Analysis of Non-Newtonian Unsteady Thin Film MHD Flow on a Vertical Moving and Oscillating Belt

Magnetohydrodynamic (MHD) thin film flow of an unsteady third order fluid is considered. The non linear partial differential equations are solved analytically by using Optimal Homotopy Asymptotic Method (OHAM) and Adomian Decomposition Method (ADM). The Comparison of both methods is analyzed numerically and graphically. The effects of model parameters have also been studied. KEYWORDS: Unsteady thin film flows, MHD, Lifting, Drainage, Third order fluid, OHAM and ADM. I INTRODUCTION Non-Newtonian thin film flows have large applications in a number of technological processes including production of polymer films or thin sheets. In physical configuration of non-Newtonian fluids it is complicated to clarify their mechanical manners by a particular constitutive equation. For this reason, a great variety of constitutive equations have been proposed (1). Siddiqui et al. (2) studied the thin film flow of Sisko fluid and Oldroyd 6-constant fluid on a vertical moving belt. The nonlinear equations governing the flow solved using homotopy perturbation method. Volume flux and average velocity are also calculated. Hayat and Sajid (3) investigated the comparison between Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM) for thin film flow of non-Newtonian fluids on moving belt. Nargis and Tahir (4) investigated the thin film flow of a third order fluid in two cases when the fluid moves down an inclined plane and moves on a moving belt. The volume flux and average film velocity are also discussed. There is no single constitutive equation that can be used to examine all the non-Newtonian fluids, different linear and non linear equations have been proposed. A third grade fluid is a subclass of non-Newtonian fluid and its governing non-linear equation has successfully studied and treated in many literatures. Ariel (5) discussed the steady and laminar flow of a third grade fluid through a porous flat channel. The flow is governed by a non-linear boundary value problem and different numerical methods are developed to obtain the appropriate solution. Sahoo et al. (6) investigated the non-Newtonian boundary layer flow and heat transfer over an exponentially stretching sheet with uniform transverse magnetic field. They find the combined effects of the partial slip and the third grade fluid parameters on the velocity profile. Islam et al.(7, 8) discussed the unsteady second grade fluid flow between wire and die. The problem is solved by OHAM and the ideas of OHAM extend not for the solution of linear and non-linear differential equations but also can be applied for linear and non- linear partial differential equations. The study of unsteady magnetohydrodynamics (MHD) thin film flows has received substantial concentration in the past due to its applications in the field of engineering, polymer industry and petroleum industries. TazaGul et al. (9) investigated the heat transfer analysis in electrically conducting third grade thin film fluid. They studied the combined effect of heat and MHD on the velocity field and the effects of model parameters on velocity, skin friction and temperature variation. Khan et al. (10-12) discussed the solution of the unsteady flow of an incompressible, electrically conducting third grade fluid bounded by porous plate using homotopy analysis methods (HAM). The analytical solutions are shows through graph. Ali et al. (13) investigated the numerical solution of electrically conducting fluid flow and heat transfer over porous stretching sheet. The governing non-linear partial differential equations of motion have been numerically solved by Method of Stretching Variables. The effects of physical parameters Magnetic parameter, Grashof number, Prandlt number and injection parameter S have been observed on velocity, temperature distributions. Idrees et al. (14) studied the low of incompressible fluid between two parallel plates and the governing fourth order non linear differential equation is solved by using Optimal Homotopy Asymptotic Method. This method is effective, sampler and easier. Yongqi and Wu (15) discussed the unsteady flow of an incompressible fourth grade fluid in a uniform magnetic field and the unsteady flow is induced by oscillating two-dimensional infinite porous plate. They compared the flow behavior of the fourth-grade non-Newtonian fluid with the Newtonian fluid. Aiyesimi et al. (16) investigated thin film flow of an MHD third grade fluid down an inclined plane. The solutions of