IJAAMMInt. J. Adv. Appl. Math. and Mech. 6(1) (2018) 49 – 64 (ISSN: 2347-2529)

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 porousplate in a rotating system with hall current, heat source and suction

Research Article

Laisa Mahtarin Ivaa, Mohammad Sanjeed Hasanb, Sanjit Kumar Paulc, Rabindra Nath Mondala, ∗

a Department of Mathematics, Jagannath University, Dhaka-1100, Bangladeshb Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Gopalganj, Bangladeshc Department of Basic Sciences & Humanities, University of Asia Pacific, Dhaka, Bangladesh

Received 19 May 2018; accepted (in revised version) 20 August 2018

Abstract: Unsteady free convection heat and mass transfer flow past a semi-infinite vertical porous plate has been studied nu-merically by using an explicit finite difference method. First, thermal boundary layer equations have been derivedfrom the Navier-Stokes equation and concentration equation by boundary layer technique, and the equations havebeen made non-dimensionalised by using non-dimensional variables. The equations are then solved by using explicitfinite difference method. The solution of heat and mass transfer flow is studied to examine the velocity, temperatureand concentration distribution and it is found that the solution is dependent on several governing parameters, in-cluding 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, tem-perature and concentration profiles for various parameters entering into the problem are separately discussed andshown graphically.

MSC: 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 vis-cous fluid close to the surface. The boundary layer equations are especially interesting from a physical point of viewbecause they have the capacity to admit a large number of invariant solutions i.e. basically closed-form solutions. Inthe present context, invariant solutions are meant to be a reduction to a simpler equation such as an ordinary differ-ential equation. Prandtl’s boundary layer equations admit more and different symmetry groups or simply symmetriesare invariant transformations which do not alter the structural form of the equation under investigation. The problemof heat transfer in the boundary layer on a continuous moving surface has many practical applications in manufactur-ing processes in industry. Mechanics of non-linear fluids present a special challenge to engineers, physicists and themathematician. The non-linearity can manifest itself into a variety of ways. The heat transfer in boundary layer flowover a flat plate is very much important in practical point of view. In recent years, considerable attention has beengiven to study the heat transfer characteristics in boundary layer flow of a Newtonian fluid past a flat plate becauseof 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

B0 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 porousplate with the effect of thermal diffusion. Alam et al. [2] studied numerical simulation of the combined free and forcedconvection and mass transfer flow past a vertical porous plate in a porous medium with heat generation and thermaldiffusion. Das et al. [3] investigated the effects of mass transfer flow past an impulsively started infinite vertical platewith constant heat flux and in the presence of chemical reaction. Anjali devi and Uma devi [4] applied Runge-Kuttabased shooting method to study radiation effect on steady laminar hydromagnetic flow of a viscous, Newtonian andelectrically 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 applicationsin industrial manufacturing processes such as plasma studies, petroleum industries, MHD power generator coolingof clear reactors, boundary layer control in aerodynamics and crystal growth. Many authors have studied the effectsof magnetic field on mixed, natural and force convection heat and mass transfer problems. Indeed, MHD laminarboundary layer behavior over a semi-infinite vertical plate is a significant type of flow having considerable practicalapplications in chemical engineering, electrochemistry and polymer processing. This problem has also an importantbearing 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 inves-tigators of whom Debnath et al. [5], Raptis and Perdikis [6] are worth mentioning. Hossain [7] studied the effects ofviscous and Joule heating on the flow of viscous incompressible fluid past a semi-infinite plate in presence of a uni-form transverse magnetic field. The combined effects of forced and natural convection heat transfer in the presence ofa transverse magnetic field from vertical surfaces are also studied by many researchers. In recent years, considerableattention has been given to study the heat transfer characteristics in boundary layer flow of a Newtonian fluid past aflat plate because of its extensive application in production engineering. Chen [8] investigated the momentum, heatand mass transfer characteristics of MHD natural convection flow over a permeable, inclined surface with variablewall 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 ver-tical 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. 6(1) (2018) 49 – 64 51

flow of a viscoelastic fluid adjacent to a vertical porous plate. Pal and Talukdar [11] analysed the combined effects ofmixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conductingfluid past a vertical permeable surface embedded in a porous medium. Hayat et al. [12] analysed a mathematicalmodel in order to study the heat and mass transfer characteristics in mixed convection boundary layer flow abouta linearly stretching vertical surface in a porous medium filled with a viscoelastic fluid, by taking into account theDufour and Soret effects. Alam et al. [13] applied similarity transformations to investigate heat and mass transfercharacteristics of MHD free convection of steady flow of an incompressible electrically conducting fluid over an in-clined 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 numeri-cal differential equations, which aim at the numerical solution of ordinary, partial differential and thermal boundarylayer equations respectively. The idea is to replace the derivatives appearing in the differential equation by finite dif-ferences that approximate them. Common applications of the finite difference method are in computational scienceand engineering disciplines, such as thermal engineering, fluid mechanics, etc. The unsteady solution of thermalboundary 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-dimensionalboundary layer flow. A general, implicit, numerical, marching procedure is presented for the solution of parabolic par-tial differential equations, with special reference to those of the boundary layer. The main novelty lies in the choiceof a grid which adjusts its width so as to conform the thickness of the layer in which significant property gradientsare present. The non-dimensional stream function is employed as the independent variable across the layer. In thepresent study, finite difference scheme is used to find out the solution of heat and mass transfer flow past a verticalplate, 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 inclinedplate 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 porousinclined plate in the presence of thermal radiation with variable suction and thermophoresis. Vasu et al. [17] investi-gated radiation and mass transfer effects on transient free convection flow of a dissipative fluid past an semi-infinitevertical plate with uniform heat and mass flux. Seth et al. [18] studied unsteady hydromagnetic couette flow of aviscous incompressible electrically conducting fluid in a rotating system in the presence of an inclined magnetic fieldtaking Hall current into account. They showed that hall current and rotation tend to accelerate fluid velocity in boththe primary and secondary flow directions while Magnetic field has a retarding influence on the fluid velocity in boththe primary and secondary flow directions. Hasanuzzaman et al. [19] studied the similarity solution of unsteadycombined free and force convective laminar boundary layer flow about a vertical porous surface with suction andblowing. Alam et al. [20] applied shooting method along with sixth order Runge-Kutta integration scheme to solvean unsteady two-dimensional MHD free convective and heat transfer flow of a viscous incompressible fluid alonga vertical porous flat plate with internal heat generation/absorption. They showed that the velocity profile and thetemperature profiles are much pronounced with the increasing value of heat generation but the opposite effect wasobserved for suction parameter and Prandlt number. Recently, Sheri et al [21] applied finite element method to studytransient 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 ofan incompressible viscous and electrically conducting dusty fluid past a continuously moving inclined plate by usingimplicit finite difference method. In this paper, our objective is to study the roll of magnetic field on MHD fluid flowthrough a semi-infinite rotating vertical porous plate with hall current and heat source. The investigation has beenmade for solving the system of nonlinear partial differential equations. For this purpose an explicit finite differencetechnique has been used for which non-similar solutions of the coupled non-linear partial differential equations aresought. Therefore, it is necessary to investigate, in detail, the distributions of primary velocity, secondary velocity andtemperature 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 andthe Y-axis normal to it. Initially we consider that the plate as well as the fluid is at the same temperature and the con-centration level everywhere in the fluid is same. Also it is considered that the fluid and the plate is at rest after that theplate is to be moving with a constant velocity in its own plane and instantaneously at time t > 0, the species concentra-tion 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 thespecies far away from the plate respectively. The physical model of the study is shown in Fig. 1. Within the frameworkof the above stated assumptions with reference to the generalized equations described before the equation relevantto the transient two dimensional problems are governed by the following system of coupled non-linear differentialequations.

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∂

2u

∂y2 + gβ (T −T∞)+ gβ∗ (C −C∞) − υ

ku − σB0

2

ρ(1+m2)(u +mw) (2)

∂w

∂t+u

∂w

∂x+ v

∂w

∂y= υ∂

2w

∂y2 + σB02

ρ(1+m2)(mu −w)− υ

kw (3)

Energy equation:

∂T

∂t+u

∂T

∂x+ v

∂T

∂y= K

ρCp

∂2T

∂y2 +Q (T∞−T ) (4)

Concentration equation:

∂C

∂ t+u

∂C

∂x+ v

∂C

∂y= D

∂2C

∂y2 (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 → Tw , C →Cw at y = 0

u = 0, v = 0, T → Tw , C →Cw as y →∞

(7)

where x, y are the Cartesian coordinate system, u, v are the velocity components along the x- and y-directions re-spectively, T and C are, respectively the temperature and concentration of the fluid, T∞and C∞ are respectively thetemperature and concentration of the fluid in the static flow far away from the boundary. In this formulations, g isthe local acceleration due to gravity; υ is the kinematic viscosity, α is the thermal conductivity, k is the heat diffusivitytemperature,β is the coefficient volume for thermal expansion,β∗is the coefficient volume expansion for concentra-tion, D is the coefficient of mass diffusivity,

Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 6(1) (2018) 49 – 64 53

3. Mathematical formulation

Since the solutions of the governing Eqs. (1)-(5) under the initial condition (6) and boundary conditions (7) will bebased on a finite difference method, it is required to make the said equations dimensionless. For this purpose, we nowintroduce the following dimensionless variables:

y ′ = yU0

υ, U = u

U0, W = w

U0, τ= tU 2

0

υ,

T = T −T∞Tw −T∞

, C = C −C∞Cw −C∞

, S = v0

U0(suction parameter)u

Using this relation we have the following derivatives,

∂u

∂t= U 3

0

v

∂U

∂τ,

∂u

∂y= U 2

0

v

∂U

∂y ′ ,∂2u

∂y2 = U 30

v2

∂2U

∂y ′2

∂w

∂t= U 3

0

v

∂W

∂τ,

∂w

∂y= U 2

0

v

∂W

∂y ′ ,∂2w

∂y2 = U 30

v2

∂2W

∂y ′2

∂T

∂t= U 2

0 (Tw −T∞)

v

∂T

∂τ,

∂T

∂y= U0 (Tw −T∞)

v

∂T

∂y ′∂2T

∂y2 = U 20 (Tw −T∞)

v2

∂2T

∂y ′2

∂C

∂t= U 2

0 (Cw −C∞)

v

∂C

∂τ,

∂C

∂y= U0 (Cw −C∞)

v

∂C

∂y ′∂2C

∂y2 = U 20 (Cw −C∞)

v2

∂2C

∂y ′2

Now substituting the above derivatives into the Eqs. (1)-(5) and simplifying we obtain the following nonlinear coupledpartial differential equations in terms of dimensionless variables.

∂U

∂τ=Gr T +GmC + ∂2U

∂y ′2 +S∂U

∂y ′ −M

1+m2 (U +mW )

∂W

∂τ= ∂2W

∂y ′2 +S∂W

∂y ′ +M

1+m2 (mU −W )

∂T

∂τ= 1

Pr

∂2T

∂y ′2 +S∂T

∂y ′ −αT

∂C

∂τ= 1

Sc

∂2C

∂y ′2 +S∂C

∂y ′

where, Gr = gβ(Tw−T∞)υU0

3 is the Grashof number, Gm = g∗β(Cw−C∞)υU0

3 is the modified Grashof number, M = σB 20υ

U 20ρ

is the

magnetic parameter, Pr = υρCp

K is the Prandlt number, Sc = υD is the Schmidt number, α = Qυ

U 20

is the heat source

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 →∞

4. Numerical calculations

In this section, we attempt to solve the governing second order nonlinear coupled dimensionless partial differ-ential equations with the associated initial and boundary conditions. From the concept of the above discussion, forsimplicity the explicit finite difference method has been used to solve the Eqs. (1)-(5) subject to the conditions givenby (6) and (7). To obtain the difference equations the region of the flow is divided into a grid of lines parallel to X andY axes, where X -axis is taken along the plate and Y - axis is normal to the plate. Here we consider the plate of heightXmax (= 200) i.e. X varies from 0 to 200 and regard Ymax (= 20) as corresponding to Y →∞ i.e. Y varies from 0 to 20.Again, we consider the plate of height Xmax (= 100) i.e. X varies from 0 to 100 and regard Ymax (= 20) for Pr = 0.71 ascorresponding to Y →∞ i.e. Y varies from 0 to 20. We consider that m = 400,n = 400 in the X and Y axes grid spacesfor 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 shownin Fig. 2.

It is assumed that ∆X ,∆Y are the constant mesh sizes along the X - and Y -directions respectively, and are takenas ∆X = 0.5 (0 6 x 6 200) and ∆Y = 0.05 (0 6 y 6 20) . However, in the case of Pr = 0.71, ∆X and ∆Y are taken as

54 MHD free convection heat and mass transfer flow over a vertical porous plate...

Fig. 2. The finite difference space grid

(a) (b)

Fig. 3. Primary velocity profile U due to change of magnetic parameter M for fixed hall current m, (a) m = 0.40 (b) m = 0.60

∆X = 0.5 (0 6 x 6 100) and ∆Y = 0.1 (0 6 y 6 20) with the smaller time step ∆τ= 0.001 . Now using the explicit finitedifference approximation we get,(

∂U

∂τ

)i , j

= U ′i , j −Ui , j

∆τ,

(∂U

∂X

)i , j

= Ui , j −Ui−1, j

∆X, (8)

(∂U

∂Y

)i , j

= Ui , j+1 −Ui , j

∆Y,

(∂V

∂Y

)i , j

= Vi , j −Vi , j−1

∆Y, (9)

(∂T

∂τ

)i , j

=T ′

i , j − Ti , j

∆τ;

(∂T

∂X

)i , j

= Ti , j − Ti−1, j

∆X;

(∂T

∂Y

)i , j

= Ti , j+1 − Ti , j

∆Y(10)

(∂C

∂τ

)i , j

=C ′

i , j − Ci , j

∆τ;

(∂C

∂X

)i , j

= Ci , j − Ci−1, j

∆X;

(∂C

∂Y

)i , j

= Ci , j+1 − Ci , j

∆Y(11)

Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 6(1) (2018) 49 – 64 55

(a) (b)

Fig. 4. Primary velocity profile due to change of magnetic parameter M and hall current m, (a) m = 0.80 (b) M = 0.40

(a) (b)

Fig. 5. Primary velocity profile due to change of hall current m for fixed magnetic parameter M, (a) M = 0.60 (b) M = 0.80

(∂2U

∂Y 2

)i , j

= Ui , j+1 −2Ui , j +Ui , j−1

(∆Y )2 ;

(∂2T

∂Y 2

)i , j

= Ti , j+1 −2Ti , j + Ti , j−1

(∆Y )2 (12)

(∂2C

∂Y 2

)i , j

= Ci , j+1 −2Ci , j + Ci , j−1

(∆Y )2 (13)

Substituting the above relations into the corresponding differential equations we obtain an appropriate set of finitedifference equations as follows,

Ui , j −Ui−1, j

∆X+ Vi , j −Vi , j−1

∆Y= 0 (14)

U ′i , j −Ui , j

∆τ+Ui , j

Ui , j −Ui−1, j

∆X+Vi , j

Ui , j+1 −Ui , j

∆Y= Ui , j+1 −2Ui , j +Ui , j−1

(∆Y )2 +Gr Ti , j +GmCi , j (15)

56 MHD free convection heat and mass transfer flow over a vertical porous plate...

(a) (b)

Fig. 6. Primary velocity profile due to change of (a) Grashof number Gr, (b) Modified Grashof number Gm.

(a) (b)

Fig. 7. Primary velocity profile due to change of (a) Grashof number Gr (b) Modified Grashof number Gm.

U ′i , j =

[Ui , j+1 −2Ui , j +Ui , j−1

(∆Y )2 +Gr Ti , j +GmCi , j −Ui , jUi , j −Ui−1, j

∆X−Vi , j

Ui , j+1 −Ui , j

∆Y

]∆τ+Ui , j (16)

T ′i , j − Ti , j

∆τ+Ui , j

Ti , j − Ti−1, j

∆X+Vi , j

Ti , j+1 − Ti , j

∆Y= 1

Pr

Ti , j+1 −2Ti , j + Ti , j−1

(∆Y )2 −αTi , j (17)

C ′i , j − Ci , j

∆τ+Ui , j

Ci , j − Ci−1, j

∆X+Vi , j

Ci , j+1 − Ci , j

∆Y= 1

Sc

Ci , j+1 −2Ci , j + Ci , j−1

(∆Y )2 (18)

C ′i , j =

[1

Sc

Ci , j+1 −2Ci , j + Ci , j−1

(∆Y )2 −Ui , jCi , j − Ci−1, j

∆X−Vi , j

Ci , j+1 − Ci , j

∆Y

]∆τ+ Ci , j (19)

Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 6(1) (2018) 49 – 64 57

(a) (b)

Fig. 8. Primary velocity profile due to change of (a) Heat source parameter α (b) Suction parameter S.

(a) (b)

Fig. 9. Primary velocity profile due to change of Schmidt number Sc.

The initial and boundary conditions with the finite difference scheme are

U 0i , j = 0, V 0

i , j = 0, T 0i , j = 0, C 0

i , j = 0

U n0. j = 0, V n

0, j = 0, T n0, j = 0, C n

0, j = 0

U ni ,0 = 0, V n

i ,0 = 0, T ni ,0 = 1, C n

i ,0 = 1

U ni ,L = 0, V n

i ,L = 0, T ni ,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 nrepresents a value of time, τ = n∆τ , where n = 0,1,2,3........ . From the initial condition (20), the values of U , T , Cis known at τ = 0. During any one time step, the coefficients Ui , j and Vi , j 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 eventuallyconverge to values which approximate the steady-state solution ofEqs. (14)-(19). These converged solutions are showngraphically in the following figures.

58 MHD free convection heat and mass transfer flow over a vertical porous plate...

(a) (b)

Fig. 10. Secondary velocity profile due to change of magnetic parameter M for fixed hall current m, (a) m = 0.40 (b) m = 0.60

(a) (b)

Fig. 11. Secondary velocity profile due to change of Magnetic parameter M and Hall current, (a) m = 0.80 (b) M = 0.40

5. Results and discussion

For the Purpose of discussing the results of the problem, the approximate solutions are obtained for various param-eters. In order to analyze the physical situation of the model, we have computed the steady-state numerical valuesof the non-dimensional Primary velocity U, Secondary velocity W, Temperature T and Concentration C within theboundary layer for different values of Magnetic parameter M, Hall current m, heat source parameter , Grashof num-ber Gr , Prandtl number Pr, Scimidt number Sc with the fixed value of the modified Grashof number Gm. To get thesteady-state solution, the computations have been carried out up-to τ .We observed that the results of the computa-tions, however, changes rapidly after τ. Along with the obtained steady-state solutions, the flow behaviors in case ofcooling 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(a)-9(b). In Figs. 3-4, we show the effects onprimary velocity profile U due to the change of Magnetic parameter M for fixed Hall current m. The effects of Hallcurrent m and the Magnetic parameter M is represented in Fig. 5. We see that primary velocity U increases rapidlywith 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. 6(1) (2018) 49 – 64 59

(a) (b)

Fig. 12. Secondary velocity profile due to change of Hall current m for fixed Magnetic parameter M, (a) M = 0.60 (b) M = 0.80

(a) (b)

Fig. 13. Secondary velocity profile due to change of (a) Grashof number Gr (b) Modified Grashof number Gm.

velocity U increases which is presented in Fig. 6(a)-7(b). As seen in Fig. 8(a), the primary velocity U decreases as theHeat source parameter increases. Fig. 8(b) shows the effects of suction parameter S on primary velocity U, wherewe see that increasing the suction parameter S makes an increase to the primary velocity U rapidly, while with thedecrease of Schmidt number Sc the primary velocity U decreases slowly, which is presented in Fig. 9(a)-9(b). Hence itis concluded that the maximum primary velocity occurs in the vicinity of the plate.

The transient secondary velocity profiles have been shown in Fig. 10(a)-16(b). Figs. 10-11, we see that secondaryvelocity profile w has a substantial change due to change of magnetic parameter M for fixed hall current m. Theeffects of hall current m and the magnetic parameter M is presented in Fig. 12. As seen in Fig. 13(a)-14(b), the withthe increase of Grashof number Gr and Modified Grashof number Gm the secondary velocity profile w also increases.From Fig. 15(a), we see that secondary velocity w is decreasing when heat source parameter increases. Figure Fig. 15(b)represents the effects of Suction parameter S on secondary velocity W. Increasing of the Suction parameter S makesan increase to the secondary velocity W rapidly. As the Schmidt number decreases the secondary velocity W decreasesslowly, which are represented in Fig. 16(a)-16(b). Hence it is concluded that the maximum secondary velocity occursin the vicinity of the plate.

60 MHD free convection heat and mass transfer flow over a vertical porous plate...

(a) (b)

Fig. 14. Secondary velocity profile due to change of (a) Grashof number Gr (b) Modified Grashof number Gm.

(a) (b)

Fig. 15. Secondary velocity profile due to change of (a) Heat source parameter α (b) Prandtl number Pr

The temperature profiles have been shown in Fig. 17(a)-18(b). From Fig. Fig. 17(a), we see that temperature Tdecreases with the increase of the Suction parameter S. The effects of the Heat source parameter on the temperatureprofile T is presented in Fig. 17(b)-18(a). From these figures, it is observed that increasing the Heat source parame-ter decreases the temperature. From Fig. 18(b), we see that temperature decreases with the increase of the Prandtlnumber Pr. Next, the concentration profiles have been shown in Fig. 19(a)-20(b). From Fig. 19(a)-19(b), we see thatconcentration C decreases with the increase of suction parameter S. In Fig. 20(a)-20(b), we see that the concentrationprofiles increase with the decrease of Schmidt number Sc. However, concentration C becomes zero as Y →∞ for allparameters.

6. Concluding remarks

The present paper investigates transient heat and mass transfer flow by natural convection of an electrically con-ducting incompressible viscous fluid past an electrically non-conducting continuously moving semi-infinite verticalporous 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. 6(1) (2018) 49 – 64 61

(a) (b)

Fig. 16. Secondary velocity profile due to change of Schimdt number Sc.

(a) (b)

Fig. 17. Temperature profile due to change of (a) Suction parameter S (b) Heat source parameter α.

heat and mass fluxes.The resulting governing system of dimensionless coupled non-linear partial differential equations are solved nu-

merically by using an explicit finite difference method. The results are discussed for different values of importantparameters as Grashof number, Modified Grashof number, magnetic parameter, hall parameter, heat source param-eter, Prandtl number and the Schmidt number. Some of the important findings, obtained from the representation ofthe 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 increaseof 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 ofM, Pr and Sc and remains same with the increase of S. It is observed that increasing the heat source parameterdecreases the temperature.

62 MHD free convection heat and mass transfer flow over a vertical porous plate...

(a) (b)

Fig. 18. Temperature profile due to the change of (a) Heat source parameter (b) Prandtl number Pr

(a) (b)

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 andSc and remains unchanged with the increase of M and m.

Laisa Mahtarin Iva et al. / Int. J. Adv. Appl. Math. and Mech. 6(1) (2018) 49 – 64 63

(a) (b)

Fig. 20. Concentration profile due to change of the Schmidt number Sc.

References

[1] M. A. Sattar and M. M. Alam, Unsteady hydromagnetic free convection flow with hall current and mass transferalong an accelerated porous plate with time dependent temperature and concentration, Can. J. Phys., 70 (1992)369-382.

[2] M. S. Alam, M. M. Rahman and M. A. Samad, Numerical Study of the Combined Free-Forced Convection andMass Transfer Flow Past a Vertical Porous Plate in a Porous Medium with Heat Generation and Thermal Diffusion,Nonlinear Analysis: Modeling and Control, 11(4) (2006) 331-343.

[3] S. S. Das, V. K. Tripathy and J. K. Das, Hydrodynamics convective flow past a vertical porous plate through a porousmedium with suction and heat source. International Journal of Energy and Environment, I (2010) 467-478.

[4] S. P. Anjali Devi and R. Uma Devi, Effect of thermal radiation on hydrodynamics flow due to a porous rotatingdisk with Hall current. Journal of Applied Fluid Mechanics, 5(2) (2012) 1-7.

[5] L. Debnath, S. C. Roy and A. K. Chatterjee, Effects of hall current on unsteady hydro-magnetic flow past a porousplate in a rotating fluid system, Journal of Applied Mathematics and Mechanics, 59 (1979) 469-480.

[6] A. A. Raptis and C. P. Perdikis, Effects of mass transfer and free-convection currents on the flow past an infiniteporous plate in a rotating fluid, Astrophysics and Space Science, 84(2) (1982) 457-461.

[7] M. A. Hossain, Viscous and Joule heating effects on MHD free-convection flow with variable plate temperature,Int. J. Heat and Mass Transfer, 35(12) (1992) 3485-3487.

[8] C. H. Chen, Heat and mass transfer in MHD flow by natural convection from a permeable inclined surface withvariable wall temperature and concentration, Acta Mechanica, 172(3-4) (2004) 219-235.

[9] E. M. Abo-Eldahaband and E. M. E. Elbarbary, Hall current effect on Magneto-hydrodynamic free convection flowpast a semi-infinite vertical plate with mass transfer, Int. J. Engg. Sci., 39 (2001) 1641-1652.

[10] M. K. Chowdhury and M. N. Islam, MHD free convection flow of viscoelastic fluid past an infinite vertical porousplate, Heat Mass transfer, 36 (2000) 439-447.

[11] D. Pal and B. Talukdar, Buoyancy and chemical reaction effects on MHD mixed convection heat and mass trans-fer in a porous medium with thermal radiation and Ohmic heating, Communications in Nonlinear Science andNumerical Simulation, 15(10) (2010) 2878âAS2893.

[12] T. Hayat, M. Mustafa and I. Pop, Heat and mass transfer for Soret and DufourâAZs effect on mixed convectionboundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid, Com-munications in Nonlinear Science and Numerical Simulation, 15(5) (2010) 1183âAS1196.

[13] M. S. Alam, M. Ali, and M. D. Hossain, Heat and Mass Transfer in MHD Free Convection Flow over an InclinedPlate with Hall Current, Int. J. Engineering and Science (IJES), 2(7) (2013) 81-88.

[14] S. V. Patankar and D. B. Spalding, A finite-difference procedure for solving the equations of the two-dimensionalboundary layer, Int. J. Heat and Mass Transfer, 10(10) (1967) 1389-1411.

[15] C. Chamkha and R. Khaled, Simultaneous heat and mass transfer in free convection, Industrial Engineering

64 MHD free convection heat and mass transfer flow over a vertical porous plate...

Chemical, 49 (2001) 961-968.[16] M. S. Alam, M. M. Rahman and M. A. Sattar, Effects of variable suction and thermophoresis on steady MHD

combined free-forced convective heat and mass transfer flow over a semi-infinite permeable inclined plate inthe presence of thermal radiation, Int. J. Therm Sci., 47 (2008) 758-765.

[17] B. Vasu, V. R. Prasad and N. B. Reddy, Radiation and Mass Transfer Effects on Transient Free Convection Flow ofa Dissipative Fluid past Semi-Infinite Vertical Plate with Uniform Heat and Mass Flux, Journal of Applied FluidMechanics, 4 (2011) 15-26.

[18] G. Seth, R. Nandkeolyar and M. S. Ansari, Effects of Hall Current and Rotation on Unsteady MHD Couette Flow inthe Presence of an Inclined Magnetic Field, Journal of Applied Fluid Mechanics, 5 (2012) 67-74.

[19] M. Hasanuzzaman, B. Mandal and M. M. Touhid Hossain, A Study of Similarity Solution of Unsteady CombinedFree and Force Convective Laminar Boundary Layer Flow About a Vertical Porous Surface with Suction and Blow-ing, Annals of Pure and Applied Mathematics, 6(1) (2014) 85-97.

[20] M. S. Alam, M. M. Haque and M. J. Uddin, Unsteady MHD free convective heat transfer flow along a verticalporous flat plate with internal heat generation, Int. J. Adv. Appl. Math. and Mech. 2(2) (2014) 52 âAS 61.

[21] S. R. Sheri, R. S. Raju and S. A. Kumar, Transient MHD free convection flow past a porous vertical plate in presenceof viscous dissipation, Int. J. Adv. Appl. Math. and Mech. 2(4) (2015) 25–34.

[22] N. Pandya, Ravi Kant Yadav and A. K. Shukla, Combined effects of Soret-Dufour, Radiation and Chemical reactionon Unsteady MHD flow of Dusty fluid over inclined porous plate embedded in porous medium, Int. J. Adv. Appl.Math. and Mech. 5(1) (2017) 49–58.

Submit your manuscript to IJAAMM and benefit from:

I Rigorous peer review

I Immediate publication on acceptance

I Open access: Articles freely available online

I High visibility within the field

I Retaining the copyright to your article

—————————————————————————————

Submit your next manuscript at I editor.ijaamm@gmail.com