Finite Element Solution On Effects Of Viscous Dissipation & Diffusion Thermo On Unsteady Mhd Flow Past An Impulsively Started Inclined Oscillating Plate With Mass Diffusion &Variable Temperature

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
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Finite Element Solution On Effects Of Viscous Dissipation & Diffusion
Thermo On Unsteady Mhd Flow Past An Impulsively Started Inclined
Oscillating Plate With Mass Diffusion &Variable Temperature
B. Shankar Goud1, M.N Rajashekar2
1Department of Mathematics, JNTUH College of Engineering Kukatpally, Hyderabad- 085 , TS, India.
2Department of Mathematics, JNTUH College of Engineering Nachupally, Karimnagar -505501, TS, India.
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Abstract - The aim of this is paper is to investigate on the
effects of viscous dissipation and diffusion thermo on an
unsteady MHD flow with an inclined oscillating plate
started impulsively. The effects with a variable temperature
and mass diffusion are observed. Considered fluid is gray,
absorbing-emitting radiation, but a non-scattering medium.
Solutions of the nonlinear differential equation are obtained
by finite element method. The effects of different flow
parameters on the flow variables are discussed. The results
have been analyzed graphically.
Key Words: Unsteady, variable temperature and mass
diffusion, MHD, FEM, Viscous Dissipation
1. INTRODUCTION
The study of the hydromagnetic flow of an electrically
conducting fluid has many applications in science and
engineering problems such as magnetohydrodynamic
(MHD) generator, plasma studies, nuclear reactors,
aerodynamic heating, etc. Soundalgekar et al [1]
investigated the problem of free convection effects on
Stokes problem for a vertical plate with transverse applied
magnetic field. Elbasheshy [2] studied MHD heat and mass
transfer problem along a vertical plate under the
combined buoyancy effects on of thermal and spices
diffusion. Ibrahim [3] has investigated analytical solution
of heat and mass transfer over a permeable stretching
plate affected by chemical reaction, internal heating,
Dufor-Soret effect and Hall effect.MHD flow Past an
Impulsively started vertical plate with variable
temperature and mass diffusion was studied by Rajput
and Surender kumar [4]. Rao and Shivaiah [5] studied
chemical reaction effects on unsteady MHD flow past
semi- infinite vertical porous plate viscous dissipation. P.K
Sing [6] showed heat and mass transfer in MHD boundry
layer flow past an inclined plate with variable temperature
and mass diffusion. T.Arun kumar and L Anand Babu [7]
has analyzed the study of Radiation effect of MHD flow
past an impulsive started vertical plate with variable
temperature and uniform mass diffusion – A finite element
method.
P.Srikanth Rao and D.Mahendar [8] investigated Soret
effect on unsteady MHD free convection flow past a semi-
infinite vertical permeable moving plate.
D.Chennakesavaiah and P V Satyanarayana[9] studied the
radiation absorption and dufour effect to MHD flow in
vertical surface. Dufour effects on unsteady MHD free
convection and mass transfer flow fast through a porous
medium in slip regime with heat source/ sink was studied
by K.Sharmilaa and S.Kaleeswari[10]. K.Anitha [11] has
analyzed chemical reaction and radiation effects on
unsteady MHD natural convection flow of rotating fluid
past a vertical porous flat plate in the presence of a viscous
dissipation. The effect of Hall current on an unsteady MHD
free convective flow along a vertical plate with the thermal
radiation was studied by P.Srikanth Rao and D.Mahendar
[12].
2. MATHEMATICAL ANALYSIS
In this paper we have considered MHD flow between two
parallel electrically non conducting plates inclined at an
angle from vertical. axis is taken along the plate and
normal to it. A transverse magnetic field of uniform
strength is applied on the flow. The viscous dissipation
and induced magnetic field has been neglect due to its
small effect. Initially it has been considered that the plate
as well as the fluid is at same temperature and the
Concentration level everywhere in the fluid is same
as in stationary condition. At time 0t  , the plate starts
oscillating in its own plane with frequency  and
temperature of the plate is raised to and the
concentration level near the plate is raised linearly with
respect to time.
The flow modal is as follows:

    
2
*
2
2
0
cos
u u
g T T g C C
t t
B u
   


 
         
  
 
 
 
- (1)
22 2
2 2
m T
p s p p
D kT k T C u
t C C C C yy y


       
          
- (2)
 
2
2 r
C C
D K C C
t y

       
  
- (3)
With the corresponding initial and boundary conditions:
2
0
0
2
0
0 : 0, , ,
0 : cos , ( ) ,
( ) 0
0, ,
w
w
t u T T C C for all y
U t
t u U t T T T T
U t
C C C C at y
u T T C C as y



 
 
 

          

  
            

  
        

         
- (4)
In order to write the governing equations and the
boundary conditions in dimensionless form, the following
non dimensional quantities are introduced.
 
 
 
 
 
 
 
2
0
0 0
0
2 3
0 0
* 2
3 2
0
2
0
2
0
, , , ,
, ,Pr ,
, , ,
, ,
, ,
p
w w
w
w
m T w r
S P w
P w
U yt U u
t y u Sc
U D
cC CT T
C
T T C C k
g T T
Gr
U U
g C C B
Gm M
U U
D k C C K
Du Kr
c c T T U
U
Ec
c T T

 


  
  
   




 





 
    

     
    

  
   

   


  
 
 

  






- (5)
With the non dimensional quantities equations (1),(2) and
(3) reduces to the following dimensionless form:
 
2
2
cos r m
u u
G G C Mu
t y
 
 
   
 
- (6)
22 2
2 2
1
Pr
C u
Df Ec
t yy y
      
    
    
- (7)
2
2
1C C
KrC
t Sc y
 
 
 
- (8)
With the initial and boundary conditions in dimensionless
form are:
0, 0, 0, , 0
0 : cos , , , 0,
0, 0, 0
u C for all y t
t u t t C t at y
u C as y

 

    

     
     
- (9)
3. SOLUTION OF THE PROBLEM
Using finite element method with Crank-Nikolson
discretization taking 0.1, 0.01h k  .The element
equation for the typical element ( )e j ky y y  for the
boundary value problem can be written as:
2 ( ) ( )
( ) ( )
2
0
k
T
j
y e e
e e
y
u u
N Mu P dy
ty
  
    
 
 - (10)
Where cos cosr mP G G C  
( ) ( ) ( )
( ) ( ) ( ) ( )
0
k k k
T T T
j jj
y y ye e e
e e e e
y yy
u u u
N N dy N Mu P dy
y y t
     
       
     
 
Neglecting the first term in the above equation, we get
( ) ( )
( ) ( ) ( )
0
k
T T
j
y e e
e e e
y
u u
N N Mu P dy
y t
    
     
    

Let ( ) ( ) ( )e e e
u N  be the finite element approximation
solution ( j ky y y  ) where
( ) ( )
,
Te e
j k j kN N N u u        , ,
jk
j k
k j k j
y yy y
N N
y y y y

 
 
are the basis functions.
y yk k
j j j k j j j kj j
j k j k k kky yj j
k
y yk k
j j j k j j
j k k k k ky yj j
N N N N N N N Nu u
dy dy
N N N N N N N Nu
u
N N N N u N
M dy P dy
N N N N u N


                                      
       
      
      
 
 
Where denotes the differentiation with respect to ' 'y
and ' ' denotes the differentiation with respect to ' 't . Here
' '1 1
,j kN N
h h

  , k jh y y 
( ) ( ) ( )
( )
1 1 2 1 2 1 11
1 1 1 2 1 2 16 6 2
e e e
jj j
e
k k
k
u uul Ml Pl
l u u
u


                                    

2
( )
1 1 2 1 2 1 11 1
1 1 1 2 1 2 16 6 2
jj j
e
k k
k
u uu M P
u ul u


                                    
We write the element equation for the elements
1i iy y y   and 1i iy y y   assemble three elements
equations, we obtain
2
1
1
( )
1
1
1
1
1 1 0 2 1 0
1 1
1 2 1 1 4 1
6
0 1 1 0 1 2
2 1 0 1
1 4 1 2
6 2
0 1 2 1
i
i
iie
i
i
i
i
i
u
u
u u
l
u
u
u
M P
u
u









 
      
             
           
 
    
        
        
- (11)
Now put row corresponding to the node to zero, from the
above equation the difference scheme is
 
 
2 1 11 1( )
1 1
1 1
2 4
6
4
6
i i ii i ie
i i i
u u u u u u
l
M
u u u P
  
  
 
 
        
  
Put
2
( ) 2e
l h , 2
k
r
h
 .
 
 
2 1 11 1
1 1
1 1
2 4
6
4
6
i i ii i i
i i i
u u u u u u
h
M
u u u P
  
  
 
 
        
  
Applying the trapezoidal rule, the following equations in
Crank-Nicholson method are obtained:
1 1 1 *
1 1 2 3 1 4 1 5 6 1
n n n n n n
i i i i i iAu A u A u A u A u A u P  
         - (12)
Where
1 4
2 5
3 6
*
2 6 2 6
8 12 4 , 8 12 4
2 6 , 2 6
12 12 ( cos cosn n
r i m i
A Mk r A Mk r
A r Mk A r Mk
A Mk r A Mk r
P Pk k G G C 
     
     
     
  
Similarly applying the applying Galerikin finite element
method for equations (7) and (8) the following equations
are obtained:
1 1 1 **
1 1 2 3 1 4 1 5 6 1
n n n n n n
i i i i i iB B B B B B P       
         - (13)
1 1 1
1 1 2 3 1 4 1 5 6 1
n n n n n n
i i i i i iC C C C C C C C C C C C  
        - (14)
Where
1 4 1
2 5 2
3 6 3
5
4
6
22
**
1 2
2Pr 6 2Pr 6 2 6
8Pr 12 8Pr 12 8 4 12
2Pr 6 2Pr 6 2 6
8 4 12
2 6
2 6
12 12 Pr Pri
i
B r B r C Sc kScKr r
C Sc kScKr r
C Sc kScKr r
C Sc kScKr r
C u
P Pk k Df Ec
yy
      
      
      
  
  
  
   
         
Here ,h k are the mesh sizes along y  direction and t 
direction respectively. Index refers to the space and
refers to the time. In equation (12), (13) and (14), taking
( ) and using (9), the following systems of
equations are obtained:
( ) - (15)
Where ’s are the matrices of order and ’s column
matrices having componenets. The solutions of the
above system of equations are obtained by using Thomas
algorithm for the velocity( ), temperature( ),
concentration( ). Also numerical solutions are obtained
by C-program. Computations are carried out until the
steady state is reached. In order to prove the convergence
of the Galerkin finite element method, the computations
are carried out for slight changed values of ,h k by
running same C program, no significant changes was
observed in the values of velocity( ), temperature( ),
concentration( ). Hence, the finite element method is
stable and convergent.
4. RESULTS AND DISCUSSION
In order to assess the effects of the dimensionless thermo
physical parameters on the regme calculations have been
carried out on velocity, temperature, concentration fields
for various physical parameters like, mass Grashof
number (Gm), thermal Grashof number (Gr), magnetc field
parametrt (M), Dufour number (Df), Prandtl number
problem (Pr), Schmidt number (Sc) and time (t).The
results are represented through graphs in figures 1 to 16
for various parameters
Figure 1 show that the velocity profile for different angle
of inclination (α) of the oscillating plate. The numerical
results shows that the effect of increasing values of angle
of inclination result in decreasing velocity.

Fig .1.Velocity profile for different values of α
Fig.2.Velocity profile for different values of Gm
Fig. 3.Velocity profile for different values of Gr
Figure 2 describes that an increase in mass Grashof
number cause an increase in velocity of the fluid under
consideration. It is observed that from figure 3, the
velocity is increased when the thermal Grashof number is
increased.
Fig.4. Velocity profile for different values of M
Fig. 5.Velocity profile for different values of Df
Fig.6. Velocity profile for different values of ωt
Figure 4 details the effect of increasing the values of
magnetic parameter M resulting a decrease in velocity.
From Figure 5 show that the decrease in velocity is caused
by increase in Dufour number
0
4
8
12
16
20
0 0.4 0.8 1.2 1.6 2
Pr = 0.71, Sc = 2.01, Df = 1.5, Gm = 1000, Gr = 10,
t = 0.2, M = 2, ωt = 300
α = 150, 300, 600
u
y
0
4
8
12
16
0 0.4 0.8 1.2 1.6 2
Pr = 0.71, Sc = 2.01, Df = 1.5, α = 300,
Gr = 10, t = 0.2, M = 2, ωt = 300
Gm = 500, 800, 1000
u
y
0
5
10
15
20
25
30
35
0 0.4 0.8 1.2 1.6 2
Pr = 0.71, Sc = 2.01, Df = 1.5, α = 300,
Gm = 1000, t = 0.2, M = 2, t = 300
Gr = 10,100,500
0
4
8
12
16
20
0 0.4 0.8 1.2 1.6 2
Pr = 0.71, Sc = 2.01, Df = 1.5, α = 300,
Gm = 1000, t = 0.2, Gr = 10, t = 300
M = 1, 3, 5
u
y
0
4
8
12
16
0 0.4 0.8 1.2 1.6 2
Pr = 0.71, Sc = 2.01, M = 2, α = 300,
Gm = 1000, t = 0.2, Gr = 10, ωt = 300
Df = 0.5, 8, 10u
y
0
4
8
12
16
0 0.4 0.8 1.2 1.6 2
Pr = 0.71, Sc = 2.01, M = 2, α = 30,
Gm = 1000, t = 0.2, Gr = 10, Df = 1.5
ωt = 300, 900, 1800
y
u

Fig.7.Velocity profile for different values of Pr
Fig.8.Velocity profile for different values Sc
Fig.9.Velocity profile for different values of t
From figure 9, when time is increased then the velocity
increased. Figures 10 to 13 describes that the temperature
profile for parameters Dufour number, Prandtl number,
Schmidt number, time respectively.
Fig.10.Temperature profile for different values of Df
Fig.11.Temperature profile for different values of Pr
Fig.12.Temperature profile for different of Sc
An increase in the above mentioned parameters lead to
increased temperature. Figure 14-16 displays the effects
of the Dufour number, Prandtl number and time on
concentration profiles. We observe that concentration
profiles increases with increasing Df, Pr, t.
0
4
8
12
16
0 0.4 0.8 1.2 1.6 2
Pr = 0.71, 4.5, 7
Sc = 2.01, M = 2, α =3 00, Gm = 1000,
u
y
0
2
4
6
8
10
12
14
16
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Pr = 0.71, M = 2, α = 300 , Gm = 1000,
t = 0.2, Gr = 10, ωt = 300, Df = 1.5
Sc = 2.01, 2.10, 2.20
0
4
8
12
16
0 0.4 0.8 1.2 1.6 2
t = 0.15, 0.18, 0.2
Pr = 0.71, Sc = 2.01, M = 2, α = 300,
u
y
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Df = 0.5, 1, 1.5
Pr = 7, Sc = 2.01, M = 2, α = 300, Gm = 1000, Gr = 10,
ωt = 300, t = 0.2
y

0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2

y
Df = 0.2, Sc = 2.01, M = 2, α = 30, Gm = 1000, Gr = 10,
t = 300, t = 0.2
Pr = 0.71, 4.5, 7
0
0.2
0.4
0.6
0.8
0 0.4 0.8 1.2 1.8
Pr = 7, Df = 0.2, M = 2, α = 30, Gm = 1000,
Gr = 10, Wt = 30, t = 0.2
Sc = 2.01, 3, 4

y

Fig.13.Temperature profile for different values of t
Fig. 14. Concentration profile for different values of Df
Fig.15. Concentration profile for different values of Pr
Observation of Figure 6 depicts that an increase in phase
angle, decreases velocity. Figure 7 demonstrates that the
increase in velocity with an increase in Prandtl number.
Figure 8 depicts an increase in velocity due to an increase
in Schmidt number.
Fig.16. Concentration profile for different values of t
5. CONCLUSIONS
In this article a mathematical model has been presented
for the effects of the viscous dissipation and diffusion
thermo on unsteady magnetohydrodynamics flow past an
impulsively started inclined oscillating plate with mass
diffusion and variable temperature. Solutions for the
model have been derived by finite element method. The
conclusions of the study are as follows:
1. Velocity increases with the increase in mass
Grashof number, Prandtl number, Schmidt
number and time.
2. Velocity decrease with increase in the angle of
inclination of plate, thermal Grasof number , the
magnetic field, Dufour number and phase angle.
3. Temperature profiles increases with the
increase in Dufour number , Prandtl number,
Schmidt number and time.
REFERENCES
[1] Soundalgekar VM.,Gupta Sk.,Birajdar NS “Effect of
mass transfer and free effects on MHD Stokes
problem for a vertical plate ,Nucl Eng Res, 53,pp.309-
46,1979.
[2] ElbasheshyEMS “Heat and mass transfer problems
along a vertical plate and concentration in the
presence of magnetic field”, International journal of
Engineering Science, 34(5), pp. 15-22, 1997.
[3] Ibrahim a.abdallah. “Analytical solution of heat and
mass transfer over a permeable stretching plate
affected by chemical reaction, internal heating, Dufor-
soret effect and Hall effect”, Thermal Science: 13(2),
pp. 183-197, 2009.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Pr = 0.71, Df = 0.2, M = 2, α = 300,
Gm = 1000, Gr = 10, ωt = 300, Sc = 2.01
t = 0.1, 0.15, 0.2

y
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.6 1 1.4 1.8
Pr = 0.71, Sc = 2.01, M = 2 , α = 300,
Gm = 1000, Gr = 10, ωt = 30,0 t = 0.2
Df = 0.5, 1, 1.5C
y
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Pr = 0.71, 4.5, 7
Df = 0.2, Sc = 2.01, M = 2, α= 300,
Gm = 1000, Gr = 10, ωt = 30, t = 0.2
C
y
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
C
y
Pr = 0.71, Df = 0.2 , M = 2, α = 300, Gm = 1000, Gr = 10,
ωt = 300, Sc = 2.01
t = 0.1, 0.15, 0.2

[4] U.S. Rajput and Surender kumar “MHD flow Past an
Impulsively started vertical plate with variable
Temperature and Mas Duffusion”, Applied
Mathematical Science, 5(3),pp.149-157,2011.
[5] Rao and Shivaiah “Studied chemical reaction effects
on unsteady MHD flow past semi- infinite vertical
porous plate viscous dissipation”, Appl.Math.Mech-
Engl.Ed., 34(8), pp. 1065-1078, 2011.
[6] P.K Sing “ Heat and mass transfer in MHD boundry
layer flow past an inclined plate with variable
temperature and mass diffusion”, International Journal
of Scientific &Engineering Research ,3(6), pp.1-11,
2012.
[7] T.Arun kumar and L Anand Babu “Study of Radiation
effect of MHD flow past an impulsive started vertical
plate with variable temperature and uniform mass
Diffusion – A finite element method”, Ind.J.Sci.and tech.
1(3), pp.3-9, 2013.
[8] P.Srikanth Rao and D.Mahendar “Soret effect on
unsteady MHD free convection flow past a semi-
infinite vertical permeable moving plate”,
International of Mathematical Archive, 5(8), pp.235-
245, 2014.
[9] D. Chenna Kesavaiah and P V Satyanarayan “Radiation
absorption and dufour effects to mhd flow in vertical
surface”,Global journal of engineering ,design &
technology, 3(2) , pp.51-57, 2014.
[10] K.Sharmilaa and S.Kaleeswari “Dufour effects on
unsteady MHD free convection and mass transfer flow
fast through a porous medium in slip regime with heat
source/ sink”, International Journal of scientific
Engineering and Applied Science(IJSEAS),1(6), pp.307-
320,2015
[11] K.Anitha “Chemical reaction and radiation effects on
unsteady MHD natural convection flow of rotating
fluid past a vertical porous flat plate in the presence of
a viscous dissipation”, International Journal of Science
and research (IJSR), 4(3),2015.
[12] P.Srikanth Rao and D.Mahendar “The effect of Hall
current on an unsteady MHD free convective flow
along a vertical plate with the thermal radiation”, IOSR
Journal of Mathematics, 11(6), pp.122-141, 2015.

Finite Element Solution On Effects Of Viscous Dissipation & Diffusion Thermo On Unsteady Mhd Flow Past An Impulsively Started Inclined Oscillating Plate With Mass Diffusion &Variable Temperature

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Finite Element Solution On Effects Of Viscous Dissipation & Diffusion Thermo On Unsteady Mhd Flow Past An Impulsively Started Inclined Oscillating Plate With Mass Diffusion &Variable Temperature