Magneto convective flowand heat transfer of two immiscible fluids between vertical wavy wall and a parallel flat wall

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 777
MAGNETO-CONVECTIVE FLOWAND HEAT TRANSFER OF TWO IMMISCIBLE FLUIDS BETWEEN VERTICAL WAVY WALL AND A PARALLEL FLAT WALL Mahadev M Biradar Associate Professor, Dept. of Mathematics Basaveshwar Engineering College (Autonomous) Bagalkot, Karnataka INDIA 587 102 Abstract Convective flow and heat transfer between vertical wavy wall and a parallel flat wall consisting of two regions, one filled with electrically conducting and other with viscous fluid is analyzed. Governing equation of motion have been solved by linearization technique. Results are presented for various parameters such as Hartmann number, Grashof number, viscosity ratio, width ratio, conductivity ratio and source or sink. The effect of all the parameters except the Hartmann number and source or sink remains same for two viscous immiscible fluids. The effect of Hartmann number is to decrease the velocity at the wavy and flat wall. The suppression near the flat wall compared to wavy wall is insignificant. The velocity is large for source compared to sink for equal and different wall temperature. Keywords: Convection, vertical, wavy wall, immiscible
-----------------------------------------------------------------------***----------------------------------------------------------------------- 1. INTRODUCTION Many transport process exists in natural and industrial applications in which the transfer of heat and mass occurs simultaneously as a result of buoyancy effect of thermal diffusion. Natural convection heat transfer plays an important role in the electronic components cooling since it has desirable characteristics in thermal equipments design; absence of mechanical or electromagnetic noise; low energy consumption, very important in portable computers; and reliability, since it has no elements to fail. The optimization of the heat transfer has increasingly importance in electronic packaging due to the higher heat densities and to the electronic components and equipments miniaturization Sathe et.al. (1998). In spite of the abundant results about natural convection in electronic packaging, works dealing with heat transfer maximization is scarce in literature Landon et.al (1999) Da Silva et.al., (2004). This is, probably, due to the non-linear nature and to the difficult of natural convection simulation.
The corrugated wall channel is one of several devices employed for enhancing the heat transfer efficiency of industrial transport process. The problem of viscous flow in wavy channels was first treated analytically by Burns and Parks (1967) who expressed the stream function as a Fourier series under the assumption of stokes flow. Wang and Vanka (1995) determined the rates of heat transfer for flow through a periodic array of wavy passages. They observed that in the steady-flow regime, the average Nusselt numbers for the wavy-wall channel were only slightly larger than those for a parallel-plate channel. The problem of natural or mixed convection along a sinusoidal wavy surface extended previous work to complex geometries Yao (1983), Moulic et.al., (1989,1989) and has received considerable attention due to its relevance to real geometries. An example of such geometry is a “roughened” surface that occurs often in problem involving the enhancement of heat transfer. The flow and heat transfer of electrically conducting fluids in channels under the effect of a transverse magnetic field occurs in MHD-generators, pumps, accelerators, nuclear-reactors, filtration, geo-thermal system and others. Recently there are experimental and theoretical studies on hydromagnetic aspects of two fluid flows available in literature. Lohrasbi and Sahai (1998) dealt with two-phase magnetohydrodynamic (MHD) flow and heat transfer in a parallel plate channel. Two-phase MHD flow and heat transfer in an inclined channel is investigated by Malashetty and Umavathi (1997). Recently Malashetty et.al, (2000, 2001) analyzed the problem of fully developed two fluid magnetohydrodynamic flows with and without applied electric field in an inclined channel. Chamakha (2000) considered the steady, laminar flow of two viscous, incompressible electrically conducting and heat generating or absorbing immiscible fluids in an infinitely – long, impermeable parallel-plate channel filled with a uniform porous medium.

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In spite of the numerous applications of corrugated walls,
much work is not seen in literature. Hence it is the objective of
the present work, is to study the problem of flow and heat
transfer between vertical wavy wall and a parallel flat wall
consisting of two regions, one filled with electrically
conducting fluid and second with electrically non-conducting
fluid.
2. MATHEMATICAL FORMULATION
Fig-1: Physical Configuration
Consider the channel as shown in figure 4.1, in which the X
axis is taken vertically upwards and parallel to the flat wall
while the Y axis is taken perpendicular to it in such a way that
the wavy wall is represented by Y h * cos KX ( )    1 and
the flat wall by ( ) Y h 2  . The region
1 0  y  h is occupied
by viscous fluid of density
1  , viscosity
1  , thermal
conductivity
1 k and the region
2 h  y  0 is occupied
another viscous fluid of density
2  , viscosity
2  , thermal
conductivity
2 k . The wavy and flat walls are maintained at
constant and different temperatures Tw and T1 respectively.
We make the following assumptions:
(i) that all the fluid properties are constant except
the density in the buoyancy-force term;
(ii) that the flow is laminar, steady and two-dimensional;
(iii) that the viscous dissipation and the work done by
pressure are sufficiently small in comparison
with both the heat flow by conduction and the
wall temperature;
(iv) that the wavelength of the wavy wall, which is
proportional to 1/K, is large.
Under these assumptions, the equations of momentum,
continuity and energy which govern steady two-dimensional
flow and heat transfer of viscous incompressible fluids are
Region –I
(1) (1) 1
(1) (1) (1) (1) 2 (1) (1) U U P
U V U g
X Y X
  
    
       
    
(2.1)
(1) (1) 1
(1) (1) (1) (1) 2 (1) V V P
U V V
X Y Y
 
    
      
    
(2.2)
0
1 1






Y
V
X
U ( ) ( )
(2.3)
   
(1) (1)
(1) 1 (1) (1) (1) 2 (1) 1
p
T T
C U V k T Q
X Y

   
     
   
(2.4)
Region –II
(2) (2) 2
(2) (2) (2) (2) 2 (2)
(2) 2 (2)
0
U U P
U V U
X Y X
g B U
 
 
    
      
    
 
(2.5)
(2) (2) 2
(2) (2) (2) (2) 2 (2) V V P
U V V
X Y Y
 
    
      
    
(2.6)
0
2 2






Y
V
X
U ( ) ( )
(2.7)
   
(2) (2)
(2) 2 (2) (2) (2) 2 (2) 2
p
T T
C U V k T Q
X Y

   
     
   
(2.8)
Where the superscript indicates the quantities for regions I and
II, respectively. To solve the above system of equations,
one needs proper boundary and interface conditions. We
assume 1
p C =   2
p C
The physical hydro dynamic conditions are
(1) U  0 (1) V  0 , at (1) * Y  h  cosKX
(2) U  0 (2) V  0, at (2) Y  h
(1) (2) U U (1) (2) V V , at Y  0
  2
( 2 )
1
( 1 )
X
V
Y
U
X
V
Y
U









 








  atY  0 (2.9)

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The boundary and interface conditions on temperature are
(1)
W T  T at (1) * Y  h  cosKX
(2)
1 T  T at (2) Y  h
(1) (2) T  T at Y  0
1 2
(1) (2) T T T T
k k
Y X Y X
       
      
       
atY  0 (2.10)
The conditions on velocity represent the no-slip condition and
continuity of velocity and shear stress across the interface. The
conditions on temperature indicate that the plates are held at
constant but different temperatures and continuity of heat and
heat flux at the interface.
The basic equations (4.2.1) to (4.2.8) are made dimensionless
using the following transformations
    ( 1 )
( 1 )
( 1 ) X ,Y
h
1
x, y  ;     ( 1 )
( 1 )
( 1 )
( 1 ) U,V
h
u,v


    ( 2 )
( 2 )
( 2 ) X ,Y
h
1
x, y  ;     ( 2 )
( 2 )
( 2 )
( 2 ) U,V
h
u,v


 
  W S
s
( 1 )
( 1 )
T T
T T


  ;  
  W S
s
( 2 )
( 2 )
T T
T T


 
(1)
(1)
2
(1)
(1)
P
P
h



 
 
 
; (2)
(2)
2
(2)
(2)
P
P
h



 
 
 
(2.11)
Where Ts is the fluid temperature in static conditions.
Region –I
(1) (1) 1 2 (1) 2 (1)
(1) (1) (1)
2 2
u u P u u
u v G
x y x x y

    
     
    
(2.12)
(1) (1) 1 2 (1) 2 (1)
(1) (1)
2 2
v v P v v
u v
x y y x y
    
    
    
(2.13)
0
y
v
x
u ( 1 ) ( 1 )






(2.14)
y Pr x y Pr
v
x
u
( ) ( ) ( )
( )
( )
( )     

 


 













2
1 2 1
2
1 2
1
1
1 1
(2.15)
Region –II
(2) (2) 2 2 (2) 2 (2)
(2) (2)
2 2
2 2 3 (2) 2 2 (2)
u u P u u
u v
x y x x y
 m r h G M mh u
    
    
    
 
(2.16)
(2) (2) 2 2 (2) 2 (2)
(2) (2)
2 2
v v P v v
u v
x y y x y
    
    
    
(2.17)
0
y
v
x
u ( 2 ) ( 2 )






(2.18)
(2) (2) 2 (2) 2 (2)
(2) (2)
2 2
2
Pr
(2.19)
Pr
km
u v
x y x y
mh Q
   

     
    
       

The dimensionless form of equation (2.9) and (2.10) using
(2.11) become
(1) u  0 ;
(1) v  0 at y  1 cos x
(2) u  0 ;
(2) v  0 at y  1
(1) 1 (2)
u u
rmh
 ;
(1) 1 (2)
v v
rmh
 at y  0
  2
2 2
1
x
v
y
u
rm h
1
x
v
y
u
 


 







  


 







at y  0
(2.20)
(1)  1 at y  1 cos x
(2)   at y  1
(1) (2)   at y  0

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1 2
h y x
k
x y  


 







  


 






   
at y  0 (2.21)
3. SOLUTIONS
The governing equations (2.12) to (2.15), (2.16) to (2.19)
along with boundary and interface conditions (2.20) to (2.21)
are two dimensional, nonlinear and coupled and hence closed
form solutions can not be found. However approximate
solutions can be found using the method of regular
perturbation. We take flow field and the temperature field to
be
Region-I
u x, y u ( y ) u x, y ( 1 )
1
( 1 )
0
( 1 )  
v x, y v x, y ( 1 )
1
( 1 ) 
    (1) (1) (1)
0 1 P x, y  P (x)  P x, y
x, y ( y ) x, y ( 1 )
1
( 1 )
0
( 1 )    (3.1)
Region -II
u x, y u ( y ) u x, y ( 2 )
1
( 2 )
0
( 2 )  
v x, y v x, y ( 2 )
1
( 2 ) 
    (2) (2) (2)
0 1 P x, y  P (x)  P x, y
x, y ( y ) x, y ( 2 )
1
( 2 )
0
( 2 )    (3.2)
Where the perturbations
     
1 1 1 , ,
i i i
u v P and
 
1
i
 for
i 1,2 are small compared with the mean or the zeroth order
quantities. Using equation (4.3.1) and (4.3.2) in the equation
(2.12) to (2.15), (2.16) to (2.19) become
3.1 Zeroth Order
Region-I
( 1 )
2 0
( 1 )
0
2
G
dy
d u
   (3.3)


 
2
1
0
2
dy
d ( )
(3.4)
Region-II
(2)
0
(2) 2 2 3
0
2 2
2
(2)
0
2
mh M u  m r h G
dy
d u
   (3.5)
k
Qh
dy
d ( ) 2
2
2
0
2  
 (3.6)
3.2 First Order:
Region-I
(1) (1) 1 2 (1) 2 (1)
(1) 1 (1) 0 1 1 1 (1)
0 1 2 2 1
u du P u u
u v G
x dy x x y

   
     
   
(3.7)
(1) 1 2 (1) 2 (1)
(1) 1 1 1 1
0 2 2
v P v v
u
x y x y
   
   
   
(3.8)
0
y
v
x
u
2
( 1 )
1
( 1 ) 2
1 





(3.9)
(1) (1) 2 (1) 2 (1)
(1) 1 (1) 0 1 1
0 1 2 2 Pr
d
u v
x dy x y
       
    
    
(3.10)
Region-II
(2) (2) 2 2 (2) 2 (2)
(2) 1 (2) 0 1 1 1
0 1 2 2
2 2 3 (2) 2 2 (2)
1 1
u du P u u
u v
x dy x x y
 m r h G mh M u
   
    
   
 
(3.11)
(2) 2 2 (2) 2 (2)
(2) 1 1 1 1
0 2 2
v P v v
u
x y x y
   
   
   
(3.12)
0
y
v
x
u
2
( 2 )
1
( 2 ) 2
1 





(3.13)
(2) (2) 2 (2) 2 (2)
(2) 1 (2) 0 1 1
0 1 2 2 Pr
d km
u v
x dy x y
       
    
    
(3.14)

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Where    1
1 0 S P P
C
x
 


and    2
2 0 S P P
C
x
 


taken equal to zero (see Ostrach, 1952). With the help of (3.1)
and (3.2) the boundary and interface conditions (2.20) to
(2.21) become
  (1)
0 u 1  0 at y  1
  (2)
0 u 1  0 at y 1
( ) ( ) u
rmh
u 2
0
1
0
1
 at y  0
(1) (2)
0 0
2 2
du 1 du
dy rm h dy
 at y  0 (3.15)
  (1)
0  1 1 at y  1
  (2)
0  1  at y 1
( ) ( 2 )
0
1
0   at y  0
(1) (2)
0 0 d k d
dy h dy
 
 at y  0 (3.16)
 
 
 
1
(1) 0 (1)
1 1 1 ; 1 0
du
u Cos x v
dy
      at y  1
    (2) (2)
1 1 u 1  0; v 1  0 at y 1
( ) ( ) u
rmh
u 2
1
1
1
1
 ;
( ) ( ) v
rmh
v 2
1
1
1
1
 at y  0
 


 













x
v
y
u
x rm h
v
y
u ( ) ( ) ( ) ( 2 )
1
2
1
2 2
1
1
1
1 1
at y  0
(3.17)
 
1
(1) 0
1 1
d
Cos x
dy

     at y  1
  (2)
1  1  0 at y 1
( ) ( 2 )
1
1
1   at y  0
 


 













h y x
k
y x
( ) ( ) ( ) ( 2 )
1
2
1
1
1
1
1    
at y  0
(3.18)
Introducing the stream function 1  defined by
y
u
( )
( )


 
1
1 1
1

;
x
v
( )
( )



1
1 1
1

(3.19)
And eliminating
1
1 P and
2
1 P from equation (3.7), (3.8) and
(3.11), (3.12) we get
Region-I
3 (1) 3 (1) (1) 2 (1)
(1) 1 1 1 0
0 2 3 2
4 (1) 4 (1) 4 (1) (1)
1 1 1 1
2 2 4 4 2
d u
u
x y x x dy
G
x y x y y
  
   
    
    
     
   
  
    
(3.20)
(1) (1) (1) 2 (1) 2 (1)
(1) 1 1 0 1 1
0 2 2 Pr
d
u
x x dy x y
        
    
     
(3.21)
Region-II
3 (2) 3 (2) (2) 2 (2) 4 (2)
(2) 1 1 1 0 1
0 2 3 2 2 2
4 (2) 4 (2) (2) 2
1 1 2 2 3 1 2 2 1
4 4 2
2
d u
u
x y x x dy x y
m r h G mh M
x y y y
   
   

     
    
       
   
   
   
(3.22)
(2) (2) (2) 2 (1) 2 (1)
(2) 1 1 0 1 1
0 2 2 Pr
d km
u
x x dy x y
        
    
     
(3.23)
Assuming
       1 , , i i x i x y e y    
        1 ,
i i x i x y e t y   
for i = 1,2 (3.24)
From which we infer
        1 1 , ,
i i x i u x y e u y  
        1 1 ,
i i x i v x y e v y  
for i = 1,2 (3.25)

__________________________________________________________________________________________
And using (3.24) in (3.20) to (3.23), we get
Region-I
 
 
(1)
2 (1)
(1) 2 0
0 2
(1)
2 2 2
iv d u
i u
dy
dt
G
dy
     
   
 
     
 
   
(3.26)
(1)
2 0
0 Pr
d
t t i u t
dy

  
 
     
 
(3.27)
Region-II
 
 
(2)
2 (2)
(2) 2 0 2 2
0 2
(2)
2 2 2 2 3 2 (3.28)
iv d u
i u mh M
dy
dt
m r h G
dy
      
    
 
       
 
   
(2)
2 0
0
Pr d
t t i u t
km dy

  
 
     
 
(3.29)
Below we restrict our attention to the real parts of the solution
for the perturbed quantities
     
1 1 1 , ,
i i i   u and
 
1
i
v for
i 1,2
The boundary conditions (3.17) and (3.18) can be now written
in terms of
(1) 1
1 0 ;
du
Cos x
y dy



 

(1)
1 0
x



at y  1
(2)
1 0;
y



(2)
1 0
x



at y 1
y rmh y
( ) ( )




 2
1
1
1  1 
;
x rmh x
( ) ( )




 2
1
1
1  1 
at y  0
 


 













2
2
1
2
2
2
1
2
2 2 2
1
1
2
2
1
1
2 1
y x rm h y x
( ) ( ) ( ) ( )    
at y  0 (3.30)
1
(1) 0 d
t
dy

  at y  1
(2) t  0 at y 1
1 2
t  t at y  0 for i 1
1 2
dt dt
dy dy
 at y  0 (3.31)
If we consider small values of  then substituting
        2  
0 1 2 ,
i i i i
  y         
        2  
0 1 2 ,
i i i i
t  y  t t  t     
for i 1,2 (3.32) in to (3.26) to (3.31) gives, to order of  ,
the following sets of ordinary differential equations and
corresponding boundary and interface conditions
Region-I
4 (1) (1)
0 0
4
d dt
G
dy dy

 (3.33)
2 (1)
0
2 0
d t
dy
 (3.34)
(1)
4 (1) 2 2 (1)
1 0 0 1
4 0 2 2 0
d d d u dt
i u G
dy dy dy dy
 

 
    
 
,(3.35)
2 (1) (1)
1 0
2 0 0 0
Pr
d t d
i u t
dy dy


 
   
 
(3.36)
Region-II
4 (2) 2 (2) (2)
0 2 2 0 2 2 3 0
4 2
d d dt
M mh m r h G
dy dy dy
 
  
(3.37)
2 (2)
0
2 0
d t
dy
 (3.38)

__________________________________________________________________________________________
(2)
4 (2) 2 (2) 2 2
1 2 2 1 0 0
4 2 0 2 2 0
(2)
2 2 3 1 (3.39)
d d d d u
M mh i u
dy dy dy dy
dt
m r h G
dy
  


 
    
 

2 (2) (2)
1 0
2 0 0 0
d t Pr d
i u t
dy km dy


 
   
 
(3.40)
and
(1) 1
0 0 d du
Cos x
dy dy

  ;
(1)
0   0 at y  1
(2)
0 0;
d
dy


(2)
0   0 at y 1
(1) (2)
0 0 d 1 d
dy rmh dy
 
 ; ( ) ( )
rmh
2
0
1
0
1
   at y  0
2 (1) 2 (2)
0 0
2 2 2 2
d 1 d
dy rm h dy
 
 ; ( ) ( )
rm h
2
2 2 0
1
0
1
  
at y  0 (3.41)
(1)
1 0
d
dy

 ;
(1)
1   0 at y  1
(2)
1 0;
d
dy


(2)
1   0 at y 1
(1) (2)
1 1 d 1 d
dy rmh dy
 
 ; (1) (2)
1 1
1
rmh
   at y  0
2 (1) 2 (2)
1 1
2 2 2 2
d 1 d
dy rm h dy
 
 ; (1) (2)
1 2 2 1
1
rm h
  
at y  0 (3.42)
 
1
1 0
0
d
t
dy

  at y  1
2
0 t  0 at y 1
(1) (2)
0 0 t  t at y  0
(1) (2)
0 0 dt k dt
dy h dy
 at y  0 (3.43)
(1)
1 t  0 at y  1
(2)
1 t  0 at y 1
1 2
1 1 t  t at y  0 for i 1
1 2
1 1 dt dt
dy dy
 at y  0 (3.44)
3.3 Zeroth-Order Solution (Mean Part)
The solutions to zeroth order differential Eqs. (3.3) to (3.6)
using boundary and interface conditions (3.15) and (3.16) are
given by
Region-I
1 2
2
3
3
2
4
1
1
0 u l y l y l y A y A ( )     
1 2
2
1
1
0 d y C y C ( )    
Region-II
(2) 2
0 1 2 1 2 3 u  B cosh ny  B sinh ny  s y  s y  s
(2) 2
0 2 3 4   f y C y C
The solutions of zeroth and first order of  are obtained by
solving the equation (3.33) to (3.40) using boundary and
interface conditions (3.41) and (3.42) and are given below
(1) 4 3 3 4 2
0 4 5 6 6 2
A A
  l y  y  y  A y  A
(2) 2
0 3 4 5 6 4   B cosh ny  B sinh ny  B y  B  s y
(2)
0 7 8 t  C y C
  (1) 10 9 8 7 6 5 4
1 20 21 22 23 24 25 26
7 3 8 2
9 10 6 2
i l y l y l y l y l y l y l y
A A
y y A y A
       
   
(1)
0 5 6 t  C y C

__________________________________________________________________________________________
  (1) 7 6 5 4 3 2
1 8 9 10 11 12 13
6 8
t i Pr d y d y d y d y d y d y
q y q
     
 


(2)
1 7 8 7 10
3 3 2
29 30 31
2
32 33 34
6 5
35 36 37 38
4 3 2
39 40 41
cosh sinh
cosh sinh cosh
sinh cosh sinh
cosh sinh
B ny B ny B y B
i s y ny s y ny s y ny
s y ny s y ny s y ny
s ny s ny S y S y
S y S y S y
     
 
   
   
 


(2)
1 11 12 13
5 4 3 2
14 15 16 17 18
5 7
Pr
cosh sinh cosh
sinh
t i f y ny f y ny f ny
km
f ny f y f y f y f y
q y q
  
    
 
The first order quantities can be put in the forms
u sin x cos x i r       1
v sin x cos x r i        1
t cos x t sin x r i      1
Region-I
      1 1 1
1 0 1 0 1 cos sin r r i i u    x       x    
    (1) 1 1
1 0 1 0 1 cos sin i i r r v    x     x  
    (1) 1 1
1 0 1 0 1 cos sin r r i i    x t t   x t t
Region-II
    (2) 2 2
1 0 1 0 1 cos sin r r i i u    x       x    
    (2) 2 2
1 0 1 0 1 cos sin i i r r v    x     x  
    (2) 2 2
1 0 1 0 1 cos sin r r i i    x t t   x t t
4. RESULTS AND DISCUSSION
4.1 Discussion of the Zeroth Order Solution:
The effect of Hartmann number M on zeroth order velocity is
to decrease the velocity for   0, 1, but the suppression
is more effective near the flat wall as M increases as shown in
figure 2.
The effect of heat source   0 or sink   0 and in the
absence of heat source or sink   0 , on zeroth order
velocity is shown in figure 3 for   0,1. It is observed
that heat source promote the flow, sink suppress the flow, and
the velocity profiles lie in between source or sink for  0 .
We also observe that the magnitude of zeroth order velocity is
optimum for  1 and minimal for   1, and profiles
lies between   1 for   0 . The effect of source or sink
parameter  on zeroth order temperature is similar to that on
zeroth order velocity as shown in figure 4. The effect of free
convection parameter G, viscosity ratio m, width ratio h, on
zeroth order velocity and the effect of width ratio h,
conductivity ratio k, on zeroth order temperature remain the
same as explained in chapter-III
The effect of free convection parameter G on first order
velocity is shown in figure.5. As G increases u1 increases near
the wavy and flat wall where as it deceases at the interface and
the suppression is effective near the flat wall for   0,1.
The effect of viscosity ratio m on u1 shows that as m increases
first order velocity increases near the wavy and flat wall, but
the magnitude is very large near flat wall . At the interface
velocity decreases as m increases and the suppression is
significant towards the flat wall as seen in figure 6 for
  0,1. The effect of width ratio h on first order velocity
u1 shows that u1 remains almost same for h<1 but is more
effective for h>1. For h = 2, u1 increases near the wavy and
flat wall and drops at the interface, for   0,1 as seen in
figure 7.
The effect of Hartmann number M on u1 shows that as M
increases velocity decreases at the wavy wall and the flat wall
but the suppression near the flat wall compared to wavy wall
is insignificant and as M increases u1 increases in magnitude
at the interface for   0,1 as seen in figure 8. The effect
of  on u1 is shown in figure 9, which shows that velocity is
large near the wavy and flat wall for heat source   5 and is
less for heat sink  5. Similar result is obtained at the
interface but for negative values of u1. Here also we observe
that the magnitude is very large near the wavy wall compared
to flat wall.

__________________________________________________________________________________________
The effect of convection parameter G, viscosity ratio m, width
ratio h, thermal conductivity ratio k on first order velocity v1 is
similar as explained in chapter III. The effect of Hartmann
number M is to increase the velocity near the wavy wall and
decrease velocity near the flat wall for  0,1, whose
results are applicable to flow reversal problems as shown in
figure.10. The effect of source or sink on first order velocity v1
is shown in figure 11. For heat source, v1 is less near wavy
wall and more for flat wall where as we obtained the opposite
result for sink i.e. v1 is maximum near wavy wall and
minimum near the flat wall, for  0 the profiles lie in
between   5
ratio h, thermal conductivity ratio k, Hartmann number and
source and sink parameter  on first order temperature are
shown in figures 12 to 17. It is seen that G, m, h, and k
increases in magnitude for values of  0,1. It is seen that
from figure 16 that as M increases the magnitude of 1 
decreases for   0,1. Figure 17 shows that the magnitude
of  is large for heat source and is less for sink whereas 1 
remains invariant for   0 .
ratio h, and thermal conductivity ratio k, on total velocity
remains the same as explained in chapter III. The effect of
heat source or sink on total velocity shows that U is very large
for   5 compared to   5 and is almost invariant for
  0 as seen figure 18, for all values of .
The effect of Grashof number G, viscosity ratio m, shows that
increasing G and m suppress the total temperature but the
supression for m is negligible as seen in figures.19 and 20.
The effects of width ratio h and conductivity ratio k is same as
explained in chapter-III. The effect of source or sink
parameter on total temperature remains the same as that on
total velocity as seen figure.21.
Where
1 d   / 2 ; 2
1 f  Qh / k ; 2 1f  f / 2
 
k h
d f h
C

   
  1 2
3
1 ; 4 3 2 C   C  f ;
2 4 C  C 1 2 1 C  C 1 d ;
12
1
1
Gd
l   ;
6
1
2
GC
l   ;
2
2
3
GC
l  
2
2
2 2 3
1 n
m r h Gf
s

 ;
2
3
2 2 3
2 n
m r h GC
s


n
m r h GC
n
m r h Gf
s 2
4
2 2 3
4
2
2 2 3
3
2 



24
5
4
GC
l  ;
2 2 3
7
4 2
m r h .G C
s
n

 
2 2
2
5
6
rm h
n
l

 ;
rmh
n
l
4
6   ;
rmh
l
4
7   ;
8 2 2
6
rm h
l 
2 2 1 2 3
4
9 1 4 8 6 4
6
2 2 l l l
rm h
s
l  A  l    
  
  
10 7 6
7 5
cosh sinh cosh
sinh sinh cosh
l n n n l n n l
l n n l n n n
   
 
    11 7 6 8 l  l ncosh n l l sinh n  ncosh n
    12 9 4 7 6 l  l sinh n ncosh n  s l ncosh n l
2 1 5 4 d  l C  l ; 3
3 2 5 1 6 1 4 6
A
d l C l C C l

   
4 1 3
4 3 5 2 8 2 6
A C A
d l C l C

    ;
1 4
5 1 5 3 6 4 2
C A
d  AC  l C  A 
6 2 5 1 6 6 1 5 d  A C  AC  A C A ;
7 2 6 1 6 d  A C C A 8 2d  d / 42 ;
9 3d  d / 30 ; 10 4 d  d / 20
11 5 d  d /12 ; 12 6 d  d / 6 ;
13 7 d  d / 2 3 1 7 3 1 f  BC  B f ;
4 2 7 4 1 f  B C  B f ; 5 1 8 3 3 f  BC  B C
6 2 8 4 3 f  B C  B C ; 7 1 7 4 1 f  s C  s C
8 2 7 1 8 5 1 4 3 f  s C  s C  B f  s C ;
9 3 7 2 8 6 1 5 3 f  s C  s C  B f  B C

__________________________________________________________________________________________
10 3 8 3 6 f  s C C B ;
2
11 3 f  f / n ;
2
12 4 f  f / n 4 5
13 3 2 2
f f
f
n n
 
   
 
;
3 6
14 3 2 2
f f
f
n n
 
   
 
15 7 f  f / 20
16 8 f  f /12; 17 9 f  f / 6 ;
18 10 f  f / 2
  1 8 9 10 11 12 3 q  Pr d  d  d  d  d  d
    11 13 12 14
2
15 16 17 18
Pr f f cosh n f f sinh n
q
km f f f f
    
         
  3 13
Pr
q f
km
 ;   4 14 11
k Pr
q f n f
h km
 
  1 2 3 4
5
q q q q h
q
k h
  


; 5
6 4
kq
q q
h
  ;
  7 5 2 q   q  q ;   8 6 1 q  q  q ; 2
13 Pr
6
d
l  G ;
3
14 1 3 2 4 1 3 2 4 12 2 6 Pr
5
d
l  l A  l l  l A  l l G
4
15 1 4 2 3 3 4 1 4 2 3 3 4 12 6 2 Pr
4
d
l  l A  l A  l l  l A  l A  l l G
16 2 4 3 3 1 4 1 5
3 5
2 4 3
12 12
3 Pr
3 3
l l A l A Al l A
A d
l A l G
    
 
17 3 4 1 3 2 4 1 6
6
2 5 3 4
12 12
6 Pr
2
l l A A A A l l A
d
l A l A G
    
 
18 1 4 2 3 2 6 3 5 7 l  A A  A A  6l A  2l A GPr d ;
19 2 4 3 6 6 l  A A  2l A Gq
13
20 5040
l
l  ; 14
21 3024
l
l  ; 15
22 1680
l
l  ;
16
23 840
l
l  ; 17
24 360
l
l  ; 18
25 120
l
l  ;
19
26 24
l
l  ; 2 2
5 1 3 1 4 s  s B n  B s n ;
2 2
6 1 4 2 4 s  s B n  B s n ;
2 2 2 2 3
7 2 3 1 5 12
Pr
s s B n B B n m r h f
km

 
    
 
2 2 2 2 3
8 2 4 2 5 11
Pr
s s B n B B n m r h f
km

 
    
 

 
2 2 2
9 3 3 1 6 1 4 1 3
2 2 3
11 14 1 4
2
Pr
2
s s B n B B n B s n s B
m r h f f n B s
km

    

  

 
2 2 2
10 3 4 2 6 2 4 1 4
2 2 3
12 13 2 4
2
Pr
2
s s B n B B n B s n s B
m r h f f n B s
km

    

  
2 2 3
11 15
Pr
s 5 f m r h G
km
  ; 2 2 3
12 16
Pr
s 4 f m r h G
km
 
2 2 3
13 17
Pr
s 3 f m r h G
km
  ;
2 2 3
14 18 1 5 2 4
Pr
s 2 f m r h G 2s B 2s s
km
   
2 2 3
15 1 6 1 4 5 3 4 s  2s B  2s s m r h Gq  2s s ; 6
16 6
s
s
n
 ;
5
17 6
s
s
n
 ; 5 8
18 2 4 4
s s
s
n n
   ; 6 7
19 2 4 4
s s
s
n n
   ;
6 7 10
20 3 2 2 4 4
s s s
s
n n n
  
5 8 9
21 3 2 2 4 4
s s s
s
n n n
   ; 5 8
22 4 3 8 8
s s
s
n n
   ;
6 7
23 4 3 8 8
s s
s
n n
   ; 11
24 2
s
s
n
  ; 12
25 2
s
s
n
  ;

__________________________________________________________________________________________
11 13
26 4 2
12 s s
s
n n
   ; 12 14
27 4 2
6s s
s
n n
  
11 13 15
28 6 4 2
24 s 2 s s
s
n n n
    ;
16
29 2
s
s
n
 ;
17
30 2
s
s
n
 ; 17 17 18
31 3 4 2
3s 3s s
s
n n n
    ;
16 16 19
32 3 4 2
3s 3s s
s
n n n
   
16 16 17 19 20
33 4 5 4 3 2
6s 12s 6s 4s s
s
n n n n n
     ;
16 17 17 18 21
34 4 5 4 3 2
6s 12s 6s 4s s
s
n n n n n
    
16 17 17 18 21 22
35 5 5 6 4 3 2
12 s 6s 18s 6s 2s s
s
n n n n n n

     
17 16 16 19 20 23
36 5 5 6 4 3 2
12 s 6s 18s 6s 2s s
s
n n n n n n

     
24
37 30
s
s  ; 25
38 20
s
s  ; 26
39 12
s
s  ; 27
40 6
s
s 
28
41 2
s
s 
-1.0 -0.5 0.0 0.5 1.0
0
2
4
6
8
10



Region-I (wavy wall) Region-II (flat wall)
M = 6
M = 4
M = 2
y
= -1.0
= 0.0
= 1.0
Fig. 2 Zeroth order velocity profiles for different values of Hartmann number M
u
0
-1.0 -0.5 0.0 0.5 1.0
-10
-5
0
5
10



 = 5
 = 0
 = -5
Region-II (flat wall)
Region-I (wavy wall)
y
= -1.0
= 0.0
= 1.0
Fig. 3 Zeroth order velocity profiles for different values of 
u
0
-1.0 -0.5 0.0 0.5 1.0
-9
-6
-3
0
3
6
9



 = 5
 = 0
 = -5
y
= -1.0
= 0.0
= 1.0
Fig. 4 Zeroth order temperature profiles for different values of 


-1.0 -0.5 0.0 0.5 1.0
-0.16
-0.08
0.00
0.08
0.16



G = 15
G = 10
G = 5
u
1
Fig. 5 First order velocity profiles for different values of Grashof number G
= -1.0
= 0.0
= 1.0
y

__________________________________________________________________________________________
-1.0 -0.5 0.0 0.5 1.0
-0.04
0.00
0.04
0.08
y


 = -1.0
= 0.0
= 1.0
2.0
2.0
m = 2.0
m = 1.0
m=0.1
Fig. 6 First order velocity profiles for different values of viscosity ratio m
u
1
-1.0 -0.5 0.0 0.5 1.0
-0.6
-0.3
0.0
0.3
0.6
y
= -1.0
= 0.0
= 1.0



h = 2.0
u
1
Fig. 7 First order velocity profiles for different values of width ratio h
-1.0 -0.5 0.0 0.5 1.0
-0.04
-0.02
0.00
0.02
y
= -1.0
= 0.0
= 1.0



M = 6
M = 4
M = 2
u
1
Fig. 8 First order velocity profiles for different values of M
-1.0 -0.5 0.0 0.5 1.0
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
= -1.0
= 0.0
= 1.0



 = 5
 = -5
 = 0
y
u
1
Fig. 9 First order velocity profiles for different values of 
-1.0 -0.5 0.0 0.5 1.0
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.010
y
= -1.0
= 0.0
= 1.0



M = 6
M = 4
M = 2
Fig. 10 First order velocity profiles for different values of M
v
1
-1.0 -0.5 0.0 0.5 1.0
-0.015
-0.010
-0.005
0.000
0.005
0.010
y



= -1.0
= 0.0
= 1.0



Fig. 11 First order velocity profiles for different values of 
v
1
-1.0 -0.5 0.0 0.5 1.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0


 = -1.0
= 0.0
= 1.0
15
15
10
10
G = 5
G = 10
G = 15


Fig. 12 First order temperature profiles for different values of G
y
-1.0 -0.5 0.0 0.5 1.0
-0.16
-0.12
-0.08
-0.04
0.00
y


 = -1.0
= 0.0
= 1.0
2.0
2.0
1.0
0.1
0.1
m = 0.1
m = 1.0
m = 2.0


Fig. 13 First order temperature profiles for different values of m

__________________________________________________________________________________________
-1.0 -0.5 0.0 0.5 1.0
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
y
h = 2.0


 = -1.0
= 0.0
= 1.0
h = 0.1,0.5
Fig. 14 First order temperature profiles for different values of h

1
-1.0 -0.5 0.0 0.5 1.0
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0 k = 1
k = 0.5
k = 0.1
y



= -1.0
= 0.0
= 1.0


Fig. 15 First order temperature profiles for different values of k
-1.0 -0.5 0.0 0.5 1.0
-0.16
-0.14
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00



= -1.0
= 0.0
= 1.0
y
6
6
4
4
2
2
M = 6
M = 4
M = 2
Fig. 16 First order temperature profiles for different values of M


-1.0 -0.5 0.0 0.5 1.0
-0.16
-0.14
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
y



= -1.0
= 0.0
= 1.0









Fig. 17 First order temperature profiles for different values of 
-1.0 -0.5 0.0 0.5 1.0
-8
-4
0
4
8
12


 = -1.0
= 0.0
= 1.0
 = 5
 = 0
 = -5.0
y
Fig. 18 Total solution of velocity profiles for different values of 
U
-1.0 -0.5 0.0 0.5 1.0
-2
0
2
4
6
8
10



= -1.0
= 0.0
= 1.0
m = 0.1,1.0 ,2.0

Fig. 20 Total solution of temperature profiles for different values of m
y
-1.0 -0.5 0.0 0.5 1.0
0
2
4
6
8
G = 5, 10, 15



= -1.0
= 0.0
= 1.0
y
Fig. 19 Total solution of temperature profiles for different values of Grashof number G

-1.0 -0.5 0.0 0.5 1.0
-10
-8
-6
-4
-2
0
2
4
6
8
10



 = 5
 = 0
 = -5
y
= -1.0
= 0.0
= 1.0
Fig. 21 Total solution of temperature profiles for different values of 


__________________________________________________________________________________________
REFERENCES:
[1]. Sathe S., Sammkia B., (1998) A review of recent developments in same practical aspects of air-cooled electronic packages, Trans. ASME. J. Heat Transfer vol. 120 pp 830-839.
[2]. London M.D., Campo A., (1999) Optimal shape for laminar natural convective cavities containing air and heated from the side, Int. Comm. Heat mass Transfer vol. 26, pp 389-398.
[3]. Da silva A.K, Lorente S, Bejan A,(2004) Optimal distribution of discrete heat sources on a wall with natural convection, Int. J. Heat Mass Transfer vol. 47 pp 203-214.
[4]. Burns J.C, Parks T, J. (1967) Fluid Mech.29 405-416
[5]. Wang G. Vanka P. (1995) Convective heat transfer in periodic wavy passages. Int. J. Heat Mass Transfer vol. 38(17) pp 3219
[6]. Yao L.S., (1983) Natural convection along a vertical wavy surface, ASME J. Heat Transfer vol. 105 pp 465- 468.
[7]. Moulic S. G., Yao L.S., (1989) Mixed convection along a wavy surface, ASME J. Heat transfer vol.111 pp 974-979.
[8]. Lohrasbi. J and Sahai.V (1988) Magnetohydrodynamic Heat Transfer in two-phase flow between parallel plates. Applied scientific Research vol. 45 pp. 53-66.
[9]. Malashetty M.S. and Umavathi J.C., (1997) Two- phase Magnetohydrodynamic flow and heat transfer in an inclined channel, Int. J. Multiphase Flow, vol. 22, pp 545-560.
[10]. Malashetty M. S, Umavathi J. C. and Prathap Kumar (2000) Two-fluid magneto convection flow in n inclined channel” I.J. Trans phenomena, vol. 3. Pp 73- 84.
[11]. Malashetty M. S, Umavathi J. C, Prathap Kumar J. (2001) Convection magnetohydrodynamic two fluid flow and heat transfer, vol.37, pp 259-264.
[12]. Chamkha Ali. J, (2000) Flow of two- immiscible fluids in porous and nonporous channels, Journal of Fluids Engineering, vol. 122, pp 117-124.

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