Unsteady flow due to a contracting cylinder in a nanofluid using Buongiorno’s model Khairy Zaimi a , Anuar Ishak b , Ioan Pop c,⇑ a Institute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia b School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia c Department of Mathematics, Babes ß-Bolyai University, 400084 Cluj-Napoca, Romania article info Article history: Received 14 July 2013 Received in revised form 31 July 2013 Accepted 19 September 2013 Available online 19 October 2013 Keywords: Unsteady flow Nanofluids Dual solutions Contracting cylinder abstract The unsteady flow and heat transfer of a nanofluid over a contracting cylinder is studied. Using a simi- larity transformation, the unsteady Navier–Stokes equations are transformed into a system of ordinary differential equations, which are then solved numerically using a shooting method. The effects of the unsteadiness parameter and the Brownian motion parameter on the flow field and heat transfer charac- teristics are analyzed and graphically presented. Dual solutions are found to exist for a certain range of the unsteadiness parameter. It is observed that the skin friction coefficient, the Nusselt number and the Sherwood number decrease with increasing values of the unsteadiness parameter. The magnitude of the Nusselt number and the Sherwood number decrease as the Brownian motion parameter increases. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction The study of fluid flow due to a stretching/shrinking surface has received significant interest among researchers over the past few years. This growing interest is accelerated because of their great practical applications. For example, in engineering and industrial processes, these flows have been used in the manufacturing and extraction of polymer and rubber sheets, wire drawing and glass-fiber production, etc. It is worth pointing out that the investigations of flow over the shrinking sheet have been studied by Miklavc ˇic ˇ and Wang [1], Wang [2], Ishak et al. [3], Fang et al. [4], Bhattacharyya et al. [5] and Faraz et al. [6], among others. Miklavc ˇic ˇ and Wang [1] are the first who investigated the viscous flow over a shrinking surface. The flow is quite different from the stretching case where the sheet is shrunk towards a slot. As mentioned by Goldstein [7], this new type of shrinking flow is essentially a backward flow and the mass suction is needed to maintain the flow. For this flow configuration, the fluid loses any memory of the perturbation introduced by the leading edge, say the slot. As a result, the flow generated due to a shrinking surface produces a distinct physical behavior compared to the forward stretching surface. Different aspects of flow over a shrinking surface have been ex- plored by a great number of researchers. However, little attention has been drawn on both the steady and unsteady flow over the stretching/shrinking cylinder. It seems that Lok and Pop [8] were the first who obtained triple solutions for the steady stagnation point flow of a viscous and incompressible fluid over a permeable shrinking circular cylinder. Recently, Wan Zaimi et al. [9] have studied the unsteady viscous flow due to a permeable shrinking cylinder by means of transformations introduced by Fang et al. [10,11]. They discovered that dual solutions exist in a certain range of suction and unsteadiness parameters. Most recently, Rohni et al. [12] presented a numerical solution to the problem of the unsteady flow over a shrinking sheet with mass suction in a nanofluid using the nanofluid model proposed by Buongiorno [13]. Nanofluids which comprised of a mixture of nanoparticles and the base fluid e.g. water, ethylene glycol and oil were first introduced by Choi [14]. This mixture has good stability and rheological properties which increases the thermal conductivity, and with no penalty in the pressure drop [15]. Thus, the introduction of nanofluids is to overcome the problem of low thermal conductivity of the base flu- ids which leads to poor heat transfer capability. Motivated by the above-mentioned investigations, the present paper looks into the problem of unsteady flow over an imperme- able contracting cylinder in a nanofluid using the nanofluid model proposed by Buongiorno [13]. In this study, the effects of Brownian motion parameter and unsteadiness parameter on the flow field and heat transfer characteristics are thoroughly examined. The coupled system of ordinary differential equations obtained is solved numerically using a shooting method. From the practical point of view, the study of flow over an expanding/contracting 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.09.047 ⇑ Corresponding author. Tel.: +40 264 594 315; fax: +40 264 591 906. E-mail addresses: anuar_mi@ukm.my (A. Ishak), popm.ioan@yahoo.co.uk (I. Pop). International Journal of Heat and Mass Transfer 68 (2014) 509–513 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt