MHD Boundary Layer Flow and Heat Transfer Over a Continuous
Moving Wavy Surface in Porous Media
F. M. Hady1, R. A. Mohamed2 and A. Mahdy2*
1
2
Department of Mathematics, Faculty of Science, Assiut University, Egypt
Department of Mathematics, Faculty of Science, South Valley University(Qena), Egypt
2*
E-mail mahdy4@yahoo.com
Abstract
Nonsimilarity solutions for the boundary layer flow and heat transfer on a continuous
moving wavy porous surface immersed in a quiescent electrically conducting fluidsaturated porous medium with a constant transverse magnetic field are obtained.
Details results for the velocity f  and temperature θ fields are presented, as well as
the results for the skin-friction coefficient C f and the local Nusselt number Nu .
These numerical results are given for different values of the amplitude of the wavy
surface a , magnetic parameter M , inverse Darcy number Da 1 , and Prandtl number
Pr , then these results are illustrated graphically.
1
Introduction
Boundary layer flow and heat transfer within fluid-saturated porous media has attracted
considerable because of its importance in geophysics, oil recovery techniques, thermal insulation
engineering, packed-bed catalytic reactors, and heat storage beds. A number of similarity
solutions and numerical studies of magnetohydrodynamic (MHD) flow and heat transfer in
porous media have been presented. A comprehensive survey of relevant papers may be found in
the recent monograph by Nield and Bejan (1). Most of the studies included there refer to bodies of
relatively simple geometry, such as flat plates, cylinders and spheres.
Sakiadis
(2,3)
initially
studied flow and heat transfer over a continuously moving surface with a constant speed.
Extensions of this problem were reported previously by many investigators (see, for instance,
Tsuo et al.
(4)
, Jacobi
(5)
. Chen and Strobel
(6)
, Ali
(7)
, Chamkha
(8)
( who studied the same
problem in the case of flat plate) and Cobble (9)) Obtained similarity solutions for MHD flow over
semi-infinite porous plate with suction or injection being imposed at the plate surface. Other
works dealing with magnetic field effects on this type of problem can be found in the papers by
(10)
Gupta
, and Watanabe
(11)
. Rossow
(12)
analyzed hydrodynamic behavior of boundary layers
along a flat plate in the presence of a constant transverse magnetic field. The most of previous
studies have centered on those cases where the thermal boundary conditions allow the use of
similarity transformations to reduce the governing boundary layer equations to a system of
ordinary differential equations. In general this means that the heated surface is plane. However, a
number of papers which study the effects of such nonuniformities on thermal boundary layer are
those of Moulic and Yao
al.
(13)
, and Yao(14), Bourhan Tashtoush and Al-Odat
, Jer-Huan Jan et
(16)
, and an experimental study were obtained for the problem of natural convection along a
vertical wavy surfaces by Rahman
(17)
. In the parallel field of thermal boundary layers in porous
media, which is of interest here, see the papers Rees and Pop
2
(15)
(18,19,20)
, Ching-Yang Cheng
(21)
.
Finally Hossain and Pop
(22)
have studied the problem of MHD boundary layer flow and heat
transfer along a continuous moving wavy surface, and in this paper we extend this problem to the
case of porous media.
Now the purpose of this paper is to study the MHD boundary layer flow and heat transfer
over a continuously moving wavy surface in an electrically conducting fluid at rest embedded in
a fluid-saturated porous medium using Darcy law. The applied magnetic field is assumed to be
uniform and the magnetic Reynolds number is assumed to be small so that the induced magnetic
field can be neglected. In addition, it is assumed that there is no external electric field and that the
electric field due to polarization of charges is negligible. Numerical results are presented for a
selection of parameter sets consisting of the surface wave amplitude a and a nondimensional
numbers and heat transfer characteristics are discussed in detail.
Mathematical Formulation
Consider the problem of steady, laminar, incompressible two-dimensional boundary-layer
flow of an electrically-conducting fluid over a wavy surface immersed in a porous medium
moving tangentially from left to right with a constant speed U w in the presence of magnetic field,
and is maintained at a uniform temperature Tw . Far above the wavy plate, the fluid is stationary
and is kept at a temperature T where Tw > T . The porous medium is assumed to be uniform
(that is, it has a constant porosity and permeability) and all thermophysical properties are
assumed constant. The wavy surface is electrically insulated and a constant magnetic field B0
normal to the surface is imposed. Fig.(1) illustrates the geometry and coordinate system, which
is fixed in space. We assumed that the surface profile could be described by
yˆ  ωˆ ( xˆ )  aˆ sin ( π xˆ/ )
(1)
3
where â is the amplitude of the surface wave,  is a characteristic length scale associated with the
waves. In the present problem the magnetic Reynolds number is assumed to be small so that the
induced magnetic field can be neglected compared to the applied field. Under these assumptions
with the usual Boussinesq approximation, the governing boundary-layer equations that are based
on the balance laws of mass, linear momentum and energy for this investigation can be written
as:
 uˆ
 υˆ

 0
ˆ
x
 yˆ
(2)
uˆ
  2 uˆ
 uˆ
 uˆ
1  Ρˆ
 2 uˆ
 υˆ
 
 ν

  xˆ 2
 xˆ
 yˆ
ρ  xˆ
 yˆ 2

2
 ν
  uˆ  σ 0 B0 uˆ
 κ
ρ

(3)
uˆ
 2 υˆ
 υˆ
 υˆ
1  Ρˆ
2 υˆ
 υˆ
 
 ν

  xˆ 2
 xˆ
 yˆ
ρ  yˆ
 yˆ 2

 ν
  υˆ
 κ

(4)
uˆ
T
T
ν  2 T
2 T
 υˆ


2
 xˆ
 yˆ
Pr   xˆ
 yˆ 2




(5)
where û and υ̂ are the volume-averaged velocity components in the x̂ and ŷ directions,
respectively. T , P̂ is the volume-averaged temperature and Pressure. Properties ν , ρ and α are
the solution viscosity, density, and the equivalent thermal of the fluid, respectively. Furthermore,
σ 0 , B0 and k are the fluid electrical conductivity, magnetic induction and the permeability of
the saturated porous medium, respectively.
The appropriate boundary conditions for the velocity and temperature of this problem are:
yˆ  ωˆ (xˆ) :
uˆ  U w , υˆ  0 ,
T  Tw
yˆ   :
uˆ  0 , Pˆ  Pˆ ,
T  T
(6)
Now, introducing the following dimensionless variables,
~  ωˆ / 
( x~ , y~ )  ( xˆ , yˆ )/  ,
a~  aˆ/  ,
ω
~ , υ~ )  ( uˆ , υˆ ) /U , T  T   T  T θ ,
(u
w

w

4
2 ~
Ρˆ  Pˆ  ρU w
Ρ,
(7)
Then equations (2)-(5) can be nondimensionalized and turn into the following form
~
~
~
u
u
Ρ
~ ~
~
~
~
u ~  υ ~   ~  Re 1 2 u
 Da 1  M u
x
y
x
(8)
~
 υ~
 υ~
Ρ
~
~
~
u ~  υ ~   ~  Re 1 2 υ~  Da 1 υ~
x
y
y
(9)
~2
~  Θ  υ~  Θ  Pr 1 Re 1 
u
Θ
~
~
x
y
(10)

~
where  2 (  2
 x~2  2
Cartesian coordinates, M 

~2 ) is the two-dimensional Laplacian operator expressed in
y
σ 0 B02 
,
ρ Uw
Da 1 
U 
ν
and Re  w ,
ν
kUw
are the magnetic field
parameter, the inverse Darcy number and the Reynolds number respectively.
Using the following dimensionless parameters
x  x~ ,
u  u~ ,
~
Ρ Ρ,
y  ( y~  ω ) Re 1 2 ,
υ  ( υ~  ω u~ ) Re 1 2 ,
(Note that
ω 
~
dω
dω
d2 ω
 

, ω
~
dx
dx
d x2
(11)
)
and after ignoring the small order terms in Re (after allowing Re   ), the dimensionless
governing equations become
u
u
u
Ρ
 υ

 ω Re 1
x
y
x
 υ
υ
ω   u
 υ
y
 x
2

 u 2   Re 1
  ω




(12)
Ρ
2 u
 ω 1  ω 2
 ω Da 1 u
y
 y2
(13)
Ρ
2 u
 1  ω 2

y
 y2
2


 Da
1
M u
Equation (13) indicates that the pressure gradient along the y -direction is
O Re1 2 ,which implies that the lowest order pressure gradient along x-direction can be
determined from the inviscid flow solution. For our problem the inviscid flow field is at rest and
5
hence  P  x  0 and elimination of  P  y between eq.(12) and (13) leads to the following
boundary layer equations for the present problem:
1  ω   yu
2

M
  Da 1 

1  ω 2

u
u
u
 υ

x
y
u
Θ
Θ
2 Θ
 υ
 Pr 1 ( 1  ω 2 )
x
y
 y2
2
2

 
 u  ωω u 2

1  ω 2

(14)
(15)
The corresponding boundary conditions for the present problem then turn into
y 0 :
u  1,
Θ  1,
y  :
u 0,
Θ 0,
(16)
To solve Equations (14) and (15) with the boundary conditions (16) we introduce
the following transformations to reduce the governing equations to a convenient form:
Ψ  x1
2
f (x,η) ,
η  x 1 2 y,
Θ  Θ (x,η) ,
(17)
where  is the stream-function that satisfied continuity equation and is defined by
u 
Ψ
,
 y
Ψ
 x
υ  
(18)
Thus the boundary layer equations are written as:
1  ω  f   21 f f   x  Da
2
1

1  ω  Pr
2
1
Θ  

M
1  ω 2

 
 f   x ω ω f 2

1  ω 2

 Θ
1
f
f Θ   x  f 
 Θ
2

x
x

(19)



(20)
The boundary conditions to be satisfied are that
6
f (x , 0)  0,
f  (x , 0) 1 ,
f  (x ,  )  0 ,
Θ (x ,  )  0
Θ (x , 0)  1
(21)
Of special significance for this type of flow and heat transfer situation are the skin-friction
coefficient C f and the Nusselt number Nu , These physical quantities are defined, respectively,
as follows:
Cf 
τˆw
,
ρ U w2
Nu 
xˆ qˆ w
k  Tw  T

(22)
,
Where the skin-friction τˆw and the heat flux q̂ w at the wall are given by
  uˆ
 υˆ
τˆw  μ 

ˆ

y

xˆ

,

ω

n  
,
2


1

ω



,
yˆ  ωˆ (xˆ)
1
1  ω 2




is
qˆ w   k n   T ,
the
unit
vector
(23)
normal
to
the
wavy
plate.
Using Equations (7), (11), and (16) we get the skin-friction coefficient C f and the local
Nusselt number Nu from the following expressions:
C f Re 1x 2 
1  ω 
2

f  ( x ,0 )
Nu Rex-1 2   1  ω 2
where Re x 
U w xˆ
ν

1 2
Θ  (x,0)
(24)
(25)
is the local Reynolds number.
7
Results and Discussion
The nonlinear-coupled ordinary differential equations (19)-(20) and the associated
boundary conditions (21) were solved numerically and the fourth-order Runge Kutta method
and the implicit-finite difference method are appropriate for the solution of this problem. Many
numerical results were obtained throughout the course of this work and a representative set of
graphical results for the velocity and temperature fields as well as the skin-friction coefficient and
Nusselt number are presented, discussed and shown in Figs. (2) through (15)
Representative velocity f  and temperature θ profiles are shown in Figs. (2) through
(5) exhibiting the effects of the wave amplitude a and of the magnetic field parameter M
where results are given for Pr = 0.7, Da 1 = 0.1, and because of the behavior of these profiles at
crest (x = 0.5) and trough (x = 1.5) position is very similar, thus here we will consider the case of
x = 0.5 (crest). Figs. (2) and (3) show typical profiles for both the velocity and temperature.
Increases in the amplitude of the waviness a have a tendency to increase both of f  and θ . On
the other hand, Figs. (4) and (5) illustrated that the velocity profiles decreases, while the
temperature profiles increases owing to the increase of magnetic field M .
Figs. (6) and (7) present the influence of the porous medium inverse Darcy number
Da 1 on the velocity and temperature profiles. The parameter Da 1 represent resistance to flow
since it restrict the motion of the fluid along the wavy surface and therefore, it has the same effect
of the magnetic field M , namely, decreasing the fluid velocity f  and increasing its temperature
θ
as shown in Figs.(6) and (7).
Fig. (8) presents the influence of the Prandtl number Pr on the temperature profiles.
Since the flow problem is uncoupled from the thermal problem, changes in the value of Pr will
not affect the fluid velocity (and for this reason we did not show the velocity profiles). We can
8
see from this figure that increasing the Prandtl number Pr tends to increase of the temperature
profiles.
Figs. (9) through (15) depict effects of the wave amplitude a , magnetic field parameter
M inverse Darcy number Da 1 , and Prandtl number Pr on the skin-friction coefficient C f and
the Nusselt number Nu , respectively. The reductions in the velocity profiles and the increases in
the temperature profiles as either of M and Da 1 increase cause the slopes of the velocity and
temperature profiles to increase and decrease, respectively. This has the direct effect of increasing
C f and decreasing Nu as clearly shown in Figs. (11) through (14). Moreover it is observed
from these figures that the skin-friction coefficient C f and the local Nusselt number Nu vary
periodically in the direction of x when a  0 (wavy surface), from Figs. (11) and (13) we see
that the skin-friction coefficient is greater than to that corresponding to a flat surface ( a  0 ).
Fig. (15) illustrates the effect of Prandtl number Pr to the local Nusselt number Nu and
its clearly that as Pr increases the local Nusselt number Nu increase.
9
Fig. (1) Schematic of the problem
10
1.0
Da
Pr
M
x
0.8
-1
= 0.1
= 0.7
= 0.1
= 0.5
0.6
f'
a =
0.4
0.3
0.1
0.2
0.0
0.0
0
1
2
3
4
5
6
7

Fig. (2) Velocity profiles against η for different values of
the wave amplitude a
11
1.0
-1
Da = 0.1
Pr = 0.7
M = 0.1
x
= 0.5 (crest)
0.8
a =
0.2
0.4
0.1
0.2
0.0

0.6
0.0
0
1
2
3
4
5
6

Fig. (3) Temperature profiles against η for different values of
the wave amplitude a
12
1.0
Da
Pr
a
x
0.8
-1
=
=
=
=
0.1
0.7
0.1
0.5
f'
0.6
0.4
M =
0.2
0.0
0.2
0.5
0.0
0
1
2
3

Fig. (4) Velocity profiles against η for different values of
the magnetic field parameter M
13
1.0
-1
Da = 0.1
Pr = 0.7
= 0.1
a
= 0.5
x
0.8
0.6
M =

0.5
0.4
0.2
0.0
0.2
0.0
0
2
6
4
8
10

Fig. (5) Temperature profiles against η for different values of
the magnetic field parameter M
14
1.0
Pr
a
M
x
0.8
Da
-1
= 0.7
= 0.1
= 0.1
= 0.5
=
0.6
f '
0.0
0.4
0.3
0.2
1.0
0.0
0
1
2
3
4
5
6
7

Fig. (6) Velocity profiles against η for different values of
the inverse Darcy number Da 1
15
1.0
Pr
x
M
a
0.8
=
=
=
=
0.7
0.5
0.1
0.1

0.6
-1
Da =
0.4
1.0
0.5
0.1
0.2
0.0
0
2
4
6
8
10

Fig. (7) Temperature profiles against η for different values of
the inverse Darcy number Da 1
16
1.0
x
M
a
Pr =
0.8
0.7
= 0.5
= 0.2
= 0.1
-1
Da = 0.1
0.6

1.0
0.4
0.2
10
0
1
2
3
4

Fig. (8) Temperature profiles against η for different values of
the Prandtl number Pr
17
3.5
Pr
= 0.7
a = 0.2
-1
Da = 0.1
M = 0.1
3.0
2.5
Cf Re
1/2
2.0
1.5
0.1
1.0
0.0
0.5
0.0
0
1
2
3
4
5
x
Fig. (9) Skin-friction coefficient against x for different
values of the wave amplitude a
18
0.90
-1
Da = 0.1
Pr = 0.7
M = 0.1
0.75
a = 0.2
0.60
NuRe
-1/2
0.1
0.45
0.30
0.15
0.00
0
1
2
3
4
5
x
Fig. (10) Local Nusselt number against x for different
values of the wave amplitude a
19
Da
Pr
a
2.0
-1
= 0.1
= 0.7
= 0.1
M = 0.5
= 0.2
1.6
-Cf Rex
1/2
= 0.0
1.2
0.8
0.4
0.0
0
1
2
3
4
x
Fig. (11) Skin-friction coefficient against x for different
values of the magnetic field parameter M
20
5
Pr
a
0.6
Da
= 0.7
= 0.1
-1
M =
= 0.1
0.2
-1/2
0.5
NuRe
0.5
0.0
0.4
0.3
0.2
0
1
2
3
4
5
x
Fig. (12) Local Nusselt number against x for different
values of the magnetic field parameter M
21
2.8
Pr = 0.7
a = 0.1
M = 0.1
2.4
Da
-1
= 1.0
2.0
-Cf Re
1/2
0.5
1.6
0.1
1.2
0.0
0.8
0.4
0.0
0
1
3
2
4
5
x
Fig. (13) Skin-friction coefficient against x for different
values of the inverse Darcy number Da 1
22
a = 0.1
Pr = 0.7
M = 0.0
0.525
NuRe
-1/2
0.450
0.375
-1
Da =
1.0
0.300
0.5
0.1
0.225
0
1
2
3
4
5
x
Fig. (14) Local Nusselt number against x for different
values of the inverse Darcy number Da 1
23
1.6
Pr = 10.0
1.2
NuRe
-1/2
Da
a
M
0.8
-1
= 0.1
= 0.1
= 0.2
Pr = 1.0
0.4
Pr = 0.7
0.0
0
1
2
3
4
5
x
Fig. (15) Local Nusselt number against x for different
values of the Prandtl number Pr
24
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(22). M. A. Hossain and I. Pop, Arch. Mech. Engng., 48, 813 (1996).
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Nomenclature
a
Amplitude of the wavy surface
Greek symbols
B0
Magnetic induction

Kinematics coefficient of viscosity
Cf
Skin-friction coefficient

Fluid density
Da
Darcy number

Stream function
f
nonsimilarity function

Dimensionless temperature
g
Gravitational acceleration
0
fluid electrical conductivity

Characteristic reference wavelength

Porous medium permeability
M
Magnetic field parameter

coordinate of the wavy surface
Nu
Nusselt number

Laplacian operator
P
Pressure of the fluid
Pr
Prandtl number
Superscripts
qw
wall heat flux
^
Dimensional quantity
Re
Reynolds number
'
differentiation with respect to η
Re x
Local Reynolds number
T
Temperature
Subscripts
u, 
Dimensionless velocities

Conditions far away from the surface
Uw
Constant velocity
w
wall surface
x, y
Dimensionless coordinates
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