Scientific Research

An Academic Publisher

Series Solution of Non-Similarity Boundary-Layer Flow in Porous Medium ()

This paper aims to present complete series solution of non-similarity boundary-layer flow of an incompressible viscous fluid over a porous wedge. The corresponding nonlinear partial differential equations are solved analytically by means of the homotopy analysis method (HAM). An auxiliary parameter is introduced to ensure the convergence of solution series. As a result, series solutions valid for all physical parameters in the whole domain are given. Then, the effects of physical parameters γ and Prandtl number Pr on the local Nusselt number and momentum thickness are investigated. To the best of our knowledge, it is the first time that the series solutions of this kind of non-similarity boundary-layer flows are reported.

Keywords

Share and Cite:

*Applied Mathematics*, Vol. 4 No. 8A, 2013, pp. 127-136. doi: 10.4236/am.2013.48A018.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | H. Blasius, “Grenzschichten in Füssigketiten mit Kleiner Reibung,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 56, 1908, pp. 1-37. |

[2] | V. M. Falkner and S. W. Skan, “Some Approximate So lutions of the Boundary-Layer Equations,” Philosophical Magazine, Vol. 12, 1931, pp. 865-896. |

[3] | D. R. Hartree, “On an Equation Occurring in Falkner and Skan’s Approximate Treatment of the Equations of the Boundary Layer,” Proceedings of the Cambridge Philo sophical Society, Vol. 33, No. 2, 1937, pp. 223-239. doi:10.1017/S0305004100019575 |

[4] | E. R. G. Eckert, “Die Berechnung des Warmeuberganges in der Laminaren Grenzschicht um Stromter Korper,” VDI-Forschungsheft, Vol. 416, 1942, pp. 1-24. |

[5] | H.-T. Lin and L.-K. Lin, “Similarity Solutions for Lami nar Forced Convection Heat Transfer from Wedges to Fluids of Any Prandtl Number,” International Journal of Heat and Mass Transfer, Vol. 30, No. 6, 1987, pp. 1111-1118. doi:10.1016/0017-9310(87)90041-X |

[6] | E. M. A. Elbashbeshy and M. F. Dimian, “Effect of Ra diation on the Flow and Heat Transfer over a Wedge with Variable Viscosity,” Applied Mathematics and Computa tion, Vol. 132, No. 2-3, 2002, pp. 445-454. doi:10.1016/S0096-3003(01)00205-3 |

[7] | J. C. Y. Koh and J. P. Hartnett, “Skin-Friction and Heat Transfer for Incompressible Laminar Flow over Porous Wedges with Suction and Variable Wall Temperature,” International Journal of Heat and Mass Transfer, Vol. 2, No. 3, 1961, pp. 185-198. doi:10.1016/0017-9310(61)90088-6 |

[8] | C. H. Hsu, C. H. Chen and J. T. Teng, “Temprature and Flow Fields for the Flow of a Second Grade Fluid Past a Wedge,” International Journal of Non-Linear Mechanics, Vol. 32, No. 5, 1997, pp. 933-946. |

[9] | E. Magyari and B. Keller, “Exact Solutions for Self-Simi lar Boundary Layer Flows Induced by Permeable Stretch ing Walls,” European Journal of Mechanics: B/Fluids, Vol. 19, No. 1, 2000, pp. 109-122. doi:10.1016/S0997-7546(00)00104-7 |

[10] | K. R. Rajagopal and A. S. Gupta, “An Exact Solution for the Flow of a Non-Newtonian Fluid Past an Infinite Po rous Plate,” Meccanica, Vol. 19, No. 2, 1984, pp. 158-160. doi:10.1007/BF01560464 |

[11] | M. A. Hossain, M. Z. Munir, M. S. Hafiz and H. S. Tak har, “Flow of Viscous Incompressible Fluid with Tem perature Dependent Viscosity Past a Permeable Wedge with Uniform Surface Heat Flux,” Heat and Mass Trans fer, Vol. 36, No. 4, 2000, pp. 333-341. doi:10.1007/s002310000079 |

[12] | A. M. Lyapunov, “General Problem on Stability of Mo tion (English Translation),” Taylor and Francis, London, 1992. |

[13] | A. V. Karmishin, A. I. Zhukov and V. G. Kolosov, “Me thods of Dynamics Calculation and Testing for Thin Walled Structures,” Mashinostroyenie, Moscow, 1990. |

[14] | G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Boston, 1994. doi:10.1007/978-94-015-8289-6 |

[15] | D. Cimpean, J. H. Merkin and D. B. Ingham, “On a Free Convection Problem over a Vertical Flat Surface in a Po rous Medium,” Transport Porous, Vol. 64, No. 3, 2006, pp. 393-411. doi:10.1007/s11242-005-5236-y |

[16] | M. Massoudi, “Local Non-Similarity Solutions for the Flow of a Non-Newtonian Fluid over a Wedge,” Interna tional Journal of Non-Linear Mechanics, Vol. 36, No. 6, 2001, pp. 961-976. doi:10.1016/S0020-7462(00)00061-5 |

[17] | K. J. Wanous and E. M. Sparrow, “Heat Transfer for Flow Longitudinal to a Cylinder with Surface Mass Transfer,” Journal of Heat Transfer, Vol. 87, No. 2, 1965, pp. 317-319. doi:10.1115/1.3689103 |

[18] | D. Catherall, K. Stewartson and P. G. Williams, “Viscous Flow Past a Flat Plate with Uniform Injection,” Proceed ings of the Royal Society A, Vol. 284, No. 1398, 1965, pp. 370-396. doi:10.1098/rspa.1965.0069 |

[19] | E. M. Sparrow and H. S. Yu, “Local Non-Similarity Ther mal Boundary-Layer Solutions,” Journal of Heat Trans fer: Transactions of the ASME, Vol. 93, No. 4, 1971, pp. 328-334. |

[20] | C. Wang, S. J. Liao and J. M. Zhu, “An Explicit Analytic Solution for Non-Darcy Natural Convection over Hori zontal Plate with Surface Mass Flux and Thermal Disper sion Effects,” Acta Mechanica, Vol. 165, No. 3-4, 2003, pp. 139-150. doi:10.1007/s00707-003-0039-0 |

[21] | T. Hayat, M. Khan and M. Ayub, “On the Explicit Ana lytic Solutions of an Oldroyd 6-Constant Fluid,” Interna tional Journal of Engineering Science, Vol. 42, No. 2, 2004, pp. 123-135. doi:10.1016/S0020-7225(03)00281-7 |

[22] | S. J. Liao, “A New Branch of Solutions of Boundary Layer Flows over an Impermeable Stretched Plate,” In ternational Journal of Heat and Mass Transfer, Vol. 48, No. 12, 2005, pp. 2529-2539. doi:10.1016/j.ijheatmasstransfer.2005.01.005 |

[23] | S. J. Liao and E. Magyari, “Exponentially Decaying Bound ary Layers as Limiting Cases of Families of Algebraically Decaying Ones,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 57, No. 5, 2006, pp. 777-792. doi:10.1007/s00033-006-0061-x |

[24] | S. P. Zhu, “A Closed-Form Analytical Solution for the Valuation of Convertible Bonds with Constant Dividend Yield,” ANZIAM Journal, Vol. 47, No. 4, 2006, pp. 477-494. doi:10.1017/S1446181100010087 |

[25] | S. P. Zhu, “An Exact and Explicit Solution for the Valua tion of American Put Options,” Quantitative Finance, Vol. 6, No. 3, 2006, pp. 229-242. doi:10.1080/14697680600699811 |

[26] | M. Yamashita, K. Yabushita and K. Tsuboi, “An Analytic Solution of Projectile Motion with the Quadratic Resis tance Law Using the Homotopy Analysis Method,” Jour nal of Physics A, Vol. 40, No. 29, 2007, pp. 8403-8416. doi:10.1088/1751-8113/40/29/015 |

[27] | H. Song and L. Tao, “Homotopy Analysis of 1D Unsteady, Nonlinear Groundwater Flow through Porous Media,” Journal of Coastal Research, Vol. 50, 2008, pp. 292-295. |

[28] | W. H. Cai, “Nonlinear Dynamics of Thermal-Hyraulic Net works,” Ph.D. Thesis, University of Notre Dame, 2006. |

[29] | H. Xu and I. Pop, “Homotopy Analysis of Unsteady Bound ary-Layer Flow Started Impulsivley from Rest along a Symmetric Wedge,” Journal of Applied Mathematics and Mechanics, Vol. 88, No. 6, 2008, pp. 507-514. |

[30] | N. Kousar and S. J. Liao, “Series Solution of Non-Simi larity Boundary-Layer Flows over a Porous Wedge,” Trans port in Porous Media, Vol. 83, No. 2, 2010, pp. 397-412. |

[31] | S. J. Liao, “An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations,” Commu nications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 8, 2010, pp. 2003-2016. |

[32] | S. J. Liao, “Beyond Perturbation: Introduction to the Ho motopy Analysis Method,” Chapman and Hall/CRC Press, Boca Raton, 2003. doi:10.1201/9780203491164 |

[33] | S. J. Liao, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992. |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.