International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:10 No:02 71
109602-3737 IJBAS-IJENS © April 2010 IJENS I J E N S
MHD Heat Transfer Mixed Convection Flow Along a
Vertical Stretching Sheet in Presence of Magnetic
Field With Heat Generation
M. Mohebujjaman1, Tania S. Khaleque
2 and M.A. Samad
2, b
1Department of Textile Engineering, Southeast University, Banani, Dhaka-1213, Bangladesh
2Department of Mathematics, Dhaka University, Dhaka-1000, Bangladesh
E-mail: [email protected] b Corresponding author
Abstract-- The steady two-dimensional MHD heat transfer
mixed convection flow of a viscous incompressible fluid near a
stretching permeable sheet in presence of a uniform magnetic
field with heat generation is considered when the buoyancy force
assists or opposes the flow. The equations governing the flow and
temperature fields are reduced to a system of coupled non-linear
ordinary differential equations. These non-linear differential
equations are integrated numerically by using Nachtsheim-
Swigert [1] shooting iteration technique along with sixth order
Runge-Kutta integration scheme. Critical values of buoyancy
parameter are obtained for vanished shear stress. The numerical
results are benchmarked with the earlier study by Mohamed Ali
and Fahd Al-Yousef [2] and found to be in excellent agreement.
Finally the effects of the pertinent parameters which are of
physical and engineering interest are presented in tabular form.
Index Term-- boundary layer, buoyancy force, electric
conductivity, variable wall temperature.
NOMENCLATURE
0B Uniform magnetic field strength xNu Local Nusselt number
wf Suction parameter Pr Prandtl number
M Magnetic field parameter Q Dimensionless heat source parameter
g Acceleration due to gravity 0Q Heat source parameter
pC Specific heat at constant pressure wq Local heat flux
fC Skin friction coefficient wv Suction velocity
f Dimensionless stream function Buoyancy parameter
wT Temperature of fluid at the sheet
T Temperature of fluid within the boundary layer
T
Temperature of fluid far away from the plate
u
Component of velocity in the x -direction
v Component of velocity in the y -direction x Coordinate along the sheet
y Coordinate normal to the sheet
Greek Symbols
Electric conductivity
Similarity parameter Step size
Dimensionless temperature Coefficient of volume expansion
Thermal conductivity Density of the fluid
Coefficient of dynamic viscosity Coefficient of kinematic viscosity
I. INTRODUCTION
Both the hydromagnetic flow and heat transfer in a viscous
incompressible fluid over a moving continuous stretching
surface is a significant type of flow has considerable practical
applications in industries and engineering. For example,
materials manufactured by extrusion processes, E.G. Fisher
[3], and heat-treated materials traveling between a feed roll
and a wind-up roll or on a conveyor belt possess the
characteristics of a moving continuous surface. Many
metallurgical processes involve the cooling of continuous
strips or filaments by drawing them through a quiescent fluid
and that in the process of drawing, these strips are sometimes
stretched. Mention may be made of wire drawing, annealing,
tinning of copper wires, crystal growing, spinning of
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filaments, continuous casting, glass fiber production and paper
production, T. Atlan, S. Oh, H. Gegel [4], Z. Tadmor and I.
Klein [5]. In all the cases the properties of the final product
depend to a great extent on the rate of cooling and the
processes of stretching as explained by M.V. Karwe and Y.
Jaluria [6], [7]. By drawing such strips or filaments in an
electrically conducting fluid subjected to a magnetic field, the
rate of cooling can be controlled and a final product of desired
characteristics can be achieved. Another interesting
application of hydromagnetics to metallurgy lies in the
purification of molten metals from nonmetallic inclusions by
the application of a magnetic field. The study of heat transfer
has become important industrially for determining the quality
of the final product. The dynamics of the boundary layer flow
over a moving continuous solid surface originated from the
pioneer work of B.C. Sakiadis [8] who developed a numerical
solution for the boundary layer flow field of a stretched
surface, many authors have attacked this problem to study the
hydrodynamic and thermal boundary layers, M.E. Ali [9],
[10], E. Magyari and B. Keller [11], [12].Due to entrainment
of ambient fluid, this boundary layer flow situation is quite
different from the classical Blasius problem of boundary flow
over a semi-infinite flat plate. Suction or injection of a
stretched surface was studied by L.E. Erickson, L.T. Fan and
V.G. Fox [13] and V.G.Fox, L.E. Erickson and L.T. Fan [14]
for uniform surface velocity and temperature and investigated
its effects on the heat and mass transfer in the boundary layer.
J.B. McLeod and K.R. Rajagopal [15] have investigated the
uniqueness of the flow of a Navier Stokes fluid due to a linear
stretching boundary. A. Raptis and C. Perdikis [16] have
studied the viscous flow over a non-linearly stretching sheet in
the presence of a chemical reaction and magnetic field. C.K.
Chen and M.I. Char [17] have studied the suction and
injection on a linearly moving plate subject to uniform wall
temperature and heat flux and the more general case using a
power law velocity and temperature distribution at the surface
was studied by M.E. Ali [18]. E. Magyari, M.E. Ali and B.
Keller [19] have reported analytical and computational
solutions when the surface moves with rapidly decreasing
velocities using the self-similar method. The study of heat
generation or absorption in moving fluids is important in
problems dealing with chemical reactions and these concerned
with dissociating fluids. Possible heat generation effects may
alter the temperature distribution; consequently, the particle
deposition rate in nuclear reactors, electronic chips and semi
conductor wafers. K. Vajravelu and A. Hadjinicolaou [20]
studied the heat transfer characteristics in the laminar
boundary layer of a viscous fluid over a stretching sheet with
viscous dissipation or frictional heating and internal heat
generation. Laminar mixed convection boundary layers
induced by a linearly stretching permeable surface was studied
by Mohamed Ali and Fahh Al-Yousef [2].
In the present study we aim to extend the analysis
by Mohamed Ali and Fahh Al-Yousef [2] considering a
uniform magnetic field which is normal to the stretching
surface with heat generation, which have been of interest
to the engineering community and to the investigators
dealing with the problem in geophysics, astrophysics,
electrochemistry and polymer processing.
II. MATHEMATICAL ANALYSIS
A steady-state two-dimensional motion of mixed convection
boundary layer flow from a vertically upward moving
permeable stretching sheet through a quiescent incompressible
fluid with suction or injection at the surface is considered. The
stretching sheet coincides with the plane 0y as shown in
fig. 1.
Fig. 1. Coordinate system, boundary
conditions and steady
boundary layers adjacent to the stretching wall issuing from a
narrow linear slot and
moving in the positive x direction.
A uniform magnetic field of strength 0B is imposed normal to
the stretching sheet, where the flow is confined to 0y . For
incompressible viscous fluid milieu with constant properties
using Boussinesq approximation, the equations governing this
convective flow are
0
y
v
x
u (1)
uB
y
u)TT(g
y
uv
x
uu
2
0
2
2
(2)
)TT(c
Q
y
T
cy
Tv
x
Tu
pp
0
2
2
(3)
subject to the following boundary conditions:
yTT,u
yxCTTTT
y),x(vv,xUuu
nw
wm
w
at
at
at
0
0
00
(4)
It should be mentioned that, positive or negative m indicate
that the sheet is accelerated or decelerated from the extruded
slit respectively. The x coordinate is directed along the
continuous stretching sheet and points in the direction of
motion, measured from the point where the sheet originates,
and the y coordinate is measured perpendicular to x axis
and to the direction of the slot ( z axis) where the continuous
stretching plane issues. In order to allow for fluid suction or
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injection through the sheet, the sheet is regarded to be porous
with a transpired velocity )x(vw . Positive or negative wv
imply suction or injection at the surface respectively. u and
v are the velocity components in the x and y directions
respectively. is the kinematic viscosity, g is the
acceleration due to gravity, is the volumetric coefficient of
thermal expansion, T and
T are the fluid temperature within
the boundary layer and in the free-stream respectively, is
the electric conductivity, 0
B is the uniform magnetic field
strength (magnetic induction), is the density of the fluid,
is the thermal conductivity of the fluid, 0
Q is the volumetric
rate of heat generation/absorption, w
T is the uniform wall
temperature. In order to obtain a solution of equations (1)-(4),
we introduce the following similarity variables
),(fxUu 'm 0 )(xCTT n (5)
x
m
Rem
x
y
x
xUmy
2
1
2
1 0
(6)
'
m
fm
fm
xm
Uv
2
1
2
1
1
22
1
0 (7)
where 'f and are the dimensionless velocity and
temperature respectively, and is the similarity variable.
Substitution in the governing equations gives rise to the
following two-point boundary-value problem.
01
2
1
2
1
2 2
m
fm
M)f(
m
mfff '''''''
(8)
01
2
1
2
m
Qpr)f
m
nf(pr ''''
(9)
The last term in equation (8) is due to the buoyancy force and
2
0
12
U
xCg mn
which serves as the buoyancy parameter,
when 0 the governing equations reduce to those of forced
convection limit. On the other hand, if is of a significantly
greater order of magnitude than one, the buoyancy force
effects will predominate and the flow will essentially be free
convective and combined convective flow exists when
).(O 1 Third term in equation (8) is due to the magnetic
field and 0
12
0
U
xBM
m
serves as the magnetic field
parameter. A consideration of equation (8) shows that and
M are functions of .x Therefore, the necessary and sufficient
conditions for the similarity solutions to exist is
that 12 mn , 01 m for f and be expressed as
function of alone. Which implies 1m and 1n .
Furthermore, if these conditions are not satisfied, local
similarity solutions are obtained, W.M. Kays and M.E.
Crawford [21], A. Mucoglu and T.S. Chen [22]. In this paper
local similarity solutions are also found for 1m at 1n
and 0 . The transformed boundary conditions are:
as,,f
at,ff,f w
00
011 (10)
where1
2
0
1
mU
xvf
m
ww
is the suction ( or injection )
parameter,
pcpr is the Prandtl number and
0
0
Uc
p is the heat source/sink parameter. The
parameters of engineering interest for the present problem are
the skin friction coefficient and local Nusselt number which
indicate physically wall shear stress and local wall heat
transfer rate respectively. From equation (5) we write
x
''m Rem
x.)(fxU
y
u
y
u
2
110
)(fRem
xU ''
x
m 2
11
0
At the stretching sheet i.e. at 0y ,
)(fRem
xUy
u ''
x
m
wall
02
11
0
, hence the
expression for wall shear stress can be developed from the
similarity solution in the form
)(fRem
xUy
u ''x
m
wall
w 02
110
The wall shear stress can be expressed in a dimensionless
form which is known as the skin friction coefficient is given
by 2
2
1w
wf
u
C
which
implies
x
''
m
''x
m
fRe
)(fm
xU
)(fRem
xU
C
02
12
2
1
02
1
220
10
)(fm
ReC ''xf 0
2
12
(11)
and for 1m which is given by ).(fReC ''xf 02
The local wall heat flux )q( w may be written by Fourier’s
law as
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x'n
y
w Rex
m)(xC
y
Tq
1
2
10
0
Hence
the local Nusselt number is obtained as
)(Rem
xC
qxNu '
xn
wx 0
2
1
which implies
)(m
Re
Nu '
x
x 02
1
(12)
and for 1m it is given by ).(Re
Nu '
x
x 0 The numerical
values proportional to fC and xNu , corresponding to 1m
are shown in Table III-Table IV.
III. NUMERICAL COMPUTATION The numerical solutions of the non-linear differential equation
(8)-(9) under the boundary conditions (10) have been
performed by applying a shooting method namely
Nachtsheim and Swigert [1] iteration technique (guessing the
missing values) along with sixth order Runge-Kutta
integration scheme. The boundary conditions, equation (10),
associated with the non-linear ordinary differential equations
(8)-(9) are the two-point asymptotic class, that is, values of
the dependent variable are specified at two different values of
the independent variable. Specification of an asymptotic
boundary condition implies that the first derivative (and
higher derivatives of the boundary layer
equation, if exists) of the dependent variable approaches zero
as the outer specified value of the independent variable is
approached. For the method of numerically integrating a two-
point asymptotic boundary-value problem of the boundary-
layer type, the initial-value method is similar to an initial
value problem. Thus, it is necessary to estimate as many
boundary conditions at the surface as were (previously) given
at infinity. The governing differential equations are then
integrated with these assumed surface boundary conditions. If
the required outer boundary condition is satisfied, a solution
has been achieved. However, this is not generally the case.
Hence, a method must be devised to estimate logically the
new surface boundary conditions for the next trial integration.
Asymptotic boundary value problems such as those governing
the boundary-layer equations are further complicated by the
fact that the outer boundary condition is specified at infinity.
In the trial integration, infinity is numerically approximated
by some large value of the independent variable. There is no a
priori general method of estimating these values. Selecting
too small a maximum value for the independent variable may
not allow the solution to asymptotically converge to the
required accuracy. Selecting a large value may result in
divergence of the trial integration or in slow convergence of
surface boundary conditions. Selecting too large a value of
the independent variable is expensive in terms of computer
time. Nachtsheim-Swigert [1] developed an iteration method
to overcome these difficulties. In equation (10) there are two
asymptotic boundary conditions and, hence, two unknown
surface conditions such as )(''f 0 and )(' 0 . Within the
context of the initial-value method and the Nachtsheim-
Swigert [1] iteration technique, the outer boundary conditions
may be functionally represented as
4100 j,))('),(''f(Y)(Y jjmaxj , (13)
Where ''fY,Y,'fY 321 and 'Y 4 . The last two of
these represent asymptotic convergence criteria.
Choosing 10 g)(f '' , 20 g)(' and expanding in a first-
order Taylor’s series after using equation (13), we obtain
41
2
1
1
j
,gg
Y)(Y)(Y j
i i
j
maxC,jmaxj (14)
where the subscript 'C' indicates the value of the function at
max determined from the trial integration. Solution of these
equations in a least-squares sense required determining the
minimum value of
Error
4
1
2
j
j (15)
Now differentiating (15) with respect to ig 21,i we obtain
4
1
0
j i
j
jg
. (16) Substituting equation
(14) into (16) after some algebra, we obtain
,bga
k
iiik
2
1
(17)
where
21
4
1
4
1
,k,i
,g
YYb,
g
Y
g
Ya
j i
j
C,ji
j k
j
i
j
ik
(18)
Now solving the system (17) using Cramer’s rule, we obtain
the missing (unspecified) values of ig as
iii ggg . (19)
Thus, adopting the numerical technique aforementioned along
with the sixth order Runge-Kutta-Butcher initial value solver,
the solutions of the equations (8)-(9) with boundary
conditions (10) are obtained as a function of the coordinate
for various values of the parameters. We have chosen a step
size 0010. to satisfy the convergence criterion of 610
in all cases. The value of
was found to each iteration loop
by
. The maximum value of
to each group of
parameters Pr,,M,f w and Q determined when the values
of the unknown boundary conditions at 0 not change to
successful loop with error less than 610 . In order to verify
the effects of the step size , we ran the code for our model
with three different step sizes as ,.0050 001.0 ,
00010. and in each case we found excellent agreement
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:10 No:02 75
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among them. The figures 2(a) and 2(b) show the velocity and
temperature profiles for different step sizes.
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
f'
Curves
---------
O
0.005
0.001
0.0001
Fw=-0.6, Pr=0.72, M=0.5, Q=0.5,
(a)
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Curves
----------
O 0.001
0.0001
Fw=-0.6, Pr=0.72, M=0.5, Q=0.5,
(b)
Fig. 2. Velocity and temperature profiles for different step size : (a)
Velocity and (b) Temperature.
To assess the accuracy of the present code, we compare the
critical values of the buoyancy parameter s.)crt(
corresponding to vanished shear stresses at the surface with
those of Mohamed Ali and Fahh Al-Yousef [2] by
setting 0M , 0Q .
Table I demonstrate the comparison of the data produced by the present code and those of M. Ali (2002). In fact, the
results show a close agreement, and hence justify the use of the present code.
TABLE I
wf
1n , 1m
720.pr 03.pr 010.pr
M. Ali et al. Present M. Ali et al. Present M. Ali et al. Present
0.6 3.011 3.0102 6.2075 6.2142 13.335 13.4967
0.4 2.7250 2.72220 5.2069 5.2211 10.339 10.4267
0.2 2.4743 2.4697 4.397 4.3881 7.9374 7.9577
0.0 2.264 2.2499 3.7238 3.6989 ------- ---------
-0.2 2.078 2.0599 3.1447 3.1366 ------- ---------
-0.4 1.912 1.8968 2.698 2.6842 ------- ---------
-0.6 1.7735 1.7575 -------- 2.3248 ------- ---------
IV. RESULT AND DISCUSSION
For the purpose of discussing the result, the numerical
calculations are presented in the form of non-dimensional
velocity and temperature profiles. Numerical computations
have been carried out for different values of the Prandtl
number )pr( , magnetic field parameter )M( , buoyancy
parameter )( , heat source/sink parameter )Q( and suction (
or injection ) parameter ).f( w These are chosen arbitrarily
where 710.pr corresponds physically to air at C20 ,
01.pr corresponds to electrolyte solution such as salt water
and 07.pr corresponds to water. Equations (8) and (9)
were solved numerically subject to the boundary conditions
given in equation (10) for ,m 1 60.f w to 60. with a
step of ,.20 ,.pr 720 ,3 10 and for temperature exponent
,n 1 0 and .1 In Fig.3 we have plotted the dimensionless
velocity and temperature profiles for ,m 1 ,n 1 ,.f w 60
60. and for ..pr 720 showing the effects of the buoyancy
parameter . It can be seen that the velocity increases but
temperature decreases near the stretching sheet as we increase
the buoyancy parameter . It is also clear that the velocity
gradient at the surface increases from negative value to a
positive value. The velocity gradient at the surface is positive,
which signify that the stretching sheet velocity is smaller than
that of the adjacent fluid velocity. On the other hand negative
velocity gradient at the surface indicates that stretching sheet
velocity is greater than the adjacent fluid velocity. The
specific critical value of s.)crt( ,.1600822 3780413. are
for zero velocity gradient when ,.f w 60 60. respectively
as shown in Fig.3(a), where the surface shear stress are
vanished. As we increase the velocity gradient at the
surface increases for both injection and suction, as well as
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they are almost identical which indicates that the flow is
predominated by the buoyancy effects. Since the velocity
gradient at the surface increases as we increase , so the
shear stress at the surface increases. It is also observed from
the Fig. 3(a) that, the hydrodynamic boundary layer thickness
is greater for 60.f w than for 60.f w since the earlier
for injection while the later is for suction.
Fig.3 (b) shows the temperature profiles for the same
parameters used in Fig.3 (a). It is clear that as increases the
thermal boundary layer thickness decreases for both suction
and injection. In this Fig. we see that the temperature gradient
as well as thermal boundary layer thickness are decreasing as
we increase . The temperature gradient is always negative
which means that heat is transferred from the sheet to the
ambient medium. Hence, the heat transferred rate from the
sheet to the ambient medium is getting large as increases
for both suction and injection. The thermal boundary layer
thickness is smaller for 60.f w than for 60.f w where
the earlier one is for suction while the later one is for
injection.
0 0.5 10
2
4
6
8
10
12
14
16
18
20
f'fw= - 0.6
fw= 0.6
2500.0250.025.02.160082
2500.0250.025.03.37804
(a)
0 1 2 3 40
0.25
0.5
0.75
1
fw= -0.6
fw= 0.6
2500.0250.025.02.160082
2500.0250.025.03.378041
(b)
Fig. 3. Dimensionless downstream velocity and temperature profiles
corresponding to ,m 1 1n and ..pr 720 showing the buoyancy
effects; (a) velocity and (b) temperature.
Moreover, the temperature profiles for suction are squeezed
together with reduced thermal boundary layer thickness than
that for injection therefore, suction enhances the heat transfer
coefficient from the stretched surface than injection.
We have delineated, In Fig4, the dimensionless velocity and
temperature profiles showing the effect of the Prandtl number
Pr when ,m 1 60.fw and 025. but 1n and
.0 1m indicates that the sheet is stretching linearly. We see
from Fig4 (a) and Fig4 (b) that the velocity and temperature
decrease rapidly as we increase Prandtl number Pr and the
temperature exponent n . The figures also indicate that
increasing Prandtl number Pr
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
n= -1
n = 0
n= -1
n = -1
n = 0
n = -1
n =0
pr = 0.72
pr = 3.0
pr = 10.0
f 'm =1, fw = 0.6,
(a)
0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n=0
n=-1
n=-1
n=0
n=-1
n=0
pr = 0.72
pr = 3.0
pr = 10.0
m =1, fw = 0.6,
(b)
Fig. 4. Dimensionless downstream velocity and temperature profiles
corresponding to ,m 1 ,n 0 1 and 025. showing the Prandtl
number effects; (a) velocity and (b) temperature.
and the exponent n reduce the momentum and thermal
boundary layer thickness that in turn reduce the shear stress
and increase the heat transfer coefficient at the surface
respectively. The velocity gradient at the surface is positive
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for both 1n and 0 , when 720.Pr , 03. and 010. , which
means that the adjacent fluid velocity is higher than the
stretching sheet velocity. The velocity of the fluid within the
boundary layer, when ,n 1 is greater than that of the fluid
velocity when 0n (surface temperature is uniform), for
each value of the Prandtl number Pr . For example at
250. when 720.Pr and ,n 0 the velocity is %.9314
and temperature is %.718 less with in the boundary layer than
that of the velocity when 1n . On the other hand the
velocity reduces %.461 and temperature reduces %.1180 if
we increase Pr from 720. to 010. when 1 , 60.fw
and 025. .The momentum boundary layer thickness
remains same for both 1n and 0n for a fixed value
of Pr . But the difference of the velocities between these two
situations, that is 1n and 0n for each Prandtl number
decreases as we increase Pr . The specific critical (not shown
in figure) value of 5969520.PrPr )crt( is for zero
velocity gradient where the surface shear stress is vanished.
The dimensionless velocity and temperature profiles are
presented in Fig.5 for different values of the magnetic field
parameter when ,m 1 60.fw , 025. but 1n and
.0 It is clear from Fig.5 that velocity decreases but the
temperature increases as we increase the magnetic field
parameter M for both cases when 1n and .0 The
magnetic field lines act like strings and tend to retard the
motion of the fluid. The consequence of which is to increase
the temperature of the fluid. It is also noticeable that for a
fixed value of the magnetic field parameter, the velocity and
temperature corresponding to case of 0n are lower
compare to the case of 1n .
0 0.5 1 1.50
0.5
1
1.5
2
2.5
f '
M0.00.51.01.52.0
n=0
n= -1
m=1, fw= 0.6,
(a)
0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
m=1, fw= 0.6,
n=0
n= -1
M2.01.51.00.50.0
M2.01.51.00.50.0
(b)
Fig. 5. Dimensionless downstream velocity and temperature profiles
corresponding to ,m 1 ,n 0 1 and 025. showing the effects of
the magnetic field parameter; (a) velocity and (b) temperature.
That is, the thickness of the momentum and thermal boundary
layer are higher for the case 1n than for the case 0n .
The velocity gradients are positive but the temperature
gradients are negative for different situations of both cases.
Fig. 6 depicts the effects of the heat source parameter on
the velocity and temperature profiles. We see that velocity and
temperature increase rapidly as we increase the heat source
parameter Q . The figures also indicate that increasing heat
source parameter Q , increase the momentum and thermal
boundary layer thickness that in turn increase the shear stress
and reduce the heat transfer coefficient at the surface
respectively.
0 0.5 1 1.5 20
0.25
0.5
0.75
1
1.25
1.5
1.75
2
fw= 0.6, M=0.5, Pr= 0.72, m=1.0, n= -1,
Q=0.0, 0.5, 1.0, 1.5, 2.0
f'
(a)
0 1 2 30
1
2
3
4
5
fw= 0.6, M=0.5, Pr= 0.72, m=1.0, n= -1,
Q=0.0, 0.5, 1.0, 1.5, 2.0
(b)
Fig. 6. Dimensionless downstream velocity and temperature profiles
corresponding to 01.m , 01.n showing the effects of the heat source
parameter Q ; (a) velocity and (b) temperature.
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The critical (not shown in figure) value of the heat source
parameter is 6560.QQ )crt( which is for drag less
velocity profile. It is clear that the velocity and temperature
gradients at the surface increase from negative to positive
value. The specific critical value of 399760.QQ )crt( is
for zero temperature gradient at the surface. Critical values of
buoyancy parameter s.)crt( for which shear stress at the
surface become zero when
,m 1 1n , ,.pr 720 ,.03 ,.010 ,.M 50 50.Q and for
various values of wf are shown in the following table.
Table II
,m 1 1n ,m 1 1n
wf 720.pr 03.pr 010.pr 720.pr 03.pr
0.6
3.378041
6.87491
15.18558
2.26354
4.96301
0.4
3.093619
5.8113
11.78121
1.93646
3.791265
0.2
2.845829
4.91055
8.98045
1.637226
2.74961
0.0
2.631609
4.16187
6.76494
1.3622
1.813537
-0.2
2.447956
3.55106
5.094422
1.10724
0.96093
-0.4
2.2918
3.06359
3.90799
0.86808
-----------
-0.6
2.16008
2.68366
3.11479
0.64065
-----------
TABLE III
NUMERICAL VALUES PROPORTIONAL TO fC
AND xNu FOR DIFFERENT VALUES OF
,Q
ANDn .
n fC xNu Q n
fC xNu
2500.0
1.0
-1.0
271.6381653
357.2735710
-3.9996076
-0.3222749
0.0
1.0
-1.0
-0.4577762
-0.1787497
-1.1735897
-0.4334700
250.0
1.0
-1.0
45.7724434
60.6641322
-2.3664587
-0.2413562
0.5
1.0
-1.0
-0.3425068
0.1279294
-0.9615057
0.1244760
25.0
1.0
-1.0
6.3019204
8.8952453
-1.4629240
-0.1085501
1.0
1.0
-1.0
-0.1609941
0.7442207
-0.6775980
1.3320847
2.5
1.0
-1.0
-0.3425068
0.1279294
-0.9615057
0.1244760
1.5
1.0
-1.0
0.1199152
1.7995660
-0.2610622
3.8806777
1.0
1.0
-1.0
-0.9892398
-0.7398144
-0.8191061
0.2837815
2.0
1.0
-1.0
0.4895942
3.2751343
0.3368945
8.3870186
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:10 No:02 79
109602-3737 IJBAS-IJENS © April 2010 IJENS I J E N S
TABLE IV
NUMERICAL VALUES PROPORTIONAL TOfC
AND xNu FOR DIFFERENT VALUES OF M , Pr AND
n .
M n fC xNu Pr n
fC xNu
0.0
1.0
-1.0
-0.0861588
0.4192974
-1.0091940
0.0946265
0.72
1.0
-1.0
-0.3425068
0.1279294
-0.9615057
0.1244760
0.5
1.0
-1.0
-0.3425068
0.1279294
-0.9615057
0.1244760
1.0
1.0
-1.0
-0.4855775
-0.0624840
-1.2034948
0.0049507
1.0
1.0
-1.0
-0.5701267
-0.1323773
-0.9168410
0.1560966
3.0
1.0
-1.0
-0.9567890
-0.7418091
-2.7618639
-1.0987332
1.5
1.0
-1.0
-0.7751737
-0.3667972
-0.8747137
0.1893687
7.0
1.0
-1.0
-1.2204772
-1.1310145
-5.5449711
-3.4881805
2.0
1.0
-1.0
-0.9619481
-0.5794918
-0.8345793
0.2243773
10.0
1.0
-1.0
-1.3000409
-1.2432359
-7.5168460
-5.2959597
V. CONCLUSION
A mathematical model has been derived for MHD heat
transfer mixed convection flow in the presence of
magnetic field with heat generation. A benchmarked
numerical solution has been obtained to the transformed
boundary layer equations using Nachtsheim-Swigert
(1965) shooting iteration technique along with sixth-order
Runge-Kutta integration scheme. The numerical code has
been verified by comparison with previous computation by
Mohamed Ali et al. (2002), for the absence of magnetic
field and heat generation. We see that velocity increases
but the temperature decreases as we increase the buoyancy
parameter both in the cases of suction or injection. But the
suction always stabilizes both the velocity and temperature
fields. If we increase Prandtl number, velocity and
temperature profiles decrease. The temperature index
n has great effect on both velocity and temperature
profiles and increasing it reduces the shear stress and but
increases the heat transfer coefficient at the surface.
Magnetic field parameter has significant effect on velocity
profiles. Velocity and temperature profiles increase rapidly
with the increase of the heat source parameter therefore
using heat source parameter the velocity and temperature
fields can be controlled.
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