《高等传热学》课程教学资源（文献资料）Boundary-layer flow of a nanofluid past a stretching sheet.pdf

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

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

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

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

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