Spectral-Homotopy Perturbation Method for Solving Governing MHD Jeffery-Hamel Problem

We present a new modification of the homotopy perturbation method (HPM) for solving nonlinear boundary value problems. The technique is based on the standard homotopy perturbation method and blending of the Chebyshev pseudospectral methods. The implementation of the new approach is demonstrated by solving the MHD Jeffery-Hamel flow and the effect of MHD on the flow has been discussed. Comparisons are made between the proposed technique, the previous studies, the standard homotopy perturbation method, and the numerical solutions to demonstrate the applicability, validity, and high accuracy of the presented approach. The results demonstrate that the new modification is more efficient and converges faster than the standard homotopy perturbation method at small orders. The MATLAB software has been used to solve all the equations in this study. 1. Introduction The incompressible viscous fluid flow through convergent-divergent channels is one of the most applicable cases in fluid mechanics, civil, environmental, mechanical, and biomechanical engineering. The mathematical investigations of this problem were pioneered by Jeffery [1] and Hamel [2]. They presented an exact similarity solution of the Navier-Stokes equations in the special case of two-dimensional flow through a channel with inclined plane walls meeting at a vertex and with a source or sink at the vertex and have been extensively studied by several authors and discussed in many textbooks, for example, [3, 4]. In the Ph.D. thesis [5] we find that Jeffery-Hamel flow used as asymptotic boundary conditions to examine a steady of two-dimensional flow of a viscous fluid in a channel. But, here certain symmetric solutions of the flow has been considered by Sobey and Drazin [6]. Although asymmetric solutions are both possible and of physical interest. The classical Jeffery-Hamel problem was extended by Axford [7] to include the effects of an external magnetic field on an electrically conducting fluid; in MHD Jeffery-Hamel problems there are two additional nondimensional parameters that determine the solutions, namely, the magnetic Reynolds number and the Hartmann number. Most scientific problems such as Jeffery-Hamel flows and other fluid mechanic problems are inherently in form of nonlinear differential equations. Except a limited number of these problems, most of them do not have exact solution and some of the solved by numerical methods. Therefore, these nonlinear equations should be solved using other methods. Therefore, many different methods have recently introduced some ways to obtain analytical