Alexandria Engineering Journal (2016) 55, 2321–2331

HO ST E D BY

Alexandria University

Alexandria Engineering Journal

www.elsevier.com/locate/aejwww.sciencedirect.com

ORIGINAL ARTICLE

Buoyancy induced MHD transient mass transfer

flow with thermal radiation

E-mail address: saheel_nazib@yahoo.com

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

http://dx.doi.org/10.1016/j.aej.2016.05.0071110-0168 � 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

N. Ahmed

Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India

Received 22 February 2016; revised 19 April 2016; accepted 11 May 2016

Available online 16 June 2016

KEYWORDS

Gray fluid;

Thermal radiation;

Free convection;

Hh functions

Abstract The problem of a transient MHD free convective mass transfer flow past an infinite ver-

tical porous plate in presence of thermal radiation is studied. The fluid is considered to be a gray,

absorbing-emitting radiating but non-scattered medium. Analytical solutions of the equations gov-

erning the flow problem are obtained. The effects of mass transfer, suction, radiation and the

applied magnetic field on the flow and transport characteristics are discussed through graphs.� 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

MHD is the science of motion of electrically conducting fluidsin presence of magnetic field. It concerns with the interaction

of magnetic field with the fluid velocity of electrically conduct-ing fluid. MHD generators, MHD pumps and MHD flowmeters are some of the numerous examples of MHD princi-ples. Dynamo and motor are classical examples of MHD prin-

ciple. Convection problems of electrically conducting fluid inpresence of magnetic field have got much importance becauseof its wide applications in Geophysics, Astrophysics, Plasma

Physics, Missile technology, etc. MHD principles also find itsapplications in Medicine and Biology.

The natural flow arises in fluid when the temperature as

well as species concentration change causes density variationleading to buoyancy forces acting on the fluid. Free convectionis a process of heat or mass transfer in natural flow. The heat-ing of rooms and buildings by use of radiator is an example of

heat transfer by free convection.

On the other hand, the principles of mass transfer are rele-

vant to the working of systems such as a home humidifierand the dispersion of smoke released from a chimney into theenvironment. The evaporation of alcohol from a container is

an example of mass transfer by free convection. Radiation isalso a process of heat transfer through electromagnetic waves.Radiative convective flows are encountered in countless indus-

trial and environment processes such as heating and coolingchambers, evaporation from large open water reservoirs, astro-physical flows and solar power technology. Due to importance

of the above physical aspects, several authors have carried outmodel studies on the problems of free convective hydrodynamicand magneto-hydrodynamic flows of incompressible viscouselectrically conducting fluids under different flow geometries

and physical conditions taking into account of thermal radia-tion. Some of them are Mansour [1], Ganesan and Loganathan[2], Mbeledogu et al. [3], Makinde [4], Samad and Rahman [5],

Orhan and Ahmet [6], Prasad et al. [7],Takhar et al. [8], Geb-hart et al. [9], Ali et al. [10], Hossain et al. [11], Hossain et al.[12], Ghaly [13], Muthucumaraswamy and Janakiraman [16],

Muthucumaraswamy and Sivakumar [17], Ahmed and Dutta[18], Ahmed [19], Muthucumaraswamy et al. [20]. The effectof thermal radiation together with first order chemical reaction

Nomenclature

a suction parameter~B magnetic induction vector or magnetic flux densityB0 strength of the applied magnetic field,

½MT�1Q�1�;TeslaCp specific heat at constant pressure, ½L2T�2h�1�; Joule

kg K

C species concentration, ½ðMOLÞL�3�; Kmolm3

C1 species concentration far away from the plate,½ðMOLÞL�3�; Kmol

m3

D molar mass diffusivity, ½L2T�1�, m2 s�1

~E � electric field~g gravitational acceleration vector

g acceleration due to gravity, ½LT�2�, m s�2;~J, current density vector;j~Jj magnitude of current density, ½QL�2T�1�,

Ampere/m2;

k thermal conductivity, ½MLT�3h�1�, Wm K;

k�, mean absorption coefficient, ðL�1Þ, 1=m;L constant of proportion for temperature gradient,

½T�mh�L* constant of proportion for concentration gradient,

½ðMOLÞL�3T�m�M magnetic parameter (square of the Hartmann

number)m nondimensional nonnegative integer constant;

Pr Prandtl numberp pressure, ½ML�1T�2�, Newton=m2 (Pascal)~q fluid velocity vector

N, radiation parameter~qr, radiative flux vectorqr radiative flux, ½MT�3�, W=m2;Sc, Schmidt number;t, time, ½T�, second (s)

T1 temperature far away from the plate, ½h�, KT fluid temperature,½h�, Ku X-component of ~q, ½LT�1�, m=sðx; y; zÞ Cartesian coordinates, ½L�, meter (m)g non-dimensional y-coordinateq fluid density, ½ML�3�, kg=m3

q1 density far away from the plate, ½ML�3�, kg=m3

l dynamic of viscosity, ½ML�1T�1�, kg=ms

t kinematic viscosity, ½L2T�1�, m2 s�1

�u viscous dissipation of energy per unit volume,

½ML�1T�3�, J=m3sr electrical conductivity, ½M�1L�3TQ2�, ðohm�

meter�1

r� Stefan–Boltzmann constant, ½MT�3h�4�, Wm2K4

b volumetric co-efficient of thermal expansion, ½h��1,1K

�b volumetric co-efficient for mass transfer, ½MOL��1,1

Kmol

k the ratio�bL�bL

fðgÞ non-dimensional temperaturewðgÞ non-dimensional concentration

2322 N. Ahmed

has been investigated by Rajesh [21] adopting an implicit finitedifference scheme of Crank-Nicolson method. Several authorshave adopted the finite element method for performing model

studies to investigate the effects of thermal radiation on theheat and mass transfer characteristics in different hydro-magnetic and hydrodynamic flows under different physical

and geometrical conditions, the names of whom Sheri et al.[22], Sheri and Rao [23] and Sivaiah [24] are worth mentioning.Perdikis and Rapti [14] have obtained an analytical solution to

the problem of an MHD unsteady free convective flow past aninfinite vertical porous plate in the presence of radiation withtime dependent suction. In the work of Perdikis and Rapti[14], the mass transfer effect is not taken into account. Further

their investigation ignored the effects of the physical parame-ters on the transport characteristics which seem to be veryimportant from the physical point of view.

In the present work, an attempt has been made to general-ize the problem investigated by Perdikis and Rapti [14], tostudy the mass transfer effect together with the effects of differ-

ent physical parameters on the flow and the transport charac-teristics. It is seen that the results of the present work for somelimiting cases are in excellent agreement with those of Perdikis

and Rapti [14].

2. Mathematical analysis

The equations governing the motion of an incompressible, vis-cous, electrically conducting and radiating fluid in the presence

of a magnetic field having constant mass diffusivity are asfollows:

Continuity equation:

~r �~q ¼ 0 ð1ÞMagnetic field continuity equation:

~r � ~B ¼ 0 ð2ÞOhm’s law:

~J ¼ rð ~E � þ~q� ~BÞ ð3ÞMHD momentum equation with buoyancy force:

q@~q

@tþ ð~q � ~rÞ~q

� �¼ � ~rpþ ~J� ~Bþ q~gþ lr2~q ð4Þ

Energy equation:

qCp

@T

@tþ ð~q: ~rÞT

� �¼ kr2Tþ �uþ

~J2

r� ~r �~qr ð5Þ

Species continuity equation:

@C

@tþ ð~q � ~rÞC ¼ Dr2C ð6Þ

All the physical quantities are defined in the Nomenclature.We now consider the unsteady MHD two-dimensional free

convective mass transfer flow of an incompressible viscous andelectrically conducting fluid bounded by an infinite verticalporous plate. A magnetic field of constant flux density is

Buoyancy induced MHD transient mass transfer flow 2323

assumed to be applied normal to the plate directed into thefluid region. The fluid is a gray, emitting and absorbing radiat-ing, but non-scattered medium, and the Rosselend approxima-

tion is used to describe the radiative heat flux in the energyequation.

In order to make the mathematical model of the present

work idealized, the present investigation is restricted to the fol-lowing assumptions.

I. All the fluid properties are considered constants exceptthe influence of the density with temperature and con-centration in the buoyancy force.

II. The viscous and Ohmic dissipations of energy are

negligible.III. The magnetic Reynolds number is small.IV. The plate is electrically non-conducting.

V. The radiation heat flux in the direction of the platevelocity is considered negligible in comparison with thatin the normal direction.

VI. No external electric field is applied for which the polar-

ization voltage is negligible leading to ~E� ¼ ~O.

We introduce a rectangular Cartesian coordinate systemðx; y; zÞ with X� axis along the plate in the upward verticaldirection, Y� axis normal to the plate and directed into the

fluid region and Z� axis along the width of the plate. Let

~q ¼ ðu; v; 0Þ denote the fluid velocity, ~B ¼ ð0;B0; 0Þ be the mag-

netic flux density and ~qr ¼ ð0; qr; 0Þ be the radiation flux at thepoint ðx0; y0; z0; t0Þ in the fluid.

Eq. (1) yields

@v

@y¼ 0 ð7Þ

On the basis of the assumption (VI), infinite plate assump-tion, and by the virtue of Eq. (3), the momentum Eq. (4) splits

to the following equations:

q@u

@tþ v

@u

@y

� �¼ l

@2u

@y2� qg� @p

@x� rB2

0u ð8Þ

@p

@y¼ 0 ð9Þ

Eq. (9) shows that p is independent of y indicating the factthat the pressure inside the boundary layer is the same as thepressure outside the boundary layer along a normal to the

plate and due to this fact, Eq. (8) in the free stream takes theform:

@p

@x¼ �q1g ð10Þ

The elimination of the term @p@x

from Eqs. (8) and (10) gives,

q@u

@tþ v

@u

@y

� �¼ l

@2u

@y2þ ðq1 � qÞg� rB2

0u ð11Þ

The equation of state on the basis of classical Boussinesqapproximation is

q1 ¼ q 1þ bðT� T1Þ þ �bðC� C1Þ� � ð12Þ

On unification of Eqs. (11) and (12), we establish thefollowing non-linear partial differential equation:

@u

@tþ v

@u

@y¼ t

@2u

@y2þ gbðT� T1Þ þ g�bðC� C1Þ � rB2

0u

qð13Þ

The assumption (II) leads the energy Eq. (5) to reduce to

the form

qCp

@T

@tþ v

@T

@y

� �¼ k

@2T

@y2� @qr

@yð14Þ

The reduced form of the mass diffusion Eq. (6) is as givenbelow:

@C

@tþ v

@C

@y¼ D

@2C

@y2ð15Þ

Eq. (7) leads that v is function of time only and we take the

suction velocity v as follows:

v ¼ �att

� 1=2

ð16Þ

In Eq. (16), the negative sign signifies the fact that the direc-

tion of the suction velocity is toward the plate.The radiation heat flux by using the Rosseland approxima-

tion, is given by

qr ¼ � 4r�

3k�@T 4

@yð17Þ

We further impose the following restrictions:

I: The plate temperature and the temperature away from

the plate are proportional to tm

II: The plate concentration and the concentration awayfrom the plate are proportional to tm, where m is anon-negative integer constant.

III: The plate as well as the fluid far away from the plate is atrest

IV: The difference between the fluid temperature T and T1is very small.

Under these assumptions, the appropriate boundary condi-

tions for the velocity, temperature and concentration fields aredefined as

uð0; tÞ ¼ 0; uð1; tÞ ¼ 0

Tð0; tÞ ¼ T1 þ Ltm;Tð1; tÞ ! T1Cð0; tÞ ¼ C1 þ L�tm;Cð1; tÞ ! C1

9>=>; ð18Þ

On the basis of the restriction (IV), we may expand

fðTÞ ¼ T 4 in Taylor’s series about T1 as described below:

fðTÞ ¼ T 4 ¼ fðT1Þ þ ðT� T1Þf 0ðT1Þ þ � � � ð19ÞBy neglecting the higher powers T� T1 in (19), we obtain

T 4 ¼ 4T 31T� 3T 4

1 ð20ÞNow Eq. (17), accomplished by Eq. (20) transforms to

qr ¼ � 16r�T 31

3k�@T

@yð21Þ

On the use of the relation (21), Eq. (14) becomes

@T

@tþ v

@T

@y¼ k

qCp

@2T

@y2þ 16r�T3

13qCpk

�@2T

@y2ð22Þ

On substitution of the expression for v from Eq. (16), Eqs.(13), (22) and (15) respectively transforms to

2324 N. Ahmed

@u

@t� a

tt

� 1=2 @u

@y¼ t

@2u

@y2þ gbhþ g�bu� rB2

0u

qð23Þ

@h@t

� att

� 1=2 @h@y

¼ 1

qCp

kþ 16r�T31

3k�

� �@2h@y2

ð24Þ

@u@t

� att

� 1=2 @u@y

¼ D@2u@y2

ð25Þ

where h ¼ T� T1 and u ¼ C� C1The boundary conditions to be satisfied by Eqs. (23), (24),

and (25) now become

uð0; tÞ ¼ 0; uð1; tÞ ¼ 0

hð0; tÞ ¼ Ltm; hð1; tÞ ! T1uð0; tÞ ¼ L�tm;uð1; tÞ ! C1

9>=>; ð26Þ

Proceeding with the analysis, we introduce the followingnon-dimensional variables and similarity parameters to nor-malize the flow model.

Pr ¼ lCp

k; M ¼ rB2

0t

q; Sc ¼ t

D; N ¼ kk�

4r�T31;

k ¼�bL�

bL; g ¼ y

2ffiffiffiffitt

p ð27Þ

All the above quantities are defined in the Nomenclature as

mentioned earlier.We define the velocity, temperature and concentration

fields as

uðg; tÞ ¼ Lgbtmþ1�uðg; tÞ ð28Þ

hðg; tÞ ¼ LtmfðgÞ ð29Þ

uðg; tÞ ¼ L�tmwðgÞ ð30Þwhere �uðg; tÞ ¼ u0ðgÞ þMu1ðgÞ þM2u2ðgÞ þ � � �

For small values of M, the substitutions of Eqs. (27)–(30)into Eqs. (23)–(25) and the comparison of the coefficients of

the like powers of M, give the following differential equations:

w00ðgÞ þ 2Scðgþ aÞw0ðgÞ � 4mScwðgÞ ¼ 0 ð31Þ

f00ðgÞ þ 2cðgþ aÞf0ðgÞ � 4mcfðgÞ ¼ 0 ð32Þ

u000ðgÞ þ 2ðgþ aÞu00ðgÞ � 4ðmþ 1Þu0ðgÞ ¼ �4fðgÞ � 4kwðgÞð33Þ

u001ðgÞ þ 2ðgþ aÞu01ðgÞ � 4ðmþ 2Þu1ðgÞ ¼ 4u0ðgÞ ð34Þ

u002ðgÞ þ 2ðgþ aÞu02ðgÞ � 4ðmþ 3Þu2ðgÞ ¼ 4u1ðgÞ ð35Þwhere a dash denotes differentiation with respect to g, andc ¼ 3NPr

3Nþ4.

u0 ¼

A3Hh2mþ2

ffiffiffi2

pn

� �þ B3Hh2mþ2

ffiffiffiffiffi2c

pn

� �þ C3Hh2mþ2

ffiffiffiffiffiffiffiffi2Sc

pn

� �;

A4Hh2mþ2

ffiffiffi2

pn

� �þ B4Hh2mffiffiffi2

pn

� �þ C3Hh2mþ2

ffiffiffiffiffiffiffiffi2Sc

pn

� �; c ¼

A5Hh2mþ2

ffiffiffi2

pn

� �þ B3Hh2mþ2

ffiffiffiffiffi2c

pn

� �þ C4Hh2mffiffiffi2

pn

� �; c –

B5 Hh2mffiffiffi2

pn

� �� Hh2mffiffi2

pað Þ

Hh2mþ2

ffiffi2

pað ÞHh2mþ2

ffiffiffi2

pn

� �� �; c ¼ 1;Sc ¼ 1

8>>>>>><>>>>>>:

The corresponding boundary conditions (26) reduce to

wð0Þ ¼ 1; wð1Þ ! 0

fð0Þ ¼ 1; fð1Þ ! 0

u0ð0Þ ¼ 0; u0ð1Þ ! 0

u1ð0Þ ¼ 0; u1ð1Þ ! 0

u2ð0Þ ¼ 0; u2ð1Þ ! 0

9>>>>>>>>>=>>>>>>>>>;

ð36Þ

3. Method of solutions

To solve Eqs. (31)–(35) subject to the conditions (36), we con-sider the following transformations:

n ¼ gþ a; �n ¼ffiffiffiffiffiffiffiffi2Sc

pn; �q ¼

ffiffiffiffiffi2c

pn; e ¼

ffiffiffi2

pno

ð37Þ

In the aid of the transformations (37), Eqs. (31)–(35) in new

system are

d2w

dn2þ n

dw

dn� 2mw ¼ 0 ð38Þ

d2f

dq2þ q

df

dq� 2mf ¼ 0 ð39Þ

d2u0de2

þ edu0de

� 2ðmþ 1Þu0 ¼ �2fðgÞ � 2kwðgÞ ð40Þ

d2u1de2

þ edu1de

� 2ðmþ 2Þu1 ¼ 2u0 ð41Þ

d2u2de2

þ edu2de

� 2ðmþ 3Þu1 ¼ 2u1 ð42Þ

The transformed boundary conditions become

wffiffiffiffiffiffiffiffi2Sc

pa

� � ¼ 1; wð1Þ ! 0

fffiffiffiffiffi2c

pa

� � ¼ 1; fð1Þ ! 0

u0ffiffiffi2

pa

� � ¼ 0; u0ð1Þ ! 0

u1ffiffiffi2

pa

� � ¼ 0; u1ð1Þ ! 0

u2ffiffiffi2

pa

� � ¼ 0; u2ð1Þ ! 0

9>>>>>>>>>>=>>>>>>>>>>;

ð43Þ

The solutions of Eqs. (38)–(42) under the conditions (43)are

w ¼ A1Hh2mffiffiffiffiffiffiffiffi2Sc

pn

� ð44Þ

f ¼ A2Hh2mffiffiffiffiffi2c

pn

� ð45Þ

c – 1;Sc– 1

1;Sc – 1

1;Sc ¼ 1 ð46Þ

u1 ¼

A7Hh2mþ4

ffiffiffi2

pn

� �� A3Hh2mþ2

ffiffiffiffiffi2c

pn

� �þ B6Hh2mþ4

ffiffiffiffiffi2c

pn

� �þ C6Hh2mþ4

ffiffiffiffiffiffiffiffi2Sc

pn

� �c – 1;Sc – 1

A8Hh2mþ4

ffiffiffi2

pn

� �� A4Hh2mþ2

ffiffiffi2

pn

� �þ B7Hh2mffiffiffi2

pn

� �þ C7Hh2mþ4

ffiffiffiffiffiffiffiffi2Sc

pn

� �c ¼ 1;Sc–1

A9Hh2mþ4

ffiffiffi2

pn

� �� A5Hh2mþ2

ffiffiffi2

pn

� �þ B8Hh2mþ4

ffiffiffiffiffi2c

pn

� �þ C8Hh2mffiffiffi2

pn

� �; c – 1;Sc ¼ 1

EHh2mþ4

ffiffiffi2

pn

� �þ FHh2mþ2

ffiffiffi2

pn

� �þ GHh2mffiffiffi2

pn

� �; c ¼ 1;Sc ¼ 1

8>>>><>>>>:

ð47Þ

u2 ¼

A11Hh2mþ6

ffiffiffi2

pn

� �� A7Hh2mþ4

ffiffiffi2

pn

� �þ A3

2Hh2mþ2

ffiffiffi2

pn

� �þ B11Hh2mþ6

ffiffiffiffiffi2c

pn

� �þ C11Hh2mþ6

ffiffiffiffiffiffiffiffi2Sc

pn

� �; c – 1;Sc – 1

A12Hh2mþ6

ffiffiffi2

pn

� �� A8Hh2mþ4

ffiffiffi2

pn

� �þ B12Hh2mþ2

ffiffiffi2

pn

� �þ C12Hh2mffiffiffi2

pn

� �þ E1Hh2mþ6

ffiffiffiffiffiffiffiffi2Sc

pn

� �; c ¼ 1;Sc– 1

A13Hh2mþ6

ffiffiffi2

pn

� �� A9Hh2mþ4

ffiffiffi2

pn

� �þ B13Hh2mþ2

ffiffiffi2

pn

� �þ C13Hh2mþ6

ffiffiffiffiffi2c

pn

� �þ E2Hh2mðffiffiffi2

pnÞ; c– 1;Sc ¼ 1

A14Hh2mþ6

ffiffiffi2

pn

� �� EHh2mþ4

ffiffiffi2

pn

� �þ F1Hh2mþ2

ffiffiffi2

pn

� �þ G1Hh2mffiffiffi2

pn

� �; c ¼ 1;Sc ¼ 1

8>>>><>>>>:

ð48Þ

Buoyancy induced MHD transient mass transfer flow 2325

where the function ‘Hhn’ is defined as follows (Jeffrey andJeffrey [15]):

HhnðxÞ ¼R1x

ðu�xÞnn!

e�12u

2du; n 2 Z; n P 0

ð�1Þn�1 ddx

� ��n�1e�

12x

2; n 2 Z; n < 0

(

And the constants involved in the above solutions aredefined by

A1 ¼ 1

Hh2mffiffiffiffiffiffiffiffi2Sc

pa

� � ; A2 ¼ 1

Hh2mffiffiffiffiffi2c

pa

� � ; B3 ¼ 2A2

1� c;

C3 ¼ 2kA1

1� c

A3 ¼ �B3Hh2mþ2

ffiffiffiffiffi2c

pa

� �þ C3Hh2mþ2

ffiffiffiffiffiffiffiffi2Sc

pa

� �Hh2mþ2

ffiffiffi2

pa

� � ;

A4 ¼ �B4Hh2mffiffiffi2

pa

� �þ C3Hh2mþ2

ffiffiffiffiffiffiffiffi2Sc

pa

� �Hh2mþ2

ffiffiffi2

pa

� �

B4 ¼ 1

Hh2mffiffiffi2

pa

� � ; C4 ¼ kB4; B5 ¼ ð1þ kÞB4;

B6 ¼ 2B3

c� 1; C6 ¼ 2C3

Sc� 1

A5 ¼ �B3Hh2mþ2

ffiffiffiffiffi2c

pa

� �þ C4Hh2mffiffiffi2

pa

� �Hh2mþ2

ffiffiffi2

pa

� � ;

A6 ¼ �B5Hh2mffiffiffi2

pa

� �Hh2mþ2

ffiffiffi2

pa

� �

A11 ¼ 1

2Hh2mþ6

ffiffiffi2

pa

� � 2A7Hh2mþ4

ffiffiffi2

pa

� �� A3Hh2mþ2

ffiffiffi2

pa

� �� 2B11

2C11Hh2mþ6

ffiffiffiffiffiffiffiffi2Sc

pa

� �"

A12 ¼ 1

Hh2mþ6

ffiffiffi2

pa

� � A8Hh2mþ4

ffiffiffi2

pa

� �� B12Hh2mþ2

ffiffiffi2

pa

� �� C12Hh

E1Hh2mþ6

ffiffiffiffiffiffiffiffi2Sc

pa

� �"

A7 ¼A3Hh2mþ2

ffiffiffi2

pa

� �� B6Hh2mþ4

ffiffiffiffiffi2c

pa

� ��C6Hh2mþ4

ffiffiffiffiffiffiffiffi2Sc

pa

� �Hh2mþ4

ffiffiffi2

pa

� �

A8 ¼A4Hh2mþ2

ffiffiffi2

pa

� �� B7Hh2mffiffiffi2

pa

� �� C7Hh2mþ4

ffiffiffiffiffiffiffiffi2Sc

pa

� �Hh2mþ4

ffiffiffi2

pa

� �

A9 ¼A5Hh2mþ2

ffiffiffi2

pa

� �� B8Hh2mþ4

ffiffiffiffiffi2c

pa

� �� C8Hh2mffiffiffi2

pa

� �Hh2mþ4

ffiffiffi2

pa

� �

E ¼ � B5

2Hh2mþ4

ffiffiffi2

pa

� �Hh2m

ffiffiffi2

pa

� � ;F ¼ B5

Hh2mffiffiffi2

pa

� �Hh2mþ2

ffiffiffi2

pa

� �

G ¼ � B5

2Hh2mffiffiffi2

pa

� �Hh2m

ffiffiffi2

pa

� � ;

B7 ¼ �B4

2; B8 ¼ 2B3

c� 1; C7 ¼ 2C3

Sc� 1; C8 ¼ �C4

2

B11 ¼ 2B6

c� 1; C11 ¼ 2C6

Sc� 1; E1 ¼ 2C7

Sc� 1;

C12 ¼ �B7

3; B12 ¼ A4

2

Hh2mþ6

ffiffiffiffiffi2c

pa

� �� #

2m

ffiffiffi2

pa

� ��#

B13 ¼ A5

2; C13 ¼ 2B8

c� 1; E2 ¼ �C8

3; F1 ¼ �F

2; G1 ¼ �G

3

A13 ¼ 1

Hh2mþ6

ffiffiffi2

pa

� � A9Hh2mþ4

ffiffiffi2

pa

� �� B13Hh2mþ2

ffiffiffi2

pa

� �� C13Hh2mþ6

ffiffiffiffiffi2c

pa

� ��E2Hh2m

ffiffiffi2

pa

� �" #

A14 ¼ 1

Hh2mþ6

ffiffiffi2

pa

� � EHh2mþ4

ffiffiffi2

pa

� � F1Hh2mþ2

ffiffiffi2

pa

� � G1Hh2m

ffiffiffi2

pa

� h i

2326 N. Ahmed

4. Coefficient of skin friction

The viscous drag at the plate per unit area in the direction of

the plate velocity is quantified by the Newton’s law of viscosityin the form:

�s ¼ l@u

@y

�y¼0

¼ Lgbqffiffiffiffitt

ptm

2

@�u

@g

�g¼0

ð49Þ

The coefficient of the skin friction at the plate is as follows:

s ¼ 2s

Lgbqffiffiffiffitt

ptm

¼ @�u

@g

�g¼0

¼ u00ð0Þ þMu00ð0Þ þM2u00ð0Þ þ � � �¼ s0 þMs1 þM2s2 þ � � � ð50Þ

where

s0 ¼

� ffiffiffi2

pA3Hh2mþ1

ffiffiffi2

pa

� �þ ffiffiffic

pB3Hh2mþ1

ffiffiffiffiffi2c

pa

� �þ ffiffiffiffiffiSc

pC3Hh2m

�; c–1;Sc–1

� ffiffiffi2

pA4Hh2mþ1

ffiffiffi2

pa

� �þ B4Hh2m�1

ffiffiffi2

pa

� �þ ffiffiffiffiffiSc

pC3Hh2mþ

�; c ¼ 1;Sc–1

� ffiffiffi2

pA5Hh2mþ1

ffiffiffi2

pa

� �þ ffiffiffic

pB3Hh2mþ1

ffiffiffiffiffi2c

pa

� �þ C4Hh2m��

; c–1;Sc ¼ 1

� ffiffiffi2

pB5 Hh2m�1

ffiffiffi2

pa

� �� Hh2mffiffi2

pað Þ

Hh2mþ2

ffiffi2

pað ÞHh2mþ1

ffiffiffi2

pa

� �� �; c ¼

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

s1 ¼

� ffiffiffi2

p A7Hh2mþ3

ffiffiffi2

pa

� �� ffiffiffic

pA3Hh2mþ1

ffiffiffiffiffi2c

pa

� �þ ffiffiffi

cp

B6Hh2mþ3

ffiffiffiffiffi2c

pa

� �þ ffiffiffiffiffiSc

pC6Hh2mþ4

ffiffiffiffiffiffiffiffi2Sc

pa

� �" #

; c

� ffiffiffi2

p A8Hh2mþ3

ffiffiffi2

pa

� �� A4Hh2mþ1

ffiffiffi2

pa

� �þB7Hh2m�1

ffiffiffi2

pa

� �þ ffiffiffiffiffiSc

pC7Hh2mþ3

ffiffiffiffiffiffiffiffi2Sc

pa

� �" #

; c ¼ 1

� ffiffiffi2

p A9Hh2mþ3

ffiffiffi2

pa

� �� A5Hh2mþ1

ffiffiffi2

pa

� �þffiffiffic

pB8Hh2mþ3

ffiffiffiffiffi2c

pa

� �þ C8Hh2m�1

ffiffiffi2

pa

� �" #

; c – 1;Sc ¼

� ffiffiffi2

pEHh2mþ3

ffiffiffi2

pa

� �þ FHh2mþ1

ffiffiffi2

pa

� �þ GHh2m�1

ffiffiffi2

pa

� �� �;

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

s2 ¼

� ffiffiffi2

pA11Hh2mþ5

ffiffiffi2

pa

� ��A7Hh2mþ3

ffiffiffi2

pa

� �þA3

2Hh2mþ1

ffiffiffi2

pa

� �þh� ffiffiffi

2p

A12Hh2mþ5

ffiffiffi2

pa

� ��A8Hh2mþ3

ffiffiffi2

pa

� �þB12Hh2mþ1

ffiffiffi2

pa

� �h� ffiffiffi

2p

A13Hh2mþ5

ffiffiffi2

pa

� ��A9Hh2mþ3

ffiffiffi2

pa

� �þB13Hh2mþ1

ffiffiffi2

pa

� �h� ffiffiffi

2p

A14Hh2mþ5

ffiffiffi2

pa

� ��EHh2mþ3

ffiffiffi2

pa

� �þF1Hh2mþ1

ffiffiffi2

pa

� �þh

8>>>>>>><>>>>>>>:

5. Coefficient of rate of heat transfer

The heat flux q� from the plate to the fluid is quantified by the

Fourier law of conduction in the form

q� ¼ �j�@T@y

�y¼0

¼ � j�L

2ffiffiffiffitt

p tmf0ð0Þ; j� ¼ kþ 16r�T31

3k�ð54Þ

The coefficient of the rate of heat transfer from the plate tothe fluid in terms of the Nusselt number is given by

Nu ¼ q�kL2ffiffiffitt

p tm¼ �f0ð0Þ ¼

ffiffiffiffiffi2c

pA2Hh2m�1

ffiffiffiffiffi2c

pa

� ð55Þ

þ1

ffiffiffiffiffiffiffiffi2Sc

pa

� ��

1

ffiffiffiffiffiffiffiffi2Sc

pa

� ��

1

ffiffiffi2

pa

� ��

1;Sc ¼ 1

ð51Þ

– 1;Sc– 1

;Sc– 1

1

c ¼ 1;Sc ¼ 1

ð52Þ

ffiffiffic

pB11Hh2mþ5

ffiffiffiffiffi2c

pa

� �þ ffiffiffiffiffiffiSc

pC11Hh2mþ5

ffiffiffiffiffiffiffiffi2Sc

pa

� �i; c–1;Sc–1

þC12Hh2m�1

ffiffiffi2

pa

� �þ ffiffiffiffiffiffiSc

pE1Hh2mþ5

ffiffiffiffiffiffiffiffi2Sc

pa

� �i; c¼ 1;Sc–1

þ ffiffiffic

pC13Hh2mþ5

ffiffiffiffiffi2c

pa

� �þE2Hh2m�1

ffiffiffi2

pa

� �i; c–1;Sc¼ 1

G1Hh2m�1

ffiffiffi2

pa

� �i; c¼ 1;Sc¼ 1

ð53Þ

Buoyancy induced MHD transient mass transfer flow 2327

6. Coefficient of mass transfer

The mass flux Mw at the plate is determined by the Fick’s lawof mass diffusion

Mw ¼ �D@C

@y

�y¼0

¼ �DLkb

2ffiffiffiffitt

p tmw0ð0Þ ð56Þ

The coefficient of mass transfer at the plate in terms ofSherwood number is given by (see Fig. 1)

Sh ¼ 2ffiffiffiffitt

pMw

�bDLkbtm

¼ �w0ð0Þ ¼ffiffiffiffiffiffiffiffi2Sc

pA1Hh2m�1

ffiffiffiffiffiffiffiffi2Sc

pa

�

¼ �@/@y

�y¼0

ð57Þ

Figure 2 Concentration versus g for m = 1, a= 0.5.

7. Results and discussion

In order to get clear insight of the physical problem, numericalcomputations from the analytical solutions for representative

velocity field, temperature field, concentration field, and thecoefficient of skin friction, the coefficient of the rate of heattransfer in terms of Nusselt number and the rate of mass trans-fer in terms of Sherwood number at the plate have been carried

out by assigning some arbitrarily chosen specific values to thesimilarity parameters such as magnetic parameterM (square ofthe Hartmann number), radiation parameter N, suction

parameter a, the constant m, the constant ratio k and the nor-mal coordinate g. Throughout our investigation, the value ofthe Prandtl number Pr has been chosen to be 0.71 which cor-

responds to air as the numerical computations are concerned.The values of the Schmidt number Sc are taken as 0.22, 0.60,0.96, and 1.42. We recall that the aforesaid values of the Sch-

midt number Sc represent respectively the Hydrogen, Steam,Carbon dioxide and Chlorine diffused in dilute mixture withair. The numerical results computed from the analytical solu-tions of the problem have been illustrated in Figs. 2–19.

The variations in the dimensionless concentration wðgÞ ver-sus the normal coordinate g under the influence of the Schmidtnumber Sc, the constant ‘m’ and suction parameter ‘a’ are

demonstrated in Figs. 2–4. All the three figures uniquely indi-cate the comprehensive fall in wðgÞ for increasing the values ofSc, m and a. It is inferred that an increase in the Schmidt num-

ber Sc means a decrease in mass diffusivity as the definition of

Figure 1 Flow configuration.

Sc is concerned. That is to say that there is a substantial dropin concentration for high mass diffusivity. This observation isconsistent with the physical fact that when mass (solute) dif-

fuses in the solvent at a high rate, the concentration level ofthe medium of diffusion gets enhanced. Further the three fig-ures as expected show that wðgÞ decreases asymptotically asthe normal coordinate g increases.

The velocity profiles under the influence of the parametersm; k;N;Sc;M and a, the suction parameter are exhibited inFigs. 5–10. It is observed from Fig. 6 that the fluid flow is

accelerated due to the effect of the parameter k, while anincrease in each of the values of the parameters m and Sccauses the flow to decelerate considerably as seen in Figs. 5

and 8. Fig. 7 establishes the fact that the thermal radiationhas also some contributions in controlling the growth of thethickness of the velocity boundary layer to some extent. It is

inferred from Fig. 9 that an increase in the magnetic parameterM has an inhibiting effect on the fluid velocity to some extent.The fluid velocity u gets continuously reduced with increasingM. In other words, the imposition of the transverse magnetic

field causes the flow to retard slowly and steadily. This phe-nomenon has an excellent agreement with the physical factthat the Lorentz force that appears due to interaction of the

transverse magnetic field with the fluid velocity acting as aresistive force to the fluid flow which serves to decelerate theflow. The variation in fluid velocity under the effect of suction

is presented in Fig. 10. It is observed in this figure that likemagnetic field, the effect of suction also results in a substantialdecrease in the fluid velocity. As such the imposition of the mag-netic field as well as the suction is an effective regulatory mech-

anism for the flow regime. All Figs. 5–10 uniquely establish thefact that the fluid velocity first increases in a thin layer adjacentto the plate and thereafter it decreases asymptotically indicat-

ing the fact that the buoyancy force has a significant effect onthe flow near the plate and its effect gets nullified far awayfrom the plate.

Figs. 11–15 correspond to the coefficient of skin friction s atthe plate against the suction parameter a under the influence ofm; k;N;Sc;M. A trend of decay in the coefficient of skin fric-

tion s is clearly marked in Figs. 11 and 15 due to increasingm and Sc, thereby reducing the viscous drag at the plate. Onthe other hand, the frictional resistance of the fluid at the plateis seen to be enhanced under the effect radiation, imposition of

Figure 3 Concentration versus g for Sc = 0.60, a= 0.5.

Figure 4 Concentration versus g for Sc = 0.60, m= 1.

Figure 5 Velocity versus g for N= 3, Pr = 0.71, a= 0.5,

Sc = 0.60, k= 0, M= 0.01.

Figure 6 Velocity versus g for N = 3, Pr = 0.71, a= 0.5,

Sc = 0.60, m = 1, M= 0.01.

Figure 7 Velocity versus g for m = 1, Pr = 0.71, a= 0.5,

Sc = 0.60, k= 3, M= 0.01.

Figure 8 Velocity versus g for N = 3, Pr = 0.71, a= 0.5,

m= 1, k= 3, M = 0.01.

2328 N. Ahmed

Figure 9 Velocity versus g for N= 3, Pr= 0.71, a= 0.5,

Sc = 0.60, k= 0, m = 1.

Figure 10 Velocity versus g for N= 3, Pr = 0.71, m = 1,

Sc = 0.60, k= 3, M= 0.01.

Figure 11 Skin friction versus a for N = 3, Pr = 0.71,

Sc = 0.60, k= 1, M= 0.01.

Figure 12 Skin friction versus a for N = 3, Pr = 0.71,

Sc = 0.60, m = 1, M= 0.01.

Figure 13 Skin friction versus a for m = l, Pr= 0.71,

Sc = 0.60, k= 3, M= 0.01.

Figure 14 Skin friction versus a for N = 3, Pr = 0.71,

Sc = 0.60, k= 3, m = 1.

Buoyancy induced MHD transient mass transfer flow 2329

Figure 15 Skin friction versus a for N = 3, Pr = 0.71,

M= 0.01, k = 3, m = 1.

Figure 16 Nusselt number versus a for N = 3. Pr = 0.71.

Figure 17 Nusselt number versus a for m = 1, Pr = 0.71.

Figure 18 Sherwood number versus a for m = 1.

Figure 19 Sherwood number versus a for Sc = 0.60.

Figure 20 Velocity versus g (Figure No. 6 of Perdikis and Rapti

[14]).

2330 N. Ahmed

Buoyancy induced MHD transient mass transfer flow 2331

the transverse magnetic field and for increasing values of k asobserved from Figs. 12–14. An interesting behavior of the skinfriction s at the plate against the suction parameter a is

observed in Fig. 14. This figure indicates that in the absenceof the magnetic field, the drag force due to viscosity gets reducedunder the effect of suction whereas in the presence of magnetic

field, this behavior takes a reverse turn.The effects of the parameters m, a and k on the Nusselt

number Nu are displayed in Figs. 16 and 17. These figures sim-

ulate that an increase in each of the values of the above param-eters leads the rate of heat transfer from the plate to the fluidto increase substantially. Figs. 18 and 19 present how the ratemass transfer at the plate in terms of Sherwood number Sh is

affected by the Schmidt number Sc, the suction parameter aand parameter m . Like Nusselt number, the Sherwood num-ber is also enhanced comprehensively under the influence of

the aforesaid parameters. Further, it is noticed that the effectof radiation on the rate of heat transfer, and the effect of massdiffusion on the Sherwood number are more pronounced for large

suction.

8. Comparison of results

In order to highlight the accuracy of the numerical computa-tions from the analytical solutions in the present work, oneof the results of the present study for a special case has been

compared with those of Perdikis and Rapti [14]. ComparingFigs. 5 and 20 (Fig. 6 of the work done by Perdikis and Rapti[14]), we see that the two figures are almost identical as thebehavior of the velocity field against the normal coordinate gunder the influence of the parameter m is concerned. There isan excellent agreement between the results of the present workand those of Perdikis and Rapti [14].

Acknowledgment

The author is highly thankful to CSIR-HRDG for funding thisresearch work under Research Grant-in-aid No. 25(0209)/12/EMR-II.

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