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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print),ISSN 0976 6359(Online) Volume 3, Issue 2, May-August (2012), IAEME
573
COMBINED HEAT AND MASS TRANSFER IN MHD THREE-DIMENSIONAL
POROUS FLOW WITH PERIODIC PERMEABILITY & HEAT ABSORPTION
Dr P.Ravinder Reddy1
Dr K.Srihari2
Dr S. Raji Reddy2
1Department of Mechanical Engineering, CBIT, Gandipet,Hyderabad, (A.P), India
500 075,+91-040-23518467,email:[email protected], fax:+91-08413-234155
2Department of Mathematics, Mahatma Gandhi institute of technology
Gandipet, Hyderabad, (A.P), India, 500075.
ABSTRACT
The paper analyzed the effects of mass transfer and heat sink on three-dimensional free
convective heat transfer flow through a highly porous medium with periodic permeability, in the presence
of transfers magnetic field. Assuming the free stream velocity to be uniform, solutions of governing
equations of motion are obtained, using finite deference technique, which is more economical from
computational view point. The results obtained for the velocity, temperature, concentration, skin friction,
rate of heat and mass transfer are discussed and analyzed through graphs, to observe the effects of various
flow parameters. It is found that the concentration of the species is higher for small values of Sc and
lower for larger values of Sc. Also it is found that heat sink and magnetic field reduces the velocity of the
fluid while heat transfer coefficient increases in the presence of heat absorption parameter.
Key words
Volumetric rate of heat absorption; Magnetic field; Porous medium; Periodic permeability; Finite
deference technique.
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING ANDTECHNOLOGY (IJMET)
ISSN 0976 6340 (Print)
ISSN 0976 6359 (Online)
Volume 3, Issue 2, May-August (2012), pp. 573-593
IAEME: www.iaeme.com/ijmet.html
Journal Impact Factor (2012): 3.8071 (Calculated by GISI)
www.jifactor.com
IJMET
I A E M E
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INTRODUCTION
Many natural phenomena and engineering applications are susceptible to magneto hydrodynamics
(MHD) analysis. From technological point of view, magneto-hydrodynamic flow finds application in thefields of stellar and planetary magneto-spheres, aeronautics, meteorology, solar physics, cosmic fluid
dynamics, chemical engineering, electronics, and induction flow metry, MHD generators, MHDaccelerators, construction of turbine and other centrifugal machines. Due to its increasing importance invarious technical applications using magneto hydrodynamic effect, it is desirable to extend many of the
available hydrodynamic solutions for those cases when the viscous fluid is electrically conducting.
Also,in the recent years, the flows through porous medium are of principal interest because theseare quite prevalent in nature. Such Flows have many scientific and engineering applications, viz., in the
fields of agricultural engineering to study the under ground water resources, seepage of water in river beds;
in chemical engineering for filtration and purification processes In view of these applications, a series ofinvestigations have been made by Raptis et al. [1-3] in to the steady flow past a vertical wall. Raptis [4]
studied the unsteady flow through porous medium bounded by an infinite porous plate subjected to a
constant suction and variable temperature. Raptis and Perdikis [5] further studied the problem of freeconvective flow through a porous medium bounded by a vertical porous plate with constant suction where
the free stream velocity oscillates in time about a constant mean value.
In all the studies mentioned above the permeability of the porous medium has been assumed asconstant. In fact, a porous material containing the fluid is a non-homogeneous medium and there can be
numerous in homogeneities present in a porous medium. Therefore, the permeability of the porous medium
may not necessarily be constant. Sing and Suresh Kumar [6] have analyzed a free convective twodimensional unsteady flow through a highly porous medium bounded by an infinite vertical porous plate
when the permeability of the medium fluctuates in time about a constant mean. Most of the investigators
have restricted themselves to two-dimensional flows only by assuming either constant or time dependent
permeability of the porous medium. However, there may arise situations where the flow field may beessentially three dimensional, for example, when variation of the permeability distribution is transverse to
the potential flow. The effect of such a transverse permeability distribution of the porous medium boundedby horizontal flat plate has been studied by Sing and Verma [7] and Singh et al [8]. Recently, Singh and
Sharma [9] studied the effect of transverse periodic variation of the permeability on the heat transfer and
the free-convective of a viscous incompressible fluid through a highly porous medium bounded by a
vertical porous plate. In addition to this, more recently, Jain et al [10] studied the effects of periodictemperature and periodic permeability on three-dimensional free convective flow through porous medium
in slip flow regime. But, in all the above mentioned three-dimensional studies, the effects of mass transfer
and heat sink in the presence of magnetic field have not been studied.
Coupled heat and mass transfer phenomenon in porous media is gaining attention due to itsinteresting applications. Processes involving in heat and mass transfer in porous media are oftenencountered in the chemical industry, in reservoir engineering in connection with chemical recovery
process, in the study dynamics of hot and salty springs of a sea. Underground spreading of chemical and
other pollutants, grain storage and evaporation cooling. Also, the propagation of thermal energy in the
presence of heat sink have great applications in various fields of energy, atomic pollutions, space scienceand in engineering and technology there are occasions where the heat sink is needed to maintain desired
heat transfer.
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In view of these applications, the aim of the present investigation is to study the effects of masstransfer and heat absorbing sink on three-dimensional free convective heat transfer flow with periodic
permeability, under the influence of transfers magnetic field. In the present work, the effects of different
flow parameters encountered in the equations were also studied. In the above stated three-dimensionalstudies, a series expansion method was employed to solve the fluid flow problem. But, the present
problem has been solved numerically, using finite difference method, because it is more economical
from computational view point.
Mathematical analysis:
We now consider the flow of a viscous fluid through a highly porous medium bounded by an infinite
vertical porous plate with constant suction. The plate is lying vertically on the x*-z
*plane with x
*-axis
taken along the plate in the upward direction. The y*-axis is taken normal to the plane of plate and
directed into the fluid flowing laminarly with a uniform free stream velocity U. A magnetic field ofuniform strength is applied normal to the flow, along *y -axis. The permeability of the porous medium is
assumed to be of the form.
)/cos1()(
*
*
0**
Lz
KzK
+= (1)
Where *0K is the mean permeability of the medium. L is the wavelength of the permeability
distribution and (
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Thus, denoting velocity components by *** ,, wvu in the directions of *** ,, zyx respectively and
the temperature by the T*
and concentration by C*, the flow through a highly porous medium is governed
by following equations:
0*
*
*
*
=
+
z
w
y
v(2)
*
2
0*
*2*
*2
2*
*2*****
*
**
*
** )()()( u
BUu
Kz
u
y
uCCgTTg
z
uw
y
uv
+
++=
+
(3)
*
*2*
*2
2*
*2
*
*
*
**
*
** 1
v
Kz
v
y
v
y
p
z
vw
y
vv
+
+
=
+
(4)
*
2
0*
*2*
*2
2*
*2
*
*
*
**
*
** 1 w
Bw
Kz
w
y
w
z
p
z
ww
y
wv
+
+
=
+
(5)
)(2*
*2
2*
*2
*
**
*
**
+
=
+
TTQ
z
T
y
T
C
k
z
Tw
y
Tv
p(6)
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+
=
+
2*
*2
2*
*2
*
**
*
**
z
C
y
CD
z
Cw
y
Cv (7)
The boundary conditions of the problem are:
********,,0,,0;0 ww CCTTwVvuy ======
********* ,,,0,; CCTTppwUuy (8)
where *wT and*
wC are the temperature and concentration of the plate,*
T and*
C are the temperature and
concentration of the fluid far away from the plate, *p is a constant pressure in the free stream and V>0
is a constant and the negative sign indicates that suction towards the plate.
Introducing the following non dimensional quantities:
,*
L
yy = ,*
L
zz = ,*
U
uu = ,*
V
vv =
,*
V
ww = ,
2
*
U
pp
= ,
**
**
=
TT
TT
w
**
**
=
CC
CC
w
(9)
in Eq. (2) to (7)., the following equations are obtained:
0=
+
z
w
y
v(10)
uM
K
zu
z
u
y
uGmGr
z
uw
y
uv
ReRe
)cos1()1(
Re
1ReRe
2
02
2
2
2
+
+
++=
+
(11)
( )
02
2
2
2
Re
cos1
Re
1
K
vz
z
v
y
v
y
p
z
vw
y
vv
+
+
+
=
+
(12)
( )w
M
K
wz
z
w
y
w
z
p
z
ww
y
wv
ReRe
cos1
Re
12
02
2
2
2
+
+
+
=
+
(13)
Szyz
wy
v
+
=
+
2
2
2
2
PrRe1 (14)
+
=
+
2
2
2
2
Re
1
zySczw
yv (15)
where
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( )2
**
UV
TTgGr w
=
(Grashof number)
2
***
UV
CCgG w
m
=
(Modified Grashof number)
=
VLRe (Reynolds number),
k
Cp=Pr (Prandtl number)
DSc
= (Schmidt number),
2
*
00
L
KK = (Permeability parameter)
LBM 0= (Magnetic parameter),
2QLS = (Heat absorption parameter)
The corresponding boundary conditions reduce to
;0=y ,0=u ,1=v ,0=w ,1= 1= (16)
;y ,1u ,1w , pp ,0 0
In order to solve the problem we assume the solutions of the following form because the
amplitude ( )1 is very small:
( ) ( ) ( ) ( ) ...,,, 22
10 +++= zyuzyuyuzyu
( ) ( ) ( ) ( ) ...,,, 22
10 +++= zyvzyvyvzyu
( ) ( ) ( ) ( ) ...,,, 22
10 +++= zywzywywzyw (17)
( ) ( ) ( ) ( ) ...,,, 22
10 +++= zypzypypzyp
( ) ( ) ( ) ( ) ...,,, 22
10 +++= zyzyyzy
( ) ( ) ( ) ( ) ...,,, 22
10 +++= zyzyyzy
When ,0= the problem is reduced to the two-dimensional free convective flow through a porous
medium with constant permeability which is governed by following equations:
00 =dy
dv(18)
00
20
20
0
2002
02
1ReRe
1Re
KGmGru
KM
dy
duv
dy
ud=
+ (19)
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0PrRe 00
02
02
=
Sdy
dv
dy
d(20)
0Re
0
02
02
=
dy
d
Scvdy
d
(21)
The corresponding boundary conditions become
;0=y ,00 =u ,10 =v ,10 = 10 = (22)
;y ,10 u ,0 pp ,00 00
The solutions of Eq. (18) to (21). under the boundary conditions (22) are given by
( )
yScyryReGmeGreGmGru
Re
10100.11
++= (23)
rye
=0 (24)
ySce
Re0
= (25)
with
,10 =v 00 =w and = pp0 (26)
where
+
=
0
22
2
01
Re
Re
KMrr
,2
4PrRePrRe 22 Sr
++=
( )
+
=
0
22
2
11
1Re
Re
KMScSc
,0
22 1
4
Re
2
Re
KMR +++=
When ,0 substituting (17) in Eq. (10) to (15). and comparing the coefficients of identical power
of , neglecting the higher powers of , the following equations are obtained with the help of Eq.
(26).:
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011 =
+
z
w
y
v(27)
( )
1
2
0
10
2
12
2
12
111
10
1 ReRe
cos1
Re
1ReRe u
M
K
uzu
z
u
y
uGG
y
u
y
uv
+
+
++=
(28)
( )
0
121
2
21
211
Re
cos
Re
1
K
zv
z
v
y
v
y
p
y
v
+
+
=
(29)
1
2
0
121
2
21
211
ReReRe
1w
M
K
w
z
w
y
w
z
p
y
w
+
+
=
(30)
12
12
2
12
101
PrRe
1
S
zyyy
v
+
=
(31)
+
=
21
2
21
210
1Re
1
zyScyyv (32)
The corresponding boundary conditions are:
;0=y ,01 =u ,01 =v ,01 =w ,01 = 01 = (33)
;y ,01 u ,01 w ,01 p ,01 01
Eq. (27) to (32). are the partial differential equations, which describe free convective three-
dimensional flow. In order to solve these equations we shall first consider (27), (29) and (30), being
independent of the main flow component 1u , temperature field 1 and concentration field 1 . In the
following form, 11 , wv and 1p are assumed:
( ) ( ) zyvzyv cos, 111 = (34)
( ) ( ) zyvzyw
sin1
, 111 = (35)
( ) ( ) zypzyp cos, 111 = (36)
Where the prime in )(11 yv denotes the differentiation with respect to .y Expressions for ( )zyv ,1
and ( )zyw ,1 have been chosen so that the equation of continuity (28) is satisfied. Substituting the
expressions (34), (35) & (36) in (29) and (30), the following differential equations can be obtained:
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( ) ( )1Re
1
Re
111
01111
21111 +++= v
Kvvpv (37)
11
2
0
1111
1111
11
ReReRe
1v
M
K
vv
vp
v
+=
(38)
Eliminating the terms 1111 ,pp in Eq. (37) and (38)., the following is obtained.
0Re21
Re0
2
110
24
112
112
0
21111 =
+
++
+++
Kv
Kvv
KMvv iv (39)
The corresponding boundary conditions become
0:
0,0:0
11
1111
=
===
vy
vvy. (40)
In order to solve the differential Eq. (28), (31) and (32). for 11 ,u and 1 respectively,
the following are assumed
zyuzyu cos)(),( 111 = (41)
zyzy cos)(),( 111 = (42)
zyzy cos)(),( 111 = . (43)
Substituting the above equations in (28), (31) and (32), the following equations can be obtained:
0011
2111
201111
2
0
21111
1ReReRe
1Re K
uGGuvuKMuu
+=
+++ (44)
011112
1111 PrRePrRe =+ v (45)
011112
1111 ReRe =+ vScSc (46)
with corresponding boundary conditions
0,0,0:0 111111 ==== uy (47)
0,0,0: 111111 uy .
Substituting the following finite difference formulae
h
iviviv
2
)1()1()( 111111
+=
2
11111111
)1()(2)1()(
h
iviviviv
++=
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3
1111111111
2
)2()1(2)1(2)2()(
h
iviviviviv
+++=
4
111111111111
)2()1(4)(6)1(4)2()(
h
iviviviviviv iv
++++=
in Eq. (39)., we get
02)2()1()()1()2(0
42
115114113112111 =+++++K
hivAivAivAivAivA
(48)
where
hA Re21 +=
232
0
222 Re2
12Re28 +
++++= h
KMhhA
++
+++=
0
2442
0
223 22
1412
Kh
KMhA
232
0
224 Re2
12Re28
+++= h
KMhhA
hA Re25 = .
Substitution of similar finite difference formulae in Eq. (44) to (46)., the following equations are
obtained:
)()1()()1( 115111111 iBiuAiuBiuA =++ (49)
)()1()()1( 321 iDiDiDiD =++ (50)
)()1()()1( 321 iEiEiDiE =++ (51)
where RandAA 1051 ,,, have already been defined and
hD PrRe21 +=
22
2 24 hD +=
hD PrRe23 =
ihr eivPhiD
PrRe11
2 )()Re(2)( =
SchE Re21 +=
hScE Re23 =
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ihSceivSchiE Re112 )()Re(2)( =
+++= 2
0
22
1
124
KMhB
( ) )(2
)()(Re)(2)()()( 30
2
12
211 iBK
hiGiGhiBivRiB e ++=
where ( ) ihRihScih eGGRSceGeGiB ++= 1RePrRe)( 110Re
11
PrRe
02
( ) ihRihScih eGGeGeGiB ++= 1)( 110Re
11
PrRe
03 .
Eq. (48), (49), (50) and (51). have been solved by Gauss-seidel iteration methodfor velocity,
temperature and concentration. Also, numerical solutions for these equations have been obtained, using
C-Program. To prove convergence of finite difference scheme, the computation is carried out for
slightly changed value of h , running same program. Negligible change is observed in the values
andu, and also after each cycle of iteration the convergence checking is performed, i.e.
81 10+
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( )0
**
*
=
=
=
ypwp
uyLVC
k
TTVC
qN
0
110
cosPrRe
1
=
+
= yz
dy
d
dy
d
(53)
Sherwood - number
Knowing the concentration field, the expression for the rate of mass transfer in terms of modified
Nusselt number is given by
( )0
**
*
1
=
=
=
yw
uyVL
D
CCV
DqN
0
110
cosRe
1
=
+
=y
zdy
d
dy
d
Sc (54)
Results and discussion:
In order to get the physical insight of the problem, numerical calculations are carried out for
different flow parameters such as Heat absorption Parameter S, Grashof number Gr, Modified Grashof
number Gm, Magnetic parameter M, Permeability Parameter K0, Reynolds number Re, Prandtl number
(Pr) and Schmidt number (Sc) are studied. During the course of numerical calculations, the values of Pr
are chosen to be 0.71 &1.0 corresponding to air and electrolytic solution. But the propagation of
thermal energy through electrolytic solution in the presence of heat absorbing sink and magnetic field
has wide range of applications in chemical, aeronautical engineering and atomic propulsion space
science.
Fig. 1. shows the effect of free convection parameter Gr on velocity field u for cooling of the
plate both in the presence and absence of heat absorption parameter. It is observed that the velocity
increases due to greater cooling of the channel (as Gr increases). Further, it is interesting to note that the
velocity of fluid decreases in the presence of heat absorption parameter. In Fig. 2. the effects of Gm,
K on velocity field u has been exhibited by the curves both in presence and absence of heat absorption.
It is observed that an increase in Gm and K leads to an increase in the velocity, but it decreases in the
presence of heat absorption. This is in good agreement with the physical fact that heat sink decreases
the velocity of the fluid.
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Fig. 3. and Fig. 4. are drawn for various values S and M respectively; on velocity field u. it is
observed from these figures that the velocity of the fluid decreases with the increase of heat absorption
and magnetic parameter. Fig. 5. reveals that an increase in Pr and Sc decreases the velocity when in
the presence of heat absorption parameter.
Fig. 6. display the effects of Re and Pr on temperature profile in the presence of heat
absorption. From this it is evident that the fluid temperature decreases due to increase in the Pr. This is
in agreement with the physical fact that the thermal boundary layer thickness decreases with increase in
Pr. The reason underlying such a behavior is that the high prandtl number fluid has a low thermal
conductivity. This results in the reduction of the thermal boundary layer thickness. This figure also
shows that an increase in heat absorption parameter and Reynolds number leads to decrease in the
temperature of the fluid. Further, it is interesting to note that the effect of heat absorption on
temperature is more significant than in the case of velocity field.
Fig. 7. depicts the species concentration for different gases like Hydrogen (H2: Sc=0.22),
Oxygen (O2: Sc=0.66) and methanol (Sc=1.0) at a temperature 250C and 1 atmospheric pressure. The
values of Schmidt number (Sc) are chosen to represent the most common diffusing chemical species
which are of interest. A comparison of the curves in the figure shows a decrease in concentration
distribution with an increase in Schmidt number because the smaller values of Sc are equalent to
increasing chemical molecular diffusivity (D). Hence, the concentration of the species is higher for
small values of Sc and lower for larger values of Sc. This figure also shows that an increase in Re,
decreases the concentration field.
In Fig. 8. the non-dimensional skin-friction coefficient plotted against the Reynolds number for
different values of Gr, Gm and S. it is evident from this figure that the skin-friction increases with the
increase of Gr and Gm while it decreases in the presence of heat absorption parameter S. Fig. 9. is
plotted for non-dimensional heat transfer coefficient versus Reynolds number for different values of S
and Pr .It is observed that the heat transfer coefficient increases in the presence of heat absorption but, it
decreases with the increase of Pr. In Fig. 10. variation of the non-dimensional mass transfer
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coefficient is plotted against the Reynolds number for various values of Sc. It has been observed that
the mass transfer coefficient decreases as the value of Sc decreases.
CONCLUSIONS
The following conclusions have been drawn from the above results:
1. The effect of a heat absorbing sink on steady incompressible three-dimensional fluid flow
through a highly porous medium is to suppress the velocity and temperature fields, which is turn,
causes the enhancement of the heat transfer coefficient.
2. The velocity of a fluid decreases in the presence magnetic field. This due to the fact that
magnetic field reduces the velocity field.
3. The effect of heat absorption parameter on temperature field is more significant than in the case
of velocity field.
4. The concentration of the species is higher for small values of Sc and lower for larger values of
Sc.
5. This problem has been solved numerically, using finite difference technique and the results
obtained are in good agreement with the experimental results, as valid in the literature.
Nomenclature
G Acceleration due to gravity
Coefficient of volumetric thermal expansion
* Coefficient of mass expansion
*p Pressure
Density
Kinematics viscosity
Viscosity
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k
Thermal conductivity
Cp Specific heat at constant pressure
D Concentration diffusivity
*wC Concentration of the plate
*
wT Temperature of the plate,
*
T Temperature of the fluid far away from the plate
*
C Concentration of the fluid far away from the plate
Gr Grashof number
Gm Modified Grashof number
Re Reynolds numberB0 Magnetic field component
Sc Schmidt number
K0 Permeability parameter
M Magnetic parameter
Q Volumetric rate of Heat absorption
S Heat absorption parameter
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print),ISSN 0976 6359(Online) Volume 3, Issue 2, May-August (2012), IAEME
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0
1
2
3
4
5
6
7
8
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5 2
____ S=0.0
------- S=2.0Gr=15.0
Gr=1.0
Gr=10.0
Gr=5.0
Fig.1- Effect of Gr on velocity field u when Gm=1.0,
Re=5.0, M=1.0, KO=1.0, Pr=0.71, Sc=0.66, =0.1 andZ=0.0
Fig.2-Effects of Gm, K0 and S on velocity field u when
Gr=1.0, M=1.0, Re=5.0, Pr=0.71, Sc=0.66, =0.1 and Z=0.0
Gm K0 S
1) 1.0 1.0 0.0
2) 1.0 1.0 2.0
3) 1.0 4.0 0.0
4) 3.0 1.0 2.0
1
4
3
2
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0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.5 1 1.5 2
Fig.3- Effect of S on velocity field u when Gr=1.0, Gm=1.0,
Re=5.0, K0=1.0, M=1.0, Pr=0.71, Sc=0.66, =0.1 and Z=0.0
Fig.4-Effect of M on velocity field u when Gr=1.0, Gm=1.0,
Re=5.0, K0 =1.0, Pr=0.71, Sc=0.66, S=1.0, =0.1 andZ=0.0
S=2.0
S=3.0
S=1.0S=0.0
M=0.0
M=1.0
M=2.0
M=3.0
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0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4.
Pr Sc S
1) 0.71 0.66 0.0
2) 0.71 0.66 1.0
3) 7.0 0.66 1.0
4) 0.71 2.62 1.0
Fig.5-Effects of Pr and Sc on velocity field u when Gr=1.0,
Gm=1.0, Re=5.0, K0=1.0, M=1.0, =0.1 and Z=0.0
Fig.6-Effects of Re, Pr and S on Temperature field
when M=1.0, K0=1.0, =0.1 and Z=0.0
1
2
34
1
2
3
5
4
Re Pr S
1) 2.0 0.71 0.0
2) 2.0 0.71 2.03) 5.0 0.71 2.0
4) 5.0 1.0 2.0
5) 10.0 0.71 2.0
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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print),ISSN 0976 6359(Online) Volume 3, Issue 2, May-August (2012), IAEME
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0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4
Re Sc
1) 5.0 0.22
2) 5.0 0.663) 5.0 1.0
4) 10.0 0.22
0
1
2
3
4
5
6
7
8
9
0 5 10 15 20 25 30
Skin-frictio
n
Gr=1.0,Gm=1.0,S=0.0
Gr=1.0,Gm=1.0,S=2.0
Gr=5.0,Gm=1.0,S=2.0
Gr=1.0,Gm=5.0,S=2.0
Fig.7- Effects of Re and Sc on Concentration field
when M=1.0, Ko=1.0, =0.1 and Z=0.0
Fig.8-Effect of S, Gr and Gm on Skin-friction Coefficient
when M=1.0,Ko=1.0, Pr=0.71, Sc=0.66, =0.1 and Z=0.0
1
2
3
4
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30
S=0.0
S=2.0
Pr=0.71
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30
Sc=0.22
Sc=0.66
Sc=1.0
Fig.9-Effect of S and Pr on Nusselt number Nu
when M=1.0 Ko=1.0 =0.1 and Z=0.0
Fig.10-Effect of Sc on Sherwood number Sh
when Re=5.0, M=1.0, Ko=1.0, =0.1 and Z=0.0
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