全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量

相关文章

更多...

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

DOI: 10.1155/2014/512702

Full-Text   Cite this paper   Add to My Lib

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. 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

References

[1]  G. B. Jeffery, “The two-dimensional steady motion of a viscous fluid,” Philosophical Magazine, vol. 6, pp. 455–465, 1915.
[2]  G. Hamel, “Spiralf?rmige Bewegungen z?her Flüssigkeiten,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 25, pp. 34–60, 1916.
[3]  L. Rosenhead, “The steady two-dimensional radial flow of viscous fluid between two inclined plane walls,” Proceedings of the Royal Society A, vol. 175, pp. 436–467, 1940.
[4]  A. McAlpine and P. G. Drazin, “On the spatio-temporal development of small perturbations of Jeffery-Hamel flows,” Fluid Dynamics Research, vol. 22, no. 3, pp. 123–138, 1998.
[5]  R. M. Sadri, Channel entrance flow [Ph.D. thesis], Department of Mechanical Engineering, the University of Western Ontario, 1997.
[6]  I. J. Sobey and P. G. Drazin, “Bifurcations of two-dimensional channel flows,” Journal of Fluid Mechanics, vol. 171, pp. 263–287, 1986.
[7]  W. I. Axford, “The magnetohydrodynamic Jeffrey-Hamel problem for a weakly conducting fluid,” The Quarterly Journal of Mechanics and Applied Mathematics, vol. 14, pp. 335–351, 1961.
[8]  J. He, “A review on some new recently developed nonlinear analytical techniques,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 1, pp. 51–70, 2000.
[9]  J. H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 207–208, 2005.
[10]  S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems [Ph.D. thesis], Shanghai Jiao Tong University, 1992.
[11]  S. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004.
[12]  Q. Esmaili, A. Ramiar, E. Alizadeh, and D. D. Ganji, “An approximation of the analytical solution of the Jeffery-Hamel flow by decomposition method,” Physics Letters A: General, Atomic and Solid State Physics, vol. 372, no. 19, pp. 3434–3439, 2008.
[13]  O. D. Makinde and P. Y. Mhone, “Hermite-Padé approximation approach to MHD Jeffery-Hamel flows,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 966–972, 2006.
[14]  O. D. Makinde, “Effect of arbitrary magnetic Reynolds number on MHD flows in convergent-divergent channels,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 18, no. 5-6, pp. 697–707, 2008.
[15]  J. H. He, “Variational iteration method—a kind of non-linear analytical technique: Some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.
[16]  J. K. Zhou, Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986, (Chinese).
[17]  S. S. Motsa, P. Sibanda, F. G. Awad, and S. Shateyi, “A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem,” Computers & Fluids, vol. 39, no. 7, pp. 1219–1225, 2010.
[18]  J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
[19]  J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000.
[20]  L. N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
[21]  W. S. Don and A. Solomonoff, “Accuracy and speed in computing the Chebyshev collocation derivative,” SIAM Journal on Scientific Computing, vol. 16, no. 6, pp. 1253–1268, 1995.
[22]  A. A. Joneidi, G. Domairry, and M. Babaelahi, “Three analytical methods applied to Jeffery-Hamel flow,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3423–3434, 2010.

Full-Text

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133