Unsteady effects on mixed convection boundary layer flow from a permeable slender cylinder due to non-linearly power law stretching P.M. Patil a,⇑ , S. Roy b , I. Pop c a Department of Mathematics, JSS’s Banashankari Arts, Commerce and Shanti Kumar Gubbi Science College, Vidyagiri, Dharwad 580 004, India b Department of Mathematics, Indian Institute of Technology, Madras, Chennai 600 036, India c Faculty of Mathematics, University of Cluj, R-400082 Cluj-Napoca, CP 253, Romania article info Article history: Received 14 September 2011 Accepted 17 November 2011 Available online 25 November 2011 Keywords: Unsteady flow Mixed convection Slender cylinder Surface mass transfer Non-linearly stretching sheet abstract An unsteady mixed convection boundary layer flow over a permeable non-linearly stretching vertical slender cylinder is considered to investigate the combined effects of buoyancy force and thermal diffu- sion in presence of surface mass transfer, where the slender cylinder is in line with the flow. The unstead- iness in the flow and temperature fields is caused by the continuously non-linearly stretching vertical slender cylinder. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar transformations. The resulting system of coupled non-linear partial differential equations is solved by an implicit finite difference scheme in combination with the quasi-linearization technique. Numerical computations are performed to understand the physical situations of linear and non-linearly stretching surface for different values of parameters to display the velocity and temperature profiles graphically. The obtained results show that the buoyancy parameter k and the Prandtl number Pr enhance the skin friction coefficient and the local Nusselt number. It is also shown that the suction parameter A (>0) is to decrease the velocity and temperature profiles, but injection parameter A (<0) does the reverse. Results are compared with previously published work and are found to be in excellent agreement. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Flow over a cylinder is generally considered as two-dimensional as the radius of the cylinder is large enough compared to the boundary layer thickness. In contrast, for a slender cylinder when the radius of the cylinder is same as the order of the boundary layer thickness, the flow is considered to be axisymmetric. In the axisymmetric flow, the governing equations consists of the trans- verse curvature term which is strong enough to induce the behav- ior of flow and temperature fields and correspondingly, the skin friction coefficient and heat transfer rates over the surface. The im- pact of transverse curvature is vital in many applications such as wire and fiber drawing wherein accurate prediction is expected and thick boundary layer can exist on slender or near slender bodies. Suction or injection (blowing) of a fluid through the stretching surface, as, for instance, in mass transfer cooling, can significantly change the flow field and, as a consequence, affects the heat transfer rate from the surface. In general suction tends to increase the skin friction and heat transfer coefficients, whereas, injection has opposite tendency [1]. Injection or withdrawal of fluid through a permeable bounding heated or cooled surface is of general interest in practical problems involving boundary layer control applications such as film cooling, polymer fiber coating, and coating of wires. Mixed convection flows, or coupled forced and natural convection flows arise in many transport processes both in natural and engineering applications. Such processes occur when the effects of buoyancy forces in forced convection or the ef- fects of forced flow in natural convection become much more sig- nificant. The interaction between flow and thermal fields due to stretching of a boundary has very important role in many practical engineering applications. Unsteady mixed convection flows do not necessarily allow similarity solutions in many physical situations. The non-similarity and unsteadiness in such flow problems may be due to the free stream velocity or due to the curvature of the body or due to the surface mass transfer or even possibly due to all these phenomenal effects. Since the mathematical difficulties involved in obtaining non-similar solutions for such problems, many researchers and scientists have confined their work either to steady non-similar flows or to unsteady semi similar or self sim- ilar flows [2–4]. Chen and Mucoglu [5] and Mucoglu and Chen [6] have investi- gated the effects of mixed convection boundary layer flow over an impermeable vertical slender cylinder due to the thermal diffusion with prescribed wall temperature and heat flux conditions, respec- tively. They obtained the solution by using the local non-similarity 0045-7930/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2011.11.008 ⇑ Corresponding author. Tel.: +91 836 2468478; fax: +91 836 2462200. E-mail address: pmpmath@gmail.com (P.M. Patil). Computers & Fluids 56 (2012) 17–23 Contents lists available at SciVerse ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid