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In this paper, the effect of heat generation and radiation parameters on MHD flow along a uniformly heated vertical flat plate in the presence of a magnetic field has been investigated numerically. The nonlinear partial differential equations,

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Desalination and Water Treatment
16 (2010) 57–65
www.deswater.com
April
1944-3994/1944-3986 © 2010 Desalination Publications. All rights reserveddoi no. 10.5004/dwt.2010.1086
Effect of thermal radiation and heat generation on MHDﬂow past a uniformly heated vertical plate
Goutam Saha
a,
*, Tamanna Sultana
b
, Sumon Saha
c
a
Department of Mathematics, University of Dhaka, Dhaka-1000, BangladeshTel.
+880 1 817 505874
; email: ranamath06@gmail.com
b
Institute of Natural Science, United International University (UIU), Dhaka-1209, Bangladesh
c
Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia
Received 8 March 2009; Accepted 13 December 2009
ABSTRACT
In this paper, the effect of heat generation and radiation parameters on MHD ﬂow along auniformly heated vertical ﬂat plate in the presence of a magnetic ﬁeld has been investigatednumerically. The nonlinear partial differential equations, governing the problem under consid-eration for this analysis are transferred to simultaneous nonlinear ordinary differential equa-tions of ﬁrst order and those are further transformed into initial value problem by applyingMulti-segment integration technique. Finally, solutions are obtained by integrating the initialvalue problem using fourth order Runge-Kutta integration scheme. Rosseland approximationis used to describe the radiative heat ﬂux in the energy equation. Comparison with previouslypublished work is performed and excellent agreement with the results is obtained. Numericalresults for the details of the temperature proﬁles are shown graphically with the variation of thegoverning parameters considering in the present problem.
Keywords
:
Electrically conducting ﬂuid; Porous medium; Radiation; Similarity solution.
1. Introduction
The importance of the radiation effect on MHD ﬂowand heat transfer problems has found increasing attentionin industries. At high operating temperature, radiationeffect can be quite signiﬁcant. Many processes in engi-neering areas occur at high temperatures and knowledgeof radiation heat transfer becomes very important for thedesign of pertinent equipment. Nuclear power plants,gas turbines and the various propulsion devices for air-craft, missiles, satellites and space vehicles are examplesof such engineering areas. Most of the existing analyticalstudies for this problem are based on the constant physi-cal properties of the ambient ﬂuid. However, it is knownthat these properties may change with temperature,especially ﬂuid viscosity. To accurately predict the ﬂowand heat transfer rates, it is necessary to take into accountthis variation of viscosity with temperature.Ostrach [1] presented the similarity solution of natu-ral convection along vertical isothermal plate. Kay [2]reported that thermal conductivity of liquids with lowPrandtl number varied linearly with temperature inrange of 0–400
°
F. Arunachalam and Rajappa [3] con-sidered forced convection ﬂow of liquid metals (havinglow Prandtl number) with variable thermal conductiv-ity and derived explicit closed form of analytical solu-tion. Chaim [4] also studied heat transfer in ﬂuid ﬂow oflow Prandtl number with variable thermal conductivity.Carey and Mollendorf [5] observed the effect of temper-ature dependent viscosity on free convective ﬂuid ﬂow.Crepeau and Clarksean [6] discussed similarity solu-tion of natural convection with internal heat generation,
*Corresponding author.
58
which decayed exponentially. Chamkha and Khaled [7]obtained similarity solution of natural convection on aninclined plate with internal heat generation or absorp-tion in presence of transverse magnetic ﬁeld.The thermal radiation of a gray ﬂuid, which is emit-ting and absorbing radiation in a non-scattering mediumhas been examined by Ali
et al.
[8], Ibrahim [9], Mansour[10], Hossain
et al.
[11] and Elbashbeshy and Dimian [12].In the aspect of convection and radiation, Viskanta andGrosh [13] considered the effects of thermal radiationon the temperature distribution and the heat transfer inan absorbing and emitting media ﬂowing over a wedge by using the Rosseland diffusion approximation. Thisapproximation leads to a considerable simpliﬁcation inthe expression for radiant ﬂux. In Viskanta and Grosh[13] and Raptis [14], the temperature differences withinthe ﬂow were assumed sufﬁciently small such that T
4
might be expressed as a linear function of temperature,i.e.T
4
≈
4(T
∞
)
3
T
−
3(T
∞
)
4
. Hossain
et al.
[15] investigated thenatural convection–radiation interaction on a boundarylayer ﬂow along a vertical plate with uniform suction.Yih [16] investigated the natural convection ﬂow of anoptically dense viscous ﬂuid over an isothermal trun-cated cone. Recently, Bataller [17] investigated radiationeffects in the laminar boundary layer about a ﬂat-plate ina uniform stream of ﬂuid. He found that as the value ofradiation parameter increases, a diminution in the ther-mal radiation’s effect occurs. Later, he studied radiationeffects for the Blasius and Sakiadis ﬂows with a convec-tive surface boundary condition [18]. A comparison ofthese two ﬂows was described in this work.Chen [19] performed an analysis to study the MHDnatural convection ﬂow over a permeable inclined sur-face with variable wall temperature and concentration.The results showed that the velocity was decreased inthe presence of a magnetic ﬁeld and with the increaseof the angle of inclination, the effect of buoyancy forcedecreased. Heat transfer rate was however increasedwhen the Prandtl number was increased. Duwairi [20]investigated the effect of viscous and Joule heating onforced convection ﬂow from radiative isothermal sur-faces. He found that the heat transfer rate was decreasedwhen the radiation parameter was increased. Duwairiand Damseh [21] also studied the convection heat trans-fer problem with radiation effects from vertical surfacefor buoyancy aiding and opposing ﬂows. They concludedthat increasing the conduction – radiation parameterdecreased the heat transfer rates for the buoyancy aidedﬂow and increased them for the buoyancy opposing ﬂow.Ibrahim
et al.
[22] investigated similarity reductions forproblems of radiative and magnetic ﬁeld effects on freeconvection and mass transfer ﬂow past a semi-inﬁniteﬂat plate. They obtained new similarity reductions andfound an analytical solution for the uniform magneticﬁeld by using Lie group method. They also presentedthe numerical results for the non-uniform magneticﬁeld. Seddeek [23] investigated effects of radiation andvariable viscosity on a MHD free convection ﬂow past asemi-inﬁnite ﬂat plate with an aligned magnetic ﬁeld inthe case of unsteady ﬂow. The effect of variable viscosityon hydromagnetic ﬂow and heat transfer past a continu-ously moving porous boundary with radiation was fur-ther studied by Seddeek [24].In the present paper, investigation is carried out forthe thermal radiation interaction of the boundary layerﬂow of electrically conducting ﬂuid past a uniformlyheated vertical plate embedded in a porous medium.The governing equations are converted into nonlinearsystem of coupled ordinary differential equations andsolved numerically using Multi-segment integrationtechnique. The normalized similarity solutions are thenobtained numerically for various parameters enteringinto the problem and discussed them from the physicalpoint of view.
2. Mathematical formulation of the problem
Let us consider a steady, two-dimensional ﬂow ofa viscous, incompressible and electrically conductingﬂuid of temperature
T
∞
past a semi-inﬁnite heated ver-tical plate having constant temperature
T
w
(where,
T
w
>
T
∞
). A magnetic ﬁeld of uniform strength, B
o
is appliedperpendicular to the plate. The magnetic Reynoldsnumber is taken to be small enough so that the inducedmagnetic ﬁeld can be neglected. The ﬂow is assumedto be in the
x
-direction, which is taken along the platein the upward direction and
y
-axis is normal to it. Theﬂow conﬁguration and the coordinate system are shownin Fig. 1. In order to consider the effect of radiation, interms of the radiative heat ﬂux, the Rosseland approxi-mation is incorporated in the energy equation. The radi-ative heat ﬂux in the
x
-direction is considered negligiblein comparison to the
y
-direction. Within the frameworkof the above-noted assumptions, it is considered that the
B
0
T
w
TU(x)T
∞
yux
υ
Fig. 1. Flow conﬁguration and coordinate system.
G. Saha et al. / Desalination and Water Treatment 16 (2010) 57–65
G. Saha et al. / Desalination and Water Treatment 16 (2010) 57–65
59
boundary layer approximations hold and the govern-ing equations relevant to the problem in the presence ofradiation are given by
∂∂+∂∂=
ux
υ
y
0(1)
uuxu y pxu yBuKUu
∂∂+∂∂= −∂∂+∂∂− − −
( )
υρνσρ ν
1
2202*
(2)
uT xT ycT yQcTT cq y
pppr
∂∂+∂∂=∂∂+ − −∂∂
∞
υκ ρ ρ ρ
220
1()(3)where
u
and
υ
are the velocity components along
x
and
y
coordinates respectively,
U
(
x
) is the free stream veloc-ity,
ν
=
μ
/
ρ
is the kinematic viscosity,
μ
is the coefﬁ-cient of dynamic viscosity,
ρ
is the mass density ofthe ﬂuid,
σ
is the electrical conductivity of the ﬂuid,
B
o
is the magnetic induction,
K
*
is the Darcy permeability,
T
is the temperature of the ﬂuid in the boundary layer,
T
∞
is the temperature of the ﬂuid outside the boundarylayer,
c
p
is the speciﬁc heat of the ﬂuid at constant pres-sure,
κ
is the thermal conductivity and
q
r
is the radiativeheat ﬂux.It is assumed that the velocity of the free stream is inthe form of
U(x) = a x + c x
2
(4)
where
a
and
c
are constants.In the free stream
u
=
U
(
x
), Eq. (2) reduces to
UUx pxBU
∂∂= −∂∂−
1
02
ρσρ
.(5)Eliminating
∂∂
px
between Eqs. (2) and (5), we obtain
uuxu yu yUUxBUuKUu
∂∂+∂∂=∂∂+∂∂+ − − −υ νσρ ν
2202
()()
*
(6)By using Rosseland approximation,
q
r
[25] for radia-tion from an optically thick layer (Ali
et al.
[8]), it can bewritten as follows
qT y
r
= −∂∂
43
4
σκ
**
, (7)where
σ
*
is the Stefan–Boltzmann constant and
κ
* is themean absorption coefﬁcient.Moreover, the temperature differences within theﬂow are assumed to be sufﬁciently small such that T
4
may be expressed as a linear function of temperature.This is accomplished by expanding T
4
in a Taylor’sseries about T
∞
and neglecting higher-order terms,thus
TTTTTTTT
44334
443
≅ + − = −
∞ ∞ ∞ ∞ ∞
().(8)By using Eqs. (7) and (8), Eq. (3) gives
uT xT ycT yQcTT T cT y
ppp
∂∂+∂∂=∂∂+ − +∂∂
∞∞
υκ ρ ρσρ κ
220322
163()
**
.(9)The corresponding boundary conditions for theabove problem are given byat y0as y
uTT uUxTT
w
= = = =→ → → ∞⎫⎬⎪⎭⎪
∞
00,,(),
υ
(10)In order to obtain a solution of Eqs. (1), (6) and (9),the following transformations are introduced:
η ν=
a y
(11a)
uaxfcxg
=′+′
()()
η η
2
(11b)
υ ν η η
ν
= − −
af cx g
a
()()2 (11c)
TTTTT cxaT
ww
= + − +⎡⎣⎢⎤⎦⎥
∞
()()()
01
2
η η
(11d)
Pc
p
=ρνκ
(Prandtl number) (11e)
N Ba
=σρ
03
(Magnetic parameter) (11f)
MaK
=ν
*
(Darcy parameter) (11g)
QQac
p
=
0
ρ
(Heat generation parameter) (11h)
KT
=
∞
κ κ σ
**
4
3
(Radiation parameter) (11i)
PnKPK
=+
334 (Radiative Prandtl number) (11j)where prime denotes differentiation with respect to
η
.In view of (11), Eq. (1) is satisﬁed identically andEqs. (6) and (9) reduce to
′′′+′′−′+ −′−′+ =
ffffNfMf
2
110()(12)
′′′+′′−′ ′+′′+ −′−′+ =
gfgfgfgNgMg
32130()(13)
′′+′+ + − =
TPnfTTTQ
0000
10()()(14)
′′+ −′+′+′+ =
TPnfTfTgTPnTQ
11101
0().(15)The corresponding boundary conditions (10) becomes
ffggTT fgTT
=′= =′= = = =′=′= = =
00000001110
0101
,,,,,,,,atas
η
η→ ∞⎫⎬⎭
. (16)
G. Saha et al. / Desalination and Water Treatment 16 (2010) 57–65
60
The nonlinear system of coupled differential Eqs.(12)–(15) together with the boundary conditions (16)are solved numerically using Multi-segment integra-tion technique. First of all, higher order nonlinear dif-ferential Eqs. (12)–(15) are converted into simultaneouslinear or nonlinear differential equations of order ﬁrstand they are further transformed into initial valueproblem by applying Multi-segment integration tech-nique (Kalnins and Lestingi [26]). Once the boundaryvalue problem is reduced to initial value problem, it isthen solved using Runge-Kutta fourth order technique(Jain [28]). Now rewrite the governing Eqs. (12)–(15)into a set of ﬁrst order ordinary differential equationsas follows:
⇒′′′′′′′′⎧⎨⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎫⎬⎪⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪
dd f f f g g gT T T T
η
0011
⎪⎪⎪=′′′′
( )
−′′+ −′
( )
−′−′′′−′′+′ ′−′′+′
f f fffNfMf g g fgfgfgN
2
1132
g gMgT PnQTfT T PnfTfTgTPnQ
−
( )
+′−′−
( )
−′
{ }
′− −′+′+′
( )
−
131
0001110
T T
1
⎧⎨⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎫⎬⎪⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪⎪⎪
(17)The fundamental set of nonlinear equations (17)together with the boundary conditions (16) has to be integrated over a ﬁnite range of the independentvariable
η
. However, the numerical integration ofthese equations is not possible beyond a very limitedrange of
η
due to the loss of accuracy in solving forthe unknown initial values, as pointed out by Sepe-toski
et al
. [27]. Thus, the multi-segment integrationtechnique developed by Kalnins and Lestingi [26] has been used in this analysis. If the fundamental variables
fffgggTTTT
,,,,,,,,,
′ ′′ ′ ′′′ ′
0011
of Eqs. (17) are repre-sented in matrix notation by [w] in a standard form asfollows:
dwdFwPQKNM
ηη=
( )
,;,,,,, (18)in which
wwwherei
iT
=⎡⎣⎤⎦≤ ≤
110and
FFFFFFFFFFFw
=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥=
123456789102
wwwwwNwMwwwwwwwwwNwMw
32213225616253455
11321
− + −
( )
− −− + − + −
( )
+ −−−
( )
−
{ }
− − + +
( )
−⎧⎨⎪⎪⎪⎪⎪
31
87181029110489
wPnQwwwwPnwwwwwwPnQw
⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎫⎬⎪⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪⎪⎪
(19)The boundary conditions Eqs. (16) can be rearrangedin the following form as follows:
A w(0) + B w(
η
→
∞
) = C,
(20)
where
A
=
10000000000100000000000100000000001000000000001000000000000100000000000000000000000000000000000000000
⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
,
B
=
00000000000000000000000000000000000000000000000000000000000000100000000000010000000000010000000000010
⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
,
C
=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
0000001110.Let us consider the initial value problem correspond-ing to boundary value problems,
dW dGwWPQKNM
ηη=
( )
,,;,,,,(21)
G. Saha et al. / Desalination and Water Treatment 16 (2010) 57–65
61
with
WIandWWwhereij
ij
0110
1010
( )
= = ≤ ≤
×
[],
,
(22)
GJwhereJ FFFFFFFFFFwwwww
0
123456789101234
( )
= =
[],,,,,,,,,,,,,
55678910
,,,,,
wwwww
⎛ ⎝ ⎜⎞ ⎠ ⎟
(23)where
I
is the Identity matrix and
J
is the Jacobianmatrix.In which,
GGGGGGGGGG
j j j j j j j j j j
12345678910
⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥=− + + +
( )
=
j j j jjjj j
wwwwwwwwNMwww
121023221313256
2
,,..,
()
j j jjjjjjj j
wwwwwwwwwwwwNMww
− + + + − + + +
()()()()
16162525343458
32
PPnQwwwwwwPnwwwwwwww
jjj j jjjj
71818102929110110
− +
()
{ }
− − +
()
+ +
(()
+ +
()
{ }
−⎧⎨⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎫⎬⎪⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪
wwwwPnQw
jjj
48489
⎪⎪⎪
(24)
3. Results and discussion
To assess the accuracy of the present code, thegraphs of similarity temperatures
T
0
(
η
)
and
T
1
(
η
)
fordifferent radiation parameter have been plotted wherethe Prandtl number and magnetic parameter are takenﬁxed at 0.71 and 0.8 respectively. These graphs are alsocompared with that of Raptis
et al.
[29] for
K
= 3 and 30.Fig. 2 shows the comparison of the temperature proﬁles
T
0
and
T
1
for
P
= 0.71 and
N
= 0.8 produced by the pres-ent code and that of Raptis
et al.
[29]. They obtained thesolution using Runge-Kutta shooting method whereasthe scheme exploited in the present paper is Multi-seg-ment integration technique. Infact, the results show aclose agreement and thus give an encouragement forthe use of the present code. Hence, the scheme used inthis paper is stable and accurate.The aim of this work is to determine the effects ofdifferent parameters on the normalized similarity tem-peratures
T
0
(
η
)
and
T
1
(
η
).
In the calculations, the val-ues of magnetic parameter
(N),
Darcy parameter
(M)
,heat generation parameter
(Q)
, Prandtl number
(P)
andradiation parameter
(K)
are chosen arbitrarily. The effectof the Darcy parameter
M
on the normalized similaritytemperatures
T
0
(
η
)
and
T
1
(
η
)
is shown in Fig. 3. Fromthis ﬁgure, it is observed that temperature proﬁles
T
0
(
η
)
increase with the increase of
M
whereas
T
1
(
η
)
decreaseat the same time. As the value of
M
increases, the resis-tance to the ﬂow also increases, which means that thetemperature ﬁeld approximates more closely to theequivalent conductive state. Fig. 4 shows the effect ofmagnetic ﬁeld parameter
(N)
on the temperature pro-ﬁles. This ﬁgure reveals that the normalized similaritytemperature
T
0
(
η
)
shows no effect with the variation ofmagnetic ﬁeld parameter but similarity temperature
T
1
(
η
)
increases with the increase of
N
. This is due to thefact that the magnetic ﬁeld tends to retard the velocityﬁeld, which in turn induces the temperature ﬁeld andthus results the increase of the temperature proﬁles. Themagnetic ﬁeld can therefore be used to control the ﬂowcharacteristics. In Fig. 5, the heat generation parameter
(Q)
is varied keeping all other parameters ﬁxed. It isfound that the similarity temperatures
T
0
(
η
)
increasemonotonically and similarity temperatures
T
1
(
η
)
alsoincrease close to the plate as
Q
increases. However, aftera short distance from the plate, the proﬁles overlap anddecrease monotonically as
Q
increases. For differentvalues of Prandtl number
(P)
, signiﬁcant changes ontemperature proﬁles are observed, which is presentedin Fig. 6. From this ﬁgure, it can be concluded that incase of cooling the plate, normalized similarity tem-perature
T
0
(
η
)
increases as
P
increases and similaritytemperatures
T
1
(
η
)
also increases close to the plate. It
Fig. 2. Comparison of the temperature proﬁles
T
0
and
T
1
for
P
= 0.71 and
N
= 0.8.

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