t ′ ρC p

2 ⎡ ∂ 2T 1 ∂q′r ⎤ υ ⎛ ∂u′ ⎞ ⎢ 2 − ⎥+ ⎜ ⎟ , k ∂y ′ ⎥⎦ C p ⎝ ∂y ′ ⎠ ⎢⎣ ∂y ′

(3)

∂C ′ ∂ 2C ′ ′). = Dm − k r′ (C ′ − C∞ ∂t ′ ∂y ′2

(4)

The boundary conditions are y ′ = 0 : u ′ = U 0 , v′ = 0, T = Tw , C ′ = C w′ ; (5)

y ′ → ∞ : u ′ = U 0 , v′ = 0, T = T∞ , C ′ = C∞′ . We introduce the dimensionless quantities:

t′ = Sc = Gr =

υt

,

y′ =

υy , U0

υ , Dm

Ec =

U 02 , C p (Tw − T∞ )

U 02

g βυ (Tw − T∞ ) U 03

,

u ′ = uU 0 ,

Gc =

v ′ = vU 0 , Pr =

μC p

g β∗υ (C w′ − C∞′ ) U 03

k ,

K′ = ,

υ2 K U 02

M2 =

θ =

,

σB02 υ ρU 02

Ω′ = ,

T − T∞ , Tw − T∞

U 02 Ω , υ

kr = C =

k r′ υ U 02

,

(6)

C ′ − C∞′ . C w′ − C∞′

With these variables, we express the basic equations (1), (2), and (4) in dimensionless form:

∂u ∂ 2u u − 2Ω v = 2 − M 2 u − + Grθ + GcC , ∂t K ∂y

(7)

1329

∂v ∂ 2v v − 2Ω u = 2 − M 2 v − , ∂t K ∂y

(8)

∂C 1 ∂ 2C = − k rC. ∂t Sc ∂y 2

(9)

We combine for convenience Eqs. (7) and (8) into a single equation, multiplying Eq. (8) by i = to Eq. (7):

−1 and adding the result

∂q ⎛ 1⎞ ∂ 2q + ⎜ 2i Ω + M 2 + ⎟ q = 2 + Grθ + GcC , q = u + iv. ∂t ⎝ K⎠ ∂y

(10)

Using the Rosseland approximation, we present the radiative heat flux as

q′r = −

4σ∗ ∂T 4 . 3α ∂y ′

(11)

It should be noted that this approximation assumes an optically thick medium. If the temperature differences within the flow are sufficiently small, Eq. (11) can be linearized by expanding T 4 in a Taylor series about T∞ , which after neglecting the higher-order terms gives

T 4 ≅ 4T∞3T − 3T∞4 .

(12)

In view of Eqs. (6), (11), and (12), we reduce Eq. (3) to 2

⎛ ∂u ⎞ ∂θ 1 + N ∂ 2θ = + Ec ⎜ ⎟ , 2 Pr ∂y ∂t ⎝ ∂y ⎠

(13)

where N = 16σ∗T∞3 (3k α) is the radiation parameter. The initial and boundary conditions are t ≤ 0 : q ( y, t ) = θ( y, t ) = C ( y, t ) = 0; t > 0 : q (0, t ) = q0 , θ(0, t ) = 1, C (0, t ) = 1,

(14)

q (∞, t ) → 0, θ(∞, t ) → 0, C (∞, t ) → 0. Solution of the Problem. Applying the Galerkin finite element method to Eq. (10) over the element ( e) for

y j ≤ y ≤ y k , we obtain yk

⎧⎪ ( e )T ⎨N ⎩⎪

∫

yj

⎡ ∂ 2 q ( e ) ∂q ( e ) ⎤⎫⎪ ( e) − − + Rq P ⎢ ⎥⎬ dy = 0, 2 ∂t ⎢⎣ ∂y ⎥⎦⎪⎭

where P = Grθ + GcC , R = 2i Ω + M +

y

⎧⎪ ∂q ( e ) ⎫⎪ k N ( e)T ⎨ ⎬ − ⎩⎪ ∂y ⎭⎪ y j

1330

yk

∫

yj

(15)

. Integrating the first term in Eq. (15) by parts gives

( e)T ∂q ( e ) ⎪⎧ ∂N + N ( e)T ⎨ y y ∂ ∂ ⎪⎩

⎛ ∂q ( e ) ⎞⎪⎫ + Rq ( e ) − P ⎟⎬dy = 0. ⎜⎜ ⎟ ⎝ ∂t ⎠⎪⎭

(16)

Neglecting the first term in Eq. (16), we have yk

∫

yj

( e)T ∂q ( e ) ⎪⎧ ∂N + N ( e)T ⎨ ∂ y ∂ y ⎩⎪

⎛ ∂q ( e ) ⎞⎪⎫ + Rq ( e ) − P ⎟⎬dy = 0 . ⎜⎜ ⎟ ⎝ ∂t ⎠⎪⎭

q ( e ) = N ( e )ϕ ( e ) be the linear piecewise approximation solution over the element ( e) , where y − yj T y − y = ⎡⎣ N j N k ⎤⎦ , ϕ ( e ) = ⎡⎣u j uk ⎤⎦ , and N j = k , Nk = are the basis functions. Then yk − y j yk − y j Let

N ( e) yk

∫

yj

⎧⎡ N 'j N 'j N 'j N k' ⎤ ⎪ ⎥ ⎨⎢ ' ' ' ' ⎪⎩⎣⎢ N j N k N k N k ⎦⎥

⎡q j ⎤⎫⎪ ⎢ ⎥⎬dy + ⎣qk ⎦⎪⎭

yk

∫

yj

y

k ⎪⎧⎡ N j N j N j N k ⎤ ⎡q j ⎤⎪⎫ ⎪⎧⎡ N j N j N j N k ⎤ ⎨⎢ ⎥ ⎢ ⎥⎬dy + R ∫ ⎨⎢ ⎥ ⎪⎣ N j N k N k N k ⎦ ⎩⎪⎣ N j N k N k N k ⎦ ⎣q k ⎦⎭⎪ yj ⎩

yk

⎡q j ⎤⎪⎫ ⎢ ⎥⎬dy = P ∫ ⎣qk ⎦⎭⎪ yj

⎡N j ⎤ ⎢ ⎥dy. ⎣N k ⎦

Simplifying this equation, we get

1 l ( e)

2

⎡ 1 −1⎤ ⎡q j ⎤ 1 ⎡2 1⎤ ⎡q j ⎤ R ⎡2 1⎤ ⎡q j ⎤ P ⎡1⎤ ⎢−1 1⎥ ⎢ ⎥ + 6 ⎢1 2⎥ ⎢q ⎥ + 6 ⎢1 2⎥ ⎢ ⎥ = 2 ⎢1⎥ . ⎣ ⎦ ⎣qk ⎦ ⎣ ⎦ ⎣ k⎦ ⎣ ⎦ ⎣qk ⎦ ⎣⎦

In the above equations primes and dots denote differentiation with respect to y and t, respectively. Assembling the element equations for two consecutive elements ( yi −1 ≤ y ≤ yi and yi ≤ y ≤ yi +1) , we obtain

1 l ( e)

2

⎡ 1 −1 0⎤ ⎡qi −1 ⎤ ⎡2 1 0⎤ ⎡q i −1⎤ ⎡2 1 0⎤ ⎡qi −1⎤ ⎡1 ⎤ ⎢−1 2 −1⎥ ⎢q ⎥ + 1 ⎢1 4 1⎥ ⎢ q ⎥ + R ⎢1 4 1⎥ ⎢q ⎥ = P ⎢2⎥ . ⎢ ⎥⎢ i ⎥ 6 ⎢ ⎥⎢ i ⎥ 6 ⎢ ⎥⎢ i ⎥ 2 ⎢ ⎥ ⎢⎣ 0 −1 1⎥⎦ ⎢⎣qi +1⎥⎦ ⎢⎣0 1 2⎥⎦ ⎢⎣q i +1⎥⎦ ⎢⎣0 1 2⎥⎦ ⎢⎣qi +1⎥⎦ ⎢⎣1 ⎥⎦

(17)

Further, assuming the row corresponding to the ith node equal to zero, we get from Eq. (17) the difference scheme with

l ( e) = h : 1 h

2

[−qi −1 + 2qi

− qi +1 ] +

1 R [q i −1 + 4q i + q i +1] + [qi −1 + 4qi + qi +1] = P. 6 6

(18)

Applying the trapezoidal rule, we obtain the system of equations according to the Crank–Nicholson method:

A1qin−+11 + A2 qin +1 + A3qin++11 = A4 qin−1 + A5qin + A6 qin+1 + P*.

(19)

Next from Eqs. (13) and (9) we derive the following equations:

B1θin−+11 + B2 θin +1 + B3θin++11 = B4 θin−1 + B5θin + B6θin+1,

(20)

C1Cin−+11 + C2Cin +1 + C3Cin++11 = C4Cin−1 + C5Cin + C6Cin+1,

(21)

where

A1 = 2 − 6r + Rk ,

A2 = 8 + 12r + 4 Rk ,

A3 = 2 − 6r + Rk ,

A4 = 2 + 6r − Rk ,

A5 = 8 − 12r − 4 Rk ,

A6 = 2 + 6r − Rk ,

1331

B1 = 2 Pr − 6(1 + N ) r,

B2 = 8 Pr + 12(1 + N ) r,

B3 = 2 Pr − 6(1 + N ) r,

B4 = 2 Pr + 6(1 + N ) r,

B5 = 8 Pr − 12(1 + N ) r,

B6 = 2 Pr + 6(1 + N ) r,

C1 = 2Sc − 6r + k rSck ,

C2 = 8Sc + 12r + 4k rSck ,

C3 = 2Sc − 6r + k rSck ,

C4 = 2Sc + 6r − k rSck ,

C5 = 8 Sc − 12r − 4k rSck ,

C6 = 2Sc + 6r − k rSck ,

P* = 12 Pk = 12kGrθ + 12kGcC. Here, r = k/h2, h and k are the mesh sizes along y and t, respectively, and the indices i and j refer to space and time. From Eqs. (19)–(21), where i = 1(1)n , we obtain the following system of equations with consideration for boundary conditions (14):

Ai X i = Bi , i = 1, 2, 3,

(22)

where Ai are matrices of order n, and Xi and Bi are column matrices with n components. We deduced the solutions of the above system for the velocity, temperature, and concentration using the Thomas algorithm, and established stability and convergence of the solutions. Knowing the velocity field, we express the skin friction at the plate as

⎛ ∂u ⎞ . τ = ⎜ ⎟ ⎝ ∂y ⎠ y = 0

(23)

Results and Discussion. We numerically calculated the dimensionless temperature θ , concentration C, and velocity q at the following values of the parameters: Gr = 1, Gc = 1, M = 1, K = 1, Pr = 0.71, N = 1, Ec = 0.001, Sc = 0.22, and kr = 1. All figures correspond to these values unless otherwise indicated. The temperature and species concentration are connected with the velocity via the Grashof number Gr and modified Grashof number Gc, as is seen from Eq. (10). The velocity profiles for various values of Gr are shown in Fig. 1a. The graphs for the velocity profiles for various values of Gc are analogous to those in Fig. 1a and are not presented. The both Grashof numbers characterize the relative effect of the thermal buoyancy force and species buoyancy force, respectively, in reference to the viscous hydrodynamic force in the boundary layer. As expected, the velocity increases with Gr and Gc. It attains a distinctive maximum value in the vicinity of the plate and then decreases approaching the free stream value. Figure 1b shows the velocity profiles for different Prandtl numbers corresponding to different substances: mercury (Pr = 0.025), air (Pr = 0.71), water (Pr = 7.00), and methanol (Pr = 11.62). It is seen that the velocity decreases with increase in the Prandtl number. Figure 2a presents the temperature profiles for different values of Pr. It is evident from the figure that the thermal boundary-layer thickness is greater for fluids with small Prandtl number. The reason is that smaller values of Pr are equivalent to an increasing thermal conductivity, and therefore for such fluids heat is able to diffuse away from the heated surface more rapidly than for fluids with higher Pr values. The effect of the thermal radiation parameter N on the velocity and temperature profiles in the boundary layer is shown in Figs. 1c and 2b, respectively. An increase in this parameter leads to a significant increase in the thermal conductivity of the fluid and in the thickness of its thermal boundary layer. An increase in the fluid temperature results in a higher fluid velocity. The velocity profiles in the presence of different foreign species with various Schmidt numbers, namely, hydrogen (Sc = 0.22), helium (Sc = 0.30), oxygen (Sc = 0.60), and steam (Sc = 0.66), are shown in Fig. 1e. It is seen that the velocity decreases as Sc increases. Figure 3a shows the concentration profiles at different Sc for the same fluids. It is observed that the concentration diminishes along the coordinate, and this change is more abrupt for steam and oxygen. Thus, hydrogen can be used for maintaining a sufficiently uniform concentration field. Figures 1d and 2c show the effect of the Eckert number Ec on the velocity and temperature profiles. It is seen that both quantities increase with Ec. 1332

Fig. 1. Velocity profiles for different values of Gr (a), Pr (b), N (c), Ec (d), Sc (e), kr (f), and Ω (g). Figures 1f and 3b show the effect of the chemical reaction parameter kr on the velocity and concentration profiles, respectively. As expected, the presence of the chemical reaction significantly affects both profiles. It should be mentioned that the case studied relates to a destructive chemical reaction. In fact, as the chemical reaction parameter increases, a considerable reduction in the velocity occurs, and the presence of the peak indicates that the maximum velocity takes place in the fluid body close to the surface, but not at the surface itself. The concentration also decreases with increase in the chemical reaction

1333

Fig. 2. Temperature profiles for different values of Pr (a), N (b), and Ec (c).

Fig. 3. Concentration profiles for different values of Sc (a) and kr (b).

parameter. It is evident that an increase in this parameter significantly alters the concentration boundary-layer thickness but does not change the momentum one. The effect of the rotation parameter Ω on the velocity profile is shown in Fig. 1g. It is seen that the velocity increases with Ω. The values of the skin friction for different values of the governing parameters are given in Table 1. It is seen from the table that an increase in the Prandtl number, Schmidt number, Eckert number, and in the magnetic parameter leads to a decrease in the value of the skin friction, whereas the latter increases with Grashof number, modified Grashof number, radiation parameter, and permeability.

1334

TABLE 1. Skin Friction at Different Values of the Governing Parameters M Ec Gr Gc Pr K N 1.0 1.0 1.0 1.0 0.71 1.0 0.001 2.0 1.0 1.0 1.0 0.71 1.0 0.001 1.0 2.0 1.0 1.0 0.71 1.0 0.001 1.0 1.0 2.0 1.0 0.71 1.0 0.001 1.0 1.0 1.0 2.0 0.71 1.0 0.001 1.0 1.0 1.0 1.0 7.0 1.0 0.001 1.0 1.0 1.0 1.0 0.71 2.0 0.001 1.0 1.0 1.0 1.0 0.71 1.0 0.01 1.0 1.0 1.0 1.0 0.71 1.0 0.001 1.0 1.0 1.0 1.0 0.71 1.0 0.001

Sc 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.22 0.30 0.22

kr 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0

τ 1.2265 2.4398 2.4215 1.1684 1.5307 1.0568 1.3602 1.2015 1.0026 1.1968

Conclusions. We have examined the unsteady hydromagnetic natural convective heat and mass transfer for a rotating Boussinesq fluid past a vertical porous plate in the presence of radiative heat transfer. Employing the finite element technique, we have solved the governing equations in the complex plane numerically. We can conclude the following: 1. The fluid velocity increases with parameters Gr, Gc, Ec, and N and decreases with increase in Pr, Sc, and kr. 2. The fluid temperature increases with N and Ec and decreases with increase in Pr. 3. The fluid concentration decreases with increase in kr and Sc. This research was supported by the University Grants Commission, New Delhi, India.

NOTATION B0, magnetic field strength; C, species concentration; Cp, specific heat at constant pressure; Dm, solutal diffusivity; Ec, Eckert number; g, acceleration due to gravity; Gr, Grashof number; Gc, modified Grashof number; kr, chemical reaction constant; K, porosity parameter; k, thermal conductivity; M, Hartmann number; N, radiation parameter; Pr, Prandtl number; q, complex velocity; qr, radiative flux; Sc, Schmidt number; t, time; T, temperature; U0, free stream velocity; u, v, velocity components; y, coordinate; α2 , absorption coefficient; β , coefficient of volume expansion due to temperature; β* , coefficient of volume expansion due to concentration; θ , dimensionless temperature; μ , permeability; ν, kinematic viscosity; ρ , fluid density; σ , electrical conductivity; σ* , Stefan–Boltzmann constant; τ, skin friction; Ω , angular velocity. Indices: w, wall conditions; ∞ , free stream.

REFERENCES 1. H. L. Agarwal, P. C. Ram, and V. Singh, Effects of Hall currents on the hydro-magnetic free convection with mass transfer in a rotating fluid, Astrophys. Space Sci., 100, 297–283 (1984). 2. A. R. Bestman and S. K. Adjepong, Unsteady hydro-magnetic free-convection flow with radiative heat transfer in a rotating fluid, Astrophys. Space Sci., 143, 217–224 (1998). 3. G. E. A. Azzam, Radiation effects on the MHD mixed free-fixed convection flow past a semi-infinite moving vertical plate for high temperature differences, Phys. Scr., 66, 71–76 (2002). 4. A. R. Bestman, Free convection heat transfer to steady radiating non-Newtonian MHD flow past a vertical porous plate, Int. J. Numer. Methods Eng., 21, 899–908 (1985). 5. A. J. Chamkha, Thermal radiation and buoyancy effects on hydro-magnetic flow over an accelerating permeable surface with heat source or sink, Int. J. Eng. Sci., 38, 1699–1712 (2000). 6. A. J. Chamkha, Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption, Int. J. Eng. Sci., 42, 217 –230 (2004). 7. C. I. Cookey, A. Ogulu, and V. B. Omubo-Pepple, Influence of viscous dissipation and radiation on unsteady MHD freeconvection flow past an infinite heated vertical plate in a porous medium with time dependent suction, Int. J. Heat Mass Transf., 46, 2305–2311 (2003). 1335

8. E. M. E. Elbarbary and N. S. Elgazery, Chebyshev finite difference method for the effect of variable viscosity on magneto micro-polar fluid flow with radiation, Commun. Heat Mass Transf., 31, 409–419 (2004). 9. K. A. Helmy, Unsteady free convection flow past a vertical porous plate, Z. Agnew. Math. Mech., 78, No. 4, 255–270 (1998). 10. F. S. Ibrahim, I. A. Hassanein, and A. A. Bakr, Unsteady magneto-hydro-dynamic micropolar fluid flow and heat transfer over a vertical porous medium in the presence of thermal and mass diffusion with constant heat source, Can. J. Phys., 82, 775–790 (2004). 11. B. K. Jha, MHD free-convection and mass transfer flow through a porous medium, Astrophys. Space Sci., 175, 283–289 (1991). 12. R. Kandasamy, K. Periasamy, and K. K. S. Prabhu, Effects of chemical reaction heat and mass transfer along a wedge with heat source and convection in the presence of suction or injection, Int. J. Heat Mass Transf., 48, 1288–1394 (2005). 13. Y. J. Kim, Unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction, Int. J. Eng. Sci., 38, 833–845 (2000). 14. A. Ogulu and C. I. Cookey, MHD free-convection and mass transfer flow with radiative heat transfer, Model. Measur. Control B, 70, No. 1–2, 31–37 (2001). 15. A. Ogulu and J. Prakash, Heat transfer to unsteady magneto-hydrodynamic flow past an infinite moving vertical plate with variable suction, Phys. Scr., 74, 232–239 (2006). 16. J. Prakash, A. Ogulu, and E. Zhandire, MHD free convection and mass transfer flow of a micro-polar thermally radiating and reacting fluid with time dependent suction, Indian J. Pure Appl. Phys., 46, 679–684 (2008). 17. A. Pantokratoras, Non-Darcian forced convection heat transfer over a flat plate in a porous medium with variable viscosity and variable Prandtl number, J. Porous Media, 10, 201–208 (2007). 18. S. Shivaiah and J. Anand Rao, Chemical reaction effect on an unsteady mhd free convection flow past an infinite vertical porous plate with constant suction, Chem. Ind. Chem. Eng. Q, 17, No. 3, 249−257 (2011). 19. S. Shivaiah and J. Anand Rao, Chemical reaction effects on unsteady MHD flow past semi-infinite vertical porous plate with viscous dissipation, Appl. Math. Mech., 32, No. 8, 1065–1078 (2011).

1336