TWO FLUID ELECTROMAGNETO CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN VERTICAL WAVY WALL AND A PARALLEL FLAT WALL

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online), Volume 6, Issue 4, April (2015), pp. 86-106 © IAEME
86
TWO FLUID ELECTROMAGNETO CONVECTIVE FLOW
AND HEAT TRANSFER BETWEEN VERTICAL WAVY
WALL AND A PARALLEL FLAT WALL
Mahadev M Biradar
Associate Professor, Department of Mathematics,
Basaveshwar Engineering College (Autonomous) Bagalkot Karnataka INDIA 587103
ABSTRACT
The mixture of viscous and magneto convective flow and heat transfer between a long
vertical wavy wall and a parallel flat wall in the presence of applied electric field parallel to gravity ,
magnetic field normal to gravity in the presence of source or sink is investigated. The non-linear
equations governing the flow are solved using the linearization technique. The effect of Grash of
number and width ratio is to promote the flow for both open and short circuits. The effect of
Hartmann number is to suppress the flow, the effect of source is to promote and the effect of sink is
to suppress the velocity for open and short circuits. Conducting ratio decreases the temperature
where as width ratio increases the temperature.
Keywords: Convection, Vertical Wavy, Electrically Conducting Fluid
1. INTRODUCTION
From a technological point of view the free convection study of fluids with known physical
properties, which are confined between two parallel walls, is always important, for it can reveal
hitherto-unknown properties of fluids of practical interest. In such buoyancy driven flows the exact
governing equations are unwieldy so that recourse to approximations such as Boussinesq’s is
generally called for. Assuming the linear density temperature variation (Boussinesq’s approximation)
Ostrach [1] studied the laminar natural convection heat transfer in fluids with and without heat
sources, in channels with constant wall temperature.
Previous studies of natural convection heat and mass transfer have focused mainly on a flat
plate or regular ducts. Botfemanne [2] has considered simultaneous heat and mass transfer by free
convection along a vertical flat plate only for steady state theoretical solutions with Pr = 0.71 and Sc
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING
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ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 6, Issue 4, April (2015), pp. 86-106
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87
= 0.63. Callahan and Marner [3] studied the free convection with mass transfer on a vertical flat plate
with Pr = 1 and a realistic range of Schmidt number.
It is necessary to study the heat and mass transfer from an irregular surface because irregular
surfaces are often present in many applications. It is often encountered in heat transfer devices to
enhance heat transfer. For examples, flat-plate solar collectors and flat-plate condensers in
refrigerators. The natural convection heat transfer from an isothermal vertical wavy surface was first
studied by Yao [4, 5, 6] and using an extended Prantdl’s transposition theorem and a finite-difference
scheme. He proposed a simple transformation to study the natural convection heat transfer from
isothermal vertical wavy surfaces, such as Sinusoidal surface.
The fluid mechanics and the heat transfer characteristics of the generator channel or
significantly influenced by the presence of the magnetic field. Shail [7] studied the problem of
Hartmann flow of a conducting fluid in a horizontal channel of insulated plates with a layer of non-
conducting fluid over lying a conducting fluid in two fluid flow and predicted that the flow rate can
be increases to the order of 30% for suitable values of ratio of viscosities, width of the two fluids
with appropriate Hartmann number.
Hartmann [8] carried out the pioneer work on the study of steady magnetohydrodynamic
channel flow of a conducting fluid under a uniform magnetic field transverse to an electrically
insulated channel wall. Later Osterle and Young [9] investigated the effect of viscous and Joule
dissipation on hydromagnetic force convection flows and heat transfer between two vertical plates
with transverse magnetic field under short circuit condition. Poots [10] treated the open circuit case
with and without heat sources, the two bounding plates being maintained at different temperatures,
Umavathi [11] studied the effect of viscous and ohmic dissipation on magnetoconvection in a
vertical enclosure for open and short circuits, recently Malashetty et.al. [11] studied the
magnetoconvection of two-immiscible fluids in vertical enclosure.
Electric current in semi-conducting fluids such as glass and electrolyte generate Joulean
heating Song [12], other application include those involving exothermic and endothermic chemical
reaction and dealing with dissociating fluids Vajravelu and Nayefeh [13].
All the reference mentioned above dealt with vertical parallel walls. Since flow past wavy
boundaries are encountered in many situation such as the rippling melting surface, physiological
applications and many more as mention above, motivated us to study flow and heat transfer of the
mixture of viscous and electrically conducting fluid in the presence of applied electric and magnetic
fields between vertical wavy walls and a parallel flat wall.
2. MATHEMATICAL FORMULATION
Fig. 1. Physical Configuration

88
Consider the channel as shown in Fig. 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 KXcos*hY )(
ε+−= 1
and the flat wall by )(
hY 2
= . The region ( )1
0 y h≤ ≤ is
occupied by viscous fluid of density 1ρ , viscosity 1µ , thermal conductivity 1k and the region
( )2
0h y≤ ≤ is occupied another viscous fluid of density 2ρ , viscosity 2µ , thermal conductivity 2k .
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
ρ µ ρ
 ∂ ∂ ∂
+ = − + ∇ − 
∂ ∂ ∂ 
(1)
( )1(1) (1)
(1) (1) (1) (1) 2 (1)V V P
U V V
X Y Y
ρ µ
 ∂ ∂ ∂
+ = − + ∇ 
∂ ∂ ∂ 
(2)
0
11
=
∂
∂
+
∂
∂
Y
V
X
U )()(
(3)
( )
(1) (1)
1(1) (1) (1) (1) 2 (1)
1p
T T
C U V k T Q
X Y
ρ
 ∂ ∂
+ = ∇ + 
∂ ∂ 
(4)
Region –II
( )2(2) (2)
(2) (2) (2) (2) 2 (2) (2) 2 (2)
0 0 0
U U P
U V U g B U B E
X Y X
ρ µ ρ σ σ
 ∂ ∂ ∂
+ = − + ∇ − − − 
∂ ∂ ∂ 
(5)
( )2(2) (2)
(2) (2) (2) (2) 2 (2)V V P
U V V
X Y Y
ρ µ
 ∂ ∂ ∂
+ = − + ∇ 
∂ ∂ ∂ 
(6)
0
22
=
∂
∂
+
∂
∂
Y
V
X
U )()(
(7)
( )
(2) (2)
2(2) (2) (2) (2) 2 (2)
2p
T T
C U V k T Q
X Y
ρ
 ∂ ∂
+ = ∇ + 
∂ ∂ 
(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
pC = ( )2
pC

89
The physical hydro dynamic conditions are
(1)
0U = (1)
0V = , at (1) *
Y h cosKXε= − +
(2)
0U = (2)
0,V = at (2)
Y h=
(1) (2)
U U= (1) (2)
,V V= at 0Y =
( ) ( )2
)2(
1
)1(
X
V
Y
U
X
V
Y
U






∂
∂
+
∂
∂
=





∂
∂
+
∂
∂
µµ at 0Y = (9)
The boundary and interface conditions on temperature are
(1)
WT T= at (1) *
Y h cosKXε= − +
(2)
1T T= at (2)
Y h=
(1) (2)
T T= at 0Y =
( ) ( )1 2
(1) (2)T T T T
k k
Y X Y X
∂ ∂ ∂ ∂   
+ = +   
∂ ∂ ∂ ∂   
at 0Y = (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 (1) to (8) are made dimensionless using the following transformations
( ) ( ) )1(
)1(
)1(
Y,X
h
1
y,x = ; ( ) ( ) )1(
)1(
)1(
)1(
V,U
h
v,u
ν
=
( ) ( ) )2(
)2(
)2(
Y,X
h
1
y,x = ; ( ) ( ) )2(
)2(
)2(
)2(
V,U
h
v,u
ν
=
( )
( )SW
s
)1(
)1(
TT
TT
−
−
=θ ;
( )
( )SW
s
)2(
)2(
TT
TT
−
−
=θ
(1)
(1)
2(1)
(1)
P
P
h
ν
ρ
=
 
 
 
;
(2)
(2)
2(2)
(2)
P
P
h
ν
ρ
=
 
 
 
(11)
where Ts is the fluid temperature in static conditions.
Region –I
( ) (1)1(1) (1) 2 2 (1)
(1) (1) (1)
2 2
u u P u u
u v G
x y x x y
θ
∂ ∂ ∂ ∂ ∂
+ = − + + +
∂ ∂ ∂ ∂ ∂
(12)
( ) (1)1(1) (1) 2 2 (1)
(1) (1)
2 2
v v P v v
u v
x y y x y
∂ ∂ ∂ ∂ ∂
+ = − + +
∂ ∂ ∂ ∂ ∂
(13)

90
0
y
v
x
u )1()1(
=
∂
∂
+
∂
∂
(14)
PryxPry
v
x
u
)()()(
)(
)(
)( αθθθθ
+








∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
2
121
2
21
1
1
1 1
(15)
Region –II
( ) (2)2(2) (2) 2 2 (2)
(2) (2) 2 2 3 (2) 2 2 (2) 2 3 2
2 2
u u P u u
u v m r h G M mh u m h rM E
x y x x y
β θ
∂ ∂ ∂ ∂ ∂
+ = − + + + − −
∂ ∂ ∂ ∂ ∂
(16)
( ) (2)2(2) (2) 2 2 (2)
(2) (2)
2 2
v v P v v
u v
x y y x y
∂ ∂ ∂ ∂ ∂
+ = − + +
∂ ∂ ∂ ∂ ∂
(17)
0
y
v
x
u )2()2(
=
∂
∂
+
∂
∂
(18)
Pr
Qmh
yxPr
km
y
v
x
u
)()()(
)(
)(
)( αθθθθ 2
2
222
2
22
2
2
2
+








∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
(19)
The dimensionless form of equation (9) and (10) using (11) become
(1)
0u = ; (1)
0v = at 1 cosy xε λ= − +
(2)
0u = ; (2)
0v = at 1y =
(1) (2)1
u u
rmh
= ; (1) (2)1
v v
rmh
= at 0y =
( ) ( )2
22
1
x
v
y
u
hrm
1
x
v
y
u






∂
∂
+
∂
∂
=





∂
∂
+
∂
∂
at 0y = (20)
(1)
1θ = at 1 cosy xε λ= − +
(2)
θ θ= at 1y =
(1) (2)
θ θ= at 0y =
( ) ( )21
xyh
k
xy 





∂
∂
+
∂
∂
=





∂
∂
+
∂
∂ θθθθ
at 0y = (21)
3. SOLUTIONS
The governing equations (12) to (15), (16) to (19) and (24) to (25) along with boundary and
interface conditions (20) to (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

91
Region-I
( ) ( )y,xu)y(uy,xu )1(
1
)1(
0
)1(
ε+=
( ) ( )y,xvy,xv )1(
1
)1(
ε=
( ) ( )(1) (1)(1)
0 1, ( ) ,P x y P x P x yε= +
( ) ( )y,x)y(y,x )1(
1
)1(
0
)1(
θεθθ += (22)
Region -II
( ) ( )y,xu)y(uy,xu )2(
1
)2(
0
)2(
ε+=
( ) ( )y,xvy,xv )2(
1
)2(
ε=
( ) ( )(2) (2)(2)
0 1, ( ) ,P x y P x P x yε= +
( ) ( )y,x)y(y,x )2(
1
)2(
0
)2(
θεθθ += (23)
where the perturbations ( ) ( ) ( )
1 1 1, ,
i i i
u v P and ( )
1
i
θ for 1,2i = are small compared with the mean or the
zeroth order quantities. Using equation (5.3.1) and (5.3.2) in the equation (12) to (15), (16) to (19)
and (24) to (25) become
Zeroth order:
Region-I
)1(
02
)1(
0
2
G
dy
ud
θ−= (24)
α
θ
−=2
1
0
2
dy
d
)(
(25)
Region-II
)2(
0
322232)2(
0
22
2
)2(
0
2
θβ GhrmEMrhmuMmh
dy
ud
−=− (26)
k
hQ
dy
d
)( 2
2
2
0
2
αθ −
= (27)
First order:
Region-I
( )1(1)(1) 2 (1) 2 (1)
(1) (1) (1)01 1 1 1
0 1 12 2
duu P u u
u v G
x dy x x y
θ
∂ ∂ ∂ ∂
+ = − + + +
∂ ∂ ∂ ∂
(28)
( )1(1) 2 (1) 2 (1)
(1) 1 1 1 1
0 2 2
v P v v
u
x y x y
∂ ∂ ∂ ∂
= − + +
∂ ∂ ∂ ∂
(29)

92
0
y
v
x
u
2
)1(
1
2)1(
1
=
∂
∂
+
∂
∂
(30)
(1)(1) 2 (1) 2 (1)
(1) (1) 01 1 1
0 1 2 2
Pr
d
u v
x dy x y
θθ θ θ ∂ ∂ ∂
+ = + 
∂ ∂ ∂ 
(31)
Region-II
( )2(2)(2) 2 (2) 2 (2)
(2) (2) 2 2 3 (2) 2 2 (2)01 1 1 1
0 1 1 12 2
duu P u u
u v m r h G mh M u
x dy x x y
β θ
∂ ∂ ∂ ∂
+ = − + + + −
∂ ∂ ∂ ∂
(32)
( )2(2) 2 (2) 2 (2)
(2) 1 1 1 1
0 2 2
v P v v
u
x y x y
∂ ∂ ∂ ∂
= − + +
∂ ∂ ∂ ∂
(33)
0
y
v
x
u
2
)2(
1
2)2(
1
=
∂
∂
+
∂
∂
(34)
(2)(2) 2 (2) 2 (2)
(2) (2) 01 1 1
0 1 2 2
Pr
d km
u v
x dy x y
θθ θ θ ∂ ∂ ∂
+ = + 
∂ ∂ ∂ 
(35)
where ( ) ( )( )1
1 0 SP P
C
x
∂ −
=
∂
and ( ) ( )( )2
2 0 SP P
C
x
∂ −
=
∂
taken equal to zero (see Ostrach, 1952). With
the help of (5.3.1) and (5.3.2) the boundary and interface conditions (20) to (21) become
( )(1)
0 1 0u − = at 1−=y
( )(2)
0 1 0u = at 1y =
)()(
u
rmh
u 2
0
1
0
1
= at 0=y
(1) (2)
0 0
2 2
1du du
dy dyrm h
= at 0=y (36)
( )(1)
0 1 1θ − = at 1−=y
( )(2)
0 1θ θ= at 1y =
)()( 2
0
1
0 θθ = at 0=y
(1) (2)
0 0d dk
dy h dy
θ θ
= at 0=y (37)
( )
( )
( )
1
(1) (1)0
1 11 ; 1 0
du
u Cos x v
dy
λ− = − − = ; at 1−=y
( ) ( )(2) (2)
1 11 0; 1 0u v= = ; at 1=y

93
)()(
u
rmh
u 2
1
1
1
1
= ; )()(
v
rmh
v 2
1
1
1
1
= ; at 0=y






∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
x
v
y
u
hrmx
v
y
u )()()()( 2
1
2
1
22
1
1
1
1 1
at 0=y (38)
( )
( )1
(1) 0
1 1
d
Cos x
dy
θ
θ λ− = − at 1−=y
( )(2)
1 1 0θ = at 1=y
)()( 2
1
1
1 θθ = at 0=y






∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
xyh
k
xy
)()()()( 2
1
2
1
1
1
1
1 θθθθ
at 0=y (39)
Introducing the stream function 1ψ defined by
y
u
)(
)(
∂
∂
−=
1
11
1
ψ
;
x
v
)(
)(
∂
∂
=
1
11
1
ψ
(40)
and eliminating ( )1
1P and ( )2
1P from equation (5.3.7), (5.3.8) and (5.3.11), (5.3.12) we get
Region-I
2 (1)3 (1) 3 (1) (1) 4 (1) 4 (1) 4 (1) (1)
(1) 01 1 1 1 1 1 1
0 2 3 2 2 2 4 4
2
d u
u G
x yx y x dy x y x y
ψ ψ ψ ψ ψ ψ θ ∂ ∂ ∂ ∂ ∂ ∂ ∂
+ − = + + − 
∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ 
(41)
(1)(1) (1) 2 (1) 2 (1)
(1) 01 1 1 1
0 2 2
Pr
d
u
x x dy x y
θθ ψ θ θ ∂ ∂ ∂ ∂
+ = + 
∂ ∂ ∂ ∂ 
(42)
Region-II
2 (2)3 (2) 3 (2) (2) 4 (2) 4 (2) 4 (2) (2) 2
(2) 2 2 3 2 201 1 1 1 1 1 1 1
0 2 3 2 2 2 4 4 2
2
d u
u m r h G mh M
x y x x dy x y x y y y
ψ ψ ψ ψ ψ ψ θ ψ
β
 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
+ − = + + − − 
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 
(43)
(2)(2) (2) 2 (1) 2 (1)
(2) 01 1 1 1
0 2 2
Pr
d km
u
x x dy x y
θθ ψ θ θ ∂ ∂ ∂ ∂
+ = + 
∂ ∂ ∂ ∂ 
(43)
We assume the stream function in the form
( ) ( ),yey,x xi
ψεψ λ
=1 ( ) ( ),ytey,x xiλ
εθ =1 (44)
From which we infer
( ) ( ),yuey,xu xi
11
λ
ε= ( ) ( ),yvey,xv xi
11
λ
ε= (45)
and using (5.3.20) in (5.3.16) and (5.3.19) we get

94
Region-I
( ) ( )
(1)2 (1) (1)
(1) 2 2 20
0 2
2iv d u dt
i u G
dydy
ψ λ ψ λ ψ ψ λ ψ λ ψ
 
′′ ′′− − − − − = 
 
(46)
(1)
2 0
0Pr
d
t t i u t
dy
θ
λ λ ψ
 
′′ − = + 
 
(47)
Region-II
( ) ( )
(2)2 (2) (2)
(2) 2 2 2 2 2 2 2 30
0 2
2iv d u t
i u mh M m r h G
ydy
ψ λ ψ λ ψ ψ ψ λ ψ λ ψ β
  ∂
′′ ′′ ′′− − − − − − = 
∂ 
(48)
(2)
2 0
0
Pr d
t t i u t
km dy
θ
λ λ ψ
 
′′ − = + 
 
(49)
Below we restrict our attention to the real parts of the solution for the perturbed quantities
( ) ( ) ( )
1 1 1, ,
i ii
uψ θ and ( )
1
i
v for 1,2i =
The boundary conditions (5.3.17) and (5.3.18) can be now written in terms of ( )
1
i
ψ as
( )1(1)
01
;
du
Cos x
y dy
ψ
λ
∂
= −
∂
(1)
1
0
x
ψ∂
=
∂
; at 1−=y
(2)
1
0;
y
ψ∂
=
∂
(2)
1
0
x
ψ∂
=
∂
; at 1=y
yrmhy
)()(
∂
∂
=
∂
∂ 2
1
1
1 1 ψψ
;
xrmhx
)()(
∂
∂
=
∂
∂ 2
1
1
1 1 ψψ
; at 0=y
2 (1) 2 (1) 2 (2) 2 (2)
1 1 1 1
2 2 2 2 2 2
1
x y rm h x y
ψ ψ ψ ψ ∂ ∂ ∂ ∂
− = − 
∂ ∂ ∂ ∂ 
at 0=y
( ) ( ) ( )
( ) ( ) ( )
2 (1) (1) 3 (1) 3 (1)
1 1 11 1 1 1
0 0 12 3
2 (2) (2) 3 (2) 3 (2)
2 2 23 2 21 1 1 1
0 0 12 3 2 3
1
y
y
u u Gr
xy x x y y
u u Gr h m r
rm h xy x x y y
ψ ψ ψ ψ
θ
ψ ψ ψ ψ
β θ
∂ ∂ ∂ ∂
− + + + −
∂ ∂ ∂ ∂
 ∂ ∂ ∂ ∂
= − + + + − 
∂ ∂ ∂ ∂ 
(50)
( )
( )1
(1) 0
cos
d
t x
dy
θ
λ= − at 1−=y
(2)
0t = at 1=y
( ) ( )1 2
t t= at 0=y
( ) ( ) ( ) ( )1 1 2 22 2 2 2
2 2 2 2
d t d t d t d tk
hdx dy dx dy
 
 + = +
 
 
at 0=y (51)

95
If we consider small values of λ then substituting
( )
( ) ( ) ( ) ( )2
0 1 2,
i i i i
yψ λ ψ λψ λ ψ= + + + − − − −
( )
( ) ( ) ( ) ( )2
0 1 2,
i i i i
t y t t tλ λ λ= + + + − − − − for 1,2i = (52)
In to (5.3.26) to (5.3.31) gives, to order of λ , the following sets of ordinary differential
equations and corresponding boundary and conditions
Region-I
4 (1) (1)
0 0
4
d dt
G
dydy
ψ
= (53)
2 (1)
0
2
0
d t
dy
= (54)
(1)2 24 (1) (1)
0 01 1
0 04 2 2
d d ud dt
i u G
dydy dy dy
ψψ
ψ
 
− − = 
 
, (55)
(1)2 (1)
01
0 0 02
Pr
dd t
i u t
dydy
θ
ψ
 
= + 
 
(56)
Region-II
4 (2) (2)
2 2 30 0
4
d dt
m r h G
dydy
ψ
β= (57)
2 (2)
0
2
0
d t
dy
= (58)
(2)2 24 (2) 2 (2) (2)
2 2 2 2 30 01 1 1
0 04 2 2 2
d d ud d dt
M mh i u m r h G
dydy dy dy dy
ψψ ψ
ψ β
 
− = − + 
 
(59)
(2)2 (2)
01
0 0 02
Pr dd t
i u t
km dydy
θ
ψ
 
= + 
 
(60)
with boundary conditions
( )1(1)
0 0d du
Cos x
dy dy
ψ
λ= ; (1)
0 0ψ = ; at 1−=y
(2)
0
0;
d
dy
ψ
= (2)
0 0ψ = at 1=y
(1) (2)
0 01d d
dy rmh dy
ψ ψ
= ; )()(
rmh
2
0
1
0
1
ψψ = at 0=y
2 (1) 2 (2)
0 0
2 2 2 2
1d d
dy rm h dy
ψ ψ
= ;
)()(
hrm
2
022
1
0
1
ψψ = at 0=y (61)

96
(1)
1
0
d
dy
ψ
= ; (1)
1 0ψ = at 1−=y
(2)
1
0;
d
dy
ψ
= (2)
1 0ψ = at 1=y
(1) (2)
1 11d d
dy rmh dy
ψ ψ
= ; (1) (2)
1 1
1
rmh
ψ ψ= at 0=y
( ) ( )
2 (1) 2 (2)
1 22 21 1
1 12 2 2 2
1d d
dy rm h dy
ψ ψ
λ ψ λ ψ+ = + ;
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
1 11 1 1 1(1) 2
0 0
2 22 2 2 2(2) 2 2 2 3
0 02 3
1
y y y yyy
y y y yyy
i u i u Gt
i u i u G m r h t
rm h
λ ψ λψ λ ψ ψ
λ ψ λψ λ ψ ψ β
− + − + −
= − + − + −
at 0=y (62)
( )
( )1
1 0
0
d
t
dy
θ
= − at 1−=y
( )2
0 0t = at 1=y
(1) (2)
0 0t t= at 0=y
( ) ( )
(1) (2)
1 20 0dt dtk
i t i t
dy h dy
λ λ
 
+ = + 
 
at 0=y (63)
(1)
1 0t = at 1−=y
(2)
1 0t = at 1=y
( ) ( )1 2
1 1t t= at 0=y
( ) ( )1 2
1 1dt dt
dy dy
= at 0=y (64)
Zeroth-order solution (mean part)
The solutions to zeroth order differential Eqs. (5.3.3) to (5.3.6) using boundary and interface
conditions (5.3.15) and (5.3.16) are given by
Region-I
21
2
2
3
1
)1(
0 AyAylylu +++=
21
)1(
0 CyC +=θ
Region-II
(2) 3
0 1 2 1 1 2cosh sinhu B ny B ny s y s y s= + + + +
43
)2(
0 CyC +=θ

97
The solutions of zeroth and first order of λ are obtained by solving the equation (5.3.33) to
(5.3.40) using boundary and interface conditions (5.3.41) and (5.3.42) and are given below
(1) 4 3 23 4
0 4 5 6
6 2
A A
l y y y A y Aψ = + + + +
(2) 2
0 3 4 5 6 4cosh sinhB ny B ny B y B s yψ = + + + +
(2)
0 7 8t C y C= +
( )(1) 10 9 8 7 6 5 4 3 27 8
1 20 21 22 23 24 25 26 9 10
6 2
A A
i l y l y l y l y l y l y l y y y A y Aψ = + + + + + + + + + +
( )(1) 7 6 5 4 3 2
1 8 9 10 11 12 13 6 8Prt i d y d y d y d y d y d y q y q= + + + + + + +
(
)
(2) 3 3 2
1 7 8 7 10 29 30 31
2
32 33 34 35 36
6 5 4 3 2
37 38 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 hny s ny
S y S y S y S y S y
ψ = + + + + + +
+ + + + + +
+ + + +
(
)
(2) 5 4
11 12 13 14 15 161
3 2
17 18 5 7
Pr
cosh sinh cosh sinht i f y ny f y ny f ny f ny f y f y
km
f y f y q y q
= + + + + +
+ + + +
The first order quantities can be put in the forms
xcosxsinu ri λψλψ ′−′=1
xcosxsinv ir λψλλψλ ′−′=1
xsintxcost ir λλθ −=1
Region-I
( )
( )( )
( )( )1 11
1 0 1 0 1cos sinr r i iu x xλ ψ λψ λ ψ λψ′ ′ ′ ′= − + + +
( )( )
( )( )1 1(1)
1 0 1 0 1cos sini i r rv x xλ λ ψ λψ λ λ ψ λψ= − + − +
( )( )
( )( )1 1(1)
1 0 1 0 1cos sinr r i ix t t x t tθ λ λ λ λ= + − +
Region-II
( )( )
( )( )2 2(2)
1 0 1 0 1cos sinr r i iu x xλ ψ λψ λ ψ λψ′ ′ ′ ′= − + + +
( )( )
( )( )2 2(2)
1 0 1 0 1cos sini i r rv x xλ λ ψ λψ λ λ ψ λψ= − + − +
( )( )
( )( )2 2(2)
1 0 1 0 1cos sinr r i ix t t x t tθ λ λ λ λ= + − +
(1)
0 5 6t C y C= +

98
Skin friction and Nusselt number
The shearing stress yxτ at any point in the fluid is given in nondimensional form, by
=





= yxyx
h
τ
υρ
τ 2
2
( ) ( ) ( )`yveiyueyu xixi
110
λλ
λεε +′+′=
yxτ at the wavy wall ( )xcoshy )(
λε+−= 1
and at the flat wall 1=y are given by
( )
( )
( )11(1)
0 10 Re i x
e u uλ
τ τ ε  ′′ ′= + + 
( ) ( )22(2)
10 Re i x
e uλ
τ τ ε  ′= +  
Where Re represent the real part of
( )1
1 2 3 10 4 3 2l l l Aτ = − + − +
( )2
1 2 1 20 sinh cosh 2B n n B n n s sτ = + + +
( ) ( ) ( )
( )( )11 1 1
0 0 0 11 cos cos sinr iu x u x xτ λ ψ λ λψ λ′ ′′ ′′ ′′= + + − +
( ) ( )
( )( )22 2
0 0 11 cos sinr iu x xτ ψ λ λψ λ′ ′′ ′′= + − +
The nondimensional Nusselt number is given by
( ) ( )0 1
i xd
Nu y e y
dy
λθ
θ ε θ′ ′= = +
At the wavy wall and flat fall wall Nu takes the form
( )
( )
( )11(1)
0 10 Re i x
Nu Nu e tλ
ε θ ′′ ′= + + 
( ) ( )22(2)
10 Re i x
Nu Nu e tλ
ε  ′= +  
where
( )1
1 10 2Nu d C= − +
( )2
2 30 2Nu f C= +
Pressure drop:
The fluid pressure ( ),P x y ( ( )0P x = constant) at any point ( ),x y as
( ),
P P
P x y dP dx dy
x y
∫ ∫
 ∂ ∂
= = + ∂ ∂ 

99
( )
( )
( )
exp
. , Re ii x
i e P x y L i z y
λ
ε
λ
 
− =  
 
(65)
where L is an arbitrary constant and
( ) ( ) ( )2
0 0Z y i u u Gtψ λ ψ λ ψ ψ′′′ ′ ′ ′= − − − −
( ) ( ) ( ) ( ) ( )( ) ( )2
0 1 0 1 0 0 1 0 0 1 0 1Z y i u u G t tψ λψ λ ψ λψ λ ψ λψ ψ λψ λ′′′ ′′′ ′ ′ ′ ′ ′= + − + − + − + − +
Equation (5.3.45) can be rewritten as
( ) ( ) ( ) ( ) ( )( )
^
, ,1 / Re 1i x
P P x y P x ie z y zλ
ε λ  = − = − 
( ) ( ) ( )( )Re 1P i Cos x iSin x z y z
ε
λ λ
λ
 = + − 
( ) ( ) ( )( )Re 1i Cos x Sin x z y z
ε
λ λ
λ
 = − −  (66)
where
^
P has been named the pressure drop since it indicates the difference between the present at
any point y in the flow field and that at the flat wall, with x fixed. The pressure drops
^
P at
1
0
2
x andλ π= have been named
^
0P and
^
2
Pπ
and their numerical values for several sets of values of
the non-dimensional parameters G have been evaluated. In what follows we record the qualitative
differences in the behavior of the various flow and heat-transfer characteristics which show clearly
the effects of the wavy wall of the channel under consideration.
4. RESULTS AND DISCUSSION
Discussion of the Zeroth Order Solution
The effect of Hartmann number M on zeroth order velocity is to decrease the velocity for
0, 1= ±θ , but the suppression is more effective near the flat wall as M increases as shown in figure
2. The effect of heat source ( )0α > or sink( )0α < and in the absence of heat source or sink ( )0α = ,
on zeroth order velocity is shown in figure 3 for 0, 1θ = ± . 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=α . We
also observe that the magnitude of zeroth order velocity is optimum for 1θ = and minimal for
1θ = − , and profiles lies between 1= ±θ for 0θ = . The effect of source or sink parameter α 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, 1θ = ± . 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, 1θ = ± . The effect of width ratio h on first

100
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, 1θ = ± 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, 1θ = ± as seen in figure 8. The
effect of α on u1 is shown in figure.9, which shows that velocity is large near the wavy and flat wall
for heat source 5α = and is less for heat sink 5α = − . 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. The effect of Hartmann number M is to increase the
velocity near the wavy wall and decrease velocity near the flat wall for 0, 1θ = ± , 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 0α = the profiles lie in between 5α = ±
The effect of convection parameter G, viscosity ratio m, width ratio h, thermal conductivity
ratio k, Hartmann number and source and sink parameter α 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, 1θ = ± . It is
seen that from figure 16 that as M increases the magnitude of 1θ decreases for 0, 1θ = ± . Figure17
shows that the magnitude of α is large for heat source and is less for sink where as 1θ remains
invariant for 0α = .
The effect of convection parameter G, viscosity ratio m, width ratio h, and thermal
conductivity ratio k, on total velocity remains the same . The effect of heat source or sink on total
velocity shows that U is very large for 5α = compared to 5α = − and is almost invariant for 0α =
as seen figure 18, for all values ofθ .
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. The effect of source or sink parameter on total
temperature remains the same as that on total velocity as seen figure 21.
-1.0 -0.5 0.0 0.5 1.0
0
5
10
15
20
25
30
Gr=15.0
Gr=10.0
Gr=5.0
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E= 1.0
Fig2 ZerothordersolutionofvelocityprofilesfordifferentvaluesofGrashoffnumberGr
u0
-1.0 -0.5 0.0 0.5 1.0
0
2
4
6
8
10 m=1.0
m=0.5
m=0.1
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
Fig3 Zerothordersolutionofvelocityprofilesfordifferentvaluesofviscosityratiom
u0
-1.0 -0.5 0.0 0.5 1.0
0
15
30
45
60
h=0.1,0.5,2.0
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
Fig4 Zerothordersolutionofvelocityprofilesfordifferentvaluesofwidthratioh
u0

101
-1.0 -0.5 0.0 0.5 1.0
0
10
20
30
k=0.5
k=1.0
k=0.1 Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E= 1.0
Fig5 Zerothordersolutionofvelocityprofilesfordifferentvaluesofk
u0
-1.0 -0.5 0.0 0.5 1.0
0
4
8
M=6.0
M=4.0
M=2.0
θR
=-1.0
θR
=0.0
θR
=1.0
u0
Fig.6 ZerothordersolutionofvelocityprofilesfordifferentvaluesofHartmannnumberM
Region-II(flatwall)Region-I(wavywall)
-1.0 -0.5 0.0 0.5 1.0
-10
-5
0
5
10
α=-5.0
α=5.0
α=0.0
θR
=-1.0
θR
=0.0
θR
=1.0
u0
Fig7Zerothordersolutionofvelocityprofilesfordifferentvaluesofα
-1.0 -0.5 0.0 0.5 1.0
0
10
20
h=2.0
h=0.5
h=0.1
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
Fig8Zerothordersolutionofvelocityprofilesfordifferentvaluesofwidthratioh
θ0
-1.0 -0.5 0.0 0.5 1.0
0
10
20
30
40
k=0.5
k=1.0
k=0.1
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
Fig9Zerothordersolutionofvelocityprofilesfordifferentvaluesofk
θ0
-1.0 -0.5 0.0 0.5 1.0
-5
0
5
α=-5.0
α=5.0
α=0.0
θR
=-1.0
θR
=0.0
θR
=1.0
θ0
Fig11Zerothordersolutionofvelocityprofilesfordifferentvaluesofα
-1.0 -0.5 0.0 0.5 1.0
-2
-1
0
1
2
Gr=15.0
Gr=10.0
Gr=5.0
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E= 1.0
Fig12 FirstordersolutionofvelocityprofilesfordifferentvaluesofGrashoffnumberGr
U1
-1.0 -0.5 0.0 0.5 1.0
-2
-1
0
1
h=0.1,0.5,2.0
y
E=-1.0
E= 0.0
E= 1.0
Fig14 Firstordersolutionofvelocityprofilesfordifferentvaluesofwidthratioh
u1
-1.0 -0.5 0.0 0.5 1.0
-30
-20
-10
0
10
20
30
Fig15Firstordersolutionofvelocityprofilesfordifferentvaluesofk
k=0.5
k=1.0
k=0.1
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
u1

102
-1.0 -0.5 0.0 0.5 1.0
-0.2
0.0
0.2
M=6.0M=4.0
M=2.0θR
=-1.0
θR
=0.0
θR
=1.0
u1
Fig16FirstordersolutionofvelocityprofilesfordifferentvaluesofHartmannnumberM
-1.0 -0.5 0.0 0.5 1.0
-0.2
0.0
0.2
α=-5.0
α=5.0
α=0.0
θR
=-1.0
θR
=0.0
θR
=1.0
u1
Fig17Firstordersolutionofvelocityprofilesfordifferentvaluesofα
-1.0 -0.5 0.0 0.5 1.0
-0.04
-0.02
0.00
0.02
Fig21Firstordersolutionofvelocityprofilesfordifferentvaluesofk
k=0.5
k=1.0
k=0.1
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
v1
-1.0 -0.5 0.0 0.5 1.0
-0.01
0.00
0.01
Fig22FirstordersolutionofvelocityprofilesfordifferentvaluesofHartmannnumberM
M=6.0
M=4.0
M=2.0
E=-1.0
E= 0.0
E= 1.0
v1
-1.0 -0.5 0.0 0.5 1.0
-0.01
0.00
0.01
α=-5.0
α=5.0
α=0.0
E=-1.0
E=0.0
E=1.0
V1
Fig23Firstordersolutionofvelocityprofilesfordifferentvaluesofα
-1.0 -0.5 0.0 0.5 1.0
-3
-2
-1
0
k=0.5
k=1.0
k=0.1
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
Fig27Firstordersolutionoftemperatureprofilesfordifferentvaluesofk
θ1
-1.0 -0.5 0.0 0.5 1.0
-0.12
-0.06
0.00
Fig28FirstordersolutionoftemperatureprofilesfordifferentvaluesofHartmannnumberM
M=6.0
M=4.0
M=2.0
E=-1.0
E=0.0
E=1.0
θ1
-1.0 -0.5 0.0 0.5 1.0
-0.15
-0.10
-0.05
0.00
α=-5.0
α=5.0
α=0.0
E=-1.0
E=0.0
E=1.0
θ1
Fig29Firstordersolutionoftemperatureprofilesfordifferentvaluesofα
-1.0 -0.5 0.0 0.5 1.0
0
5
10
15
20
25
30
Gr=15.0
Gr=10.0
Gr=5.0
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E= 1.0
Fig30 TotalsolutionofvelocityprofilesfordifferentvaluesofGrashoffnumberGr
U

103
-1.0 -0.5 0.0 0.5 1.0
-12
0
12
24
36
m=1.0
m=0.5
m=0.1
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
Fig31Totalsolutionofvelocityprofilesfordifferentvaluesofviscosityratiom
U
-1.0 -0.5 0.0 0.5 1.0
0
15
30
45
60
Fig32 Totalsolutionofvelocityprofilesfordifferentvaluesofwidthratioh
y
E=-1.0
E= 0.0
E=1.0
U
-1.0 -0.5 0.0 0.5 1.0
-15
0
15
30
45
k=0.5
k=1.0
k=0.1 Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E= 1.0
Fig33 Totalsolutionofvelocityprofilesfordifferentvaluesofk
U
-1.0 -0.5 0.0 0.5 1.0
0
4
8
M=6.0
M=4.0
M=2.0
E=-1.0
E=0.0
E=1.0
U
Fig34TotalsolutionofvelocityprofilesfordifferentvaluesofHartmannnumberM
-1.0 -0.5 0.0 0.5 1.0
-10
-5
0
5
10
α=-5.0
α=5.0
α=0.0
E=-1.0
E=0.0
E=1.0
U
Fig35Totalsolutionofvelocityprofilesfordifferentvaluesofα
-1.0 -0.5 0.0 0.5 1.0
-2
0
2
4
6
8
Gr=15.0
Gr=10.0
Gr=5.0
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
Fig36 TotalsolutionoftemperatureprofilesfordifferentvaluesofGrashoffnumberGr
θ
-1.0 -0.5 0.0 0.5 1.0
-2
0
2
4
6
8
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E= 1.0
Fig37 Total solutionoftemperatureprofilesfordifferentvaluesofviscosityratiom
θ
-1.0 -0.5 0.0 0.5 1.0
0
10
20
Fig38 Totalsolutionoftemperatureprofilesfordifferentvaluesofwidthratioh
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E= 1.0
θ
-1.0 -0.5 0.0 0.5 1.0
0
10
20
30
40
Fig39Totalsolutionoftemperaturreprofilesfordifferentvaluesofk
k=0.5
k=1.0
k=0.1
Region-II(flatwall)
Region-I(wavywall)
y
E=-1.0
E= 0.0
E=1.0
θ

104
-1.0 -0.5 0.0 0.5 1.0
-2
0
2
4
6
8
M=6.0M=4.0
M=2.0
E=-1.0
E= 0.0
E= 1.0
θ
Fig40TotalsolutionoftemperatureprofilesfordifferentvaluesofHartmannnumberM
-1.0 -0.5 0.0 0.5 1.0
-10
-5
0
5
α=-5.0
α=5.0
α=0.0
E=-1.0
E=0.0
E=1.0
θ
Fig41Totalsolutionoftemperatureprofilesfordifferentvaluesofα
-4 -2 0 2 4 6
-6
-3
0
3
6
NuW
0
NuW
0
Nu1
0
Nu1
0
Fig.FirstorderNusseltnumbder at y=-1.0andy=1.0
τ
-4 -2 0 2 4 6
-100
-50
0
50
100
Fig.ZerothorderNusseltnumbery=-1andy=1.0
K=0.1,0.5
Nu
-4 -2 0 2 4 6
-25
-20
-15
-10
-5
0
5
10
15
20
Gr=5.0
Fig.ZerothorderNusseltnumbder at y=-1.0andy=1.0
Nu1
0
NuW
0
Nu
-4 -2 0 2 4 6
-100
-50
0
50
100
150
K=0.1,0.5
Nu0
Fig.ZerothorderNusseltnumbery=-1andy=1.0
-3 0 3 6
-6
-3
0
3
6
h=0.5
Nu1
0
Nu1
0
NuW
0
NuW
0
Fig.ZerothorderNusseltnumbder at y=-1.0andy=1.0
-4 -2 0 2 4 6
-15
-10
-5
0
5
10
15
20
τ
τ
0
w
τ
0
1
E=-1.0
E=1.0
Fig.Firstorderskinfrictionprofilesfordifferentvaluesofm
-4 -2 0 2 4 6
-50
-40
-30
-20
-10
0
10
20
30
40
50
τ
τ
0
w
τ
0
w
τ
0
w
τ
0
1
τ
0
1
τ
0
1
E=-1.0
E=1.0
Fig.FirstorderskinfrictionprofilesfordifferentvaluesofGr

105
-4 -2 0 2 4 6
-60
-40
-20
0
20
40
60
τ
0
1
τ
0
1
τ
0
1
τ
0
w
τ
0
w
E=-1.0
E=1.0
Fig.ZerothorderskinfrictionprofilesfordifferentvaluesofGr
τ
0
w
τ0
-4 -2 0 2 4 6
-6
-3
0
3
6
9
τ
τ
0
w
τ
0
1
E=-1.0
E=1.0
Fig.Firstorderskinfrictionprofilesfordifferentvaluesofh
-4 -2 0 2 4 6
-6
-4
-2
0
2
4
6
τ0
τ
0
w
τ
0
w
τ
0
1
τ
0
1
E=-1.0
E=1.0
Fig.Zerothorderskinfrictionprofilesfordifferentvaluesofh
-4 -2 0 2 4 6
-15
-10
-5
0
5
10
τ
0
1
τ
0
w
τ
0
w
E=-1.0
E= 1.0
Fig.Zerothorderskinfrictionprofilesfordifferentvaluesofm
τ
0
w
τ0
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
-30
-25
-20
-15
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
Gr=15.0
Gr=10.0
Gr=5.0
Gr=15.0
Gr=10.0
Gr=5.0
P^
π/2X102P^
π/2X102
E=-1.0
E=1.0
Fig.PressuredropprofilesfordifferentvaluesofGr
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
-20
-10
0
10
20
0.0 0.2 0.4 0.6 0.8 1.0
-30
-15
0
15
30
α=-5.0
α=0.0
α=5.0
τ
Fig.Pressuredrpoprofilesfordifferentvaluesofα
α=-5.0
α=0.0
α=5.0
E=-1.0
E= 1.0
REFERENCE
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2. F.A Bottemanne., Theoretical solution of simultaneous heat and mass transfer by free
convection about a vertical flat plate, Appl. scient. Res. 25, 137 (1971)
3. G.D Callahan,. W.J Marner, Transient free convection with mass transfer on an isothermal
vertical flat plate, Int. J. Heat Mass Transfer 19 165 (1976)
4. L.S Yao, Natural convection along a wavy surface, ASME J. Heat Transfer 105, 465(1983).
5. L.S Yao, A note on Prandtl transposition theorem, ASME J. Heat transfer 110, 503(1988).
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974 (1989)
7. Shial R. (1973) on laminar two-phase flow in magneto hydrodynamics, Int. J. Engng. Sci,
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8. Hartrmann J. (1937) Hg-Dyanamics I Theory of the laminar flow of an electrically
conductive liquid in a homogeneous magnetic field, kgl Danskevidenskab, selkab mal Fys
Medd, 15.
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TWO FLUID ELECTROMAGNETO CONVECTIVE FLOW AND HEAT TRANSFER BETWEEN VERTICAL WAVY WALL AND A PARALLEL FLAT WALL

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