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Solution of the Magnetohydrodynamics Jeffery-Hamel Flow Equations by the Modified Adomian Decomposition Method

Published online by Cambridge University Press:  21 July 2015

Lei Lu ,
Junsheng Duan  and
Longzhen Fan
Lei Lu
Affiliation:
School of Sciences, Shanghai Institute of Technology, Shanghai 201418, China School of Management, Fudan University, Shanghai 200433, China
Junsheng Duan*
Affiliation:
School of Sciences, Shanghai Institute of Technology, Shanghai 201418, China
Longzhen Fan
Affiliation:
School of Management, Fudan University, Shanghai 200433, China
*
*Corresponding author. Email: [email protected] (L. Lu), [email protected] (J. S. Duan)
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Abstract

In this paper, the nonlinear boundary value problem (BVP) for the Jeffery-Hamel flow equations taking into consideration the magnetohydrodynamics (MHD) effects is solved by using the modified Adomian decomposition method. We first transform the original two-dimensional MHD Jeffery-Hamel problem into an equivalent third-order BVP, then solve by the modified Adomian decomposition method for analytical approximations. Ultimately, the effects of Reynolds number and Hartmann number are discussed.

Keywords

76W0534B15Jeffery-Hamel flowmagnetohydrodynamicsnonlinear differential equationAdomian decomposition methodAdomian polynomials
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 5 , October 2015 , pp. 675 - 686
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Jeffery, G. B., The two-dimensional steady motion of a viscous fluid, Philos. Mag., 6 (1915), pp. 455–465.CrossRefGoogle Scholar
[2]Hamel, G., Spiralformige bewgungen zaher flussigkeiten, Jahresbericht der Deutschen Math. Vereinigung, 25 (1916), pp. 34–60.Google Scholar
[3]Rosenhead, L., The steady two-dimensional radial flow of viscous fluid between two inclined plane walls, Proc. Roy. Soc. A, 175 (1940), pp. 436–467.Google Scholar
[4]Batchelor, K., An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967.Google Scholar
[5]Hamadiche, M., Scott, J. and Jeandel, D., Temporal stability of Jeffery-Hamel flow, J. Fluid. Mech., 268 (1994), pp. 71–88.Google Scholar
[6]Fraenkel, L. E., Laminar flow in symmetrical channels with slightly curved walls I: on the Jeffery-Hamel solutions for flow between plane walls, Proc. Roy. Soc. A, 267 (1962), pp. 119–138.Google Scholar
[7]Makinde, O. D. and Mhone, P. Y., Hermite-Padé approximation approach to MHD Jeffery-Hamel flows, Appl. Math. Comput., 181 (2006), pp. 966–972.Google Scholar
[8]Holmes, M. H., Introduction to Perturbation Methods (2nd edn), Springer, New York, 2013.Google Scholar
[9]He, J. H., A review on some new recently developed nonlinear analytical techniques, Int. J. Nonlinear. Sci. Numer. Simul., 1 (2000), pp. 51–70.Google Scholar
[10]Lakestani, M., Razzaghi, M. and Dehghan, M., Numerical solution of the controlled Duffing oscillator by semi-orthogonal spline wavelets, Phys. Scripta, 74 (2006), pp. 362–366.Google Scholar
[11]Jafari, H., Borhanifar, A. and Karimi, S. A., New solitary wave solutions for the bad Boussinesq and good Boussinesq equations, Numer. Methods Partial Differential Equations, 25 (2009), pp. 1231–1237.Google Scholar
[12]Baleanu, D., Wu, G. C. and Duan, J. S., Some analytical techniques in fractional calculus: realities and challenges, in: Machado, J.A.T., Baleanu, D. and Luo, A. C.J. (Eds), Discontinuity and Complexity in Nonlinear Physical Systems, pp. 35–62, Springer, Cham/Heidelberg/New York, 2014.CrossRefGoogle Scholar
[13]Adomian, G., Stochastic Systems, Academic, New York, 1983.Google Scholar
[14]Adomian, G., Nonlinear Stochastic Operator Equations, Academic, Orlando, 1986.Google Scholar
[15]Adomian, G., Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic, Dordrecht, 1989.CrossRefGoogle Scholar
[16]Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, 1994.Google Scholar
[17]Wazwaz, A. M., Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing, and Springer-Verlag, Berlin, 2009.Google Scholar
[18]Wazwaz, A. M., Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education Press, Beijing, and Springer-Verlag, Berlin, 2011.CrossRefGoogle Scholar
[19]Serrano, S. E., Engineering Uncertainty and Risk Analysis: A Balanced Approach to Probability, Statistics, Stochastic Modeling, and Stochastic Differential Equations (2nd revised edn), HydroScience, Ambler, 2011.Google Scholar
[20]Adomian, G. and Rach, R., Inversion of nonlinear stochastic operators, J. Math. Anal. Appl., 91 (1983), pp. 39–46.Google Scholar
[21]Duan, J. S., Rach, R., Baleanu, D. and Wazwaz, A. M., A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Fract. Calc., 3 (2012), pp. 73–99.Google Scholar
[22]Rach, R., A bibliography of the theory and applications of the Adomian decomposition method, Kybernetes, 41 (2012), pp. 1087–1148.CrossRefGoogle Scholar
[23]Rach, R., A new definition of the Adomian polynomials, Kybernetes, 37 (2008), pp. 910–955.Google Scholar
[24]Rach, R., A convenient computational form for the Adomian polynomials, J. Math. Anal. Appl., 102 (1984), pp. 415–419.Google Scholar
[25]Wazwaz, A. M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111 (2000), pp. 53–69.Google Scholar
[26]Abdelwahid, F., A mathematical model of Adomian polynomials, Appl. Math. Comput., 141 (2003), pp. 447–453.Google Scholar
[27]Abbaoui, K., Cherruault, Y. and Seng, V., Practical formulae for the calculus of multivariable Adomian polynomials, Math. Comput. Model., 22 (1995), pp. 89–93.Google Scholar
[28]Zhu, Y., Chang, Q. and Wu, S., A new algorithm for calculating Adomian polynomials, Appl. Math. Comput., 169 (2005), pp. 402–416.Google Scholar
[29]Biazar, J., Ilie, M. and Khoshkenar, A., An improvement to an alternate algorithm for computing Adomian polynomials in special cases, Appl. Math. Comput., 173 (2006), pp. 582–592.Google Scholar
[30]Azreg-Aïnou, M., A developed new algorithm for evaluating Adomian polynomials, CMES Comput. Model. Eng. Sci., 42 (2009), pp. 1–18.Google Scholar
[31]Duan, J. S., Recurrence triangle for Adomian polynomials, Appl. Math. Comput., 216 (2010), pp. 1235–1241.Google Scholar
[32]Duan, J. S., An efficient algorithm for the multivariable Adomian polynomials, Appl. Math. Comput., 217 (2010), pp. 2456–2467.Google Scholar
[33]Duan, J. S., Convenient analytic recurrence algorithms for the Adomian polynomials, Appl. Math. Comput., 217 (2011), pp. 6337–6348.Google Scholar
[34]Abbaoui, K. and Cherruault, Y., Convergence of Adomian’s method applied to differential equations, Comput. Math. Appl., 28 (1994), pp. 103–109.Google Scholar
[35]Abbaoui, K. and Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29 (1995), pp. 103–108.Google Scholar
[36]Abdelrazec, A. and Pelinovsky, D., Convergence of the Adomian decomposition method for initial-value problems, Numer. Methods Partial Differential Equations, 27 (2011), pp. 749–766.Google Scholar
[37]Raja, M. A. Z. and Samar, R., Numerical treatment for nonlinear MHD Jeffery-Hamel problem using neural networks optimized with interior point algorithm, Neurocomputing, 124 (2014), pp. 178–193.Google Scholar
[38]Raja, M. A.Z. and Samar, R., Numerical treatment of nonlinear MHD Jeffery-Hamel problems using stochastic algorithms, Comput. Fluids, 91 (2014), pp. 28–46.CrossRefGoogle Scholar
[39]Duan, J. S. and Rach, R., A new modification of the Adomian decomposition method for solving boundary value problems for higher oder nonlinear differential equations, Appl. Math. Comput., 218 (2011), pp. 4090–4118.Google Scholar