



































IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 

 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD 
convective Heat transfer through a porous medium in a vertical channel 

 
Prakasha.P1, Prof. K. Shivashankara2, Venuprasad K. K.3, Dhananjaiah D. S4. 

 1Department of Mathematics, Government First Grade College, Magadi, Ramanagar,  India  
E-mail id: profprakasha@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.Pete, Mandya, India  
E-mail id: kkvpmaths@gmail.com  

4Department of Mathematics, Government First Grade College K.R.Nagar, Mysuru, India  
E-mail id: dhanu2614@gmail.com 

 
 

Abstract: We analysed the unsteady MHD free connective flow through a porous medium in a 
vertical channel with the unsteadiness in the flow is due to the travelling thermal wave imposed on 
the wall y = L. The coupled MHD equations governing the flow and heat transfer have been solved 
by using a perturbation technique with the aspect ratio as perturbation parameter. The expression 
for the velocity, the temperature, the shear stress and the rate of heat transfer are derived and are 
analysed for different variations of the governing parameters G,R, and .  
Key words: Convection, Porous medium, Magneticfield and Dissipation. 
 
1. Introduction: 

 The energy crisis has been a topic of great importance in recent years all over the world 
.This has resulted in an unabated exploration for new ideas and avenues in harnessing various 
conventional energy sources like tidal waves, wind power and geothermal energy. It is well known 
that in order to harness maximal geothermal energy one should have complete and precise 
knowledge of quanta of perturbation needed to initiate convection currents in mineral fluids 
embedded in the earth’s crest enables one to use mineral energy to extract the minerals 
.Convection fluid flows generated by travelling thermal waves have also received attention due to 
applications in physical problems. The linearised analysis of these flows has shown that a 
travelling thermal wave cal generate a mean shear flow within a layer of fluid, and the induced 
mean flow is proportional to the square of the amplitude of the wave. From a physical point of 
view, the motion induced by travelling thermal waves is quite interesting as a purely fluid-
dynamical problem and can be used as a possible explanation for the observed four-day retrograde 
zonal motion of the upper atmosphere of Venus. All the above mentioned studies are based on the  
hypothesis that the effect of dissipation is neglected. This is possible in case of ordinary fluid flow 
like air and water under gravitational force .But this effect is excepted to be relevant for fluids with 
high values of the dynamic viscosity flows. In view if this, several authors notably Barletta [1,2], 
Bulent Yesilata [3] , Elhakein [4], Israel et al[5] and  Rossidischio [6] have studied the effect of 
viscous dissipation on the convective flows past an infinite vertical plate and through vertical 
channels and ducts. 
In recent years, a great deal of interest has been generated in the area of boundary layer flow and 
heat transfer of a fluid over a stretching sheet In view of its numerous and wide range of 
applications in various fields such as polymer processing industry in particular manufacturing 

IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 1

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


IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

process of artificial films, artificial fibers, and dilute polymer solutions. To be more specific, it 
may be pointed out that many metallurgical processes involve the cooling of continuous strips or 
filaments by drawing them through a quiescent fluid, and in the process of drawing, these strips are 
sometimes stretched. The heat transfer analysis over a stretching surface is of much practical 
interest due to its abundant applications such as heat-treated materials travelling between a feed 
roll and wind-up roll or materials manufactured by extrusion, glass-fiber, and paper production, 
cooling of metallic sheets or electronic chips, drawing of plastic films, liquid films in condensation 
processes. Due to the high applicability of this problem in the industrial phenomena, it has 
attracted attention of many researchers. 
The effects of the buoyancy force on the development of the velocity and thermal boundary layer 
flows over a stretching sheet were first studied by Chen[7]. Elbashbeshy and Bazid[8], Sharidan et 
al.[9], and Tsai et al.[10] obtained a similarity solution for the flow and heat transfer of a fluid over 
an unsteady stretching surface. The problem of mixedconvection adjacent to a vertical 
continuously stretching sheet in the presence of a variable magnetic field was studied by Ishak et 
al.[11]. Aziz[12] obtained the numerical solution for the laminar thermal boundary over a flat plate 
with a convective surface boundary conditions.  
The effect of viscous dissipation changes the temperature distributions by playing a role as an 
energy source, which affects the heat transfer rates. The merit of the effect of viscous dissipation 
depends on whether the plate is being cooled or heated. Chen[13] examined the  effect of 
combined heat and mass transfer on magnetohydrodynamic (MHD) free convection from a vertical 
surface with the Ohmic heating and viscous dissipation. Pal and Hiremath[14] determined the heat 
transfer characteristics in the laminar boundary layer flow over an unsteady stretching sheet placed 
in a porous medium in the presence of viscous dissipation and internal absorption or generation. 
Veena et al.[15] obtained the solutions of heat transfer in a visco-elastic fluid past a stretching 
sheet with viscous dissipation and internal heat generation.In light of the above investigations, it is 
found that these studies are restricted to the fluid flow and heat transfer problems. However, the 
fluid flow embedded with dust particles is encountered in different engineering problems 
concerned with nuclear reactor cooling, powder technology, rain erosion, paint spraying, etc. The 
important applications of dust particles in the boundary layer include soil erosion by natural winds 
and dust entrainment in a cloud during a nuclear explosion. It also occurs in awide range of 
technical processes like fluidization, flow in rocket tubes, combustion, and purification of crude 
oil. Palani and Ganesan[16] investigated the flow of dusty gas past a semi-infinite isothermal 
inclined plate. 
 
2. Mathematical formulation : 

 We consider the motion of viscous, incompressible fluid through a porous medium in a 
vertical channel bounded by flat walls . The thermal buoyancy in the flow field is created by a 
travelling thermal wave imposed on the boundary wall at y = L while the boundary at y = -L is 
maintained at constant temperature T1. The viscous and Darcy dissipations are taken into account 
to the transport of 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 equations governing the unsteady flow and heat transfer under boundary conditions in terms 
of stream function    are  
 
 

IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 2



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 







 2

2

2

0

2
0

2

0
4222 )(])()()[( 

























ky

H
TTg e

yxyyxt    (2.1) 





































































222
0

2

2

2

2
2

2

2
2

)()(

)()()(

yx
H

k

xy
Q

yxxyt
C

e

pe












         (2.2)  

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

   


L

L

ydu
L

Q
2

1
.                                                                                                        (2.3) 

The boundary conditions for the velocity and temperature fields are  
  u = 0  , v = 0  ,T=T1  on y = -L  
  )(,0,0 2 ntmxSinTTTvu e  on  y = L                                                          (2.4) 

where    u = - y , v =  x                                                                                                       (2.5)     
Introducing the non-dimensional variables in (2 .10 )- (2.12) as   

e

e

T

TT
mttLyymxx




  ,/,,/, 12                                     (2.6)   

(under the equilibrium state ))()(
2



QL
LTLTT eee   

the governing equations (2.1) & (2.2) in the non-dimensional form ( after dropping the dashes ) are  

2

2
22

1
14

1

2
12

1
),(

),(
)(

y
MD

R

G

yx
R yt


























  




                                (2.7) 

and the energy equation in the non-dimensional form is  

 

  











































































 ))()(

)()(

22221

2

2

2
22

2

22
2
1

yx
MD

xyG

EPR

yxxyt
P c













    (2.8) 

where 

 


UL
R   (Reynolds number),   

2

3



 LTg
G e
 (Grashof number) 

1k

c p
 (Prandtl number),         

   
k

L
D

2
1  (Darcy parameter), 

 
p

c
C

gL
E

3
  (Eckert number),   

IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 3



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

Lm  ( Aspect ratio) 

2m

n


   (non-dimensional thermal wave velocity),  

)(
2

222
2



 LH
M oe )( NumberHartmann  

2

2

2

2
22

1
yx 







   

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

 10,0 








yat

yx


                        (2.9) 

 1),( yx     on   y = -1      

 )(),( txSinyx       on   y = 1     

            00 



yat

y


                                 (2.10) 

The value of  on the boundary assumes the constant volumetric flow in consistent with the 
hypothesis(2.9) .Also the wall temperature varies in the axial direction in accordance with the 
prescribed arbitrary function t . 
 
3.Analysis of the flow:  
 
The perturbation analysis is carried out by assuming that the aspect ratio    to be small.  
     
We adopt the perturbation scheme and write  

   ),(),(),(),( 2
2

10 yxyxyxyx  ……………. 

   ),(),(),(),( 2
2

10 yxyxyxyx  …………………                             (3.1)    

               

On substituting ( 3.1) in (2.13) - (2.15) and separating the like powers of  the equations and 
respective conditions to the zeroth order are  

)( ,0,0,0
2

1,0 yyyyyyyyy NCGM                                             (3.2) 

0)(
)(

)( 2
,

21
2

,

2

, 





yo
c

yyo
c

yyo
G

MDPE

G

RPE
                                           (3.3) 

With    1)1()1(0               

        0, y = 0 ,  0 , x =0           at y = 1                                                                           (3.4)  

  
1)(

11





yontxSin

yon

o

o




                                                                          (3.5)                                                                                                 

and to the first order are  

IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 4



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

)( ,0,0,0,0,1
2
1,1 yyyxyyxyyyyyyyyy GM                                                           (3.6)           

).(
2

).(
2

)( ,1,0

1

,1,0

2

,0,,0,1 yy
c

yyyy
c

oxyyox
G

DPE

G

RPE
yy




                                    (3.7) 

with 
              1(+1) -  1(-1 ) = 0  
    1, y = 0 ,  1 , x = 0  at y = 1                                                                                       (3.8) 
    1(1) = 0 at y =  1                        (3.9) 
Assuming Ec<<1 to be small we take the asymptotic expansions as 

                

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

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

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

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

11101

01000

11101

01000









yxyxyx

yxyxyx

yxEcyxyx

yxEcyxyx









                                         (3.10)     

Substituting the expansions(3.10) in equations (3.2)-(3.9) and separating the like powers- of Ec we 
get the following  

10000,00 )1(,1)1(, SinDyy                                                    (3.11)            

10,0

1)1()1(,

,00,00

0000,00,00
2

1,00





yat

GM

xy

yyyyyyy




                                   (3.12) 

 0)1(, 01
2

,00

1

,00
2

,01 


 yyyyy
G

PD

G

PR
                            (3.13) 

10,0

,0)1()1(,

,01,01

0101,01,01
2

1,01





yat

GM

xy

yyyyyyy




                            (3.14) 

  0)1()( 10,00,00,00,00,10   yxxyyy                                     (3.15) 

 
10,0,0)1()1(

,)(

,10,101010

,00,00,00,00,10,10
2

1,10





yat

GM

xy

yyyxxyyyyyyyyyy




                          (3.16) 

 

0)1(,
2

2
)(

1,10,00

1

,10,00

2

,0,01,01,00,00,01,01,00,11











yy

yyyyxyyxyxxyyy

G

PD

G

PR

                     (3.17) 

  

10,0,0)1()1(

,)

(

,11,111111

,00.01,00,01

,01,00,11,00,1,1
2

1,11







yat

GM

xy

yyyxxyyy

yyyxxyyyyyyyyyy







                                 (3.18) 

 
4. Shear stress and Nusselt number 
 
The shear stress on the channel walls is given by 

                          Ly
x

v

y

u


















         

IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 5



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

which in the non- dimensional form reduces to  

                      

1
2

,11,10,01,00

2

)](([

)(

























yyyyyyyyy

xxyy

OEcEc

a

U









     

and the corresponding expressions are 

                      )()( 2
5431  OdEcddy   

                      )()( 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

wm y
Nu




        Where    




1

1

5.0 dym   

and the corresponding expressions are  

)(

)(
)(,

)(

)(
)(

6510

987
1

654

321
1

mEcmm

mmEcm
uN

mmEcm

mEcmm
uN yy

















 

  

5.  Discussion of the Numerical results:  
The aim of this analysis is to discuss the effect of the dissipation on the convective flow 

and heat transfer of a viscous fluid through a porous medium confined in a vertical channel whose 
walls a travelling thermal wave is imposed. Assuming the Eckert number Ec <<1 the coupled 
momentum and energy equations have been solved. The velocity and temperature distributions are 
analysed for different sets of the governing parameters. 
 

 
                                                                      

Fig (1) Variation of u with G                          Fig (2) Variation of u with M  
R=35, M=2, β=0.5, γ=2, x=π/4, N1=4, t=π/4          G=2x103, R=35, β=0.5, γ=2, x=π/4, N1=4, t=π/4 

 

-1 .0 -0 .5 0 .0 0 .5 1 .0

-0 .3 5

-0 .3 0

-0 .2 5

-0 .2 0

-0 .1 5

-0 .1 0

-0 .0 5

0 .0 0

 G = 1 0
3

 G = 3 x 1 0
3

 G = 5 x 1 0
3

 G = -1 0
3

 G = -3 x1 0
3

 G = -5 x1 0
3

u



-1.0 -0.5 0.0 0.5 1.0

-0.55

-0.50

-0.45

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

 M=1.5

 M=2.5

 M=3

u



IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 6



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

  

-1 .0 -0 .5 0 .0 0 .5 1 .0

-0 .4 0

-0 .3 5

-0 .3 0

-0 .2 5

-0 .2 0

-0 .1 5

-0 .1 0

-0 .0 5

0 .0 0

    

     

     

     

     

u



 
Fig (3) Variation of u with β                                  Fig (4) Variation of u with  R &  γ  

G=2x103, R=35, M=2,γ=2, x=π/4, N1=4, t=π/4                G=2x103, M=2, β=0.5, x=π/4, N1=4, t=π/4 
 

   
      

 
   
 

 
 

 
 
 
Fig (5) Variation of u with  N1                                                                               Fig (6) Variation of u with  x+ γ t 
G=2x103, R=35, M=2, β=0.5, γ=2, x=π/4, t=π/4      G=2x103, R=35, M=2, β=0.5, γ=5, N1=4, t=π/4 

 
            
Fig. (1) exhibits the variation of u with Grashof  number G.  It is found that the axial 

velocity u is completely negative for all values of G with maximum occurring at the mid plane y=0 
which drifts towards the upper plate for higher G(>0) and it drifts towards the lower boundary for 
|G|(<0).  The magnitude of ‘u’ reduces in the lower half and enhances in the upper half with an 
increase in G, while a reversed effect is observed with increase in |G|(<0).  The variation of ‘u’ 
with M shows that for lower values of the Hartman number M we find reversed flow in the vicinity 
of both the boundaries and for higher M~O(1.5) the reversed flow in the vicinity of upper 
boundary disappears and reversal flow continues in the vicinity of lower boundary and for still 
higher values of M≤3 the reversal flow disappears in the entire flow region.  For M=5.0 the 
reversal flow reappears in the vicinity of upper boundary.  This shows that under the influence of 
strong magnetic field the free convection effect do not dominate over others.  An increase in 
M~O(1.5) enhances ‘u’ in entire fluid region and for further increase in M(≥2.5) we find 
depreciation in |u| in the flow region.  From fig. (3) we find that greater the dilation lesser the 

-1 .0 -0 .5 0 .0 0 .5 1 .0

-0 .3 5

-0 .3 0

-0 .2 5

-0 .2 0

-0 .1 5

-0 .1 0

-0 .0 5

0 .0 0

 R = 3 5 ,  

 R = 7 0

 R = 1 4 0

   

   

   u



-1 .0 -0 .5 0 .0 0 .5 1 .0

-0 .4 0

-0 .3 5

-0 .3 0

-0 .2 5

-0 .2 0

-0 .1 5

-0 .1 0

-0 .0 5

0 .0 0

 x +  t=  

 x +  t=  

 x +  t= 3  

u



-1.0 -0.5 0.0 0.5 1.0

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

 N
1
=1

 N
1
=5

 N
1
=10

 N
1
=100

u



IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 7



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

magnitude of u.  An increase in the Reynolds number R leads to an enhancement |u| in the lower 
half and depreciation in the upper half.  Also an increase in the thermal wave velocity   γ(≤10) 
depreciates |u| in the lower half and enhances in the upper half and for higher γ(≥15)       a reversed 
effect is observed in the flow region (fig. (4)).  The variation of ‘u’ with radiation parameter N1 
shows a reversal flow in the mid region for smaller values of N1 and this reversal flow disappears 
for higher values of N1.  |u| reduces in the lower half and enhances in the upper half with N1(≤1.0) 
and for higher N1(≥5)  we notice a depreciation  in |u|  in the region abutting the boundaries and an 
enhancement in the mid region (fig.(5)).  The variation of u with the phase x+γt of the boundary 

temperature curve, shows that |u| enhances with 
4

7
x


  t   and for higher values of x+γt we 

notice an enhancement in the vicinity of the boundary and depreciation in the mid region. (Fig.6)  
 
 
 
 
 
 

 
 

 
 
 

 
 
 
 

 
 

Fig (7) Variation of v with G Fig                                        (8) Variation of v with M  
 R=35, M=2, β=0.5, γ=2, x=π/4, N1=4, t=π/4         G=2x103, R=35, β=0.5, γ=2, x=π/4, N1=4, t=π/4 
 
 

 
 
 
 

 
 

 
 

 
 
 

-1.0 -0.5 0.0 0.5 1.0

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

 G=10
3

 G=3x10
3

 G=5x10
3

 G=-10
3

 G=-3x10
3

 G=-5x10
3

v



-1.0 -0.5 0.0 0.5 1.0

-0 .20

-0 .15

-0 .10

-0 .05

0.00

0.05

0.10

0.15

0.20

0.25
 R =35, 

 R =70

 R =140

 

 

 

v



-1.0 -0.5 0.0 0.5 1.0

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

 M=2

 M=5

 M=10

v



-1 .0 -0 .5 0 .0 0 .5 1 .0

-0 .6

-0 .4

-0 .2

0 .0

0 .2

0 .4

0 .6
  

  

  

  

  

v



IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 8



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

 
 Fig (9) Variation of v with  β                                               Fig (10) Variation of v with  R & γ 
G=2x103, R=35, M=2, γ=2, x=π/4, N1=4, t=π/4           G=2x103, M=2, β=0.5, x=π/4, N1=4, t=π/4 

 
 

 
 
 
 
 
 
 
 

   
    

 
Fig (11) Variation of v with  N1                                                              Fig (12) Variation of v with  x+ γ t 
G=2x103, R=35, M=2, β=0.5, γ=2, x=π/4,  t=π/4     G=2x103, R=35, M=2, β=0.5, γ=5, N1=4, t=π/4    
                                             

The secondary velocity ‘v’ which arises due to the non-uniformity in the boundary has 
been depicted in figs. (7) - (12) for different G,R,M,γ,β,N1 and x+γt.  We notice that for smaller 
G(≤103) the secondary velocity in the left half and in the upper region it is directed towards the 
boundary.  For higher G(≥3x103)  the fluid in the left half is directed towards the boundary and the 
fluid in the upper half is directed towards the mid region, while for |G| (<0) the fluid in the left half 
is directed towards the mid region and the fluid in the upper half is directed towards boundary, for 
all  |G|.  

   The variation of ‘v’ with M shows that for smaller values of M~O(0.5) the fluid in entire 
flow region is directed towards the mid region, while for higher M(≥1.5) the fluid in the flow 
region is towards the boundary except in the vicinity of the left boundary is directed towards the 
mid region.  The region where the transition takes place enhances its  size with increase in M.  An 
increase in M~O(3.5) we find a retardation in |v| and for further increase in M the velocity v in the 
left region experiences a depreciation and that in the right region experiences an enhancement and 
for still higher M(≥5) we find an enhancement  in |v| in the entire flow region (fig. (8)).  From fig. 
(9), it is found that greater the dilation β~O(3.5) larger the |v| and for higher β~O(0.5) we notice a 
depreciation in |v| in the left half and enhancement in |v| in the right half and for still higher β, 
larger |v| in entire flow region.  An increase in R enhances |v|.  Also an increase in the thermal 
wave velocity γ  enhances |v|  in the upper half and reduces it in the lower half, and for further 
increase in γ we find an enhancement in |v| (fig. (10)).  The variation ‘v’ with N1 shows that for 
small values of N1 the secondary velocity is directed towards the boundary and for higher values of 
N1 we find that v is towards the mid region for all N1.  |v|  enhances with increase in N1. (fig. (11)).  

The variation ‘v’ with  phase x+γt of  the boundary  curve shows that for increasing  
4

7
x


  t  

the velocity in the left half reduces and that in the right region enhances with x+γt   and for higher 

values of 
4

11
x


  t   we find an increase in |v|. (Fig.12)     .   

-1.0 -0.5 0.0 0.5 1.0

-0.040

-0.035

-0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

 N
1
=0.5

 N
1
=1

 N
1
=5

 N
1
=10

 N
1
=100

v



-1.0 -0.5 0.0 0.5 1.0

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08
 x+ t=

 x+ t=

 x+ t=3

v



IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 9



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

 
 
 
 

 
 

 
 
 
 
 

 
Fig (13) Variation of θ with  G                                             Fig (14) Variation of θ with M 
R=35, M=2, β=0.5, γ=2, x=π/4, N1=4, t=π/4         G=2x103, R=35, β=0.5, γ=2, x=π/4, N1=4, t=π//4 

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 

 
Fig (15) Variation of θ with β                                           Fig (16) Variation of θ with R & γ  
G=2x103, R=35, M=2,  γ=2, x=π/4, N1=4, t=π/4              G=2x103, M=2, β=0.5, x=π/4, N1=4, t=π/4 

 
 

-1 .0 -0 .5 0 .0 0 .5 1 .0

-0 .2

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

1 .8

2 .0

2 .2

2 .4

2 .6

2 .8
M = 0 .5

 M =1 .5

 M =2 .5

 M =3

 M =5





-1 .0 -0 .5 0.0 0.5 1.0

-1 .0

-0 .8

-0 .6

-0 .4

-0 .2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8
R =35, R =70

 R =140

   

 





-1 .0 -0 .5 0 .0 0 .5 1 .0

-1 .0

-0 .5

0 .0

0 .5

1 .0

1 .5

 x+  t= 

 x+  t= 

 x+  t= 3  





-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

 G=10
3

 G=3x10
3

 G=5x10
3

 G=-10
3

 G=-3x10
3

 G=-5x10
3





-1.0 -0.5 0.0 0.5 1.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

 =0.1

 =0.3

 =0.5

 =0.7

 =0.9





-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

 N
1
=0.5

 N
1
=1

 N
1
=5

 N
1
=10

 N
1
=100





IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 10



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

 
 
Fig (17) Variation of θ with  N1                                                      Fig (18) Variation of θ with x+ γ t 
G=2x103, R=35, M=2, β=0.5, γ=2, x=π/4, t=π/4      G=2x103, R=35, M=2, β=0.5, γ=5, N1=4, t=π/4 
                                 

The temperature distribution (θ) for different values of G,R,M,β,N1 and x+γt  is shown in 
figs. (13) - (18).  The perturbation temperature in general is positive and hence contributes to the 
enhancement of actual temperature in the fluid region.  Fig (13) depicts behavior of θ for different 
|G|(><0).  We notice that in a dilated channel the temperature increases/decreases with |G| 
according as with G(>0) or G(<0).  An increase in M~O(2.5) we find an enhancement in θ.  For 
higher M(≥3.0) the temperature in the left half enhances and that in the right half depreciates with 
increase in M.  Also higher the dilation larger the temperature in the left region and smaller the 
temperature in the upper region (fig.(15)). An increase in R decreases θ.  The variation of ‘θ’ with 
γ shows that for γ≤5 the temperature is negative and for higher γ≤10, θ is positive.  Also an 
increase in γ≥10 leads to an enhancement in ‘θ’ and for further higher values of γ we notice a 
depreciation in θ.  The effect of radiation on ‘θ’ is shown in fig. (17).  An enhancement in the 
radiation parameter N1 results in an increase in the temperature in entire flow region.  The 

variation of θ with phase x+γt  of the boundary temperature curve  shows that for 
2

x


  t  the 

temperature is negative and for higher values of x+γt,  θ is positive.  For an increase 
2

x


  t  we 

find depreciation in θ and for higher 
2

3
x


  t ,  θ enhances in entire flow field(Fig.18).    

Table.1 Shear stress( ) at y = 1 , P=0.71 
G I II III IV V VI VII VIII 

10 3  -27.32 -54.318 -86.19 -38.688 -49.176 -27.303 -49.182 -59.912 

3x10 3  53.5904 -83.733 -106.441 -75.856 -88.87 53.501 -108.481 -127.611 

5x10 3  84.3505 -122.711 -176.981 -100.541 -123.291 93.685 -236.881 -168.051 

-10 3  -54.375 -62.796 -89.806 -43.128 -59.124 -54.393 -71.123 -66.969 

-3x10 3  -76.578 -97.488 -102.951 -83.242 -108.591 -76.647 -177.041 -87.759 

-5x10 3  -144.941 -167.721 -251.611 -159.061 -168.491 -128.711 -281.771 -196.321 

Table.2 Shear stress( ) at y = -1 P=0.71 

 

G I II III IV V VI VII VIII 

10 3  0.0857 0.5203 0.8311 0.6351 -1.1167 0.0494 -1.2606 0.5973 

3x10 3  3.6351 -2.3797 -6.0387 -2.9492 -6.9704 3.2407 -9.6896 -3.1505 

5x10 3  33.614 4.6238 -10.804 -3.5669 4.9161 32.0035 -7.2413 1.1336 

-10 3  3.1566 3.6622 3.8092 2.3599 1.8977 3.1929 2.0418 3.7117 

-3x10 3  -6.9607 -2.6153 -3.0992 -2.6016 -18.506 -6.5656 -15.785 -2.4721 

-5x10 3  -50.075 -27.973 -25.889 -19.076 -82.909 -48.462 -70.746 -26.619 

IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 11



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

 
 
 
 
 
 

Table.3 Average Nusselt Number(Nu) at y = 1 P=0.71 
 
 
 
 
 
 
 
 
 

Table..4 Average Nusselt Number (Nu) at y = -1 P=0.71 
G I II III IV V VI VII VIII 

10 3  4.1254 3.7439 3.6133 1.8222 2.2733 1.1622 1.4049 3.6995 

3x10 3  5.7579 3.9748 3.9277 1.6329 1.7193 1.4569 1.6775 3.9398 

5x10 3  2.6433 3.9312 3.8872 1.2358 -9.867 1.8127 2.0487 3.8663 

-10 3  4.1209 3.7371 3.6112 1.7882 2.2721 1.1636 1.4041 3.6961 

-3x10 3  5.7866 3.9729 3.9265 1.5031 1.7107 1.4549 1.6752 3.9381 

-5x10 3  2.6557 3.9271 3.8852 0.8944 -10.557 1.8103 2.0461 3.8624 

 
 I II III IV V VI VII VIII 

D 1  10 3  2x10 3  3x10 3  10 3  10 3  10 3  10 3  10 3  
  5 5 5 15 5 5 5 5 
  2 2 2 2 5 -2 -5 2 
M 2 2 2 2 2 2 2 4 
 

              The shear stress () and the average Nusselt number (Nu) on the boundaries  
(y= 1) have been evaluated for different parameters and are given in tables (1)-(4). The shear 
stress is positive at y = -1 and negative at y = 1. It is found that  is observed to increase with an 
increase in G fixing the other parameters. Higher  the permeability of the medium larger the shear 
stress at both the boundaries. With reference to  we find that  increases for an increase in  for 
all G (tables 1 & 2).  
 

                The average Nusselt number measures the local rate of heat transfer across the boundary. 
We find from (tables.3 & 4) that the average Nusselt number is positive at y = 1 and negative at y 
= -1 for all variations. The magnitude of Nu at y=  1 increases with an increase in G > 0 and 
decreases with G<0 fixing the other paramerters. In axial heating case  Nu decreases with R and 
enhances with D-1 while a reversed effect is observed in the case of axial cooling. The rate of heat 
transfer (Nu) declines with an increase in the amplitude  of the boundary temperature 
(tables.3&4).  

 I II III IV V VI VII VIII 

D 1  10 3  2x10 3  3x10 3  10 3  10 3  10 3  10 3  10 3  
  5 5 5 15 5 5 5 5 
  2 2 2 2 5 -2 -5 2 
M 2 2 2 2 2 2 2 4 

G I II III IV V VI VII VIII 

10 3  -0.9304 -1.2347 -1.6751 -0.7476 -1.0113 -5.6211 -1.6412 -1.3149 

3x10 3  -0.6146 -0.9777 -1.3334 -0.5734 -0.6634 -3.9777 -3.0481 -1.0412 

5x10 3  0.2125 -0.7692 -1.2206 -0.2668 0.5107 -3.5075 -2.7442 -0.8637 

-10 3  -0.9312 -1.2373 -1.6772 -0.7435 -1.0099 -3.1893 -1.6391 -1.3176 

-3x10 3  -0.6105 -0.9761 -1.3324 -0.5467 -0.6578 -5.5872 -2.0022 -1.0398 

-5x10 3  0.2239 -0.7658 -1.2185 -0.1927 0.5286 -3.9744 -2.3682 -0.8607 

IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 12



IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Prakasha.P1* 

https://ijojournals.com/                                                       Volume 07 Issue 04 || April, 2024 || 

Mathematical Analysis of Effect of Viscous dissipation on Transient MHD convective Heat transfer through a porous medium in a vertical channel 
 

 
 

 

 
6.References 
 
[1] Barletta,Antonio , Laminar mixed convection with viscous dissipation in a vertical   channel., 
Int.J.Heat  and Mass transfer,41,No.22,pp.3501-3513,(1998). 
[2] Barletta,Antonio , Combined forced and free convection with viscous dissipation in a vertical 
circular duct., Int.J.Heat and Mass Transfer,42,No.12,pp.2243-2253,(1999). 
[3]  Bulent Yesilata , Effect of viscous dissipation on polymeric flows between two    rotating 
Coaxial parallel discs., Int.Comm. Heat and Mass Transfer, 29, No.29, No.5,   
     pp.589- 600,(2002). 
[4]  El-Hakein, M.A., Viscous dissipation effects on MHD free convection flow over a non-
isothermal surface in a micropolar fluid. Int.Comm.Heat and Mass Transfer ,27, No,4 pp.581-
590,(2000). 
[5]  Israel-Cookey ,C. , Influence of viscous dissipation and radiation on unsteady MHD free 
convection flow past an infinite heated vertical plate in a porous medium with time-dependent 
suction.,Int.J.Heat and Mass Transfer,46,No.13,pp.2305-2311,(2003). 
[6] Rossie di Schio,Baletta,A,Hahne,E and Spindler,K , Analysis of the effect of viscous   
dissipation for laminar flow in Stadium –shaped ducts, Int.Comm.Heat Mass Transfer,28, No.4, 
pp.449-459,(2001). 
[7] Chen, C. H. Laminar mixed convection adjacent to vertical, continuously stretching sheets. 
Heat and Mass Transfer, 33(5-6), 471–476 (1998) 
[8] Elbashbeshy, E. M. A. and Bazid, M. A. A. Heat transfer over an unsteady stretching 
surface.Heat and Mass Transfer, 41(1), 1–4 (2004) 
[9] Sharidan, S., Mahmood, T., and Pop, I. Similarity solutions for the unsteady boundary layer 
flow and heat transfer due to a stretching sheet. International Journal of Applied Mechanics and 
Engineering, 11(3), 647–654 (2006) 
[10] Tsai, R., Huang, K. H., and Huang, J. S. Flow and heat transfer over an unsteady stretching 
surface with a non-uniform heat source. International Communications in Heat and Mass 
Transfer,35(10), 1340–1343 (2008) 
[11] Ishak, A., Nazar, R., and Pop, I. Hydromagnetic flow and heat transfer adjacent to a stretching 
vertical sheet. Heat and Mass Transfer, 44(8), 921–927 (2008) 
[12] Aziz, A. A similarity solution for laminar thermal boundary layer over a flat plate with a 
convective surface boundary condition. Communications in Nonlinear Science and Numerical 
Simulation,14(4), 1064–1068 (2009) 
[13] Chen, C. H. Combined heat and mass transfer in MHD free convection from a vertical surface 
with Ohmic heating and viscous dissipation. International Journal of Engineering Science, 
42(7),699–713 (2004) 
[14] Pal, D. and Hiremath, P. S. Computational modelling of heat transfer over an unsteady 
stretching surface embedded in a porous medium. Meccanica, 45(3), 415–424 (2010) 
 [15] Veena, P. H., Subhas-Abel, M., Rajagopal, K., and Pravin, V. K. Heat transfer in a visco-
elastic,fluid past a stretching sheet with viscous dissipation and internal heat generation. Zeitschrift 
f¨ur,Angewandte Mathematik und Physik (ZAMP), 57(3), 447–463 (2006) 
 [16] Palani, G. and Ganesan, P. Heat transfer effects on dusty gas flow past a semi-infinite 
inclined,plate. Forsch Ingenieurwes, 71, 223–230 (2007) 

IJO JOURNALS

Volume 07 | Issue 04 | April 2024 | https://ijojournals.com/index.php/m/index 13


