Effects of suction/blowing on steady boundary layer stagnation-point flow and heat transfer towards a shrinking sheet with thermal radiation Krishnendu Bhattacharyya ⇑ , G.C. Layek Department of Mathematics, The University of Burdwan, Burdwan 713104, West Bengal, India article info Article history: Received 28 April 2010 Received in revised form 4 September 2010 Accepted 10 September 2010 Available online 15 October 2010 Keywords: Boundary layer flow Stagnation-point Shrinking sheet Suction/blowing Heat transfer Thermal radiation abstract In this paper, the effects of suction/blowing and thermal radiation on steady boundary layer stagnation- point flow and heat transfer over a porous shrinking sheet are investigated. The existence of dual solutions, unique solution and non-existence of solution for self-similar equations of the flow and heat transfer are analyzed numerically. It is noted that the range of velocity ratio parameter where the solu- tion exists increases/decreases with increasing suction/blowing. With increasing suction, temperature at the wall is found to increase (decrease) for the first (second) solution. Due to increasing Prandtl number and thermal radiation parameter the thermal boundary layer thickness becomes thinner. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction The stagnation-point flow over a stretching plate is an impor- tant problem in fluid mechanics. The steady flow in the neighbour- hood of a stagnation-point was first studied by Hiemenz [1]. Chiam [2] investigated the flow around the stagnation-point over a stretching sheet and concluded that when the stretching velocity of the plate is equal to the stagnation flow velocity in the inviscid free stream, no boundary layer is formed near the plate. Mahapatra and Gupta [3] re-investigated the stagnation-point flow problem towards a stretching plate taking different stretching and stagna- tion flow velocities. They found that two different types of bound- ary layer are formed near the stretching plate depending on the ratio of the stretching and stagnation flow velocities. Layek et al. [4] studied mass and heat transfers for boundary layer stagna- tion-point flow towards a stretching sheet subjected to heat absorption/generation and suction/blowing. Nadeem et al. [5] ob- tained solutions for boundary layer flow in the region of the stag- nation-point towards a stretching sheet using homotopy analysis method (HAM). Recently, the boundary layer flow near a shrinking sheet is gi- ven significant attention due to its various engineering applica- tions. Normally, the steady flow over a shrinking sheet is not occurred. Because the vorticity generated due to the shrinking sheet is not confined inside the boundary layer and consequently a situation appears, where some other external force is needed, which helps to confine the vorticity inside the boundary layer and only then the steady flow is possible. In confining the vorticity, the most suitable external force is the suction at the sheet. The flow development around the shrinking sheet was demonstrated by Wang [6]. The existence and uniqueness of steady viscous flow due to a shrinking sheet was established by Miklavcic and Wang [7] considering the suction effect and they concluded that for some specific values of suction, dual solutions exist and also in some range of suction, no boundary layer solution is possible. But for stagnation-point flow over a shrinking sheet, the vorticity gener- ated due to shrinking sheet remains confined within the boundary layer by the stagnation flow velocity in some cases and in this regard, Wang [8] investigated the stagnation flow problem to- wards a shrinking sheet and obtained dual solutions for some val- ues of the ratio of shrinking and stagnation flow rates. After that, Hayat et al. [9] gave an analytic solution of magnetohydrodynamic (MHD) rotating flow of a second grade fluid over a shrinking sur- face using HAM. A paper was published by Hayat et al. [10] to study the MHD flow and mass transfer of an upper-convected Max- well fluid past a porous shrinking sheet with chemical reaction species. A series solution of three-dimensional MHD and rotating flow over a porous shrinking sheet was obtained by Hayat et al. [11] using HAM. Fang [12] studied the boundary layer flow over a continuously shrinking sheet with a power-law surface velocity and with mass transfer. Fang and Zhang [13] obtained a closed- form analytical solution for MHD viscous flow over a shrinking sheet subjected to applied suction through the porous sheet. The unsteady viscous flow over a continuously shrinking sheet with 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.09.043 ⇑ Corresponding author. Tel.: +91 9474634200; fax: +91 342 2530452. E-mail addresses: krish.math@yahoo.com, krishnendu.math@gmail.com (K. Bhattacharyya). International Journal of Heat and Mass Transfer 54 (2011) 302–307 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt