Magneto-Convection of Immiscible Fluids in a Vertical Channel Using Robin Boundary Conditions

J.C. Umavathi et al Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 9( Version 1), September 2014, pp.114-138
www.ijera.com 114 | P a g e
Magneto-Convection of Immiscible Fluids in a Vertical Channel Using Robin Boundary Conditions J. Prathap Kumar, J.C. Umavathi and Y. Ramarao Department of Mathematics, Gulbarga University, Gulbarga-585 106, Karnataka, India ABSTRACT The effects of viscous dissipation on fully developed two fluid magnetohydrodynamic flow in the presence of constant electric field in a vertical channel is investigated using Robin boundary conditions. The fluids in both the regions are incompressible, electrically conducting and the transport properties are assumed to be constant. The plate exchanges heat with an external fluid. Both conditions of equal and different reference temperatures of the external fluid are considered. First, the simple cases of the negligible Brinkman number or the negligible Grashof number are solved analytically. Then, the combined effects of buoyancy forces and viscous dissipation are analyzed by a perturbation series method valid for small values of perturbation parameter. To relax the condition on the perturbation parameter, the flow fields are solved by using the differential transform method. The results are presented graphically for different values of the mixed convection parameter, Hartman number, perturbation parameter, viscosity ratio, width ratio, conductivity ratio and Biot numbers for both open and short circuit. The effects of these parameters on the Nusselt number at the walls is also drawn. It is found that the solutions obtained by perturbation method and differential transform method agree very well for small values of perturbation parameter.
I. INTRODUCTION:
The interaction between the conducting fluid and the magnetic field radically modifies the flow, with attendant effects on such important flow properties as pressure drop and heat transfer, the detailed nature of which is strongly dependent on the orientation of the magnetic field. The advent of technology that involves the MHD power generators, MHD devices, nuclear engineering and the possibility of thermonuclear power has created a great practical need for understanding the dynamics of conducting fluids. Moreover, there has been interest in studying the flow of electrically conducting fluids over surfaces. On the other hand, the heat transfer rates can be controlled using a magnetic field. One of the ways of studying magnetohydrodynamic heat transfer field is the electromagnetic field, which is used to control the heat transfer as in the convection flows and aerodynamic heating. The use of electrically conducting fluids under the influence of magnetic fields in various industries has lead to a renewed interest in investigating hydrodynamic flow and heat transfer in different geometries. For example, Sparrow and Cess [1] and Umavathi [2] studied magneto convection in vertical channel in the presence of electric field. Bhargava et al. [3] have studied the effect of magnetic field on the free convection flow of a micropolar fluid between two parallel porous vertical plates. Hayat et al. [4] have studied the Hall effects on the unsteady hydrodynamic oscillatory flow of a second grade fluid. Recently, Umavathi et al. [5] numerically studied fully developed magneto convection flow in a vertical rectangular duct. There has been some theoretical and experimental work on stratified laminar flow of two immiscible liquids in the horizontal pipe (Charles and Redburger [6], Pacham and Shail [7]). The interest in this configuration stems from the possibility of reducing the power required to pump oil in a pipeline by suitable addition of water. Shail [8] investigated theoretically the possibility of using a two-fluid system to obtain increased flow rates in an electromagnetic pump. Specially, Shail [8] studied Hartmann flow of a conducting fluid which is pumped electromagnetically in a channel bounded by two parallel horizontal insulating plates of infinite extent, there being a layer of non conducting fluid between the conducting liquid and the upper channel wall.
Another physical phenomenon is the case in which the two immiscible conducting fluids flow past permeable beds. Recent advances in two-fluid flow researchers are remarkable and many things have been clarified about various phenomena in two-fluid flow. However, of course, there are still many more things to be studied in order to achieve sufficient understanding and satisfactory prediction about two-fluid flow. In particular, the microscopic structures of two-fluid flow, such as velocity and phase distributions, interfacial structures and turbulence phenomena, are quite important topics and much effort has been made in these research areas in recent years. This is partly due to scientific interest in the physical phenomenon of two fluid flow and partly due to industrial demands for more precise predictions of two fluid flow behavior in various
RESEARCH ARTICLE OPEN ACCESS

industrial devices. The coal-fired magnetohydrodynamic generator channel is subjected to an unusual severe
thermal environment. Postlethwaite and Sluyter [9] presents an overview of the heat transfer problems
associated with a MHD generator.
Malashetty and Leela [10] reported closed form solutions for the two fluid flow and heat transfer situations
in a horizontal channel for which both phases are electrically conducting. Malashetty and Umavathi [11] studied
two fluid MHD flow and heat transfer in an inclined channel in the presence of buoyancy effects for the
situations where only one of the phases is electrically conducting. Malashetty et al. [12-14] analyzed the
problem of fully developed two fluid magnetohydrodynamic flows with and without applied electric field in an
inclined channel. Umavathi et al. [15-17] studied steady and unsteady magnetohydrodynamic two fluid flow in
a vertical and horizontal channel. Prathap Kumar et al. [18, 19] also studied mixed convection of
magnetohydrodynamic two fluid flow in a vertical enclosure.
The developing flow with asymmetric wall temperature has been considered by Ingham et al. [20], with
particular reference to situations where reverse flow occurs. On the other hand, Barletta [21] and Zanchini [22]
have pointed out that relevant effects of viscous dissipation on the temperature profiles and on the Nusselt
numbers may occur in the fully developed laminar forced convection in tubes. Thus, an analysis of the effect of
viscous dissipation in the fully developed mixed convection in vertical ducts appears as interesting. Several
studies on mixed convection problems for a Newtonian fluid in a vertical channel have already been presented
in literature. In particular, some analytical solutions for the fully developed flow have been performed. The
boundary conditions of uniform wall temperatures have been analyzed by Aung and Worku [23]. The boundary
conditions of uniform wall temperatures on a wall and a uniform wall heat fluxes, have been studied by Cheng
et al. [24]. The effect of viscous dissipation on the velocity and on the temperature fields have been analyzed by
Barletta [25] for the boundary conditions of uniform wall temperatures and by Zanchini [26] for boundary
conditions of third kind.
In the past, the laminar forced convection heat transfer in the hydrodynamic entrance region of a flat
rectangular channel wall has been investigated either for the temperature boundary conditions of the first kind,
characterized by prescribed wall temperature (Stephan [27], Hwang and Fan[28]), or boundary conditions of
second kind, expressed by prescribed wall temperature heat flux (Siegal and Sparrow[29]). A more realistic
condition in many applications, however, will be the temperature boundary conditions of third kind: the local
wall heat flux is a linear function of the local wall temperature.
The differential transform scheme (DTM) is a method for solving a wide range of problems whose
mathematical models yield equations or systems of equations involving algebraic, differential, integral and
integro-differential equations (Arikhoglu and Ozkol [30] and Biazar et al. [31]). The concept of the differential
transform was first proposed by Zhou [32], and its main applications therein is solved for both linear and non-linear
initial value problems in electric circuit analysis. This method constructs an analytical solution in the form
of polynomials. It is different from the high-order Taylor series method, which requires symbolic computation
of the necessary derivatives of the data functions. With this technique, the given differential equation and related
initial and boundary conditions are transformed into a recurrence equation that finally leads to the solution of a
system of algebraic equations as coefficients of a power series solution. Therefore the differential transform
method can overcome the restrictions and limitations of perturbation techniques so that it provides us with a
possibility to analyze strongly nonlinear problems. In recent years the application of differential transform
theory has been appeared in many researches (Rashidi et al. [33], and Ganji et al. [34]).
Recently, Umavathi and Santhosh [35, 36], Umavathi and Jaweria [37] studied mixed convection in vertical
channel using boundary conditions of third kind. The aim of this paper is to extend the analysis performed by
Zanchini [26] for electrically conducting immiscible fluids.
II. MATHEMATICAL FORMULATION:
The geometry under consideration illustrated in Figure 1 consists of two infinite parallel plates maintained
at equal or different constant temperatures extending in the X and Z directions. The region h1 2  Y  0
is occupied by electrically conducting fluid of density 1  , viscosity 1  , thermal conductivity 1 k , and thermal
expansion coefficient 1  , and the region 2 0  Y  h 2 is occupied by another immiscible electrically
conducting fluid of density 2  , viscosity 2  , thermal conductivity 2 k , and thermal expansion coefficient 2  .

Figure 1 Physical configuration
The fluids are assumed to have constant properties except the density in the buoyancy term in the
momentum equation   1 0 1 1 0    1 T T  and   2 0 2 2 0    1 T T  . A constant magnetic
field of strength 0 B is applied normal to the plates and a uniform electric field 0 E is applied perpendicular to
the plates. A fluid rises in the channel driven by buoyancy forces. The transport properties of both fluids are
assumed to be constant. We consider the fluids to be incompressible and the flow is steady, laminar, and fully
developed. It is assumed that the only non-zero component of the velocity q 
is the X-component ( 1, 2) i U i .
Thus, as a consequence of the mass balance equation, one obtains
0 i U
X



(1)
so that i U depends only on Y . The stream wise and the transverse momentum balance equations yields
Region-I
   
2
1 1
1 1 0 1 2 0 0 1 0
1 1
1
0 e P d U
g T T E B U B
X dY

 
 

     

(2)
Region-II
   
2
1 2
2 2 0 2 2 0 0 2 0
2 2
1
0 e P d U
g T T E B U B
X dY

 
 

     

(3)
and Y -momentum balance equation in both the regions can be expressed as
0
P
Y



(4)
where 0 P  p  gx (assuming 1 2 p  p  p ) is the difference between the pressure and hydrostatic pressure.
On account of equation (4), P depends only on X so that equations (2) and (3) can be rewritten as
Region-I
 
2
1 1 1
1 0 2 0 0 1 0
1 1 1 1 1
1 e dP d U
T T E B U B
g dX g dY g
 
    
     (5)
Region-II
 
2
2 2 2
2 0 2 0 0 2 0
2 2 2 2 2
1 e dP d U
T T E B U B
g dX g dY g
 
    
     (6)
From equations (5) and (6) one obtains
Region-I

2
1
2
1 1
T 1 d P
X g  dX



(7)
3
1 1 1 1 2 1
3 0
1 1 1
e T d U dU
B
Y g dY g dY
 
  

  

(8)
2 4 2
1 1 1 1 2 1
2 4 0 2
1 1 1
e T d U d U
B
Y g dY g dY
 
  

  

(9)
Region-II
2
2
2
2 2
T 1 d P
X g  dX



(10)
3
2 2 2 2 2 2
3 0
2 2 2
e T d U dU
B
Y g dY g dY
 
  

  

(11)
2 4 2
2 2 2 2 2 2
2 4 0 2
2 2 2
e T d U d U
B
Y g dY g dY
 
  

  

(12)
Both the walls of the channel will be assumed to have a negligible thickness and to exchange heat by
convection with an external fluid. In particular, at 1 Y   h 2 the external convection coefficient will be
considered as uniform with the value 1 q
and the fluid in the region 1h 2  Y  0
will be assumed to have a
uniform reference temperature
q1 T . At 2 Y  h 2
the external convection coefficient will be considered as
uniform with the value 2 q and the fluid in the region 2 0  Y  h 2
will be supposed to have a uniform
reference temperature
q2 q1 T  T . Therefore, the boundary conditions on the temperature field can be expressed
as
  1
1
1
1 1 1 1
2
, 2 q
h
Y
T
k q T T X h
Y 

        
(13)
  2
2
2
2 2 2 2
2
, 2 q
h
Y
T
k q T X h T
Y 

       
(14)
On account of equations (8) and (11), equations (13) and (14) can be rewritten as
  1
3 2
1 1 0 1 1
3 1 1 1
1 1 1
, 2 e
q
d U B dU g
q T T X h
dY dY k
 
 
        at 1
2
h
Y   (15)
  2
3 2
2 2 0 2 2
3 2 2 2
2 2 2
, 2 e
q
d U B dU g
q T X h T
dY dY k
 
 
       at 2
2
h
Y  (16)
On account of Equations (5) and (6), there exist a constant Asuch that
dP
A
dX
 (17)
For the problem under examination, the energy balance equation in the presence of viscous dissipation can
be written as
Region-I
 
2 2
1 1 1 1 2
2 0 0 1
1 1 1
e
p p
d T dU
E B U
Y c dY c
 
  
 
     
  
(18)
Region-II

 
2 2
2 2 2 2 2
2 0 0 2
2 2 2
e
p p
d T dU
E B U
Y c dY c
 
  
 
     
  
(19)
From equations (9), (18), (12) and (19) allow one to obtain differential equations for i U namely
Region-I
 
4 2 2 2
1 1 1 1 2 1 0 1
4 0 0 1 2
1 1 1
e e
p
d U g dU B d U
E B U
dY c dY dY
  
  
   
             
(20)
Region-II
 
4 2 2 2
2 2 2 2 2 2 0 2
4 0 0 2 2
2 2 2
e e
p
d U g dU B d U
E B U
dY c dY dY
  
  
   
             
(21)
The boundary conditions on i U are
    1 1 2 2 U h 2 U h 2  0 (22)
together with equations (15) and (16) which on account of equations (5) and (6) can be rewritten as
   
1
1
3 2 2
1 1 0 1 1 1
3 2
1 1 2
1 1 1 1
0 0 0 1 0
1 1 1 1 1
e
y h
e
q
d U B dU q d U
dY dY k dY
q g q A
T T E B U B
k k


 
  

 
   
 
 
      
 
   
2
2
3 2 2
2 2 0 2 2 2
3 2
2 2 2
2 2 2 2
0 0 0 2 0
2 2 2 2 2
e
y h
e
q
d U B dU q d U
dY dY k dY
q g q A
T T E B U B
k k


 
  

 
   
 
 
      
 
(23)
The continuity of velocity, shear stress, and temperature and heat flux is assumed to be continuous at
the interface as follows
    1 2 U 0 U 0 ,
    1 2
1 2
dU 0 dU 0
dy dy
   ,     1 2 T 0 T 0 ,
    1 2
1 2
dT 0 dT 0
k k
dy dy
 (24)
Equations (20)-(24) determine the velocity distribution. They can be written in a dimensionless form by
means of the following dimensionless parameters
 
1
1 1
0
U
u
U
 ,
 
2
2 2
0
U
u
U
 , 1
1
1
Y
y
D
 , 2
2
2
Y
y
D
 ,
3
1 1
2
1
g TD
Gr



 ,
1
0 1
1
Re
U D

 ,
 2
1
0 1
1
U
Br
k T



Re
Gr
  , 1 0
1
T T
T




, 2 0
2
T T
T




, 2 1
T
T T
R
T



,
2 2
2 1 0 1
1
e B D
M


 ,
 
0
1
0 0
E
E
BU
 (25)
where D  2h is the hydraulic diameter. The reference velocity and the reference temperature are given by
 
2
1 1
0
1 48
AD
U

  ,
 
2
2 2
0
2 48
AD
U

  ,   1 2
0 2 1
1 2
1 1
2
q q
q q
T T
T s T T
Bi Bi
  
     
 
(26)
Moreover, the temperature difference T is given by
q2 q1 T  T T
if
q1 q2 T  T . As a consequence, the
dimensionless parameter T R can only take the values 0 or 1. More precisely, the temperature difference
ratio T R is equal to 1 for asymmetric heating i.e.
q1 q2 T  T , while T R =0 for symmetric heating i.e.
q1 q2 T  T ,
respectively. Equation (17) implies that A can be either positive or negative. If A  0, then 0
i U , Re and  are

negative, i.e. the flow is downward. On the other hand, if A  0 , the flow is upward, so that 0
i U , Re , and
 are positive. Using equations (25) and (26), equations (20)-(24) becomes
Region-I
4 2 2
1 2 1 1 2 2 2 2 2
4 2 1 1 2
d u d u du
M Br M E M u M Eu
dy dy dy
  
             
(27)
Region-II
  
4 2
2 2 2 1
4 2
2
2 2 2 2 2 2 4 2 2
2 2 2
d u d u
M h r m
dy dy
du
Br bnkh mh M r E m h u mh Eu
dy


 
 
     
 
(28)
The boundary and interface conditions becomes
    1 2 u 1 4  u 1 4  0 ,     2
1 2 u 0 mh u 0 ,
    1 2 du 0 du 0
h
dy dy
 ,
2 2 2 2 2
1 2 2 2
2 1 2 2
1 48
48
d u d u M rmh M E r
M u u M E
dy nb dy nb nb nb
   
        
 
at y  0,
3 3 2
1 2 1 2 2
3 3
d u du 1 d u M rmh du
M
dy dy nbkh dy nbk dy
  
    
 
at y  0,
2 2 3
1 1 1 2 2
2 3 1
1 1 1 4 1
1 4
48 1
2
T
y
d u M du d u R
M u s M E
dy Bi dy Bi dy Bi

   
            
   
,
2 3 2 2
2 2 2 2 2
2 3 2
2 2 1 4
2
2
1
4
48 1
2
y
T
d u d u M h r m du
M h r mu
dy Bi dy Bi dy
R
sbn M r E
Bi




 
     
 
 
     
 
(29)
Basic Idea of Differential Transformation Method (DTM)
The transformation of the kth derivative of a function in one variable is as follows:
 
 
0
1
!
k
k
y
d u y
U k
k dy

 
  
 
(30)
where u  y is the original function and U k  is the transformed function which is called the T-function. The
differential inverse transform of U k  is defined as
   
0
k
k
u y U k y


  (31)
Equation (31) implies that the concept of the differential transformation is derived from Taylor’s series
expansion (see Zhou, [32]), but the method does not evaluate the derivatives symbolically. However, relative
derivatives are calculated by iterative procedure that is described by the transformed equations of the original
functions. In real applications, the function u  y is a finite series and hence equation (31) can be written as
   
0
n
k
k
u y U k y

  (32)
and equation (31) implies that    
1
k
k n
u y U k y

 
  is neglected as it is small. Usually, the values of n are
decided by a convergence of the series coefficients. Mathematical operations performed by differential
transform method are listed in Table 1.

III. SOLUTIONS
Case-I
The solution of equations (27) and (28) using boundary and interface conditions in equation (29) in the
absence of viscous dissipation term ( Br  0) is given by
Region-I
    1 1 2 3 4 u  c c y c Cosh My c Sinh My (33)
Region-II
    2 5 6 7 1 8 1 u  c c y c Cosh p y c Sinh p y (34)
where
2 2
1 p  M h rm , and using equation (29) in equations (5) and (6), the energy balance equations
becomes
Region-I
2
1 2 2
1 2 1
1
48
d u
M u M E
dy

 
     
 
(35)
Region-II
2
2 2 2 2
2 2 2
1
48
d u
mh M r u M E r
bn dy
  
 
     
  
(36)
Using the expressions obtained in equations (33) and (34) the energy balance equations (35) and (36) becomes
Region-I
    2 2
1 1 2
1
   48M c  c y M E

(37)
Region-II
    2 2 2
2 5 6
1
48 M h m r c c y M E r
nb
      

(38)
Case-II
The solution of equation (27) and (28) can be obtained when buoyancy forces are negligible (   0)
and viscous dissipation is dominating ( Br  0 ), so that purely forced convection occurs. For this case,
solutions of equations (27) and (28), using the boundary and interface conditions given by equation (29), the
velocities are given by
Region-I
    1 1 2 3 4 u  l l y l Cosh My l Sinh My
(39)
Region-II
    2 5 6 7 1 8 1 u  l l y l Cosh p y l Sinh p y
(40)
The energy balance equations (18) and (19) in non-dimensional form can also be written as
Region-I
2 2
1 1 2 2 2 2 2
2 1 1 2
d du
Br M u M E M Eu
dy dy
   
            
(41)
Region-II
 
2 2
2 4 2 2 2 2 2 4 2 2
2 2 2 2
d du
Br mkh M rkh E m h u Eh mu
dy dy


   
              
(42)
The boundary and interface conditions for temperature are
  1 1
1 1
1 4 1
4
1 4 1
2
T
y
d Bi R s
Bi
dy Bi



 
     
 
,   2 2
2 2
1 4 2
4
1 4 1
2
T
y
d Bi R s
Bi
dy Bi



 
    
 
,

    1 2  0  0 ,
    1 2 d 0 1 d 0
dy kh dy
 
 (43)
Using equations (39) and (40) , solving equations (41) and (42) we obtain
Region-I
        
    
1 1 2 3 4
4 3 2
5 6 7 8 9 1 2
Br GCosh 2My G Sinh 2My G Cosh My G Sinh My
G yCosh My G ySinh My G y G y G y d y d
      
     
(44)
Region-II
        
    
2 10 1 11 1 12 1 13 1
4 3 2
14 1 15 1 16 17 18 3 4
Br G Cosh 2p y G Sinh 2p y G Cosh p y G Sinh p y
G yCosh p y G ySinh p y G y G y G y d y d
      
     
(44)
Perturbation Method (PM):
We solve equations (27) and (28) using the perturbation method with a dimensionless parameter  (<<1)
defined as
 Br (46)
and does not depend on the reference temperature difference T . To this end the solutions are assumed in the
form
          2
0 1 2
0
... n
n
n
u y u y  u y  u y  u y


      (47)
Substituting equation (47) in equation (27) and (28) and equating the coefficients of like powers of  to zero,
we obtain the zero and first order equations as follows:
Region-I
Zero-order equations
4 2
10 2 10
4 2 0
d u d u
M
dy dy
  (48)
First-order equations
4 2 2
11 2 11 10 2 2 2 2 2
4 2 10 10 2
d u d u du
M M E M u M Eu
dy dy dy
 
      
 
(49)
Region-II
Zero-order equations
4 2
20 2 2 2
4 2 0
d u d u
M h r m
dy dy
   (50)
First-order equations
  
4 2
21 2 2 21
4 2
2
2 2 20 2 2 2 4 2 2
20 20 2
d u d u
M h rm
dy dy
du
Br bnkh mh M r E m h u mh Eu
dy


 
 
     
 
(51)
The corresponding boundary and interface conditions given by equation (29) for the zeroth and first order
reduces to
Zeroth-order
    10 20 u 1 4  u 1 4  0 ,     2
10 20 u 0 mh u 0 ,
    10 20 du 0 du 0
h
dy dy
 ,
2 2 2 2 2
10 2 20 2
2 10 2 20
1 48
48
d u d u M rmh M E r
M u u M E
dy nb dy nb nb nb
   
        
 
at y  0,

3 3 2
10 2 10 20 20
3 3
M
  
    
 
at y  0,
2 2 3
10 10 10 2 2
2 3 10
1 1 1 4 1
1 4
48 1
2
T
y
d u M du d u R
M u s M E
dy Bi dy Bi dy Bi

   
            
   
,
2 3 2 2
20 20 20 2 2
2 3 20
2 2 1 4
2
2
1
48
4
1
2
y
T
d u d u M h r m du
M h r mu
dy Bi dy Bi dy
R
sbn M r E
Bi




 
      
 
 
    
 
(52)
First-order
    11 21 u 1 4  u 1 4  0 ,     2
11 21 u 0 mh u 0 ,
    11 21 du 0 du 0
h
dy dy
 ,
2 2 2 2
11 2 21
2 11 2 21
d u 1 d u M rmh
M u u
dy nb dy nb
  
    
 
at y  0,
3 3 2
11 2 11 21 21
3 3
M
  
    
 
at y  0,
2 2 3
11 11 11 2
2 3 11
1 1 1 4
1
0
y
d u M du d u
M u
dy Bi dy Bi dy

 
     
 
,
2 3 2 2
21 21 21 2 2
2 3 21
2 2 1 4
1
0
y
d u d u M h rm du
M h rmu
dy Bi dy Bi dy



 
     
 
(53)
Solutions of zeroth-order equations (48) and (50) using boundary and interface conditions (52) are
    10 1 2 3 4 u  z  z y  z Cosh My  z Sinh My (54)
    20 5 6 7 1 8 1 u  z  z y  z Cosh p y  z Sinh p y (55)
Solutions of first-order equations (49) and (51) using boundary and interface conditions of equation (53) are
       
       
11 9 10 11 12 10 11
2 2 4
12 13 14 15 16
3 2
17 18
u z z y z Cosh My z Sinh My k Cosh 2My k Sinh 2My
k ySinh My k yCosh My k y Cosh My k y Sinh My k y
k y k y
     
    
 
(56)
        
       

21 13 14 15 1 16 1 2 28 1 29 1
2 2 4
30 1 31 1 32 1 33 1 34
3 2
35 36
u z z y z Cosh p y z Sinh p y A k Cosh 2p y k Sinh 2p y
k ySinh p y k yCosh p y k y Cosh p y k y Sinh p y k y
k y k y
     
    
 
(57)
Using velocities given by equations (54)-(57), the expressions for energy balance equations (35) and (36)
becomes
Region-I
       
         
       
   
2 2
1 10 11 12
13 14 15
2 2 4 2 3 2 2
16 17 18
2 2 2
9 10 1 2
1
48 3 2 3 2 2
2 4 2 4
2 12 6 2
M k Cosh My M k Sinh My Mk Cosh My
Mk Sinh My k MySinh My Cosh My k MyCosh My
Sinh My k y M y k y M y k M y
M z z y M E M z z y
      

   
     
    
(58)

Region-II
        
         
       
   
2 2
2 2 1 28 1 1 29 1 1 30 1
1 31 1 32 1 1 1 33 1 1
2 2 4 2 3 2 2
1 34 1 35 1 36 1
2 2 2
1 13 14 1 5 6
1
48 3 2 3 2 2
2 4 2 4
2 12 6 2
A p k Cosh p y p k Sinh p y p k Cosh p y
bn
p k Sinh p y k p ySinh p y Cosh p y k p yCosh p y
Sinh p y k y p y k y p y k p y
p z z y M E r Er p z z y
 

     

   
     
    
(59)
Solution with differential transformation method (DTM)
Now Differential Transformation Method has been applied to solving equations (27) and (28). Taking the
differential transformation of equations (27) and (28) with respect to k , and following the process as given in
Table 1 yields:
 
    
     
      
       
2
0
2 2 2 2
0
1
4 1 2 2
1 2 3 4
1 1 1 1
2
r
s
r
s
U r M r r U r
r r r r
Br r s s U r s U s
Br M E  r M U r s U s M EU r


    
   
        
 
     
 


(60)
 
    
     
       
     
 
2 2
2 2
0
2 2 2 2 4 2
0
2 2
1
4 1 2 2
1 2 3 4
1 1 1 1
2
r
s
r
s
V r M h r m r r V r
r r r r
Br nbkh mh r s s V r s V s
M E r Er r M h r m V r s V s
M Eh r ErmV r

  



    
   
       
  


 (61)
The differential transform of the initial conditions are as follows
        3 4
1 2 0 , 1 , 2 , 3 ,
2 6
c c
U  c U  c U  U    1
2 0
c
V
mh
 ,   2 1 ,
c
V
h

   
2 12 2
2 4 2
2 1
3 1 11 2 , 3
2 6
A c
c M c nbkh
M r c nb h
V c M c A U
nb

 
              
 
(62)
Where
2
2
11
48
48
M E r
A M E
nb nb

     ,
2
12
M rhm
A
nbk

 
Using the conditions as given in equation (62), one can evaluate the unknowns 1 c , 2 c , 3 c , and 4 c . By using
the DTM and the transformed boundary conditions, above equations that finally leads to the solution of a system
of algebraic equations.
A Nusselt number can be defined at each boundary, as follows:
 
1
1 2 1
1
2 2 1 1
2
2( )
2 ( 2) (1 ) T T
Y h
h h dT
Nu
R T h T h R T dY



      
 
2
1 2 2
2
2 2 1 1
2
2( )
2 ( 2) (1 ) T T
Y h
h h dT
Nu
R T h T h R T dY



      
(63)

By employing equation (25), in equation (63) can be written as
 
1
1
2 1
1 4
(1 )
1 4 ( 1 4) (1 ) T T
y
h d
Nu
R R dy

 



     
 
2
2
2 1
1 4
(1 1 )
1 4 ( 1 4) (1 ) T T
y
h d
Nu
R R dy

 



     
. (65)
IV. RESULTS AND DISCUSSION
In this section the fluid flow and heat transfer results for electrically conducting immiscible fluid flow in
vertical channel are discussed in the presence of an applied magnetic field 0 B normal to gravity and applied
electric field 0 E perpendicular to 0 B including the effects of both viscous and Ohmic dissipations. Robin
boundary conditions for equal and unequal wall temperatures have been incorporated for the boundary
conditions. The governing equations which are highly non-linear and coupled are solved by the well known
perturbation method using the product of mixed convection parameter  and Brinkman number Br as
perturbation parameter. The solutions obtained by perturbation method cannot be used for large values of
perturbation parameters . However this condition on  is relaxed by finding the solutions of the basic
equations using DTM which is a semi analytical method. The electric field load parameter E  0 corresponds
to short circuits configuration and E  0 corresponds to open circuit configuration and the values of E may be
taken as positive or negative depending on the polarity of 0 E .
In the absence of viscous dissipation (Br  0) and mixed convection parameter , exact solutions can be
obtained. The flow field for asymmetric heating and Br  0 is shown in figure 2. This figure indicates that
there is a flow reversal near the cold wall for  1000 and there is a symmetric profile for   0 for both
open and short circuits.
In the absence of mixed convection parameter  , plots of  for equal and unequal Biot numbers are shown
in figures 3a and 3b respectively for various values of Brinkman number. As the Brinkman number Br
increases, temperature field is enhanced for both equal and unequal Biot number. The magnitude of
enhancement at the cold wall is very large for unequal Biot numbers when compared with equal Biot numbers
for all values of electric field load parameter E . Similar nature was also observed by Zanchini [26] for one fluid
model for short circuits and in the absence of Hartman number.
The effect of  and  on the flow field is shown in figures 4a and 4b for open circuit. It is seen that the
velocity and temperature are increasing functions of for upward flow, velocity is a decreasing function of 
and temperature is an increasing function of  for downward flow. The enhancement of flow for   0
implies that a greater energy generated by viscous dissipation yields a greater fluid temperature and as a
consequence a stronger buoyancy force occurs. One can also reveal from figures 4a and 4b that the solution
agree very well between perturbation method and differential transform method for   0 and differs slightly
for   2but becomes large for   4for both buoyancy assisting (  0) and buoyancy opposing
(  0) flows. Further there is a flow reversal at the cold wall for   500 and at the hot wall
for  500 . The effects of  and  on the flow was also the similar result observed for one fluid model
for purely viscous fluid by Barletta [25] for isothermal wall conditions.
To understand the effects of Hartman number M and electric field load parameter E , the plots u and 
are drawn in figures 5a and 5b for equal Biot numbers. It is seen that for both open and short circuits the effects
of M is to de-accelerate the flow. This is a classical Hartman result. The plots of u and  for variations of
viscosity ratio m is shown in figures 6a and 6b for variations of E . It is seen that as m increases velocity
increases in region-I and decrease in region-II for both open and short circuits. Flow reversal is observed at the
cold wall and the intensity of reversal flow increases for decreasing values of m . The slope at the interface
suddenly drops for values of m 1 due to the condition imposed at the interface
2
1 2 (u  mh u ) . Figure 6b
shows that there is no effect of either m or E on the temperature field. The figures 7a and 7b refer to the

influence of width ratio h on the velocity and temperature for open and short circuits. As h increases flow
decreases in both the regions for open and short circuits. However the temperature filed is not affected by E .
Further the flow reversal is also observed for variations of h at the cold wall and the intensity of reversal flow
increases as h increases. One can observe from figures 6b and 7b that there is no drop of the slope at the
interface on that temperature owing to the continuity of heat flux at the interface    1 2 d dy  k d dy .
Figures 8a and 8b exhibit the effect of thermal conductivity ratio k on the velocity and temperature fields. The
effect of k is similar to the effect of h i.e. both the velocity and temperature decreases for increase in the value
of k in both regions. Here also there is a flow reversal at the cold wall and the intensity of reversal flow
increases with increase in k for all values of E .
The plots of u and  are drawn in figure 9a and 9b for unequal Biot numbers with T R (asymmetric wall
heating condition) for both open and short circuits. It is seen from figure 9a that there is no flow reversal either
at cold wall or at the hot wall as observed for equal Biot numbers for all values of E . However the effect of 
on u and  for unequal Biot numbers remains the same for equal Biot numbers. The magnitude of temperature
at the left wall for unequal Biot numbers is very large when compared to equal Biot numbers for all values of E .
Considering symmetric wall heat condition ( 0) T R  , the plots of u and  are drawn and shown in
figures 10a, b and 11a, b for equal and unequal Biot numbers for both open and short circuits. It is seen from
these four figures that both u and  are increasing functions of  for E  0,1. Further the temperature
profiles are symmetric for equal Biot numbers and the magnitude of temperature at the left wall is very large for
unequal Biot numbers which was the similar nature observed for asymmetric wall heat conditions.
Figures 12a, b shows the plots of 1 Nu and 2 Nu for   500,100,300 versus  for both open and
short circuits. These figures tells that 1 Nu is an increasing function of 
while 2 Nu
is decreasing function of
 for both open and short circuits. Further the effects of  on 1 Nu and 2 Nu is stronger for lower values of 
for buoyancy assisting flow. In order to compare the present results with the earlier published work the values of
viscosity ratio, width ratio and conductivity ratio taken as 1. The effects of Biot numbers on symmetric wall heat
conditions and Nusselt numbers in the absence of Hartman number and electric field load parameter are the
similar result observed by Zanchini [26] for one fluid model.
Tables 2-4 are the velocity and temperature solutions obtained by PM and DTM for symmetric and
asymmetric wall heating conditions varying the perturbation parameter  for equal and unequal Biot numbers.
In Table 2, it is seen that in the absence of perturbation parameter, the PM and DTM solutions are equal for both
the velocity and temperature fields. When the perturbation parameter  is increased   2 , it is seen that the
PM and DTM solutions do not agree. Similar nature is also observed in Table 3 and 4 for PM and DTM
solutions. Table 2 and 3 are the solutions of velocity and temperature for asymmetric wall heating conditions
for equal and unequal Biot numbers respectively. Table 2 and 3 also reveals that the percentage of error is large
at the interface for velocity when compared with the error at the boundaries. Further the percentage of error
between PM and DTM is large for unequal Biot numbers when compared with equal Biot numbers. Table 4
displays the solutions of symmetric wall heating conditions for equal Biot numbers. The percentage of error is
less for symmetric wall heating conditions for equal Biot numbers when compared with asymmetric wall heat
conditions.
Table 1: The operations for the one-dimensional differential transform method.
Original function Transformed function
y(x)  g(x)  h(x) Y(k)  G(k)  H(k)
y(x)  g(x) Y(k)  G(k)
dx
dg x
y x
( )
( )  Y(k)  (k 1)G(k 1)
2
2 ( )
( )
dx
d g x
y x  Y(k)  (k 1)(k  2)G(k  2)

y(x)  g(x)h(x) 

 
k
l
Y k G l H k l
0
( ) ( ) ( )
m y(x)  x
1, if
( ) ( )
0, if
k m
Y k k m
k m

 
   
 
Table 2: Values of velocity and Temperature for   500 and 1, 4, 1 T R  M  E  
y
  0, 1 2 Bi  Bi 10   2, 1 2 Bi  Bi 10
PM DTM % error PM DTM % error
-0.250 0.000000 0.000000 0.00% 0.000000 0.000000 0.00%
-0.150 0.276790 0.276790 0.00% 0.370820 0.385670 1.49%
-0.050 1.035680 1.035680 0.00% 1.184140 1.207620 2.35%
0.000 1.407780 1.407780 0.00% 1.565790 1.590860 2.51%
0.050 1.675850 1.675850 0.00% 1.830670 1.855350 2.47%
0.150 1.577240 1.577240 0.00% 1.684980 1.702290 1.73%
0.250 0.000000 0.000000 0.00% 0.000000 0.000000 0.00%
Temperature
-0.250 -0.357140 -0.357140 0.00% -0.351440 -0.350540
-0.150 -0.214290 -0.214290 0.00% -0.203310 -0.201580 0.17%
-0.050 -0.071430 -0.071430 0.00% -0.056990 -0.054750 0.22%
0.000 0.000000 0.000000 0.00% 0.015240 0.017630 0.24%
0.050 0.071430 0.071430 0.00% 0.087010 0.089500 0.25%
0.150 0.214290 0.214290 0.00% 0.229940 0.232520 0.26%
0.250 0.357140 0.357140 0.00% 0.368830 0.370680 0.19%
Table 3: Values of velocity and Temperature for   500 and 1, 4, 1 T R  M  E  
y   0, 1 2 Bi  0.1,Bi 10   2, 1 2 Bi  0.1,Bi 10
-0.250 0.000000 0.000000 0.00% 0.000000 0.000000 0.00%
-0.150 0.884080 0.884080 0.00% 1.060240 1.193770 13.35%
-0.050 1.334630 1.334630 0.00% 1.575460 1.757670 18.22%
0.000 1.407780 1.407780 0.00% 1.648480 1.830280 18.18%
0.050 1.376900 1.376900 0.00% 1.599440 1.767120 16.77%
0.150 0.969960 0.969960 0.00% 1.109010 1.212970 10.40%
0.250 0.000000 0.000000 0.00% 0.000000 0.000000 0.00%
Temperature
-0.250 -0.023580 -0.023580 0.00% 0.005940 0.028420 2.25%
-0.150 -0.014150 -0.014150 0.00% 0.013640 0.034770 2.11%
-0.050 -0.004720 -0.004720 0.00% 0.019390 0.037730 1.83%
0.000 0.000000 0.000000 0.00% 0.022080 0.038840 1.68%
0.050 0.004720 0.004720 0.00% 0.024730 0.039840 1.51%
0.150 0.014150 0.014150 0.00% 0.029720 0.041100 1.14%
0.250 0.023580 0.023580 0.00% 0.032630 0.038930 0.63%
Table 4: Values of velocity and Temperature for 0, 4, 1 T R  M  E  
y   0, 1 2 Bi  Bi 10   2, 1 2 Bi  Bi 10
-0.250 0.000000 0.000000 0.00% 0.000000 0.000000 0.00%
-0.150 0.927020 0.927020 0.00% 0.978550 0.985610 0.71%
-0.050 1.355770 1.355770 0.00% 1.432300 1.442850 1.06%
0.000 1.407780 1.407780 0.00% 1.487380 1.498360 1.10%

0.050 1.355770 1.355770 0.00% 1.432300 1.442850 1.06%
0.150 0.927020 0.927020 0.00% 0.978550 0.985610 0.71%
0.250 0.000000 0.000000 0.00% 0.000000 0.000000 0.00%
Temperature
-0.250 0.000000 0.000000 0.00% 2.325800 2.629610 30.38%
-0.150 0.000000 0.000000 0.00% 3.518370 3.993670 47.53%
-0.050 0.000000 0.000000 0.00% 3.704640 4.220740 51.61%
0.000 0.000000 0.000000 0.00% 3.712810 4.232370 51.96%
0.050 0.000000 0.000000 0.00% 3.704640 4.220740 51.61%
0.150 0.000000 0.000000 0.00% 3.518370 3.993670 47.53%
0.250 0.000000 0.000000 0.00% 2.325800 2.629610 30.38%
V. Conclusions
The problem of steady, laminar mixed convective flow in a vertical channel filled with electrically
conducting immiscible fluid in the presence of viscous and Ohmic dissipation is analyzed using Robin boundary
conditions. The governing equations were solved analytically using perturbation method valid for small values
of perturbation parameter and by differential transform method valid for all values of governing parameters.
The following conclusions were drawn.
[1] The flow at each position was an increasing function of  for upward flow and decreasing function of 
for downward flow.
[2] The Hartman number suppresses the flow for symmetric and asymmetric wall heating conditions for all the
governing parameters for both open and short circuits.
[3] Flow reversal was observed for asymmetric wall heating for equal Biot numbers and there is no flow
reversal for unequal Biot numbers.
[4] The viscosity ratio increases the flow increases in region-I and decreases in region-II for equal Biot
numbers. The width ratio and conductivity ratio suppress the flow in both the regions for equal Biot
numbers.
[5] The Nusselt number at the cold wall was increasing function of  and decreasing function of  at the
hot wall.
[6] The flow profiles for short circuit lie in between open circuit for positive and negative values of electric
field load parameter.
[7] The percentage of error between PM and DTM agree very well for small values of perturbation parameter.
[8] Fixing equal values for viscosity, width and conductivity for fluids in both the regions and in the absence of
Hartman number and electric field load parameter we get back the results of Zanchini [26] for one fluid
model.
VI. ACKNOWLEDGMENT:
One of the authors JPK greatly acknowledge the funding support from the Major Research project
supported by University Grants Commission India (Grant No. 41-774/2012 (SR)).
REFERENCES
[1] Sparrow EM, Cess, RD (1961) Effect of magnetic field on free convection heat transfer. Int J Heat
Mass Transfer 3: 267–274.
[2] Umavathi JC (1996) A note on magneto convection in a vertical enclosure. Int J Nonlinear Mechanics
31:371-376.
[3] Bhargava R, Kumar L, Takhar HS (2003) Numerical solution of free convection MHD micropolar fluid
flow between two parallel porous vertical plates. Int J Engng Sci 41: 123-136.
[4] Hayat T, Wang Y, Hutter K (2004) Hall effects on the unsteady hydromagnetic oscillatory flow if a
second-grade fluid. Int J Non-Linear Mech 39: 1027–1037.
[5] Umavathi JC, Liu IC, Prathap Kumar J, Pop I (2011) Fully developed magnetoconvection flow in a
vertical rectangular duct. Heat Mass Transfer 47: 1-11.
[6] Charles ME, Redburger (1962) On laminar two-phase flows in magnetohydrodynamics. Can J Chem
Eng 40: 70.
[7] Pacham BA, Shail R (1971) Stratified laminar flow of two immiscible fluids. Proc Camb Phil Soc 69:
443-448.
[8] Shail R (1973) On laminar Two-phase flow in magnetohydromagnetics. Int J Eng Sci 11:1103-1108.

[9] Postlethwaite AW, Sluyter MM (1978) Magnetohydrodynamic heat transfer problems and overview. Mech Eng 100: 29-32.
[10] Malashetty MS, Leela V (1992) Magnetohydrodynamic heat transfer in two phase flow. Int J Eng Sci 30: 371-377.
[11] Malashetty MS, Umavathi JC (1997) Two phase magnethydrodynamic flow and heat transfer in an inclined channel. Int J Multi Phase Flow 23: 545-560.
[12] Malashetty MS, Umavathi JC, Prathap Kumar J (2000) Two fluid magnetoconvectionflow in an inclined channel. Int J Transport phenomena 3: 73-84.
[13] Malashetty MS, Umavathi JC, Prathap Kumar J (2001) Convective magnetohydrodynamic two fluid flows and heat transfer in an inclined channel. Heat and Mass Transfer 37: 259-264.
[14] Malashetty MS, Umavathi JC, Prathap Kumar J (2006) Magnetoconvection of two immiscible fluids in vertical enclosure. Heat Mass Transfer 42: 977-993.
[15] Umavathi JC, Sridhar KSR (2009) Hartman two fluid Poisuille-Couette flow in an inclined channel. Int J Appl Mech Eng 14: 539-557.
[16] Umavathi JC, Liu IC, Prathap Kumar J (2010) Magnetohydrodynamic Poiseuille couette flow and heat transfer in an inclined channel. Journal of Mechanics 26.
[17] Umavathi JC, Prathap kumar J, Shekar M (2010) Mixed convective flow of immiscible viscous fluids confined between a long vertical wavy wall and a parallel flat wall. International journal of engineering science and technology 2:256-277.
[18] Prathap kumar J, Umavathi JC, Basavaraj M Biradar (2011) Mixed convection of magneto hydrodynamic and viscous fluid in a vertical channel. International Journal of Non-Linear Mechanics 46: 278–285.
[19] Prathap kumar J, Umavathi JC, Basavaraj M Biradar (2012) Two-fluid mixed magnetoconvection flow in a vertical enclosure. Journal of Applied Fluid Mechanics 5:11-21.
[20] Ingham DB, Keen DJ, Heggs PJ (1988) Flows in a vertical channel with asymmetric wall temperatures and including situations where reverse flows occurs. ASME J Heat Transfer 110: 910- 917.
[21] Barletta A (1997) Fully developed laminar forced convection in circular ducts for power law fluids with viscous dissipation. Int J Heat and Mass Transfer 40:15-26.
[22] Zanchini E (1997) Effect of viscous dissipation on the asymptotic behavior of laminar forced convection in circular tubes. Int J Heat and Mass Transfer 40:169-178.
[23] Aung W, Worku G (1986) Theory of fully developed combined convection including flow reversal. ASME Journal of Heat Transfer 108:485–588.
[24] Cheng CH, Kou HS, Huang WH (1990) Flow reversal and heat transfer of fully developed mixed convection in vertical channels. Journal of Thermo physics and Heat Transfer 4: 375–383.
[25] Barletta A (1998) Laminar mixed convection with viscous dissipation in a vertical channel. Int J Heat and Mass Transfer 41:3501–3513.
[26] Zanchini E (1998) Effect of viscous dissipation on mixed convection in a vertical channel with boundary conditions of the third kind. Int J Heat and Mass Transfer 41:3949–3959.
[27] Stephan K (1959) Warmenbergang a Druckabfall Bei nicht ausgebildeter laminarstromung in Rohren und in ebenen spalten. Chemie Ingeniear Technik 31:773–778.
[28] Hwang CL, Fan LT (1964) Finite difference analysis of forced convection heat transfer in entrance region of a flat rectangular duct. Applied Scientific Research 13: 401–422.
[29] Siegal R, Sparrow EM (1959) Simultaneous development of velocity and temperature distributions in a flat duct with uniform wall heating. American Institute of Chemical Engineers Journal 5:73–75.
[30] Arikoglu A, Ozkol I (2006) Solution of differential-difference equations by using differential transforms method. Appl Math Comp 181: 153–62.
[31] Biazar J, Eslami M (2010) Analytic solution for Telegraph equation by differential transform method . Physics Letter A 374: 2904–2906.
[32] Zhou JK (1986) Differential transformation and its applications for electrical circuits (in Chinese) Huazhong University Press, Wuhan, China.
[33] Rashidi MM, Laraqi N, Sadri SM (2010) A novel analytical solution of mixed convection about an inclined flat plate embedded in a porous medium using the DTM-Padé. International Journal of Thermal Sciences 49: 2405–2412.
[34] Ganji DD, Rahimi M, Rahgoshay M, Jafari M (2011) Analytical and numerical investigation of fin efficiency and temperature distribution of conductive, convective and radiative straight fins. Heat Transfer – Asian Research 40: 233-245.

[35] Umavathi JC, Santhosh Veershetty (2012a) Non-Darcy mixed convection in a vertical porous
channel with boundary conditions of third kind. Trans Porous Med 95:111–131.
[36] Umavathi JC, Santhosh Veershetty (2012b) Mixed convection of a permeable fluid in a vertical
channel with boundary conditions of a third kind. Heat Transfer Asian Research 41.
[37] Umavathi JC, Jaweria Sultana (2011) Mixed convection flow of conducting fluid in a vertical channel
with boundary conditions of third kind in the presence of heat source/sink. Int J Energy & Technology
3:1–10.
-0.2 -0.1 0.0 0.1 0.2
-0.6
0.0
0.6
1.2
1.8
2.4
Region-I Region-II
1000
 = 0
u
y
Figure 2. Plots of u versus y for different values of 
M = 2, R
T
= 1, Br = 0
Bi
1
= Bi
2
= 10, Er = 1
r = 1
E = -1
E = 0
E = 1
-0.2 -0.1 0.0 0.1 0.2
0
1
2
3
4
5
-0.2 -0.1 0.0 0.1 0.2
0
2
4
6
8
10
E = -1
E = 0
E = 1
2
Br = 0

Figure 3a. Plots of  versus y for different
values of Br
y
M = 2,  = 0,r = 1, Er = 1
R
T
= 1, Bi
1
= Bi
2
= 10
Region-I Region-II
Figure 3b. Plots of  versus y for different
values of Br
E = -1
E = 0
E = 1
Br = 0
2
y

Region-I Region-II
M = 2,  = 0, r = 1
Er = 1,R
T
= 1, Bi
1
= 1
Bi
2
= 10

-0.2 -0.1 0.0 0.1 0.2
0
1
2
-0.04 -0.02 0.00 0.02 0.04
-0.08
-0.04
0.00
0.04
0.08
0.12
-0.2 -0.1 0.0 0.1 0.2
-0.4
-0.2
0.0
0.2
0.4
Region-I Region-II
Figure 4b. Plots  versus y for different
values of 
PM
DTM
-4
-2
 = 0
4
2
 = 0
 = -500
 = 500
M= 4, E= -1, R
T
= 1
Bi
1
= Bi
2
= 10,
Figure 4a. Plots u versus y for different
values of 
u
y
 = 0, + 2, + 4
PM
DTM
 = 0, + 2, + 4
Region-I Region-II
M = 4, E = -1, R
T
= 1
Bi
1
= Bi
2
= 10

y
-0.2 -0.1 0.0 0.1 0.2
-0.5
0.0
0.5
1.0
1.5
2.0
-0.2 -0.1 0.0 0.1 0.2
-0.4
-0.2
0.0
0.2
0.4
values of M
E = -1
E = 0
E = 1
Figure 5a. Plots of u versus y for different
values of M
 = 500, Bi
1
= Bi
2
= 10
R
T
= 1,  = 0.1,r = Er = 1
Region-I Region-II
8
M = 4
u
y
M = 4,8
Region-I Region-II

 = 500, Bi
1
= Bi
2
= 10
R
T
= 1,  = 0.1,r = Er = 1
E = -1
E = 0
E = 1
y

-0.2 -0.1 0.0 0.1 0.2
-0.5
0.0
0.5
1.0
1.5
2.0
-0.2 -0.1 0.0 0.1 0.2
-0.4
-0.2
0.0
0.2
0.4
E = 1
E = 0
E = -1
3
m= 1
2
Region-I Region-II
values of m
Bi
1
= Bi
2
= 10, R
T
= 1
M= 4,  = 500,  = 0.1
 r = Er = 1
u
y
Bi
1
= Bi
2
= 10, R
T
= 1
M= 4,  = 500,  = 0.1
 r = Er = 1
E = 1
E = 0
E = -1
Region-I
y
Region-II
m= 1,2,3

values of m
-0.2 -0.1 0.0 0.1 0.2
-2
-1
0
1
2
-0.2 -0.1 0.0 0.1 0.2
-0.4
-0.2
0.0
0.2
0.4
E = 1
E = 0
E = -1
Region-I Region-II
3
2
h=1
values of h
 = 500,  = 0.1, R
T
= 1
Bi
1
= Bi
2
= 10, M = 4
 r = Er = 1
u
y
 = 500,  = 0.1, R
T
= 1
Bi
1
= Bi
2
= 10, M = 4
 r = Er = 1

E = 1
E = 0
E = -1
Region-I Region-II
values of h
3
2
h=1
y

-0.2 -0.1 0.0 0.1 0.2
-0.4
-0.2
0.0
0.2
0.4
-0.2 -0.1 0.0 0.1 0.2
-2
-1
0
1
2
E = 1
E = 0
E = -1
Bi
1
= Bi
2
= 10, R
T
= 1
M= 4,  = 500,  = 0.1
 r = Er = 1
Region-I Region-II
3
2
k = 1

values of k
y
Bi
1
= Bi
2
= 10, R
T
= 1
M= 4,  = 500,  = 0.1
 r = Er = 1
E = 1
E = 0
E = -1
3
2
k = 1
Region-I Region-II
values of k
u
y
-0.2 -0.1 0.0 0.1 0.2
0.0
0.4
0.8
1.2
1.6
2.0
-0.2 -0.1 0.0 0.1 0.2
-0.02
0.00
0.02
0.04
0.06
values of 
-2,  - 500
0,
 - 500
 = 2,  = 500
E = -1
E = 0
E = 1
Region-I Region-II
0,  500
M = 4, R
T
= 1,r = 1, Er = 1
Bi
1
= 0.1, Bi
2
= 10
values of 
y
u
-2,  -500
2,  500
0,  +500
M = 4, R
T
= 1,r = 1,
Er = 1, Bi
1
= 0.1, Bi
2
= 10
E = -1
E = 0
E = 1
Region-I Region-II
y


-0.2 -0.1 0.0 0.1 0.2
0.0
0.4
0.8
1.2
1.6
2.0
-0.2 -0.1 0.0 0.1 0.2
0
2
4
6
8
10
12
14
16
 = 4
 = 0
M = 4, R
T
= 0,  = 500
Bi
1
= Bi
2
= 10,  r = 1
Er = 1
values of 
Region-I Region-II
E = -1
E = 0
E = 1
u
y
M = 4, R
T
= 0,  = 500
Bi
1
= Bi
2
= 10,  r = 1
Er = 1
 = 4
 = 0
E = -1
E = 0
E = 1
Region-I Region-II
values of 

y
-0.2 -0.1 0.0 0.1 0.2
0
5
10
15
20
25
30
-0.2 -0.1 0.0 0.1 0.2
0.0
0.4
0.8
1.2
1.6
2.0
M = 4, R
T
= 0,  = 500
Bi
1
= 0.1, Bi
2
= 10
 = 2
 = 0
Region-I Region-II
values of 
E = -1
E = 0
E = 1

y
 = 2
 = 0
E = -1
E = 0
E = 1
Region-I Region-II
M = 4, R
T
= 0,  = 500
Bi
1
= 0.1, Bi
2
= 10, r = 1
Er = 1
values of 
u
y

1 2 3 4 5
4
5
6
1 2 3 4 5
1
2
3
4
Figure 12. Plots of Nu
1
and Nu
2
versus | for different values of 
R
T
= 1, Bi
1
= Bi
2
= 10
M = 4
 = 100
 = 500
 = -300 E = 1
E = 0
E = -1
Nu
1
 = 100
 = 500
 = -300
R
T
= 1, Bi
1
= Bi
2
= 10
M = 4
| |
E = 1
E = 1
E = 0
Nu
2
E = -1
E = -1
E = -1
NOMENCLATURE:
A constant used in equation (17)
Bi1, Bi2 Biot number   i i i q D k
0 B applied magnetic field
Br Brinkman number
    2
1
1 0 1  U k T
0 E dimensional applied electric field
E dimensionless electric field load parameter
    1
0 0 0 E B U
p c specific heat at constant pressure
g acceleration due to gravity
Gr Grashof number   3 2
1 1 1 g h T 
k ratio of thermal conductivities   1 2 k k
h width ratio   2 1 h h
M Hartman number   2 2
e 0 1 1  B D 
1 2 Nu , Nu Nusselt numbers
p non-dimensional pressure gradient
P difference between pressure and hydrostatic pressure

1 2 q ,q external heat transfer coefficients
Re Reynolds number
    1
1 0 1 DU 
T R temperature difference ratio
T temperature
1 2
, q q T T reference temperatures of the external fluid
0 T reference temperature
i U velocity component in the X -direction
 
0
i U reference velocity   2 48 i i  AD 
i u dimensionless velocity in the X -direction
X stream wise coordinate
x dimensionless stream wise coordinate
Y transverse coordinate
y dimensionless transverse coordinate
GREEK SYMBOLS
1  , 2  thermal diffusivities in region-I and region-II
1  , 2  thermal expansion coefficient in region-I and region-II
T reference temperature difference   q2 q1 T T
 perturbation parameter
1 2  , dimensionaless temperatures in region-I and region-II
1 2  , viscosities of the fluids in region-I and region-II
1 2  , kinematic viscosities of the fluids in region-I and region-II
e1  , e2  electrical conductivities of the fluid in region-I and region-II
 r ratio of electrical conductivities   e2 e1  
1 2  , density of fluids in region-I and region-II
 dimensionless mixed convection parameter Gr Re
SUBSCRIPTS
1 and 2 reference quantities for Region-I and II, respectively.
APPENDIX:
2 2 2 2 2 2 2 2 2 2
1 2 3 4 5 , , , , , r r r r A  M h  m A  h bnk A  mh A  E E M A  M h  m
2 2
2 2 2 2
6 7 8
2
2 , , , r
r r r
M h m
A M h EE m A A M h m
Bi

      
2 2 2 2
2 9 10
2
4
(1 ) 48 , , ,
2
T
r r r
bnR s
BC E EM A mh A M h m
Bi
 

     
2 2
2
11 12 13 1
48 1
48 , , , [ 4] r r r E EM M h m
A M E A A cm Cosh M
nb nb nbkh nbk
 
       
1 1 8
1 2 2 1 1 2 7 [ 4], cosh( ), sinh( ), ,
4 4 4
p p A
sm  Sinh M cm  sm  p  A p  A 

3 3
1 2 2 1 2 2
3 1 2 7 1 2 8 2 4 1 2 7 1 2 8 2
2 2
, ,
p sm p cm
p p cm A p sm A cm p p sm A p cm A sm
Bi Bi
       
2
1 10 10 3 1
5 6 7 12 1 13 1 8 9 2
1
1 1
, , , ( ), ,
4
p A A BC
p p p A p A p p p
bn bn bn Bi M
         
13 7
10 8 11 2 12 2 13 11 14 12 1
1
, , , , ,
4
A p
p p p p p p h p p p h
M M
          
14
15 16 17 10 11 15 10 12 18 10 11 16 2
13 13
, , , ,
p M
p p p p p p p p p p p p sm
p p
       
15
19 2 20 2 15 4 21 8 11 15 8 12 22 8 11 16 , , , ,
4
p
p   sm p  p p  p p  p p p  p p p  p p p
2 2 9 17 18
23 8 11 15 24 8 11 16 25 26 27
1 1 1
, , , , ,
p p p
p M p p p p M p p p p p p
cm cm cm
       
2
11 9 23
28 21 26 29 22 27 30 9 25 31 32
6 6
, , , , ,
A M p p
p p p p p p p p p p p
p p

        
24 16
33 35 32 19 36 33 37 34 2 38 8 32 20
6
, , , , ,
4
p p
p p p p p p p p cm p A p p
p
         
39 8 33 2 16 40 8 34 3 41 8 31 2 42 28 9 32 p  A  p  p p , p  A p  p , p  A p  BC , p  p  A p ,
45 44
43 29 9 33 44 9 34 9 45 30 9 31 46 47
43 43
, , , , ,
p p
p p A p p A p A p p A p p p
p p
          
42
48 49 31 36 46 50 35 36 48 51 37 36 47
43
, , , ,
p
p p p p p p p p p p p p p
p
       
2
52 38 39 48 53 39 47 40 54 39 46 41 1 p  p  p p , p  p p  p , p  p p  p , f  2M E,
49 53 51 54 49 50
8 7 8 4 46 47 7 48 8
50 53 51 52 51 51
, , ,
p p p p p p
z z z z p p z p z
p p p p p p

       

5 31 32 8 33 4 34 7 3 25 26 8 27 4 6 15 8 16 4 z  p  p z  p z  p z , z  p  p z  p z , z  p z  p z ,
2 11 6 12 8 1 9 2 8 z  p z  p z , z  p  z p ,
2 2 2 2 2 2 2
1 3 4 2 3 4 3 2 4 1 3 1 3 4 3 4 k  z M  z M , k  2z z M , k  2z z M  2M z z  f z , k  2z z M ,
2 2 2 2 2
5 3 2 1 4 1 4 6 2 4 7 2 8 1 2 1 2 k  2z z M  2z z M  f z , k  2z z M , k  M z , k  2z z M  f z ,
2 2 2 2 2 1 2 3 6
9 2 1 1 1 10 4 11 4 12 3 4
5
, , , ,
12 12 2 4
k k k k
k z M E M z f z k k k
M M M M
       
5 4 6 4 7 8
13 3 4 14 3 15 3 16 2 17 2
5
, , , , ,
2 4 4 4 12 6
k k k k k k
k k k k k
M M M M M M
       
7 9 2 2 2
18 4 2 19 5 7 5 8 20 3 7 8 1 ( ), , 2 ,
2
k k
k k A z A z k A z z p
M M
     
21 3 6 8 1 5 5 7 6 7 22 5 7 6 23 3 6 7 1 5 5 8 6 8 k  2A z z p  2A z z  A z , k  2A z z , k  2A z z p  2A z z  A z ,
2 2 2 19
25 5 6 26 5 5 6 6 6 27 3 6 4 5 5 6 5 28 4
1
, 2 , , ,
12
k
k A z k A z z z A k A z A A z A z k
p
       
20 21 24 23 22 24 22
29 4 30 3 4 31 3 4 32 3 33 3
1 1 1 1 1 1 1
5 5
, , , , ,
12 2 4 2 4 4 4
k k k k k k k
k k k k k
p p p p p p p
      

25 26 25 27
34 2 35 2 36 4 2
1 1 1 1
, , ( ),
12 6 2
k k k k
k k k
p p p p
      
12 13 14 15 16 17 18
2 10 11 1 1 1 1 4 cosh( ) sinh( ) ,
2 2 4 4 16 16 4 64 16
M M k k k k k k k
f  k  k  sm  cm  cm  sm   
1 1 30 31 32 33 34 35 36
3 2 28 29 2 2 2 2 4 [ cosh( ) sinh( ) ],
2 2 4 4 16 16 4 64 16
p p k k k k k k k
f  A k  k  sm  cm  cm  sm   
2 2 2 2 2 2
16 16 17 17 16 16 17 17 18 18
4 18
1 1 1 1 1
2 2
15 1 13 1 14 1 2
14 1 14 1 12 1 10
1 1 1
2
11
15 1
1
3 6 3 6 3
2
4 2 256 16 64 16 16 2
6 2
2 2 2 3 cosh( )
2
6
cosh( ) 2 2
2
k k k k k M k M k M k M k M k M
f k
Bi Bi Bi Bi Bi
k Mcm k M cm k M cm M
k cm k cm k Mcm k M
Bi Bi Bi
k M M
k sm k
Bi
           
      
  
2 2
14 12 15
13 1 14 1 1 1 1
1 1 1
2 2
2
11 5
1
6 2
3 sinh( ),
2 4
k M k M k M
Msm k Msm sm sm sm
Bi Bi Bi
M M M
k M ff
Bi
   
 
    
 
34 35 36 31 32 1 30 33 1
5 2 8 2 2 28 2 29
34 35 36 32 30 1 33 1 1
2 7 31 2 29 1
33 31 1 32 1 1
30 2 28 1
[ cosh( ) ( ) sinh( )]
256 64 16 4 16 2 4 16 2
3
[ ( ) 2 cosh( )
16 16 2 2 4 16 2
( ) 2 sinh( )]
2 4 16 2
k k k k k p k k p
f A A cm cm k sm k
k k k k k p k p p
A A k cm k p
k k p k p p
k sm k p
        
       
     34 35
2 36 32 30 1
2
33 1 2 1 30 2 33 2
33 1 2 28 1 33 31 1 32 1 1 1 2
2 3 3
2 1 2 32 1 30 1 33 1
29 1 2 34 35 33 1 31 1 2
2
3
29 1
3 3
[ 2 (2 2
4 2
) 4 cosh( ) (2 2 )
16 2 4 16
1
4 sinh( )] [ [6 6 (6 3 3 )
2 2 4 16
8 c
k k
A k k k p
k p p k k
k p cm k p k k p k p p p sm
p k p k p k p
k p A k k k p k p cm
Bi
k p
    
        
       
3 3
1 2 33 2 31 1 32 1 3 1
32 1 30 1 1 2 28 1
3
osh( ) (6 3 ) 8 sinh( )]]
2 2 4 16 2
p k k p k p p
 k p  k p  p   sm  k p
3 3
8 1 2 2 1
6 7 7 7 1 2 8 1 2 8 8 1 2 7 1 2
2 2
, ( ) ( ) , ( ) ( )
4
A p p
f A f A p cm A p sm f A p cm A p sm
Bi Bi
         
2
9 10 2 9 28 10 13 11 2 31 29 1 11 14 18 12 10
2
10 2 28 2 2 10 10 1
32 36 30 1 28 1 12 13
, 2 ( 2 ), 2 2 2 3
(2 2 2 4 ), , ,
f k A A k f k k M A h k k p f k k k M k M
A A k A A A p
k k k p k p f f
bn bn bn bn bn
         
      
2 3 2
14 17 15 13 11 13 2 31 29 1 12 2 35 33 1 31 1
3 3 4 5
29 1 15 13 1 12 1 16 2 17 2 18 17 19 2 16
6 6 2 6 ( 2 ) (6 6 3
1
8 ), , , , ,
4
f k k M k M k M A A k k p A A k k p k p
f ff
k p f A p A p f f f f f f f
M M
        
           
2 2 1 1 19
20 9 16 21 11 16 22 17 23 24 25
18 18 18
, , , , , ,
cm sm f
f f f f f M f f M f f f f
f f f
          
26 17 23 27 17 25 20 28 10 25 29 24 30 22 25 21 f 1 f f , f  f f  f , f  f  f , f  f M, f  f f  f ,
2 2 2 8
31 22 23 32 22 24 33 14 25 34 23 35 24 36 6 , , , , ,
4
A
f  f f f  f f f  f M f f  M f f  M f f    f
37 8 2 8 38 8 2 7 39 8 3 5 40 9 2 9 41 9 2 f  A cm  f , f  A sm  f , f  A f  f , f  A cm  A , f  A sm ,
12
42 17 24 43 9 3 27 44 30 12 3 45 46 12 2 13 , , , , ,
4
f
f  f f f  A f  f f  f  f f f   f   f cm  f

37 38 39 9 48 9 49
47 12 2 48 49 50 51 40 52 41
36 36 36
, , , , , ,
4 4
f f f A f A f
f f sm f f f f f f f
f f f
           
9 50
53 43 54 28 50 55 48 56 49 1 57 44 45 50 , , , , ,
4
A f
f   f f  f  hf f  hf f  hf  hp f  f  f f
58 46 45 48 59 47 45 49 60 33 13 50 61 13 48 62 15 13 49 f  f  f f , f  f  f f , f  f  A f , f  A f , f  f  A f ,
51 52 53 42
63 64 65 66 67 54 23 65 68 55 23 63
26 26 26 26
, , , , , ,
f f f f
f f f f f f f f f f f f
f f f f
           
69 56 23 64 70 29 23 66 71 57 31 65 72 58 31 63 f  f  f f , f  f  f f , f  f  f f , f  f  f f ,
67 68 69
77 62 34 64 78 35 34 66 79 80 81
70 70 70
, , , , ,
f f f
f f f f f f f f f f f
f f f
         
  84 16 82 83 85 82 86
86 76 78 80 87 77 78 81 15 16
83 83 84 86 83 87
, , , ,
f z f f f f f
f f f f f f f f z z
f f f f f f

       

12 79 80 15 81 16 11 65 63 15 64 16 66 12 14 50 48 15 49 16 z  f  f z  f z , z  f  f z  f z  f z , z  f  f z  f z ,
14
13 3 2 15 2 16 10 25 23 11 24 12 9 16 17 10 , ,
4
z
z f cm z sm z z f f z f z z f f z
 
          
 

Magneto-Convection of Immiscible Fluids in a Vertical Channel Using Robin Boundary Conditions

Recommended

Recommended

More Related Content

What's hot

What's hot (18)

Viewers also liked

Viewers also liked (20)

Similar to Magneto-Convection of Immiscible Fluids in a Vertical Channel Using Robin Boundary Conditions

Similar to Magneto-Convection of Immiscible Fluids in a Vertical Channel Using Robin Boundary Conditions (20)

Recently uploaded

Recently uploaded (20)

Magneto-Convection of Immiscible Fluids in a Vertical Channel Using Robin Boundary Conditions