Boundary layer flow of nanofluid over an exponentially stretching surface.

0.37 MB·English

by  Nadeem, Sohail

NadeemandLeeNanoscaleResearchLetters2012,7:94
http://www.nanoscalereslett.com/content/7/1/94
NANO IDEA Open Access
Boundary layer flow of nanofluid over an
exponentially stretching surface
Sohail Nadeem1* and Changhoon Lee2
Abstract
The steady boundary layer flow of nanofluid over an exponential stretching surface is investigated analytically. The
transport equations include the effects of Brownian motion parameter and thermophoresis parameter. The highly
nonlinear coupled partial differential equations are simplified with the help of suitable similarity transformations.
The reduced equations are then solved analytically with the help of homotopy analysis method (HAM). The
convergence of HAM solutions are obtained by plotting h-curve. The expressions for velocity, temperature and
nanoparticle volume fraction are computed for some values of the parameters namely, suction injection parameter
a, Lewis number Le, the Brownian motion parameter Nb and thermophoresis parameter Nt.
Keywords: nanofluid, porous stretching surface, boundary layer flow, series solutions, exponential stretching
1 Introduction fundamental properties with the large enhancement in
During the last many years, the study of boundary layer thermal conductivity as compared to the base fluid [23].
flow and heat transfer over a stretching surface has In this study, we have discussed the boundary layer
achieved a lot of success because of its large number of flow of nanofluid over an exponentially stretching sur-
applications in industry and technology. Few of these face with suction and injection. To the best of our
applications are materials manufactured by polymer knowledge, the nanofluid over an exponentially stretch-
extrusion, drawing of copper wires, continuous stretch- ing surface has not been discussed so far. However, the
ing of plastic films, artificial fibers, hot rolling, wire present paper is only a theoretical idea, which is not
drawing, glass fiber, metal extrusion and metal spinning checked experimentally. The governing highly nonlinear
etc. After the pioneering work by Sakiadis [1], a large partial differential equation of motion, energy and nano-
amount of literature is available on boundary layer flow particle volume fraction has been simplified by using
of Newtonian and non-Newtonian fluids over linear and suitable similarity transformations and then solved ana-
nonlinear stretching surfaces [2-10]. However, only a lytically with the help of HAM [24-39]. The convergence
limited attention has been paid to the study of exponen- of HAM solution has been discussed by plotting h-
tial stretching surface. Mention may be made to the curve. The effects of pertinent parameters of nanofluid
works of Magyari and Keller [11], Sanjayanand and have been discussed through graphs.
Khan [12], Khan and Sanjayanand [13], Bidin and Nazar
[14] and Nadeem et al. [15,16]. 2 Formulation of the problem
More recently, the study of convective heat transfer in Consider the steady two-dimensional flow of an incom-
nanofluids has achieved great success in various indus- pressible nanofluid over an exponentially stretching sur-
trial processes. A large number of experimental and the- face. We are considering Cartesian coordinate system in
oretical studies have been carried out by numerous such a way that x-axis is taken along the stretching sur-
researchers on thermal conductivity of nanofluids face in the direction of the motion and y-axis is normal
[17-22]. The theory of nanofluids has presented several to it. The plate is stretched in the x-direction with a
velocity U = U exp (x/l). defined at y = 0. The flow
w 0
and heat transfer characteristics under the boundary
*Correspondence:[email protected] layer approximations are governed by the following
1DepartmentofMathematics,Quaid-i-AzamUniversity,45320,Islamabad
equations
44000,Pakistan
Fulllistofauthorinformationisavailableattheendofthearticle
©2012NadeemandLee;licenseeSpringer.ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommons
AttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproductionin
anymedium,providedtheoriginalworkisproperlycited.
NadeemandLeeNanoscaleResearchLetters2012,7:94 Page2of6
http://www.nanoscalereslett.com/content/7/1/94
∂u ∂v where
+ =0, (1)
∂x ∂y Nt=DB((ρρcc))pf (Cw−C∞), Nb=TD∞T((ρρcc))pf(Tw−υT∞), Le=DυB, Pr=αυ.
∂u ∂u ∂2u
u +v =ν , (2) The physical quantities of interest in this problem are
∂x ∂y ∂y2
the local skin-friction coefficient C, Nusselt number
f
(cid:2) (cid:3) (cid:4) (cid:5) Nux and the local Sherwood number Shx, which are
u∂T +v∂T =α∂2T + (ρc)p D ∂C∂T + DT ∂T 2 , (3) defined as
∂x ∂y ∂y2 (ρc)f (cid:3)B∂y ∂y(cid:4) T∞ ∂y Cfx=ρτUw02|ye=20lx, Nux=−(Tw−x(cid:10)T∞)∂∂Ty(cid:13)(cid:13)(cid:13)(cid:13)y=0, Shx=−(Cw−(cid:10)xC∞)∂∂Cy(cid:13)(cid:13)(cid:13)(cid:13)y=0, (11)
u∂C +v∂C =D ∂2C + DT ∂2T , (4) √2ReCfx=f(cid:4)(cid:4)(0), Nux/(cid:14)2Rex=− 2xlθ(cid:4)(0), Shx/(cid:14)2Rex=− 2xlg(cid:4)(0),
∂x ∂y B∂y2 T∞ ∂y2 where Re = U x/ν is the local Renolds number.
x w
where (u, v) are the velocity components in (x, y)
directions, r is the fluid density of base fluid, ν is the 3 Solution by homotopy analysis method
f
kinematic viscosity, T is the temperature, C is the nano- For HAM solutions, the initial guesses and the linear
particle volume fraction, (rc) is the effective heat capa- operators L (i = 1 - 3) are
p i
city of nanoparticles, (rc) is the heat capacity of the
fluid, a = k/(rc) is the thefrmal diffusivity of the fluid, f0(η)=1−vw−e−η, θ0(η)=e−η, g0(η)=e−(η1,2)
f
D is the Brownian diffusion coefficient and D is the (cid:6) (cid:7) (cid:6) (cid:7)
thBermophoretic diffusion coefficient. T L1 f =f(cid:4)(cid:4)(cid:4)−f(cid:4), L2(θ)=θ(cid:4)(cid:4)−θ, L3 g =g(cid:4)(cid:4)−(1g3.)
The corresponding boundary conditions for the flow
The operators satisfy the following properties
problem are
(cid:15) (cid:16)
(cid:6) (cid:7) L c e−η+c eη+c =0, (14)
u=Uw(x)=U0exp x/l , v=−β(x), T=Tw, C=Cw at y=0, (5) 1 1 2 3
u=0, T=T∞ C=C∞ asy→∞, (cid:15) (cid:16)
in which U0 is the reference velocity, b(x) is the suc- L2 c4e−η+c5eη =0, (15)
tion and injection velocity when b(x) > 0 and b(x) < 0,
(cid:15) (cid:16)
respectively, Tw and T∞ are the temperatures of the L3 c6e−η+c7eη =0, (16)
sheet and the ambient fluid, Cw, C∞ are the nanoparti-
cles volume fraction of the plate and the fluid, in which C to C are constants. From Equations (7)
1 7
respectively. to (9), we can define the following zeroth-order defor-
We are interested in similarity solution of the above mation problems
boundary value problem; therefore, we introduce the fol- (cid:6) (cid:7) (cid:17) (cid:6) (cid:7) (cid:18) (cid:17) (cid:6) (cid:7)(cid:18)
lowing similarity transformations 1−p L1 fˆ η,p −f0(η) =ph¯1H1N˜1 fˆ η,p ,(17)
(cid:10)
uη==Uy(cid:10)0eUxp0(cid:8)exxlp(cid:9)(cid:8)f(cid:4)x(η(cid:9)),, θv==−T−vT2U∞l0e,xp(cid:8)g2x=l(cid:9)C(cid:11)f−(ηC)∞+η.f(cid:4)(η)(cid:12), (6) (cid:6)1−p(cid:7)L2(cid:17)θˆ(cid:6)η,p(cid:7)−θ0(η)(cid:18)=ph¯2H2N˜2(cid:17)θˆ(cid:6)η,p(cid:7)(cid:18)(,18)
2vl 2l Tw−T∞ Cw−C∞ (cid:6) (cid:7) (cid:15) (cid:6) (cid:7) (cid:16) (cid:15) (cid:6) (cid:7)(cid:16)
1−p L gˆ η,p −g (η) =ph¯ H N˜ gˆ η,p ,(19)
Making use of transformations (6), Eq. (1) is identi- 3 0 3 3 3
cally satisfied and Equations (2)-(4) take the form
(cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7)
fˆ 0,p =−v , fˆ(cid:4) 0,p =1, fˆ(cid:4) ∞,p =0, (20)
fηηη+ffηη−2fη2 =0, (7) w
(cid:6) (cid:7) (cid:6) (cid:7)
(cid:6) (cid:7) θˆ 0,p =1, θˆ ∞,p =0, (21)
θηη+Pr fθη−fηθ +Nbθηgη+Ntθη2 =0 (8)
(cid:6) (cid:7) (cid:6) (cid:7)
(cid:6) (cid:7) gˆ(cid:4) 0,p =1, gˆ ∞,p =0. (22)
Nt
gηη+Le fgη−fηg + θηη =0, (9)
Nb In Equations (17)-(22), ħ , ħ , andħ denote the non-
1 2 3
zero auxiliary parameters, H , H and H are the non-
1 2 3
f =−vw, fη =1, θ =1, g=1at η=0, zero auxiliary function (H1 = H2 = H3 = 1) and
(10)
fη →0, θ →0, g→0 asη→∞,
NadeemandLeeNanoscaleResearchLetters2012,7:94 Page3of6
http://www.nanoscalereslett.com/content/7/1/94
N˜ (cid:17)fˆ(cid:6)η,p(cid:7)(cid:18)= ∂3f −2(cid:3)∂f (cid:4)2+f ∂2f , (23) L2(cid:15)θm(η)−χmθm−1(η)(cid:16)=h¯2Rˇ2m(η), (36)
1 ∂η3 ∂η ∂η2
(cid:15) (cid:16)
N˜2(cid:17)θˆ(cid:6)η,p(cid:7)(cid:18)=∂∂η2θ2+Pr(cid:19)f∂∂ηθ −∂∂ηfθ+Nb∂∂θη∂∂ηg+Nt(cid:3)∂∂θη(cid:4)2(cid:20), (24) Lf3(0g)m=(ηf)(cid:4)−(0χ)m=gmf(cid:4)−(1∞(η))==0,h¯3Rˇ3m(η), ((3378))
m m m
(cid:3) (cid:4)
N˜3(cid:15)gˆ(cid:6)η,p(cid:7)(cid:16)= ∂∂η2g2 +Le f∂∂ηg −∂∂ηfg+NNbtθηη +NNbt∂∂η2θ2.(25) θm(0)=θm(∞)=0, (39)
Obviously g(cid:4) (0)=g (∞)=0, (40)
m m
fˆ(η,0)=f0(η), fˆ(η,1)=f(η), (26) where
(cid:22)
0, m≤1,
θˆ(η,0)=θ0(η), θˆ(η,1)=θ(η), (27) χm = 1, m>1. (41)
gˆ(η,0)=g (η), gˆ(η,1)=g(η). (28)
0 m(cid:21)−1 m(cid:21)−1
When p varies from 0 to 1, then fˆ(cid:6)η,p(cid:7), θˆ(cid:6)η,p(cid:7), Rˇ1m(η)=fm(cid:4)(cid:4)(cid:4)−1(η)+ fm−1−kfk(cid:4)(cid:4)−2 fm(cid:4)−1−kfk(cid:4),(42)
(cid:6) (cid:7)
k=0 k=0
gˆ η,p vary from initial guesses f (h), θ (h) andg (h)
0 0 0
to the final solutions f (h), θ (h) andg (h), respectively. m(cid:21)−1(cid:11) (cid:12)
Considering that the auxiliary parameters ħ , ħ and ħ Rˇ2m(η)=θm(cid:4)(cid:4)−1+Pr fm−1−kθk(cid:4)−fm(cid:4)−1−kθk+Nbθm(cid:4)−1−kgk(cid:4)+Ntθm(cid:4)−1−kθk(cid:4) , (43)
1 2 (cid:6) (cid:7)3 k=0
are so properly chosen that the Taylor series of fˆ η,p,
(cid:6) (cid:7) (cid:6) (cid:7)
θdˆinηg,pparaanmdegˆterη,cponevxeprgaendatedpw=it1h, hreesnpceecEtqtouaatnionems (b1e7d)-- Rˇ3m(η)=gm(cid:4)(cid:4)−1+Lem(cid:21)−1(cid:11)fm−1−kgk(cid:4) −fm(cid:4)−1−kgk(cid:12)+NNbtθm(cid:4)(cid:4)−1. (44)
k=0
(19) become
Employing MATHEMATICA, Equations (35)-(40)
(cid:6) (cid:7) (cid:21)∞ have the following solutions
fˆ η,p =f0(η)+ fm(η)pm, (29) (cid:2) (cid:19) (cid:20)(cid:5)
m=1 f(η)=(cid:21)m∞=0fm(η)=Mli→m∞ (cid:21)mM=0 a0m,0+M(cid:21)n=+11e−nη m(cid:21)=Mn−1m(cid:21)+k=10−nakm,nηk , (45)
(cid:6) (cid:7) (cid:21)∞
θˆ η,p =θ0(η)+ θm(η)pm, (30) (cid:21)∞ (cid:2)M(cid:21)+2 (cid:19) M(cid:21)+1 m(cid:21)+1−n (cid:20)(cid:5)
m=1 θ(η)=m=0θm(η)=Mli→m∞ n=1e−nη m=n−1 k=0 Akm,nηk , (46)
(cid:6) (cid:7) (cid:21)∞ (cid:2) (cid:19) (cid:20)(cid:5)
gˆ η,p =g0(η)+ gm(η)pm, (31) g(η)=(cid:21)∞ g (η)= lim M(cid:21)+2e−nη M(cid:21)+1 m(cid:21)+1−nFk ηk ,(47)
m=1 m M→∞ m,n
m=0 n=1 m=n−1 k=0
(cid:6) (cid:7)(cid:13)
1 ∂mfˆ η,p (cid:13)(cid:13) in which a0m,0, akm,n, Akm,n, Fmk,n are the constants and the
fm(η)= m! ∂pm (cid:13)(cid:13) , (32) numerical data of above solutions are shown through
p=0 graphs in the following section.
(cid:6) (cid:7)(cid:13)
1 ∂mθˆ η,p (cid:13)(cid:13) 4 Results and discussion
θm(η)= m! ∂pm (cid:13)(cid:13) , (33) The numerical data of the solutions (45)-(47), which is
p=0 obtained with the help of Mathematica, have been dis-
(cid:6) (cid:7)(cid:13) cussed through graphs. The convergence of the series
(cid:13)
1 ∂mgˆ η,p (cid:13) solutions strongly depends on the values of non-zero
gm(η)= m! ∂pm (cid:13)(cid:13) . (34) auxiliary parameters ħi (i = 1, 2, 3, h1 = h2 = h3), which
p=0 can adjust and control the convergence of the solutions.
Therefore, for the convergence of the solution, the ħ-
The mth-order problems are defined as follow
curves is plotted for velocity field in Figure 1. We have
(cid:15) (cid:16)
L1 fm(η)−χmfm−1(η) =h¯1Rˇ1m(η), (35) found the convergence region of velocity for different
values of suction injection parameter v . It is seen that
w
NadeemandLeeNanoscaleResearchLetters2012,7:94 Page4of6
http://www.nanoscalereslett.com/content/7/1/94
Figure1h-Curveforvelocity. Figure 3 Variation of temperature for different values of Pr
whenLe=2,h=-0.1,Nt=Nb=0.5,v =1.
w
with the increase in suction parameter v , the conver-
w
gence region become smaller and smaller. Almost simi-
lar kind of convergence regions appear for temperature
and nanoparticle volume fraction, which are not shown
here. The non-dimensional velocity f′ against h for var-
ious values of suction injection parameter is shown in
Figure 2. It is observed that velocity field increases with
the increase in v . Moreover, the suction causes the
w
reduction of the boundary layer. The temperature field
θ for different values of Prandtle number Pr, Brownian
parameter Nb, Lewis number Le and thermophoresis
parameter Nt is shown in Figures 3, 4, 5 and 6. In Fig-
ure 3, the temperature is plotted for different values of
Pr. It is observed that with the increase in Pr, there is a
very slight change in temperature; however, for very Figure 4 Variation of temperature for different values of Nb
large Pr, the solutions seem to be unstable, which are whenLe=2,h=-0.1,Nt=0.5,v =1,Pr=2.
w
not shown here. The variation of Nb on θ is shown in
Figure 4. It is depicted that with the increase in Nb, the
temperature profile increases with the increase in Nt;
temperature profile increases. There is a minimal change
however, the thermal boundary layer thickness reduces.
in θ with the increase in Le (see Figure 5). The results
The nanoparticle volume fraction g for different values
remain unchanged for very large values of Le. The
of Pr, Nb, Nt and Le is plotted in Figures 7, 8, 9 and 10.
effects of Nt on θ are seen in Figure 6. It is seen that
It is observed from Figure 7 that with the increase in
Figure2Velocityfordifferentvaluesofsuctionandinjection Figure 5 Variation of temperature for different values of Le
parameter. whenh=-0.1,Nt=Nb=0.5,v =1,Pr=2.
w
NadeemandLeeNanoscaleResearchLetters2012,7:94 Page5of6
http://www.nanoscalereslett.com/content/7/1/94
Figure 6 Variation of temperature for different values of Nt Figure 9 Variation of nanoparticle fraction g for different
whenLe=2,h=-0.1,Nb=0.5,vw=1,Pr=2. valuesofLewhenPr=2,h=-0.1,Nt=0.5,vw=1,Nb=0.5.
Figure 10 Variation of nanoparticle fraction g for different
valuesofNtwhenLe=2,h=-0.1,Nt=0.5,v =1,Pr=2.
w
Figure 7 Variation of nanoparticle fraction g for different
Nb, g decreases and boundary layer for g also decreases.
valuesofNbwhenLe=2,h=-0.1,Nt=0.5,v =1,Pr=2.
w The effects of Pr on g are minimal. (See Figure 8). The
effects of Le on g are shown in Figure 9. It is observed
that g decreases as well as layer thickness reduces with
the increase in Le. However, with the increase in Nt, g
increases and layer thickness reduces (See Figure 10).
Acknowledgements
ThisresearchwassupportedbyWCU(WorldClassUniversity)program
throughtheNationalResearchFoundationofKorea(NRF)fundedbythe
MinistryofEducation,ScienceandTechnologyR31-2008-000-10049-0.
Authordetails
1DepartmentofMathematics,Quaid-i-AzamUniversity,45320,Islamabad
44000,Pakistan2DepartmentofComputationalScienceandEngineering,
YonseiUniversity,Seoul,Korea
Authors’contributions
SNdonethemajorpartofthearticle;however,thefundingand
computationalsuggestionsandproofreadinghasbeendonebyCL.All
authorsreadandapprovedthefinalmanuscript.
Figure 8 Variation of nanoparticle fraction g for different
valuesofPrwhenLe=2,h=-0.1,Nt=0.5,v =1,Nb=0.5.
w
NadeemandLeeNanoscaleResearchLetters2012,7:94 Page6of6
http://www.nanoscalereslett.com/content/7/1/94
Competinginterests 24. LiaoSJ:BeyondPerturbationIntroductiontoHomotopyAnalysisMethodBoca
Thisisjustthetheoreticalstudy,everyexperimentalistcancheckit Raton:Chapman&Hall/CRCPress;2003.
experimentallywithourconsent. 25. AbbasbandyS:Theapplicationofhomotopyanalysismethodto
nonlinearequationsarisinginheattransfer.PhysLettA2006,
Received:26July2011 Accepted:30January2012 360:109-113.
Published:30January2012 26. AbbasbandyS:Homotopyanalysismethodforheatradiationequations.
IntCommunHeatMassTransf2007,34:380-387.
27. AbbasbandyS,TanY,LiaoSJ:Newton-homotopyanalysismethodfor
References
nonlinearequations.ApplMathComput2007,188:1794-1800.
1. SakiadisBC:Boundarylayerbehavioroncontinuoussolidsurfaces:
28. AbbasbandyS:Approximatesolutionforthenonlinearmodelof
IBoundarylayerequationsfortwodimensionalandaxisymmetricflow.
diffusionandreactioninporouscatalystsbymeansofthehomotopy
AIChEJ1961,7:26-28.
analysismethod.ChemEngJ2008,136:144-150.
2. LiuIC:Flowandheattransferofanelectricallyconductingfluidof
29. AbbasbandyS:SolitonsolutionsfortheFitzhugh-Nagumoequationwith
secondgradeoverastretchingsheetsubjecttoatransversemagnetic
thehomotopyanalysismethod.ApplMathModel2008,32:2706-2714.
field.IntJHeatMassTransf2004,47:4427-4437.
30. TanY,AbbasbandyS:HomotopyanalysismethodforquadraticRicati
3. VajraveluK,RollinsD:Heattransferinelectricallyconductingfluidovera
differentialequation.CommNon-linearSciNumerSimm2008,13:539-546.
stretchingsurface.IntJNon-LinearMech1992,27(2):265-277.
31. AlomariAK,NooraniMSM,NazarR:Adaptationofhomotopyanalysis
4. VajraveluK,NayfehJ:Convectiveheattransferatastretchingsheet.Acta
methodforthenumeric-analyticsolutionofChensystem.Commun
Mech1993,96(1-4):47-54.
NonlinearSciNumerSimulat2008,doi:10.1016/j.cnsns.2008.06.011.
5. KhanSK,SubhasAbelM,SonthRaviM:ViscoelasticMHDflow,heatand
32. RashidiMM,DomairryG,DinarvandS:Approximatesolutionsforthe
masstransferoveraporousstretchingsheetwithdissipationofenergy
Burgerandregularizedlongwaveequationsbymeansofthehomotopy
andstresswork.IntJHeatMassTransf2003,40:47-57.
analysismethod.CommunNonlinearSciNumerSimul2009,14:708-717.
6. CortellR:Effectsofviscousdissipationandworkdonebydeformation
33. ChowdhuryMSH,HashimI,AbdulazizO:Comparisonofhomotopy
ontheMHDflowandheattransferofaviscoelasticfluidovera
analysismethodandhomotopy-perturbationmethodforpurely
stretchingsheet.PhysLettA2006,357:298-305.
nonlinearfin-typeproblems.CommunNonlinearSciNumerSimul2009,
7. DandapatBS,SantraB,VajraveluK:Theeffectsofvariablefluidproperties
14:371-378.
andthermocapillarityontheflowofathinfilmonanunsteady
34. SamiBatainehA,NooraniMSM,HashimI:Onanewreliablemodification
stretchingsheet.IntJHeatMassTransf2007,50:991-996.
ofhomotopyanalysismethod.CommunNonlinearSciNumerSimul2009,
8. NadeemS,HussainA,MalikMY,HayatT:Seriessolutionsforthe
14:409-423.
stagnationflowofasecond-gradefluidoverashrinkingsheet.Appl
35. SamiBatainehA,NooraniMSM,HashimI:Modifiedhomotopyanalysis
MathMechEnglEd2009,30:1255-1262.
methodforsolvingsystemsofsecond-orderBVPs.CommunNonlinearSci
9. NadeemS,HussainA,KhanM:HAMsolutionsforboundarylayerflowin
NumerSimul2009,14:430-442.
theregionofthestagnationpointtowardsastretchingsheet.Comm
36. SamiBatainehA,NooraniMSM,HashimI:SolvingsystemsofODEsby
NonlinearSciNumerSimul2010,15:475-481.
homotopyanalysismethod.CommunNonlinearSciNumerSimul2008,
10. AfzalN:Heattransferfromastretchingsurface.IntJHeatMassTransf
13:2060-2070.
1993,36:1128-1131.
37. SajidM,AhmadI,HayatT,AyubM:Seriessolutionforunsteady
11. MagyariE,KellerB:Heatandmasstransferintheboundarylayeronan
axisymmetricflowandheattransferoveraradiallystretchingsheet.
exponentiallystretchingcontinuoussurface.JPhysDApplPhys1999,
CommunNonlinearSciNumerSimul2008,13:2193-2202.
32:577-785.
38. NadeemS,HussainA:MHDflowofaviscousfluidonanonlinear
12. SanjayanandE,KhanSK:Onheatandmasstransferinaviscoelastic
porousshrinkingsheetwithHAM.ApplMathMechEnglEd2009,30:1-10.
boundarylayerflowoveranexponentiallystretchingsheet.IntJTherm
39. NadeemS,AbbasbandyS,HussainM:Seriessolutionsofboundarylayer
Sci2006,45:819-828.
flowofaMicropolarfluidnearthestagnationpointtowardsashrinking
13. KhanSK,SanjayanandE:Viscoelasticboundarylayerflowandheat
sheet.ZNaturforch2009,64a:575-582.
transferoveranexponentialstretchingsheet.IntJHeatMassTransf2005,
48:1534-1542.
doi:10.1186/1556-276X-7-94
14. BidinB,NazarR:Numericalsolutionoftheboundarylayerflowoveran
Citethisarticleas:NadeemandLee:Boundarylayerflowofnanofluid
exponentiallystretchingsheetwiththermalradiation.EurJSciRes2009,
overanexponentiallystretchingsurface.NanoscaleResearchLetters2012
33:710-717. 7:94.
15. NadeemS,HayatT,MalikMY,RajputSA:Thermalradiationseffectsonthe
flowbyanexponentiallystretchingsurface:aseriessolution.Zeitschrift
furNaturforschung2010,65a:1-9.
16. NadeemS,ZaheerS,FangT:Effectsofthermalradiationsonthe
boundarylayerflowofaJeffreyfluidoveranexponentiallystretching
surface.NumerAlgor2011,57:187-205.
17. BachokN,IshakA,PopI:boundaryLayerflowofnanofluidovera
movingsurfaceinaflowingfluid.IntJThermSci2010,49:1663-1668.
18. ChoiSUS:Enhancingthermalconductivityoffluidswithnanoparticle.In
DevelopmentsandApplicationsofNon-NewtonianFlowsEditedby:Siginer
DA,WangHP1995,66:99-105,ASMEFED231/MD.
19. KhanaferK,VafaiK,LightstoneM:Buoyancydrivenheattransfer Submit your manuscript to a
enhancementinatwodimensionalenclosureutilizingnanofluids.IntJ journal and benefi t from:
HeatMassTransf2003,46:3639-3653.
20. MakindeOD,AzizA:Boundarylayerflowofananofluidpasta 7 Convenient online submission
stretchingsheetwithaconvectiveboundarycondition.IntJThermSci
7 Rigorous peer review
2011,50:1326-1332.
21. BayatJ,NiksereshtAH:Investigationofthedifferentbasefluideffectson 7 Immediate publication on acceptance
thenanofluidsheattransferandpressuredrop.HeatMassTransf, 7 Open access: articles freely available online
doi:10.1007/s00231-011-0773-0. 7 High visibility within the fi eld
22. HojjatM,EtemadSG,BagheriR:Laminarheattransferofnanofluidina 7 Retaining the copyright to your article
circulartube.KoreanJChemEng2010,27(5):1391-*1396.
23. FanJ,WangL:Heatconductioninnanofluids:structure-property
correlation.IntJHeatMassTransf2011,54:4349-4359. Submit your next manuscript at 7 springeropen.com