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

Ahmed A Khidir

doi:10.1155/2014/512702

Abstract

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.

Highlights

The incompressible viscous fluid flow through convergentdivergent channels is one of the most applicable cases in fluid mechanics, civil, environmental, mechanical, and biomechanical engineering
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
It can be seen from this table that the approximate solution of MHD Jeffery-Hamel flows obtained by spectral-homotopy perturbation method (SHPM) is very accurate and it is converges much more rapidly to the numerical result compared to the differential transformation method (DTM), homotopy perturbation method (HPM), and homotopy analysis method (HAM)

Summary

Introduction

The incompressible viscous fluid flow through convergentdivergent 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 NavierStokes 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]. Asymmetric solutions are both possible and of physical interest

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Results

Conclusion