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Effects of Thermal Radiation and Viscous Dissipation on Magnetohydrodynamic Stagnation Point Flow and Heat Transfer of Nanofluid Towards a Stretching Sheet

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Delivered by Publishing Technology to: Guest UserIP: 49.206.99.255 On: Sat, 16 Nov 2013 19:05:29Copyright: American Scientific Publishers
A RT I C L E
Copyright © 2013 by American Scientiﬁc Publishers All rights reserved.Printed in the United States of America
Journal of Nanoﬂuids
Vol. 2, pp. 283–291, 2013
(www.aspbs.com/jon)
Effects of Thermal Radiation and ViscousDissipation on MagnetohydrodynamicStagnation Point Flow and Heat Transfer of Nanoﬂuid Towards a Stretching Sheet
Yohannes Yirga
∗
and Bandari Shankar
Department of Mathematics, Osmania University, Hyderabad 500007, Andhra Pradesh, India
In this paper, the effects of thermal radiation and viscous dissipation on magnetohydrodynamic (MHD) stag-nation point ﬂow and heat transfer of nanoﬂuids towards a stretching sheet are investigated numerically. Theeffects of Brownian motion and thermophoresis on the ﬂow and heat transfer were considered. The systemof partial differential equations governing the ﬂow was reduced to a system of non linear ordinary differen-tial equations by using similarity transformations. The transformed equations were numerically solved by anexplicit ﬁnite difference scheme known as the Keller box method. The velocity, temperature, and concentrationproﬁles were obtained and utilized to compute the skin-friction coefﬁcient, the local Nusselt number, and localSherwood number for different values of the governing parameters. The study reveals that the skin friction andheat transfer rate at the surface increases with the magnetic parameter when the free stream velocity exceedsthe stretching velocity, i.e.,
>
1, and decrease when
<
1. It is also found that the local Sherwood number increases with velocity ratio parameter, Brownian motion parameter, and Lewis number. In addition to this, thethermal boundary layer become thicker by increasing the value of the Eckert number and becomes thinner byincreasing the thermal radiation parameter. A comparison of the numerical results of the present study withpreviously published data revealed an excellent agreement.
KEYWORDS:
Thermal Radiation, Viscous Dissipation, Magnetohydrodynamic, Stagnation Point, Nanoﬂuids.
1. INTRODUCTION
The study of the stagnation point ﬂow and heat transferof a viscous incompressible ﬂuid has been the concern of many researchers because of its wide applications in man-ufacturing and natural process which include cooling of electronic devices by fans, cooling of nuclear reactors dur-ing emergency shutdown, cooling of an inﬁnite metallicplate in a cooling bath, textile and paper industries, solarcentral receivers exposed to wind currents, MHD genera-tors, plasma studies and blood ﬂow problems.The classical two dimensional stagnation point ﬂow ona ﬂat plate was ﬁrst studied by Hiemenz in 1911. Sincethen many researchers have extended the idea to differ-ent aspect of the stagnation point ﬂow problems. In 1936,Homann extended the Hiemenz ﬂow to the axisymmetricthree-dimensional case. Ariel
1
studied the inﬂuence of anexternal magnetic ﬁeld on Hiemenz ﬂow. The effect of
∗
Author to whom correspondence should be addressed.Email: yohannesalbin@yahoo.comReceived: 7 June 2013Accepted: 23 June 2013
uniform suction on Homann problem where the ﬂat plateis oscillating in its own plane is considered by Weidmanand Mahalingam.
2
The two dimensional MHD steady stag-nation point ﬂow towards a stretching surface was ana-lyzed by Mahaparta and Gupta,
3
Ishak
4
and Ishak et al.
5
Their results showed that the heat transfer rate at the sur-face increases with the magnetic parameter when the freestream velocity exceeds the stretching velocity and theopposite is observed when the free stream velocity is lessthan the stretching velocity.Fluid heating and cooling are important in manyindustries such as power, manufacturing, transportation,and electronics. Effective cooling techniques are greatlyneeded for cooling any sort of high-energy device. Com-mon heat transfer ﬂuids such as water, ethylene glycol,and engine oil have limited/poor heat transfer capabilitiesdue to their low heat transfer properties. In contrast, met-als have thermal conductivities up to three times higherthan these ﬂuids, so it is natural that it would be desiredto combine the two substances to produce a heat transfermedium that behaves like a ﬂuid, but has the thermal con-ductivity of a metal. A lot of experimental and theoretical
J. Nanoﬂuids 2013, Vol. 2, No. 4
2169-432X/2013/2/283/009 doi:10.1166/jon.2013.1070
283
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Effects of Thermal Radiation and Viscous Dissipation on MHD Stagnation Point Flow and Heat Transfer of Nanoﬂuid
Yirga and Shankar
A R T I C L E
researches have been made to improve the thermal con-ductivity of these ﬂuids.In 1993, during an investigation of new coolants andcooling technologies at Argonne national laboratory inU.S, Choi invented a new type of ﬂuid called Nanoﬂuid.
6
Nanoﬂuids are ﬂuids that contain small volumetric quan-tities of nanometer-sized particles, called nanoparticles.The nanoparticles used in nanoﬂuids are typically madeof metals, oxides, carbides, or carbon nanotubes. Commonbase ﬂuids include water, ethylene glycol and oil. Nanoﬂu-ids commonly contain up to a 5% volume fraction of nanoparticles to see effective heat transfer enhancements.Nanoﬂuids are studied because of their heat transfer prop-erties: they enhance the thermal conductivity and convec-tive properties over the properties of the base ﬂuid. Typicalthermal conductivity enhancements are in the range of 15–40% over the base ﬂuid and heat transfer coefﬁ-cient enhancements have been found up to 40%.
7
Increas-ing in thermal conductivity of this magnitude cannot besolely attributed to the higher thermal conductivity of theadded nanoparticles, and there must be other mechanismsattributed to the increase in performance.After the pioneer investigation of Choi, thriving experi-mental and theoretical researches were undertaken to dis-cover and understand the mechanisms of heat transfer innanoﬂuids. The knowledge of the physical mechanismsof heat transfer in nanoﬂuids is of vital importance asit will enable the exploitation of their full heat transferpotential. Masuda et al.
8
observed the characteristic fea-ture of nanoﬂuid is thermal conductivity enhancement.This observation suggests the possibility of using nanoﬂu-ids in advanced nuclear systems.
9
A comprehensive sur-vey of convective transport in nanoﬂuids was made byBuongiorno,
10
who says that a satisfactory explanation forthe abnormal increase of the thermal conductivity and vis-cosity is yet to be found. He focused on further heat trans-fer enhancement observed in convective situations. Prabhatet al.
11
studied the convective heat transfer enhancementin nanoﬂuids. They showed that signiﬁcant deviationsbetween the data and the predictions of the heat transfercorrelations can occur in the laminar ﬂow regime, partic-ularly in the entrance region; the enhancement becomesmore pronounced at higher Reynolds number and higherparticle concentration. Kuznetsov and Nield
12
have exam-ined the inﬂuence of nanoparticles on natural convec-tion boundary-layer ﬂow past a vertical plate using amodel in which Brownian motion and thermophoresis areaccounted for. The authors have assumed the simplest pos-sible boundary conditions, namely those in which boththe temperature and the nanoparticle fraction are constantalong the wall. Nield and Kuznetsov
13
14
have studied theCheng and Minkowycz
15
problem of natural convectionpast a vertical plate in a porous medium saturated by ananoﬂuid. The model used for the nanoﬂuid incorporatesthe effects of Brownian motion and thermophoresis forthe porous medium. Nadeem and Lee
16
used the homo-topy analysis method to study the boundary layer ﬂow of nanoﬂuid over an exponentially stretching sheet. Kumariand Gorla,
17
studied the mixed convective boundary layerﬂow over a vertical plate embedded in a porous mediumsaturated with a nanoﬂuid. In addition to this, Gorla andKumari,
18
studied the mixed convection ﬂow of a Non-Newtonian fanoﬂuid over a non-linearly stretching sheet.Recently, Gorla and Llapa,
19
studied the heat transfer ina nano thin-ﬁlm region of an evaporating meniscus; Gorlaand Kumari,
20
also studied the combined convection on avertical cylinder in a non-Newtonian nanoﬂuid. The mag-netic ﬁeld effects on free convection ﬂow of a nanoﬂuidpast a vertical semi-inﬁnite ﬂat plate has been discussedby Hamad et al.
21
and Mahajan et al.
22
studied the convec-tion in magnetic nanoﬂuids. Haddad et al.
23
experimentallyinvestigated natural convection in nanoﬂuds by consideringthe role of thermophoresis and Brownian motion in heattransfer enhancement. They indicated that neglecting therole of Brownian motion and thermophoresis deterioratethe heat transfer and this deterioration elevated by increas-ing the volume fraction of nanoparticles. Anbuchezhian
24
studied the thermoporesis and Brownian motion effects onboundary layer ﬂow of nanoﬂuid in the presence of ther-mal stratiﬁcation due to solar energy. In addition to thesestudies, several researchers have investigated the ﬂow andheat transfer of nanoﬂuids.
25–35
An adequate understanding of radiative heat transferin ﬂow processes is very important in engineering andindustries, especially in the design of reliable equipments,nuclear plants, gas turbines and various propulsion devicesfor aircraft, missiles, satellites and space vehicles. Thermalradiation effects are extremely important in the context of ﬂow processes involving high temperature. The effects of thermal radiation on the boundary layer ﬂow have alsobeen considerably researched.
36–40
However, all the papers in the literature are restrictedto the study of boundary layer ﬂow of a nanoﬂuid pasta differently stretching sheet. The stagnation point ﬂowof nanoﬂuids has been given less attention. Moreover,partly or completely the effects of thermal radiation, vis-cous dissipation, magnetic ﬁeld, Bronian motion and ther-mophoresis are neglected in many of the studies conductedon stagnation point ﬂow and heat transfer of nanoﬂu-ids. A few recent studies focused on a stagnation pointﬂow. Recently, Mustafa et al.
41
used the homotopy analysismethod to study the Stagnation-point ﬂow of a nanoﬂuidtowards a stretching sheet; Bachok et al.
42
studied theStagnation-point ﬂow over a stretching/shrinking sheet ina nanoﬂuid. They analyzed the effects of the solid vol-ume fraction and the type of the nanoparticles on the ﬂuidﬂow and heat transfer characteristics of a nanoﬂuid over astretching/shrinking sheet. Very recently, Wubshet et al.
43
studied the MHD stagnation point ﬂow and heat transferdue to nanoﬂuid towards a stretching sheet.
284
J. Nanoﬂuids, 2, 283–291, 2013
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Yirga and Shankar
Effects of Thermal Radiation and Viscous Dissipation on MHD Stagnation Point Flow and Heat Transfer of Nanoﬂuid
A RT I C L E
The aim of the present paper is to study the effects of thermal radiation and viscous dissipation on MHD stag-nation point ﬂow and heat transfer of a nanoﬂuid towardsa stretching sheet by taking the brownian motion andthermophoresis in to account. The effects of the gov-erning parameters such as magnetic parameter, velocityratio parameter, Prandtl number, thermal radiation param-eter, Echart number, Lewis number, Brownian motionparameter, and thermophoresis parameter on the ﬂow andheat transfer characteristics are investigated. The combinedeffect of all the above mentioned parameters has not beenreported so far in the literature, which makes the presentpaper unique.The governing highly nonlinear partial differential equa-tions of momentum, energy and nano particle volumefraction has been simpliﬁed by using suitable similar-ity transformations and then solved numerically with thehelp of a powerful, easy to use method called the Kellerbox method. This method has already been successfullyapplied to several non linear problems corresponding toparabolic partial differential equations. As discussed inRef. [44] the exact discrete calculus associated with theKeller-box scheme is shown to be fundamentally differentfrom all other mimic numerical methods. The box-schemeof Keller,
44
is basically a mixed ﬁnite volume method,which consists in taking the average of a conservation lawand of the associated constitutive law at the level of thesame mesh cell. The paper is outlined as follows.In Section 2 the problem is formulated and similaritytransformation has been used to reduce the formulatedsystem of partial differential equation in to a non lin-ear ordinary differential equations. In Section 3 a numeri-cal solution using Keller box method has been discussed.In Section 4 the numerical results are discussed in detailgraphically and in table form. Finally, in Section 5, con-cluding remarks are given.
2. MATHEMATICAL FORMULATION
Consider a steady two-dimensional stagnation point ﬂowof a nanoﬂuid towards a stretching sheet kept at a constanttemperature
T
w
and concentration
w
. The ambient tem-perature and concentration far away from the sheet respec-tively, are
T
and
. The ﬂow is subjected to a constanttransverse magnetic ﬁeld of strength
B
0
which is assumedto be applied in the positive
y
-direction, normal to the sur-face. The induced magnetic ﬁeld is assumed to be smallcompared to the applied magnetic ﬁeld and is neglected.It is further assumed that the base ﬂuid and the suspendednanoparticles are in thermal equilibrium and no slip occursbetween them. We choose the coordinate system such that
x
-axis is along the stretching sheet and
y
-axis is normalto the sheet. The physical ﬂow model and coordinate sys-tem is shown in Figure 1. Under the above approximationand using the usual boundary layer approximations, the
Fig. 1.
Physical ﬂow model and coordinate systems.
governing equation of the conservation of mass, momen-tum, energy and nanoparticles fraction for nanoﬂuids in thepresence of magnetic ﬁeld, radiation and viscous dissipa-tion towards a stretching sheet can be written in Cartesiancoordinates
x
and
y
as.
40
43
45
dudx
+
dvdx
=
0 (1)
uux
+
vuy
=
U
U
y
+
v
nf
2
uy
2
+
B
20
x
nf
U
−
u
(2)
uT x
+
vT y
=
nf
2
T y
2
+
u
nf
c
p
nf
uy
2
+
c
p
s
c
p
f
×
D
B
yT y
+
D
T
T
T y
2
−
1
c
p
nf
q
r
y
(3)
ux
+
vy
=
D
B
2
y
2
+
D
T
T
2
T y
2
(4)Where
u
and
v
are the velocity components in the
x
and
y
directions respectively,
T
is the temperature of the nanoﬂuid,
is the nanoparticle volume fraction,
q
r
is the radiative heat ﬂux;
U
B
0
c
p
s
c
p
f
D
B
and
D
T
are the free stream velocity, electrical conduc-tivity, magnetic ﬁeld, heat capacity of the nanoparticle,heat capacity of the base ﬂuid, the Brownian diffu-sion and thermophoretic diffusion coefﬁcient respectively,and
v
nf
nf
nf
nf
K
nf
and
c
p
nf
are the kinematicviscosity, thermal diffusivity, density, viscosity, thermalconductivity, and heat capacitance of the nanoﬂuid, respec-tively, which are given as:
24
45
v
f
=
f
f
v
nf
=
nf
nf
nf
=
k
nf
c
p
nf
nf
=
1
−
f
+
s
nf
=
f
1
−
2
5
c
p
nf
=
1
−
c
p
f
+
c
p
s
k
nf
k
f
=
k
s
+
2
k
f
−
2
k
f
−
k
s
k
s
+
2
k
f
+
K
f
−
k
s
(5)
J. Nanoﬂuids, 2, 283–291, 2013
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Effects of Thermal Radiation and Viscous Dissipation on MHD Stagnation Point Flow and Heat Transfer of Nanoﬂuid
Yirga and Shankar
A R T I C L E
In which
v
f
f
f
k
f
are the kinematic viscosity, vis-cosity, density and thermal conductivity of the base ﬂuidrespectively, and
s
k
s
c
p
s
are the density, thermalconductivity and heat capacitance of the nanoparticlerespectively.Using the Rosseland approximation for radiation, theradiative heat ﬂux is simpliﬁed as:
q
r
=−
4
∗
3
k
∗
T
4
y
(6)Where
∗
and
k
∗
are the Stefan-Boltzmann constant and themean absorption coefﬁcient, respectively. The temperaturedifferences within the ﬂow are assumed to be sufﬁcientlysmall so that
T
4
may be expressed as a linear function of temperature
T
using a truncated Taylor series about the freestream temperature
T
and neglecting higher-order terms,weget:
T
4
4
T
3
T
−
3
T
4
(7)If we take
N
r
=
k
nf
k
∗
/
4
∗
T
3
as a radiation parameter,then making use of Eqs. (3), (6) and (7), becomes:
uT x
+
vT y
=
nf
k
0
2
T y
2
+
nf
c
p
nf
uy
2
+
c
p
s
c
p
f
D
B
yT y
+
D
T
T
T y
2
(8)Where
K
0
=
3
N
r
/
3
N
r
+
4
. It is worth mentioning herethat as
N
r
→
(i.e.,
k
0
→
1), we get the classical solu-tion to the energy Eq. (3) without the inﬂuence of thermalradiation.The associated boundary conditions are
u
=
U
w
=
ax v
=
0
T
=
T
w
=
w
at
y
=
0
u
=
U
=
bx v
=
0
T
→
T
→
as
y
→
(9)In which
x
and
y
represent coordinate axes along thecontinuous surface in the direction of motion and nor-mal to it, respectively.
U
w
T
w
and
w
are the refer-ence velocity of the sheet, temperatures of the sheet andnanoparticles volume fraction of the sheet, respectively,and
U
T
and
are the free stream velocity, ambi-ent temperature and nanoparticle volume fraction far awayfrom the sheet.Introducing the following similarity transformations
=
y
av
f
u
=
axf
v
=−√
av
f
f
=
T
−
T
T
w
−
T
g
=
−
w
−
(10)The equation of continuity (1) is automatically satisﬁedand Eqs. (2), (4) and (8) reduce to the following ordinaryequations
f
+
ff
−
f
2
+
M
−
f
+
2
=
0 (11)
1
+
43
N
r
+
Pr
f
+
Ecf
2
+
Nb
g
+
Nt
2
=
0 (12)
g
+
Lefg
+
NtNb
=
0 (13)With boundary conditions
f
0
=
0
f
0
=
1
0
=
1
g
0
=
1
f
=
=
0
g
=
0
as
→
(14)Where prime denote differentiation with respect to thesimilarity variable
.
f
and
g
represent the dimen-sion less velocity, temperature and particle concentration,respectively.
M
Pr
N
r
EcNbNt
and
Le
denote themagnetic parameter, velocity ratio parameter, Prandtl num-ber, radiation parameter, Eckert number, Brownian motionparameter, thermophoresis parameter, and Lewis number,respectively. These governing parameters are given as
M
=
B
20
nf
=
U
U
w
ba
Pr
=
v
f
f
N
r
=
k
∗
k
nf
4
∗
T
3
Ec
=
U
2
w
c
p
T
w
−
T
Nb
=
D
B
c
p
s
c
p
f
w
−
v
nf
Nt
=
D
T
T
c
p
s
c
p
f
T
w
−
T
v
nf
Le
=
v
f
D
B
(15)The quantities of practical interest in this study are thelocal skin friction coefﬁcient
C
f
, the local Nusselt num-ber
Nu
x
and the local Sherwood number
Sh
x
, which aregiven as
40
43
√
Re
x
C
f
=
nf
f
f
0
=
1
1
−
2
5
f
0
Nu
x
√
Re
x
=−
k
nf
k
f
+
43
N
r
0
Sh
x
√
Re
x
=−
g
0
(16)Where Re
x
=
U
w
/v
f
x
is the local Reynolds number.
3. NUMERICAL SOLUTION
The non linear boundary value problem represented byEqs. (11)–(13) and (14) is solved numerically using theKeller box method. In solving the system of non lin-ear ordinary differential Eqs. (11)–(13) together with theboundary condition (14) using the Keller box method thechoice of an initial guess is very important. The successof the scheme depends greatly on how much good thisguess is to give the most accurate solution. Thus, an initialguess of
f
0
=
1
−
e
−
+
12
2
0
=
e
−
g
0
=
e
−
(17)
286
J. Nanoﬂuids, 2, 283–291, 2013
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Yirga and Shankar
Effects of Thermal Radiation and Viscous Dissipation on MHD Stagnation Point Flow and Heat Transfer of Nanoﬂuid
A RT I C L E
These choices have been made based on the convergencecriteria together with the boundary conditions in consid-eration. As in Cebeci and Pradshaw,
46
the values of thewall shear stress, in our case
f
0
is commonly used asa convergence criteria. This is because in the boundarylayer ﬂow calculations the greatest error appears in thewall shear stress parameter. In the present study this con-vergence criteria is used. In this study a uniform grid of size
=
0
1 is chosen to satisfy the convergence criteriaof 10
−
5
, which gives about a four decimal places accuracyfor most of the prescribed quantities.
4. RESULTS AND DISCUSSION
The transformed non linear Eqs. (11)–(13) subjected tothe boundary condition (14) was solved numerically usingKeller box method, which is described in Cebeci andBradshaw.
46
The velocity, temperature, and concentrationproﬁles were obtained and utilized to compute the skin-friction coefﬁcient, the local Nusselt number, and localSherwood number in Eq. (18). The results of the veloc-ity proﬁles, temperature proﬁle and concentration proﬁlefordifferent values of the governing parameters viz., mag-netic parameter
M
, velocity ratio parameter
, Prandtlnumber
Pr
, radiation parameter
N
r
, viscous dissipationparameter Eckert number
Ec
, Brownian motion parameter
Nb
, thermophoresis parameter
Nt
, and Lewis number
Le
are presented in graphs, while the values of the skin fric-tion coefﬁcient, Nusselt number and local Sherwood num-ber for some values of the parameters are presented ingraphs and tables.To validate the accuracy of our results a comparisonhas been made with previously reported works by Mustafaet al.
41
and Wubshet et al.
43
The comparisons are found tobe in an excellent agreement as shown in Tables I and II.Table I depicts the values of the local skin friction coef-ﬁcients at the surface
−
f
0
for various values of thevelocity ratio parameter
, when
M
=
0. Table II con-tains the values of the local Nusselt number
−
0
whichshows the heat transfer rate at the surface, for some valuesof the Prandtl number
Pr
and velocity ratio parameter
,when
M
=
0,
N
r
→
,
Ec
=
0,
Nt
=
0,
Nb
→
0 and
Le
=
2
0.
Table I.
Comparison of values of local friction coefﬁcient
f
0
withMustafa et al.
41
and Wubshet et al.
43
for some values of
, when
M
=
0,
N
r
→
,
Ec
=
0,
Nt
=
0,
Nb
→
0, and
Le
=
2
0.
Mustafa
35
Wubshet et al.
37
Present study0.01
−
0
99823
−
0
9980
−
0
99870.1
−
0
96954
−
0
9694
−
0
96970.2
−
0
91813
−
0
9181
−
0
91840.5
−
0
66735
−
0
6673
−
0
66762.0 2
01767 2
0175 2
02013.0 4
72964 4
7292 4
7393
Table II.
Comparison of values of local friction coefﬁcient
−
0
withMustafa et al.
41
and Wubshet et al.
43
for some values of
and
Pr
, when
M
=
0,
N
r
→
,
Ec
=
0,
Nt
=
0,
Nb
→
0, and
Le
=
2
0.
Pr
Mustafa
35
Wubshet et al.
37
Present study1.0 0
1 0
6022 0
602156 0
60200
2 0
6245 0
624467 0
62440
5 0
6924 0
692460 0
69261.5 0
1 0
7768 0
776802 0
77680
2 0
7971 0
797122 0
79720
5 0
8648 0
864771 0
8652
Figures 2 and 3 show the effects of the magnetic param-eter
M
and velocity ratio parameter
on the ﬂow ﬁeldvelocity
f
. Figure 2, shows the effect of the magneticparameter
M
on the ﬂow ﬁeld velocity
f
for three dif-ferent values of the velocity ratio parameter
, i.e.,
=
0
=
1
and
=
3. When
=
0
the velocity ﬁeld andthe boundary layer thickness decrease with an increasein
M
. When
=
1, the variation of
M
has no inﬂuenceon the ﬂow velocity. On the other hand when
=
3, theﬂow velocity increases and the boundary layer thicknessdecreases with an increase in
M
.Figure 3, shows the effect of the velocity ratio parame-ter
on the ﬂow ﬁeld velocity
f
. It can be observedthat when the stretching velocity of the sheet exceeds thefree stream velocity (i.e.,
<
1), the ﬂow velocity and theboundary layer thickness increases with an increase in
.Moreover, when
>
1
the ﬂow velocity increases andthe boundary layer thickness decreases with an increase in$
\
lambda$. Furthermore, from Figures 2 and 3 it can beobserved that, when
>
1, the ﬂow has boundary layerstructure but when
<
1, the ﬂow has an inverted bound-ary layer structure.Figures 4–7 demonstrate the effects of various param-eters on the nanoﬂuid temperature proﬁle
. Figure 4,shows the effects of the Prandtl number
Pr
on temperatureof the nanoﬂuid. The temperature of the nanoﬂuid at a
00.511.522.533.5400.511.522.53
η
f
′
(
η
)
M = 0, 1, 5, 10
λ
= 3.0
λ
= 1.0
λ
= 0.0M = 0, 1, 5, 10M = 0, 1, 5, 10
Fig. 2.
Velocity proﬁle
f
for different values of
M
, when
Pr
=
1
0,
Nr
=
3,
Ec
=
1
0,
Nt
=
0
=
Nb
=
0
5, and
Le
=
2
0,
=
0
0
1
0
3
0.
J. Nanoﬂuids, 2, 283–291, 2013
287

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