A Modified Homotopy Perturbation Transform Method for Transient Flow of a Third Grade Fluid in a Channel with Oscillating Motion on the Upper Wall

A new analytical algorithm based on modified homotopy perturbation transform method is employed for solving the transient flow of third grade fluid in a porous channel generated by an oscillating upper wall. This method incorporates the He’s polynomial into the HPM, combined with Laplace transform. Comparison with HPM and OHAM analytical solutions reveals that the proposed algorithm is highly accurate. This proves the validity and great potential of the proposed algorithm as a new kind of powerful analytical tool for transient nonlinear problems. Graphs representing the solutions are discussed, and appropriate conclusions are drawn. 1. Introduction The equations describing the motion of non-Newtonian fluids are strongly of nonlinear higher order than the Navier-Stokes equation for Newtonian fluids. These nonlinear equations form a very complex structure, with a small number of exact solutions. Mostly, numerical methods have largely been used to handle these equations. The class of problems with known exact solution is related to the problem for infinite flat plate. The related studies in the recent years are as follows: Fakhar et al. [1] examine the exact unsteady flow of an incompressible third grade fluid along an infinite plane porous plate. They obtained results by applying a translational type of symmetries combined with finite difference method. Danish and Kumar [2] analysed a steady flow of a third grade between two parallel plates using similarity transformation. Abdulhameed et al. [3] consider an unsteady viscoelastic fluid of second grade for an infinite plate. They applied Laplace transform together with the regular perturbation techniques to obtain the exact solution. Ayub et al. [4] analysed the problem of steady flow of a third grade fluid for an infinite plate porous plate using homotopy analysis method (HAM). Homotopy perturbation method developed by He [5] for solving linear and nonlinear initial-boundary value problem merges two techniques, the perturbation and standard homotopy. Recently, the homotopy perturbation method has been modified by some scientists to obtain more accurate results and rapid convergence and also to reduce the amount of computation. Ghorbani [6] introduced He’s polynomials based on homotopy perturbation method for nonlinear differential equations. The homotopy perturbation transform method (HPTM) introduced by Khan and Wu [7] is a combination of the homotopy perturbation method and Laplace transform method that is used to solve various types of linear and nonlinear systems of partial differential equations. The