Homotopy Analysis for laminar thermal boundary layer over a flat plate with a convective surface boundary condition

M.S. Abdelmeguid

doi:https://doi.org/10.14445/22315373/IJMTT-V37P512

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 37 | Number 2 | Year 2016 | Article Id. IJMTT-V37P512 | DOI : https://doi.org/10.14445/22315373/IJMTT-V37P512

Homotopy Analysis for laminar thermal boundary layer over a flat plate with a convective surface boundary condition

M.S. Abdelmeguid

Citation :

M.S. Abdelmeguid, "Homotopy Analysis for laminar thermal boundary layer over a flat plate with a convective surface boundary condition," International Journal of Mathematics Trends and Technology (IJMTT), vol. 37, no. 2, pp. 78-83, 2016. Crossref, https://doi.org/10.14445/22315373/IJMTT-V37P512

Abstract

In this paper, we will study the resulting thermal similarity equation for laminar flow and heat transfer between two separated fluids for various Prandtl numbers and a range of values characterizing the hot fluid convection process. The problem of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid has been solved analytically using homotopy analysis method (HAM) and numerically using Matlab bvp4c numerical routine. Velocity and temperature distributions were numerically discussed and presented in the graphs.Convergence of the HAM solution is checked. The effects of various Prandtl numbers and a range of values of the parameter characterizing the hot fluid convection process for similarity energy equation are considered. The temperature and heat transfer characteristics of the Blasius flow have been investigated if the convective heat transfer of the fluid heating the plate on its lower surface is proportional to . The comparison between analytical and numerical results has an excellent agreement with previously published works.

Keywords

Thermal; HAM; Convective; Boundary layer

References

[1] Aziz A. A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun Nonlinear Sci Numer Simul 2009;14: 1064– 1068.
[2] Shokouhmand H., Pakdaman M. F., Kooshkbaghi M., A similarity solution in order to solve the governing equations of laminar separated fluids with a flat plate. Commun Nonlinear Sci Numer Simul 2010;15: 3965–3973.
[3] Magyari E., Comment on ‘‘A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition” by Aziz A. Commun Nonlinear Sci Numer Simul 2011;16: 599–601.
[4] Liao SJ. The proposed homotopy analysis technique for the solution of non-linear problems. PhD thesis, Shanghai Jiao Tong University; 1992.
[5] Liao SJ. An approximate solution technique not depending on small parameters: a special example. Int J Non-linear Mech 1995;303:371–80.
[6] Liao SJ. Boundary element method for general non-linear differential operators. Eng Anal Bound Elem 1997;202:91–9.
[7] Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003.
[8] Liao SJ, Cheung KF. Homotopy analysis of non-linear progressive waves in deep water. J Eng Math 2003;45(2):103–16.
[9] Liao SJ. On the homotopy analysis method for non-linear problems. Appl Math Comput 2004;47(2):499–513.
[10] Hayat T, Khan M, Ayub M. On the explicit analytic solutions of an Oldroyd 6 – constant fluid. Int J Eng Sci 2004;42:123–35. [11] Hayat T, Khan M, Ayub M. Couette and Poisevill flow of an Oldroyd 6 – constant fluid with magnetic field. J Math Appl 2004;298:225–44.
[12] Mehmood Ahmer, Ali Asif. Analytic solution of generalized three-dimensional flow and heat transfer over a stretching plane wall. Int Commun Heat Mass Transfer 2006;33(10):1243–52.
[13] Liao SJ. Notes on the homotopy analysis method: Some definitions and theorems. Commun Non-linear Sci Numer Simulat. 2009;14:983–97.
[14] Fakhari A, Domairry G. Ebrahimpour, approximate explicit solutions of non-linear BBMB equations by homotopy analysis method and comparison with the exact solution. Phys Lett A 2007;368:64–8.
[15] Domairry G, Nadim N. Assessment of homotopy analysis method and homotopy-perturbation method in non-linear heat transfer equation. Int Commun Heat Mass Transfer 2008;35:93– 102.
[16] Domairry G, Mohsenzadeh A, Famouri M. The application of homotopy analysis method to solve non-linear differential equation governing Jeffery–Hamel flow. Commun Non-linear Sci Numer Simulat 2009;14(1):85–95.
[17] Domairry G, Fazeli M. Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity. Commun Non-linear Sci Numer Simulat 2009;14(2):489–99.
[18] Ziabakhsh Z, Domairry G, Esmaeilpour M. 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 Non-linear Sci Numer Simulat 2009;14:1284–94.
[19] Mehmood A, Ali A, Shah T. Heat transfer analysis of unsteady boundary layer flow by homotopy analysis method. Commun Non-linear Sci Numer Simulat 2008;13(5):902–12.
[20] Hayat T, Sajid M, Ayub M. On explicit analytic solution for MHD pipe flow of a fourth grade fluid. Commun Non-linear Sci Numer Simulat 2008;13(4):745–51.
[21] Tan Y, Abbasbandy S. Homotopy analysis method for quadratic Riccati differential equation. Commun Non-linear Sci Numer Simulat 2008;13(3):539–46.
[22] Ali A, Mehmood A. Homotopy analysis of unsteady boundary layer flow adjacent to permeable stretching surface in a porous medium. Commun Nonlinear Sci Numer Simulat 2008;13(2):340–9.
[23] Hayat T, Sajid M, Ayub M. A note on series solution for generalized Couette flow. Commun Non-linear Sci Numer Simulat 2007;12(8):1481–7.
[24] Abdelmeguid M.S., Homotopy Analysis of a hydromagnetic viscous fluid over a non-linear stretching and shrinking sheet, ICMIS Luxor 2013.
[25] 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 2007;50:75–84.
[26] Hang X, Liao SJ, Pop I. Series solutions of unsteady threedimensional MHD flow and heat transfer in the boundary layer over an impulsively stretching plate. Eur J Mech B/Fluids 2007;26:15–27.
[27] Zhu J, Zheng LC, Zhang XX. Analytic solution of stagnation-point flow and heat transfer over a stretching sheet by means of homotopy analysis method. Appl Math Mech 2009;30:432–42.
[28] Ali A, Mehmood A. Homotopy analysis of unsteady boundary layer flow adjacent to permeable stretching surface in a porous medium. Commun Nonlinear Sci Numer Simul 2008;13:340–9.
[29] Ziabakhsh Z, Domairry G. Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method. Commun Nonlinear Sci Numer Simul 2009;14:1868–80.