×

zbMATH — the first resource for mathematics

Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by homotopy analysis method. (English) Zbl 1163.34307
Summary: The similarity transform for the steady three-dimensional problem of a condensation film on an inclined rotating disk gives a system of nonlinear ordinary differential equations which are analytically solved by applying a newly developed method namely the homotopy analysis method (HAM). The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. The velocity and temperature profiles are shown and the influence of the Prandtl number on the heat transfer and the Nusselt number is discussed in detail. The validity of our results is verified by numerical results.

MSC:
34A34 Nonlinear ordinary differential equations and systems, general theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Nayfeh, A.H., Introduction to perturbation techniques, (1979), Wiley
[2] Rand, R.H.; Armbruster, D., Perturbation methods, bifurcation theory and computer algebraic, (1987), Springer · Zbl 0651.34001
[3] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992
[4] 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
[5] Liao, S.J., An explicit, totally analytic approximation of Blasius viscous flow problems, Internat. J. non-linear mech., 34, 759-778, (1999) · Zbl 1342.74180
[6] 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
[7] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman and Hall/CRC Press Boca Raton
[8] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl. math. comput., 147, 499-513, (2004) · Zbl 1086.35005
[9] Liao, S.J., Comparison between the homotopy analysis method and homotopy perturbation method, Appl. math. comput., 169, 1186-1194, (2005) · Zbl 1082.65534
[10] Sajid, M.; Hayat, T.; Asghar, S., Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt, Nonlinear dynam., 50, 27-35, (2007) · Zbl 1181.76031
[11] Allan, F.M., Derivation of the Adomian decomposition method using the homotopy analysis method, Appl. math. comput., 190, 6-14, (2007) · Zbl 1125.65063
[12] 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
[13] Abbasbandy, S., Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chem. eng. J., 136, 144-150, (2008)
[14] F.M. Allan, Construction of analytic solution to chaotic dynamical systems using the homotopy analysis method, Chaos Solitons Fractals (in press). Corrected Proof, Available online 9 August 2007
[15] Allan, F.M.; Syam, M.I., On the analytic solution of non-homogeneous Blasius problem, J. comput. appl. math., 182, 362-371, (2005) · Zbl 1071.65108
[16] Hayat, T.; Javed, T., On analytic solution for generalized three-dimensional MHD flow over a porous stretching sheet, Phys. lett. A, 370, 243-250, (2007) · Zbl 1209.76024
[17] Sajid, M.; Siddiqui, A.M.; Hayat, T., Wire coating analysis using MHD Oldroyd 8-constant fluid, Int. J. eng. sci., 45, 381-392, (2007)
[18] Hayat, T.; Sajid, M., Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet, Int. J. heat mass transfer, 50, 75-84, (2007) · Zbl 1104.80006
[19] Hayat, T.; Ahmed, Naveed; Sajid, M.; Asghar, S., On the MHD flow of a second grade fluid in a porous channel, Comput. math. appl., 54, 407-414, (2007) · Zbl 1123.76072
[20] Hayat, T.; Sajid, M.; Ayub, M., A note on series solution for generalized Couette flow, Commun. nonlinear sci. numer. simul., 12, 1481-1487, (2007) · Zbl 1114.76057
[21] X.J. Ran, Q.Y. Zhu, Y. Li, An explicit series solution of the squeezing flow between two infinite plates by means of the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul. (in press). Corrected Proof, Available online 2 August 2007
[22] Bouremel, Yann, Explicit series solution for the glauert-jet problem by means of the homotopy analysis method, Commun. nonlinear sci. numer. simul., 12, 714-724, (2007) · Zbl 1115.76065
[23] M.M. Rashidi, G. Domairry, S Dinarvand, Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul. (in press). Corrected Proof, Available online 17 October 2007
[24] Z. Ziabakhsh, G. Domairry, Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul. (in press). Corrected Proof, Available online 18 January 2008 · Zbl 1156.76425
[25] Cheng, J.; Liao, S.J.; Mohapatra, R.N.; Vajravelu, K., Series solutions of nano boundary layer flows by means of the homotopy analysis method, J. math. anal. appl., 343, 233-245, (2008) · Zbl 1135.76016
[26] Zhang, T.T.; Jia, L.; Wang, Z.C.; Li, X., The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels, Phys. lett. A, 372, 3223-3227, (2008) · Zbl 1220.76025
[27] Sparrow, E.M.; Gregg, J.L., A theory of rotating condensation, J. heat trans., 81, 113-120, (1959)
[28] von Karman, T., Uber laminare und turbulente reibung, Z. angew. math. mech., 1, 233-252, (1921) · JFM 48.0968.01
[29] Beckett, P.M.; Hudson, P.C.; Poots, G., Laminar film condensation due to a rotating disk, J. eng. math., 7, 63-73, (1973) · Zbl 0248.76047
[30] Chary, S.P.; Sarma, P.K., Condensation on a rotating disk with constant axial suction, J. heat trans., 98, 682-684, (1976)
[31] Jensen, K.F., Flow phenomena in chemical vapor deposition of thin films, Ann. rev. fluid. mech., 23, 197-232, (1991)
[32] Wang, C.Y., Condensation film on an inclined rotating disk, Appl. math. model., 31, 1582-1593, (2007)
[33] He, J.H., A coupling method for homotopy technique and perturbation technique for nonlinear problem, Intnat. J. non-linear mech., 35, 37-43, (2000) · Zbl 1068.74618
[34] He, J.H., Homotopy perturbation method for solving boundary value problems, Phys. lett. A, 350, 87-88, (2006) · Zbl 1195.65207
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.