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(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

* 

https://ijojournals.com/                                                                Volume 07 Issue 03 || March, 2024 || 

Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

Mathematical study of MHD Convective flow with chemical reaction 

through a porous medium in a vertical wavy channel 

Venuprasad K. K.1, Prof. K. Shivashankara2, Dhananjaiah D. S3, Prakasha.P4 

1Department of Mathematics, Government First Grade College K.R.Pete, Mandya, India  

E-mail id: kkvpmaths@gmail.com  

 2Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore, India  

E-mail id: drksshankara@gmail.com 
3Department of Mathematics, Government First Grade College K.R.Nagar, Mysuru, India  

E-mail id: dhanu2614@gmail.com 
4Department of Mathematics, Government First Grade College, Madagi, Ramnagar,  India  

E-mail id: profprakasha@gmail.com  

Abstract: 

In this work, we examine how transitory free connective heat and mass transfer flow over a porous 
media in a vertical wavy channel is impacted by chemical reactions and radiation absorption. The 
oscillatory flux in the flow zone is the cause of the unsteadiness in the flow. The Laplace 
transformation approach is used to get the analytical solutions for the governing equations. Analysis 
is done on the profiles of temperature, concentration, and velocity. The governing parameters are 
varied, and the resulting expressions for the velocity, concentration, shear stress, and rate of heat 
and mass transfer are examined. 

Key words: MHD, Wavy channel, Chemical reaction. 

1. Introduction 

The use of non-Newtonian liquids in engineering and industry is required due to their 

increasing significance. It is remarkable for its many applications in numerous areas, which include 

food processing, lubricant performance, plastic manufacture, and/or biological liquid transportation. 

Numerous fluids, including diluted polymer solutions, slurry flows, industrial oil, and numerous 

flow issues caused by different mechanical and/or thermal boundary conditions have been 

discussed, as well as the second graded fluid preserves these fluids. For the second-graded fluids, 

Tan and Masuoka [1] discovered the Stokes first difficulties, whereas Rashidi et al. [2] addressed 

the second-order fluids' unstable compressible flows. Hayat and associates. [3] investigated by the 

fluids of second grade with variable free stream and unstable stagnation point flow. 

A branch of fluid dynamics called magnetohydrodynamics (MHD) examined how 

electrically conducting fluids interact with one another in a magnetic field. Many research projects 

in the field of MHD have been carried out during the course of the few decades that have preceded 

them, following Hartmann's well-established work [4]. flow in metalized fluid channels subjected 

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mailto:kkvpmaths@gmail.com
mailto:drksshankara@gmail.com
mailto:dhanu2614@gmail.com
mailto:profprakasha@gmail.com


IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

* 

https://ijojournals.com/                                                                Volume 07 Issue 03 || March, 2024 || 

Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

to an external magnetic field. The parabolic movement has several uses, such as solar cookers, solar 

concentrators, and parabolic through star collectors. There are several uses for solar cookers with 

parabolic concentrator models, including roasting, baking, and distilling. Applications for the solar 

concentrator model included increasing evaporation rates in dissipation streams, food dispensing, 

and generating drinking water from both seawater and saltwater. Murthy et al. [5] examined 

through the assessments of temperature exchanger units' thermal characteristics for parabolic 

 Diffusion-thermo effect, radiating-absorptions, Hall, and ion slip influences on the MHD 

liberated convection gyratory flows of the nanofluids passing the semi-infinite permeable inspiring 

plate with the constant temperature sources were recently investigated by Krishna and Chamkha 

[6]. Krishna et al. [7] investigated the effects of radiating and Hall currents on the unstable MHD 

freed central heating flows into the perpendicular channel/duct packed by the absorbent media. 

Krishna and Chamkha [8] took into consideration the temperature generating/absorption, thermo-

diffusions on the unsteady complimentary convection MHD flows of radiation, and the chemically 

reactive second-grade liquid passing past an unbounded perpendicular plate during the absorbent 

media in addition to taking the Hall current into account. 

 For the few decades prior, convective heat transportations in a permeable medium have 

piqued intense curiosity. Numerous thermal engineering functions in a range of constraints, such as 

geophysics, thermal and insulation engineering, the model of crowded sphere beds, electronic 

system cooling, chemical catalytic reactor, ceramic processes, granular insulations and grains 

storage device fibers, gasoline reservoirs, coal combustion engines, groundwater pollution, and 

filtration processes, all stimulate its interests.  

       The partial differential equations recurrently appear in the lot of areas of the natural and 

physical disci- plines. They described dissimilar physical organisms, ranges from gravitational to 

fluid dynamics and had been utilized to solve the problem by the physical and chemical sciences, 

mathematical bio-sciences, solid mechanical engineering knowledge, etc. Soundalgekar and Takhar 

ini- tially, deliberated the consequence of radiation for the natural convective flow of the gasses 

over a semi infinite plate with numerical modeling. Takhar et al.[9] explored the impact of radiation 

on MHD free convective flow past semi-infinite vertical plate. Currently, Hossain et al.[10] exposed 

the effects of radiation on combined convective flow during an absorbent plate. Muthucumarswamy 

and Kumar[11] explored the heat radiation influences on affecting never-ending vertical plate with 

variable heat and mass diffusions. 

2.Formulation of the Problem 

          We consider the effect of chemical reaction on the unsteady motion of viscous, fluid  through 
a porous medium in a vertical  channel bounded by wavy walls . The thermal buoyancy in the flow 
field is created by an oscillatory flux in the fluid region. The walls are maintained at constant 
temperature and concentration. The  Boussinesq approximation is used  so that the density variation 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

* 

https://ijojournals.com/                                                                Volume 07 Issue 03 || March, 2024 || 

Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

will be considered only in the buoyancy force. The viscous and Darcy dissipations are neglected in 
comparison with heat by conduction and convection in the energy equation. Also the Kinematic 
viscosity ,the thermal conducting k are treated as constants. We choose a rectangular Cartesian 
system 0 ( x  ,y )  with x-axis in the vertical direction and y-axis normal to the walls. The walls of 

the channel  are  at )(
L

x
Lfy


 .     

      The flow is maintained by an oscillatory volume flux for which a characteristic velocity is 
defined as 

                      



fL

fL

ti dyu
L

ekq
1

)1(  .                                                       (1) 

The boundary conditions for the velocity and temperature fields are  

             u = 0  , v = 0  ,T=T1   ,C=C1                on )(
L

x
Lfy


  

             22,0,0 CCTTvu               on  )(
L

x
Lfy


         (2) 

 
In view of the continuity equation we define the stream function  as 
                         u = - y , v =  x                                                              (3)            
The equations governing the flow, heat and mass transfer  in terms of the Stokes steam function  
are 

     







2
0

0
4222

)()(

)(])()()[(







k
CCg

TTg

y

yxyyxt

         (4) 

                                                                                                                             

     )()()( 1
2

eofpe CCQTTQTk
y

T

xx

T

yt

T
C 





















 
        (5) 

 )()( 1
2

1 oCCkCD
y

C

xx

C

yt

C






















 
                 (6) 

 
Introducing the non-dimensional variables in (4 )-(6) as   

     
21

2

21

2 ,,/,,/,/
cC

CC
C

TT

TT
ttLyyLxx









                                (7)   

     
the governing equations in the non-dimensional form ( after dropping the dashes ) are  

     


 214
2

22 ))(()
),(

),(
)(( 




 DNC

R

G

yx
R yyt                                  (8) 

    CQ
yxxyt

P 1
22 )( 

























                                (9) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

* 

https://ijojournals.com/                                                                Volume 07 Issue 03 || March, 2024 || 

Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

KCC
y

C

xx

C

yt

C
Sc 





















 22 )(


        (10) 

where 

 


UL
R          (Reynolds number),  

2

3



 LTg
G e
 (Grashof number) 

f

p

k

c
 ( Prandtl number), 

1D
Sc


 (Schmidt Number), 

pf Ck

QL2

 (Heat source parameter),
1

2
1

D

LK
K       (Chemical reaction parameter), 






2
2 L
       (Wormsely Number),

k

L
D

2
1  (Darcy parameter),  

)(

)(

21

2
211

1
TTk

LCCQ
Q

f 


 (Radiation absorption parameter) 

 
The corresponding boundary conditions are  
           1)1()1(      

 10,0 











at

yx
         (11) 

            
10,0),(

11,1),(









onCyx

onCyx
   

            00,0 











at

y

C

y
                 (12) 

 
The value of  on the boundary assumes the constant volumetric flow in consistent with the 
hyphothesis (1) .Also the wall temperature varies in the axial direction in accordance with the 
prescribed arbitrary function t . 
 
3. METHOD OF SOLUTION 

   The main aim of the analysis is to discuss the perturbations created over a combined free and 
forced convection flow due to traveling thermal wave imposed on the boundaries. The perturbation 
analysis is carried out by assuming that the aspect ratio    to be small.    
   
Introduce the transformation such that  

 
xx

xx








  ,  then  )1()( O

x
O

x










  

For small values of <<1 ,the flow develops slowly with axial gradient of order  

And hence we take )1(O
x





 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

* 

https://ijojournals.com/                                                                Volume 07 Issue 03 || March, 2024 || 

Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

Using the above transformation the equations(8 - 10) reduces to  




 214
1

2
12

1
2 ))(()

),(

),(
)(( 




 DNC

R

G

yx
R yyt               (13) 

    CQ
x

N
yyxxyt

P 12

2

2
2

2

2
2

1 )()( 





































                (14) 

KCC
y

C

xx

C

yt

C
Sc 





















 2
1

2 )(


       (15) 

where 

 
2

2

2

2
22

1
yx 







   

Introducing the transformation 
)(xf

y
  the equations(13-15) reduce to  







 

221

3
4

2
22

)(

))(()
),(

),(
)((

FfD

NC
R

Gf
F

x

F
FRf t









                      (16) 

                                            

    CQF
xx

f
t

P 1
22 )(( 































                     (17) 

KCCF
C

xx

C
f

t

C
Sc 





















 22 )((







                         (18) 

Where 

 
2

2

2

2
22














x
F  

 We adopt the perturbation scheme and write  

 
.

..............)),,(.),,((),,(),,(),,( 1100  txketxtxketxtx itit 
 

.

..............)),,(.),,((),,(),,(),,( 1100  txketxtxketxtx itit 

.

..............)),,(.),,((),,(),,(),,( 1100  txCketxCtxCketxCtxC itit 
  (19)          

On substituting (19) in (16) - (18) and separating the like powers of  the equations and respective 
conditions to the zeroth order are 

                   )()()( ,0,0

3

,0
22

1,0   NC
R

Gf
fM y                               (20) 

       
010

2
, )( CQfo   

          (21)   

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

* 

https://ijojournals.com/                                                                Volume 07 Issue 03 || March, 2024 || 

Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

      0)( 0
2

,  CKScfCo                              (22) 

with 
         0(+1)-0(-1) = 1 ,  
        0,  = 0 ,  0 , x =0           at  = 1                                       (23)   

      
10,0

11,1

0

0









onC

onC

o

o

                               (24)   

 

010
22

,0 )( CQfiP            (25) 

 

 0)( 22
,0  oCfKScC                    (26) 

   

))(())(( .0,0

3

,0
222

1,0   CN
R

Gf
fiM       (27) 

0)1(0)1(  oo C  

0)1(,0)1(1)1()1( ,,  xoooo         (28) 

The first order equations are 

  ))(())(()( ,0,0,0,0,1,1

3

,1
22

1,1   xxRfCN
R

Gf
fM           (29) 

    11,0,,01
2 )()()(

,1
CQPRff oxox   

                 (30)   

    )()()( ,0,,01
2

,1 oxox CCScfCKScfC
y  


                          (31)        

)

)(())(())((

,0,0,0,0,0,0

,0,0,1,1

3

,1
222

1,1









xxx

xRfCN
R

Gf
fiM




         (32) 

  

11,0,,0

,0,,01
22

)

()())((
,1

CQ

PRffiP

oxox

oxxo














                 (33)   

)

()())((

,0,,0

,0,,01
22

,1










oxox

oxxo

CC

CCScfCScfiKC




                    (34)                 

          
with 
                  1(+1) -  1(-1 ) = 0  
                 1,  = 0 ,  1 , x = 0  at  = 1                                                 (35) 
               1(1) = 0 C1(1) = 0        

     0)1(0)1( 11  C  

0)1(,0)1(1)1()1( ,1,111  x                  (36)  

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

* 

https://ijojournals.com/                                                                Volume 07 Issue 03 || March, 2024 || 

Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

The equations (20)- (22), (25)-(27)&(29)-(34) are solved analytically subject to the relevant 
boundary conditions. For sake brevity we are not presenting the solutions. 

 
4.Nusselt number and Sherwood number 

                      )()( 2
8761  OdEcddy   

The local rate of heat transfer coefficient Nusselt number (Nu) on the walls has been calculated 
using the formula  

1)(
1







 



 y
Nu

wm

 where  



1

1

5.0  dm  

and the corresponding expressions are 

  
)1(

)(
)(

))((

)(
)( 108

1
119

1








 

mm

dd
uN

txSin

dd
uN








 ,  

where 1514 ddm    

                  
The local rate of mass transfer coefficient Sherwood Number (Sh) on the walls has been calculated 
using the formula  

1)(
1







 y

wm y

C

CC
Sh  where   




1

1

5.0 dyCCm  

and the corresponding expressions are 

 
)1(

)(
)(

)(

)(
)( 75

1
64

1






 

mm C

dd
Sh

C

dd
Sh


              

 
5.Results and discussion 
 
 This investigation focuses on the impact of heat generation and thermo-diffusion on the unstable 

free convection MHD gy- rated flow of radiation and chemical reactive second order fluid across an 

unbounded perpendicular plate during absorbent medium. The analytical solutions for the 

governing equations are obtained by the application of the Laplace transformation procedure. The 

profiles of concentration, temperature, and velocity are analyzed graphically. For quite a few 

quantities of the magnetic field parameter M, chemical reaction parameter Kr, temperature 

generating and/or absorbing parameters, it is represented the second-grade fluid velocity,  and 

concentration distributions. Magnetic field parameter M, chemical reaction parameter Kr,. We 

fiXed M = 0.5, K = 0.5, Pr = 0.71, R = 2, Gr = 10, Gm = 5, Sr = 0.1, H = 2, = 0.5, and t = 0.5 for 

computational purposes and drew the profiles with each parameter adjusted across the range. The 

concentration, temperature, and velocity profiles are shown in Figs. 1-4 to be less than those for 

isothermal temperature and ramped surface concentration in the case of ramped wall temperature. 

The magnetic domain parameter in the liquid flow generates an electrical field. Consequently, it 

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(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

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Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

was deduced that when isothermal temperature and ramping surface concentration are present 

simultaneously, the magnetic field lowers both of them. A fluid's velocity is expected to be slowed 

down by the addition of a magnetic field, which will increase the resistive model forces (Lorentz 

forces) acting on the fluid in the boundary layers.  

Fig.1.The velocity profile for u against M 

  Fig. 2.  The velocity Profile for v against M 

 

Figs.1 and 2  has been shown that, the intensity of the magnetic field has reducing effects on 

velocity profiles for together heated circum- stances. It is anticipated to the information that, the 

representing of magnetic domain parameter produces electrical field in the liquid flow. This implied 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

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Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

that, the magnetic field has reducing effect for together ramped wall temperature with ramped 

surface concentration as well as isothermal temperature with ramped surface concentration. It is ex- 

pected to the information that, the application of the magnetic field to fluid give augment to the 

resistive model forces (Lorentz forces) on the fluid in the boundary layers, this slow down the 

movement of the fluid. 

Fig.3 The velocity Profile for u against Kr. 

 

 

  

 

 

 

  

 

 

Fig.4 The velocity Profile for v against Kr. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

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Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

Additionally, it is noted that as the second grade parameter rises, the border layer widths decrease. 
As seen in Figs. 3 and 4, chemical reactions have a slowing effect on the velocity of liquid flows in 
combined thermal cases. 
 

Fig.5  The ConcentrationProfile for Kr. 

 

 As shown in Fig. 5, the chemical reactions have a decreasing effect on the concentration profiles 

and liquid flow velocity for the combined thermal case. The destructive reactions Kr > 0 have been 

demonstrated to cause falls into the concentration field, which worsens the effects of buoyant forces 

because of the concentration gradient.  

The flows domain is then narrowed down.depending on the Nusselt number, producing, 
absorbing, and/or radiating parameters H and R. When the temperature producing and/or absorbing 
parameter H and the Prandtl number Pr increase, the Nusselt number Nu increases. Conversely, 
when the radiating parameter R increases, the Nusselt number Nu decreases for both ramping wall 
temperature and isothermal plate. The Nusselt number decreases with increasing time for an 
isothermal plate and increases with ramping wall temperature. 

 
 
 
 
 
 

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(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

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Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

 
 
 
 

Table.1 The Nusselt number 

Pr R H t Ramped Temperature Isothermal Temperature 

0.71 2 2 0.2 0.732255 1.779785 

3    0.846796 2.077621 

7    1.247212 2.379774 

 5   0.694214 1.510547 

 8   0.657995 1.227544 

  -2  0.647895 1.513705 

  5  0.846778 1.950562 

   0.5 0.874546 1.469785 

   0.8 0.958589 1.394958 

  

The effects of temperature producing and/or absorbing parameter H, radiating parameter R, and Pr 
on the Nusselt number were shown in Table.1. When the temperature producing and/or absorbing 
parameter H and the Prandtl number Pr increase, the Nusselt number Nu increases. Conversely, 
when the radiating parameter R increases, the Nusselt number Nu decreases for both ramping wall 
temperature and isothermal plate. The Nusselt number decreases with increasing time for an 
isothermal plate and increases with ramping wall temperature. 

 

 

 

 

 

 

 

 

 

 

 

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(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

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Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

Table.2 The Shear stresses 

 

 

 

 

M K α Kr Sr Gr Gm H R Ramped Temperature Isothermal Plate 

0.5 0.5 0.1 2 0.1 5 2 2 2 
x  y  x  y  

0.8         2.032214 0.075785 1.624789 0.79889 

1         1.613058 0.086898 1.335469 0.828547 

 1        1.269592 0.093578 1.105014 0.932554 

 1.5        1.934796 0.053478 1.449635 0.732969 

  1       1.705478 0.042502 1.304789 0.621559 

  1.5       2.602466 0.086895 1.738966 0.931748 

   3      3.735896 0.113547 1.880254 1.109589 

   4      2.335895 0.083874 1.978801 0.998478 

    0.5     2.968747 0.115748 2.965479 1.398041 

    1     1.939745 0.049411 1.075884 0.479952 

     8    1.749985 0.042115 0.330856 0.239658 

     10    1.93211 0.053041 1.602147 0.767587 

      5   1.703522 0.046874 1.580145 0.741847 

      8   1.976547 0.064306 1.612265 0.790895 

       -5  1.869954 0.049447 1.601458 0.767014 

       5  1.939665 0.060256 1.565595 0.701748 

        5 2.33565 0.086874 1.749854 0.901849 

        8 2.105452 0.083289 1.738859 0.998954 

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(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

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Table.3 The Sherwood number(Pr=0.710,R=2.0,H=-2.0) 

Sr Kr Sc t Ramped Temperature Isothermal Temperature 

0.1 2 0.22 0.2 0.430478 0.545478 

0.5    0.347254 0.462254 

1    0.292854 0.407854 

 3   0.492785 0.607785 

 4   0.570699 0.685699 

  0.3  0.403895 0.518895 

  0.6  0.370548 0.485548 

   0.5 0.500897 0.615897 

 
This is scrutinized from Table 2 that, it is notified that, for together ramped wall temperature and 

isothermal plate, the stress components τx as well as τy enhances by an increasing in second graded 

fluid parameter α, chemical reacting parameter Kr, temperature generations and/or absorptions H 

and thermal radiation parameter R, as well as it reduces by an increasing in the permeability 

parameter K, thermal-diffusion (Soret) parameters Sr, thermal Grashof numbers Gr and mass 

Grashof quantity Gm. This is also found that by an increasing in the intensity of the magnetic fields 

then the stress components τx retards and the component τy boosting up for together ramped wall 

and isothermal plate. Finally, the Sherwood number Sh is reduced with an increasing in the Soret 

number Sr as well as Schmidt number, and it is increasing with an increasing in chemically reacting 

parameter Kr and certain instant of time for together ramped wall temperature and an isothermal 

plate (Table 3). 

6.References 

[1]W. Tan, T. Masuoka, Stokes’ first problem for a second grade fluid in a porous half- space with 

heated boundary, Int. J. Non- Linear Mech. 40 (2005) 515–522. 

[2] M.M. Rashidi, S.A. Majid, A. Mostafa, Application of homotopy analysis method to the 

unsteady squeezing flow of a second-grade fluid between circular plates, Math. Probl. Eng. 18 

(2010), 706840. 

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(ISSN: 2992-4421 )                                                                                                               Venuprasad K. K.1

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Mathematical study of MHD Convective flow with chemical reaction through a porous medium in a vertical wavy channel 

 

 

 

[3]. Hayat, M. Qasim, S.A. Shehzad, A. Alsaedi, Unsteady stagnation point flow of second grade 

fluid with variable free stream, Alexandria Eng. J. 53 (2014) 455–461. 

[4].J. Hartmann, Hg-dynamics I theory of the laminar flow of an electrically conductive liquid in a 

homogenous magnetic field, Det Kongelige Danske Videnskabernes Selskab Mathematisk-fysiske 

Meddelser 15 (1937) 1–27. 

[5] V.V.S. Murty, A. Gupta, N. Mandloi, A. Shukla, Evaluation of thermal performance of heat 

exchanger unit for parabolic solar cooker for off-place cooking, Indian J. Pure Appl. Phys. 45 

(2007) 745–748. 

 [6]M.V. Krishna, A.J. Chamkha, Hall and ion slip effects on MHD rotating boundary layer flow of 

nanofluid past an infinite vertical plate embedded in a porous medium, Results in Physics 15 

(2019), 102652, https://doi.org/10.1016/j. rinp.2019.102652. 

[7] M.V. Krishna, G.S. Reddy, A.J. Chamkha, Hall effects on unsteady MHD oscillatory free 

convective flow of second grade fluid through porous medium between two vertical plates, Physics 

of Fluids 30 (2018), 023106, https://doi.org/10.1063/ 1.5010863. 

[8] M.V. Krishna, M.V. Chamkha, Hall effects on unsteady MHD flow of second grade fluid 

through porous medium with ramped wall temperature and ramped surface concentration, Physics 

of Fluids 30 (2018), 053101, https://doi.org/10.1063/ 1.5025542. 

[9] Tahkar HS, Gorla SR and Soundalgekar VM. Short commu-nication radiation effects on MHD 

free convection flow of a gas past a semi-infinite vertical plate. Int J Numer Methods Heat Fluid 

Flow 1996; 6: 77–83. 

[10]Hossain AM, Alim MA and Rees DAS. The effect of radi ation on free convection from a porous 

vertical plate. Int J Heat Mass Transf 1999; 42: 181–191. 

[11] Muthucumarswamy R and Kumar GS. Heat and mass transfer effects on moving vertical plate 

in the presence of thermal radiation. Theoret Appl Mach 2004; 31: 35–46. 

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