×

Investigation of a powerful analytical method into natural convection boundary layer flow. (English) Zbl 1221.76145

Summary: The natural convection boundary layer flow modeled by a system of nonlinear differential equations is considered. By means of similarity transformation, the non-linear partial differential equations are reduced to a system of two coupled ordinary differential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). Convergence of the obtained series solution is discussed. Finally some figures are illustrated to show the accuracy of the applied method and assessment of various prandtl numbers on the temperature and the velocity is undertaken.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76E06 Convection in hydrodynamic stability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Liao SJ. A kind of linearity-invariance under homotopy and some simple applications of it in mechanics, Report No. 520, Institute of Shipbuilding, University of Hamburg; 1992.; Liao SJ. A kind of linearity-invariance under homotopy and some simple applications of it in mechanics, Report No. 520, Institute of Shipbuilding, University of Hamburg; 1992.
[2] 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
[3] 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
[4] Liao, S. J., Series solutions of unsteady boundary-layer flows over a stretching flat plate, Stud Appl Math, 17, 239-264 (2006) · Zbl 1145.76352
[5] Liao, S. J.; Magyari, E., Exponentially decaying boundary- layers as limiting cases of families of algebraically decaying ones, Z Angew Math Phys, 57, 777-792 (2006) · Zbl 1101.76056
[6] Wang, Z. K.; Cao, T. A., An introduction to homotopy methods (1991), Chongqing Publishing House: Chongqing Publishing House Chongqing
[7] Duld, A., Lectures on algebraic topology (1972), Springer-Verlag: Springer-Verlag Berlin Heidelberg
[8] Nash, C.; Sen, S., Topology and geometry for physicists (1983), Academic Press, Inc.: Academic Press, Inc. London · Zbl 0529.53001
[9] Ortega, J. M.; Rheinboldt, W. C., Iterative solution of non-linear equations in several variables (1970), Academic press: Academic press New York · Zbl 0241.65046
[10] Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University; 1992.; Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University; 1992.
[11] 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
[12] Abbasbandy, S., Homotopy analysis method for heat radiation equations, Int Com Heat Mass Transfer, 34, 380-387 (2007)
[13] Domairry, G.; Nadim, N., Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation, Int Commun Heat mass transfer, 35, 1, 93-102 (2008)
[14] Domairry, G.; Mohsenzadeh, A.; Famouri, M., The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery-Hamel flow, Commun Nonlin Sci Num Sim, 14, 1, 85-95 (2009) · Zbl 1221.76056
[15] Fakhari, A.; Domairry, G.; Ebrahimpour, M., Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution, Phys Lett A, 368, 1-2, 64-68 (2007) · Zbl 1209.65109
[16] Hayat, T.; Khan, M.; Asghar, S., Magneto hydrodynamic flow of an oldroyd 6-constant fluid, Appl Math Comput, 155, 417-425 (2004) · Zbl 1126.76388
[17] Hayat, T.; Sajid, M., An analytic solution for thin film flow of a forth grade fluid down a vertical cylinder, Phys Lett A, 361, 316-322 (2007) · Zbl 1170.76307
[18] Liao, S. J., Beyond perturbation: introduction to the homotopy analysis method (2003), Chapman & Hall/CRC Press: Chapman & Hall/CRC Press Boca Raton
[19] Liao, S. J., J Fluid Mech, 385, 101 (1999)
[20] Sajid, M.; Hayat, T.; Asghar, S., on the analytic solution of the steady flow of a forth grade fluid, Phys Lett A, 355, 18-26 (2006)
[21] 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
[22] Hayat, T.; Abbas, Z.; Sajid, M.; Asghar, S., The influence of thermal radiation on MHD flow of a second grade fluid, Int J Heat Mass Transfer, 50, 931-941 (2007) · Zbl 1124.80325
[23] Hayat, T.; Sajid, M.; Pop, I., Three-dimensional flow over a stretching surface in a viscoelastic fluid, Nonlinear Anal: Real World Applicat, 9, 1811-1822 (2008) · Zbl 1154.76315
[24] Mehmood, Ahmer; Ali, Asif, Int Commun Heat mass transfer, 33, 10, 1243-1252 (2006)
[25] Bejan, A., Convection heat transfer (1995), Wiley: Wiley New York
[26] Kays, W. M.; Crawford, M. E., Convection heat and mass transfer (1993), Mc Craw-Hill: Mc Craw-Hill New York
[27] Liao, S. J.; Tan, y., A general approach to obtain series solutions of nonlinear differential equations, Stud Appl Math, 119, 297-340 (2007)
[28] Liao, S. J., Notes on the homotopy analysis method: some definitions and theorems, Commun Nonlinear Sci Numer Simulat, 14, 983-997 (2009) · Zbl 1221.65126
[29] Liao, S. J., An explicit totally analytic approximation of Blasius’ viscous flow problems, Int J Non-Linear Mech, 34, 4, 759-778 (1999) · Zbl 1342.74180
[30] 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
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.