Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane A.M. Siddiqui a , R. Mahmood b, * , Q.K. Ghori b a Department of Mathematics, Pennsylvania State University, York Campus, 1031 Edgecomb Avenue, York, PA 17403, USA b Department of Mathematics, COMSATS Institute of Information Technology, H-8/I- Islamabad, Pakistan Accepted 14 May 2006 Communicated by Prof. M.S. El Naschie Abstract The present paper analyses the thin film flow problem with a third grade fluid on an inclined plane. The governing non-linear equation is solved for the velocity field using the traditional perturbation technique as well as the recently introduced homotopy perturbation method and the results are compared. The expressions for volume flux and average film velocity are also given. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction In the literature there are very few exact solutions of the Navier–Stokes equations and even these become rare when constitutive equations of non-Newtonian fluids are used. This is because the Navier–Stokes equations are highly non- linear. Perturbation techniques [1,2] are widely applied for obtaining approximate solutions to these equations involv- ing a small parameter e. These techniques are so powerful that sometimes the parameter e is artificially introduced into a problem having no parameter and then finally set equal to unity to recover the solution of the original problem. Recently, He [3–5] proposed a new perturbation method which is, infact, a coupling of the traditional perturbation method and homotopy as used in topology. This is the homotopy perturbation method (HPM). In his several papers He applied this method to discuss non-linear boundary value problems [6] as well as non-linear problems on bifurcation [7,8], asymptotology [9],wave equation [10] and osciliator with discontinuities [11]. Because of the success of homotopy perturbation method, Abbasbandy [12,13] used it for Laplace transform, Cveticanin [14] applied it to study pure non- linear differential equations and El-Shahed [15] applied this technique to the integro-differential equation for Volterra’s model. Turning to fluid mechanics, Siddiqui et al [16,17] have invoked this method for solving non-linear problems involving non-Newtonian fluids. 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.026 * Corresponding author. E-mail addresses: ams5@psu.edu (A.M. Siddiqui), mahmoodrashid2000@yahoo.com (R. Mahmood), ghori@comsats.edu.pk (Q.K. Ghori). Chaos, Solitons and Fractals 35 (2008) 140–147 www.elsevier.com/locate/chaos