×

zbMATH — the first resource for mathematics

Study on nonlinear Jeffery-Hamel flow by He’s semi-analytical methods and comparison with numerical results. (English) Zbl 1189.65298
Summary: The problem of Jeffery-Hamel flow is presented and the variational iteration method and the homotopy perturbation method are employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. Comparisons are made between the Numerical solution (NM) and the results of the He’s variational iteration method (VIM) and He’s homotopy perturbation method (HPM). The results reveal that these methods are very effective and simple and can be applied for other nonlinear problems.

MSC:
65N99 Numerical methods for partial differential equations, boundary value problems
35Q35 PDEs in connection with fluid mechanics
PDF BibTeX Cite
Full Text: DOI
References:
[1] Jeffery, G.B., The two-dimensional steady motion of a viscous fluid, Phil. mag., 6, 455-465, (1915) · JFM 45.1088.01
[2] Hamel, G., Spiralförmige bewgungen Zäher flüssigkeiten, Jahresber. Deutsch. math.-verein., 25, 34-60, (1916) · JFM 46.1255.01
[3] Rosenhead, L., The steady two-dimensional radial flow of viscous fluid between two inclined plane walls, Proc. R. soc. A, 175, 436-467, (1940) · Zbl 0025.37501
[4] Batchelor, K., An introduction to fluid dynamics, (1967), Cambridge University Press · Zbl 0152.44402
[5] Reza M. Sadri, Channel entrance flow, Ph.D. Thesis, Department of Mechanical Engineering, The University of Western Ontario, 1997
[6] Sobey, I.J.; Drazin, PG., Bifurcations of two-dimensional channel flows, J. fluid mech., 171, 263-287, (1986) · Zbl 0609.76050
[7] Hamadiche, M.; Scott, J.; Jeandel, D., Temporal stability of jeffery – hamel flow, J. fluid mech., 268, 71-88, (1994) · Zbl 0809.76039
[8] Fraenkel, L.E., Laminar flow in symmetrical channels with slightly curved walls. I: on the jeffery – hamel solutions for flow between plane walls, Proc. R. soc. lond. A, 267, 119-138, (1962) · Zbl 0104.42403
[9] Makinde, O.D.; Mhone, P.Y., Hermite – padé approximation approach to MHD jeffery – hamel flows, Appl. math. comput., 181, 966-972, (2006) · Zbl 1102.76049
[10] Schlichting, Hermann, Boundary-layer theory, (2000), McGraw-Hill Press New York · Zbl 0096.20105
[11] Rathy, R.K., An introduction to fluid dynamics, (1976), Oxford and IBH Pl New Delhi · Zbl 0118.21703
[12] McAlpine, A.; Drazin, P.G., On the spatio-temporal development of small perturbations of jeffery – hamel flows, Fluid dyn. res., 22, 123-138, (1998) · Zbl 1051.76554
[13] He, J.H., Variational iteration method—A kind of non-linear analytical technique: some examples, Int. J. non-linear mech., 34, 699-708, (1999) · Zbl 1342.34005
[14] He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl. math. comput., 114, 115-123, (2000) · Zbl 1027.34009
[15] He, J.H.; Wu, X.H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos solitons fractals, 29, 108-113, (2006) · Zbl 1147.35338
[16] He, J.H., Variational iteration method-some recent results and new interpretations, J. comput. appl. math., 207, 3-17, (2007) · Zbl 1119.65049
[17] He, J.H., Homotopy perturbation technique, Comput. methods appl. mech. engrg., 178, 257-262, (1999) · Zbl 0956.70017
[18] He, J.H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. non-linear mech., 35, 37-43, (2000) · Zbl 1068.74618
[19] He, J.H., Homotopy perturbation method: A new nonlinear analytical technique, Appl. math. comput., 135, 73-79, (2003) · Zbl 1030.34013
[20] He, J.H.; Wu, X.H., Exp-function method for nonlinear wave equations, Chaos solitons fractals, 30, 700-708, (2006) · Zbl 1141.35448
[21] He, J.H.; Wu, X.H., Variational iteration method: new development and applications, Comput. math. appl., 54, 881-894, (2007) · Zbl 1141.65372
[22] He, J.H., Variational approach for nonlinear oscillators, Chaos solitons fractals, 34, 1430-1439, (2007) · Zbl 1152.34327
[23] He, J.H., Addendum: new interpretation of homotopy perturbation method, Int. J. mod. phys. B, 20, 1141-1199, (2006)
[24] He, J.H., Homotopy perturbation method: A new nonlinear analytical technique, J. appl. math. comput., 135, 73-79, (2000) · Zbl 1030.34013
[25] Ganji, D.D.; Afrouzi, G.A.; Talarposhti, R.A., Application of he’s variational iteration method for solving the reaction – diffusion equation with ecological parameters, Comput. math. appl., 54, 1010-1017, (2007) · Zbl 1267.65151
[26] Ganji, D.D.; Tari, H.; Jooybari, M.B., Variational iteration method and homotopy perturbation method for nonlinear evolution equations, Comput. math. appl., 54, 1018-1027, (2007) · Zbl 1141.65384
[27] Ganji, D.D.; Sadighi, A., Exact solutions of nonlinear diffusion equations by variational iteration method, Comput. math. appl., 54, 1112-1121, (2007) · Zbl 1145.35311
[28] Ganji, D.D.; Sadighi, A., Solution of the generalized nonlinear Boussinesq equation using homotopy perturbation and variational iteration methods, Int. J. nonlinear sci. numer. simul., 8, 435-444, (2007) · Zbl 1120.65108
[29] Gorji, M.; Ganji, D.D.; Soleimani, S., New application of he’s homotopy perturbation method, Int. J. nonlinear sci. numer. simul., 8, 319-328, (2007)
[30] Ganji, D.D.; Sadighi, A., Application of he’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, Int. J. nonlinear sci. numer. simul., 7, 411-418, (2006)
[31] Tari, H.; Ganji, D.D.; Rostamian, M., Approximate solutions of \(K(2, 2)\), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method, Int. J. nonlinear sci. numer. simul., 8, 203-210, (2007)
[32] Yusufoğlu, E., Homotopy perturbation method for solving a nonlinear system of second order boundary value problems, Int. J. nonlinear sci. numer. simul., 8, 353-358, (2007)
[33] Xu, L., The variational iteration method for fourth order boundary value problems, Chaos solitons fractals, (2007)
[34] Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. nonlinear sci. numer. simul., 7, 27-34, (2006) · Zbl 1401.65087
[35] Biazar, J.; Ghazvini, H., He’s variational iteration method for solving hyperbolic differential equations, Int. J. nonlinear sci. numer. simul., 8, 311-314, (2007) · Zbl 1193.65144
[36] Xu, L., Variational approach to solitons of nonlinear dispersive \(K(m, n)\) equations, Chaos solitons fractals, 37, 137-143, (2008) · Zbl 1143.35361
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.