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On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets. (English) Zbl 1221.76213
Summary: Unsteady MHD flow of a Maxwellian fluid above an impulsively stretched sheet is studied under the assumption that boundary layer approximation is applicable. The objective is to find an analytical solution which can be used to check the performance of computational codes in cases where such an analytical solution does not exist. A convenient similarity transformation has been found to reduce the equations into a single highly nonlinear PDE. Homotopy analysis method (HAM) will be used to find an explicit analytical solution for the PDE so obtained. The effects of magnetic parameter, elasticity number, and the time elapsed are studied on the flow characteristics.

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
76A05 Non-Newtonian fluids
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[1] Tanner, R.I., Engineering rheology, (1985), Clarendon Press Oxford · Zbl 1171.76300
[2] Bird, R.B.; Armstrong, R.C.; Hassager, O., Dynamics of polymeric liquids, Fluid mechanics, vol. 1, (1987), Wiley New York
[3] Crochet, M.J.; Davies, A.R.; Walters, K., Numerical simulation of non-Newtonian flow, (1984), Elsevier Amsterdam · Zbl 0583.76002
[4] Larson, R.G., Constitutive equations for polymer melts and solutions, (1988), Butterworths Boston
[5] Sadeghy, K.; Najafi, A.H.; Saffaripour, M., Sakiadis flow of an upper-convected Maxwell fluid, Int J non-linear mech, 40, 1220, (2005) · Zbl 1349.76081
[6] Sadeghy, K.; Hajibeygi, H.; Taghavi, S.M., Stagnation-point flow of upper-convected Maxwell fluids, Int J non-linear mech, 41, 1242, (2006)
[7] Bhatnagar, R.K.; Gupta, G.; Rajagopal, K.R., Flow of an Oldroyd-B fluid due to a stretching sheet in the presence of a free stream velocity, Int J non-linear mech, 30, 3, 391, (1995) · Zbl 0837.76009
[8] Hagen, T.; Renardy, M., Boundary layer analysis of the phan-thien-tanner and giesekus model in high weissenberg number flow, J non-Newton fluid mech, 73, 181, (1997)
[9] Renardy, M., High weissenberg number boundary layers for the upper-convected Maxwell fluid, J non-Newton fluid mech, 68, 125, (1997)
[10] Aliakbar, V.; Alizadeh-Pahlavan, A.; Sadeghy, K., The influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets, Cnsns, 14, 779, (2009)
[11] Djukic, D.S., Hiemenz magnetic flow of power-law fluids, J appl mech trans ASME, 41, 822, (1974)
[12] Anderson, H.I.; Bech, K.H.; Dandapat, B.S., Magnetohydrodynamic flow of a power-law fluid over a stretching sheet, Int J non-linear mech, 27, 929, (1992) · Zbl 0775.76216
[13] Sadeghy, K.; Khabazi, N.; Taghavi, S.M., Magnetohydrodynamic (MHD) flows of viscoelastic fluids in converging/diverging channels, Int J eng sci, (2007) · Zbl 1213.76247
[14] Liao, S.J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J fluid mech, 488, 189, (2003) · Zbl 1063.76671
[15] Alizadeh-Pahlavan, A.; Aliakbar, V.; Vakili-Farahani, F.; Sadeghy, K., MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method, Cnsns, 14, 473, (2009)
[16] Hilton, P.J., An introduction to homotopy theory, (1953), Cambridge University Press · Zbl 0051.40302
[17] Liao SJ. The proposed homotopy analysis techniques for the solution of nonlinear problems. PhD dissertation. Shanghai: Shanghai Jiao Tong University; 1992 [in English].
[18] Liao, S.J., A second-order approximate analytical solution of a simple pendulum by the process analysis method, J appl mech trans ASME, 59, 970, (1992) · Zbl 0769.70017
[19] Liao, S.J., A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int J non-linear mech, 32, 815, (1997) · Zbl 1031.76542
[20] Liao, S.J., An explicit totally analytic approximation of Blasius viscous flow problems, Int J non-linear mech, 34, 759, (1999) · Zbl 1342.74180
[21] Liao, S.J., Beyond perturbation: introduction to homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton
[22] Liao, S.J., A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate, J fluid mech, 385, 101, (1999) · Zbl 0931.76017
[23] Liao, S.J.; Campo, A., Analytic solutions of the temperature distribution in Blasius viscous flow problems, J fluid mech, 453, 411, (2002) · Zbl 1007.76014
[24] Liao, S.J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J fluid mech, 488, 189, (2003) · Zbl 1063.76671
[25] Liao, S.J., An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate, Cnsns, 11, 326, (2006) · Zbl 1078.76022
[26] Xu, H.; Liao, S.J.; Pop, I., Series solutions of unsteady three-dimensional MHD flow and heat transfer in the boundary layer over an impulsively stretching plate, Euro J mech B - fluids, 26, 15, (2007) · Zbl 1105.76061
[27] Xu, H.; Liao, S.J., Series solutions of unsteady magnetohydrodynamic flows of non-Newtonian fluids caused by an impulsively stretching plate, J non-Newton fluid mech, 159, 46, (2005) · Zbl 1195.76069
[28] Cheng, J.; Liao, S.J.; Pop, I., Analytic series solution for unsteady mixed convection boundary layer flow near the stagnation point on a vertical surface in a porous medium, Trans porous med, 61, 365, (2005)
[29] Abbas Z, Hayat, T, Sajid M, Asghar S. Unsteady flow of a second grade fluid film over an unsteady stretching sheet, Math Comput Model. doi:10.1016/j.mcm.2007.09.015. · Zbl 1145.76317
[30] ZHU, S.P., A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield, Anziam j, 47, 477, (2006) · Zbl 1147.91336
[31] ZHU, S.P., An exact and explicit solution for the valuation of American put options, Quant finan, 6, 229, (2006) · Zbl 1136.91468
[32] Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Stud appl math, 119, 297, (2007)
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