Analysis of three dimensional couette flow and heat

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
__________________________________________________________________________________________
Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 85
ANALYSIS OF THREE DIMENSIONAL COUETTE FLOW AND HEAT
TRANSFER IN POROUS MEDIUM BETWEEN TWO PERMEABLE
PLATES WITH SINUSOIDAL TEMPERATURE AND MAGNETIC FIELD
V.P.Rathod1
, Ravi.M2
1
CNCS, Department of Mathematics, Haramaya University
2
Asst.Professor, Department of Mathematics, Govt First Grade College, Raichur, Karnataka State, India
Abstract
This paper constitutes the analysis of three dimensional couette flow and heat transfer through porous medium under magnetic field.
There are two permeable plates in the porous medium, viz., stationery plate and moving plate. The stationery plate is maintained at
sinusoidal surface temperature while the moving plate is kept at uniform injection velocity under isothermal condition. The
expressions for velocity and temperature fields are obtained. With this, skin-friction components and Nusselt number at both the plates
will be derived and are plotted against Reynolds number.
Keywords: Three dimensional, Heat transfer, porous medium, Permeable plates, Sinusoidal, Magnetic field
--------------------------------------------------------------------***----------------------------------------------------------------------
1. INTRODUCTION
The unsteady laminar flow of an incompressible viscous
electrically conducting second order fluid between infinite
parallel plates subject to a transverse magnetic field is
investigated by Lalitha Jayaraman and Ramanaiah [9]. Raptis
et al, [5,4,2,13] have discussed the free convection flow
through a porous medium bounded by an infinite plate, when
there is no stream velocity. Raptis and Perdikis [12] have
studied the effect of free convection and mass transfer flow
through a porous medium bounded by infinite vertical porous
plates, when there is a free stream velocity. Analysis of three
dimensional couette flow and heat transfer in porous medium
between two permeable plates with sinusoidal temperature is
studied by Tak and Vyas [7]. Raptis and Perdikis [11] have
studied the combined free and forced convective flow through
a porous medium bounded by a semi-infinite vertical porous
plate. Gersten and Gross [6] have studied the effect of
transverse sinusoidal suction velocity on flow and heat
transfer along an infinite vertical porous wall. Raptis and
Takhar [1] have studied the forced flow of a viscous
incompressible fluid through the forced flow of a viscous
incompressible fluid through a highly porous medium
bounded by a semi-infinite vertical plate in the presence of
mass transfer. Oscillatory flow through a porous medium by
the presence of free convective flow is studied by Raptis,
Perdikis, and C.P. Magneto hydro dynamic free convective
effect for an incompressible viscous fluid past an infinite
limiting surface is studied by Raptis and Tzivandis[3]. The
MHD flow and heat transform in a channel with porous walls
of different permeability has been investigated by the method
of quasi linearization by Rama Bhargava and Meena Rani
[14]. Soundalgekar and Bhat[8] have studied an approximate
analysis of an oscillatory MHD channel flow and heat transfer
under transverse magnetic field.
In the present problem, by taking viscous dissipation in to
account, the three dimensional couette flow and heat transfer
in a porous medium under magnetic field are analysed .The
stationery plate is subjected to both sinusoidal temperature and
suction velocity while the moving plate is isothermal with
uniform injection velocity.
2. MATHEMATICAL ANALYSIS
Consider the three dimensional couette flow through porous
medium between two infinite porous plates under magnetic
field. So that, the stationery plate is kept along x-z plane and
the other plate is at a distance c from the stationery plate
which is moving with velocity U along x-direction. The
stationery plate is kept at sinusoidal temperature and suction
velocity, varying in z-direction while the moving plate is
subjected to the uniform injection velocity under isothermal
condition. Taking viscous dissipation in to account, the flow
and heat transfer are governed by the following equations:
0
v w
y z
 
 
 
 
 
(1)

__________________________________________________________________________________________
2 2
2 2
2
0
v +w = ( )
u u u u
y z y z
Bu
u
K



   
 
   



   

   
 
(2)
2 2
2 2
2
0
1
w = - ( )
v v p v v
v
y z y y z
Bv
v
K




    
 
    



    
  
    
 
(3)
2 2
2 2
2
0
1
v +w ( )
w w p w w
y z z y z
Bw
w
K




    
 
    



    
   
    
 
(4)
2 2
2 2
[v +w ]= [ ]p
T T T T
C K
y z y z


   
 
   

   
 
   

(5)
Where
2 2 2
2 2
2 [( ) ( ) ] [( )
( ) ( ) ]
v w u
y z y
w w u
y z z
  
  

  
  
  
  
  
  
  
  
  
(6)
Where u*
,v*
,w*
are respectively the velocity components in the
directions of x, y and z axes. ,
K
,T
, p
,  , pC , K,  , and 0B are respectively the
kinematic viscosity, permeability of porous medium,
temperature, pressure, density, Specific heat of the fluid at
constant pressure, thermal conductivity, viscosity of the fluid,
electrical conductivity, uniform magnetic field of the fluid
concerned.
The boundary conditions:
0u
 , v =-V(1+ cos )
z
c




, 0w
 ,
1T =T (1+ cos )
z
c
 

, 0P
 at 0y

u U
 , v =-V
, 0w
 , 2T =T
, p =
V c
K


, T2>T1 at
y c
 (7)
Where 1 ,U, V are constants with dimension of velocity
and c,T1 are constants with dimensions of length and
temperature respectively.
Introducing the following non-dimensional quantities:
y=
y
c

, z=
z
c

, u=
u
U

, v=
v
U

, w=
w
U

, 2
p=
p
U

,
1
2 1
T T
T T





,
pC
P
K



, 2
K
K
c

 ,
V
U
  ,
2
2 1
E=
( )p
U
C T T
,
d
RK

 , 1
2 1
a=
T
T T
and
2
0
M=
B c
U


(8)
Where R, P, K, and E are respectively, Reynolds number,
Prandtl number, permeability parameter, suction parameter
and Eckert number
With this, the equations (1) to (5) reduce to
0
v w
y z
 
 
 
(9)
2 2
2 2
1
( ) -Mu
u u u u u
v w
y z R y z RK
   
   
   
(10)
Where
2 2
2 2
1
( )
Mv
v v P v v
v w
y z y R y z
v
RK
    
    
    
 
(11)
Uc
R



__________________________________________________________________________________________
2 2
2 2
1
( )
-Mw
w w P w w
v w
y z z R y z
w
RK
    
    
    

(12)
(13)
Where
2 2 2 2 2
2{( ) +( ) }+( ) +( ) +( )
v w u w w u
y z y y z z

     
 
     
with boundary conditions:
y=0: u=0, v=- (1+ cos )z   , w=0, p=0 and
za  cos (14)
y=1: u=1, v=- , w=0, p=d, 1
In order to solve above equations (10) to (13), it is assumed
that
0 1u(y,z)=u (y)+ ( , )u y z (15)
0 1v(y,z)=v (y)+ ( , )v y z
w(y, z)= w1(y, z)
0 1p(y,z)=p (y)+ (y,z)p
0 1( , ) ( ) ( , )y z y y z   
Since, amplitude )1( of sinusoidal suction velocity is
small compared to its wavelength.
Employing equations (15) in equations (9) to (14) and taking
0  , the unperturbed quantities satisfies the following
equations:
0v 0  (16)
0)( 0
0
00  Mu
K
u
uRu  (17)
MdP 0 (18)
0)(
2
000  uEPPR  (19)
y=0: u 0 =0, 0v =- , 0 =0 , p0=0,
y=1: u 0 =1, 0v =- , 0 1  , p0=d (20)
Where prime denotes the derivatives with respect to y. The
solutions of equations (16) to (19) satisfying the boundary
conditions (20) can be expressed as follows:
0v = - (21)
1 2
1 2
0u =
m y m y
m m
e e
e e


(22)
0P =(d+M )y (23)
And
1
( )
1 2
2 1 2
2
1
0 1 2 2
1
2 ( )
2 1 2
2
2 1 2 1 2
e [
( ) 2(2 )
2
]
2(2 ) ( ) ( )
PR y
m y
m m
m y m m y
meEP
c c
e e m PR
m e m m e
m PR m m PR m m



 


   
 
 
   
(24)
Where
2 2
1
1 1
m = [-R + 4( )
2
R M
K
   
2 2
2
1 1
= [-R - 4( )
2
m R M
K
   
 1
1 2
2 2
2
1
1 1 2
1
2
2 1 2
2
2 1 2 1 2
1
[ {
( ) 2(2 )
( ) 2 ( )
}-1]
2(2 ) ( ) ( )
mPR
m mPR
m m mPR PR
m e eEP
c
e e e m PR
m e e m m e e
m PR m m PR m m


 

 

 
 

 
 
 
 
   
2 2
2 2
1
( )
E
v w
y z PR y z R
   

   
   
   

__________________________________________________________________________________________
1
1 2
2 1 2
2
1
2 2
1
2
2 1 2
2
2 1 2 1 2
( 1)1
[ {
1 ( ) 2(2 )
( 1) 2 ( 1)
}+1]}
2(2 ) ( ) ( )
m
m mPR
m m m
m eEP
c
e e e m PR
m e m m e
m PR m m PR m m


 



 
  
 
 
   
Also, the perturbed quantities satisfies the following.
1 1
0
v w
y z
 
 
 
(25)
2 2
0 1 1 1 1
1 12 2
1
[ ]
u u u u u
v Mu
y y R y z RK

   
    
   
(26)
2 2
1 1 1 1 1
12 2
1
[ ]
v p v v v
Mv
y y R y z RK

   
      
   
(27)
2 2
1 1 1 1 1
12 2
1
[ ]
w p w w w
Mw
y z R y z RK

   
      
   
(28)
And
2 2
0 01 1 1 1
1 2 2
1 2
[ ] . .
u uE
v
y y PR y z R y y
  

    
    
     
(29)
y=0;
1 0u  , 1 cosv z   , 1 0w  , 1 cosa z  , 1 0p 
y=1 ; 1 0u  1 0v  1 0w  , 1 0  , 1 0p  (30)
Equations (25) to (29) are linear partial differential equations,
which describes perturbed three dimentional flow due to
variation of suction velocity stationery surface temperature
along z-direction. The form of suction velocity and surface
temperature suggests the following forms of 1u , 1v , 1w , 1P and
1 :
1 2( )cosu u y z (31)
(32)
(33)
1 2( )cosp p y z (34)
1 2( )cosy z   (35)
The expression for 1v and 1w have been chosen so that the
equation of continuity (25) is satisfied. The equations (27) and
(28) , being independent of the main flow and the temperature
field, can be solved first. Therefore, substituting (32), (33)
and (34) in equations (27) and (28), the following ordinary
simultaneous differential equations are obtained:
2 2
2 2 2 2( ) ( 1 ) ( )kv RK v k RKM v RK P         (36)
2
2 2 2 2( ) ( 1 )kv RK v k RKM v RKP        (37)
With corresponding boundary conditions:
2 2
2 2
0: , 0,
1: 0, 0
y v v
y v v
    
  
(38)
Where a prime denotes derivative with respect to y.
The solution of system of equations (36) and (37) is
substituted in equations (32) to (34) to obtain the values of 1v
, 1w and 1p :
1 2
1 1 2 3 4[ ]cosm y m y y y
v c e c e c e c e z 

  
    (39)
1 2
1 1 1 2 2 3 4
1
[ ]sinm y m y y y
w c m e c m e c e c e z 
  

   
     (40)
1 2
1 1 22
3 4
1
[
]cos
m y m y
y y
P A e A e
RK
A e A e z 


 

  
  
(41)
Where
2 2 2
1
1
4( )
2
R R RM
Km
      
 
And
1 2( ) sv v y co z
1 2
1
( )sinw v y z

 

__________________________________________________________________________________________
2 2 2
2
1
4( )
2
R R RM
Km
      
 
31 2 4
1 2 3 4, , ,
dd d d
c c c c
d d d d
   
1 2
1 2
1 2
1 2
1 1 1 1
d=
m m
m m
e e e e
m m
m e m e e e
 
 
 
 
  
  
  
  
;
2
2
1
2
2
1 1 1
0
d =
0
0
m
m
e e e
m
m e e e
 
 

 
 
 
 

 
 
;
1
1
2
1
1
1 1 1
0
d =
0
0
m
m
e e e
m
m e e e
 
 

 
 
 
 

 
 
;
1 2
1 2
3
1 2
1 2
1 1 1
0
d =
0
0
m m
m m
e e e
m m
m e m e e





  
  

  
  
;
1 2
1 2
4
1 2
1 2
1 1 1
0
d =
0
0
m m
m m
e e e
m m
m e m e e





 
 

 
 
3 2 2
1 1 1 1 1 1 1= ( 1 )A Kc m RK c m k RKM c m        ,
3 2 2
2 2 2 2 2 2 2( 1 )A Kc m RK c m k RKM c m        
,
3 2 2
3 3 3 3( 1 )A Kc RK c k RKM c          ,
3 2 2
4 4 4 4( 1 )A Kc RK c k RKM c          
Now, for the main flow and the temperature field, Substitute
the expressions (31) and (35) in equations (26) & (29), we get,
2
2 2 2 2 0( ) ( 1 )Ku RK u k RKM u RKv u        (42)







20022
2
22 2)( uPEuPRvPR  (43)
With boundary conditions.
2 2
2 2
0: 0,
1: 0, 0
y u a
y u


  
  
(44)
Solving (42) & (43) with boundary conditions (44) &
substituting the solutions in equations (31) & (35), the
expressions for 1 1u and can be given as:
1 2 1 1
1 2
1 2 1 1
2 1 2 2 2
2
( )
1 5 6 1
( ) ( ) ( )
2 3 4
( ) ( ) ( )
5 6 7
( )
8
[ {
}]cos
m y m y m m y
m m
m m y m y m y
m m y m m y m y
m y
R
u D e D e B e
e e
B e B e B e
B e B e B e
B e z
 



  
  
   

  

  
  

(45)
2
1
2 1
1
2
1 2
1
( )
[( )
( )
( ) +f(y)]cos
a
a y
a a
a
a y
a a
G e F a
e
e e
G e F a
e z
e e


 


 


(46)
Where
)1()()( 2
11
2
11
11
1
RKMKmmRKmmK
mKc
B







)1()()( 2
21
2
21
12
2
RKMKmmRKmmK
mKc
B








__________________________________________________________________________________________
)1()()( 2
1
2
1
13
3
RKMKmRKmK
mKc
B



)1()()( 2
1
2
1
14
4
RKMKmRKmK
mKc
B



1 2
5
2 2
2 1 2 1( ) ( ) ( 1 )
Kc m
B
K m m RK m m K RKM 

      
)1()()( 2
22
2
22
22
6
RKMKmmRKmmK
mKc
B







)1()()( 2
2
2
2
23
7
RKMKmRKmK
mKc
B



)1()()( 2
2
2
2
24
8
RKMKmRKmK
mKc
B



))((
)(
2112
2
5 mmmm
m
eeee
AeBR
D


 

;
))((
)(
2121
1
6 mmmm
m
eeee
AeBR
D


 

87654321 BBBBBBBBA 
1 1 1 2 1 1
2 1 2 2 2 2
( ) ( ) ( ) ( )
1 2 3 4
( ) ( ) ( ) ( )
5 6 7 8
m m m m m m
m m m m m m
B B e B e B e B e
B e B e B e B e
 
 
    
    
   
   
2 2
2
1
1
( ) ( )
2 4
R R
m RM
K
 

      ,
)
1
(
4
)
2
( 2
22
2 RM
K
RR
m 



1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
F A A A A A A A A A A
A A A A A A A A A A
          
         
1 2
1 1 1 2
2 1 2 2 1
2 1 2
1 2 1 1 2 2
( ) ( ) ( )
1 2 3
( 2 ) ( 2 )( )
4 5 6
( 2 ) ( 2 ) ( 2 )
7 8 9
( 2 ) ( 2 ) ( 2 )
10 11 12
( ) ( )
13 14
( ) m PR y m PR y PR y
m m y m m yPR y
m m y m m y m y
m y m y m y
m m m y m m m
f y Ae A e A e
A e A e A e
A e A e A e
A e A e A e
A e A e
   
 

  
   
   
   
    
    
   
  
  
  
  1 2
1 2 1 1 1 2
2 1 2 2
( )
15
( ) ( ) ( )
16 17 18
( ) ( )
19 20
y m m y
m m y m m y m m y
m m y m m y
A e
A e A e A e
A e A e


 
     
   

  
 
1 2
1 1 1 2 2 1 2 2
1 2 1 2
1 2 1 1 2 2 1 2
( ) ( ) ( ) ( )
1 2 3 4
( 2 ) ( 2 ) ( 2 ) ( 2 )
5 6 7 8
( 2 ) ( 2 ) ( 2 ) ( 2 )
9 10 11 12
( ) ( ) ( )
13 14 15
G= m PR m PR PR PR
m m m m m m m m
m m m m
m m m m m m m m
Ae A e A e A e
A e A e A e A e
A e A e A e A e
A e A e A e
A
     
   

     
      
     
      
   
   
   
  
 1 2 1 1 1 2
2 1 2 2
( ) ( ) ( )
16 17 18
( ) ( )
19 20
m m m m m m
m m m m
e A e A e
A e A e
     
   
 
 
2
4)( 2222
1
 

RPPR
a ;
2
4)( 2222
2
 

RPPR
a
2
1
2
1
21
22
1
)()( 







PRmPRPRm
ccRP
A ;
2
2
2
2
22
22
2
)()( 







PRmPRPRm
ccRP
A ;
22
23
22
3
)()( 




PRPRPR
ccRP
A ;
2 2
4 2
4 2 2
( ) ( )
P R c c
A
PR PR PR

     


   
;

__________________________________________________________________________________________
1 2
1 1 1
5 2
1
1 1 1 2 2
1 1 1 1
(
( ) 2
1
2 ( ))( )
( 2 ) ( 2 )
m m
EPRm Pc m
A
e e m PR
B m m
m m PR m m

 

 
 
    
;
1 2
2 1 2
6 2
2
5 2 1 2 2
1 2 1 2
(
( ) 2
1
2 ( ))( )
( 2 ) ( 2 )
m m
EPRm Pc m
A
e e m PR
B m m
m m PR m m

 

 
 
    
;
1 2
1 2 1
7 2
1
2 1 2 2 2
2 1 2 1
(
( ) 2
1
2 ( ))(
( 2 ) ( 2 )
m m
EPRm Pc m
A
e e m PR
B m m
m m PR m m

 

 
 
    
1 2
2 2 2
8 2
2
6 2 2 2 2
2 2 2 2
(
( ) 2
1
2 ( ))( )
( 2 ) ( 2 )
m m
EPRm Pc m
A
e e m PR
B m m
m m PR m m

 

 
 
    
1 2
3 11
9 2
1
3 1 2 2
1 1
(
( ) 2
1
2 ( ))( )
( 2 ) ( 2 )
m m
Pc mEPRm
A
e e m PR
B m
m PR m


   

 
 
   
1 2
3 22
10 2
2
7 2 2 2
2 2
(
( ) 2
1
2 ( ))( )
( 2 ) ( 2 )
m m
Pc mEPRm
A
e e m PR
B m
m PR m


   

 
 
   
1 2
1 4 1
11 2
1
4 1 2 2
1 1
(
( ) 2
1
2 ( ))( )
( 2 ) ( 2 )
m m
EPRm Pc m
A
e e m PR
B m
m PR m


   

 
 
     
1 2
2 4 2
12 2
2
8 2 2 2
2 2
(
( ) 2
1
2 ( ))( )
( 2 ) ( 2 )
m m
EPRm Pc m
A
e e m PR
B m
m PR m


   

 
 
     
1 2
1 1 2
13 1 5 2 12
1 2
2 1 1 1
2 2
1 2 1 1 2 1
2
( ( )
( ) ( )
( ))
1
( )
( ) ( )
m m
Pc m mEPR
A m B m m
e e m m PR
m B m m
m m m PR m m m

 
  
  
 
      
1 2
2 1 2
14 1 6 2 22
1 2
2 2 1 2
2 2
1 2 2 1 2 2
2
( ( )
( ) ( )
( ))
1
( )
( ) ( )
m m
Pc m mEPR
A m B m m
e e m m PR
m B m m
m m m PR m m m

 
  
  
 
      
1 2
3 1 2
15 1 7 22
1 2
2 3 1
2 2
1 2 1 2
2
( ( )
( ) ( )
( ))
1
( )
( ) ( )
m m
Pc m mEPR
A m B m
e e m m PR
m B m
m m PR m m



   
  
  
 
     
1 2
4 1 2
16 1 8 22
1 2
2 4 1
2 2
1 2 1 2
2
( ( )
( ) ( )
( ))
1
( )
( ) ( )
m m
Pc m mEPR
A m B m
e e m m PR
m B m
m m PR m m



   
  
  
 
     
1 2
1 1 5
17 2 2
1 1 1 1
2 1
( )
( ) ( )m m
EPm m D
A
e e m m PR m m 

 

    
1 2
1 2 6
18 2 2
1 2 1 2
2 1
( )
( ) ( )m m
EPm m D
A

 

    
1 2
2 1 5
19 2 2
2 1 2 1
2 1
( )
( ) ( )m m
EPm m D
A

 

    
1 2
2 2 6
20 2 2
2 2 2 2
2 1
( )
( ) ( )m m
EPm m D
A

 

    

__________________________________________________________________________________________
3. RESULTS AND DISCUSSION
Knowing velocity and temperature fields, the important
characteristic parameters namely, skin-friction components
x and z along main flow and transverse direction, non-
dimensional rate of heat transfer (Nusselt number) at both the
plates can be calculated.
1 2
1 2
1 2
1 5 2 6
1 1 1 1 2 2
1 3 1 4 2 1 5
2 2 6 2 7 2 8
[
{( ) ( )
( ) ( ) ( )
( ) ( ) ( ) }]cos
x m m
m m
m m
m D m D
e e
R
m m B m m B
e e
m B m B m m B
m m B m B m B z
 
 
  
 
 



  

   

     
     
(47)
2 21
1 1 2 2
2 2
3 4
( ) [
]sin
z
z
d dw
c m c m
v dy
c c z
 
 
 
  

     
 
(48)
3.1 Nusselts Number at the Stationary Plate:
*
*
* 0
2 1
( )y
c T
Nu
T T y

 



1 2
2 1
2 1 1 2
2 2
1 2
2 2
1 2
1 2
1 2
1 2
Nu=-[ ( ) (
( ) 2 2
2
( )
( ) ( )
{( ) ( ) (0)}cos ]
m m
a a
a a a a
m mEP
PR c
e e m PR m PR
m m
m m PR
G e F a G e F a
a a f z
e e e e

 



  
  
 
 
   
 
 
(49)
3.2 Nusselt Number at the Moving Plate
*
*
2 1
( )
* y c
c T
Nu
T T y 

 
 
[
1
1 2
2 1 2
2 1
1 2
2 1 1 2
22
( ) 1
2 2
1
2 ( )2
2 1 2
2 1 2
1 2
Nu = - [ ( ) (
( ) 2
2
2 ( )
( ) ( )
{( ) ( )
(1)}cos ]
m
PR
m m
m m m
a a
a a
a a a a
m eEP
PR c e
e e m PR
m e mm e
m PR m m PR
G e F a G e F a
ae a e
e e e e
f z



 




 
 
 
  
   
 
 

(50)
The velocity components u, v and w in representative plane
z=0 and z=1/2,are plotted against y in figures 1,2 and 3
respectively, for various values of Reynolds number R,
Suction parameters  and permeability parameters K at
constant magnetic field M. It is observed from these figures,
that u and w both increase as R,  or K. Also , when R or K
increase v increases.
In figure4, the temperature distribution function  is plotted
against y, in representative plane z=0 at constant magnetic
field. From the fig, temperature increases as Eckert number E
or Prandtl number P increases but the same decreases as
 increases.
The absolute values of the skin friction components x in plane
z=0 and z in plane z=1/2are plotted against Reynolds number
R at constant magnetic field M in figures 5 and 6 respectively,
for various values of  and K. It is noted that both x and
z increase as Reynolds number R increases or  increases.
Further, permeability parameter K increases x increases
whereas z decreases.
Fig7.shows the variation of Nusselt number lNul, at the plate
y=1, against R at constant magnetic field M. Here, lNul
decreases significantly with R. Also when  increases lNul
increases.
The velocity components u, v and w in representative plane
z=0 and z=1/2,are plotted against y in figures 8,9 and 10
respectively, for the fixed values of Reynolds number R,
Suction parameters  and permeability parameters K at
different magnetic fields. It is observed from these figures,
that if Magnetic field parameter M increases, both u and v
decreases, but w increases. As R increases, u, v and w
increases.
In figure11, the temperature distribution function  is plotted
against y, in representative plane z=0 for different magnetic
0 2
0 0( ) ( ) cosx
x
d du du
z
u dy dy

  


  

__________________________________________________________________________________________
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0 M=0.2;
R  K
B 50 0.01 50
C 50 0.02 50
D 50 0.03 50
E 50 0.04 50
F 60 0.05 1000
G 300 0.01 50
H 400 0.01 50
I 1000 0.01 50
Fig1.velocity component u against y in plane z=0 for 
I
H
G
F
E
D
C
B
u
y
0.0 0.2 0.4 0.6 0.8 1.0
0.000
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
M=0.2;
R  K
B 50 0.01 50
C 50 0.02 50
D 50 0.03 50
E 50 0.04 50
F 300 0.01 50
G 300 0.01 50
H 400 0.01 50
I 1000 0.01 50
Fig.3.Velocity component w against y in plane z=1/2 for 
I
H
G
F
E
D
C
B
w*10
y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
M=0.2;
Fig.4.Temperature  against y in plane z=0 for
R=50,,K=10,and a=0.78
I
H
G
F
E
D C
B
P E
B 1.51 0.1
C 2.00 0.1
D 2.50 0.1
E 3.00 0.1
F 3.00 0.05
G 4.00 0.05
H 5.00 0.05
I 6.00 0.05

y
0 20 40 60 80 100
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
Fig.5.Skin-friction coeffecient x
against Reynolds-
number R in plane z=0 for =0.1
M=0.2;
G
F
E
D C
B
 K
B 0.05 1
C 0.05 5
D 0.05 100
E 0.02 100
F 0.03 100
G 0.04 100
x
R
0.0 0.2 0.4 0.6 0.8 1.0
-0.210
-0.208
-0.206
-0.204
-0.202
-0.200
Fig2.Velocity component v against y in plane z=0
for ,
D
C
B
M=0.2;
R K
B 50 10
C 100 50
D 500 100
v
y
fields. As y and M increases  also increases for fixed Eckert
number E, Prandtl number P, Reynolds number R, Suction
parameters  and permeability parameters K.
The absolute values of the skin friction components x in plane
z=0 and z in plane z=1/2 are plotted against Reynolds
number R in figures 12 and 13 respectively, for fixed values of
 and K at different magnetic fields. It is noted that x
increases and z decreases as Reynolds number R increases,
while x and z decreases, as M increases for fixed  , K.
Figure14.shows the variation of Nusselt number |Nu|, at the
plate y=1, against R. Here, |Nu| increases significantly with R.
Also, |Nu| increases as M increases.

__________________________________________________________________________________________
0 20 40 60 80 100
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
-0.035
-0.040
-0.045
F
E
G
D
C
BM=0.2; K
B 0.05 1
C 0.05 6
D 0.05 100
E 0.02 100
F 0.03 100
G 0.04 100
Fig.6.Skin-friction coeffecient z
against Reynolds-
number R in plane z=1/2 for =0.1
z
R
0 20 40 60 80 100
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
Sl.No.  K P E
I 0.02 10 0.44 0.01
II 0.04 10 0.44 0.01
III 0.01 50 0.71 0.01
IV 0.02 50 0.71 0.01
V 0.05 10 0.78 0.05
VI 0.05 100 0.78 0.05
M=0.2;
Fig.7.Nusselt number |Nu| against Reynolds-
number R inplane z=0 for =0.1
VI
V
IV
III
II
I
|Nu|
R
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
M=0
M=0.4
M=0.2
M=0.0
M=0.2
M=0.4
Fig.8.Velocity component u against y in
plane z=0 for =0.1,R=50,=0.01,k=50
u
y
0.0 0.2 0.4 0.6 0.8 1.0
-0.210
-0.208
-0.206
-0.204
-0.202
-0.200
M=0.0
M=0.2
M=0.4
Fig.9.velocity component v against y in plane z=0
for =0.2,=0.05,R=500,K=100
M=0.2
M=0.4
M=0
v
y
0.0 0.2 0.4 0.6 0.8 1.0
0.000
-0.001
-0.002
-0.003
-0.004
-0.005
M=0.0
M=0.2
M=0.4
M=0
M=0.4
M=0.2
Fig.10.Velocity component w against y in
plae z=1/2 for =0.1,R=50,=0.02,K=50
w*10
y
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Fig.11.Temperature function against y in
plane z=0 R=50,=0.05,K=10,,a=0.78, P=1.51,E=0.1
M=0.0
M=0.2
M=0.4
M=0.4
M=0.2
M=0

y

__________________________________________________________________________________________
0 20 40 60 80 100
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
M=0.0
M=0.2
M=0.4
Fig.12.Skin-friction coefficient x
against
Reynolds number R in plane z=0 for =0.1,,K=1
M=0.4
M=0.2
M=0
x
R
0 20 40 60 80 100
-0.054
-0.052
-0.050
-0.048
-0.046
-0.044
-0.042
-0.040
-0.038
-0.036
-0.034
-0.032
-0.030
-0.028
-0.026
-0.024
-0.022
-0.020
-0.018
-0.016
M=0.4
M=0.2
M=0
Fig.13.Skin-friction coefficient z
against Reynolds number
in plane z=1/2 for ,=0.05,K=100
M=0.0
M=0.2
M=0.4
z
R
0 20 40 60 80 100
-0.38
-0.37
-0.36
-0.35
-0.34
-0.33
-0.32
-0.31
M=0.0
M=0.2
M=0.4
M=0.0
M=0.4
M=0.2
Fig.14.Nusselt number |Nu|against Reynolds
number R in plane z=0 for ,,
K=10,P=0.44,E=0.01,y=1,a=0.78
|Nu|
R
REFERENCES
[1]. A. A. Raptis and H. S. Takhar, 1986, Combined mass
transfer and forced flow through a porous medium, Int.
Comm. Heat and Mass Transfer, 13, pp. 599-603.
[2]. A. A. Raptis, N. G. Kafousias and C. V. Massalas, 1982,
Free convection and mass transfer flow through a porous
medium bounded by an infinite vertical porous plate with
constant heat flux, ZAMM, 62, pp. 489-491.
[3]. A. Raptis and G. Tzivanidis, 1983, Magnetohydrodynamic
free convective effect for an incompressible viscous fluid past
an infinite limiting surface, Astro Physics and Space Science,
94(2), pp. 311-317.
[4]. A. Raptis, C. Perdikis and G. Tzivanidis, 1976, Free
convection flow through a porous medium bounded by a
vertical surface, Journal of Physics D:Applied Physics,
14(7)L99.
[5]. A. Raptis, G.Tzivanidis and N. G. Kafousias, 1981, Free
convection and mass transfer flow through a porous medium
bounded by an infinite vertical limiting surface with constant
suction, Letters in Heat and Mass Transfer, 8, pp. 417-424.
[6]. K. Gersten and J. F. Gross, 1974, Three dimensional flow
and heat transfer, ZAMP, 25, pp. 399-408.
[7]. S. S. Tak and M. K. Vyas, 2006, Analysis of three
dimensional couette flow and heat transfer in porous medium
between two permeable plates with sinusoidal temperature,
Ultra Science, 18(3)M, pp. 489-500.
[8]. V. M. Soundalgekar and J. P. Bhat, 1971, Oscillatory
MHD channel flow and heat transfer, Ind. J. Pure. Appl.
Math., 15, pp. 819-828.
[9]. Lalitha Jayaraman and G. Ramanaiah, 1984, Second-order
fluid-transient MHD couette flow with constant stress at upper
plate, Indian Journal of Pure and Applied Mathematics, 15(8),
pp. 927-934.
[10]. Raptis, A.A., Perdikis, C.P., 1985 Oscillatory flow
through a porous medium by the presence of free convective
flow, Int. J. Engng. Sci., pp. 23, 51-55.
[11]. A. A. Raptis and C. P. Pfrdikis, 1988, Combined free and
forced convection flow through a porous medium,
International Journal of Energy Research, 12(3), pp. 557-560.
[12]. A. A. Raptis and C. Perdikis, 1987, Mass transfer and
free convection flow through a porous medium, Energy
Research, 11, pp. 423-428.
[13]. A. A. Raptis, 1983, Unsteady free convective flow
through a porous medium, International Journal of
Engineering Science, 21(4), pp. 345-348.
[14]. Rama Bhargava and Meena Rani, 1984, MHD flow and
heat transfer in a channel with porous walls of different
permeability(eng), Indian Journal of Pure and Applied
Mathematics, 15 (4), pp. 397-408.

Analysis of three dimensional couette flow and heat

More Related Content

What's hot (19)

Viewers also liked (20)

Similar to Analysis of three dimensional couette flow and heat (20)

More from eSAT Publishing House (20)

Recently uploaded (20)

Analysis of three dimensional couette flow and heat