MHD free convection heat and mass transfer flow over a vertical porous plate in a rotating system with hall current, heat source and suction

Int. J. Adv. Appl. Math. and Mech. 5(4) (2018) 49 64 (ISSN: 2347-2529) IJAAMM Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics MHD free convection heat and mass transfer flow over a vertical porous plate in a rotating system with hall current, heat source and suction Research Article Laisa Mahtarin Iva a, Mohammad Sanjeed Hasan b, Sanjit Kumar Paul c, Rabindra Nath Mondal a, a Department of Mathematics, Jagannath University, Dhaka-1100, Bangladesh b Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Gopalganj, Bangladesh c Department of Basic Sciences & Humanities, University of Asia Pacific, Dhaka, Bangladesh Received 19 May 2018; accepted (in revised version) 20 August 2018 Abstract: MSC: Unsteady free convection heat and mass transfer flow past a semi-infinite vertical porous plate has been studied numerically by using an explicit finite difference method. First, thermal boundary layer equations have been derived from the Navier-Stokes equation and concentration equation by boundary layer technique, and the equations have been made non-dimensionalised by using non-dimensional variables. The equations are then solved by using explicit finite difference method. The solution of heat and mass transfer flow is studied to examine the velocity, temperature and concentration distribution and it is found that the solution is dependent on several governing parameters, including the magnetic parameter, heat source parameter, Grashof number, modified Grashof number, hall parameter, Prandtl number, Schmidt number, wall temperature and concentration exponent. The effects on the velocity, temperature and concentration profiles for various parameters entering into the problem are separately discussed and shown graphically. 76Dxx 76Sxx Keywords: Boundary layer Heat transfer Finite difference Unsteady solutions 2018 The Author(s). This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/3.0/). 1. Introduction The boundary layer plays a vital role in many aspects of fluid mechanics because they describe the motion of a viscous fluid close to the surface. The boundary layer equations are especially interesting from a physical point of view because they have the capacity to admit a large number of invariant solutions i.e. basically closed-form solutions. In the present context, invariant solutions are meant to be a reduction to a simpler equation such as an ordinary differential equation. Prandtl s boundary layer equations admit more and different symmetry groups or simply symmetries are invariant transformations which do not alter the structural form of the equation under investigation. The problem of heat transfer in the boundary layer on a continuous moving surface has many practical applications in manufacturing processes in industry. Mechanics of non-linear fluids present a special challenge to engineers, physicists and the mathematician. The non-linearity can manifest itself into a variety of ways. The heat transfer in boundary layer flow over a flat plate is very much important in practical point of view. In recent years, considerable attention has been given to study the heat transfer characteristics in boundary layer flow of a Newtonian fluid past a flat plate because of its extensive application in production engineering. Sattar and Alam [1] presented unsteady free convection and Corresponding author. E-mail address(es): rnmondal71@yahoo.com (Rabindra Nath Mondal). 49

50 MHD free convection heat and mass transfer flow over a vertical porous plate... Nomenclature Gr Grashof number Gm Modified Grashof number Sc Schmidt number B 0 Magnetic field M Magnetic parameter R Rotation Pr Prandtl number Re Reynolds number S Suction parameter t Time parameter J Current density Nu Nusselt number L Length of boundary layer u, v Velocity components in the x- and y-direction respectively Greek symbols α Heat source parameter τ Shear stress Differential operator δ Boundary layer thickness µ Coefficient of viscosity υ Coefficient of kinematic viscosity ρ Density of the fluid in the boundary layer Subscripts w Condition at wall Condition at at infinity mass transfer flow of a viscous, incompressible and electrically conduction fluid past a moving infinite vertical porous plate with the effect of thermal diffusion. Alam et al. [2] studied numerical simulation of the combined free and forced convection and mass transfer flow past a vertical porous plate in a porous medium with heat generation and thermal diffusion. Das et al. [3] investigated the effects of mass transfer flow past an impulsively started infinite vertical plate with constant heat flux and in the presence of chemical reaction. Anjali devi and Uma devi [4] applied Runge-Kutta based shooting method to study radiation effect on steady laminar hydromagnetic flow of a viscous, Newtonian and electrically conducting fluid past a porous rotating disk taking into account the Hall current. Magneto-hydrodynamics (MHD) flow problems have become considerable attention to the researchers because of its significant applications in industrial manufacturing processes such as plasma studies, petroleum industries, MHD power generator cooling of clear reactors, boundary layer control in aerodynamics and crystal growth. Many authors have studied the effects of magnetic field on mixed, natural and force convection heat and mass transfer problems. Indeed, MHD laminar boundary layer behavior over a semi-infinite vertical plate is a significant type of flow having considerable practical applications in chemical engineering, electrochemistry and polymer processing. This problem has also an important bearing on metallurgy where MHD techniques have recently been used. Considering the aspects of rotational flows, model studies were carried out on MHD free convection and mass transfer flows in a rotating system by many investigators of whom Debnath et al. [5], Raptis and Perdikis [6] are worth mentioning. Hossain [7] studied the effects of viscous and Joule heating on the flow of viscous incompressible fluid past a semi-infinite plate in presence of a uniform transverse magnetic field. The combined effects of forced and natural convection heat transfer in the presence of a transverse magnetic field from vertical surfaces are also studied by many researchers. In recent years, considerable attention has been given to study the heat transfer characteristics in boundary layer flow of a Newtonian fluid past a flat plate because of its extensive application in production engineering. Chen [8] investigated the momentum, heat and mass transfer characteristics of MHD natural convection flow over a permeable, inclined surface with variable wall temperature and concentration, taking into consideration the effects of ohmic heating and viscous dissipation. Abo-Eldahab and Elbarbary [9] studied the hall current effects on MHD free-convection flow past a semi-infinite vertical plate with mass transfer. Chowdhury and Islam [10] presented a theoretical analysis of a MHD free convection

Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 5(4) (2018) 49 64 51 flow of a viscoelastic fluid adjacent to a vertical porous plate. Pal and Talukdar [11] analysed the combined effects of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid past a vertical permeable surface embedded in a porous medium. Hayat et al. [12] analysed a mathematical model in order to study the heat and mass transfer characteristics in mixed convection boundary layer flow about a linearly stretching vertical surface in a porous medium filled with a viscoelastic fluid, by taking into account the Dufour and Soret effects. Alam et al. [13] applied similarity transformations to investigate heat and mass transfer characteristics of MHD free convection of steady flow of an incompressible electrically conducting fluid over an inclined plate under the influence of an applied uniform magnetic field taking into account the effects of Hall current. Finite difference method is an explicit numerical method used mostly in numerical analysis, especially in numerical differential equations, which aim at the numerical solution of ordinary, partial differential and thermal boundary layer equations respectively. The idea is to replace the derivatives appearing in the differential equation by finite differences that approximate them. Common applications of the finite difference method are in computational science and engineering disciplines, such as thermal engineering, fluid mechanics, etc. The unsteady solution of thermal boundary layer equation is one of the most interesting choices to the researcher by using finite difference method. Patankar and Spalding [14] performed a finite-difference procedure for solving the equations of the two-dimensional boundary layer flow. A general, implicit, numerical, marching procedure is presented for the solution of parabolic partial differential equations, with special reference to those of the boundary layer. The main novelty lies in the choice of a grid which adjusts its width so as to conform the thickness of the layer in which significant property gradients are present. The non-dimensional stream function is employed as the independent variable across the layer. In the present study, finite difference scheme is used to find out the solution of heat and mass transfer flow past a vertical plate, and examine the velocity, temperature and concentration distribution characteristics. Chamkha and Khaled [15] investigated the problem of coupled heat and mass transfer by hydromagnetic free convection from an inclined plate in the presence of internal heat generation or absorption, and similarity solutions were presented. Alam et al. [16]analyzed 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. Vasu et al. [17] investigated radiation and mass transfer effects on transient free convection flow of a dissipative fluid past an semi-infinite vertical plate with uniform heat and mass flux. Seth et al. [18] studied unsteady hydromagnetic couette flow of a viscous incompressible electrically conducting fluid in a rotating system in the presence of an inclined magnetic field taking Hall current into account. They showed that hall current and rotation tend to accelerate fluid velocity in both the primary and secondary flow directions while Magnetic field has a retarding influence on the fluid velocity in both the primary and secondary flow directions. Hasanuzzaman et al. [19] studied the similarity solution of unsteady combined free and force convective laminar boundary layer flow about a vertical porous surface with suction and blowing. Alam et al. [20] applied shooting method along with sixth order Runge-Kutta integration scheme to solve an unsteady two-dimensional MHD free convective and heat transfer flow of a viscous incompressible fluid along a vertical porous flat plate with internal heat generation/absorption. They showed that the velocity profile and the temperature profiles are much pronounced with the increasing value of heat generation but the opposite effect was observed for suction parameter and Prandlt number. Recently, Sheri et al [21] applied finite element method to study transient MHD free convection flow past a vertical porous plate in the presence of viscous dissipation. Very recently, Pandya et al. [22] investigated effects of Soret-Dufour, radiation and chemical reaction on an unsteady MHD flow of an incompressible viscous and electrically conducting dusty fluid past a continuously moving inclined plate by using implicit finite difference method. In this paper, our objective is to study the roll of magnetic field on MHD fluid flow through a semi-infinite rotating vertical porous plate with hall current and heat source. The investigation has been made for solving the system of nonlinear partial differential equations. For this purpose an explicit finite difference technique has been used for which non-similar solutions of the coupled non-linear partial differential equations are sought. Therefore, it is necessary to investigate, in detail, the distributions of primary velocity, secondary velocity and temperature distributions across the boundary layer in addition to the suction parameter and hall current. 2. Mathematical model and governing equations By introducing Cartesian coordinate system, the X-axis is chosen along the plate in the direction of the flow and the Y-axis normal to it. Initially we consider that the plate as well as the fluid is at the same temperature and the concentration level everywhere in the fluid is same. Also it is considered that the fluid and the plate is at rest after that the plate is to be moving with a constant velocity in its own plane and instantaneously at time t > 0, the species concentration and the temperature of the plate are raised to and respectively, which are thereafter maintained constant, where, are species concentration and temperature at the wall of the plate and, are the concentration and temperature of the species far away from the plate respectively. The physical model of the study is shown in Fig. 1. Within the framework of the above stated assumptions with reference to the generalized equations described before the equation relevant to the transient two dimensional problems are governed by the following system of coupled non-linear differential equations.

52 MHD free convection heat and mass transfer flow over a vertical porous plate... Fig. 1. The physical model and coordinate system Continuity equation: u x + v y = 0 (1) Momentum equation: u t + u u x + v v y = υ 2 u y 2 + g β(t T ) + g β (C C ) υ k u σb 2 0 ρ(1 + m 2 (u + mw) (2) ) w t + u w x + v w y = υ 2 w y 2 + σb 2 0 ρ(1 + m 2 ) (mu w) υ k w (3) Energy equation: T t + u T x + v T y = K 2 T ρc p y 2 +Q (T T ) (4) Concentration equation: C t + u C x + v C y = D 2 C y 2 (5) with the corresponding initial and boundary conditions: At t = 0 u = 0, v = 0, T T C C everywhere (6) t > 0, u = 0, v = 0, T T, C C at x = 0 u = 0, v = 0, T T w, C C w at y = 0 u = 0, v = 0, T T w, C C w as y (7) where x, y are the Cartesian coordinate system, u, v are the velocity components along the x- and y-directions respectively, T and C are, respectively the temperature and concentration of the fluid, T and C are respectively the temperature and concentration of the fluid in the static flow far away from the boundary. In this formulations, g is the local acceleration due to gravity; υ is the kinematic viscosity, α is the thermal conductivity, k is the heat diffusivity temperature,β is the coefficient volume for thermal expansion,β is the coefficient volume expansion for concentration, D is the coefficient of mass diffusivity,

3. Mathematical formulation Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 5(4) (2018) 49 64 53 Since the solutions of the governing Eqs. (1)-(5) under the initial condition (6) and boundary conditions (7) will be based on a finite difference method, it is required to make the said equations dimensionless. For this purpose, we now introduce the following dimensionless variables: y = yu 0 υ, U = u U 0, W = w U 0, τ = tu 2 0 υ, T = T T T w T, C = C C C w C, S = v 0 U 0 Using this relation we have the following derivatives, (suction parameter)u u t = U 0 3 U v τ, w t = U 0 3 W v τ, u y = U 0 2 U v y, T t = U 0 2 (T w T ) T v τ, C t = U 0 2 (C w C ) C v τ, w y = U 0 2 W v y, 2 u y 2 = U 3 0 v 2 2 U y 2 2 w y 2 = U 3 0 v 2 2 W y 2 T y = U 0 (T w T ) T v y C y = U 0 (C w C ) C v y 2 T y 2 = U 0 2 (T w T ) 2 T v 2 y 2 2 C y 2 = U 0 2 (C w C ) 2 C v 2 y 2 Now substituting the above derivatives into the Eqs. (1)-(5) and simplifying we obtain the following nonlinear coupled partial differential equations in terms of dimensionless variables. U τ = G r T +G m C + 2 U y 2 + S U y W τ = 2 W y 2 + S W y + T τ = 1 2 T P r y 2 + S T y α T C τ = 1 2 C S c y 2 + S C y M (mu W ) 1 + m2 M (U + mw ) 1 + m2 where, Gr = g β(t w T )υ U 3 is the Grashof number, Gm = g β(c w C )υ 0 U 3 is the modified Grashof number, M = σb 0 2υ 0 U0 2ρ is the magnetic parameter, Pr = υρc p K is the Prandlt number, S c = υ D parameter and m = Hall current. T = 1, C = 1, U = 1, y = 0, W = 0, at y = 0 t 0, T = 0, C = 0, U = 0, y = 0, W, at y is the Schmidt number, α = Qυ U 2 0 is the heat source 4. Numerical calculations In this section, we attempt to solve the governing second order nonlinear coupled dimensionless partial differential equations with the associated initial and boundary conditions. From the concept of the above discussion, for simplicity the explicit finite difference method has been used to solve the Eqs. (1)-(5) subject to the conditions given by (6) and (7). To obtain the difference equations the region of the flow is divided into a grid of lines parallel to X and Y axes, where X -axis is taken along the plate and Y - axis is normal to the plate. Here we consider the plate of height X max (= 200) i.e. X varies from 0 to 200 and regard Y max (= 20) as corresponding to Y i.e. Y varies from 0 to 20. Again, we consider the plate of height X max (= 100) i.e. X varies from 0 to 100 and regard Y max (= 20) for P r = 0.71 as corresponding to Y i.e. Y varies from 0 to 20. We consider that m = 400,n = 400 in the X and Y axes grid spaces for Pr = 1.0and Pr = 7.0. For Pr = 0.71, we have taken m = 200 and n = 200 in the X - and Y -axes grid spaces as shown in Fig. 2. It is assumed that X, are the constant mesh sizes along the X - and Y -directions respectively, and are taken as X = 0.5 (0 x 200) and = 0.05 (0 y 20). However, in the case of Pr = 0.71, X and are taken as

54 MHD free convection heat and mass transfer flow over a vertical porous plate... Fig. 2. The finite difference space grid Fig. 3. Primary velocity profile U due to change of magnetic parameter M for fixed hall current m, m = 0.40 m = 0.60 X = 0.5 (0 x 100) and = 0.1 (0 y 20) with the smaller time step τ = 0.001. Now using the explicit finite difference approximation we get, ( ) U = U ( ) U U, = U U i 1,j, (8) τ τ X X ( ) U Y = U +1 U, ( ) V Y = V V 1, (9) ( ) T τ = T T τ ; ( ) T X = T T i 1,j ; X ( ) T Y = T +1 T (10) ( ) C τ = C C ; τ ( ) C X = C C i 1,j ; X ( ) C Y = C +1 C (11)

Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 5(4) (2018) 49 64 55 Fig. 4. Primary velocity profile due to change of magnetic parameter M and hall current m, m = 0.80 M = 0.40 Fig. 5. Primary velocity profile due to change of hall current m for fixed magnetic parameter M, M = 0.60 M = 0.80 ( 2 ) U Y 2 = U +1 2U +U 1 ( ) 2 ; ( 2 ) T Y 2 = T +1 2 T + T 1 ( ) 2 (12) ( 2 ) C Y 2 = C +1 2 C + C 1 ( ) 2 (13) Substituting the above relations into the corresponding differential equations we obtain an appropriate set of finite difference equations as follows, U U i 1,j X + V V 1 = 0 (14) U U τ +U U U i 1,j X +V U +1 U = U +1 2U +U 1 ( ) 2 +Gr T +Gm C (15)

56 MHD free convection heat and mass transfer flow over a vertical porous plate... Fig. 6. Primary velocity profile due to change of Grashof number Gr, Modified Grashof number Gm. Fig. 7. Primary velocity profile due to change of Grashof number Gr Modified Grashof number Gm. [ ] Ui U,j +1 2U +U 1 U U i 1,j U +1 U = ( ) 2 +Gr T +Gm C U V τ +U (16) X T T τ +U T T i 1,j X +V T +1 T = 1 Pr T +1 2 T + T 1 ( ) 2 α T (17) C C τ +U C C i 1,j X +V C +1 C = 1 S c C +1 2 C + C 1 ( ) 2 (18) [ C = 1 Sc ] C +1 2 C + C 1 C C i 1,j C +1 C ( ) 2 U V τ + C (19) X

Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 5(4) (2018) 49 64 57 Fig. 8. Primary velocity profile due to change of Heat source parameter α Suction parameter S. Fig. 9. Primary velocity profile due to change of Schmidt number Sc. The initial and boundary conditions with the finite difference scheme are U 0 = 0, V 0 = 0, T 0 = 0, C 0 = 0 U n 0.j = 0, V n 0,j = 0, T n 0,j = 0, C n 0,j = 0 U n i,0 = 0, V n i,0 = 0, T n i,0 = 1, C n i,0 = 1 U n i,l = 0, V n i,l = 0, T n i,l = 0, C n i,l = 0 (20) Here the subscripts i and j designate the grid points with x- and y-coordinates respectively, and the superscript n represents a value of time, τ = n τ, where n = 0,1,2,3.... From the initial condition (20), the values of U, T, C is known at τ = 0. During any one time step, the coefficients U and V appearing in Eqs. (14)-(19) are treated as constants. Then at the end of any time-step τ, the temperature T, the concentration C, the new velocity U, the new induced field V at all interior nodal points may be obtained by successive applications of Eqs. (14)-(19). This process is repeated in time and provided that the time-step is sufficiently small, U, V, T, C should eventually converge to values which approximate the steady-state solution ofeqs. (14)-(19). These converged solutions are shown graphically in the following figures.

58 MHD free convection heat and mass transfer flow over a vertical porous plate... Fig. 10. Secondary velocity profile due to change of magnetic parameter M for fixed hall current m, m = 0.40 m = 0.60 Fig. 11. Secondary velocity profile due to change of Magnetic parameter M and Hall current, m = 0.80 M = 0.40 5. Results and discussion For the Purpose of discussing the results of the problem, the approximate solutions are obtained for various parameters. In order to analyze the physical situation of the model, we have computed the steady-state numerical values of the non-dimensional Primary velocity U, Secondary velocity W, Temperature T and Concentration C within the boundary layer for different values of Magnetic parameter M, Hall current m, heat source parameter, Grashof number Gr, Prandtl number Pr, Scimidt number Sc with the fixed value of the modified Grashof number Gm. To get the steady-state solution, the computations have been carried out up-to τ.we observed that the results of the computations, however, changes rapidly after τ. Along with the obtained steady-state solutions, the flow behaviors in case of cooling problems are discussed graphically. The profiles of the transient primary velocity, transient secondary velocity, temperature and concentration versus Y are illustrated in Figs. 3-9 The transient primary velocity profiles have been shown in Fig. 3-9. In Figs. 3-4, we show the effects on primary velocity profile U due to the change of Magnetic parameter M for fixed Hall current m. The effects of Hall current m and the Magnetic parameter M is represented in Fig. 5. We see that primary velocity U increases rapidly with the increase of Hall current. With the increase of Grashof number Gr and modified Grashof number Gm, primary

Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 5(4) (2018) 49 64 59 Fig. 12. Secondary velocity profile due to change of Hall current m for fixed Magnetic parameter M, M = 0.60 M = 0.80 Fig. 13. Secondary velocity profile due to change of Grashof number Gr Modified Grashof number Gm. velocity U increases which is presented in Fig. 6-7. As seen in Fig. 8, the primary velocity U decreases as the Heat source parameter increases. Fig. 8 shows the effects of suction parameter S on primary velocity U, where we see that increasing the suction parameter S makes an increase to the primary velocity U rapidly, while with the decrease of Schmidt number Sc the primary velocity U decreases slowly, which is presented in Fig. 9-9. Hence it is concluded that the maximum primary velocity occurs in the vicinity of the plate. The transient secondary velocity profiles have been shown in Fig. 10-16. Figs. 10-11, we see that secondary velocity profile w has a substantial change due to change of magnetic parameter M for fixed hall current m. The effects of hall current m and the magnetic parameter M is presented in Fig. 12. As seen in Fig. 13-14, the with the increase of Grashof number Gr and Modified Grashof number Gm the secondary velocity profile w also increases. From Fig. 15, we see that secondary velocity w is decreasing when heat source parameter increases. Figure Fig. 15 represents the effects of Suction parameter S on secondary velocity W. Increasing of the Suction parameter S makes an increase to the secondary velocity W rapidly. As the Schmidt number decreases the secondary velocity W decreases slowly, which are represented in Fig. 16-16. Hence it is concluded that the maximum secondary velocity occurs in the vicinity of the plate.

60 MHD free convection heat and mass transfer flow over a vertical porous plate... Fig. 14. Secondary velocity profile due to change of Grashof number Gr Modified Grashof number Gm. Fig. 15. Secondary velocity profile due to change of Heat source parameter α Prandtl number Pr The temperature profiles have been shown in Fig. 17-18. From Fig. Fig. 17, we see that temperature T decreases with the increase of the Suction parameter S. The effects of the Heat source parameter on the temperature profile T is presented in Fig. 17-18. From these figures, it is observed that increasing the Heat source parameter decreases the temperature. From Fig. 18, we see that temperature decreases with the increase of the Prandtl number Pr. Next, the concentration profiles have been shown in Fig. 19-20. From Fig. 19-19, we see that concentration C decreases with the increase of suction parameter S. In Fig. 20-20, we see that the concentration profiles increase with the decrease of Schmidt number Sc. However, concentration C becomes zero as Y for all parameters. 6. Concluding remarks The present paper investigates transient heat and mass transfer flow by natural convection of an electrically conducting incompressible viscous fluid past an electrically non-conducting continuously moving semi-infinite vertical porous plate under the action of strong magnetic field taking into account the induced magnetic field with constant

Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 5(4) (2018) 49 64 61 Fig. 16. Secondary velocity profile due to change of Schimdt number Sc. Fig. 17. Temperature profile due to change of Suction parameter S Heat source parameter α. heat and mass fluxes. The resulting governing system of dimensionless coupled non-linear partial differential equations are solved numerically by using an explicit finite difference method. The results are discussed for different values of important parameters as Grashof number, Modified Grashof number, magnetic parameter, hall parameter, heat source parameter, Prandtl number and the Schmidt number. Some of the important findings, obtained from the representation of the results, are listed below: 1. The primary velocity increases with the increase of (time), Gr and Gm while it decreases with the increase of M, m, Pr and Sc. On the other hand, the primary velocity decreases as the heat source parameter increases. 2. The Secondary velocity increases with the increase of τ (time), Gr, Gm and m while it decreases with the increase of M, Pr and Sc. However, the secondary velocity decreases as the heat source parameter increases. 3. The Temperature increases with the increase of (time), Gr, Gm and m while it decreases with the increase of M, Pr and Sc and remains same with the increase of S. It is observed that increasing the heat source parameter decreases the temperature.

62 MHD free convection heat and mass transfer flow over a vertical porous plate... Fig. 18. Temperature profile due to the change of Heat source parameter Prandtl number Pr Fig. 19. Concentration profile due to change of the suction parameter S. 4. The Concentration increases with the increase of τ (time), while it decreases with the increase of Gr, Gm, Pr and Sc and remains unchanged with the increase of M and m.

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