TECHNICAL NOTE 1 An analytic approximation of wire coating analysis for third-grade magneto-hydrodynamic flow I Shafieenejad 1∗ , Ni Moallemi 2 , Na Moallemi 3 , and A B Novinzadeh 1 1 Department of Aerospace Engineering, K N Toosi University of Technology, Tehran, Iran 2 Department of Mechanical Engineering, Iran University of Science andTechnology,Tehran, Iran 3 Department of Electrical Engineering, International Division of Shiraz University, Iran The manuscript was received on 7 April 2009 and was accepted after revision for publication on 11 May 2009. DOI: 10.1243/09544062JMES1663 Abstract: In this article, homotopy perturbation method (HPM) is applied to solve non-linear problems for third-grade fluid. The fluid is electrically conducting in the presence of a uniform applied magnetic field, and wire coating by withdrawal from a bath of third-grade fluid is inves- tigated. Although traditional HPM is divergent for higher-order approximation to obtain enough accuracy in this work, a new method on boundary conditions is proposed to solve this boundary value problem by HPM and convergence of the solution is clearly discussed. Results reveal that this method is very effective and simple to overcome the divergence of the HPM solution. The influence of the non-Newtonian parameter on the velocity, volume flowrate, radius of coated wire, shear stress at the surface of the wire, and the force required to pull the wire is seen and discussed. Keywords: homotopy perturbation method, magneto-hydrodynamic, wire coating, third-grade flow 1 INTRODUCTION Because of their extensive use in industry, non- Newtonian fluids have gained considerable impor- tance in the last few years. Various workers in the field cite a wide variety of applications in rheological prob- lems in biological sciences, geophysics, and chemical and petroleum industries. Several models have been proposed and the third-grade model is one of them. Such studies take into account the non-Newtonian effects for unidirectional flow in unsteady state. In a wire coating operation, the wire is put through a bath of the coating liquid such as polymer melt and then through a die that wipes the liquid and leaves a coating of the desired thickness. In a way it is the motion of wind that passes through the polymer and draws it through the die. The flow in the die is a drag flow with constant cross-section analogous to the case of axial annular drag flow or to the plane Couette flow [1]. Very little work has been conducted regarding the wire coating problem. The basic development of a ∗ Corresponding author: Department of Aerospace Engineering, K N Toosi University of Technology, Tehran, Iran. email: shafiee_iman@yahoo.com wire coating modelling in a viscous fluid is given in the books by Denn [2] and Middleman [3]. Based on power law fluid, Akter and Hashmi [4, 5] developed a mathematical model for wire coating and investigated the effect of the change in viscosity during drawing of the wire coating process. This provides the motivation for the present study where we consider a model of a third-grade fluid. The study of flow for an electrically conducting fluid has applications in many engineer- ing problems such as magneto-hydrodynamic (MHD) power generators, MHD pumps, accelerators, plasma studies, geothermal energy extractions, polymer tech- nology, petroleum industry, purification of crude oil and fluid droplets and sprays, the flows of liquid state metals and alloys, cooling of continuous strips and filaments drawn through a quiescent fluid, purifica- tion of molten metals from non-metallic inclusions, flows in biomechanics, insulation systems, enhanced oil recovery, and so on. Many asymptotic techniques, including the homo- topy analysis method [6], the variational iteration method [7, 8], and Adomian’s decomposition method [9–11], were used to handle strongly non-linear sys- tems. The homotopy perturbation method (HPM) was first proposed by He [12–15] for solving differ- ential and integral equations. The method, which is JMES1663 Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science