B. Seshaiah * S.V.K.Varma
* M.C.Raju
**
* Department of Mathematics, S.V.University Tirupati, A.P, India. Email: seshu.maths@gmail.com,svijayakumarvarma@yahoo.co.in
** Department of Mathematics, Annamacharya Institute of technology and Sciences,
Rajampet (autonomous), A.P, India. Email: mcrmaths@yahoo.co.in
ABSTRACT:
The objective of the present study is to investigate the effects of chemical reaction and radiation on unsteady MHD free convection fluid flow embedded in a porous medium with time dependant suction with temperature gradient heat source are studied. A variable time-dependent suction velocity v
  
V
0

1
 
A e n
 t


is considered normal to the flow. The plate is taken along x

- axis in vertical upward direction against the gravitational field. y

- axis is taken normal to the flow in the direction of applied transverse magnetic field. Further, due to semi-infinite plane surface assumption the flow variables are the functions of y

and t
 only. Employing a Perturbation technique, the solutions are obtained for Velocity, Temperature, Concentration, Skin-friction,
Nusselt number and Sherwood number in terms of the parameters, like, Prandtl number(Pr), Schmidt parameter(S c
), magnetic field parameter(M), heat source parameter(Q), chemical reaction parameter( k r
), mass Grashof number(G m
) and thermal Grashof number(G r
) respectively, and are discussed through graphs and tables.
Keywords : Chemical reaction, radiation,MHD, free convection, porous medium and heat source.
1.
INTRODUCTION:
The study of radiative heat and mass transfer in convective flow is found to be most important in industrial and technologies. The applications are often found in situations such as fiber and granules insulations, geothermal systems in heating and cooling of chambers. Further the magneto convection plays an important role in the magnetic separation of molecular semi conducting materials. MHD flows assumes greater significance in several biological and engineering systems when the flow is considered over a permeable boundary. When mass transfer takes place in a fluid at rest, the mass is transferred purely by molecular diffusion resulting from the concentration gradient. A number of investigations have been carried out with combined heat and mass transfer under the assumption of different physical real life situations. Due to the prime importance of heat and mass transfer involving in chemical reaction, industrial process, and the problem received considerable attention in recent years. In the last few years, many investigations [1-12], have been carried out in regard
1

to the present work. Chamkha [2] studied, thermal radiation and buoyancy effects on
Hydro-magnetic flow over an accelerating permeable surface with heat source or sink.
Radiation and mass transfer effects on unsteady MHD free convective fluid flow embedded in porous medium with heat generation / absorption was studied by Shanker et al. [13]. Unsteady MHD free convection oscillatory couette flow through a porous medium with periodic wall temperature was considered by Raju and Varma [14].
Radiation and mass transfer effects on a free convection flow through a porous medium bounded by a vertical surface were studied by Raju et al. [15-16]. Ravikumar et al. [17] investigated magnetic field effect on transient free convection flow through porous medium past an impulsively started vertical plate with fluctuating temperature and mass diffusion. Chemical reaction and radiation effects on MHD free convection flow through a porous medium bounded by a vertical surface with constant heat and mass flux was studied by Reddy et al. [18].
This paper deals with the effects of thermal radiation, time-dependant suction and chemical reaction on two dimensional MHD free convective Boussinesq fluid flow, over a semi-infinite vertical plate, moving exponentially with time, in the presence of temperature gradient heat source under the influence of applied transverse magnetic field normal to the flow. The Problem is governed by the system of coupled partial differential equations, and employing a perturbation technique the solutions are obtained.
2.
NOMENCLATURE:
A Suction parameter
C Specific heat at constant pressure p
D Molar diffusivity g Acceleration due to gravity
G m
Mass Grashof number
G r
Thermal Grashof number
K Permeability of porous medium k r
Chemical reaction rate constant
M Magnetic parameter
n
Constant exponential index
N r
Thermal radiation parameter
N u
Nusselt number
P r q r
Prandtl number

Radiative flux vector
Q Heat source parameter
S c
Schmidt parameter t Time
T Dimensionless temperature u , v
U
0
Velocity components
Mean velocity
2

x , y Cartesian coordinates
Greek symbols:

Coefficient of volume expansion due to temperature

*
Coefficient of volume expansion due to concentration v

Kinematic viscosity
Thermal conductivity

Density
Sub scripts: w

Condition at the wall
Free stream conditions, primes denote dimensional quantities
3.
MATHEMATICAL FORMULATION:
Let us consider a problem of unsteady two-dimensional, laminar, boundary layer flow of a viscous, incompressible, electrically conducting fluid along a semiinfinite vertical plate in the presence of thermal and concentration buoyancy effects. A variable time-dependent suction velocity v
  
V
0

1
 
A e n
 t


is considered normal to the flow. The plate is taken along gravitational field. y
 x

- axis in vertical upward direction against the
- axis is taken normal to the flow in the direction of applied transverse magnetic field. Further, due to semi-infinite plane surface assumption the flow variables are the functions of y

and t
 only.
Figure-3: Physical configuration and coordinate system
The following assumptions are made in obtaining the governing equations. i.
The viscous dissipation is neglected ii.
The Joule dissipation is neglected. iii.
The induced magnetic field is assumed to be negligible as the magnetic
Reynolds number of the flow is taken to be very small.
3

iv.
The plate is taken along x

- axis in vertical upward direction against to the gravitational field. y

- axis is taken normal to the flow in the direction of applied transverse magnetic field. v.
Due to semi-infinite plane surface assumption the flow variables are the functions of y

and t
 only.
Then under above assumptions and by usual Boussinesq’s approximation, the unsteady flow is governed by the following set of equations.
 v

 y


0
(1)
 u

 t

 v

 u

 y

 
 2 u

 y
 2
 g
 
T

T


 g

*

C

C




B
0
2
 u
 
 k
 u

(2)

T
 t

 v


T
 y

 
 2 T
 y

2

1
 c p
 q
 y r


Q
 c p

 y

T

T


(3)

C
 t

 v


C
 y

 
 2
C
 y

2
 k
 r
2
C
(4)
By using Rosselant approximation as in Raptis, Ogulu and Makinde, we can write the radiative flux q r as: q r

4

3 k
*
* 

T y
4

(5)
It is assumed that the temperature differences within the flow are sufficiently small so that
4
T can be expanded in a Taylor series about the free stream temperature
T

so that after rejecting higher order terms:
T
4 
4 T
3

T

3 T
4

(6)
The equation of energy after submission of equations (5) & (6) can now be written as

T
 t

 v


T
 y

 k
 c p
 2
T
 y

2

16

3
 c p
*
T
3
 k
*
 2
T
 y

2
(7)
From equation (1) we have
 v

 y


0
 v
  f ( t ) . Here, we can see that the suction velocity is a function of time t
 only. So, it is assumed in the form Cookey et al. [4]. v
  f ( t )
 
V
0

1
 
A e n
 t


Where, A is the suction parameter and

A

( 8 )
1 . Here the minus sign indicates that the suction is towards the plate.
The following non-dimensional parameters are introduced in the equations (2), (4) &
(7) to get dimensionless form. y

U y
0
 v
, t

2
U t
0

, v
P r

 k
, Sc
 v
D
,
 
T

T

T

T w
,
 


 w
, u
 u
U
0

,
4

G r

M
 

T


U
0
3


2
B v
0

U
2
0
, k r
2
, G m



2 k v r ,
U
0
2
N r g
 *


C


,
U
0
3 n
 vn

,
U
0
2

16

* 3
T

3
* k k
, Q

Q
_


C U p 0
2
K
 v
2
0
2
,
On introducing equation (9) into equations (2), (4) & (7), we obtain the following governing equations in dimensionless form:
(9)
 u
 t


1
 
Ae n t
  u
 y
 
 t


1
 
Ae n t
  
 y

 2 u
 y
2

G r
 
G m
 
M

1 k

 u
(10)

 

1
P r
N r


 2

 y
2

Q
 
 y
(11)
 
 t


1
 
Ae n t
  
 y

1
S c
 2

 y
2
 k r
2

(12)
The corresponding initial and boundary conditions are: u

1
  e n t ,
 
1
  e n t ,
 
1
  e n t on y

0 u

0 ,
 
0 ,
 
0 on y
 
(13)
METHOD OF SOLUTION:
The equations (10) to (12) are coupled non-linear partial differential equations whose solutions in closed-form are difficult to obtain, to solve these coupled non-linear partial differential equations, we assume that following Soundalgekar (1975), the unsteady flow is superimposed on the mean study flow, so that in the neighborhood of the plate, we have: u
 u
0 y
 
  
0
  
0

 e u
1 y
 o (

2 )
 e n t

1
( )
 o (

2 )
 e n t

1
( )
 o (

2 )
(14)
We now substitute equation (14) in equations (10) to (12) and equating harmonic and non-harmonic terms, neglecting higher order terms in

, we obtain :
(15) u
0
   
0
 
G r

0

G m

0 u
1
 u

1
( M
 
)
1
 
G r

1

G m

1


0
 h hQ
1
 
( Qh
 h )
)
 
0

0

1
  nh

1
 
Ah
 
0

Au
0

(16)
(17)
(18)

0
 
S c
 
0

1
 
S c
 
1

 r c


0

S c n
 k r
2
0


1
 
AS c
 
0
(19)
(20)
5

where, M
 
M

1 k
, h

1

P r
N r
and primes indicate differentiation with respect to y.
The boundary conditions are: u
0

1, u
1

1,

0

1,

1

1,

0

1,

1 u
0

0, u
1

0,

0

0,

1

0,

0


1 on y
0,

1

0

0 as y
  u
0
( y )

B
6 e
 m
5 y 
B
4 e
 m
3 y 
B
5 e
 m
1 y
(21)
(22) u
1
( y )

B
12 e
 m
6 y 
B
7 e
 m
1 y 
B
8 e
 m
2 y 
B
9 e
 m
3 y 
B
10 e
 m
4 y 
B
11 e
 m
5 y
(23)

0
( y )
 e
 m
3 y
( 24 )

1

0
( y )
( y )


( 1
 e
 m
1 y
B
3
) e
 m
4 y

1
( y )

B
1 e
 m
1 y 

B
3 e
 m
3 y
B
2 e
 m
2 y
( 25 )
( 26 )
(27)
Such that the velocity, temperature and concentration distributions can be expressed as: u ( y , t )

( y , t )




B
6  e
B
12
 m
3 e y e
 m
5

 m
6 y
 y

B
4 e
 m
3 y 
B
5 e
 m
1 y




( 1
B
7
 e
B
 m
1 y
3
) e

 m
4
B
8 y  e
 m
2 y
B
3 e

 m
3 y
B
9 e
 e n t
 m
3 y 
B
10 e
 m
4 y 
B
11 e
 m
55 y
 e n t
(28)
(29)
SKIN-FRICTION:

( y , t )
 e
 m
1 y  

B
1 e
 m
1 y
Skin-friction coefficient (
 e
 m
2 y
 e n t
B
2

) at the plate is given by
( 30 )

 

 du dy

 y

0
⟹


 m
5
B
6
 m
3
B
4
 m
1
B
5

  e n t




 m
6
B
12
 m
1
B
7
 m
2
B
8
 m
3
B
9
 m
4
B
10
(31)
 m
5
B
11




(32)
NUSSELT NUMBER:
The rate of heat transfer coefficient (Nu) at the plate is given by
Nu


 d
 dy

 y

0
(33)
Using equations (29), Nusselt number (Nu) is derived as :
Nu
  m
3
  
 m
4
( 1

B
3
)
 m
3
B
3
 e n t (34)
6

SHERWOOD NUMBER:
The rate of mass transfer coefficient (Sh) at the plate is given by
Sh


 d
 dy

 y

0
(35)
From equations (30), we obtain Sherwood number (Sh) as follows:
Sh
  m
1
 

 m
1
B
1
 m
2
B
2
 e n t
(36)
5. RESULTS AND DISCUSSION:
In order to get a physical insight of the problem, numerical computations are carried out for the non-dimensional velocity( u
), temperature(

), Concentration(

),
Skinfriction, Nusselt number and Sherwood number are in terms of the parameters
P r ,
S c ,
M , Q , k , N r , k r ,
A , G m and G r respectively. The values of Prandtl number are chosen such that for Air( P r

0 .
71 ) , Electrolytic solution( P r

1 .
00 ) , water( P r

7 .
0 ) and water at hydrogen

S c

S c

1 .
0

0
4 C(

0 .
22
P r


11 .
4 ) . The values of Schmidt number are chosen so that for
, water-vapor
and propyl benzene

S c


S c

2 .
62

0 .
60

, Ammonia

S c

0 .
78

, methanol
. Figures(1) to (3) display the effects of
Prandtl number ( P r
), Thermal radiation parameter( N r
) and Heat source parameter( Q ) on temperature distribution respectively. An increase in the prandtl number( P r
) is observed to decrease in temperature boundary layer and an increase in suction parameter(A) shows a negligible effect on the temperature field. An increase in the thermal radiation parameter ( N r
) results to increase in the thermal boundary layer, while an increase in heat source parameter ( Q ) results to decrease in the thermal boundary layer.
Figures(4) & (5) depict the effect of the chemical reaction rate constant ( k r
) and
Schmidt number( S ) on the species concentration. It is observed that an increase in c
Schmidt number ( S c
) or chemical reaction rate constant ( k r
)
results in decrease of concentration and concentration boundary layer. Again, from figure(1) & (4) we observe that suction parameter(A) has negligible effect on temperature and concentration fields. Figures(6) to (9) shows the effects of material parameters such as
P r ,
S c ,
M , Q , k , N r , k r ,
A , G r and G when the plate is cooled by free convection m currents ( G r
>0). It is observed that an increase in Prandtl number or Schmidt number leads to reduction in the velocity field, also an increase in thermal radiation parameter
( N r
) results to increase in the velocity field, while increase in rate of chemical reaction constant leads to decrease in velocity and there is a negligible effect of suction parameter(A) on velocity field, in the presence of increase in chemical reaction constant.
7

Figure(10) depicts the effects of Magnetic field parameter(M) and permeability of porous medium(K) on the velocity field. An increase in Magnetic field parameter(M) for fixed permeability of porous medium(K) leads to decrease in the velocity field, while an increase in permeability of porous medium (K) for fixed Magnetic field parameter(M) results to increase in the velocity for cooling of the plate. From figure(9) it is also observed that with an increase in suction parameter(A) leads to decrease in the velocity for cooling of the plate. Figure (11) & (12) displays the effects of thermal
Grashoff number and mass Grashoff number on velocity field. An increase in thermal
Grashoff number or in mass Grashoff number results in increase in the velocity field for cooling of the plate. Figure(13) shows, an increase in heat source parameter(Q) results to decrease in the velocity. Figures(14) to (20) describes the effects of material parameters P r ,
S c ,
M , Q , k , N r , k r ,
A , G r
and G m
, when the plate is heated by free-convection currents ( G r

0 ). It is observed that with an increase in heat source parameter or Prandtl number or Schmidt number or Magnetic field parameter the velocity increases, whereas it decreases with increasing values of thermal radiation parameter or thermal Grashoff number. Nusselt number(Nu), which measures the rate of heat transfer at the plate y=0, is shown in Table(1) for different values of Prandtl number( P r
), Heat source parameter(Q) and thermal radiation
( N r
) respectively. It is found that an increase in the Prandtl number( P r
) leads to decrease in the rate of heat transfer ,while it decreases as heat source parameter (Q) increases and thermal radiation( N r
) decreases.
Sherwood number (Sh), which measures the rate of mass transfer at the plate y=0, is shown in Table(2) for different values of Schmidt number and chemical reaction rate constant respectively. It is observed that Sherwood number decreases with an increasing values of Schmidt number (Sc) or chemical reaction rate constant( k r
).
The numerical values of Skinfriction are presented in Table(3) & (4) for the both cases cooling ( G r
> 0) and heating ( G r
< 0) of the plate respectively. It is observed that, an increase in M, P r
, S c and k r leads to decrease in the value of skin-friction, while it increases with increasing values of Q , k , N r ,
G r and G for m
G r
> 0(cooling of the plate). For G r
< 0 (heating of the plate), Skinfriction increases with an increasing values of P r
, M, G r and G and it decreases as m
S c
, Q , N r
, k r
& K increases.
8

1.4
1.2
Pr=0.71, A=0.3
Pr=1.00, A=0.3
Pr=3.00, A=0.3
Pr=0.71, A=1.0
1.4
1.2
Nr=0.0
Nr=0.1
Nr=0.2
Nr=0.3
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0.5
1 1.5
2 2.5
y
3 3.5
4 4.5
5
0
0 0.5
1 1.5
y
2 2.5
3 3.5
Figure 1: Temperature profiles for different Pr Figure 2: Temperature profiles for
4 different Nr
1.4
1.2
Q=0.0
Q=0.5
Q=1.0
Q=1.5
1
0.4
0.2
0.8
0.6
0
0 1 2 y
3 4 5 6
Figure 3: Effect of Q on Temperature field when Pr=0.71, A=0.3, Nr=0.1
.
9

1.4
1.2
Kr=0.0, A=0.3
Kr=1.0, A=0.3
Kr=2.0, A=0.3
Kr=2.0, A=1.0
1.4
1.2
Sc=0.22
Sc=0.60
Sc=0.78
Sc=1.00
Sc=2.62
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 1 2 3 4 y
5 6 7 8 9 10
0
0 1 2 3 4 y
5 6 7 8 9
Figure 4: Concentration profiles Figure 5: Concentration profiles
10
4
3.5
Pr=0.71
Pr=1.00
Pr=7.00
Pr=11.4
3
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 y
6 7 8 9 10
Figure 6: Effect of Pr on velocity field for cooling of the plate when
Gr=10, Gm=5, Sc=0.22, Q=0.1, Nr=0.1, Kr=0.5,
M=0.5, k=1.0, A=0.3 and t=1.0
10

4
3.5
3
2.5
2
1.5
1
Sc=0.22
Sc=0.60
Sc=0.78
Sc=1.00
Sc=2.62
0.5
0
0 1 2 3 4 5 6 7 8 9 10
Y
Figure 7: Effect of Sc on velocity field for cooling of the plate Gr=10, Gm=5,
Pr=0.71, Q=0.1, Nr=0.1, Kr=0.5, M=0.5, k=1.0,
A=0.3 and t=1.0
3.5
3
2.5
2
4.5
4
1.5
1
0.5
Nr=0.0
Nr=0.2
Nr=0.4
Nr=0.6
0
0 1 2 3 4
Y
5 6 7 8 9 10
Figure 8: Effect of Nr on velocity field for cooling of the plate Gr=10, Gm=5,
Pr=0.71, Sc=0.22,Q=0.1, Nr=0.1, Kr=0.5, M=0.5, k=1.0, A=0.3 and t=1.0
11

3.5
3
4.5
4
1
0.5
2.5
2
1.5
Kr=0.0, A=0.3
Kr=2.0, A=0.3
Kr=4.0, A=0.3
Kr=4.0, A=1.0
0
0 1 2 3 4 5
Y
6 7 8 9 10
Figure 9: Effect of kr on velocity field for cooling of the plate Gr=10, Gm=5, Pr=0.71,
Sc=0.22,Q=0.1, Nr=0.1, M=0.5, k=1.0, A=0.3 and t=1.0.
3.5
3
M=1.0,k=1.0
M=1.5,K=1.0
M=2.0,K=1.0
M=2.5,K=1.0
M=1.0,K=2.0
2.5
2
1.5
1
0.5
0
0 1 2 3 4 5 6 7 8 9 10
Y
Figure 10: Effect of M and k on velocity field for cooling of the plate Gr=10, Gm=5,
Pr=0.71, Sc=0.22,Q=0.1, Nr=0.1, Kr=0.5, e=0.02, n=0.5, A=0.3 and t=1.0
12

2
1
4
3
6
5
Gr=5
Gr=10
Gr=15
Gr=20
0
0 1 2 3 4 5
Y
6 7 8 9 10
Figure 11: Effect of Gr on velocity field for cooling of the plate Q=0.1, Gm=5,
Pr=0.71, Sc=0.22, Nr=0.1, Kr=0.5, e=0.02, n=0.5,
A=0.3, M=0.5, k=1.0 and t=1.0
6
5
4
3
2
1
9
8
7
Gm=5
Gm=10
Gm=15
Gm=20
0
0 1 2 3 4 5
Y
6 7 8 9 10
Figure 12: Effect of Gm on velocity field for cooling of the plate Gr=10, Pr=0.71,
Sc=0.22,Q=0.1, Nr=0.1, Kr=0.5, e=0.02, n=0.5,
A=0.3, M=0.5,k=1.0 and t=1.0
13

4
3.5
3
2.5
2
Q=0.0
Q=0.05
Q=0.10
Q=0.15
1.5
1
0.5
0
0 1 2 3 4 5
Y
6 7 8 9 10
Figure 13: Effect of Q on velocity field for cooling of the plate Gr=10, Gm=5, Pr=0.71,
Sc=0.22, Nr=0.1, Kr=0.5, e=0.02, n=0.5, A=0.3,
M=0.5, k=1.0 and t=1.0
2
Q=0.0
Q=0.1
Q=0.2
Q=0.3
1
0
-1
-2
-3
-4
0 1 2 3 4 5 6 7 8 9 10
Y
Figure 14: Effect of Q on velocity field for cooling of the plate
Gr=-10, Gm=-5, Pr=0.71, Sc=0.22, Nr=0.1, Kr=0.5, e=0.02, n=0.5, A=0.3, M=0.5, k=1.0 and t=1.0
14

0.5
0
1.5
1
-0.5
-1
-1.5
-2
-2.5
-3
Pr=0.71
Pr=1.00
Pr=7.00
Pr=11.4
-3.5
0 1 2 3 4 5 6 7 8 9 10
Y
Figure 15: Effect of Pr on velocity field for cooling of the plate when Gr=10, Gm=-5, Sc=0.22, Q=0.1, Nr=0.1,
Kr=0.5, M=0.5, k=1.0, A=0.3 and t=1.0
1.5
1
0.5
Sc=0.22
Sc=0.60
Sc=0.78
Sc=1.00
Sc=2.62
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
0 1 2 3 4 5 6 7 8 9 10
Y
Figure 16: Effect of Sc on velocity field for cooling of the plate
Gr=-10, Gm=-5, Pr=0.71, Q=0.1, Nr=0.1, Kr=0.5,
M=0.5, k=1.0, A=0.3 and t=1.0
15

2
1
0
-1
-2
-3
Nr=0.0,A=0.3
Nr=0.1,A=0.3
Nr=0.2,A=0.3
Nr=0.3,A=0.3
Nr=0.3,A=1.0
-4
0 1 2 3 4 5 6 7 8 9 10
Y
Figure 17: Effect of Nr on velocity field for cooling of the plate
Gr=-10, Gm=-5, Pr=0.71, Q=0.1, Nr=0.1, Kr=0.5,
M=0.5, k=1.0, A=0.3 and t=1.0
2
Kr=0.0,A=0.3
Kr=0.2,A=0.3
Kr=0.4,A=0.3
Kr=0.4,A=1.0
1
0
-1
-2
-3
-4
0 1 2 3 4
Y
5 6 7 8 9 10
Figure 18: Effect of kr on velocity field for cooling of the plate
Gr=-10, Gm=-5, Pr=0.71, Sc=0.22,Q=0.1, Nr=0.1,
M=0.5, k=1.0, A=0.3 and t=1.0.
16

-0.5
-1
-1.5
-2
-2.5
0.5
0
1.5
1
M=1.0,K=1.0
M=2.0,K=1.0
M=3.0,K=1.0
M=4.0,K=1.0
M=4.0,K2.0
-3
0 1 2 3 4 5 6 7 8 9 10
Y
Figure 19: Effect of M and k on velocity field for cooling of the plate Gr=-10, Gm=-5, Pr=0.71, Sc=0.22,Q=0.1,
Nr=0.1, Kr=0.5, e=0.02, n=0.5, A=0.3 and t=1.0
1.5
1
Gr=-10
Gr=-15
Gr=-20
Gr=-25
0.5
0
-0.5
-2
-2.5
-3
-1
-1.5
-3.5
0 1 2 3 4 5 6 7 8 9 10
Y
Figure 20: Effect of Gr on velocity field for cooling of the plate
Gr=-10, Gm=5, Pr=0.71, Sc=0.22,Q=0.1, Nr=0.1,
Kr=0.5, e=0.02, n=0.5, A=0.3 and t=1.0
Table.1: Nusselt number
Pr Q Nr Nusselt Number
0.71 0.1 0.2 -0.686434826809087
17

3.00
0.71
0.71
0.1
0.05
0.1
0.2 -2.875386218081379
0.2 -0.656027106142078
0.1 -0.748218439916505
Sc
0.22
0.60
0.22
Table.2: Sherwood number
Kr
0.5
Sherwood number
-0.386934345219000
0.5
1.0
-0.825297440539457
-0.615083743154758
Pr Sc
Table.3: Skin-friction for cooling of the plate (Gr > 0 )
Q Nr Kr M k Gr Gm Skin-friction
0.71 0.22 0.05 0.1 0.5 0.5 1.0 10.0 5.0 9.245208880171708
7.00 0.22 0.05 0.1 0.5 0.5 1.0 10.0 5.0 3.779571982295740
0.71 0.60 0.05 0.1 0.5 0.5 1.0 10.0 5.0 8.116737163053905
0.71 0.22 0.1 0.1 0.5 0.5 1.0 10.0 5.0 9.102473801566175
0.71 0.22 0.05 0.2 0.5 0.5 1.0 10.0 5.0 9.512798717555882
0.71 0.22 0.05 0.1 1.0 0.5 1.0 10.0 5.0 8.571928491215347
0.71 0.22 0.05 0.1 0.5 1.0 1.0 10.0 5.0 7.792824488976461
0.71 0.22 0.05 0.1 0.5 0.5 2.0 10.0 5.0 11.412767986662217
0.71 0.22 0.05 0.1 0.5 0.5 1.0 20.0 5.0 16.075070381094058
0.71 0.22 0.05 0.1 0.5 0.5 1.0 10.0 10.0 13.555768749314174
Table.4 : Skin-friction for heating of the plate (Gr < 0)
Pr Sc Q Nr Kr M K Gr Gm Skin-friction
0.71 0.22 0.05 0.1 0.5 0.5 1.0 -10.0 5.0 -4.414514121672991
7.00 0.22 0.05 0.1 0.5 0.5 1.0 -10.0 5.0 1.051122776202977
0.71 0.60 0.05 0.1 0.5 0.5 1.0 -10.0 5.0 -5.542985838790796
0.71 0.22 0.10 0.1 0.5 0.5 1.0 -10.0 5.0 -4.271779043067458
0.71 0.22 0.05 0.2 0.5 0.5 1.0 -10.0 5.0 -4.682103959057165
0.71 0.22 0.05 0.1 1.0 0.5 1.0 -10.0 5.0 -5.087794510629355
0.71 0.22 0.05 0.1 0.5 1.0 1.0 -10.0 5.0 -4.435299653160232
0.71 0.22 0.05 0.1 0.5 0.5 2.0 -10.0 5.0 -4.386570210179798
0.71 0.22 0.05 0.1 0.5 0.5 1.0 -20.0 5.0 -11.244375622595344
0.71 0.22 0.05 0.1 0.5 0.5 1.0 -10.0 10.0 -0.103954252530525
6.
CONCLUSIONS:
The effects of chemical reaction and radiation on unsteady MHD free convection fluid flow embedded in a porous medium with time dependant suction with temperature gradient heat source are studied. Employing a Perturbation technique, the solutions are obtained for Velocity, Temperature, Concentration, Skin-friction, Nusselt number and Sherwood number in terms of the parameters, like, Prandtl number(Pr),
18

Schmidt parameter(S c
), magnetic field parameter(M), heat source parameter(Q), chemical reaction parameter( k r
), mass Grashof number(G m
) and thermal Grashof number(G r
) respectively, and are discussed through graphs and tables.
An increase in the Prandtl number ( P r
) is observed to decrease in temperature boundary layer and suction parameter(A) has negligible effect on the temperature field.
An increase in the thermal radiation parameter( N r
) results in increase in the thermal boundary layer, while an increase in heat source parameter( Q ) results in decrease in the thermal boundary layer. An increase in Prandtl number(P r
) or Schmidt number(S c
) leads to reduction in the velocity field, also increase in thermal radiation parameter( N r
) results an increase in the velocity, while increase in the rate of chemical reaction constant( k r
) leads to decrease in the velocity field and there is a negligible effect of suction parameter(A) on the velocity field. An increase in magnetic field parameter (M) for fixed permeability of porous medium ( K ) leads to reduction in the velocity field, while an increase in permeability of porous medium ( K ) for fixed magnetic field parameter(M) results to increase in velocity for cooling of the plate. It is observed that with an increase in heat source parameter(Q) or prandtl number(P r
) or Schmidt number(S c
) or Magnetic field parameter(M) the velocity increases, whereas it decreases with an increasing values of radiation parameter(R) or thermal Grashof number(G r
) for heating of the plate. It is found that an increase in the prandtl number( P r
) leads to decrease in the rate of heat transfer ,while it decreases as heat source parameter(Q) increases and thermal radiation parameter( N r
) decreases.
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