Generalize couette flow by assume solution

1. Generalized Couette Flow
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2. It is the special case of flow because it is formed
from couette and poiseuille flow.
The flow in bounded in between two parallel
plates at y=0 and y=h and it is initially at rest.
The fluid is also magnetically conducted and
pass through porous media. The flow is due to
pressure gradient as well as of motion of upper
plate. The governing equation for this flow is
given as
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3. ----- (1)
And the boundary are
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22
0
2
1 Bu u p
u u
t y x k


 
  
   
  
0
(0, ) 0
( , )
u t
u h t U



4. Where
*
0
*
0
*
0
,
,
,
u
u
u
x
x
x
y
y
y



* 0
*
0
0
,
,
R ,e
tu
t
d
pd
p
u
u d





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5. 2 2
0
2
,
1
,
1
,
a B
M
k
k a
H M
k




 
M is the MHD
v/k is the porous media
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6. Put all these dimensional less
values in equation(1) we get
After simplification we get
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2 0* 2 * *
* *0 0 0 0 0
* 2 2 * 2 *
U U U U Uu u p
u u
d t d y d x k

  

  
   
  
* 2 * *
*
* 2 * *
u u p
Hu
t y x
  
  
  

7. To make it more simplify we Drop
the sign of * we get
---------------(2)
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2
2
u u p
Hu
t y x
  
  
  

8. Assume solution for this
phenomena is
---------(3)
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( , ) Re{ ( ) }iwt
U y t F y e
iwtp
e
x


 


9. Put all these values in equation(2)
we get
After applying derivative we get
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2
2
Re [Re{ ( ) }] [Re{ ( ) }] [Re{ ( ) }]iwt iwt iwt iwt
F y e F y e e M F y e
t y

 
  
 
2
2
{ ( )} [Re ]( ( ))
d
F y iw M F y
dy
   

10. After the arranging of above
equation we have
(4)
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2
2
{ ( )} [Re ]( ( ))
d
F y iw M F y
dy
   

11. And the transform boundary
conditions are:
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0
(0) 0
(1)
f
f U



12. The auxiliary equation of (4) is
The solution is
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2
(Re ) 0m iw M  
1 2cosh Re sinh Recy c iw My c iw My   

13. For particular solution we have
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'
''
0
0
p
p
p
y A
y
y




14. Put these values in equation(4) we get
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0 [Re ]
[Re ]
iw M A
A
iw M


   



15. The general solution is
Apply the boundary conditions in above
equation we get
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1 2cosh Re sinh Re
[Re ]
y c iw My c iw My
iw M

    

1
0
2
Re
cosh Re
Re Re
sinh Re
c
iw M
U iw M
iw M iw Mc
iw M

 
 

  
 


16. Repalce and
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2c1c
0
cosh Re
Re
cosh Re
Re Re sinh Re
sinh Re
[Re ]
y iw My
iw M
U iw M
iw M iw M iw My
iw M
iw M

 

   

  
  


