Steady circulatory flow of a second grade fluid about a rotating porous cylinder M. Emin Erdog ˘an, C. Erdem _ Imrak * Istanbul Teknik Universitesi, Makina Fakültesi, 34439 Gümüs ßsuyu, _ Istanbul, Turkey article info Keywords: Second grade fluid Porous cylinder Circulation Perturbation solution Exact solution abstract An exact solution for the circulatory flow of an incompressible second grade fluid about a rotating porous cylinder is given. The solution is expressed in terms of the confluent hyper- geometric functions and it is valid for all values of the cross-Reynolds number, the elastic number and the ratio of the circulation at infinity to that on the surface of the cylinder. The velocity, the vorticity and the torque exerted by the fluid on the cylinder are calculated. It is shown that there are some discrepancies between the results obtained by the exact solu- tion and those obtained by the perturbation solution which is valid for small values of the elastic number. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction An exact solution for steady circulatory flow about a rotating circular cylinder with suction is given. Exact solutions are very important for many reasons. Exact solutions provide a standard for checking the accuracies of many approximate meth- ods such as numerical or empirical. Although computer techniques make the complete integration by the governing equa- tions for second grade fluids feasible, the accuracy of the results can be established by a comparison with an exact solution. The exact solution given in this paper is connected with flows over porous materials. The flow of fluids over porous materials has many applications in practice such as the boundary layer control. Practical applications of the problems considered in this paper have been discussed in [1]. The problem considered in this paper is an extension of the circulatory flow of a viscous fluid about a rotating circular cylinder with suction to the flow of a second grade fluid. The flow of a viscous fluid has been investigated by Preston [1]. Unsteady circulatory flow of a viscous fluid about a circular cylinder with suction was studied by Nanda [2]. The investiga- tion of this flow due to the confinement of the vorticity can be found in [3]. Similar problem in an annulus have been inves- tigated by many authors [2,4–6] for both Newtonian and non-Newtonian fluids. The extension of the problem considered in [1] for a viscous flow to the non-Newtonian case has been investigated by Roy [7], assuming that the solution of the govern- ing equation can be given by a perturbation expansion in terms of the small values of the elastic number. When one uses a perturbation expansion in terms of the coefficient appearing in the high-order derivative of the governing equation, the no-slip condition is sufficient. However, the governing equation is a third order differential equation and one needs an additional condition. Thus unless additional condition is prescribed over no-slip condition this will be a parametric solution. A critical review on the boundary conditions, existence and uniqueness of the solution has been given by Rajagopal [8]. The condition used by many authors is that when the viscoelastic parameter goes to zero, the solution should give the Newtonian case [9–13]. In this paper, the additional condition is derived from the properties of the confluent hypergeometric function. 0020-7225/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2010.06.007 * Corresponding author. Tel.: +90 212 243 47 71; fax: +90 212 245 07 95. E-mail address: imrak@itu.edu.tr (C.E. _ Imrak). International Journal of Engineering Science 48 (2010) 1225–1232 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci