Stagnation point flow and heat transfer over a non-linearly moving flat plate in a parallel free stream with slip Natalia C. Rosßca a , Alin V. Ros ßca b , Ioan Pop a,⇑ a Department of Mathematics, Faculty of Mathematics and Computer Science, Babes ß-Bolyai University, R-400084 Cluj-Napoca, Romania b Department of Statistics, Forecasts and Mathematics, Faculty of Economics and Business Administration, Babes ß-Bolyai University, 400084 Cluj-Napoca, Romania article info Article history: Received 14 August 2013 Received in revised form 16 October 2013 Accepted 20 October 2013 Available online 30 October 2013 Keywords: Boundary layer Non-linear moving flat plate Parallel free stream Slip effects abstract An analysis is presented for the steady boundary layer flow and heat transfer of a viscous and incompressible fluid in the stagnation point towards a non-linearly moving flat plate in a parallel free stream with a partial slip velocity. The governing partial differential equa- tions are converted into nonlinear ordinary differential equations by a similarity transfor- mation, which are then solved numerically using the function bvp4c from Matlab for different values of the governing parameters. Dual (upper and lower branch) solutions are found to exist for certain parameters. Particular attention is given to deriving numerical results for the critical/turning points which determine the range of existence of the dual solutions. A stability analysis has been also performed to show that the upper branch solu- tions are stable and physically realizable, while the lower branch solutions are not stable and, therefore, not physically possible. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction The boundary layer flow due to a continuously moving solid surface has received considerable attention since the pio- neering study by Sakiadis [1,2]. The principal reason for interest in this problem is its practical relevance in various manu- facturing processes in industry such as the cooling of an infinite metallic plate in a cooling bath, the boundary layer along material handling conveyers, the boundary layer along a liquid film in condensation processes. Apart from this, the mathe- matical model considered in this context has significance in studying several problems of engineering, meteorology, ocean- ography, design of supersonic and hypersonic flights, etc. Sakiadis [1,2] showed that the partial differential equation of motion governing the steady and laminar flow caused by a continuously moving solid plate can be reduced by a similarity transformation to an ordinary differential equation. This equation is identical to the momentum equation for the boundary layer flow that develops between a constant fluid stream and a stationary flat plate first considered by Blasius [3]. However, the boundary conditions are different and so is the resulting velocity profile. Unlike Blasius [3] flow, the continuous moving surface sucks the ambient fluid and pumps it again in the downstream direction. Dual solutions were found when the plate advances toward the oncoming stream. The boundary layer problem due to Sakiadis [1,2] has been extended in a variety of ways during the subsequent decades. Klemp and Acrivos [4,5] studied the motion induced by impermeable finite and semi-infinite flat plates moving at constant velocity beneath a uniform mainstream. Later, Hussaini and Lakin [6] showed that the solutions for such boundary layer problems exist only up to a certain critical value of a moving parameter. Further, Hussaini et al. [7] considered the problem studied by Klemp and Acrivos [4] with a view to obtaining analyticity of the solutions. Riley and Weidman [8] analyzed the 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.10.019 ⇑ Corresponding author. Tel.: +40 722218681; fax: +40 264405300. E-mail address: popm.ioan@yahoo.co.uk (I. Pop). Commun Nonlinear Sci Numer Simulat 19 (2014) 1822–1835 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns