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MHD Boundary Layer Flow of Heat and Mass Transfer in the presence of
Heat Generation/Absorption, radiation and chemical reaction of a
temperature dependent Viscous Fluid Past a time dependent permeable
Vertical Plate under Oscillatory Suction Velocity
Adebile E.A1 and Sogbetun L.O
2
Department of Mathematical Sciences
Federal University of Technology, Akure
Abstract
In this research, the researchers studied and made analysis on the MHD boundary layer flow of a
variable viscous fluid over a vertical porous plate in a porous medium of time dependent
permeability in the presence of radiation and chemical reaction under oscillatory suction velocity
taking into account the heat generation/absorption and reaction parameter effects. A time
dependent suction was assumed and the radiative flux was described using the differential
approximation for radiation. The governing system of partial differential equations was
linearised using asymptotic techniques. Computational results and graphical representations
showing the effects of the governing model parameters were made and found to be in good
agreement with those in the literature.
Keywords: heat generation/absorption, reaction parameter, MHD, vertical porous plate,
viscous fluid, suction velocity
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1.0 Literature review
Interest on MHD flow through a porous plate by researchers has tremendouslyincreased over the
years. This is due to the fact that heat and mas transfer through porous media occur in many
engineering, geophysical and biological applications. For instance, permeable porous plates are
used in the filtration processes and also for heated body to keep its temperature constant and to
make the heat insulation of the surfaces more effective. Many investigators have considered
works on the unsteady oscillatory free convective flow through porous media because of its
importance in chemical engineering; turbo machinery and aerospace technology.
Kumar et al [2] in his work,examined an unsteady oscillatory laminar free convective fluid
through a porous medium along a porous hot vertical plate with time dependent suction in the
presence of heat source/sink. In another development, Soundalgekar et al [3],analysed free
convective effects on the oscillatory flow past an infinite vertical porous plate with constant
suction. Singh et al 43], and Venkateswarlu and Rao [5] in their researches studied the effects of
permeability variation and oscillatory suction velocity on free convection and mass transfer flow
of a viscous fluid past an infinite vertical porous plate in the presence of a uniform transverse
magnetic field. Okedoye et al [6 ], on the other hand, researched on the unsteady
magnetohydrodynamic heat and mass transfer in MHD flow of an incompressible, electrically
conducting, viscous fluid past an infinite vertical porous plate along with porous medium of time
dependent permeability under oscillatory suction velocity normal to the plate.
All these aforementioned references did not considered flows involving effects of heat
generation/absorption and reaction parameter on the oscillatory suction velocity in the presence
of temperature dependent viscosity while such flows are encountered in various fields.Adebile
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and Sogbetun[1] very recently investigated MHD flow of a fluid with temperature dependent
viscosity but considered the steady situationuner the influence of constant suction velocity.In the
present study, we extend the previos work by investiging the effects of heat
generation//absorption and reaction parameter on the unsteady MHD boundary layer flow of a
variable viscous fluid over a vertical porous plate in a porous medium of time dependent
permeability in the presence of radiation and chemical reaction under oscillatory suction
velocity. The permeability of the porous medium is considered to be ''
10
' tineKtK and the
suction velocity is assumed to be ''
10
' tinevtv where 00v and 1 is a positive
constant.
1.1 Nomenclature
:u Velocity along x coordinate :'T Non dimensional fluid temperature
:v Velocity along y coordinate :'C Non dimensional species concentration
:g Acceleration due to gravity :T Fluid temperature
:'U Non dimensional fluid velocity : Reaction parameter
:wT Ambient temperature :* Stefan- Boltzmann constant
:C Species concentration :B Coefficient of mass expansion
:wC Ambient species concentration :B Coefficient of thermal expansion
:0B Transverse magnetic field :w Ambient density
: Skin-friction coefficient : Electrical conductivity
: Heat generation/absorption coefficient : Density of the fluid
:k Thermal conductivity :Sc Schmidt number
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:pc Specific heat at constant pressure :D Molar diffusivity
:A Pre-exponential factor :rcG Mass grashof number
:0v Normal velocity at the plate :rG Thermal grashof number
:*k Mean absorption coefficient : Delta, 10 o
:M Hartmann number : Epsilon, 10 o
:Nu Nuselt number :Sh Sherwood number
: Angular velocity :t Time
:Pr Prandtl number : Fluid viscosity
2.0 Mathematical Formulation
A magnetohydrodynamic flow of viscous, incompressible, electrically conducting fluid past an
infinite vertical plate in a porous medium under suction velocity is considered. The x- axis is
taken along the plate in the direction of the flow and y- axis normal to it. A uniform magnetic
field is applied normal to the direction of the flow. It is assumed that the magnetic Reynold
number is less than unity so that the induced magnetic field is neglected in comparison to the
applied magnetic field. We further assumed that all the fluid properties are constant except that
of the influence of density variation with temperature. Thus, the basic flow in the medium is
entirely due to buoyancy force caused by temperature difference between the wall and the
medium. Initially at 0t , the plate as well as fluid is assumed to be at the same temperature and
the concentration of species is very low so that the Soret and Dofour effect are neglected [6].
When 0t , the temperature of the plate is instantaneously raised (or lowered) to '
wT and the
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concentration of species is raised (or lowered) to '
wC . Under the above assumptions and taking
the usual Boussinesq’s approximation into account, the governing equations for momentum,
energy and concentration are presented below:
0'
'
dy
dv
(2.1)
CCAy
CD
y
Cv
t
C '
2'
'2
'
''
'
'
(2.2)
TTy
T
c
k
y
Tv
t
T
p
t ''
2'
'2
'
''
'
'
(2.3)
'2
0
'
'''*
'
'
''
''
'
'
1
1 UB
ek
UCCgTTg
y
U
yy
Uv
t
Uiwt
(2.4)
The boundary conditions are:
0'U TTTeT w
iwt1'
ww
iwt CCCeC 1'at 0'y
0'U , TT ' CC '
as 'y (2.5)
The suction velocity from equation (2.1) is assumed to be )1(0
' iwtevv where
00v and
1 is positive constant.
Introducing the following non dimensional quantities:
'
0 yvy
f
tvt
'2
0 0
'
v
UU
2
0
'4
v
nw
TT
TT
w
'
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CC
CCC
w
'
2
2
0
'vkk
2
0v
A
3
0
*
v
TTgG w
rD
Sc
t
P
k
cPr
0
0
v
BM
2
0
'
v and
3
0v
CCgG w
rc
Below are the non-dimensional governing equations for momentum, energy and concentration of
the unsteady state and their boundary conditions:
Cy
C
Scy
Ce
t
C iwt
2
211
4
1
(2.6)
2
2
Pr
11
4
1
yye
t
iwt (2.7)
UMeky
U
yCGG
y
Ue
t
Uiwtrcr
iwt 2
1
11
4
1
(2.8)
The relevant boundary conditions in dimensionless form are:
,0U ,1 iwte iwteC 1 on 0y
,0U ,0 0C as y (2.9)
The fluid viscosity was assumed to obey the Reynolds model [7]
e (2.10)
Where , is a parameter depending on the nature of the fluid. Using equation (2.10) in equation
(2.8), we obtain
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UMeky
Ue
yCGG
y
Ue
t
Uiwtrcr
iwt 2
1
11
4
1
(2.11)
3.0 Method of Solution
To solve equations (2.6), (2.7) and (2.11), we seek an asymptotic expansion about for our
dependent variables of the form
.....0, 2
10
iwteUyUtyU (3.1a)
.....0, 2
10
iwteyty (3.1b)
.....0, 2
10
iwteCyCtyC (3.1c)
Corresponding to the species equation we have
00
'
0
''
0 CScScCC (3.2)
,100C 00 yC as y
'
01
'
1
''
14
1ScCCiwScScCC
(3.3)
,101C
01 yC as y
Corresponding to the energy equation we have
0PrPr 0
'
0
''
0 (3.4)
100 00 as y
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'
01
'
1
''
1 Pr4
1PrPr iw (3.5)
101 01 as y
Corresponding to the momentum equation we have
000
2
0
'
0
'
0
1CGGUM
kUUe
yrcr (3.6)
,000U ,00U
'
0111
2
0
1
'
1
'
1
1
4
1UCGGUM
kiwUUUe
yrcr (3.7)
,001U 01U
Solving equations (3.2)-(3.5) and substitute the results into (3.1b) and (3.1c), we have
iwtnyxyny eeaeaeyC 108 (3.8)
iwtmyymy eeaeaety 53, (3.9)
Where
44
2
1 2 iwSSSx ccc ccc SSSn 4
2
1 2
4
2
10iw
SnSn
nSa
cc
c
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108 1 aarrr PPPm 4
2
1 2
44
2
1 2 iwPPP rrr
4
2
5iw
PmPm
mPa
rr
r
53 1 aa
To solve equations (3.6)-(3.7), we make use of the following transformation:
Let o where 1 . Thus we assumed the following:
tohUUU ......01000 (3.10)
tohUUU ......11101
(3.11)
Substitute equations (3.12)-(3.14) into equations (3.6)-(3.7) and compile the order of .
We have
0000
2
0
00
2
00
21
CGGUMkdy
dU
dy
Udrcr
(3.12)
0)0(00U 0)(00U
01
01
2
0
01
2
01
2
00 UMkdy
dU
dy
Ud
dy
dU
dy
d (3.13)
0)0(01U 0)(01U
dy
dUCGGUM
kiw
dy
dU
dy
Udrcr
00
1110
2
0
10
2
10
21
4
1
(3.14)
,0010U 010U
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dy
dUUM
kiw
dy
dU
dy
Ud
dy
dUe
dy
d
dy
dU
dy
d iwt 01
11
2
0
11
2
11
2
10
1
10
0
1
4
1
,0011U 011U (3.15)
Based on the solutions obtained in equations (3.12) and (3.15), we calculate equation (3.1a) to be
iwt
yyynyxyy
ymymnymxmyymymy
ynyxymyyy
ynymyymnmyymynymyy
e
eaeaeaeaeaea
eaeaeaeaeaeaea
eaeaeaeaeaea
eaeaeaeaeaeaeaeaeaeatyU
43424140
2
3938
373635
2
34333230
292827262523
22212019
2
181715141311,
(3.16)
Where
2
0
1411
2
1M
k
2
0
1411
2
1M
kccc SSSn 4
2
1 2
rrr PPPm 42
1 2
44
2
1 2 iwPPP rrr
2
0
1
4411
2
1M
k
iw
2
0
1
4411
2
1M
k
iw
44
2
1 2 iwSSSx ccc
2
0
2
13
1M
kmm
Ga r
2
0
2
14
1M
knn
Ga rc
141311 aaa
2
0
2
1155
17
1M
kmm
aeameama
iwtiwt
2
0
2
2
135
18
124
21
Mk
mm
maeaa
iwt
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2
0
2
1455
19
1M
kmnmn
namneameana
iwtiwt
2
0
2
113
20
1M
k
eaaa
iwt
2
0
2
313
21
1M
kmm
meaama
iwt
2
0
2
314
22
1M
knn
enaana
iwt
22212019181715 aaaaaaa
2
0
2
3
25
1
4M
k
iw
aGa rt
2
0
2
135
26
1
4M
k
iwmm
maaGa rt
2
0
2
8
27
1
4M
k
iwxx
aGa rc
2
0
2
1410
28
1
4M
k
iwnn
naaGa rc
2
0
2
11
29
1
4M
k
iw
aa
292827262523 aaaaaa
2
0
2
235
32
1
4
1
Mk
iwmm
ameaa
iwt
2
0
2
2125326525
33
1
4M
k
iwmm
aamemaameaama
iwtiwt
2
0
2
18
2
265
34
1
424
21
Mk
iwmm
amaeaa
iwt
2
0
2
275
35
1
4
1
Mk
iwmxmx
xaxmeaa
iwt
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2
0
2
1952828
36
1
4M
k
iwmnmn
aenaamnnamna
iwt
2
0
2
1729529
37
1
4M
k
iwmm
aameaama
iwt
2
0
2
323
38
1
4M
k
iw
eaaa
iwt
2
0
2
3
2
25
39
1
424
2
Mk
iw
eaaa
iwt
2
0
2
327
40
1
4M
k
iwxx
exaaxa
iwt
2
0
2
22328
41
1
4M
k
iwnn
aenaana
iwt
2
0
2
20329
42
1
4M
k
iw
aeaaa
iwt
2
0
2
15
43
1
4M
k
iw
aa
43424140393837363534333230 aaaaaaaaaaaaa
Skin-friction coefficient at the plate is
iwt
y
e
aanaxaaa
mamnamxamamamaa
anaxamaaa
namaamnamamaanamaay
U
434241403938
37363534333230
292827262523
22212019181715141311
0
2
2
2
(3.17)
Heat transfer coefficient uN at the plate is
ti
y
u emaamy
N 53
0 (3.18)
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Mass transfer coefficient hS at the plate is
ti
y
h enaxany
CS 108
0
(3.19)
4.0 Discussion of Results
In order to investigate the effect of various varying parameters on the flow behaviour and the
temperature distribution within the boundary layer, computational results were obtained for
various values of ,,,,,,,, 0kSGGM crcr and t with fixed values for rP and cS . These
parameters were assigned the following values ,2.0,2.0,0.1,2.1,5.0 0kGGM rcr
2.0,2.0,5.0 and 2t except where stated otherwise while the values of rP and
cS were taken to be 0.71 and 0.6 respectively for plasma. It should be noted that 0,0 and
0 represent destructive, no and generative chemical reactions respectively. Also,
0,0 and 0 indicates heat absorption, no heat generation/absorption and heat
generation respectively. The figures are presented in 3-dimensional figures.
From equation (2.10), we could see that increase in viscosity parameter leads to decrease in
viscosity.
Numerical values of skin friction are showed in Table 4.1. We observed that an increase in
viscosity parameter, mass Grashof number or the thermal Grashof number increases skin friction
whereas increase in magnetic parameter or reaction parameter leads to a decrease in skin friction
coefficient.
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Table 4.2 displayed the effects of viscosity parameter on the dimensionless velocity. It is
discovered thatincrease in viscosity parameter bring about increase in the fluid velocity near the
plate only while reverse is the case as we move away from the plate. The fluid velocity increased
and reached its maximum value at very short distance from the plate and then decreased to zero.
Figures 4.1 - 4.10, show the effect of the varying parameters on the velocity field. It is observed
that maximum velocity occurs in the body of the fluid close to the surface. Figures 4.1-4.2
highlight the effects of delta and epsilon on the fluid velocity. It could be seen that increase
in delta or epsilon increases velocity. We observed in figure 4.3 that increase in destructive
chemical reaction 0 parameter reduces the velocity field while increase in the generative
chemical reaction 0 parameter increases the velocity. We displayed the effect of on
velocity in figure 4.4; it is shown that increases in brings about increase in velocity field.
Furthermore, we investigate the effect of Hartmann number M on the fluid velocity in figure 4.5.
We discovered that increase in Hartmann number M reduces the velocity field as a result of an
opposing force (Lorentz force). Also, figure 4.6 shows that increase in permeability increases the
velocity. The effectsof mass and thermal Grashof numbers on the velocity field are shown in
fugures 4.7 and 4.8 respectively. We discovered that velocity increases as either mass or thermal
Grashofnumber increases. Figure 4.9 and 4.10 show that increase in and t increase the
velocity.
Figure 4.11 show the effect of chemical reaction parameters on concentration field. It is observed
that for a generative chemical reaction, there exist oscillations in the field away from the surface.
This brings about the presence of minimum and maximum concentration in the field which
however less than the surface concentration. For destructive chemical reaction, the boundary
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layer reduces as the reaction parameters increases. Also there is reduction in concentration field
as reaction parameter increases positively.
In figure 4.13, we display the concentration field as a function of ty, , it could be seen that
concentration decreases as the flow progresses and decreases faster as we move away from the
boundary. While concentration is displayed as a function of t, in figure 4.12. Oscillation is
observed along axiswith a steep decrease in the field as y increases.
We displayed in figure 4.14- 4.16, the temperature profile for various values of parameters under
consideration.
It could be seen from figure 4.14 that heat absorption 0 resulted in decreases in the fluid
body temperature, while heat 0 increases the fluid body temperature which leads to
presence of extremes temperature in the body of the fluid greater than the surface temperature.
In figure 4.16 and 4.15, we show the temperature profile as a function of ty, and t,
respectively. It is observed that the temperature decreases as y increases, and oscillatory field along t and
axis which is more pronounced at the initial stage continuous as y increases.
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02
46
810
-0.2
-0.1
0
0.1
0.2
-0.1
0
0.1
0.2
0.3
ydelta
02
46
810
-0.2
-0.1
0
0.1
0.2
-0.1
0
0.1
0.2
0.3
yepsilon
02
46
810
-2
-1
0
1
2
-0.1
0
0.1
0.2
0.3
yphi
02
46
810
-2
-1
0
1
2
-0.1
0
0.1
0.2
0.3
yreaction parameter
Table 3.10: Skin-friction coefficient when 0 Table 3.11: Velocity yU distribution for various values of
Fig 3.1: Velocity field as function of ,y
Fig 3.2: Velocity field as function of ,y
y 2.0U 1.0U 0.0U 1.0U 2.0U
0 0 0 0 0 0
2 0.0945 0.0943 0.0941 0.0939 0.0937
4 -0.0085 -0.0086 -0.0086 -0.0086 -0.0087
6 -0.0261 -0.0261 -0.0261 -0.0261 -0.0261
8 -0.0134 -0.0134 -0.0134 -0.0134 -0.0134
10 -0.0008 -0.0008 -0.0008 -0.0008 -0.0008
rG rcG 1.0 0k 1.0 0.0 2.0
-2.0 1.0 0.5 -0.5 0.2 -0.2 -0.2664 -0.2900 -0.3373
-1.0 1.0 0.5 -0.5 0.2 -0.2 0.0740 0.0779 0.0857
0.0 1.0 0.5 -0.5 0.2 -0.2 0.4143 0.4458 0.5087
1.2 1.0 0.5 -0.5 0.2 -0.2 0.8228 0.8873 1.0164
1.2 -1.0 0.5 -0.5 0.2 -0.2 -0.0059 -0.0043 -0.0011
1.2 0.0 0.5 -0.5 0.2 -0.2 0.4084 0.4415 0.5076
1.2 1.0 0.0 -0.5 0.2 -0.2 0.8405 0.9067 1.0392
1.2 1.0 1.0 -0.5 0.2 -0.2 0.7759 0.8359 0.9560
1.2 1.0 0.5 -1.0 0.2 -0.2 0.7981 0.8612 0.9872
1.2 1.0 0.5 1.0 0.2 -0.2 0.7169 0.7776 0.8988
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02
46
810
-2
0
2
4
-0.2
0
0.2
0.4
0.6
yGrc
02
46
810
-2
0
2
4
-0.2
0
0.2
0.4
0.6
yGrt
02
46
810
1214
-4
-2
0
2
4
-2
-1
0
1
yk0
02
46
810
0
1
2
3
4
-0.1
0
0.1
0.2
0.3
yM
02
46
810
-10
-5
0
5
10
-0.1
0
0.1
0.2
0.3
yw
02
46
810
0
1
2
3
4
-0.1
0
0.1
0.2
0.3
yt
Fig 3.3: Velocity field as function of ,y Fig 3.4: Velocity field as function of ,y
Fig 3.5: Velocity field as function of My,
Fig 3.6: Velocity field as function of 0,ky
Fig 3.7: Velocity field as function of rcGy, Fig 3.8: Velocity field as function of rtGy,
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02
46
810
-10
-5
0
5
10
-0.5
0
0.5
1
1.5
yw
02
46
810
-2
-1
0
1
2
-0.5
0
0.5
1
yu
02
46
810
0
1
2
3
4
-0.5
0
0.5
1
1.5
yt
02
46
810
-10
-5
0
5
10
0
0.5
1
1.5
yw
02
46
810
-2
-1
0
1
2
-0.5
0
0.5
1
yphi
02
46
810
0
1
2
3
4
0
0.5
1
1.5
yt
Fig 3.9: Velocity field as function of ,y
Fig 3.10: Velocity field as function of ty,
Fig 3.11: Concentration field as function of ,y Fig 3.12: Concentration field as function of ty,
Fig 3.13: Concentration field as function of ,y Fig 3.14: Temperature field as function of ,y
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Fig 3.15: Temperature field as function of ,y Fig 3.16: Temperature field as function of ty,
Numerical Method
Applying Crank-Nicolson formula to equations (2.6), (2.7) and (2.11), we have
jijijiji DCRCQCP ,31,131,31,13 (5.1)
jijijiji DRQP ,21,121,21,12 (5.2)
jijijiji DURUQUP ,11,111,11,11 (5.3)
With corresponding conditions (2.9) becoming
0,0 jU jiwt
j e1,0 jiwt
j eC 1,0
0, jU 0, j 0, jC (5.4)
Where
f
t
fSc
rS 13
y
t
f
f
fSc
rP
42
23
fSc
rQ 13
fSc
r
y
t
f
fR
24
23
jijijiji CRCSCPD ,13,3,13,3
yf
tf
f
rRP
4Pr2
1 2
'
2Pr
11
'
2f
rRQ
Pr2
1
4
'
22
f
rR
y
t
f
fR
f
t
f
rRS
Pr
11
'
2
jijijiji CRCSCPD ,12,2,12,2
y
f
f
t
f
rdP
42
2
'
11
f
rdQ
'
11 1
y
f
f
t
f
rdR
42
2
'
11
2
'
11 1 d
f
t
f
rdS
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43,11,1,11,1 ddf
tURUSUPD jijijiji
Thus, y and t are constant mesh sizes along y and t directions respectively. We need a
scheme to find single values at next level time in terms of known values at an earlier time level.
A forward difference approximation for the first order partial derivatives of ,C and U with
respect to t and y and a central difference approximation for the second order partial derivative
of ,C and U with respect to t and y are used. We used the following transformations
4
1f jiwt
ef 12 ed1
2
0
2
1M
kd rtGd3 CGd rt4 1
2
'
y
tr
We have converted partial differential equations (2.6), (2.7) and (2.11) that hold everywhere in
some domain into a system of simultaneous linear equations (5.1)-(5.3) to get approximate
solutions. The corresponding code (programme) is written in Mathlab for calculating numerical
solutions for concentration, temperature and velocity.
5.1 Tables and Graphical Presentations: Numerical Solution
To ensure the validity of our analytical solutions, we have compared our numerical solutions
with the exact solutions for concentration, temperature and velocity for some variation
parameters affecting fluid profile. These parameters are assigned the values
3.0,0.1,5.0,1.0,1.0,5.0,1.0,0.1,0.5,0.1 AtnRGGM rcr in case 5.
The corresponding code (programme) is written in Mathlab for calculating both the exact and
numerical solution. In table 5.1 - 5.3, the comparison between analytical values and numerical
values for concentration, temperature and velocity are presented. The error differences are
reasonable and passable. The closeness of curves corresponding to both exact and numerical
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0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
numerical
analytical
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
numerical
analytical
0 2 4 6 8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
numerical
analytical
solutions in all the cases considered in figure 5.1-5.3 further confirm the accuracy of our method
of solution.
Table 5.1: Comparison between exact and Table 5.2: Comparison between exact and
numerical values for concentration numerical values for temperature
Table 5.3: Comparison between exact and
numerical values for velocity
Figure 5.1: Concentration profiles
Figure 5.2: Temperature profiles Figure 5.3: Velocity profile
y Analytical Numerical Error
0 1 1 0
2 0.2529 0.2524 0.0005
4 0.064 0.0637 0.0003
6 0.0162 0.0161 0.0004
8 0.0041 0.0041 0
10 0.001 0.001 0
y Analytical Numerical Error
0 1 1 0
2 0.1746 0.1739 0.0007
4 0.0305 0.0303 0.0002
6 0.0053 0.0053 0
8 0.0009 0.0009 0
10 0.0002 0.0002 0
y Analytical Numerical Error
0 0 0 0
2 0.0837 0.1015 0.0178
4 0.0185 0.0222 0.0037
6 0.0042 0.005 0.0008
8 0.0010 0.0011 0.0001
10 0.0002 0.0003 0.0001
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REFERENCE
[1] Adebile ,E .A.and Sogbetun MHD Flow of A Non-Newtonian Fluid With Temperature
Dependent Viscosity Past A Vertical Plate In The Presence Of Radiative Heat Flux And
Chemical Reaction. International Journal of Engineering Research & Technology (IJERT) Vol.
1 Issue 10,
[2] A. Kumar, B. Chand and Kaushik, On Unsteady Oscillatory Laminar Free Convection
Flow of an Electrically Conducting Fluid through Porous Medium along a Porous Hot
Vertical Plate with Time Dependent Suction in the Presence of Heat Source/ Sink, J. of
Acad Math, vol. 24, 339 – 354, 2002.
[3] V. M. Soundalgekar, Free Convection Effects on the Oscillatory Flow Past an Infinite
Vertical Porous Plate with Constant Suction 1, Proc. Royal Soc. London A 333, 25-36,
1973.
[4] A.K. Singh, A.K. Singh and N.P. Singh, Heat and Mass Transfer in MHD Flow of a
Viscous Fluid Past a Vertical Plate under Oscillatory Suction Velocity, Ind. J. of Pure
Appl. Math, vol. 34, 429-442, 2003.
[5] K. Venkateswarlu and J. A. Rao, Numerical Solution of Heat and Mass Transfer in MHD
Flow of a Viscous Fluid Past a Vertical Plate under Oscillatory Suction Velocity, IE(I)
Journal-MC, 206 -212, 2005.
[6] Okedoye, A. M. and Bello, O. A., MHD Flow of a Uniformly Stretched Vertical
Permeable Surface under Oscillatory Suction Velocity, J. of the Nigerian Association of
Mathematical Physics vol. 13, 211-220, 20008.
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[9]. Massoudi, M. and Phuoc, T. X., Flow of a Generalized Second Grade Non-Nowtonian
Fluid with Variable Viscosity, Continum Mech. Thermodyn. 16, 529-538, 2004.
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