Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition Anuar Ishak School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia Faculty of Industrial Science and Technology, Universiti Malaysia Pahang, 26300 Gambang, Pahang, Malaysia article info Keywords: Boundary layer Convective boundary condition Heat transfer Permeable surface Similarity solution abstract The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is con- sidered. Similarity solutions for the flow and thermal fields are possible if the mass transpi- ration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x 1/2 , where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary dif- ferential equations, before being solved numerically. The effects of the governing parame- ters on the flow and thermal fields are thoroughly examined and discussed. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction The term ‘‘similarity solution” in fluid mechanics was first introduced by Blasius [1] when solving an application problem of Prandtl’s boundary layer theory. The idea is to simplify the governing equations by reducing the number of independent variables, by a coordinate transformation. Analogous to dimensional analysis, instead of parameters, like the Reynolds num- ber, the coordinates themselves are collapsed into dimensionless groups that scale the velocities [2]. The terminology ‘‘sim- ilarity” is used because, despite the growth of the boundary layer with distance x from the leading edge, the velocity profile u/U 1 remains geometrically similar. The same concept was then extended to the temperature profile. However, not all prob- lems admit similarity solutions, since they depend on various factors, such as the surface geometries, boundary conditions, and the surface heating conditions. The heat transfer part of the above problem was solved by Pohlhausen [3], by assuming uniform plate temperature. Recently, Aziz [4] and Magyari [5] studied the similar problem, but with convective boundary condition. Aziz demonstrated that similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is proportional to x 1/2 . He reported the results for Prandtl number Pr = 0.1, 0.72 and 10. However, the numerical results reported in [4] for Pr = 0.1 are not enough accurate, owing to the small boundary layer thickness set in all the computations. It is well known that the Prandtl number Pr is a ratio of viscous to conduction effects, the lower the Prandtl number, the thicker the thermal boundary layer [6]. The objective of the present study is to extend the work of Aziz [4], by introducing the effects of suction and injection on the flat surface, besides giving accurate numerical results for Pr = 0.1. The process of suction and injection (blowing) has its importance in many engineering applications such as in the design of thrust bearing and radial diffusers, and thermal oil recovery. Suction is applied to chemical processes to remove reactants. Blowing is used to add reactants, cool the surfaces, prevent corrosion or scaling and reduce the drag (see Labropulu et al. [7]). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.06.026 E-mail address: anuarishak@yahoo.com Applied Mathematics and Computation 217 (2010) 837–842 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc