zbMATH — the first resource for mathematics

The application of homotopy analysis method to thin film flows of a third order fluid. (English) Zbl 1146.76588
Summary: The aim of the current article is to provide the analytic solutions to two thin film flows of a third order fluid. These are: (i) when the fluid moves on a belt and (ii) when the fluid moves down an inclined plane. Both problems have been solved using homotopy analysis method (HAM). These problems were already solved by M. Siddiqui et al. [Thin film flow of a third grade fluid on a moving belt by He’s homotopy perturbation method. Int. J. Non-Linear Sci. Numer. Simul. 7, 1–8 (2006); Chaos Solitons Fractals 35, No. 1, 140–147 (2008; Zbl 1135.76006)] using homotopy perturbation method (HPM) and traditional perturbation technique. With the help of two examples, it is shown that HPM is a special case of HAM. It has been noted that the solution up to second order is not enough in the case of flow on a moving belt. It is explicitly proved that the solutions of the flow down an inclined plane given in [Zbl 1135.76006] are divergent and hence have no meanings. The variation of velocity field corresponding to pertinent flow parameters is graphically presented and discussed.

MSC:
 76A20 Thin fluid films 34A45 Theoretical approximation of solutions to ordinary differential equations 76A05 Non-Newtonian fluids
Full Text:
References:
 [1] Siddiqui, A.M.; Mahmood, R.; Ghori, Q.K., Thin film flow of a third grade fluid on a moving belt by he’s homotopy perturbation method, Int J non-linear sci numer simul, 7, 1-8, (2006) · Zbl 1187.76622 [2] Siddiqui AM, Mahmood R, Ghori QK. Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. Chaos, Solitons & Fractals, in press, doi:10.1016/j.chaos.2006.05.026. · Zbl 1135.76006 [3] Liao SJ. On the proposed homotopy analysis technique for nonlinear problems and its applications. PhD dissertation, Shanghai Jio Tong University; 1992. [4] Liao, S.J., Beyond perturbation: introduction to homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton [5] Liao, S.J., A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J fluid mech, 385, 101-128, (1999) · Zbl 0931.76017 [6] Liao, S.J.; Campo, A., Analytic solutions of the temperature distribution in Blasius viscous flow problems, J fluid mech, 453, 411-425, (2002) · Zbl 1007.76014 [7] Liao, S.J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J fluid mech, 488, 189-212, (2003) · Zbl 1063.76671 [8] Liao, S.J.; Cheung, K.F., Homotopy analysis of nonlinear progressive waves in deep water, J eng math, 45, 105-116, (2003) · Zbl 1112.76316 [9] Hayat, T.; Khan, M.; Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta mech, 168, 213-232, (2004) · Zbl 1063.76108 [10] Liao, S.J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transfer, 48, 2529-2539, (2005) · Zbl 1189.76142 [11] Liao, S.J., An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate, Comm non-linear sci numer simul, 11, 326-339, (2006) · Zbl 1078.76022 [12] Wu, W.; Liao, S.J., Solving solitary waves with discontinuity by means of the homotopy analysis method, Chaos, solitons & fractals, 26, 177-185, (2005) · Zbl 1071.76009 [13] Wu, Y.Y.; Wang, C.; Liao, S.J., Solving the one loop solution of the Vakhnenko equation by means of the homotopy analysis method, Chaos, solitons & fractals, 23, 1733-1740, (2005) · Zbl 1069.35060 [14] Sajid, M.; Hayat, T.; Asghar, S., On the analytic solution of steady flow of a fourth grade fluid, Phys lett A, 355, 18-24, (2006) [15] Abbas, Z.; Sajid, M.; Hayat, T., MHD boundary layer flow of an upper-convected Maxwell fluid in a porous channel, Theor comput fluid dyn, 20, 229-238, (2006) · Zbl 1109.76065 [16] Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys lett A, 360, 109-113, (2006) · Zbl 1236.80010 [17] Hayat, T.; Khan, M.; Ayub, M., Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field, J math anal appl, 298, 225-244, (2004) · Zbl 1067.35074 [18] Tan, Y.; Xu, H.; Liao, S.J., Explicit series solution of travelling waves with a front of Fisher equation, Chaos, solitons & fractals, 31, 462-472, (2007) · Zbl 1143.35313 [19] Zhu, S.P., An exact and explicit solution for the valuation of American put options, Quant fin, 6, 229-242, (2006) · Zbl 1136.91468 [20] Zhu, S.P., A closed-form analytical solution for the valuation of convertable bonds with constant dividend yield, Anzian J, 47, 477-494, (2006) · Zbl 1147.91336 [21] He, J.H., A coupling method for homotopy technique and perturbation technique for nonlinear problem, Int J non-linear mech, 35, 37-43, (2000) · Zbl 1068.74618 [22] Fitzpatrick, P.M., Advanced calculus, (1996), PWS Publishing Company
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.