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 nanoBoundary-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 electro￾lyte, etc. In industry, polymer sheets and filaments are manufac￾tured 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 solu￾tion of the two-dimensional Navier–Stokes equations has been ob￾tained 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 flu￾ids, 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, numer￾ous 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 nano￾particles to conventional heat transfer liquids increased the ther￾mal 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, nano￾technology is considered by many to be one of the significant forces that drive the next major industrial revolution of this cen￾tury. 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 exper￾imental studies on nanofluids include thermal conductivity [14], convective heat transfer [15–19]. A comprehensive survey of con￾vective 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