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The three-dimensional flow past a stretching sheet and the homotopy perturbation method. (English) Zbl 1138.76029
Summary: We obtain an approximate analytical solution for steady laminar three-dimensional flow of an incompressible viscous fluid past a stretching sheet using the homotopy perturbation method proposed by J.-H. He [Comput. Methods Appl. Mech. Eng. 178, No. 3–4, 257–262 (1999; Zbl 0956.70017)]. The flow is governed by a boundary value problem (BVP) consisting of a pair of nonlinear differential equations. The solution is simple yet highly accurate and compares favorably with exact solutions obtained early in the literature. The methodology presented in the paper is useful for solving BVPs consisting of more than one differential equation.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
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[1] He, J.H., A new approach to nonlinear partial differential equations, Commun. nonlinear sci. numer. simul., 2, 230-235, (1997)
[2] He, J.H., Newton-like method for solving algebraic equations, Commun. nonlinear sci. numer. simul., 3, 106-109, (1998) · Zbl 0918.65034
[3] He, J.H., An approximation solution technique depending upon an artificial parameter, Commun. nonlinear sci. numer. simul., 3, 92-97, (1998) · Zbl 0921.35009
[4] He, J.H., Homotopy perturbation technique, Comput. methods appl. mech. engrg., 178, 257-262, (1999) · Zbl 0956.70017
[5] He, J.H., A coupling method of homotopy and perturbation technique for nonlinear problems, Int. J. nonlinear mech., 35, 37-43, (2000) · Zbl 1068.74618
[6] He, J.H., Homotopy perturbation method, A new nonlinear analytic technique, Appl. math. comput., 135, 73-79, (2003) · Zbl 1030.34013
[7] He, J.H., A simple perturbation approach to Blasius equation, Appl. math. comput., 140, 217-222, (2003) · Zbl 1028.65085
[8] He, J.H., Comparison of homotopy perturbation and homotopy analysis method, Appl. math. comput., 156, 527-539, (2004) · Zbl 1062.65074
[9] He, J.H., The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. math. comput., 151, 287-292, (2004) · Zbl 1039.65052
[10] He, J.H., Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. nonlinear sci. numer. simul., 6, 207-208, (2005) · Zbl 1401.65085
[11] He, J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos solitons fractals., 26, 695-700, (2005) · Zbl 1072.35502
[12] He, J.H., Homotopy perturbation method for solving boundary value problems, Phys. lett. A, 350, 87-88, (2006) · Zbl 1195.65207
[13] He, J.H., Some asymptotic methods for strongly nonlinear equations, Internat. J. modern phys. B, 20, 1141-1199, (2006), New interpretation of homotopy perturbation method, Internat. J. Modern Phys. B 20 (2006) 2561-2568 · Zbl 1102.34039
[14] He, J.H., Non-perturbative methods for strongly nonlinear problems, (2006), dissertation de-Verlag im Internet GmbH Berlin
[15] Crane, L.J., Flow past a stretching sheet, Zamp, 21, 645-647, (1970)
[16] Gupta, P.S; Gupta, A.S., Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. chem. eng., 55, 744-746, (1977)
[17] Andersson, H.I., MHD flow of a viscoelastic fluid past a stretching surface, Acta mech., 95, 227-230, (1992) · Zbl 0753.76192
[18] Troy, W.C.; Overman, E.A.; Ermountrout, G.B.; Keener, J.P., Uniqueness of flow of a second order fluid past a stretching sheet, Quart. appl. math., 44, 753-755, (1987) · Zbl 0613.76006
[19] Chang, W.-D., The uniqueness of the flow of a viscoelastic fluid over a stretching sheet, Quart. appl. math., 47, 365-366, (1989) · Zbl 0683.76012
[20] Ariel, P.D., On the second solution of flow of viscoelastic fluid over a stretching sheet, Quart. appl. math., 53, 629-632, (1995) · Zbl 0841.76006
[21] Ariel, P.D., MHD flow of a viscoelastic fluid past a stretching sheet with suction, Acta mech., 105, 49-56, (1994) · Zbl 0814.76086
[22] Wang, C.Y., Flow due to a stretching boundary with partial slip — an exact solution of the navier – stokes equations, Chem. eng. sci., 57, 3745-3747, (2002)
[23] Wang, C.Y., The three dimensional flow due to a stretching flat surface, Phys. fluids, 27, 1915-1917, (1984) · Zbl 0545.76033
[24] Ariel, P.D., Computation of flow of viscoelastic fluids by parameter differentiation, Internat. J. numer. methods fluids, 15, 1295-1312, (1992) · Zbl 0825.76541
[25] Ariel, P.D., Axisymmetric flow of a second grade fluid past a stretching sheet, Internat. J. engrg. sci., 39, 529-553, (2001) · Zbl 1210.76127
[26] P.D. Ariel, Computation of MHD flow due to moving boundary, Trinity Western University, Department of Mathematical Sciences Technical Report -MCS-2004-01, 2004
[27] Samuel, T.D.M.A.; Hall, I.M., On the series solution to the laminar boundary layer with stationary origin on a continuous moving porous surface, Proc. camb. phil. soc., 73, 223-229, (1973) · Zbl 0255.76083
[28] Ariel, P.D.; Hayat, T.; Asghar, S., Homotopy perturbation method and axisymmetric flow over a stretching sheet, Int. J. nonlinear sci. numer. simul., 7, 399-406, (2006)
[29] Ariel, P.D., Generalized three-dimensional flow due to a stretching sheet, Zamm, 83, 844-852, (2003) · Zbl 1047.76019
[30] Ackroyd, J.A.D., A series method for the solution of laminar boundary layers on moving surfaces, Zamp, 29, 729-741, (1978) · Zbl 0399.76043
[31] Ariel, P.D., On computation of the three-dimensional flow past a stretching sheet, Appl. math. comput., 188, 1244-1250, (2007) · Zbl 1114.76056
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