Mathematical Analysis for Optically Thin Radiating/ Chemically Reacting Fluid in a Darcian Porous Regime

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 6 (017), pp Research India Publications Mathematical Analysis for Optically Thin Radiating/ Chemically Reacting Fluid in a Darcian Porous Regime Nava Jyoti Hazarika 1 and Sahin Ahmed 1 Department of Mathematics, Tyagbir Hem Baruah College, Jamugurihat, Sonitpur , Assam, India. Department of Mathematics, Rajiv Gandhi University, Rono Hills, Itanagar, Arunachal Pradesh-79111, India. Abstract In this paper, we analyzed an unsteady MHD flow of two-dimensional, laminar, incompressible, Newtonian, electrically-conducting and radiating fluid along a semi-infinite vertical permeable moving plate with periodic heat and mass transfer by taking into account the effect of viscous dissipation in presence of chemical reaction. A uniform magnetic field is applied transversely to the porous plate. The plate moves with a constant velocity in the direction of the fluid flow while the free stream velocity follows an exponentially increasing small perturbation law subject to a constant suction velocity to the plate. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and nonharmonic functions. Numerical evaluation of the analytical results are performed and graphical results for velocity, temperature and concentration profiles within the boundary layer and the tabulated results for the Skinfriction co-efficient, Nusselt number and Sherwood number are presented and discussed. It is seen that, an increase in chemical reaction parameter leads to decrease both fluid velocity as well as concentration. Moreover, the skinfriction has been depressed by the influence of chemical reaction parameter, where as the rate of heat transfer is escalated. The present model has several important applications such as dispersion of chemicals contaminants, superconvecting geothermics, geothermal energy extractions and plasma physics. Keywords: Thin gray gas; Dispersion of chemicals contaminants; Viscous dissipation; MHD; Darcian regime; skin-friction. Corresponding author: Sahin Ahmed

2 1778 Nava Jyoti Hazarika and Sahin Ahmed 1. INTRODUCTION The study of heat and mass transfer to chemical reacting MHD free convection flow with radiation effects on a vertical plate has received a growing interest during the last decades. Accurate knowledge of the overall convection heat transfer has vital importance in several fields such as thermal insulation, dying of porous solid materials, heat exchangers, stream pipes, water heaters, refrigerators, electrical conductors and industrial, geophysical and astrophysical applications such as polymer production, manufacturing of ceramic, packed-bed catalytic reactor, food processing, cooling of nuclear reactor, enhanced oil recovery, underground energy transport, magnetized plasma flow, high speed plasma wind, cosmic jets and stellar system. For some industrial application such as glass production, furnace design, propulsion systems, plasma physics and spacecraft re-entry aerothermodynamics which operate at higher temperatures and radiation effect can also be significant. Consolidated effects of heat and mass transfer problems are of importance in many chemical formulations and reactive chemicals. Therefore, considerable attention had been paid in recent years to study the influence of the participating parameters on the velocity fields. More such engineering application can be seeing in electrical power generation system when the electrical energy is extracted directly from a moving conducting fluid. There has been a renewed interest in studying Magnetohydrodynamic (MHD) flow and heat transfer in porous and non-porous media due to the effect of magnetic fields on the boundary layer flow control and on the performance of many systems using electrically conducting fluids. In addition, this type of flow finds applications in many engineering problems such as MHD generators, plasma studies, nuclear reactors and geothermal energy extractors. Chamkha [1] presented an unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption. An analysis of an unsteady MHD convective flow past a vertical moving plate embedded in a porous medium in the presence of transverse magnetic field a reported by Kim []. Singh [3] studied the effects of mass transfer on free convection in MHD flow of viscous fluid. Ahmed [4] looked the effects of unsteady free convective MHD flow through a porous medium bounded by an infinite vertical porous plate. Raptis [5] studied mathematically the case of unsteady two-dimensional natural convective heat transfer of an incompressible, electrically conducting viscous fluid in a highly porous medium bound by an infinite vertical porous plate. Soundalgekar [6] obtained approximate solutions for the two-dimensional flow an incompressible, viscous fluid past an infinite porous vertical plate with constant suction velocity normal to the plate, the difference between the temperature of the plate and the free stream is moderately large causing the free convection currents. Recently, free convective fluctuating MHD flow through porous media past a vertical porous plate with variable temperature and heat source was studied by Acharya et al. [7]. Rao et al. [8] was discussed the heat transfer on steady MHD rotating flow through porous medium in a parallel plate channel. Pattnaik and Biswal [9] studied the analytical solution of MHD free convective flow through porous media with time dependent temperature and concentration. More recently, Hazarika and Ahmed [10]

3 Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1779 have investigated the analytical study of unsteady MHD chemically reacting fluid over a vertical porous plate in a Darcian porous Regime. Chemical reaction effects on MHD free convective flow through porous medium with constant suction and heat flux has discussed by Seshaiah and Varma [11]. All the above investigations are restricted to MHD flow and heat transfer problems only. However of the late the effects of radiation on MHD flow, heat and mass transfer have becomes more important industrially. The radiation flows of an electrically conducting fluid with high temperature, in the presence of magnetic fields, are encountered in electrical power generation, astrophysical flows, solar power technology, space vehicle re-entry, nuclear engineering applications and other industrial areas. Radiative heat and mass transfer play an important role in manufacturing industries for the design of fins, steel rolling, nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering applications. Radiation effects on mixed convection along an isothermal vertical plate were studied by Hossain and Takhar [1]. Prasad et al. [13] studied the radiation and mass transfer effects on unsteady MHD free convection flow past a vertical porous plate embedded in porous medium. Zueco and Ahmed [14] proposed the mixed convection MHD flow along a porous plate with chemical reaction in presence of heat source. The transient MHD free convective flow of a viscous, incompressible, electrically conducting, gray, absorbing-emitting, but not scattering, optically thick fluid medium which occupies a semi-infinite porous region adjacent to an infinite hot vertical plate moving with constant velocity was presented by Ahmed and Kalita [15]. The effects of chemical reaction as well as magnetic field on the heat and mass transfer of Newtonian twodimensional flow over an infinite vertical oscillating plate with variable mass diffusion investigated by Ahmed and Kalita [16]. Recently, Ahmed [17] presented the effects of conduction-radiation, porosity and chemical reaction on unsteady hydromagnetic free convection flow past an impulsively started semi-infinite vertical plate embedded in a porous medium in presence of thermal radiation. The thermal radiation and Darcian drag force MHD unsteady thermal-convection flow past a semiinfinite vertical plate immersed in a semi-infinite saturated porous regime with variable surface temperature in the presence of transversal uniform magnetic field have been discussed by Ahmed et al. [18]. Radiation and mass transfer on unsteady MHD convective flow past an infinite vertical plate in presence of Dufour and Soret effects studied by Vedavathi et al. [19]. Ahmed et al. [0] investigated the effects of chemical reaction and viscous dissipation on MHD heat and mass transfer flow through Perturbation method. In all these investigations, the viscous dissipation is neglected. Gebhart [1] had shown the importance of viscous dissipative heat in free convection flow in the case of isothermal and constant heat flux at the plate. Soundalgekar [] analyzed the viscous dissipative heat on the two-dimensional unsteady free convective flow past an infinite vertical porous plate when the temperature oscillates in time and there is constant suction at the plate. Prasad and Reddy [3] had discussed about the Radiation and Mass transfer effects on an unsteady MHD convection flow with viscous

4 1780 Nava Jyoti Hazarika and Sahin Ahmed dissipation. Cookey et al. [4] had investigated the influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction. Recently, radiation effects on an unsteady MHD convective flow past a vertical plate in porous medium with viscous dissipation analyzed by Gudagani et al. [5]. In this paper the effects of chemical reaction and thermal radiation of optically thin gray gas on a mixed convective boundary layer flow of an electrically conducting fluid over an semi-infinite porous surface embedded in a Darcian porous regime in presence of viscous dissipative heat is investigated. The governing equations are solved by using a regular perturbation theory.. MATHEMATICAL ANALYSES In this flow model, we consider two-dimensional unsteady hydromagnetic laminar mixed convective boundary layer flow of a viscous, incompressible, electrically conducting and radiating fluid in an optically thin environment, past a semi-infinite vertical permeable moving plate embedded in a Darcian porous medium, in presents of thermal and concentration buoyancy effects with chemical reaction of first order. The x-axis is taken in the upward direction along the plate and y-axis normal to it. A uniform magnetic field is applied in the direction perpendicular to the plate. The transverse applied magnetic field and magnetic Reynolds number are assumed to be very small, so that the induced magnetic field is negligible. Also, it is assumed that there is no applied voltage, so that the electric field is absent. The concentration of the diffusing species in the binary mixture is assumed to be very small in comparison with the other chemical species which are present, and hence the Soret and Dufour effects are negligible. Further, due to semi-infinite plane surface assumption, the flow variables are functions of normal distance y and t only. Now, under the usual Boussinesq s approximation, the governing boundary layer equations are: v y = 0 (1) u v + v t y = 1 p ρ x + ν u y + gβ T(T T ) + gβ C (C C ) ( ν κ + σb 0 ) u () ρ T t + v T y = k [ T ρc p y 1 q k y ] + ν c p ( u y ) q y 3α q 16σ 3 T αt y = 0 (4) C t + v C y = D C y C r(c C ) (5) (3)

5 Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1781 The third and fourth terms on the right hand side of momentum Eq. () denote the thermal and concentration buoyancy effects respectively. The second and third terms on right hand side of energy Eq. (3) represent the radiative heat flux and viscous dissipation respectively. Also the second term on right hand of concentration Eq. (5) represents the chemical reaction effect. The permeable plate moves with a constant velocity in the direction of fluid flow and the free steam velocity follows the exponentially increasing small perturbation law. In addition, it is assumed that the temperature and concentration at the wall as well as the suction velocity are exponentially varying with time. Eq. (4) is the differential approximation for radiation and the radiative heat flux q satisfies this non-linear differential equation. The boundary conditions for the velocity, temperature and concentration fields are: { u = u p, T = T w + ε(t w T )e nt, C = C w + ε(c w C )e nt at y = 0 } (6) u = U = U 0 (1 + εe nt ), T T, C C as y It is clear from the equation (1) that the suction velocity at the plate is either a constant or function of time only. Hence, the suction velocity normal to the plate is assumed in the form: v = V 0 (1 + εae nt ) (7) The negative sign indicates that the suction is towards the plate. Outside the boundary layer, Eq. () gives: 1 p ρ x = du dt + ν κ U + σ ρ B 0 U (8) Since the medium is optically thin with relatively low density and α 1, the radiative heat flux given by Eq. (3), in the spirit of Cogley et al. [] becomes: q y = 4α (T T ) where B is Planck s function. where α = δλ B 0 T, (9) In order to write the governing equations and boundary conditions in dimensionless form, the following non-dimensional quantities are introduced.

6 178 Nava Jyoti Hazarika and Sahin Ahmed u = u U 0, v = v V 0, θ = T T T w T, φ = y = V 0y ν, U = U, U p = u p, t = t V 0 U 0 ν C C C w C, Pr = νρc p k, Sc = ν D, M = σb 0 U 0 n ν n = V, K = K V 0 0 ν, C r = νc r ν ρv 0, Gr = νβ Tg(T w T ) U 0 V 0 Gm = νβ Cg(C w C ) U 0, Ec = U 0 V 0 C p (T w T ), R = α (T w T ), ρc p ku 0 { }, V 0, (10) In view of Eqs. (4) and (7) (10), Eqs. (), (3) and (5) reduce to the following dimensionless form: u t (1 + εaent ) u y = du dt + u y + Grθ + Gmφ + N(U u) (11) θ t (1 + εaent ) θ y = 1 θ Pr [ y R θ] + Ec ( u y ) (1) φ t (1 + εaent ) φ y = 1 φ Sc y C rφ (13) where N = M + K 1 The corresponding dimensionless boundary conditions are: { u = U p, θ = 1 + εe nt, φ = 1 + εe nt, at y = 0 u = U = 1 + εe nt, θ 0, φ 0 as y } (14) SOLUTION OF THE PROBLEM The Eqs. (11-13) are coupled, non-linear partial differential equations and these cannot be solved in closed-form. However, these equations can be reduced to a set of ordinary differential equations, which can be solved analytically. This can be done by representing the velocity, temperature and concentration of the fluid in the neighbourhood of the plate as: u(y, t) = u 0 (y) + εe nt u 1 (y) + 0(ε ) + { θ(y, t) = θ 0 (y) + εe nt θ 1 (y) + 0(ε ) + φ(y, t) = φ 0 (y) + εe nt φ 1 (y) + 0(ε ) + } (15)

7 Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1783 Substituting Eq. (15) in Eqs. (11-13) and equating the harmonic and non-harmonic terms, and neglecting the higher order terms of 0(ε ), we obtain: u 0 (y) + u 0 (y) Nu 0 (y) = N Grθ 0 (y) Gmφ 0 (y) (16) u 1 (y) + u 1 (y) (N + n)u 1 (y) = (N + n) Au 0 (y) Grθ 1 (y) Gmφ 1 (y) (17) θ 0 (y) + Pr θ 0 (y) R θ 0 (y) = PrEc [u 0 (y)] (18) θ 1 (y) + Pr θ 1 (y) (R + npr)θ 1 (y) = PrA θ 0 (y) PrEc u 0 (y)u 1 (y) (19) φ 0 (y) + Sc φ 0 (y) Sc Crφ 0 (y) = 0 (0) φ 1 (y) + Sc φ 1 (y) Sc(n + Cr)φ 1 (y) = ASc φ 0 (y) (1) where prime denotes ordinary differentiation with respect to y. The corresponding boundary conditions can be written as: { u 0 = U p, u 1 = 0, θ 0 = 1, θ 1 = 1, φ 0 = 1, φ 1 = 1 at y = 0 u 0 = 1, u 1 = 1, θ 0 0, θ 1 0, φ 0 0, φ 1 0 as y } () The Eqs. (16) (1) are still coupled and non-linear, whose exact solutions are not possible. So we expand u 0, u 1, θ 0, θ 1, φ 0, φ 1 in terms of Ec in the following form, as the Eckert number is very small for incompressible flows. F(y) = F 0 (y) + Ec F 1 (y) + 0(Ec ) (3) where F stands for any u 0, u 1, θ 0, θ 1, φ 0, φ 1. Substituting Eq. (3) in Eqs. (16) (1), equating the co-efficient of Ec to zero and neglecting the terms in Ec and higher order, we get the following equations: The zeroth order equations are: u 01 (y) + u 01 (y) Nu 01 (y) = N Gr θ 01 (y) Gm φ 01 (y) (4) u 0 (y) + u 0 (y) Nu 0 (y) = Gr θ 0 (y) Gm φ 0 (y) (5) θ 01 (y) + Pr θ 01 (y) R θ 01 (y) = 0 (6) θ 0 (y) + Pr θ 0 (y) R θ 0 (y) = Pr[u 01 (y)] (7) φ 01 (y) + Sc φ 01 (y) Sc Cr φ 01 (y) = 0 (8) φ 0 (y) + Sc φ 0 (y) Sc Cr φ 0 (y) = 0 (9)

8 1784 Nava Jyoti Hazarika and Sahin Ahmed and the respective boundary conditions are: { u 01 = U p, u 0 = 0, θ 01 = 1, θ 0 = 0, φ 01 = 1, φ 0 = 0 at y = 0 u 01 1, u 0 0, θ 01 0, θ 0 0, φ 01 0, φ 0 0 at y } (30) The first order equations are: u 11 (y) + u 11 (y) (N + n)u 11 (y) = { (N + n) Gr θ 11(y) Gm φ 11 (y) A u 01 (y) } (31) u 1 (y) + u 1 (y) (N + n)u 1 (y) = Gr θ 1 (y) Gm φ 1 (y) A u 0 (y) (3) θ 11 (y) + Pr θ 11 (y) N 1 θ 11 (y) = PrA θ 01 (y) (33) θ 1 (y) + Pr θ 1 (y) N 1 θ 1 (y) = PrA θ 0 (y) Pr u 01 (y)u 11 (y) (34) φ 11 (y) + Sc φ 11 (y) Sc(n + Cr)φ 11 (y) = ASc φ 01 (y) (35) φ 1 (y) + Sc φ 1 (y) Sc(n + Cr)φ 1 (y) = ASc φ 0 (y) (36) where N 1 = R + npr. and respective boundary conditions are: u { 11 = 0, u 1 = 0, θ 11 = 1, θ 1 = 0, φ 11 = 1, φ 1 = 0 at y = 0 u 11 1, u 1 0, θ 11 0, θ 1 0, φ 11 0, φ 1 0 at y } (37) Solving Eqs. (4) (9) under the boundary conditions in Eq. (30) and Eqs. (31) - (36) under the boundary conditions in Eq. (37) and using Eqs. (15) and (3), we obtain the Velocity, Temperature and Concentration distributions in the boundary layer as: u(y, t) = { + εe nt P 3 e m 3y + P 1 e m y + P e m 1y + 1 +Ec { J 8e m 3y + J 1 e m y + J e m 3y + J 3 e m y + J 4 e m 1y +J 5 e (m +m 3 )y + J 6 e (m 1+m )y + J 7 e (m 1+m 3 )y } [ +Ec { G 6e m6y + G 1 e m5y + G e my + G 3 e m 4y +G 4 e m1y + G 5 e m3y } + 1 L 19 e m6y + L 1 e m5y + L e my + L 3 e m 3y +L 4 e my + L 5 e m1y + L 6 e (m +m 3 )y + L 7 e (m 1+m )y { +L 8 e (m 1+m 3 )y + L 9 e (m 3+m 6 )y + L 10 e (m 3+m 5 )y +L 11 e (m 3+m 4 )y + L 1 e (m +m 6 )y + L 13 e (m +m 5 )y +L 14 e (m +m 4 )y + L 15 e (m 1+m 6 )y + L 16 e (m 1+m 5 )y +L 17 e (m 1+m 4 )y + L 18 e m 3y }]}

9 Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1785 e my + Ec { S 7e my + S 1 e m3y + S e my + S 3 e m 1y +S 4 e (m +m 3 )y + S 5 e (m 1+m )y + S 6 e (m 1+m 3 )y } {D e m5y + D 1 e my } + R 17 e m5y + R 1 e my + R e m3y + R 3 e m y θ(y, t) = +R 4 e m1y + R 5 e (m +m 3 )y + R 6 e (m 1+m )y +εe nt Ec +R 7 e (m 1+m 3 )y + R 8 e (m 3+m 6 )y + R 9 e (m 3+m 5 )y +R 10 e (m 3+m 4 )y + R 11 e (m +m 6 )y + R 1 e (m +m 5 )y + R 13 e (m +m 4 )y + R 14 e (m 1+m 6 )y { [ { +R 15 e (m 1+m 5 )y + R 16 e (m 1+m 4 )y }]} φ(y, t) = e m 1y + εe nt {Z e m 4y + Z 1 e m 1y } The Skin-friction, Nusselt number and Sherwood number are important physical parameters for this type of boundary layer flow. THE SKIN FRICTION Knowing the velocity field, the Skin-friction at the plate can be obtained, which in non-dimensional form is given by: C f = τ w = ( u ρu 0 V 0 y ) = ( u 0 y + u εent 1 y ) y=0 y=0 m 3 P 3 m P 1 m 1 P + Ec { m 3J 8 m J 1 m 3 J m J 3 m 1 J 4 } (m + m 3 )J 5 (m 1 + m )J 6 (m 1 + m 3 )J 7 ( m 6 G 6 m 5 G 1 m G m 4 G 3 m 1 G 4 m 3 G 5 ) m 6 L 19 m 5 L 1 m L m 3 L 3 m L 4 m 1 L 5 = +εe nt (m + m 3 )L 6 (m 1 + m )L 7 (m 1 + m 3 )L 8 +Ec (m 3 + m 6 )L 9 (m 3 + m 5 )L 10 (m 3 + m 4 )L 11 (m + m 6 )L 1 (m + m 5 )L 13 (m + m 4 )L 14 [ [ { (m 1 + m 6 )L 15 (m 1 + m 5 )L 16 (m 1 + m 4 )L 17 m 3 L 18 }]] RATE OF HEAT TRANSFER Knowing the temperature field, the rate of heat transfer co-efficient can be obtained, which in the non-dimensional form, in terms of the Nusselt number is given by: Nu = x ( T ) y y=0 T w T = NuRe 1 x = ( θ y ) = ( θ 0 y + θ εent 1 y ) y=0 y=0

10 1786 Nava Jyoti Hazarika and Sahin Ahmed m + Ec { m S 7 m 3 S 1 m S m 1 S 3 (m + m 3 )S 4 } (m 1 + m )S 5 (m 1 + m 3 )S 6 ( m 5 D m D 1 ) m = 5 R 17 m R 1 m 3 R m R 3 m 1 R 4 +εe nt (m + m 3 )R 5 (m 1 + m )R 6 (m 1 + m 3 )R 7 +Ec (m 3 + m 6 )R 8 (m 3 + m 5 )R 9 (m 3 + m 4 )R 10 (m + m 6 )R 11 (m + m 5 )R 1 (m + m 4 )R 13 [ [ { (m 1 + m 6 )R 14 (m 1 + m 5 )R 15 (m 1 + m 4 )R 16 }]] where Re x = V 0x ν is the local Reynolds number. RATE OF MASS TRANSFER Knowing the concentration field, the rate of mass transfer co-efficient can be obtained, which in the non-dimensional form, in terms of the Sherwood number is given by: Sh = x VALIDITY ( C ) y y=0, C w C ShRe 1 x = ( C y ) = ( C 0 y + C εent 1 y ) y=0 y=0 = [ m 1 + εe nt ( m 4 Z m 1 Z 1 )] When Cr = 0, the present paper reduces to the work which was done by Prasad and Reddy [3]. Table 1: Comparison of the present results with those of Prasad and Reddy [3] with effects of Gr and Gm on Cf when Gr=.0, Gm=1.0, Pr=0.71, Sc=0.6, M=1.0, R=0.5, K=0.5, n=0.1, Up=0.5, A=0.5, Cr=0., t=1.0, Ec=0.001, Ԑ= Gr Gm Prasad and Reddy [3] Effects of Gr on Cf Effect of Gm on Cf Effects of Gr on Cf Present work Effects of Gm on Cf

11 Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1787 The Table 1 shows that the accuracy of the present model in comparison with the previous model studied by Prasad and Reddy [3] and this comparison is validated the present study. RESULTS AND DISCUSSION The formulation of the problem that accounts for the effects of radiation and viscous dissipation on the flow of an incompressible viscous chemically reacting fluid along a semi-infinite, vertically moving porous plate embedded in a porous medium in the presence of transverse magnetic field was accomplished. Following Cogley et al. [] approximation for the radiative heat flux in the optically thin environment, the governing equations on the flow field were solved analytically, using a perturbation method and the expressions for the velocity, temperature, concentration, Skin-friction, Nusselt number and Sherwood number were obtained. In order to get a physical insight of the problem, the above physical quantities are computed numerically for different values of the governing parameters viz. Thermal Grashof number Gr, the Solutal Grashof number Gm, Radiation parameter R, Magnetic parameter M, Permeability parameter K, Plate velocity Up, Prandtl number Pr, Schmidt number Sc, Eckert number Ec and Chemical reaction Cr. Figure 1 shows the typical velocity profiles in the boundary layer for various values of the thermal Grashof number. It is observed that an increase in Gr, leads to a rise in the values of the velocity due to enhancement in the buoyancy force. Here, the positive values of Gr correspond to cooling of the plate. In addit0ion, it is observed that the velocity increases rapidly near the wall of the porous plate as Grashof number increases and then decays to the free stream velocity. Figure depicts the typical velocity profiles in the boundary layer for distinct values of the solutal Grashof number Gm. The velocity distribution attaints a distinctive maximum value in the region of the plate surface and then decrease properly to approach the free stream value. As expected, the fluid velocity increases and the peak value becomes more distinctive due to increase in the buoyancy force represented by Gm. For different values of thermal radiation parameter R on the velocity and temperature profiles are shown in Figure 3 and 4. It is noticed that an increase in the radiation parameter results a decrease in the velocity and temperature within the boundary layer, as well as decreased the thickness of the velocity and temperature boundary layers. The effect of magnetic field on velocity profiles in the boundary layer is depicted in Figure 5. It is obvious that the existence of the magnetic field is to decrease the velocity in the momentum boundary layer because the application of the transverse magnetic field results in a resisting type of force called Lorentz force, which results in reducing the velocity of the fluid in the boundary layer. Figure 6 shows the effect of the permeability of the porous medium parameter K on the velocity distribution. It is found that the velocity increases with an increase in K.

12 1788 Nava Jyoti Hazarika and Sahin Ahmed The velocity distribution across the boundary layer for several values of plate moving velocity Up in the direction of the fluid flow is depicted in Figure 7. Although we have different initial plate moving velocities, the velocity decreases to a constant value for given material parameters. Figure 8 and 9 shows the behaviour velocity and temperature for different values of Prandtl number Pr. The numerical results show the effect of increasing values of Prandtl number results in the decreasing velocity. From Figure 9, it is observed that an increase in the Prandtl number results a decrease in the thermal boundary layer thickness and in general lower average temperature within the boundary layer. The reason is that smaller values of Pr are equivalent to increase in the thermal conductivity of the fluid and therefore heat is able to diffuse away from the heated surface more rapidly for higher values of Pr. Hence in the case of smaller Prandtl numbers as the thermal boundary layer is thicker and the rate of heat transfer is reduced. Figure 10 and 11 shows the effects of Schmidt number on the velocity and concentration respectively. As the Schmidt number increases, the concentration decreases. This causes the concentration buoyancy effects to decrease yielding a reduction in the fluid velocity. Reductions in the velocity and concentration distributions are accompanied by simultaneous reductions in the velocity and concentration boundary layers. The effects of chemical reaction on velocity and concentration are depicted by Figure 1 and 13. It is noticed that an increase in the chemical reaction parameter results a decrease in the velocity and concentration within the boundary layer. Table -5, represents the effects of Eckert number and Chemical reaction on the velocity u, temperature Ө, Skin-friction Cf, Nusselt number Nu and Sherwood number Sh. The effects of viscous dissipation parameter i.e. the Eckert number on the velocity and temperature are shown in Table and 3. It is revealed that velocity and temperature profiles scores grow with the increase of the Eckert number Ec. Eckert number, physically is a measure of frictional heat in the system. Hence the thermal regime with large Ec values is subjected to rather more frictional heating causing a source of rise in the temperature. To be specific, the Eckert number Ec signifies the relative importance of viscous heating to thermal diffusion. Viscous heating may serve as energy source to modify the temperature regime respectively. It is observed from Table 4, when Eckert number increases the Skin-friction increases and Nusselt number decreases. However, from Table 5, it can be seen that as the Chemical reaction increases, the Skin-friction decreases and Sherwood number increases.

13 Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1789

14 1790 Nava Jyoti Hazarika and Sahin Ahmed

15 Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1791 Table : Effects of Ec on velocity (u) when Gr=.0, Gm=.0, Pr=0.71, Sc=0.6, M=1.0, R=0.5, K=0.5, n=0.1, Up=0.5, A=0.5, Cr=0., t=1.0, Ԑ= y Ec=0 Ec=0.1 Ec=0. Ec=

16 179 Nava Jyoti Hazarika and Sahin Ahmed Table 3: Effects of Ec on temperature (Ө) when Gr=.0, Gm=.0, Pr=0.71, Sc=0.6, M=1.0, R=0.5, K=0.5, n=0.1, Up=0.5, A=0.5, Cr=0., t=1.0, Ԑ= y Ec=0 Ec=0.1 Ec=0. Ec= Table 4: Effects of Ec on Cf and NuRex -1. Reference values in the figure 14 and 15: Ec C f NuRe x Table 5: Effects of Cr on Cf and NuRex -1. Reference values in the figure 1 and 13: Cr C f NuRe x CONCLUSIONS The governing equations for unsteady MHD convective heat and mass transfer flow past a semi-infinite vertical permeable moving plate embedded in a porous medium with radiation and viscous dissipation effects were formulated.chemical reaction effects is also included in the present work. The plate velocity is maintained at constant value and the flow is subjected to a transverse magnetic field. The present investigation brings out the following conclusions of physical interest on the velocity, temperature and concentration distribution of the flow field. It is found that when thermal and solutal Grashof number is increased, the thermal and concentration buoyancy effects are enhanced and thus the fluid velocity increased. However, the presence of radiation effects caused reductions in the fluid temperature, which resulted in decrease in the fluid velocity.

17 Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1793 It is observed that the existence of magnetic body force and chemical reaction decreases the fluid velocity. The permeability parameter and plate velocity have the influence of increasing the fluid velocity. As Prandtl number increased the velocity and temperature are both decreased. When Schmidt number increased, the concentration level decreased resulting in decreased fluid velocity. In presence of Eckert number both velocity and temperature increased. NOMENCLATURE u, v Velocity components in x, y directions respectively, t Time, p Pressure, g Acceleration due to gravity, κ Permeability of porous medium, T Temperature of the fluid in the boundary layer, T Temperature of the fluid far away from the plate, C Species concentration in the boundary layer, C Species concentration in the fluid far away from the plate, B o c p k q σ D Magnetic induction, Specific heat at constant pressure, Thermal conductivity, Radiative heat flux, Stefan-Boltzmann constant, Mass diffusivity and C r u p T w C w U U 0 n Chemical reaction. Plate velocity, Temperature of the plate, Concentration of the plate, Free stream velocity, Constant, Constant

18 1794 Nava Jyoti Hazarika and Sahin Ahmed A V 0 Real positive constant Non-zero positive constant GREEK SYMBOL ρ Density, β T β C ν σ α Thermal expansion co-efficient, Concentration expansion co-efficient, Kinematic viscosity, Electrical conductivity of the fluid, Fluid thermal diffusivity, Ԑ small such that Ԑ 1 APPENDIX m 1 = Sc ScCr, m = Pr + Pr + 4R N, m 3 =, m 4 = Sc + Sc + 4Sc(n + Cr), m 5 = Pr + Pr + 4N 1, m (N + n) =, Gr P 1 = m m N, P Gm = m 1 m 1 N, P 3 = U p 1 P 1 P, J 1 = J 4 = J 6 = GrS 7 m m N, J = GrS 1 4m 3 m 3 N, J 3 = GrS 4m m N, GrS 3 4m 1 m 1 N, J GrS 4 5 = (m + m 3 ) (m + m 3 ) N, GrS 5 (m 1 + m ) (m 1 + m ) N, J GrS 6 7 = (m 1 + m 3 ) (m 1 + m 3 ) N, J 8 = (J 1 + J + J 3 + J 4 + J 5 + J 6 + J 7 ), G 1 = GrD m 5 m 5 (N + n), G = Am P 1 GrD 1 m m (N + n), G GmZ 3 = m 4 m 4 (N + n), G 4 = Am 1P GmZ 1 m 1 m 1 (N + n), G 5 = Am 3 P 3 m 3 m 3 (N + n), G 6 = (1 + G 1 + G + G 3 + G 4 + G 5 ),

19 Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1795 L 1 = L 4 = GrR 17 m 5 m 5 (N + n), L = Am J 1 GrR 1 m m (N + n), L 3 Am 3 J GrR = 4m 3 m 3 (N + n), Am J 3 GrR 3 4m m (N + n), L Am 1 J 4 GrR 4 5 = 4m 1 m 1 (N + n), A(m + m 3 )J 5 GrR 5 L 6 = (m + m 3 ) (m + m 3 ) (N + n), L 7 A(m 1 + m )J 6 GrR 6 = (m 1 + m ) (m 1 + m ) (N + n), A(m 1 + m 3 )J 7 GrR 7 L 8 = (m 1 + m 3 ) (m 1 + m 3 ) (N + n), L 9 GrR 8 = (m 3 + m 6 ) (m 3 + m 6 ) (N + n), L 10 = L 11 = L 1 = L 13 = L 14 = L 15 = L 16 = L 18 = GrR 9 (m 3 + m 5 ) (m 3 + m 5 ) (N + n), GrR 10 (m 3 + m 4 ) (m 3 + m 4 ) (N + n), GrR 11 (m + m 6 ) (m + m 6 ) (N + n), GrR 1 (m + m 5 ) (m + m 5 ) (N + n), GrR 13 (m + m 4 ) (m + m 4 ) (N + n), GrR 14 (m 1 + m 6 ) (m 1 + m 6 ) (N + n), GrR 15 (m 1 + m 5 ) (m 1 + m 5 ) (N + n), Am 3 J 8 m 3 m 3 (N + n), L GrR = (m 1 + m 4 ) (m 1 + m 4 ) (N + n), L 19 = ( 1 + L 1 + L + L 3 + L 4 + L 5 + L 6 + L 7 + L 8 + L 9 + L 10 +L 11 + L 1 + L 13 + L 14 + L 15 + L 16 + L 17 + L 18 ), S 1 = Prm 3 P 3 4m 3 Prm 3 R, S = Prm P 1 4m Prm R, S 3 = Prm 1 P 4m 1 Prm 1 R,

20 1796 Nava Jyoti Hazarika and Sahin Ahmed Prm m 3 P 3 P 1 S 4 = (m + m 3 ) Pr(m + m 3 ) R, S 5 Prm 1 m P 1 P = (m 1 + m ) Pr(m 1 + m ) R, S 6 = D 1 = Prm 3 m 1 P P 3 (m 1 + m 3 ) Pr(m 1 + m 3 ) R, S 7 = (S 1 + S + S 3 + S 4 + S 5 + S 6 ), PrAm PrAm S 7 m, D = 1 D 1, R 1 = Prm N 1 m, Prm N 1 R = PrAm 3S 1 Prm 3 G 5 P 3 4m, R 3 Prm 3 N 3 = PrAm S Prm G P 1 1 4m, Prm N 1 R 4 = PrAm 1S 3 Prm 1 G 4 P 4m 1 Prm 1 N 1, R 5 = PrA(m + m 3 )S 4 Prm m 3 (G P 3 + G 5 P 1 ) (m + m 3 ) Pr(m + m 3 ) N 1, R 6 = PrA(m 1 + m )S 5 Prm 1 m (G 4 P 1 + G P ) (m 1 + m ) Pr(m 1 + m ) N 1, R 7 = PrA(m 1 + m 3 )S 6 Prm 3 m 1 (G 4 P 3 + G 5 P ) (m 1 + m 3 ) Pr(m 1 + m 3 ) N 1, R 8 = Prm 3 m 6 G 6 P 3 Prm 3 m 5 G 1 P 3 (m 3 + m 6 ), R Pr(m 3 + m 6 ) N 9 = 1 (m 3 + m 5 ), Pr(m 3 + m 5 ) N 1 Prm 3 m 4 G 3 P 3 R 10 = (m 3 + m 4 ), R Pr(m 3 + m 4 ) N 11 1 Prm m 6 G 6 P 1 = (m + m 6 ), Pr(m + m 6 ) N 1 Prm m 5 G 1 P 1 R 1 = (m + m 5 ), R Pr(m + m 5 ) N 13 1 Prm m 4 G 3 P 1 = (m + m 4 ), Pr(m + m 4 ) N 1 Prm 1 m 6 G 6 P R 14 = (m 1 + m 6 ), R Pr(m 1 + m 6 ) N 15 1 Prm 1 m 5 G 1 P = (m 1 + m 5 ) Pr(m 1 + m 5 ) N 1 R 16 = Prm 1 m 4 G 3 P (m 1 + m 4 ) Pr(m 1 + m 4 ) N 1, R 17 = ( R 1 + R + R 3 + R 4 + R 5 + R 6 + R 7 + R 8 + R 9 +R 10 + R 11 + R 1 + R 13 + R 14 + R 15 ),

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