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1 Volume-7, Issue-, January-February 07 International Journal of Engineering and Management Research Page Number: 7-50 Effects of Radiation on MHD Boundary Layer Flow of Combined Heat and Mass Transfer over a Moving Inclined Plate in a Porous Medium with Suction and Viscous Dissipation in Presence of Hall Current and Chemical Reaction V.Subhakanthi, N. Bhaskar Reddy, Department of Mathematics, SVU, Tirupati, INDIA ABSTRACT This paper analyzes the chemical reaction and radiation effects on heat and mass flow over a moving inclined plate in porous medium with suction and viscous dissipation in presence of Hall current. A suitable similarity transformation is used to transform the nonlinear system of partial differential equations into a system of ordinary differential equations.to solve the resultant system a well tested numerical technique Runge-Kutta fourth order is used along with shooting technique. The behavior of primary and secondary velocities, temperature and concentration for variations in thermo physical parameters are presented in graphs. transfer in magnetohydrodynamic boundary layer The values of skin friction coefficient, Nusselt number and Sherwood number are also computed and are reported in tables. Keywords-- heat and mass transfer-mhd- radiationviscous dissipation-chemical reaction I. INTRODUCTION The free convection processes involving the combined mechanism of heat and mass transfer are encountered in many natural processes, in many industrial applications and in many chemical processing systems. The study of free convective mass transfer flow has become the object of extensive research as the effects of heat transfer along with mass transfer effects are dominant features in many engineering applications such as rocket nozzles, cooling of nuclear reactors, high sinks in turbine blades, high speed aircrafts, chemical devices and process equipments. The study of MHD flows have stimulated more attention due its important applications in different subject areas such as astrophysical, geophysical and engineering problems. Free convection in electrically conducting fluids through an external magnetic field has been a subject of considerable research interest of a large number of scholars for a long time due to its miscellaneous applications in the fields of nuclear reactors, geothermal engineering, liquid metals and plasma flows etc. Fluid flow control under magnetic forces is also applicable in MHD generators and a host of magnetic devices used in industries. Jha[] explained the problem of MHD free convection and mass transfer flow past an impulsively moving vertical plate through porous medium when the vertical plate moves with uniform acceleration and applied magnetic field is fixed with the moving plate. Pioneer work on convective flow in porous media are presented in the form of books and monographs by Ingham and Pop[], Ingham et al. [], Vafai[4] and Nield and Bejan [5]. A two dimensional steady MHD mixed convection and mass transfer flow over a semi- infinite porous inclined plate in the presence of thermal radiation with variable suction and thermophoresis was studied by Alam et al. [6]. Orthan Aydm and Ahmet Kaya[7] considered MHD mixed convective heat transfer flow about an inclined plate. Gnaneswara Reddy and Bhaskar Reddy[8] presented mass transfer and heat generation effects on MHD free convection flow over an inclined vertical surface in a porous medium. Recently, Hitesh Kumar[9] done his work on the heat transfer MHD boundary layer flow through a porous medium. The role of thermal radiation is of major importance in some industrial applications such as glass production and furnace design and in space technology applications, cosmical flight aerodynamics, propulsion systems, plasma physics and craft re-entry, aerothermodynamics which operate at high temperatures. Solving the governing equations become quite complicated when radiation is taken into account and hence many difficulties arise while solving such equations. Viskanta and Grosh [0] were one of the initial investigators to study the effects of thermal radiation on temperature distribution and heat transfer in an absorbing and emitting media flowing over a wedge. They used Rosseland approximation for the radiative flux vector to simplify the energy equation. Suneetha et 7 Copyright 06. Vandana Publications. All Rights Reserved.

2 al. [] studied the effects of thermal radiation on unsteady hydro magnetic free convection flow over an impulsively started vertical plate with variable surface temperature and concentration. Gnaneswara Reddy and Bhaskar Reddy[] reported the radiation and mass transfer effects on unsteady MHD free convection flow past a vertical porous plate with viscous dissipation by using finite element method. Recently, Narahari and Ishak[] carried out an analysis to study the effects of thermal radiation on unsteady free convection flow of an optically thick fluid past a moving vertical plate with Newtonian heating. Their interesting cases are impulsive movement of the plate, uniformly accelerated movement of the plate and exponentially accelerated movement of the plate. The viscous dissipation heat in the natural convective flow is important, when the flow field is of extreme size or at low temperature or in high gravitational field.. such effects are also important in geophysical flows and also in certain industrial operations and are usually characterized by the Eckert number. When the viscosity of the fluid is high, the dissipation term becomes important. For many cases, such as polymer processing which is operated at a very high temperature, viscous dissipation cannot be neglected. An extensive work on the viscous dissipative heat effects on the study free convection and on combined free and forced convection flows has been done by Ostrach [4-8]. Numerical analysis of steady non- Newtonian flows with heat transfer analysis, MHD and non- linear slip effects was examined by Ellahi and Hameed [9]. Ellahi et al. [0] explained the influence of slip on steady flows in viscous fluid with heat and mass transfer. Recently, Vajravelu[] investigated unsteady convective boundary layer flow of a viscous fluid at a vertical surface with different fluid properties. It may be noted that when the density of an electrically conducting fluid is low and /or applied magnetic field is strong (Sutton and Sherman[]), the effects of Hall current become significant. It plays an important role in determining flow features of the fluid flow problems because induces secondary flow in the fluid. Therefore it is of considerable interest to study the effects of Hall current on MHD fluid flow problems. Sato[], Sherman and Sutton[] have analyzed the Hall effects on the steady hydromagnetic flow between two parallel plates. These effects in the unsteady cases were reported by Pop[4]. Seth et al. [5] present the effects of Hall current on unsteady hydromagnetic natural convection transient flow of a viscous, incompressible, electrically conducting and heat absorbing fluid past an impulsively moving vertical plate fixed in a fluid saturated porous medium, under boussinesq approximation, taking into the effects of thermal diffusion when temperature of the plate has a temporarily ramped profile. Flow through a porous medium bounded by a vertical surface in presence of Hallcurrent was considered by Sudhakar[6]. Gopichand[7] analyzed the unsteady stretching surface in porous medium and explained the viscous dissipation and radiation effects on MHD flow over it. Another important aspect, which influences heat transfer processes is the suction /injection. It is well known that the effects of injection on the boundary layer flow area of interest in reducing the drag force. Suction and heat transfer characteristics were addressed by Youn[8]. The effects of suction or injection on the free convection boundary layers induced by a heated vertical plate fixed in a saturated porous medium with an exponential decaying heat generation were presented by Ali[9]. Suction or blowing of a fluid through the bounding surface can significantly change the flow field. In general, suction tends to increase the skin friction, whereas injection acts in the opposite manner. In many engineering activities such as in the design of thrust bearing and radial diffusers, and thermal oil recovery the process of suction/ blowing plays a significant role because of its importance. Bhattacharya [0] explained the effects of radiation and heat source/sink on unsteady MHD boundary layer flow and heat transfer over a shrinking sheet with suction /injection. The flow is permeated by an externally applied magnetic field normal to the plane of the flow in his work. The self similar equations corresponding to the velocity, temperature and concentration fields are obtained, and then solved numerically by finite difference method using quasi linearization technique. Lin et al. [] investigates study laminar boundary layer flow of power law fluids past a flat surface with suction or injection and magnetic effects. Recently, Cao et al. [] analyzed the MHD Maxwell fluid over a stretching plate with suction or injection in the presence of nano particles. Heat and mass transfer problems in the presence of chemical reaction are of importance in many processes, and have therefore received a considerable amount of attention in recent times. Possible applications can be found in processes such as drying, distribution of temperature and moisture over agricultural fields and groves of fruit trees, damage of crops due to freezing, evaporation at the surface of a water body and energy transfer in a wet cooling tower, and flow in a desert cooler. In many chemical engineering processes, chemical reactions take place between a foreign mass and the working fluid which moves due to the stretching of a surface. The order of the chemical reaction depends on several factors. One of the simplest chemical reaction is the first-order reaction in which the rate of reaction is directly proportional to the species concentration. Deka et al. [] reported the effect of first order homogeneous chemical reaction on the process of an unsteady flow over an infinite vertical plate with a constant heat and mass transfer. Muthucumaraswamy and Ganesan [4] studied the flow characteristics in an unsteady upward motion of an isothermal plate by taking chemical reaction and injection into account. Reddy et al. [5] analyzed the effects of radiation and chemical reaction on an unsteady hydromagnetic natural 8 Copyright 06. Vandana Publications. All Rights Reserved.

3 convection flow over a moving vertical plate in a porous medium. Owing to the above mentioned studies, the author made an attempt to investigate the combined effects of chemical reaction, thermal radiation and Hall current on the hydro magnetic free convective flow of heat and mass transfer over a moving inclined plate in a porous medium with suction and viscous dissipation. The governing boundary layer equations (4..) to (4..4) subject to the boundary conditions (4..5) are solved numerically by using Runge-Kutta fourth order method along with shooting technique. II. MATHEMATICAL ANALYSIS A two dimensional steady laminar MHD viscous incompressible electrically conducting and chemically reacting fluid along a moving inclined plate with an acute angle γ embedded in a porous medium, in the presence of suction is considered. x- direction is taken along the leading edge of the inclined plate and y is normal to it and extends parallel to x -axis. Let (> T ) be the uniform plate temperature, where T is the temperature of the fluid far away from the plate. Let u, v and w be the velocity components along the x and y axis and secondary velocity component along the z axis respectively in the boundary layer region. Let C be the concentration of the fluid at the surface of the plate and C be the free stream concentration. The flow is subjected to the effect of thermal radiation and a transverse magnetic field of strength B 0, which is assumed to be applied in the positive y direction, normal to the surface. The induced magnetic field is also assumed to be small compared to the applied magnetic field so it is neglected. All the fluid properties are assumed to be constant except for the density variations in the buoyancy force term of the linear momentum equation. The Hall effects and viscous dissipation are taken into account.joule heating term is neglected. The sketch of the physical configuration and coordinate system are shown in Fig. Figure Physical configuration and coordinate system w T w Under the above assumptions the boundary layer equations describing the flow field under consideration u x + are The boundary conditions for the velocity, temperature and concentration fields are u = ax, v = V, w = 0, T = T C = C w = C + bx w = T + ax, at y 0 u 0, w 0, T T, C C as y (6) where u, v and w be the velocity components along the x- axis and y- axis and secondary velocity component along the z - axis respectively in the boundary layer region. T and C are the temperature and concentration of the fluid respectively. g- the gravitational acceleration, and - the coefficients of thermal and T v y = 0 u u u u + v = υ + x y y σb gβ T 0 ( T T ) cos γ + gβ ( C C ) cos γ ( ) ( u + mw ρ + m ) k u c w w w σb0 + v = υ + mu w x y y ρ + m u T T T u + v = α + x y y υ ρc u y + c concentration expansions,γ - the acute angle or inclination parameter, B0 - the magnetic field induction, m- the hall parameter, - the kinematic viscosity, k - permeability of the porous medium, - thermal diffusivity, c - the specific heat at constant pressure, p w y _ ( ) ( ) w ρc q r y qr - the radiative heat flux, D- the mass diffusivity, k - chemical reaction rate, Tw and Cw - the temperature and concentration of the fluid at the surface of the plate, T - the temperature of the fluid far away from the plate and C - the free stream concentration. The second and third terms on the right hand side of equation (4) are the viscous dissipative heat and radiative heat flux respectively. The second term on right hand side of the equation (5) is the species chemical reaction. υ υ k p p (4) C C v x y C D k y u C C () () () (5) 9 Copyright 06. Vandana Publications. All Rights Reserved.

4 Continuity equation () is identically satisfied by the x, y, defined as stream function u, v (7) y x By using Rosseland approximation, the radiative heat flux q is given by q r r * 4 * k T y 4 (8) * * Where is the Stefan Boltzman constant and k is the mean absorption coefficient. It should be noted that by using Rosseland approximation, the present analysis is limited to optically thick fluids.if the temperature differences within the flow are sufficiently small then 4 equation (4..6)can be linearized by expanding T in a Taylor series about the free stream temperature T which after neglecting the higher order terms takes the form 4 4 T T T T (9) 4 R Pr f Ec f g 0 0 () Scf ScKr 0 (4) The corresponding boundary conditions are f F, 0 0, w g f,, at 0 f f g0 0 as (5) where prime ( ' ) denotes differentiation with respect to η. η - the similarity parameter, dimensionless stream function,, f is the - the dimensionless temperature, - the dimensionless concentration, ψ the stream function, M- the magnetic field parameter g0 - the secondary velocity parameter, G - the local thermal Grahsof number, G - the r local solutal parameter, γ Grahsof number, K- the permeability - the inclination parameter, m Hall current parameter, R- radiation parameter, Pr - the Prandtl number, Ec - the Eckert number, Sc - the Schmidt number, Kr - the chemical reaction parameter, F - the suction parameter. w c III. SOLUTION OF THE PROBLEM To transform equations () to (4) into a set of ordinary differential equations, the following similarity transformations and dimensionless variables are introduced Substituting the equations (7) to (0) into the equations () to (5) we obtain f + ff + Gr θcos γ + Gcφcos γ M mm f g kf 0 = 0 + m + m Mm g fg 0 m M f m 0 g 0 kg0 0 () () The governing boundary layer equations () to (4) subject to the boundary conditions (5) are solved numerically by using Runge-Kutta fourth order method along with shooting technique. First of all higher order non-linear differential equations () to (4) are converted into simultaneous linear differential equations of first order and they are further transformed into initial value problem by applying the shooting technique (Jain et al.[6]). The resultant initial value problem is solved by employing Runge-kutta fourth order technique. Numerical results are reported in figures for various values of the physical parameters of interest. From the process of numerical computation, the skinfriction coefficient, the Nusselt number and Sherwood number which are respectively proportional to 0 and 0 40 Copyright 06. Vandana Publications. All Rights Reserved. 0 f, are also sorted out and numerical values are presented in a tabular form. IV. RESULTS AND DISCUSSION As a result of the numerical calculations, the dimensionless velocity, temperature and concentration are obtained and their behaviour have been discussed for variations in governing parameters viz., M- the magnetic field parameter g0 - the secondary velocity parameter, G r - the local thermal Grahsof number,

5 Gc - the local solutal Grahsof number, K- the permeability parameter, γ - the inclination parameter, m Hall current parameter, R- radiation parameter,, Pr - the Prandtl number, Ec - the Eckert number, Sc - the Schmidt number, Kr - the chemical reaction parameter, F - the suction parameter. The results are presented in w Figures from Numerical results for the skin friction, Nusselt number and Sherwood number are reported in Tables and. A parametric study is carried out to demonstrate the effects of governing parameters on velocity, temperature and concentration profiles. Fig. and Fig. show the effects of thermal Grashof number Gr and solutal Grash of number Gc on the velocity respectively. As shown the velocity increases as Gr and Gc increases. Physically Gr > 0 means heating of the fluid or cooling of the boundary surface, Gr < 0 means cooling of the fluid or heating of the boundary surface and Gr = 0 corresponds to the absence of free convection current. The effect of inclination parameter on the velocity of the fluid is shown in Fig. 4. It is noticed that increasing the inclination parameter results a decrease in the velocity. Fig. 5 displays the effect of magnetic field paramater on the velocity of the fluid. The presence of a magnetic field in an electrically conducting fluid induces a force called Lorentz force, which opposes the flow. This resistive force tends to slow down the flow, so the effect of increase in M is to decrease the velocity. Fig. 6 illustrates the effect of Hall parameter m on the velocity. It is observed that the velocity of fluid increases on increasing the Hall parameter. Fig.7 shows the effect of the thermal conductivity on the velocity of the fluid. It is seen that velocity decreases on increasing the thermal conductivity. Fig. 8 represents the effect of radiation parameter on the velocity, and it is noticed that the effect of radiation parameter on the velocity of the fluid is slight. The effect of the Prandtl number on the velocity of the fluid is illustrated in Fig. 9. On increasing the Prandtl number, the velocity of the fluid flow increases. Fig. 0 depicts the effect of Eckert number on the velocity of the boundary layer. A slight change in the velocity is seen. The effect of Schmidt number on the velocity of the fluid is shown in Fig.. A slight decrease in the velocity of the fluid on increasing the Schimdt number is noticed. The effect of chemical reaction parameter on the velocity of the fluid flow is illustrated in Fig.. It is found that on increasing the chemical reaction parameter the velocity of the fluid is decreasing. Fig. shows the effect of suction parameter on the velocity. It is observed that the velocity increases on increasing the suction parameter. The effect of magnetic parameter M on the secondary velocity of the fluid is shown in Fig. 4. Increase in the secondary velocity of the fluid is observed on increasing the magnetic parameter. Fig. 5 shows the effect of radiation parameter on secondary velocity of the fluid. On increasing the radiation parameter secondary velocity of the fluid is found to be decreased. The effect of chemical reaction parameter on the secondary velocity of the fluid is shown in Fig. 6. Decrease in the secondary velocity is noticed from the figure on increasing the chemical reaction parameter. Fig. 7 and Fig. 8 illustrates the effects of thermal and mass Grashofer numbers Gr and Gc respectively on the temperature of the fluid. Decrease in the temperature is noticed. The effect of the Magnetic field parameter is shown in Fig. 9, increase in the temperature of the fluid is observed. The effects of thermal conductivity on temperature of the fluid is depicted in the Fig. 0. It is shown from the figure that temperature increases on increasing the thermal conductivity. The effects of radiation on temperature of the fluid is illustrated in Fig.. Decrease in the temperature of the fluid on increasing the radiation is observed. Effects of Prandtl number on temperature of the fluid is shown in Fig.. Increase in the temperature is noticed. Hall parameter decreases the temperature of the fluid as shown in Fig.. Effects of Eckert number on thermal boundary layer is illustrated in Fig. 4. Temperature of the fluid in the boundary layer increases on increasing the Eckert number. Chemical reaction effect on the thermal boundary layer is depicted in Fig. 5. Thermal boundary layer thickness increases on increasing the chemical reaction. Thermal boundary layer thickness is increased on increasing the inclination parameter γ as shown in Fig. 6. The effect of suction parameter on the temperature is depicted in Fig. 7. It is seen that temperature increases on increasing the suction parameter. Fig. 8 and Fig. 9 illustrate the effects of thermal and mass Grashof numbers Gr and Gc on the species concentration field. Concentration of the fluid in the boundary layer decreases on increasing Gr and Gc. Concentration of the fluid in the boundary layer increases on increasing the magnetic parameter M as shown in Fig. 0. The effect of hall parameter m on concentration field was displayed in Fig.. A decrease in the concentration is noticed on increasing the hall parameter. Fig. depicts the effect of thermal conductivity on the concentration field. Concentration of the fluid increases on increasing the thermal conductivity. The effect of Prandtl number on concentration field is illustrated in Fig.. Effect of Eckert number on concentration field was displayed in Fig. 4. Decrease in the concentration on increasing the Eckert number is noticed. Fig. 5 represents the effect of Schmidt number on concentration field. It is interesting to note that the chemical species concentration also decreases within the boundary layer with an increase in Schmidt number due to the combined effects of buoyancy forces and species molecular diffusivity. Fig. 6 depicts the influence of chemical reaction rate on concentration field. An increase in the value of chemical reaction parameter decreases the concentration of species in the boundary layer. This is due to the fact that chemical reaction in this system results in consumption of the chemical and hence results in decrease of concentration. Fig.7.demonstrates the effect of 4 Copyright 06. Vandana Publications. All Rights Reserved.

6 inclination parameter on the concentration.there is a slight change in the concentration of the fluid is observed on increasing the inclination parameter γ. Fig. 8 shows the effect of suction parameter on the concentration. It is found that concentration increases on increasing the suction. Table. I shows the comparison of results of present work with that of Ali et al.[7], and it is found that there is a good agreement. Numerical computations of skin friction coefficient, Nusselt number and Sherwood number for different values of Gr, Gc, M, m, K, Ec, Sc, γ, Fw, R and Kr =0 are reported in Table II and Table. Increasing inclination parameter decreases the velocity and increases the temperature and there is a slight change in the concentration. Secondary velocity increases on increasing the magnetic parameter, but reduces o increasing the radiation and chemical reaction parameters. Skin fraction coefficient and Nusselt number decreases where as sherewood number increases on increasing chemical reaction parameter. Skin friction coefficient and Sherwood number decreases, whereas Nusselt number increases with an increase in the radiation parameter. V. CONCLUSIONS A two dimensional steady laminar MHD viscous incompressible electrically conducting and chemically reacting fluid along a moving inclined plate with an acute angle γ embedded in a porous medium, in the presence of suction has been studied. In addition to this, thermal radition and external magnetic field strength are also considered. The governing boundary layer equations are solved numerically using well tested, highly efficient Runge-Kutta fourth order method along with shooting technique. From the present study we arrive at the following significant observations. On comparing the present results with previous work, it is found that there is a good agreement. Increasing buoyancy ratio parameters Gr and Gc increases the velocity, but decreases the temperature and concentration. Increasing the magnetic field parameter increases the temperature and the concentration, but reduces the velocity. Increasing the hall parameter enhances the velocity, but reduces the temperature and concentration. Increasing the thermal conductivity parameter rises the temperature and concentration, but decreases the velocity. Increasing the radiation parameter results a slight change in the velocity, but reduces the temperature. Increasing the prandtl number increases the velocity, temperature as well as the concentration. Increasing the Eckert number results a slight change in the velocity, increases the temperature, but decreases concentration. Increasing the Schmidt number decreases the velocity and the concentration Increasing the suction parameter results an increase in the velocity, temperature and the concentration. Increasing the chemical reaction parameter increases the temperature, but reduces the velocity and concentration Gc=0., M =,m = 0.,k =, R=, Pr =, Ec =, Sc = 0., Kr =, Gr = 0.,,, Figure : Velocity profiles for different Gr Gr= 0., M =,m = 0.,k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0 Gc = 0.,,, Figure : Velocity profiles for different Gc Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., Kr = Figure 4: Velocity profiles for different γ. 4 Copyright 06. Vandana Publications. All Rights Reserved.

7 .0 Gr= 0., Gc=0., m = 0.,k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0.0 Gr= 0., Gc=0., M =, m = 0.,k =, R=, Ec =, Sc = 0., Kr =, = 0 M =,,, 4 Pr =,.7,.7, Figure 5 :Velocity profiles for different M. 0 4 Figure 9: Velocity profiles for different Pr..0 Gr= 0., Gc=0., M =,k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0.0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =,Sc = 0., Kr =, = 0 0. m = 0.,,, 0. Ec=,,5,7 0 4 Figure 6 : Velocity profiles for different m. 0 4 Figure 0: Velocity profiles for different Ec..0 Gr= 0., Gc=0., M =,m = 0., R=, Pr =, Ec =, Sc = 0., Kr =, = 0.0 G r= 0., Gc =0., M =,m = 0., k =, R=, Pr =, Ec =, Kr =, = 0 0. k=,,,4 0 4 Figure 7: Velocity profiles for different k. 0. Sc= 0.,,.0, Figure : Velocity profiles for different Sc..0 Gr= 0., Gc=0., M =,m = 0., k =,Pr =, Ec =, Sc = 0., Kr =, = 0.0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., = 0 R =,,, Kr =,,,4 0 4 Figure 8: Velocity profiles for different R. 0 4 Figure : Velocity profiles for different Kr. 4 Copyright 06. Vandana Publications. All Rights Reserved.

8 .0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0.0 Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0 Fw = - 0.5, 0.5, Gr = 0.,,, 0 4 Figure : Velocity profiles for different Fw. 0 4 Figure 7: Temperature profiles for different Gr. 0 M= 0.5,,.5,.0 Gr= 0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0 08 g Gc = 0.,,, g Figure 4: Secondaryvelocity profiles for different M. R= 0., 4,8,0 Gr= 0., M =,m = 0., k =,Pr =, Ec =, Sc = 0., Kr =, = Figure 8: Temperature profiles for different Gc.0 0. Gr= 0., Gc=0.,m = 0., k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0 M =,,, Figure9: Temperature profiles for different M. g Figure 5: Secondaryvelocity profiles for different R. Kr=,,,0 Gr= 0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., = 0.0 Gr= 0., Gc=0., M =,m = 0., R=, Pr =, Ec =, Sc = 0., Kr =, = 0 k=,,, Figure 6: Secondary velocity profiles for different Kr. 0 4 Figure 0: Temperature profiles for different k. 44 Copyright 06. Vandana Publications. All Rights Reserved.

9 .0 Gr= 0., Gc=0., M =,m = 0., k =, Pr =, Ec =, Sc = 0., Kr =, = 0.0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., = 0 Kr= 00,,,00 R =, 5, 0, Figure : Temperature profiles for different R..0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Ec =, Sc = 0., Kr =, = Figure 5: Temperature profiles for different Kr..0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., Kr =, Pr =,.7,.7, Figure : Temperature profiles for different Pr m= 0., 0, 0, 0 Gr= 0., Gc=0., M =,k =, R=, Pr =, Ec =, Sc = 0., Kr =, = Figure 6: Temperature profiles for different γ..0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0 Fw=-,- 0.5, 0.5, 0 4 Figure : Temperature profiles for different m Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Sc = 0., Kr =, = 0 Ec=,4,6,8 Figure 7: Temperature profiles for different Fw..0 Gc=0., M =,m = 0.,k =, R=, Pr =, Ec =, Sc = 0., Kr =, = Gr=,,5,7 0 4 Figure 4: Temperature profiles for different Ec. 0 4 Figure 8: Concentration profiles for different Gr. 45 Copyright 06. Vandana Publications. All Rights Reserved.

10 .0 Gr= 0., M =,m = 0.,k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0.0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Sc = 0., Kr =, = 0 Ec=,50 0. Gc =,,5, Figure 9: Concentration profiles for different Gc. 0 4 Figure 4: Concentration profiles for different Ec..0 Gr= 0., Gc=0., m = 0.,k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0.0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Kr =, = 0 M= 0.,, 5, Sc= 0.,.0, Figure 0: Concentration profiles for different M..0 Gr= 0., Gc=0., M =,k =, R=, Pr =, Ec =, Sc = 0., Kr =, = Figure 5: Concentration profiles for different Sc..0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., = 0 0. m= 0,90 0. Kr=,5,0,5 0 4 Figure : Concentration profiles for different m. 0 4 Figure 6: Concentration profiles for different Kr..0 Gr= 0., Gc=0., M =,m = 0., R=, Pr =, Ec =, Sc = 0., Kr =, = 0.0 Gr= 0., Gc=0., M =,m = 0., k =, R=, Pr =, Ec =, Sc = 0., Kr =, k= 0.,, 5, Figure : Concentration profiles for different k. 0 4 Figure 7: Concentration profiles for different γ...0 Gr= 0., Gc=0., M =,m = 0.,.0 Pr =,,.5, k =, R=, Pr =, Ec =, Sc = 0., Kr =, = 0 Fw= -, -0.5,0.5, 0. Gr=0.,Gc=0.,M=, = 0., Pr=,R=0.5,Ec=,Sc=0., N = 0., = 45,Kr = Figure : Concentration profiles for different Pr Figure 8: Concentration profiles for different Fw. 46 Copyright 06. Vandana Publications. All Rights Reserved.

11 TABLE I Computations showing comparision of present results for f 0 and - θ ( 0) at the plate with Gr,Gc, M, m, K, Ec, Sc, γ and Fw for, R=0, Kr =0.with that of Ali[7]. Gr Gc M m K Pr Ec Sc γ de g Fw 0. Present work Ali [7] 0 f - ( 0) θ f - θ ( 0) TABLE II Variation of f 0, 0, 0 for different Gr, Gc, M, m, k, R with P r =,Ec =, Sc = 0., Kr =, γ = 0 and Fw =. Gr Gc M m k R f Copyright 06. Vandana Publications. All Rights Reserved.

12 TABLE III Variation of f 0, 0 and 0 for different Pr, Ec, Sc, kr, γ, fw with Gr = 0., Gc =0.,M =,m = 0.,k = and R=. Pr Ec Sc Kr γ fw f Copyright 06. Vandana Publications. All Rights Reserved.

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