International loumal of Heat and Mass Transfer 53 (2010)2477-2483 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer ELSEVIER journal homepage:www.elsevier.com/locate/ijhmt Boundary-layer flow of a nanofluid past a stretching sheet W.A.Khan2.I.Popb ARTICLE INFO ABSTRACT which his is the acen has odel s A sin ted whic e Prandtl Lewis number Le.Bro motion r aphical formsw umber for each Le.Nb and Nt numbers e 2010 Elsevier Ltd.All rights reserved 1.Introduction transfer fluids since the of these fluids plays The flow over a stretching surface is an important problem in ransfermediu and the heat transfer surface.Therefore.nume 的 of these fuids bysu en t partic n liquids 101.An no gy has been widely used in indu ie nan non-uniform velo ty thr mall a ount (les tha 1第b lume) hStcaewadmnenoavnpribleboundangyatow count the solid particl dispersion. After thes er th hors, nan This problem is part hat drive the next major industrial re is cer tained by thisp ork the flow urenty being expored.taimsat manipulating the structureo ids.includingo water.and ethylene glycol mixture are poor heat 14 transport in nano
Boundary-layer flow of a nanofluid past a stretching sheet W.A. Khan a,*, I. Pop b aDepartment of Engineering Sciences, PN Engineering College, National University of Sciences and Technology, PNS Jauhar, Karachi 75350, Pakistan b Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania article info Article history: Received 29 September 2009 Received in revised form 3 January 2010 Accepted 16 January 2010 Available online 18 February 2010 Keywords: Boundary layer Nanofluid Stretching sheet Brownian motion Thermophoresis Similarity solution abstract The problem of laminar fluid flow which results from the stretching of a flat surface in a nanofluid has been investigated numerically. This is the first paper on stretching sheet in nanofluids. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. A similarity solution is presented which depends on the Prandtl number Pr, Lewis number Le, Brownian motion number Nb and thermophoresis number Nt. The variation of the reduced Nusselt and reduced Sherwood numbers with Nb and Nt for various values of Pr and Le is presented in tabular and graphical forms. It was found that the reduced Nusselt number is a decreasing function of each dimensionless number, while the reduced Sherwood number is an increasing function of higher Pr and a decreasing function of lower Pr number for each Le, Nb and Nt numbers. 2010 Elsevier Ltd. All rights reserved. 1. Introduction The flow over a stretching surface is an important problem in many engineering processes with applications in industries such as extrusion, melt-spinning, the hot rolling, wire drawing, glass– fiber production, manufacture of plastic and rubber sheets, cooling of a large metallic plate in a bath, which may be an electrolyte, etc. In industry, polymer sheets and filaments are manufactured by continuous extrusion of the polymer from a die to a windup roller, which is located at a finite distance away. The thin polymer sheet constitutes a continuously moving surface with a non-uniform velocity through an ambient fluid [1]. Experiments show that the velocity of the stretching surface is approximately proportional to the distance from the orifice [2]. Crane [3] studied the steady two-dimensional incompressible boundary layer flow of a Newtonian fluid caused by the stretching of an elastic flat sheet which moves in its own plane with a velocity varying linearly with the distance from a fixed point due to the application of a uniform stress. This problem is particularly interesting since an exact solution of the two-dimensional Navier–Stokes equations has been obtained by Crane [3]. After this pioneering work, the flow field over a stretching surface has drawn considerable attention and a good amount of literature has been generated on this problem [4–9]. In recent years, some interest has been given to the study of convective transport of nanofluids. Conventional heat transfer fluids, including oil, water, and ethylene glycol mixture are poor heat transfer fluids, since the thermal conductivity of these fluids plays an important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. Therefore, numerous methods have been taken to improve the thermal conductivity of these fluids by suspending nano/micro or larger-sized particle materials in liquids [10]. An innovative technique to improve heat transfer is by using nano-scale particles in the base fluid [11]. Nanotechnology has been widely used in industry since materials with sizes of nanometers possess unique physical and chemical properties. Nano-scale particle added fluids are called as nanofluid, which is firstly utilized by Choi [11]. Choi et al. [12] showed that the addition of a small amount (less than 1% by volume) of nanoparticles to conventional heat transfer liquids increased the thermal conductivity of the fluid up to approximately two times. Khanafer et al. [13] seem to be the first who have examined heat transfer performance of nanofluids inside an enclosure taking into account the solid particle dispersion. After these authors, nanotechnology is considered by many to be one of the significant forces that drive the next major industrial revolution of this century. It represents the most relevant technological cutting edge currently being explored. It aims at manipulating the structure of the matter at the molecular level with the goal for innovation in virtually every industry and public endeavor including biological sciences, physical sciences, electronics cooling, transportation, the environment and national security. Some numerical and experimental studies on nanofluids include thermal conductivity [14], convective heat transfer [15–19]. A comprehensive survey of convective transport in nanofluids was made by Buongiorno [20] and. Kakaç and Pramuanjaroenkij [10]. Very recently, Kuznetsov and 0017-9310/$ - see front matter 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.01.032 * Corresponding author. E-mail address: wkhan_2000@yahoo.com (W.A. Khan). International Journal of Heat and Mass Transfer 53 (2010) 2477–2483 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
WA Khan L Pop/nte eat and Mes Transfer 53(2010)2477-2483 Nomenclature constant shee the etic di Lewis icle volume fraction t num numbe densti temperature at the stretching surface which nd the ossiblre accou Boundary Layer namely those in which oth th all and th etsov 22]have studied the Ch rated.The model used for the Slit resis.For the Is medium the Darcy model has peen Fig.1.Physical model and coordinate system ian cal Nusselt and local Sherwood numbers on thes e four param 贺+-0 ohpw ad they have r 微+架+(+ 2.Basic equations 贺+罗-器保别 (3) We consider the steady two-dimensional boundary layer flow -++(低吸 and x is the mea + site hat at the s 紧+-臣别别 subiect to the boundary conditions C-Cw aty=0 (6) and nand vas,see Kuznetsov and Nield Here and are the velocity components respectively.p is the fluid pressure.pris the density of the base
Nield [21] have examined the influence of nanoparticles on natural convection boundary-layer flow past a vertical plate, using a model in which Brownian motion and thermophoresis are accounted for. The authors have assumed the simplest possible boundary conditions, namely those in which both the temperature and the nanoparticle fraction are constant along the wall. Further, Nield and Kuznetsov [22] have studied the Cheng–Minkowycz [23] problem of natural convection past a vertical plate, in a porous medium saturated by a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium the Darcy model has been employed. The objective of the present study is to analyze the development of the steady boundary layer flow, heat transfer and nanoparticle fraction over a stretching surface in a nanofluid. A similarity solution is presented. This solution depends on a Prandtl number Pr, a Lewis number Le, a Brownian motion number Nb and a thermophoresis number Nt. The dependency of the local Nusselt and local Sherwood numbers on these four parameters is numerically investigated. To our best of knowledge, the results of this paper are new and they have not been published before. 2. Basic equations We consider the steady two-dimensional boundary layer flow of a nanofluid past a stretching surface with the linear velocity uw(- x) = ax, where a is a constant and x is the coordinate measured along the stretching surface, as shown in Fig. 1. The flow takes place at y 0, where y is the coordinate measured normal to the stretching surface. A steady uniform stress leading to equal and opposite forces is applied along the x-axis so that the sheet is stretched keeping the origin fixed. It is assumed that at the stretching surface, the temperature T and the nanoparticle fraction C take constant values Tw and Cw, respectively. The ambient values, attained as y tends to infinity, of T and C are denoted by T1 and C1, respectively. The basic steady conservation of mass, momentum, thermal energy and nanoparticles equations for nanofluids can be written in Cartesian coordinates x and y as, see Kuznetsov and Nield [21] and Nield and Kuznetsov [22], @u @x þ @v @y ¼ 0 ð1Þ u @u @x þ v @u @y ¼ 1 qf @p @x þ t @2 u @x2 þ @2 u @y2 ! ð2Þ u @v @x þ v @v @y ¼ 1 qf @p @y þ t @2 v @x2 þ @2 v @y2 ! ð3Þ u @T @x þ v @T @y ¼ a @2 T @x2 þ @2 T @y2 ! þ s DB @C @x @T @x þ @C @y @T @y þ DT T1 @T @x 2 þ @T @y 2 " #) ð4Þ u @C @x þ v @C @y ¼ DB @2 C @x2 þ @2 T @y2 ! þ DT T1 @2 T @x2 þ @2 T @y2 ! ð5Þ subject to the boundary conditions v ¼ 0; u ¼ uwðxÞ ¼ ax; T ¼ Tw; C ¼ Cw aty ¼ 0 u ¼ v ¼ 0; T ¼ T1; C ¼ C1 as y !1 ð6Þ Here u and v are the velocity components along the axes x and y, respectively, p is the fluid pressure, qf is the density of the base fluid, a is the thermal diffusivity, t is the kinematic viscosity, a is Nomenclature a constant C nanoparticle volume fraction Cw nanoparticle volume fraction at the stretching surface C1 ambient nanoparticle volume fraction DB Brownian diffusion coefficient DT thermophoretic diffusion coefficient f(g) dimensionless stream function K thermal conductivity Le Lewis number Nb Brownian motion parameter Nt thermophoresis parameter Nu Nusselt number Pr Prandtl number p pressure qm wall mass flux qw wall heat flux Rex local Reynolds number Shx local Sherwood number T fluid temperature Tw temperature at the stretching surface T1 ambient temperature u, v velocity components along x- and y-axes uw velocity of the stretching sheet x, y Cartesian coordinates (x-axis is aligned along the stretching surface and y-axis is normal to it) Greek symbols a thermal diffusivity /ðgÞ rescaled nanoparticle volume fraction g similarity variable h(g) dimensionless temperature t kinematic viscosity of the fluid qf fluid density qp nanoparticle mass density (qc)f heat capacity of the fluid (qc)p effective heat capacity of the nanoparticle material s ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid w stream function x Boundary Layer Stretching Sheet Force Slit y Fig. 1. Physical model and coordinate system. 2478 W.A. Khan, I. Pop / International Journal of Heat and Mass Transfer 53 (2010) 2477–2483
WA Kha Pop/joumal o Heat and Mass Trer 53(010)477-4 2479 ary con ccurate.The varia s presen ted in Tables and (b)It is obs ensionless parameters conside Pr Le Nh and Nt 0-二1-ao9 7) urther. v and N 21 y as u-a umber an also obt oresis parameter N.These linear regresion stimations can ur-NurC..N f"+-=0 (⑧) Shr=Shr CoNb+CINt Prr"+fd++N02-0 (16) (9) p+Lef6+g=0 (10 the e subject to the boundary conditions f0)=0,f0)=1,80)=1,0)=1 oEaOi6PnidPOeStesmpleinearegrcsioniomu f(o)-0,(∞)=0.)-0 (11) where primes denote differentiation with respect toand the four parameters are defined by 5 are shown in Figs. d3.sepected.tebontaare orm as in the ca se of a regular fluid.The temperature profiles con (pc)i Nt-(pe)Dr(T.-Ts) (oc)Tv 12 Present results Nb Nr =0 Wang [24]Gorla and Sidawi [25] m 911 cticali number Nu and the Sherwood number Sh which are defined as 64622 (13) Re12Nu=-(0).Re,1P2Sh=(0) 04 小究 05201 027 f()=1-e (15 Nh=02 h-0.3 Nh=0.5 Nt Shr Nr Shr Nt Shr 3.Results and discussion
a positive constant, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient and s=(qc)p/(qc)f is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid with q being the density, c is the volumetric volume expansion coefficient and qp is the density of the particles. We look for a similarity solution of Eqs. (1)–(5) with the boundary conditions (6) of the following form: w ¼ ðatÞ 1=2 xfðgÞ; hðgÞ ¼ T T1 Tw T1 ; /ðgÞ ¼ C C1 Cw C1 ; g ¼ ða=tÞ 1=2 y ð7Þ where the stream function w is defined in the usually way as u = ow/ oy and v = ow/ ox. In seeking of the similarity solution (7), we have taken into account that the pressure in the outer (inviscid) flow is p = p0 (constant). On substituting (7) into Eqs. (2)–(5), we obtain the following ordinary differential equations: f000 þ ff00 f02 ¼ 0 ð8Þ 1 Pr h00 þ f h0 þ Nb/0 h0 þ Nth02 ¼ 0 ð9Þ /00 þ Lef/0 þ NT Nb h00 ¼ 0 ð10Þ subject to the boundary conditions fð0Þ ¼ 0; f0 ð0Þ ¼ 1; hð0Þ ¼ 1; /ð0Þ ¼ 1 f0 ð1Þ ¼ 0; hð1Þ ¼ 0; /ð1Þ ¼ 0 ð11Þ where primes denote differentiation with respect to g and the four parameters are defined by Pr ¼ m a ; Le ¼ t DB ; Nb ¼ ðqcÞpDBð/w /1Þ ðqcÞf t ; Nt ¼ ðqcÞpDT ðTw T1Þ ðqcÞf T1t ð12Þ Here Pr, Le, Nb and Nt denote the Prandtl number, the Lewis number, the Brownian motion parameter and the thermophoresis parameter, respectively. It is important to note that this boundary value problem reduces to the classical problem of flow and heat and mass transfer due to a stretching surface in a viscous fluid when Nb and Nt are zero in Eqs. (9) and (10). (The boundary value problem for / then becomes ill-posed and is of no physical significance). The quantities of practical interest, in this study, are the Nusselt number Nu and the Sherwood number Sh which are defined as Nu ¼ xqw kðTw T1Þ ; Sh ¼ xqm DBðCw C1Þ ð13Þ where qw and qm are the wall heat and mass fluxes, respectively. Using variables (7), we obtain Re1=2 x Nu ¼ h0 ð0Þ; Re1=2 x Sh ¼ /0 ð0Þ ð14Þ where Rex = uw(x)x/t is the local Reynolds number based on the stretching velocity uw(x). Kuznetsov and Nield [21] referred Re1=2 x Nu and Re1=2 x Sh as the reduced Nusselt number Nur = h0 (0) and reduced Sherwood number Shr = u0 (0), respectively. It is worth mentioning that Eq. (8) with the boundary conditions (11) has the analytical solution, first obtained by Crane [3], given by fðgÞ ¼ 1 eg ð15Þ 3. Results and discussion Eqs. (8)–(10) subject to the boundary conditions (11) have been solved numerically for some values of the governing parameters Pr, Pe, Nb and Nt using an implicit finite-difference method. Neglecting the effects of Nb and Nt numbers, the results for the reduced Nusselt number h0 (0) are compared with those obtained by Wang [24], and Gorla and Sidawi [25] for different values of Pr in Table 1. We notice that the comparison shows good agreement for each value of Pr. Therefore, we are confident that the present results are very accurate. The variation of the reduced Nusselt number Nur and Sherwood number Shr with Nb and Nt for Pr = 10 and Le = 10 is presented in Tables 2(a) and (b). It is observed that Nur is a decreasing function, while Shr is an increasing function of each dimensionless parameters considered Pr, Le, Nb and Nt . Further, as in Kuznetsov and Nield [21], simple linear multiple regression estimations Nurest and Shrest of the reduced Nusselt number and reduced Sherwood number are also obtained, which incorporate the effects of Brownian motion parameter Nb and thermophoresis parameter Nt. These linear regression estimations can be written as Nurest ¼ Nur þ CbNb þ CtNt Shrest ¼ Shr þ CbNb þ CtNt ð16Þ The results of these estimations with the regression coefficients and the maximum relative errors e and c are presented in Tables 3(a) and (b). The errors e and c are calculated as e = |(NurestNu)/Nu| and c = |(ShrestSh)/Sh|, see Kuznetsov and Nield [11]. We believe that for most practical purposes the simple linear regression formulas in Eq. (16) should be adequate. Plots of the dependent similarity variables h(g) and /ðgÞ for a typical case, selected as that for Pr = 10, Le=10, Nb = 0.5 and Nt = 0.5 are shown in Figs. 2 and 3. As expected, the boundary layer profiles for the temperature function h(g) are essentially the same form as in the case of a regular fluid. The temperature profiles converge quickly than the stream function profiles. As the parameters Nt and Nb increase, the temperature increases for the specified Table 1 Comparison of results for the reduced Nusselt number h0 (0). Pr Present results Nb = Nt = 0 Wang [24] Gorla and Sidawi [25] 0.07 0.0663 0.0656 0.0656 0.20 0.1691 0.1691 0.1691 0.70 0.4539 0.4539 0.5349 2.00 0.9113 0.9114 0.9114 7.00 1.8954 1.8954 1.8905 20.00 3.3539 3.3539 3.3539 70.00 6.4621 6.4622 6.4622 Table 2 (a) Variation of Nur with Nb and Nt for Pr = 10 and Le = 10. (b) Variation of Shr with Nb and Nt when Pr = 10 and Le = 10. (a) Nb=0.1 Nb=0.2 Nb=0.3 Nb=0.4 Nb=0.5 Nt Nur Nt Nur Nt Nur Nt Nur Nt Nur 0.1 0.9524 0.1 0.5056 0.1 0.2522 0.1 0.1194 0.1 0.0543 0.2 0.6932 0.2 0.3654 0.2 0.1816 0.2 0.0859 0.2 0.0390 0.3 0.5201 0.3 0.2731 0.3 0.1355 0.3 0.0641 0.3 0.0291 0.4 0.4026 0.4 0.2110 0.4 0.1046 0.4 0.0495 0.4 0.0225 0.5 0.3211 0.5 0.1681 0.5 0.0833 0.5 0.0394 0.5 0.0179 (b) Nb=0.1 Nb=0.2 Nb=0.3 Nb=0.4 Nb=0.5 Nt Shr Nt Shr Nt Shr Nt Shr Nt Shr 0.1 2.1294 0.1 2.3819 0.1 2.4100 0.1 2.3997 0.1 2.3836 0.2 2.2740 0.2 2.5152 0.2 2.5150 0.2 2.4807 0.2 2.4468 0.3 2.5286 0.3 2.6555 0.3 2.6088 0.3 2.5486 0.3 2.4984 0.4 2.7952 0.4 2.7818 0.4 2.6876 0.4 2.6038 0.4 2.5399 0.5 3.0351 0.5 2.8883 0.5 2.7519 0.5 2.6483 0.5 2.5731 W.A. Khan, I. Pop / International Journal of Heat and Mass Transfer 53 (2010) 2477–2483 2479
2480 WA Khan I Pop/temational joumal ef Heat and Mess(1) by Eq (16)for Le-10.(b)Coefncients in Nur Nb=0.5 c. Shr c N=0.5 10 06 02 Pr,Le 1015 ted values of Nbad NtNb=0.1,0.3,0.5 o However.the thickess of th bounary the Pr=10 with the in Le=10 shows thee and le numbers on the to erature distr is oh Pr an 5 eter fig.2.Effect of Nr and Nb on ter in the dimensionless heat transfer rates was observed with the in Pr=10 Pr=10 Le=10 Nt=0.1 Nt=0.1 Nb=0.1 02 Nb=0.1,0.3,0.5 0 Le=10,20,30 05 Fig.3.ETect of No on conc ntration distribution for specified par rofiles for selected par
conditions. However, the thickness of the boundary layer for the mass fraction function /ðgÞ is found to be smaller than the thermal boundary layer thickness when Le > 1. It decreases with the increase in Nb and this decrease diminishes when Nb > 0.5. Fig. 4 shows the effects of Pr and Le numbers on the temperature distribution for the selected values of Nb and Nt parameters. It is observed that the temperature increases with the increase in both Pr and Le numbers. The effects of Le numbers on the concentration profiles for the selected parameters are shown in Fig. 5. It is clear that the concentration decreases as the Le numbers increase. The variation in dimensionless heat transfer rates vs Nt parameter is shown in Figs. 6(a) and (b). They show the effects of Pr numbers and Nb parameters on the dimensionless heat transfer rates for the same Le number. It is clear that the dimensionless heat transfer rates decrease with the increase in Nb and Nt parameters but increase with the increase in Pr numbers. However, a decrease in the dimensionless heat transfer rates was observed with the inTable 3 (a) Coefficients in Nurest and Shrest given by Eq. (16) for Le=10. (b) Coefficients in Nurest and Shrest given by Eq. (16) for Pr = 10. (a) Pr Nur Cb Ct e Shr Cb Ct c 1 0.552 0.341 0.176 0.004 2.049 0.774 0.313 0.077 2 0.786 0.727 0.371 0.016 1.941 1.013 0.131 0.098 5 0.960 1.313 0.634 0.062 1.974 0.749 0.559 0.095 10 0.810 1.324 0.617 0.110 2.301 0.237 1.112 0.111 (b) Le Nur Cb Ct e Shr Cb Ct c 5 1.057 1.624 0.812 0.114 1.093 0.940 0.877 0.151 10 0.810 1.324 0.617 0.110 2.301 0.237 1.112 0.111 15 0.708 1.184 0.535 0.105 3.066 0.702 1.158 0.111 25 0.608 1.036 0.441 0.100 4.124 1.066 1.235 0.114 η θ(η) 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Nt, Nb = 0.1, 0.3, 0.5 Pr=10 Le=10 Fig. 2. Effect of Nt and Nb on temperature distribution for specified parameters. η φ(η) 0 0.511.5 2 0 0.2 0.4 0.6 0.8 1 Nb = 0.1, 0.3, 0.5 Pr=10 Le=10 Nt=0.1 Fig. 3. Effect of Nb on concentration distribution for specified parameters. η θ(η) 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 Pr, Le = 10, 15 Nb=0.5 Nt=0.5 Fig. 4. Effect of Pr and Le on temperature distribution for selected values of Nb and Nt. η φ(η) 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Le = 10, 20, 30 Pr=10 Nt=0.1 Nb=0.1 Fig. 5. Effect of Le number on concentration profiles for selected parameters. 2480 W.A. Khan, I. Pop / International Journal of Heat and Mass Transfer 53 (2010) 2477–2483
WA Khan Pop/Heat nd Mas(010)477-43 2481 a0.55 b Nb=0.1 Pr=1 Pr=10 0.5 0.8 Le=10 Nb=0.1 Le-10 045 Nb=0.3 0.6 04 0.35 Nb=0.5 0.2 Nb=0.3 Nb=0.5 02 03 0.4 0.5 0.20.3 0.4 0.5 Nt Nt a12 b0.8 Le=5 Le=25 Pr=10 Pr=10 Nb=0.1 0.6 Nb=0.1 Nb=0.3 Nb=0.5 Nb=0.3 Nb=0.5 0.2 0.4 0.1 0.2 04 0.5 Fig 7.Effects of Nb and Le numbers on dim ionless heat transfer rates. relatively lower thermal conductivity.which reduces conduction This resultsin reduction of the thermal boundarylayer ed values of 4.Conclusions rates decn vith the increase i sfer rates is higher for smalle value for large values of Pr numbers.the ss6n防 is A similarity dimensionless massa tes is monotonic for Le of Le.Nb a nd Nt umbersare lso d in ter nian moti ion and ther ecause a higher Prand
crease in Le numbers. This is shown in Fig. 7. The change in the dimensionless heat transfer rates is found to be higher for smaller values of the parameter Nb and this change decreases with the increase of Nt. Figs. 8 and 9 show the variation in dimensionless mass transfer rates vs Nt parameter for the selected values of other parameters and the same Le number. It is clear from Fig. 8(a), that the dimensionless mass transfer rates decrease with the increase in Nt for small Pr numbers. Like dimensionless heat transfer rates, the change in the mass transfer rates is higher for smaller values of the parameter Nb and decreases with the increase in the parameter Nb. However, for large values of Pr numbers, the dimensionless mass transfer rates increase with the increase in Nt and decrease in Nb (see Fig. 8(b)). The effect of Le numbers on the dimensionless mass transfer rates is shown in Fig. 9(a) and (b) for the same values of the Pr number. The increase in dimensionless mass transfer rates is monotonic for larger Le numbers as shown in Fig. 9(b). Finally, we notice from Fig. 6 that the heat transfer at fixed values of Le, Nb and Nt increases with the Prandtl number, because a higher Prandtl number fluid has a relatively lower thermal conductivity, which reduces conduction and thereby increases the heat transfer rate at the surface of the sheet. This results in reduction of the thermal boundary layer thickness, as can be observed from Fig. 6. 4. Conclusions The problem of laminar fluid flow resulting from the stretching of a flat surface in a nanofluid has been investigated numerically first time. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. A similarity solution is presented which depends on the Prandtl number Pr, Lewis number Le, Brownian motion number Nb and thermophoresis number Nt. The variation of the reduced Nusselt and reduced Sherwood numbers with Nb and Nt for various values of Pr and Le is presented in tabular and graphical forms. Linear regression estimations of the reduced Nusselt and reduced Sherwood numbers are also obtained in terms of Brownian motion and thermophoresis parameters. It was found that the reduced Nusselt Nt Nux/Re1 x /2 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Pr = 10 Le = 10 Nb=0.1 Nb=0.3 Nb=0.5 Nt Nux/Re1 x /2 0.1 0.2 0.3 0.4 0.5 0.3 0.35 0.4 0.45 0.5 0.55 Pr = 1 Le = 10 Nb=0.1 Nb=0.3 Nb=0.5 a b Fig. 6. Effects of Nb and Pr numbers on dimensionless heat transfer rates. Nt Nux/Re1/2 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 1.2 Le = 5 Pr = 10 Nb=0.1 Nb=0.3 Nb=0.5 Nt Nux/Re1/2 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 Le = 25 Pr = 10 Nb=0.1 Nb=0.3 Nb=0.5 a b Fig. 7. Effects of Nb and Le numbers on dimensionless heat transfer rates. W.A. Khan, I. Pop / International Journal of Heat and Mass Transfer 53 (2010) 2477–2483 2481
487 WA Khan 1 Pop/temational Joumal of Heat ad Tre 53(010)477-43 a2.4 /Nb=0.1,0.3,0.5 b28 Pr=10 e=10 2.6 Nb=0.1.0.4.0. 1.8 Pr=1 2.4 Le=10 0.1 0.2 0.3 0.4 0.5 0.2 0.3 0.4 tion rates a 2 b4.6 Le=5 Le=25 1.8 Pr=10 4Pr=10 .6 14 Nb=0.1.0.2.0.3 Nb=0.2.0.3.0.4 0.5 02 0.4 0. 02 0.4 0. Fig 9.Effect of Nb and Le numbers on dimensionless concentration rates. number is a decreasing function.while the reduced Sherwood is and N easing ch value s of the parameter IS EM.Spar Acknowledgments References The Pr aC子e为 200112 1988)5 9 间S然ao之oi2 eohnc,mn ae动设Sa器份 Fluid Flow26(2005)530-546
number is a decreasing function, while the reduced Sherwood number is an increasing function of each values of the parameters Pr, Le, Nb and Nt considered. In the future, the study can be extended to different types of nanofluids as Cu, Al2O3 and TiO2. Acknowledgments The authors wish to express their very sincerely thanks to the reviewers for their valuable comments and suggestions. References [1] H.S. Takhar, A.J. Chamkha, G. Nath, Unsteady three-dimensional MHDboundary-layer flow due to the impulsive motion of a stretching surface, Acta Mech. 146 (2001) 59–71. [2] J. Vleggaar, Laminar boundary layer behaviour on continuous accelerating surface, Chem. Eng. Sci. 32 (1977) 1517–1525. [3] L.J. Crane, Flow past a stretching plate, J. Appl. Math. Phys. (ZAMP) 21 (1970) 645–647. [4] K.N. Lakshmisha, S. Venkateswaran, G. Nath, Three-dimensional unsteady flow with heat and mass transfer over a continuous stretching surface, ASME J. Heat Transfer 110 (1988) 590–595. [5] C.Y. Wang, The three-dimensional flow due to a stretching flat surface, Phys. Fluids 27 (1984) 1915–1917. [6] H.I. Andersson, B.S. Dandapat, Flow of a power-law fluid over a stretching sheet, SAACM 1 (1991) 339–347. [7] E. Magyari, B. Keller, Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls, Eur. J. Mech. B Fluids 19 (2000) 109–122. [8] E.M. Sparrow, J.P. Abraham, Universal solutions for the streamwise variation of the temperature of a moving sheet in the presence of a moving fluid, Int. J. Heat Mass Transfer 48 (2005) 3047–3056. [9] J.P. Abraham, E.M. Sparrow, Friction drag resulting from the simultaneous imposed motions of a freestream and its bounding surface, Int. J. Heat Fluid Flow 26 (2005) 289–295. [10] S. Kakaç, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187– 3196. [11] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME, FED 231/MD 66, 1995, pp. 99–105. [12] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Anomalously thermal conductivity enhancement in nanotube suspensions, Appl. Phys. Lett. 79 (2001) 2252–2254. [13] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 3639–3653. [14] H.U. Kang, S.H. Kim, J.M. Oh, Estimation of thermal conductivity of nanofluid using experimental effective particle volume, Exp. Heat Transfer 19 (2006) 181–191. [15] S.E.B. Maiga, S.J. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement by using nanofluids in forced convection flow, Int. J. Heat Fluid Flow 26 (2005) 530–546. Nt Shx/Re1/2 x 0.1 0.2 0.3 0.4 0.5 3.8 4 4.2 4.4 4.6 Nb = 0.2, 0.3, 0.4 Le = 25 Pr = 10 Nt Shx/Re1/2 x 0.1 0.2 0.3 0.4 0.5 1 1.2 1.4 1.6 1.8 2 Le = 5 Pr = 10 Nb=0.1, 0.2, 0.3 a b Fig. 9. Effect of Nb and Le numbers on dimensionless concentration rates. Nt Shx/Re1/2 0.1 0.2 0.3 0.4 0.5 2.4 2.6 2.8 Pr = 10 Le = 10 Nb=0.1, 0.4, 0.5 Nt Shx/Re1/2 0.1 0.2 0.3 0.4 0.5 1.6 1.8 2 2.2 2.4 Nb=0.1, 0.3, 0.5 Pr = 1 Le = 10 a b Fig. 8. Effects of Nb and Pr number on dimensionless concentration rates. 2482 W.A. Khan, I. Pop / International Journal of Heat and Mass Transfer 53 (2010) 2477–2483
WA Kh Pop/and Mass()477-43 248 0(2002 02967-2979 ofluids in natural convection,Int.J.Heat 1221 DA Nield.A.V.K 9cd ing ep ine 1231: cd8s6mdhahannrn品Pno2g (197 ng Free vertical stretchn suracepp Math 12020240-250 ve transport in n 251Rt
[16] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002–2018. [17] D.Y. Tzou, Thermal instability of nanofluids in natural convection, Int. J. Heat Mass Transfer 51 (2008) 2967–2979. [18] E. Abu-Nada, Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step, Int. J. Heat Fluid Flow 29 (2008) 242–249. [19] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336. [20] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [21] A.V. Kuznetsov, D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Thermal Sci. doi:10.1016/ j.ijthermalsci.2009.07.015. [22] D.A. Nield, A.V. Kuznetsov, The Cheng-Minkowycz problem for natural convective boundary layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer 52 (2009) 5792–5795. [23] P. Cheng, W.J. Minkowycz, Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res. 82 (1977) 2040–2044. [24] C.Y. Wang, Free convection on a vertical stretching surface, J. Appl. Math. Mech. (ZAMM) 69 (1989) 418–420. [25] R.S.R. Gorla, I. Sidawi, Free convection on a vertical stretching surface with suction and blowing, Appl. Sci. Res. 52 (1994) 247–257. W.A. Khan, I. Pop / International Journal of Heat and Mass Transfer 53 (2010) 2477–2483 2483
