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Solution of the Jeffery-Hamel flow problem by optimal homotopy asymptotic method. (English) Zbl 1197.76043
Summary: The present article addresses Jeffery-Hamel flow: fluid flow between two rigid plane walls, where the angle between them is \(2\alpha \). A new analytical method called the optimal homotopy asymptotic method (OHAM) is briefly introduced, and then employed to solve the governing equation. The validity of the homotopy asymptotic method is ascertained by comparing our results with numerical (Runge-Kutta method) results. The effects of the Reynolds number \((Re)\) and the angle between the two walls (\(2\alpha \)) are highlighted in the proposed work. The results reveal that the proposed analytical method can achieve good results in predicting the solutions of such problems.

76D99 Incompressible viscous fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65L99 Numerical methods for ordinary differential equations
Full Text: DOI
[1] Jeffery, G.B., The two-dimensional steady motion of a viscous fluid, Phil. mag., 6, 29, 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. royal soc. A, 175, 436-467, (1940) · Zbl 0025.37501
[4] Batchelor, G.K., An introduction to fluid dynamics, (1967), Cambridge University Press Cambridge, England · Zbl 0152.44402
[5] Sobey, I.J.; Drazin, P.G., Bifurcations of two-dimensional channel flows, J. fluid mech., 171, 263-287, (1986) · Zbl 0609.76050
[6] Hamadiche, M.; Scott, J.; Jeandel, D., Temporal stability of jeffery – hamel flow, J. fluid mech., 268, 71-88, (1994) · Zbl 0809.76039
[7] 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
[8] 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
[9] Schlichting, H., Boundary-layer theory, (1979), McGraw-Hill New York
[10] Rathy, R.K., An introduction to fluid dynamics, (1976), Oxford and IBH Pl New Delhi · Zbl 0118.21703
[11] McAlpine, A.; Drazin, P.G., On the spatio-temporal development of small perturbations of jeffery – hamel flows, Fluid dynam. res., 22, 123-138, (1998) · Zbl 1051.76554
[12] Liao, S.J., An approximate solution technique not depending on small parameters: a special example, Internat. J. non-linear mech., 303, 371-380, (1995) · Zbl 0837.76073
[13] Liao, S.J., Boundary element method for general nonlinear differential operators, Eng. anal. bound. elem., 202, 91-99, (1997)
[14] Liao, S.J.; Cheung, K.F., Homotopy analysis of nonlinear progressive waves in deep water, J. eng. math., 45, 2, 103-116, (2003) · Zbl 1112.76316
[15] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl. math. comput., 47, 2, 499-513, (2004) · Zbl 1086.35005
[16] Sajid, M.; Hayat, T., The application of homotopy analysis method to thin film flows of a third order fluid, Chaos solitons fractals, 38, 2, 506-515, (2008) · Zbl 1146.76588
[17] Hayat, T.; Khan, M.; Ayub, M., Couette and Poiseuille flow of an Oldroyd 6-constant fluid with magnetic field, J. math. anal. appl., 298, 225-244, (2004) · Zbl 1067.35074
[18] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Internat. J. engrg. sci., 42, 123-135, (2004) · Zbl 1211.76009
[19] Esmaeilpour, M.; Domairry, G.; Sadoughi, N.; Davodi, A.G., Homotopy analysis method for the heat transfer of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall, Commun. nonlinear sci. numer. simul., 15, 9, 2424-2430, (2010) · Zbl 1222.76073
[20] He, J.H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Internat. J. non-linear mech., 35, 1, 37-43, (2000) · Zbl 1068.74618
[21] He, J.H., Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. nonlinear sci. numer. simul., 6, 207-208, (2005) · Zbl 1401.65085
[22] He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos solitons fractals, 26, 695-700, (2005) · Zbl 1072.35502
[23] Esmaeilpour, M.; Ganji, D.D., Application of he’s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate, Phys. lett. A, 372, 1, 33-38, (2007) · Zbl 1217.76029
[24] Esmaeilpour, M.; Ganji, D.D.; Mohseni, E., Application of homotopy perturbation method to micropolar flow in a porous channel, J. porous media, 12, 5, 451-459, (2009)
[25] Ganji, S.S.; Ganji, D.D.; Karimpour, S.; Babazadeh, H., Applications of a modified he’s homotopy perturbation method to obtain second-order approximations of the coupled two-degree-of-freedom systems, Int. J. nonlinear sci. numer. simul., 10, 3, 303-312, (2009)
[26] Seyed Alizadeh, S.R.; Domairry, G.G.; Karimpour, S., An approximation of the analytical solution of the linear and nonlinear integro-differential equations by homotopy perturbation method, Acta appl. math., 104, 3, 355-366, (2008) · Zbl 1162.65419
[27] He, J.H., Variational iteration method — a kind of non-linear analytical technique: some examples, Internat. J. non-linear mech., 34, 699-708, (1999) · Zbl 1342.34005
[28] He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl. math. comput., 114, 115-123, (2000) · Zbl 1027.34009
[29] He, J.H., Variational theory for linear magneto-electro-elasticity, Int. J. nonlinear sci. numer. simul., 2, 4, 309-316, (2001) · Zbl 1083.74526
[30] Ganji, D.D.; Jannatabadi, M.; Mohseni, E., Application of he’s variational iteration method to nonlinear jaulent – miodek equations and comparing it with ADM, J. comput. appl. math., 207, 1, 35-45, (2007) · Zbl 1120.65107
[31] Sweilam, N.H.; Khader, M.M., Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos soliton fractals, 32, 145-149, (2007) · Zbl 1131.74018
[32] Wazwaz, A.M., The variational iteration method for exact solutions of Laplace equation, Phys. lett. A, 363, 260-262, (2007) · Zbl 1197.65204
[33] Marinca, V.; Herişanu, N., Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, Internat. commun. heat mass transfer, 35, 710-715, (2008)
[34] Marinca, V.; Herişanu, N.; Bota, C.; Marinca, B., An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Appl. math. lett., 22, 2, 245-251, (2009) · Zbl 1163.76318
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