321 a 09p

N. P. Singh, Ajay Kumar Singh, and Atul Kumar Singh
MHD FREE CONVECTION AND MASS
TRANSFER FLOW PAST A FLAT PLATE
N. P. Singh* and Ajay Kumar Singh
Department of Mathematics, C. L. Jain College, Firozabad - 283 203, India
Atul Kumar Singh
Department of Mathematics, V. S. S. D. College, Kanpur- 208 002, India
الخلاصـة:
ل ѧ ر قاب ѧ ائعٍ غي ѧ دين لم ѧ سوف نعرض في هذا البحث لمعضلة النقل الحراري ونقل الكتلة وجريانها في بع
داد، ѧ ائي الامت ѧ ودي لانه ѧ اذي، عم ѧ طح نف ѧ ول س ѧ ائع ح ѧ ري الم ѧ ائي . يج ѧ ار الكهرب ѧ ل للتي ѧ للانضغاط ، لزج وموص
ال ѧ أثير مج ѧ ت ت ѧ ر و تح ѧ صاص آبي ѧ من امت ѧ راري ض ѧ شار الح ѧ وآذلك بوجود مصدر حراري . آما نعرض للانت
صول ѧ ة . وللح ѧ زخم والطاق ѧ مغناطيس متعامد مع اتجاه الجريان . آما نقدم التحولات التشابه ية لحل معادلات ال
ات ѧ ى علاق ѧ صلنا عل ѧ على الحلول التشابهية، استخدمنا طريقة الاضطرابات لحل المعادلات التشابهيه. آما ح
رارة . ѧ ه والح ѧ آلٍ من مجال السرعه، والتوزيع الحراري، وترآيز المجال، ومعامل الجر، ومعدل انتقال الكتل
وسوف نناقش هذه النتائج من خلال الجداول الحسابية والمنحنيات لتوضيح أثر العديد من المتغيرات.
ABSTRACT
Two dimensional free convection and mass transfer flow of an incompressible,
viscous and electrically conducting fluid past a continuously moving infinite vertical
porous plate in the presence of heat source, thermal diffusion, large suction and under
the influence of uniform magnetic field applied normal to the flow is studied. Usual
similarity transformations are introduced to solve the momentum, energy and
concentration equations. To obtain local similarity solutions of the problem, the
similarity equations are solved using perturbation technique. The expressions for
velocity field, temperature distribution, concentration field, drag coefficient, rate of
heat, and mass transfer have been obtained. The results are discussed in detailed with
the help of graphs and tables to observe the effect of various parameters.
Key words: Free convection, Mass transfer, Thermal diffusion, Porous plate
AMS 1991 Subject Classification : 76W05
*Address for correspondence :
236 Durga Nagar
FIROZABAD - 283203
U. P., INDIA
Paper Received 10 December 2001; Revised 8 September 2003; Accepted 26 October 2003.
January 2007 The Arabian Journal for Science and Engineering, Volume 32, Number 1A 93

MHD FREE CONVECTION AND MASS TRANSFER FLOW PAST A FLAT PLATE
INTRODUCTION
Hansen [1] and Sanyal and Bhattacharya [2] have presented the technique to obtain similarity solutions in a
hydromagnetic flow. Ferraro and Plumpton [3] and Cramer and Pai [4] are notable authors for major contribution about
MHD free convection flows and their significant application in the field of stellar and planetary magnetospheres,
aeronautics, chemical engineering, electronics, and so on. In addition, many transport processes exist in industries and
technology where the transfer of heat and mass occurs simultaneously as a result of thermal diffusion and diffusion of
chemical species. An extensive contribution on heat and mass transfer flow has been made by Gebhart [5] to highlight
the insight on the phenomena. Gebhart and Pera [6] studied heat and mass transfer flow under various flow situations.
Thereafter, several authors, viz. Raptis and Soundalgekar [7], Agrawal et al. [8] , Jha and Singh [9], Jha and Prasad [10],
Abdusattar [11], and Soundalgekar et al. [12] have paid attention to the study of MHD free convection and mass transfer
flows.
Introducing a time dependent length scale of similarity technique, similarity solutions to study the free convection
and mass transfer flow past an impulsively started vertical porous plate in a rotating fluid in presence of large suction
have been studied by Sattar and Alam [13] based on perturbation technique demonstrated by Singh and Dixit [14]. Alam
and Sattar [15] and Singh et al. [16] modified the work of Sattar and Alam [13] for steady MHD free convection in mass
transfer flow with thermal diffusion for viscous fluid flow and flow of viscous stratified liquid respectively.
Subsequently, Singh and Singh [17] extended the problem of Singh [18] to analyze the effects of mass transfer on MHD
flow considering constant heat flux and induced magnetic field. Recently, Acharya et al. [19] have presented an analysis
to study MHD effects on free convection and mass transfer flow through a porous medium with constant suction and
constant heat flux considering Eckert number as a small perturbation parameter. This is the extension of the work of
Bejan and Khair [20] under the influence of magnetic field. More recently, Singh [21] has also studied effects of mass
transfer on MHD free convection flow of a viscous fluid through a vertical channel using the Laplace transform
technique considering symmetrical heating and cooling of channel walls.
In the above mentioned studies the heat source effect and thermal diffusion effect (commonly known as Soret effect)
is ignored, although effective cooling of electronic equipment has become warranted and cooling of electronic equipment
ranges from individual transistors to main frame computers and thermal diffusion effect has been utilized for isotopes
separation in the mixture between gases with very light molecular weight (hydrogen, helium) and medium molecular
weight (nitrogen, air) where the thermal diffusion effect is found to be of a magnitude that cannot be neglected (Eckert
and Drake [22]).
In view of the application of heat source and thermal diffusion effect, it is proposed to study two dimensional MHD
free convection and mass transfer flow past an infinite vertical porous plate taking into account the combined effect of
heat source and thermal diffusion in the presence of large suction (Singh and Dixit [14]). The similarity solutions are
obtained by employing the perturbation technique as demonstrated by Alam and Sattar [15].
BASIC EQUATIONS
Consider a steady free convection and mass transfer flow of an incompressible, electrically conducting, viscous
fluid past an electrically non-conducting continuously moving infinite vertical porous plate. Introducing a cartesian
coordinate system, x-axis is chosen along the plate in the direction of flow and y-axis normal to it. A uniform magnetic
field is applied normally to the flow region. The plate is maintained at a constant temperature Tw and the concentration
is maintained at a constant value Cw .
94 The Arabian Journal for Science and Engineering, Volume 32, Number 1A January 2007

The temperature of uniform flow is T∞ and the concentration of uniform flow is C∞ . Considering the magnetic
G
Reynold's number to be very small, the induced magnetic field is assumed to be negligible, so that B = (0,B0 (x),0)
. The
G G
equation of conservation of electric charge is ∇.J = 0
G
, where ( ) J = J x , J y , Jz
and the direction of propagation is
assumed along y-axis so that
G
J does not have any variation along y-axis so that the y derivative of J
G
namely = 0
J y
∂
∂
y
resulting in J y = constant. Also the plate is electrically non-conducting therefore the constant J y = 0 everywhere in the
flow. Considering the Joule heating and viscous dissipation terms to be negligible and that the magnetic field is not
enough to cause Joule heating, the term due to electrical dissipation is neglected in the energy equation. The density is
considered a linear function of temperature and species concentration so that the usual Boussinesq's approximation is
taken as [ { ( ) ( )}] = − T −T∞ + C −C∞ *
ρ ρ0 1 β β .
Within the frame work of delete such assumptions the equations of continuity, momentum, energy and concentration are:
v
u
∂
∂
Continuity equation : + = 0
y
x
∂
∂
(1)
u
v u
2
∂
u ∂
∂
u ϑ
β
Momentum equation : ( ) + = + g T −T∞
y
y
x
∂
∂
∂
2
B 2
( x )+ g ( C C ) u
* − − ' 0 ∞ (2)
ρ
σ
β
2
∂
T
K
v T
u T
∂
∂
∂
Energy equation : ( ) + = +Q T −T∞
y
C
y
x
p
2
∂ ρ
∂
(3)
2
2
u C M T ∂
D T
D C
v C
∂
∂
∂
∂
+ = + (4)
Concentration equation : 2
2
y
y
y
x
∂
∂
∂
The boundary conditions relevant to the problem are :
u =U0 , v = v0 (x) , T = Tw , C = C(x) at y = 0
u = 0 , v = 0 , T = T∞ , C = C∞ as y →∞ (5)
Where u, v are velocity components along x-axis and y-axis, g acceleration due to gravity, T the temperature, Tw the
wall temperature, T∞ the temperature of the uniform flow, K thermal conductivity, σ ' the electrical conductivity, DM
the molecular diffusivity, U0 the uniform velocity, C the concentration of species, C∞ the concentration of species for
uniform flow, B0(x) the uniform magnetic field, Cp the specific heat at constant pressure, Q the constant heat source
(absorption type), DT the thermal diffusivity, Cw the mean concentration, C(x) variable concentration at the plate,
v0(x) suction velocity, ρ the density, ϑ the kinematic viscosity, β the volumetric coefficient of thermal expansion,
and β * is the volumetric coefficient of thermal expansion with concentration and the other symbols have their usual
meaning.
For similarity solution, the plate concentration C(x) is considered to be
( ) ( )x
C x = C∞ + Cw − C∞
We introduce the following local similarity variables :

ψ = 2ϑxU0 f (η ),
θ η , ( ) ( )
= 0 , ( )
x
y U
2
ϑ
η
T −
T
w
∞
−
=
T T
∞
C x C
w
∞
φ η ,
=
C C
∞
−
−
Introducing the above stated similarity variables using the relations
∂ψ
= and the equation of continuity (1),
y
u
∂
we obtain :
ϑ
v = U '−
u =U0 f '(η ) and 0 ( η
f f )
x
2
Substituting u and v in Equation (1), the continuity equation is satisfied.
Introducing the above values in Equations (2) – (4), we have :
f ''' + ff '' − Mf ' + Grθ + Gmφ = 0 (6)
θ '' + Pr fθ ' − SPrθ = 0 (7)
φ '' − 2 Sc f φ' + Sc f φ ' + S Sc θ '' = 0
(8)
0
Where
C
μ
= (Prandtl number),
P p
r
K
S = xQ (Heat Source parameter),
0
2
U
ϑ
( )
= (Schmidt number), D x
S
M
c
g Tw − G T ∞
x
r
2
0
2
U
=
β
(Grashof number),
( )
0
2
' 0 2
U
M B x x
σ
= (Magnetic parameter),
ρ
−
S T T
∞
−
0 (Soret number),
∞
w
=
C C
w
( )
g * Cw − G C ∞
2
x
m
2
0
U
=
β
(Modified Grashof number)
The boundary conditions are transformed to :
f = fw , f ' = 1 , θ =1 , φ =1 at η = 0
f ' = 0 , θ = 0 , φ = 0 as η →∞ (9)
f v x x w ϑ
where ( )
= − and primes denotes the derivatives with respect to η . Here, fw > 0 denotes the injection and
0
0
2
U
fw < 0 the suction.
METHOD OF SOLUTION
To obtain the solution for large suction, we make the following transformations :
= 2 , φ (η ) f Z(ξ ) w
ξ =ηfw , f (η ) = fwX (ξ ), θ (η ) fwY(ξ )
= 2 (10)
Substituting (10) in Equations (6) – (8), we get :
X ''' + XX '' =ε (MX ' −GrY −GmZ ) (11)
Y '' + Pr XY ' =εSPrY (12)
Z '' − 2 S c ZX ' + S ' c XZ + S c S 0
Y '' = 0
(13)

The boundary conditions (9) reduce to
X = 1 , X ' =ε , Y =ε , Z =ε at ξ = 0
X ' = 0 , Y = 0 , Z = 0 as ξ →∞ (14)
⎞
⎛
= 2
1
fw
where ⎟ ⎟
ε is very small as for large suction fw >1. Hence, we can expand X, Y, and Z in terms of ε as
⎠
⎜ ⎜
⎝
follows :
( ) 1 ( ) ( ) 3
( ) 3 ....
X ξ = +εX1 ξ +ε X 2
ξ +ε X ξ + (15)
2
( ) ( ) ( ) 3
3( ) ....
Y ξ =εY1 ξ +ε Y 2
ξ +ε Y ξ + (16)
( ) ( ) ( ) 3( ) ....
2
3
Z ξ =εZ1 ξ +ε Z 2
ξ +ε Z ξ + (17)
2
Introducing X (ξ ) , Y(ξ ) , and Z(ξ ) in Equations (11) - (14) and considering up to order O(ε 3 ), we get the
following three sets of ordinary differential equations and corresponding boundary conditions :
First order O(ε ) :
0 ''1
' ''1
X + X = (18)
Y1 ''
+ PrY ' 1
= 0
(19)
''
Z + ScZ = −ScS Y (20)
0 1
'1
''1
Second order O(ε 2 ):
X + X + X X = MX −GrY −GmZ (21)
1 1
'1
''1
1
''2
' ''2
'
Y + PrY = SPrY − Pr X Y (22)
1 1 1
'2
''2
''2
0
'1
Z + ScZ = 2ScX Z − ScX Z − ScS Y (23)
1 1
'1
'2
''2
Third order O(ε 3 ):
X + X = MX − X X − X X − GrY − GmZ (24)
2 2
''1
2
''2
1
'2
''3
' ''3
'
Y + PrY = SPrY − Pr X Y − Pr X Y (25)
2 1
'2
2 1
'3''3
''3
0
'2
Z + ScZ = 2ScX Z + 2ScX Z − ScX Z − ScX Z − ScS Y (26)
1
'1
1 2
'2
2
'1
'3
''3
0 1 = X , 1 '1
X = , Y1 = 1 , Z1 = 1 at ξ = 0
0 '1
X = , Y1 = 0 , Z1 = 0 as ξ →∞ (27)
0 2 = X , 0 '2
X = , Y2 = 0 , Z2 = 0 at ξ = 0
0 '2
X = , Y2 = 0 , Z2 = 0 as ξ →∞ (28)

0 3 = X , 0 '3
X = , Y3 = 0 , Z3 = 0 at ξ = 0
0 '3
X = , Y3 = 0 , Z3 = 0 as ξ →∞ . (29)
The solutions of the above coupled equations under the prescribed boundary conditions are:
X = 1− e−ξ 1 (30)
Y = e−Prξ 1 (31)
Z1 = A1e−Prξ + (1− A1)e−Scξ (32)
X = 1 e− ξ + A + A ξ e−ξ + A e−Prξ + A e−Scξ + A (33)
( ) 7 2 8 9 10
2
2 4
Y = (A − A ξ )e−Prξ − A e−(1+Pr )ξ
2 11 12 11 (34)
Z2 = (A24 + A18ξ )e−Scξ + (B1 + A22ξ )e−Prξ
+ A e−( +Pr )ξ + A e−(1+Sc )ξ
20 21
(35)
1
2
2
X = A + A + B − A e B A ⎞
e 1
e
⎞
+ ⎛ + ⎟ ⎟
⎠
ξ ξ ξ 2 ξ 2ξ 3ξ
− − − − ⎟⎠
⎜⎝
⎛
⎜ ⎜
3 41 40 2 3
24
2
2 2
⎝
+ (A33 + A38 + A37ξ )e−Prξ + (A34 − A39 − A32ξ )e−Scξ
− A e−( +Pr )ξ − A e−(1+Sc )ξ
35 36
(36)
1
Y ⎛
A A ⎞
= − B ξ − ξ e − P r ξ + ( B + B ξ ) e − ( 1
+ P )r ξ ⎟⎠
⎜⎝
5 6
43 2
3 52 4 2
+ A e−(2+Pr )ξ
50
+ A e−(Pr +Sc )ξ 51 (37)
Z = (B + B ξ + A ξ )e−Prξ + B e− Prξ + (B + B ξ )e−(1+Pr )ξ
10 11
2
9
2
3 7 8 116
B e ( Pr )ξ (B B ξ B ξ )e Scξ (B A ξ )e 2Scξ
+ − + + + + − + + −
16 77
2
13 14 15
2
12
+ (B + B ξ )e−( +Sc )ξ + B e−( 2
+Sc )ξ + B20e−(Pr +Sc )ξ
17 18 19
. (38)
1
Introducing (15), (16), and (17) in (10), the velocity, the temperature and the concentration fields are obtained as:
3
( η ) [ ( ξ ) ε ( ) 2 ( ) ] 2
'ξ ε ξ ''1
u =U0 f =U 0
X + X + X (39)
'
θ (η ) (ξ ) ε (ξ ) ε 2
(ξ )
3 = Y1 + Y2 + Y (40)
φ (η ) (ξ ) ε (ξ ) ε 2
(ξ )
3 = Z1 + Z2 + Z (41)
The main quantities of physical interest are the local skin-friction, local Nusselt number, and the local Sherwood
number.
The equation defining the wall skin-friction is :

τ μ (42)
=0
⎞
⎟ ⎟⎠
⎛
⎜ ⎜⎝
=
u
∂
∂
y y
Thus from Equation (39), we have :
[ ''
( )] 0
τ ∝ f η η =
1 ε 1 P A S A A ε
1 1
r c A A = − + + + + + ⎡− + + 2 2
⎢⎣
[ ] 2
( ) 5 6 7 1 2
4
2
A
A
A A A
1 27 28 29
− − + + −
2 P
1
r S
c +
Pr
25 26
⎤
A (43)
( ) ( )⎥⎦
30
A S A P
− + − 1 + 1 +
32 c 37
r
−
1
c
S
The local Nusselt number denoted by Nu is :
=0
⎞
⎟ ⎟⎠
⎛
N T
⎜ ⎜⎝
= −
y
∂
u ∂
y
(44)
Hence, we have from (40) :
[ '
( )] 0
Nu ∝ θ η η =
⎡
P A A A A43
[ ] ⎢⎣
2
ε 11 12 ε
= − + − + − −
r P
r
42
⎤
⎥⎦
A
45 2
1
A A S A
+ − 49 − 50 −
51
+
P
c
r
(45)
The local Sherwood number denoted by Sh is :
=0
⎞
⎟ ⎟⎠
⎛
S C
⎜ ⎜⎝
= −
y
∂
h ∂
y
(46)
Hence, we have from (41) :
[ '
( )] 0
Sh ∝ φ η η =
( ) [ ( ) = A1 Sc − Pr − Sc +ε A18 − Pr A19 − 1+ Pr A20
( ) ] − 1+ Sc A21 + A22 + Pr A23 − Sc A24
[ ] 120 121 122
2 A S P A A +ε c − r + (47)

VERIFICATION OF THE SOLUTIONS IN SOME SIMPLE CASES
(i) If the heat source parameter S and the Soret number S0 are not taken into account in the equations, the
solutions obtained are similar to those of Alam and Sattar [15] except notations in constants.
(ii) Neglecting the heat source parameter S and taking Soret number S0 into account, the solutions obtained are
similar to those of Sattar and Alam [13] except a term on rotation parameter which also appears in their solution
due to rotating system.
(iii) Taking into account the heat source parameter S and neglecting the Soret number S0 , the solutions obtained are
similar to those of Jha and Prasad [10] except a term on permeability parameter in their solution due to porosity
of the medium.
(iv) If the heat source parameter S and Soret number S0 are ignored, the solutions reduce to those obtained by
Raptis and Soundalgekar [7], except the terms due to consideration of induced magnetic field.
(v) Ignoring the mass transfer and magnetic field, the equations resemble to those of Sattar and Kalim [23].
DISCUSSION AND CONCLUSIONS
In order to get physical insight into the problem under study, the velocity field, temperature field, concentration
field, skin-friction, rate of heat transfer, and rate of mass transfer are discussed by assigning numerical values to the
parameters encountered into the corresponding equations. To be realistic, the values of Schmidt number ( Sc ) are chosen
for hydrogen ( Sc = 0.22 ), helium ( Sc = 0.30 ), water-vapor ( Sc = 0.60 ), oxygen ( Sc = 0.66 ), and ammonia ( Sc = 0.78 ) at
25°C and one atmosphere pressure. The values of Prandtl number ( Pr ) are chosen for mercury ( Pr = 0.025 ), air
( Pr = 0.71 ), water ( Pr = 7.0 ) and water at 4°C ( Pr =11.4 ). Grashof number for heat transfer is chosen to be
Gr = 10.0, 15.0 −10.0, −15.0 and modified Grashof number for mass transfer Gm =15.0, 20.0 −15.0, − 20.0 respectively.
The values Gr > 0 , Gm > 0 correspond to cooling to the plate while the values Gr < 0 , Gm < 0 correspond to heating of
the plate. The values of magnetic parameter (M = 0.0, 0.5, 1.5 ), suction parameter ( fw = 3.0, 5.0, 7.0, 9.0 ) Soret number
( S0 = 3.0, 4.0, 8.0 ) and heat source parameter ( S = 1.0, 2.0, 4.0 ) are chosen arbitrarily.

8
6
4
2
0
-2
-4
-6
-8
Curves M S S0
I 0.0 0.0 0.0
II 0.5 0.0 0.0
III 1.5 0.0 0.0
IV 0.5 1.0 3.0
V 0.5 0.0 3.0
VI 0.5 1.0 0.0
VII 1.5 1.0 3.0
Cooling of the plate (Gr = 10.0, Gm = 15.0)
I
III
V
VII
IV
II
VI
0 1 2 3 4 5 6 7 8 9
η −−−−−−−>
u ------->
Figure 1. Velocity field for various values of M, S, and S0 (Pr = 0.71,Sc = 0.22, fw = 5.0 and U0 = 1.0).
Figure 1 represents variation in the velocity field for various values of M, S, and S0 in case of cooling
(Gr = 10.0,Gm =15.0 ) and heating (Gr = −10.0,Gm = −15.0 ) of the plate at Pr = 0.71 , Sc = 0.22 , fw = 5.0 and U0 =1.0 . It is
observed that for an externally cooled plate (i) an increase in M decreases the velocity field in the absence of S and S0 ;
(ii) an increase in M increases the velocity field in the presence of S and S0 ; (iii) an increase in S0 only increases the
velocity field while an increase in S only decreases the velocity field in the presence of M; (iv) an increase in M increases
the velocity field with constant values of S and S0 ; (v) reverse effects are observed for an externally heated plate.

8
Figure 2. Velocity field for various values of Sc, S, and S0 (Pr = 0.71, M = 0.5, fw = 5.0 and U0 = 1.0)
6
4
2
0
-2
-4
-6
-8
Curves Sc S S0
I 0.22 0.0 0.0
II 0.30 0.0 0.0
III 0.60 0.0 0.0
IV 0.22 1.0 3.0
V 0.22 0.0 3.0
VI 0.22 1.0 0.0
VII 0.30 1.0 3.0
III
VI
II
V
VII I
IV
0 1 2 3 4 5 6 7 8 9
η -------->
u -------->
Figure 2. Velocity field for various values of Sc, S, and S0 (Pr = 0.71, M = 0.5, fw = 5.0 and U0 = 1.0).
Figure 2. represents variation in the velocity field for various values of Sc , S, and S0 in case of cooling
( Gr = 10.0,Gm =15.0 ) and heating ( Gr = −10.0,Gm = −15.0 ) of the plate at Pr = 0.71 , M = 0.5 , fw = 5.0 , and U0 = 1.0 . It is
observed that for an externally cooled plate (i) an increase in Sc decreases the velocity field in the absence of S and S0 ; (ii) an
increase in S and S0 both increases the velocity field for the given value of Sc ; (iii) an increase in S0 only increases the
velocity field while an increase in S only decreases the velocity field for the given value of Sc ; (iv) An increase in Sc increases
the velocity field for the given values of S and S0 ; (v) all these effects are observed in reverse order for an externally heated
plate.

Figure 3. Velocity field for various values of Pr, S, and S0 (M = 0.5, Sc = 0.22, fw = 5.0 and U0 = 1.0)
8
6
4
2
0
-2
-4
-6
-8
Curves Pr S S0
I 0.71 0.0 0.0
II 7.00 0.0 0.0
III 0.71 1.0 3.0
IV 0.71 0.0 3.0
V 0.71 1.0 0.0
VI 7.00 1.0 3.0
III
V
II I VI IV
0 1 2 3 4 5
I IV
II
η -------->
u -------->
V
III
VI
Figure 3. Velocity field for various values of Pr, S, and S0 (M = 0.5, Sc = 0.22, fw = 5.0 and U0 = 1.0).
Figure 3 shows the variation in velocity field for various values of Pr , S, and S0 in case of cooling ( Gr = 10.0,Gm =15.0 )
and heating ( Gr = −10.0,Gm = −15.0 ) of the plate at M = 0.5 , Sc = 0.22 , fw = 5.0 , and U0 = 1.0 . It is noted that for an
externally cooled plate (i) an increase in Pr decreases the velocity field in the absence of S and S0 ; (ii) an increase in S and
S0 increases the velocity field for the given value of Pr ; (iii) an increase in S0 only increases the velocity field while an
increase in S only decreases the velocity field for the given value of Pr ; (iv) an increase in Pr increases the velocity field for
the given values of S and S0 ; (v) all these effects are observed in reverse order for an externally heated plate.

Figure 4. Velocity field for various values of fw, S, and S0 (M = 0.5, Sc = 0.22, Pr = 0.71 and U0 = 1.0)
8
6
4
2
0
-2
-4
-6
-8
Cooling of the plate (Gr = 10.0, Gm = 15.0) Curves fw S S0
I 5.0 0.0 0.0
II 7.0 0.0 0.0
III 9.0 0.0 0.0
IV 5.0 1.0 3.0
V 5.0 0.0 3.0
VI 5.0 1.0 0.0
VII 7.0 1.0 3.0
III
II
VII
VI
I
V
IV
0 1 2 3 4 5 6 7 8 9
η -------->
u --------->
Figure 4. Velocity field for various values of fw, S, and S0 (M = 0.5, Sc = 0.22, Pr = 0.71 and U0 = 1.0).
Figure 4 shows the variation in velocity field for various values of fw , S, and S0 in case of cooling ( Gr = 10.0,Gm =15.0 )
and heating ( Gr = −10.0,Gm = −15.0 ) of the plate at M = 0.5 , Sc = 0.22 , Pr = 0.71 , and U0 = 1.0 . It is noted that for
externally cooled plate (i) an increase in fw decreases the velocity field in the absence of S and S0 ; (ii) an increase in S and
S0 increases the velocity field for the given value of fw ; (iii) an increase in S0 only increases the velocity field while an
increase in S only decreases the velocity field for the given value of fw ; (iv) An increase in fw decreases the velocity field for
the given values of S and S0 both; (v) all these effects are observed in reverse order for externally heated plate.

Curves Pr S fw
I 0.71 0.0 5.0
II 7.00 0.0 5.0
III 11.4 0.0 5.0
IV 0.71 3.0 5.0
V 0.71 5.0 5.0
VI 0.71 3.0 7.0
VII 0.71 3.0 9.0
II
V
VI
VII
IV
I
III
0 1 2 3 4 5 6 7 8
η -------->
Figure 5. Temperature field for various values of Pr, S, and fw .
1.2
1
0.8
0.6
0.4
0.2
0
θ --------->
Figure 5 represents variation in the temperature field for various values of Pr , S, and fw . It is observed that an increase
in Pr , S, fw (taking into account the presence of the individual parameters or taking two or three parameters at a time)
decreases the temperature field. It is interesting to note that the temperature decreases rapidly with increase in Pr .
Curves S S0
I 0.0 0.0
II 2.0 2.0
III 5.0 2.0
IV 5.0 4.0
V 2.0 4.0
VI 0.0 4.0
VI
V
IV
I
II
III
0 1 2 3 4 5 6 7 8 9
η −−−−−−>
Figure 6. Concentration field for different values of S and S0 .
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
φ −−−−−−>
Figure 6 represents variation in the concentration field for various values of S and S0 . It is observed that an increase in S
or S0 (or both) increases the concentration field. It is also observed that the concentration field increases more rapidly in the
presence of S and S0 both in comparison to S or S0 .

Curves Sc S S0
I 0.22 0.0 0.0
II 0.30 0.0 0.0
III 0.22 0.0 2.0
IV 0.22 2.0 2.0
V 0.22 5.0 4.0
VI 0.60 5.0 4.0
VII 0.78 5.0 4.0 II
I
III
IV
V
VII VI
0 1 2 3 4 5 6 7 8 9
η --------->
Figure 7. Concentration field for various values of Sc, S, and S0 (fw = 5.0) .
2.5
2
1.5
1
0.5
0
φ -------->
Figure 7 shows variation in the concentration field for various values of Sc , S, and S0 . It is observed that (i) an increase
in Sc decreases the concentration field in absence of S and S0 ; (ii) an increase in S0 increases the concentration field in the
absence of S for the given value of Sc ; (iii) an increase in S or S0 (or both) increases the concentration field; (iv) an increase
in Sc , S, and S0 increases the concentration fields more rapidly near the plate and after attaining a maximum value it decreases
rapidly.
Figure 8. Effects of Pr, Sc, S, S0 and fw on concentration field (fw = 5.0)
Curves Pr Sc S S0 fw
I 0.71 0.22 2.0 4.0 5.0
II 7.00 0.22 2.0 4.0 5.0
III 0.71 0.66 2.0 4.0 5.0
IV 0.71 0.22 5.0 4.0 5.0
V 0.71 0.22 2.0 8.0 5.0
VI 0.71 0.22 2.0 4.0 7.0
IV
III
V
------>
φ II
I
VI
η -------> 0 1 2 3 4 5 6 7 8 9
Figure 8. Effects of Pr, Sc, S, S0 and fw on concentration field (fw = 5.0) .
2.5
2
1.5
1
0.5
0
Figure 8 shows variation in the concentration field for various values of Pr , Sc , S, S0 , and fw . It is observed that an
increase in Pr , Sc , S, or S0 increases the concentration field but an increase in fw decreases the concentration field. It is also
observed that an increase in Pr or Sc decreases the concentration field rapidly in comparison to other parameters.

2
1.5
1
0.5
0
I Hydrogen
II Helium
III Water-vapour
IV Oxygen
V Ammonia
IV
I
II
III
V
0 1 2 3 4 5 6 7 8 9
η --------->
Figure 9. Concentration field for different gases (S0 = 4.0, S = 2.0 and fw = 5.0) .
φ -------->
Figure 9 shows variation in the concentration field for the gases hydrogen, helium, water-vapor, oxygen, and ammonia at
S0 = 4.0, S = 2.0 and fw = 5.0. It is observed that an increase in Sc increases the concentration field. This indicates that
ammonia is useful if more concentration field is desired while hydrogen is useful if less concentration field is needed.
I Pr = 0.71 Air
II Pr = 0.025
III Pr = 7.00 Water
IV Pr = Water at 40C
IV
III
I
II
0 0.2 0.4 0.6 0.8 1
η ------>
Figure 10. Effects of Pr on temperature field (S = 1.0, fw = 3.0) .
1.2
1
0.8
0.6
0.4
0.2
0
θ ------->
Figure 10 shows the effects of Pr on temperature field at S = 1.0 and fw = 3.0 for air, mercury, water, and water at 4°C.
It is observed that temperature field remains almost stationary for mercury in comparison to air. The temperature field
decreases rapidly for water.

Table 1. Numerical Values of Skin-Friction (τ ) due to Cooling of the Plate
S. No. Gr Gm S S0 M Sc fw Pr τ
1 10.0 15.0 0.0 0.0 0.5 0.22 5.0 0.71 1.3944
2 10.0 15.0 1.0 3.0 0.5 0.22 5.0 0.71 3.1918
3 10.0 15.0 0.0 3.0 0.5 0.22 5.0 0.71 3.0207
4 10.0 15.0 1.0 0.0 0.5 0.22 5.0 0.71 1.3626
5 10.0 15.0 1.0 3.0 1.5 0.22 5.0 0.71 1.7858
6 10.0 15.0 1.0 3.0 0.5 0.60 5.0 0.71 2.9711
7 10.0 15.0 1.0 3.0 0.5 0.22 7.0 0.71 1.3832
8 10.0 15.0 1.0 3.0 0.5 0.22 5.0 7.00 2.5217
9 15.0 15.0 1.0 3.0 0.5 0.22 5.0 0.71 3.3238
10 10.0 20.0 1.0 3.0 0.5 0.22 5.0 0.71 4.4775
Table 1 represents the numerical values of skin-friction (τ ) at the plate due to variation in Grashof number (Gr ),
modified Grashof number (Gm ), heat source parameter (S), Soret number ( S0 ), magnetic parameter (M), Schmidt number
( Sc ), suction parameter ( fw ), and Prandtl number ( Pr ) for an externally cooled (Gr > 0 , Gm > 0 ) plate. It is observed that (i)
the presence of S and S0 both increases the skin-friction in comparison to their absence; (ii) the presence of S0 only increases
the skin-friction while the presence of S only decreases the skin-friction; (iii) an increase in M, Sc , fw , or Pr decreases the
skin-friction while an increase in Gr or Gm increases the skin friction in the presence of S and S0 .
Table 2. Numerical Values of Skin-friction (τ ) due to Heating of the Plate
S. No. Gr Gm S S0 M Sc fw Pr τ
1 –10.0 -15.0 0.0 0.0 0.5 0.22 5.0 0.71 –3.2532
2 –10.0 –15.0 1.0 3.0 0.5 0.22 5.0 0.71 –5.0506
3 –10.0 –15.0 0.0 3.0 0.5 0.22 5.0 0.71 –4.8795
4 –10.0 –15.0 1.0 0.0 0.5 0.22 5.0 0.71 –3.2214
5 –10.0 –15.0 1.0 3.0 1.5 0.22 5.0 0.71 –3.5846
6 –10.0 –15.0 1.0 3.0 0.5 0.60 5.0 0.71 –4.8298
7 –10.0 –15.0 1.0 3.0 0.5 0.22 7.0 0.71 –3.3065
8 –10.0 –15.0 1.0 3.0 0.5 0.22 5.0 7.00 –4.3805
9 –15.0 –15.0 1.0 3.0 0.5 0.22 5.0 0.71 –5.1826
10 –10.0 –20.0 1.0 3.0 0.5 0.22 5.0 0.71 –6.3363
Table 2 shows the numerical values of skin-friction due to variation in the above stated parameters for an externally
heated ( Gr < 0 , Gm < 0 ) plate. It is observed that (i) the presence of S and S0 both decreases the skin-friction in comparison to
their absence; (ii) the presence of S0 only decreases the skin-friction while the presence of S only increases the skin-friction;
(iii) an increase in M, Sc , fw , or Pr increases the skin-friction while an increase in Gr or Gm decreases the skin friction in the
presence of S and S0 .

Table 3. Numerical Values of the Rate of Heat Transfer ( Nu )
S. No. Pr S fw Nu
1 0.71 0.0 5.0 – 0.7266
2 0.71 3.0 5.0 – 0.8466
3 7.00 0.0 5.0 – 7.0351
4 7.00 3.0 5.0 – 7.1550
5 0.71 0.0 7.0 – 0.7797
6 0.71 3.0 7.0 – 0.7184
Table 3 represents the numerical values of the rate of heat transfer in terms of Nusselt number ( Nu ) due to variation in
Prandtl number ( Pr ), heat source parameter (S), and suction parameter ( fw ). It is observed that (i) in the absence of heat
source parameter, an increase in Pr decreases the rate of heat transfer while an increase in fw increases the rate of heat
transfer; (ii) in the presence of heat source parameter, an increase in fw decreases the rate of heat transfer while an increase in
Pr increases the rate of heat transfer; (iii) the rate of heat transfer decreases in presence of S in comparison to absence of S.
Table 4. Numerical Values of the Rate of Mass Transfer ( Sh )
S. No. Sc S S0 fw Pr Sh
1 0.22 0.0 0.0 5.0 0.71 –0.2448
2 0.22 1.0 3.0 5.0 0.71 0.2965
3 0.22 0.0 3.0 5.0 0.71 0.2437
4 0.60 1.0 3.0 5.0 0.71 0.8075
5 0.22 1.0 3.0 7.0 0.71 0.2730
6 0.22 1.0 3.0 5.0 7.00 4.4701
7 0.22 3.0 5.0 5.0 0.71 0.8334
Table 4 represents the numerical values of the rate of mass transfer in terms of Sherwood number ( Sh ) due to variation in
Schmidt number ( Sc ), heat source parameter (S), Soret number ( S0 ), suction parameter ( fw ), and Prandtl number ( Pr ). It is
observed that (i) the presence of S and S0 both increases the rate of mass transfer while the presence of S0 only increases the
rate of mass transfer at a slow rate; (ii) an increase in Sc or Pr increases the rate of mass transfer in the presence of S and S0
while an increase in fw decreases the rate of mass transfer; (iii) an increase in S and S0 both increases the rate of mass
transfer.
ACKNOWLEDGMENT
The authors are extremely thankful to Prof. V. M. Soundalgekar, Ex Professor of Applied Mathematics, I. I. T.,
Powai, Bombay, the learned referees, and Dr. Harry A. Mavromatis, Managing Editor, The Arabian Journal for Science
and Engineering, for improving the depth of the paper.

REFERENCES
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[3] V. C. A. Ferraro and C. Plumpton, An Introduction to Magneto Fluid Mechanics. Oxford: Clarendon Press, 1966.
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APPENDIX
A S S P
= − 0
1 , A2 =1+ M , A3 = Gr + A1Gm , ( ) A4 = 1− A1 Gm ,
c r
P −
S
r c
A A 3
, ( 1
) =
Pr Pr
( ) 1
5 −
A A 4
, 7 2 5 6
=
Sc Sc
6 −
A = 1 +M − A − A ,
A A5
8 = ,
Pr
A A6
9 = , 10 4 7 8 9
Sc
A = − 1 − A − A − A ,
A P , ( ) A12 = S + Pr ,
1
2
=
r
P
11 +
r
A13 = PrSc ( A1 − S0Pr A11 − 2S0A12 ) , 2
( ) 1
A14 = Sc 1− A ,
[ ( ) ] 11
A15 = Sc 2A1 + Pr A1 + S0 1+ Pr 2
A , A16 = Sc (1− A1)(2 + Sc ) ,
A17 = Sc S0Pr A 11
,
2
A A
= 14
A A
= 13
18 , ( ) Pr Pr Sc
Sc
−
19 ,
−
A A
=
1 1
15
A A
=
1
16
20 ( , + Pr )( + Pr −
Sc
) ( Sc
) A A
= 17
21 , +
Pr ( Pr Sc
) 22 ,
−
( )
2( )2
A A 17
S P
23
2
−
c r
= , A24 = A23 − A19 − A20 − A21 ,
P P −
S
r r c
A =1+ M + A , ( ) A26 = A2 1+ A2 + A7 − A10 ,
25 2 2
2
( ) ( )m
A27 = Pr A8 M + Pr + A11Gr + A19 − A23 G ,
A28 = A9Sc (M + Sc )+ A24Gm , A A ( Pr 2
) A11Gr A20Gm
29 = 8 −1 + − ,
A A ( Sc 2
) A21Gm
30 = 9 −1 − , A31 = A12Gr − A22Gm , A32 = A18Gm ,
A A 27
, 2 ( 1)
=
Pr Pr
33 −
2( 1)
A A 28
, ( )2
=
Sc Sc
34 −
A A 29
,
=
Pr Pr
35 +1
37 , ( )
A A
A A 30
, Pr ( Pr )
=
Sc Sc
36 +1
( )2
31
−
=
2 1
A ( )
A =
37 3 P r
−
2
, 38 −
( ) 1
P P
r r
A =
A 32 3 S c
−
2
,
39 −
( ) 1
S S
c c
( )
A = − A − A − A + A + A − P A − S A + 1
+
P A
r c r
25 26 32 33 34 35
2 2
1
3
1
40 2 2
2
2
8
( 1
S ) A 36 A 37 P A 38 S A
39
+ + + − +
c r c
,
( ) ( )
A A A A A A P A S A P A
1 1
= − + + + − − + − + − −
r c r
25 26 32 33 34 35
2 2
1
1
1
41 2 2
4
2
12
( ) ( ) 36 37 38 39
S A A P A S A
1 1
− − + − − −
c r c
,
A42 = A11S + Pr A11 + A12 + A10 , A43 = A12S − Pr A12 ,
( ) 11 11 12 7
2
A44 = SPr A11 + Pr A + Pr 1+ Pr A + Pr A − Pr A ,
1 1 A = Pr + Pr A + Pr ,
A45 = Pr 2
A 12 + Pr A 2
, ( ) 2
46 11 4
A47 = Pr A9 ,
44
48 ,
Pr
A A
+
=
1

⎞
⎟ ⎟⎠
⎛
+
⎜ ⎜⎝
+
A A
45
A A
=
2 2
46
49 , +
( Pr
) =
P
r
P
2
r P
r
1
1
50 ,
+
A A
47
[ ] 51 = , A52 = − A48 + A49 + A50 + A51 ,
Sc ( Sc +
Pr
) 53 , ( )
A S A
2 18
c
S
=
1
( +
) c
A 2 S c 2
+
S c
A
= ,
54 1
( )2
18
c
S
+
A S A
2 c
18
=
1 1
A S A
2
c
18
r r c
= ,
55 , ( )( )2
( + P )( r + P −
S
) r c
56 1 1
P P S
+ + −
( )
A S A A
2 19 −
23
c
P P S
=
1 1
A S A
2 24
c
S
=
1
57 ( )( , + r + r −
) ( ) c
c
58 ,
+
A S A
2 c
20
=
2 2
A S A
c
S
=
2
21
59 ( , + P )( ) ( ) r + P r −
S
c
c
60 ,
+
( )
( )( )
61 , A S A A A
2 1 2 −
7
c
P P S
=
1 1
( + )( ) r + r −
c
A S A A A
2 1 − 1 2 −
7
62 ,
( ) c
c
S
+
=
1
63 , ( )
A S A A
2 c
1 2
−
=
1 1
( + P )( r + P +
S
) r c
A S A A
2 1 1 2
− −
=
1 2 1
c
S S
64 ,
( + )( ) c +
c
( )
c
65 , −
A A S
=
2 2
1
( + P )( r + P −
S
) r c
A 1 A 1
S
c
S
− −
=
2 2
66 ,
( +
) c
A67 = −ScA1A8 , ( ) A68 = − 1− A1 A9 ,
[ ( ) ]
A S P A A S A A
A P S A
2 1 , ( )( ) r r c
− − = c r 1 8 +
c
1 9
( ) r r c
P P +
S
69
r c
P P S
=
4 2 2
1
70 ,
+ + +
( )
( ) c
2
A S c
A
1 −
1
A P S A A
r c
P P S
=
1 1
1 7
71 , +
S
( )( ) r r c
=
8 2
72 ,
+ + −
( )
( ) c
2
A S c
A A
1 1 7
A P S A A
r c
P P S
=
2 2
1 8
73 , S
( ) r r c
−
+
=
1
74 ,
−
( )
A A A
1 1 9
= ,
2
75
−
A S A A
c
P S
= 1 10
76 ,
−
r c
A77 = −(1− A1)A10Sc , ( )
A P S A A A A S
A P S A A
1 , ( r )( r c )
+ −
2
r c 1 9 1 8
c
( ) r r c
P P +
S
=
78
r c
P P S
=
1 1
1 2
79 ,
+ + −
( )
A P S A A + P −
S
r c 1 2
r c
2 2
= ,
80 1 1
( )2( )2
P P S
+ + −
r r c
81 , ( ) ( )
( )
( c )
2
A S c
A A
2 1 −
1
S
+
=
1
+ −
A S 2 S A 1
A
2 1
c c
= ,
82 1
( )2
2
c
S
+
( )
18
83 , c
A S A
c
S
+
=
1
A P S A −
A
r c
P P S
= 19 23
84 ,
r ( r −
c )
( )
( )
85 , − A P S A −
A
r c
P P S
=
1 1
19 23
( + r )( + r −
c )
A S P A
1 +
20
c r
P P S
=
1 1
86 ,
( + r )( + r −
c )

( )
( )
87 , A S P A
1 20
− +
c r
P P S
=
2 2
( + r )( + r −
c )
A S S A
1 +
21
c c
88 ,
( +
S
c )
=
1
( )
( c )
A S S A
1 21
− +
=
c c
89 , A90 = −ScA24 ,
S
+
2 2
2
A S A
24
−
A S c
A
= 22
91 , ( ) r r c
c
c
S
+
−
=
1
92 ,
P P −
S
A S c
A
=
1 1
22
2
A S A
c
S
−
=
1
18
93 ( , + P )( ) ( ) r + P r −
S
c
c
94 ,
+
( )
( )2
A S S A
A S A
− +
= 18
, ( ) r c
2
c c
95 1
2
c
S
+
c
P S
= 22
96 ,
−
A S A
22
c
−
A P S A
r c
22
P P S
= ( ), 2
( )( ) r r c
97
P P −
S
r r c
=
1 1
98 ,
+ + −
( )
A P S P S A
2
r r c
22
+ −
r c r c
P P S
= , A100 = A84 + A92 + A97 ,
99 1 1
( + )2( + −
)2
A101 = A85 + A86 + A93 + A99 , A102 = A83 + A88 + A91 + A95 ,
2 , [ ( ) ( )]
[ 2
]
A S S c A − P r A −
P r
A
= 52 42
( ) r r c
0 43
P P −
S
103
2
A S S A P A A
2 − 1 + 48 +
49
c r
0 45
104 ,
( 1 + P )( r 1
+ P r −
S
) c
=
( )
2
( )
105 , A S S P A
2 50
− +
=
2 2
c r
P P S
0
( + )( ) r + r −
c
2
A S S S P A
− +
= 51
c c r
P P S
106 ,
( ) r r +
c
0
107 , ( )
A S S P A
c r
P S
= 0 42
( ) r −
c
A S S P S A
2
−
r c
c r c
P S
0 42
= ,
( )2
108
−
109 , ( )
A S S A
c
P S
−
=
1
0 45
( + r −
) c
A S S P S A
2 2
r r c
− + −
= ,
c r c
P P S
0 45
110 1 1
( + )( + −
)2
A S S A
A S S A
2 , ( ) r c
−
= 0 c
43
( ) r r c
P P −
S
111
2 ,
−
= 0 c
43
P −
S
112
( )
( )2
A 2 S S 2
P S A
A S S A
− = 0 c r −
c
43
, P P S
( ) r c
113
−
r r c
c
P S
= 0 43
114 ,
−
( )
( )2
A S S P S A
2
−
r r c
c r c
P P S
0 43
115
A S S P A
c r
P S
0 43
=
2
= , −
( ) r c
116 ,
−
= , [ ( ) ( ) ]
( )
( )2
A S S P S A
2
−
r c
c r c
P S
0 43
117
−
S S A P P − A S + P −
S
c r r c r c
= ,
( )3
2
0 43
118
2
P P −
S
r r c
A A A A A A A A A A A A
= + + + + + + + + + + +
119 54 56 57 58 59 60 61 62 65 66 67
A A A A A A A A A A A
+ + + + + + + + + + +
68 69 70 71 72 73 74 75 76 78 80
A A A A A A A A A A
+ + + + + + + + + +
82 87 89 100 101 102 103 104 105 106
A A A A A A
+ + + + + +
108 110 111 113 115 118
A = A + A + A + A + A + A − A + A + A − A +
A
120 56 57 59 61 65 67 68 72 74 75 76
A A A A A A A A A A
+ + + + + + + + + +
80 87 100 101 103 104 105 108 110 111
A A A
+ + +
113 115 118

A A A A A A A A A A A A
121 = 56 + 57 + 59 + 61 + 65 + 2 67 + 69 + 70 + 72 + 74 +
76
A A A A A A A A A A
+ + + + + − + + + +
78 80 87 100 101 103 104 105 106 108
A A A A A
+ + + + +
110 111 113 115 118
A = A − A + A − A − A − 2 A − 2
A − A − A + A +
A
122 53 54 55 56 57 59 60 61 62 63 64
A A A A A A A A A A
2 2 2 2
− − − − − − + + − +
65 66 70 71 72 73 77 79 80 81
A 2 A 2 A 2
A A A A A A A
− + − − + + + + − −
82 18 87 89 90 94 96 98 101 102
A 2
A A A A A A A
− − + + − + + +
104 105 107 109 110 112 114 117
B1 = A91 − A23 , 2
B = A 25
+ A ,
B2 = A26 − 2A2 , 2
3 4
B A A43
4 = 42 + , B5 = A48 + A49 ,
Pr
45
6 ,
Pr
B A
+
=
1
B7 = A100 + A103 + A108 + A111 + A113 + A115 + A118 ,
B8 = A96 + A107 + A112 + A114 + A117 ,
B9 = A67 + A74 + A76 ,
B10 = A56 + A57 + A61 + A72 + A80 + A101 + A104 + A110 ,
B11 = A55 + A63 + A79 + A98 + A109 ,
B12 = A59 + A65 + A70 + A87 + A105 ,
B13 = A58 + A119 , B14 = 2A18 + A90 , 15 2 18
B 1 S A = c ,
B16 = A68 + A75 , B17 = A54 + A62 + A73 + A82 + A102 ,
B18 = A53 + A64 + A81 + A94 ,
B19 = A60 + A66 + A71 + A89 , and B20 = A69 + A78 + A106

321 a 09p

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