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INFLUENCE OF CHEMICAL REACTION AND HEAT SOURCE ON MHD FREE
CONVECTION BOUNDARY LAYER FLOW OF RADIATION ABSORBING
KUVSHINSHIKI FLUID IN POROUS MEDIUM
Article · January 2015
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Asian Journal of Mathematics and Computer
Research
3(2): 87-103, 2015
International Knowledge Press
www.ikpress.org
_____________________________________
*Corresponding author: ibrahimsvu@gmail.com;
INFLUENCE OF CHEMICAL REACTION AND HEAT
SOURCE ON MHD FREE CONVECTION BOUNDARY
LAYER FLOW OF RADIATION ABSORBING
KUVSHINSHIKI FLUID IN POROUS MEDIUM
S. MOHAMMED IBRAHIM1*
AND K. SUNEETHA1
1
Department of Mathematics, Priyadarshini College of Engineering & Technology, Nellore,
Andhra Pradesh, India.
AUTHORS’ CONTRIBUTIONS
This work was carried out in collaboration between both authors. Author SMI conceived of the study,
directed analyses and led the writing of the paper. Author KS assisted with the conceptualization, analysis,
and writing. Both authors contributed to the interpretation of results, and review of successive drafts of the
manuscript.
Received: 17 February 2015
Accepted: 14 March 2015
Published: 11 April 2015
_______________________________________________________________________________
ABSTRACT
The purpose of this research paper is an unsteady MHD two-dimensional free convection flow of a viscous,
incompressible, radiating, chemically reacting, radiation absorbing Kuvshinshiki fluid through a porous
medium past a semi-infinite vertical plate are investigated in presence of heat source. The non dimensional
governing equations are solved by a regular perturbation law. The expressions for velocity, temperature and
concentration fields are obtained. The effects of various physical parameters on the above flow quantities are
studied through graphs. Finally, the values of the local skin-friction coefficient, rate of heat transfer and rate
of mass transfer are also shown in tabular form.
Keywords: MHD; unsteady; boundary layer; porous medium; radiation; heat source; chemical reaction.
1 Introduction
A porous medium is characterized by a partitioning of the total volume into solid matrix and pore space,
with the latter being filled by one or more fluids. The convection problem in a porous medium has important
applications in geothermal reservoirs and geothermal extractions. The process of heat and mass transfer is
encountered in aeronautics, fluid fuel nuclear reactor, chemical process industries and many engineering
applications in which the fluid is the working medium. The wide range of technological and industrial
applications has stimulated considerable amount of interest in the study of heat and mass transfer in
convection flows. Free convective flow past a vertical plate has been studied extensively by Ostrach [1].
Comprehensive discussions and or reviews are found in literature [2-5].
Original Research Article

Ibrahim and Suneetha; AJOMCOR, 3(2): 87-103, 2015
88
The study of hydromagnetic free convection flow finds applications in science and engineering, in areas
such as geophysical exploration, solar physics and astrophysical studies. Chamkha [6] proposed the unsteady
magnetohydrodynamic convective heat and mass transfer past a semi-infinite vertical permeable moving
plate. A problem of convective heat transfer over a continuously moving plate embedded in a non-Darcian
porous medium with magnetic field was analyzed by Abo-Eldahab and Gendy [7]. Sivaiah et al. [8] studied
the heat and mass transfer effects on MHD free convective flow past a vertical porous plate.
The effects of radiation on free convection on the accelerated flow of a viscous incompressible fluid past an
infinite vertical porous plate has many important technological applications in the astrophysical, geophysical
and engineering problem. Radiation and free convection flow past a moving plate was considered by Raptis
and Perdikis [9]. Ghosh et al. [10] studied the thermal radiation effects on unsteady MHD free convective
flow along an inclined plane. Abo-Eldahab and Gendy [11] reported that thermal radiation effects on
convective heat transfer in an electrically conducting fluid past a stretching surface in presence of variable
viscosity and uniform free-stream.
In the above mentioned studies the heat source/sink effect is ignored. Due to its great applicability to
ceramic tiles production problems, the study of heat transfer in the presence of a heat source/sink has
acquired newer dimensions. A number of analytical studies have been carried made of for various forms of
heat generation in [12–15]. Singh [16] analyzed MHD free convection and mass transfer flow with heat
source and thermal diffusion. Heat source effect on MHD convective flow from a sphere to a non-Darcian
porous medium was carried by Beg et al. [17].
Chemical reactions usually accompany a large amount of exothermic and endothermic reactions. These
characteristics can be easily seen in a lot of industrial processes. Recently, it has been realized that it is not
always permissible to neglect the convection effects in porous constructed chemical reactors [18]. The
reaction produced in a porous medium was extraordinarily in common, such as the topic of PEM fuel cells
modules and the polluted underground water because of discharging the toxic substance, etc. Kandaswamy
et al. [19,20] studied the effects of chemical reaction and radiation on boundary layer flow over a porous
wedge in presence of suction or injection. The effect of chemical reaction on heat and mass transfer in a
boundary layer flow has been studied under different conditions by several researchers [21–26].
In all the above studies the fluid considered is Newtonian. Most of the practical problems involve non-
Newtonian fluids types. The study of non Newtonian fluid flow has gained the attention of engineers and
scientist in recent times due to its important application in various branches of science, engineering and
technology: particularly in chemical and nuclear industries, material processing, geophysics, and bio-
engineering. In view of these applications an extensive range of mathematical models have been developed
to simulate the diverse hydrodynamic behavior of these non-Newtonian fluids. Radiation effects on MHD
free convection flow of Kuvshinshiki fluid past a vertical porous plate in porous medium in presence of mass
transfer was studied by Vidyasagar and Ramana [27]. Gupta et al. [28] proposed the heat and mass transfer
effects on MHD free convective flow of Kuvshinshiki fluid past a vertical plate. Mohan Krishna et al. [29]
investigated the effects of chemical reaction and radiation on MHD free convective flow of Kuvshinshiki
fluid past through a vertical porous plate in presence of heat generation. The influence of the soret number
on unsteady MHD of Kuvshinshiki fluid flow past a vertical porous plate with variable suction, heat and
mass transfer was analyzed by Jimoh et al. [30]. Very recently Vidya Sagar et al. [31] studied the unsteady
MHD free convective boundary layer flow of radiation absorbing Kuvshinshiki fluid through porous
medium. Chemical reaction and Soret effects on unsteady MHD flow of a viscoelastic fluid past an
impulsively started infinite vertical plate in presence of heat generation was studied by Mohammed Ibrahim
and Suneetha [32]. MHD free convection flow of a visco-elastic (Kuvshiniski type) dusty gas through a
semi-infinite plate moving with velocity decreasing exponentially with time and radiative heat transfer was
investigated by Prakash, et al. [33]. Umamaheswar, et al. [34] studied an unsteady MHD free convective
visco-elastic fluid flow bounded by an infinite inclined porous plate in the presence of heat source.
Motivated by the above studies in this article we have analyzed the effect of heat source on an unsteady
MHD free convective boundary layer flow of radiation absorbing Kuvshinshiki fluid through porous
medium past a semi-infinite vertical plate.

89
2 Formulation of the Problem
We have considered an unsteady MHD two dimensional free convective flow of a viscous, incompressible,
radiating, chemically reacting and radiation absorbing Kuvshinski fluid through a porous medium past a
semi-infinite vertical plate in presence of heat source. The x-axis be taken along the vertical plate in the
upward direction in the direction of the flow and y- axis is taken perpendicular to it. It is assumed that,
initially, the plate and the fluid are at the same temperature
*
T and concentration
*
C in the entire of the
fluid. The effects of soret and Dofour are neglected, as the level of foreign mass is assumed to be very low.
The radiative heat flux in x-direction is considered to be negligible in comparison to that of y axis. The fluid
considered here is gray, emitting and absorbing radiation but non scattering medium. The presence of
viscous dissipation cannot be neglected and also the presence of chemical reaction of first order and the
influence of radiation absorption are considered. All the fluid properties are considered to be constant except
the influence of the density variation caused by the temperature changes, in the body force term. It is also
assumed that the induced magnetic field is neglected in comparison with applied magnetic field, as the
magnetic Reynolds number is very small. Now, under the above assumptions, the governing equations of
such type of flow, momentum, energy and species equations [31].
Continuity Equation
*
*
0
v
y



(1)
Momentum Equation
    2
2
* * 2 *
* * * * * * * * *
0
* * * * *
*
1 1
B
u u u
v g T T g C C u
t t y k t
y
 
    

 
 
    
   
         
 
   
   
    
 
(2)
Energy Equation
   
2
2 *
* * 2 * *
* * * * * *
0
1
* * * * *
*
1
1 r
p p p p p
Q
q R
T T k T u
v C C T T
t t y C C y C y C C
y


   
 
  
    
 
        
 
 
    
    
(3)
Species equation
 
2
* * 2 *
* * * *
1
* * * *
1
C C C
v D k C C
t t y y
 
   
 
    
 
  
  
(4)
The boundary conditions at the wall and in the free stream are
 
* *
*
0 1 t n
u v e
 
  ,
* *
w
T T
 ,
* *
w
C C
 at
*
y =0 (5)
*
0
u  ,
*
0
T  ,
*
0
C  as
*
y  
The equation (1) gives
*
0
v v
  (6)

90
where 0
v is the constant suction velocity. The radiative heat flux
*
r
q using the Rosseland approximation
diffusion model for radiation heat transfer is expressed as
4
* *
*
* *
4
3
r
T
q
k y

 


(7)
Where
*
 and
*
k are respectively the Stream-Boltzmann constant and the main absorption coefficient. We
assume that the temperature difference with in the flow is sufficiently small and in Taylor series about
*
T
and neglecting higher order terms, thus
4 3 4
* * * *
4 3
T T T T
 
  (8)
In view of equations (8) and (9) the equation (3) reduced to the following form
   
3
2 2
2 * *
* * 2 * * 2 *
* * * * * *
0
1
* * * * *
* *
16
1
3
p p p p p
Q
T R
T T k T u T
v C C T T
t t y C C y C k C C
y y



   

 
 
     
 
        
 
 
   
   
 
(9)
Introducing the following dimensionless variable and parameters,
 
 
 
 
 
3 * *
* * 2 * * * * * *
* *
0 0
2 * * * * * 3
0 0 0
* * * * 2 2 * 2 2
0 0 0 0
3 2 * *
0 0
2 * *
1
2 * *
0
4
, , , , , , , ,
,Pr , , , , , ,
,
w
w w
w p
p w
w
w
g T T
y v t v T T C C T
u n
u y t n R Gr
v v T T C C k k v
g C C C k v v v
Gm Sc k M Ec
v k D v C T T
R C C
Ra K
kv T T



 
 
     


  


  
 





 
       
 

      




2
1 0
0
2 2
0
,
k Qkv
r Q
v


 
(10)
Into set of equations (2) - (5), we obtain
2 2
1 1
2 2
u u u u
M u Gr Gm
t t y y
   
   
     
   
(11)
2
2 2
1
2 2
Pr Pr Pr Pr a
u
N Ec R Q
t t y y y
   
  
 
    
     
 
    
 
(12)
2 2
2 2
Sc Sc Sc KrSc
t t y y
   
 
   
   
   
(13)
Where 1 1 1 1
1 4
, 1 , 1
3
R
M M N M
k
 
     
The corresponding boundary conditions in non-dimensional form are

91
1 , 1, 1,
nt
u e
  

    at y=0,
0, 0, 0
u  
   as y   (14)
2.1 Solution of the Problem
Equations (11) - (13) are coupled, non-linear differential equations and these cannot be solved in closed-
form. However, these equations can be reduced to a set of ordinary differential equations, which can be
solved analytically. This can be done by representing the velocity, temperature and concentration of the fluid
in the neighborhood of the plate as
2
0 1
( ) ( ) 0( )..............
nt
u u y u y e
 

  
2
0 1
( ) ( ) 0( )..............
nt
y y e
   

   (15)
2
0 1
( ) ( ) 0( )..............
nt
y y e
   

  
Substituting (15) into set of equations (11)-(13) and equating the harmonic terms and non-harmonic terms,
and neglecting the higher order terms of, we obtain
0 0 1 0 0 0
u u M u Gr Gm
 
 
     (16)
1 1 2 1 1 1
u u M u Gr Gm
 
 
     (17)
2
1 0 0 0 0 0
Pr Pr ( )
N Q Ec u Ra
   
  
     (18)
1 1 1 2 0 0 1 1
Pr 2Pr
N N Ecu u Ra
   
   
     (19)
0 0 0 0
Sc ScKr
  
 
   (20)
1 1 1 1 0
Sc L
  
 
   (21)
Where prime denotes ordinary differentiation with respect to y and
2 2 2
2 1 1 2 1
, Pr Pr ,
M M n n N n n Q L nSc n Sc KrSc
   
        
The corresponding boundary conditions are
0 1 0 1 0 1
1, 1, 1, 0, 1, 0,
u u    
      at y=0
0 1 0 1 0 1
0, 0, 0, 0, 0, 0,
u u    
      as y   (22)
Solving the equations (20) and (21) subject to the corresponding boundary conditions, we obtain.
1
0
m y
e
 

1 0
  (23)

92
The set of equations (16)-(19) are still coupled and non-linear ordinary differential equations, whose exact
solutions are not possible. So we expand 0 1 0 1
, , ,
u u   in terms of Ec in the following form, as the Eckert
number is very small for incompressible flows.
To solve these equations, assuming the Eckert number E to be small, we write.
0 01 02
1 11 12
0 01 02
1 11 12
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
u y u y Ecu y
u y u y Ecu y
y y Ec y
y y Ec y
  
  
 
 
 
 
(24)
Substituting the equations (24) into equations (16)-(19), equating the coefficients of Ec to zero and
neglecting the higher order terms, we obtain
01 01 1 01 01 0
u u M u Gr Gm
 
 
     (25)
11 11 2 11 11 1
u u M u Gr Gm
 
 
     (26)
1 01 01 01 0
Pr
N Q Ra
   
 
    (27)
1 11 11 2 11 1
Pr
N N Ra
   
 
    (28)
02 02 1 02 02
u u M u Gr
 
    (29)
12 12 2 12 12
u u M u Gr
 
    (30)
 
2
1 02 02 02 01
Pr Pr
N Q u
  
  
    (31)
1 12 12 2 12 01 11
Pr 2Pr
N N u u
  
   
    (32)
The corresponding boundary conditions are
01 02 11 12
01 02 11 12
1, 0, 1, 0
1, 0, 0, 0
u u u u
   
   
   
at y=0
01 02 11 12
01 02 11 12
0, 0, 0, 0
0, 0, 0, 0
u u u u
   
   
   
as y   (33)
The analytical solutions of equations (25)-(32) under the boundary conditions (33) are given by
3 1
01 2 1
m y m y
A e Ae
  
  (34)
5 3 1
01 5 3 4
m y m y m y
u A e A e A e
  
   (35)

93
6
11
m y
u e
 (36)
7 5 3 3
1 1 2
2 2 2
02 12 6 7 8 9 10 11
m y m y m y b y
m y b y b y
A e A e A e A e A e A e A e
    
  
       (37)
8 5 6
4
12 16 13 14 15
m y b y b y
b y
A e A e A e A e
   

    (38)
9 7 5 3 3
1 1 2
2 2 2
02 24 17 18 19 20 21 22 23
m y m y m y m y b y
m y b y b y
u A e A e A e A e A e A e A e A e
    
  
        (39)
10 9 7 5 3 3
1 1 2
2 2 2
12 33 25 26 27 28 29 30 31 32
m y m y m y m y m y b y
m y b y b y
u A e A e A e A e A e A e A e A e A e
     
  
         (40)
The velocity, temperature and concentration distributions in the boundary layer become
( , )
u y t 
 
 
5 3 1
9 7 5 3 3
1 1 2
01 5 3 4
2 2 2
02 24 17 18 19 20 21 22 23
m y m y m y
m y m y m y m y b y
m y b y b y
u A e A e A e
E u A e A e A e A e A e A e A e A e
  
    
  
 
  
 
 
        
 
 
6
10 9 7 5 3 3
1 1 2
2 2 2
12 33 25 26 27 28 29 30 31 32
m y
nt
m y m y m y m y m y b y
m y b y b y
e
e
E u A e A e A e A e A e A e A e A e A e



     
  
 

 

        
 
 
(41)
 
 
3 1
7 5 3 3
1 1 2
01 2 1
2 2 2
02 12 6 7 8 9 10 11
( , )
m y m y
m y m y m y b y
m y b y b y
A e Ae
y t
E A e A e A e A e A e A e A e



 
   
  
 
 
 

 
       
 
8 5 6
4
12 16 13 14 15
m y b y b y
b y
nt
e E A e A e A e A e
    


 
    
  (42)
1
m y
e
 
 (43)
Where the expressions for the constants are given in the Appendix
The skin-friction, Nusselt number and Sherwood number are important physical parameters for this type of
boundary layer flow.
Knowing the velocity field, the skin-friction at the plate can be obtained, which in non-dimensional form if
given by
0 1
0 0
nt
y y
u u
u
Cf e
y y y

 
   
 

  
   
  
   
=
=   
5 5 3 3 1 4 9 24 17 7 18 5 3 19 20 1 1 21 2 22 3 23
2 2 2
m A m A m A Ec m A A m A m m A A m b A b A b A
           
 
6 10 33 9 25 7 26 5 27 3 28 1 29 30 1 31 2 32 3
2 2 2
nt
e m Ec m A m A m A m A m A m A A b A b A b
 
 
           
 
(44)
Knowing the temperature field, the rate of heat transfer coefficient can be obtained, which in the non-
dimensional form, in terms of the Nusselt number, is given by

94
0 1
0 0
nt
y y
Nu e
y y y
 


 
   
 

    
   
  
   
=
=
   
 
3 2 1 1 7 12 6 5 3 7 8 1 1 9 2 10 3 11
8 16 13 4 14 5 15 6
2 2 2
nt
m A m A Ec m A A m m A A m b A b A b A
e Ec m A A b A b A b
 
 
          
 
 
   
 
 
(45)
Knowing the concentration field, the rate of mass transfer coefficient can be obtained, which in the non-
dimensional form, in terms of the Sherwood number, is given by
0 1
1
0 0
nt
y y
Sh e m
y y y
 


 
   
 

     
   
  
   
(46)
3 Results and Discussion
In order to look into the physical insight of the problem, the expressions obtained in previous section are
studied with help of graphs from Figs. 1 – 9. The effects of different physical parameters viz., the thermal
Grashof number (Gr), the mass Grashof number (Gm), magnetic field parameter (M), radiation parameter
(R), radiation absorption parameter (Ra), heat Source (Q), Schmidt number (Sc) and chemical reaction (Kr)
are studied analytically by choosing arbitrary values.
Fig. 1 shows the velocity profiles for different values of thermal Grashof number Gr. It is observed that, as
Gr increases, velocity also increases. This is due to the buoyancy which is acting on the fluid particles due to
gravitational force that enhances the fluid velocity. A similar effect is identified from Fig. 2, in the presence
of mass Grashof number Gm, which also increases fluid velocity.
Typical variation of velocity profiles for various values of magnetic field parameter M is shown in Fig. 3.
From this figure it is found that velocity gets reduced by the increase of magnetic parameter M. Because the
magnetic force which is applied perpendicular to the plate, retards the flow, which is known as Lorentz
force.
The effect of Prandtl number Pr on the velocity and temperature profiles is presented in Figs. 4(a) and 4(b).
From Fig. 4(a), it is observed that velocity decreases with an increase in Prandtl number Pr. This is
physically true because, the Prandtl number is a dimensionless number which is the ratio of momentum
diffusivity (kinematic viscosity) to thermal diffusivity. In many of the heat transfer problems, the Prandtl
number controls the relative thickness of the momentum and thermal boundary layers. When Pr is small, it
means that the heat diffuses very quickly compared to the velocity (momentum), which means that in the
case of liquid metals the thickness of the thermal boundary layer is much bigger than the velocity boundary
layer. This absolutely coincides with the results that is shown in Fig. 4(b) where the thermal boundary layer
shrinks for higher values of Prandtl number Pr.
The effect of radiation absorption parameter Ra on velocity and temperature profiles was plotted in the Figs.
5(a) and 5(b). The velocity and temperature profiles increase with an increase of radiation absorption
parameter Ra which is clearly represented in Figs. 5(a) and 5(b).
The effects of radiation parameter R on the velocity and temperature profiles are presented in Figs. 6(a) and
6(b). From these figures, we noticed that as the value of radiation parameter increases, the velocity and the
temperature profiles increase. This is because the thermal radiation is associated with high temperature,
thereby increasing the temperature distribution of the fluid flow.

95
Figs. 7(a) and 7(b) depicts the velocity and temperature profiles for various values of heat source Q. From
these figures we can see that the heat generated buoyancy force increases which induce the flow rate to
increase giving rise to the increase in the velocity and temperature profiles.
For the various values of the Schmidt number Sc, the velocity, temperature and concentration profiles are
presented in Figs. 8(a) – 8(c). The Schmidt number Sc embodies the ratio of the momentum to the mass
diffusivity. This Schmidt number therefore quantifies the relative effectiveness of momentum and mass
transport by diffusion in the hydrodynamic, temperature and concentration boundary layers. It is obvious
that the effect of increasing values of Sc, results in a decreasing velocity, temperature and concentration
distributions across the boundary layer. A similar effect is identified from Figs. 9(a) - 9(c), in the presence of
chemical reaction parameter Kr, which also exhibits decreasing fluid velocity, temperature and
concentration.
Tables 1 - 3 presents the values of skin friction coefficient, Nusselt and Sherwood numbers at numerous
values of parameters. The effects of various physical parameters on skin-friction coefficient, Nuselt number
and Sherwood number are shown in Tables 1 - 3. The behavior of these parameters is self-evident from
Tables. 1, 2, 3 and hence they are not discussed any further due to brevity.
Fig. 1. The graph of u against y for varies
values of Gr
Fig. 2. The graph of u against y for varies values
of Gm
Fig. 3. The graph of u against y for varies
values of M
Fig. 4(a). The graph of u against y for varies
values of Pr
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
y
U
Gr=8,10,12,14
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
y
U
Gm=5,7,9,11
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
M=1.0,1.5,2.0,2.5
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
Pr=0.71,0.91,1.11,1.31

96
Fig. 4(b). The graph of T against y for varies
values of Pr
Fig.5(a). The graph of u against y for varies
values of Ra
values of Ra
values of R
values of R
values of Q
values of Q
values of Sc
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
Pr=0.71,0.91,1.11,1.31
0 2 4 6 8 10 12 14 16 18 20
0
0.5
1
1.5
y
U
Ra=0,1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
y
T
Ra=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
R=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
R=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
Q=0.00,0.01,0.05,0.07
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
Q=0.005,0.01,0.015,0.02
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
Sc=0.22,0.44,0.66,0.88

97
values of Sc
Fig. 8(c). The graph of C against y for varies
values of Sc
values of Kr
values of Kr
Fig. 9(c). The graph of C against y for varies values of Kr
Table 1. Effect of various physical parameters on skin-friction coefficient, Nusselt number and
Sherwood number for Pr = 0.71, R = 0.1, E = 0.01, Ra = 0.1, Q = 0.01, Sc = 0.22, Kr = 0.1
Gr Gm M K Cf Nu Sh
8 5 1.0 0.1 0.3259 0.1462 -0.2947
10 5 1.0 0.1 1.0158 0.2780 -0.2947
12 5 1.0 0.1 1.7243 0.4268 -0.2947
8 7 1.0 0.1 0.9878 0.2793 -0.2947
8 9 1.0 0.1 1.6532 0.4297 -0.22947
8 5 1.2 0.1 0.2537 0.1323 -0.2947
8 5 1.4 0.1 0.1829 0.1190 -0.2947
8 5 1.0 0.4 6.4519 3.2132 -0.2947
8 5 1.0 0.8 11.3294 7.4571 -0.2947
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
Sc=0.22,0.44,0.66,0.88
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
C
Sc=0.22,0.44,0.66,0.88
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
U
Kr=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
T
Kr=0.1,0.2,0.3,0.4
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
C
Kr=0.1,0.20.3,0.4

98
Sherwood number for Gr = 8, Gm = 5, M = 1.0, K = 0.1, Sc = 0.22, Kr = 0.1
Pr R Ra Q Cf Nu Sh
0.71 0.1 0.1 0.01 0.3259 0.1462 -0.2947
0.81 0.1 0.1 0.01 0.1853 -0.3245 -0.2947
0.91 0.1 0.1 0.01 0.1155 -0.4562 -0.2947
0.71 0.3 0.1 0.01 0.2568 -0.3725 -0.2947
0.71 0.4 0.1 0.01 0.2792 -0.3258 -0.22947
0.71 0.1 0.3 0.01 1.1333 2.5174 -0.2947
0.71 0.1 0.5 0.01 2.1960 6.1668 -0.2947
0.71 0.1 0.1 0.005 0.2894 0.0189 -0.2947
0.71 0.1 0.1 0.007 0.3022 0.0619 -0.2947
Sherwood number for Gr = 8, Gm = 5, M = 1.0, K = 0.1, Pr = 0.71, R = 0.1, E = 0.01, Ra = 0.1, Q = 0.01
Sc Kr Cf Nu Sh
0.22 0.1 0.3259 0.1462 -0.2947
0.24 0.1 0.0996 -0.8856 -0.3160
0.26 0.1 0.14454 -0.5938 -0.3371
0.22 0.4 0.0788 -0.4830 -0.4264
4 Conclusion
In this paper a theoretical study is carried out for an unsteady MHD two dimensional free convention flow of
a viscous incompressible, radiating, chemically reacting, radiation absorbing and heat source Kuvshinshiki
fluid through a porous medium past a semi-infinite vertical plate. The dimensionless equations governing the
flow are solved by perturbation technique. The fundamental parameters which were found to have an
influence on the problem under consideration are heat source, magnetic field parameter, permeability of
porous medium, radiation parameter, radiation absorption parameter, Grashof number, mass Grashof
number, Schmidt number, chemical reaction parameter and Prandtl number. The main conclusions are as
follows.
a. Velocity increases when Grashof number, mass Grashof number, radiation parameter, and heat
source, increase, whereas it decreases when there is an increase in magnetic field parameter,
Schmidt number, chemical reaction parameter, radiation absorption, Prandtl number.
b. Temperature increases with an increase in radiation parameter, radiation absorption parameter, and
heat source whereas it decreases with an increase in Prandtl number.
c. Concentration is observed to decrease when Schmidt number and chemical reaction increase.
d. Skin-friction increases with an increase in Prandtl number, permeability of porous medium, Grashof
number, mass Grashof number whereas it has reverse effect in the case of magnetic field parameter
and Schmidt number.
e. Nusselt number increases with an increase in Pradtl number, radiation absorption parameter and
heat generation, whereas it has reverse effect in the case of radiation parameter.
f. Sherwood number gets decreased when Schmidt number and chemical reaction parameter
increased.
Acknowledgement
Authors are thankful to the referee for his/her valuable suggestions, who helped to improve the quality of
this manuscript.

99
Competing Interests
Authors have declared that no competing interests exist.
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102
APPENDIX
2
1
4
,
2
Sc Sc ScKr
m
 

2
1
2
4
2
Sc Sc L
m
 
 ,
2
1
3
1
Pr Pr 4
,
2
QN
m
N
 

2
1 2
4
1
Pr Pr 4
,
2
N N
m
N
 
 1
5
1 1 4
,
2
M
m
 
 2
6
1 1 4
2
M
m
 
 ,
2
1
7
1
Pr Pr 4
,
2
QN
m
N
 

2
1 2
8
1
Pr Pr 4
,
2
N N
m
N
 
 1
1
2
1 1
1 1
,
Pr
Ra
N
A
Q
m m
N N
 

 
 

 
2 1
1 ,
A A
  2
3 2
3 3 1
,
GrA
A
m m M

 
1
4 2
1 1 1
,
GrA Gm
A
m m M


 
3 4
5 1 ,
A A A
  
 
2 2
5 5 1
6
2
5 5
1 1
Pr /
,
Pr
4 2
A m N
A
Q
m m
N N

 
 
2 2
3 3 1
7
2
3 3
1 1
Pr /
,
Pr
4 2
A m N
A
Q
m m
N N

 
 
2 2
4 1 1
8
2
1 1
1 1
Pr /
,
Pr
4 2
A m N
A
Q
m m
N N

 
 
   
5 3 3 5 1
9
2
3 5 3 5
1 1
2Pr /
,
Pr
A A m m N
A
Q
m m m m
N N

   
 
   
5 1 3 5 1
10
2
1 5 1 5
1 1
2Pr /
Pr
A Am m N
A
Q
m m m m
N N

   
 
   
4 3 3 4 1
11
2
3 4 3 4
1 1
2Pr /
Pr
A A m m N
A
Q
m m m m
N N

   
, 12 6 7 8 9 10 11,
A A A A A A A
      ,
 
   
5 5 6 1
13
2 2
5 6 6 5
1 1
2Pr /
,
Pr
A m m N
A
N
m m m m
N N

   
 
   
3 3 6 1
14
2 2
3 6 6 3
1 1
2Pr /
,
Pr
A m m N
A
N
m m m m
N N

   

103
 
   
4 1 6 1
15
2 2
1 6 6 1
1 1
2Pr /
,
Pr
A m m N
A
N
m m m m
N N

   
16 13 14 15 ,
A A A A
   12
17 2
7 7 1
,
GrA
A
m m M

 
6
18 2
5 5 1
,
4 2
GrA
A
m m M

 
7
19 2
3 3 1
,
4 2
GrA
A
m m M

 
8
20 2
1 1 1
,
4 2
GrA
A
m m M

 
9
21 2
1 1 1
,
GrA
A
b b M

 
10
22 2
2 2 1
,
GrA
A
b b M

 
11
23 2
3 3 1
GrA
A
b b M

 
,
24 17 18 19 20 21 22 23
A A A A A A A A
       ,
24
25 2
9 9 2
,
GrA
A
m m M

 
17
26 2
7 7 2
,
GrA
A
m m M

 
18
27 2
5 5 2
,
4 2
GrA
A
m m M

 
19
28 2
3 3 2
4 2
GrA
A
m m M

 
20
29 2
1 1 2
,
4 2
GrA
A
m m M

 
21
30 2
1 1 2
,
GrA
A
b b M

 
22
31 2
2 2 2
,
GrA
A
b b M

 
23
32 2
3 3 2
GrA
A
b b M

 
,
33 25 26 27 28 29 30 31 32
, ,
A A A A A A A A A
       
_______________________________________________________________________________________
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