Mixed convection stagnation flow towards a vertical shrinking sheet Siti Hidayah M. Saleh a , Norihan M. Arifin a,b , Roslinda Nazar c , Fadzilah M. Ali a,b , Ioan Pop d,⇑ a Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia b Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia c School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia d Department of Mathematics, Babes ß-Bolyai University, R-400084 Cluj-Napoca, Romania article info Article history: Received 21 December 2013 Received in revised form 20 February 2014 Accepted 21 February 2014 Available online 25 March 2014 Keywords: Mixed convection Stagnation point flow Vertical shrinking sheet Similarity solutions Numerical solution abstract The steady mixed convection stagnation-point boundary layer flow past a vertical stretching/shrinking sheet in a viscous fluid is numerically investigated. The numerical solutions are obtained by solving the similarity equations which are derived via the similarity transformation technique in order to reduce the nonlinear partial differential equations into a system of nonlinear ordinary differential equations. Numerical techniques, namely the Keller-box method, along with the shooting method are used to solve the transformed ordinary differential equations. Results have been presented and discussed for the effects of the governing parameters on the skin friction coefficient and the local Nusselt number as well as for the velocity and temperature profiles. The present results show quite interesting flow behavior for the shrinking sheet problem compared to the stretching sheet problem. On the other hand, it is also found that dual solutions exist for the present problem. The streamlines for the two dimensional flow have also been presented. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction A stagnation point occurs whenever a flow impinges on a solid object and it holds the highest pressure and the highest heat trans- fer. Hiemenz [1] was the first who investigated the two dimen- sional stagnation flow towards a stationary semi-infinite wall by using similarity transformation in order to reduce Navier–Stokes equation to a nonlinear ordinary differential equation. The exact similar solutions for the thermal field were later reported by Eckert [2]. An analysis of mixed convection near the stagnation flow of a vertical surface has been carried out by Ramachandran et al. [3] for an arbitrary variation of the surface temperature or the surface heat flux conditions. This paper has been later extended by Lok et al. [4] to the case of unsteady micropolar fluids. On the other hand, mixed convection is the combination between forced and natural or free convection flows. Both free and mixed convection flows are due to the buoyancy effects. This type of convection hap- pens due to the temperature difference between the wall and the free stream and thus changes the fluid flow and heat transfer. During the past few decades, the flows of free and mixed convec- tion have been analyzed and studied by many investigators. This is because of their importance in many applications in industrial and manufacturing processes, such as, welding, extrusion of plastics, paper drying, hot rolling, etc. Related to the mixed convection topic, we mention the papers by Seshadri et al. [5], Ishak et al. [6], Magyari and Aly [7] and many more for different situations. Nazar et al. [8] considered the unsteady mixed convection boundary layer flow near the stagnation point on a vertical surface in a porous medium. Several researchers also investigated the problem of mixed convection flow past a vertical surface and also the stagna- tion point flow for a non-Newtonian fluid. Abbas et al. [9] consid- ered a steady mixed convection boundary layer flow of an incompressible Maxwell fluid near the two dimensional stagnation point flow over a vertical stretching surface. Hayat et al. [10] con- sidered this problem for a viscoelastic fluid flow. Wang [11] studied the stagnation point flow towards a shrink- ing sheet by considering two dimensional and axisymmetric cases. He showed that the non-alignment of the stagnation flow and the shrinking sheet complicates the flow structure. Moreover, he found that solutions do not exist for large shrinking rates and may be non-unique in the two-dimensional case. Kimiaefar et al. [12] proposed an analytical approach to study the case of stagnation point flow in the vicinity of a shrinking sheet by using Homotopy Analysis Method (HAM). Recently, Ishak et al. [13] studied a prob- lem of two dimensional stagnation point flow of an incompressible fluid over a stretching vertical sheet in its own plane. Further, Bachok et al. [14] examined the steady two dimensional stagnation point flow and heat transfer from a warm, laminar liquid flow to a melting stretching/shrinking sheet. Different from the stretching http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.060 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +40 264405300. E-mail address: popm.ioan@yahoo.co.uk (I. Pop). International Journal of Heat and Mass Transfer 73 (2014) 839–848 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt