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 condi￾tions, namely those in which both the temperature and the nano￾particle 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 sat￾urated by a nanofluid. The model used for the nanofluid incorpo￾rates 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 develop￾ment of the steady boundary layer flow, heat transfer and nano￾particle fraction over a stretching surface in a nanofluid. A similarity solution is presented. This solution depends on a Pra￾ndtl number Pr, a Lewis number Le, a Brownian motion number Nb and a thermophoresis number Nt. The dependency of the lo￾cal Nusselt and local Sherwood numbers on these four parame￾ters 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 stretch￾ing surface, the temperature T and the nanoparticle fraction C take constant values Tw and Cw, respectively. The ambient values, at￾tained as y tends to infinity, of T and C are denoted by T1 and C1, respectively. The basic steady conservation of mass, momentum, thermal en￾ergy 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 nano￾particle 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