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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 953151, 9 pages
http://dx.doi.org/10.1155/2014/953151
Research Article

Revisiting Blasius Flow by Fixed Point Method

1State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi'an Jiaotong University, No. 28, Xianning West Road, Xi'an 710049, China
2School of Jet Propulsion, Beijing University of Aeronautics and Astronautics, Xueyuan Road, No. 37, Beijing 100191, China
3School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China

Received 28 October 2013; Revised 7 December 2013; Accepted 7 December 2013; Published 12 January 2014

Academic Editor: Mohamed Fathy El-Amin

Copyright © 2014 Ding Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. C. Y. Wang, “Exact solutions of the unsteady navier-stokes equations,” Applied Mechanics Reviews, vol. 42, no. 11, pp. S269–S282, 1989. View at Publisher · View at Google Scholar
  2. C. Y. Wang, “Exact solutions of the steady-state Navier-Stokes equations,” Annual Review of Fluid Mechanics, vol. 23, no. 1, pp. 159–177, 1991. View at Scopus
  3. H. Blasius, “Grenzschichten in flüssigkeiten mit kleiner reibung,” Zeitschrift für Angewandte Mathematik und Physik, vol. 56, pp. 1–37, 1908.
  4. F. M. White, Viscous Fluid Flow, McGraw-Hill, New York, NY, USA, 1991.
  5. H. Schlichting and K. Gersten, Boundary-Layer Theory, Springer, 2000.
  6. J. P. Boyd, “The Blasius function in the complex plane,” Experimental Mathematics, vol. 8, no. 4, pp. 381–394, 1999. View at Scopus
  7. J. P. Boyd, “The Blasius function: Computations before computers, the value of tricks, undergraduate projects and open research problems,” SIAM Review, vol. 50, no. 4, pp. 791–804, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. H. Weyl, “On the differential equations of the simplest boundary-layer problems,” Annals of Mathematics, vol. 43, pp. 381–407, 1942. View at Publisher · View at Google Scholar
  9. C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons Jr., “A new perturbative approach to nonlinear problems,” Journal of Mathematical Physics, vol. 30, no. 7, pp. 1447–1455, 1989. View at Scopus
  10. J. He, “Approximate analytical solution of blasius' equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 4, no. 1, pp. 75–78, 1999. View at Publisher · View at Google Scholar · View at Scopus
  11. S.-J. Liao, “An explicit, totally analytic approximate solution for Blasius' viscous flow problems,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 759–778, 1999. View at Scopus
  12. S.-J. Liao, “A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate,” Journal of Fluid Mechanics, vol. 385, pp. 101–128, 1999. View at Scopus
  13. M. Turkyilmazoglu, “A homotopy treatment of analytic solution for some boundary layer flows,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 7, pp. 885–889, 2009. View at Scopus
  14. M. Turkyilmazoglu, “An optimal variational iteration method,” Applied Mathematics Letters, vol. 24, no. 5, pp. 762–765, 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. M. Turkyilmazoglu, “An analytic shooting-like approach for the solution of nonlinear boundary value problems,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1748–1755, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. M. Turkyilmazoglu, “Convergence of the homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 12, no. 1-8, pp. 9–14, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. L. Wang, “A new algorithm for solving classical Blasius equation,” Applied Mathematics and Computation, vol. 157, pp. 1–9, 2004. View at Publisher · View at Google Scholar
  18. T. M. Shih and H. J. Huang, “Numerical method for solving nonlinear ordinary and partial differential equations for boundary-layer flows,” Numerical Heat Transfer, vol. 4, no. 2, pp. 159–178, 1981. View at Scopus
  19. T. M. Shih, “A method to solve two-point boundary-value problems in boundary-layer flows or flames,” Numerical Heat Transfer, vol. 2, pp. 177–191, 1979.
  20. S. Goldstein, “Concerning some solutions of the boundary layer equations in hydrodynamics,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 26, no. 1, pp. 1–30, 1930. View at Publisher · View at Google Scholar
  21. L. Howarth, “On the solution of the laminar boundary layer equations, proceedings of the royal society of London,” Series A-Mathematical and Physical Sciences, vol. 164, pp. 547–579, 1938. View at Publisher · View at Google Scholar
  22. T. Cebeci and H. B. Keller, “Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation,” Journal of Computational Physics, vol. 7, no. 2, pp. 289–300, 1971. View at Scopus
  23. R. Fazio, “The Blasius problem formulated as a free boundary value problem,” Acta Mechanica, vol. 95, no. 1–4, pp. 1–7, 1992. View at Publisher · View at Google Scholar · View at Scopus
  24. I. K. Khabibrakhmanov and D. Summers, “The use of generalized Laguerre polynomials in spectral methods for nonlinear differential equations,” Computers and Mathematics with Applications, vol. 36, no. 2, pp. 65–70, 1998. View at Scopus
  25. A. A. Salama, “Higher-order method for solving free boundary-value problems,” Numerical Heat Transfer, Part B, vol. 45, no. 4, pp. 385–394, 2004. View at Publisher · View at Google Scholar · View at Scopus
  26. R. Cortell, “Numerical solutions of the classical Blasius flat-plate problem,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 706–710, 2005. View at Publisher · View at Google Scholar · View at Scopus
  27. A. A. Salama and A. A. Mansour, “Fourth-order finite-difference method for third-order boundary-value problems,” Numerical Heat Transfer, Part B, vol. 47, no. 4, pp. 383–401, 2005. View at Publisher · View at Google Scholar · View at Scopus
  28. R. Fazio, “Numerical transformation methods: blasius problem and its variants,” Applied Mathematics and Computation, vol. 215, no. 4, pp. 1513–1521, 2009. View at Publisher · View at Google Scholar · View at Scopus
  29. F. Auteri and L. Quartapelle, “Galerkin-laguerre spectral solution of self-similar boundary layer problems,” Communications in Computational Physics, vol. 12, pp. 1329–1358, 2012.
  30. R. Fazio, “Scaling invariance and the iterative transformation method for a class of parabolic moving boundary problems,” International Journal of Non-Linear Mechanics, vol. 50, pp. 136–140, 2013. View at Publisher · View at Google Scholar
  31. R. Fazio, “Blasius problem and Falkner-Skan model: töpfer's algorithm and its extension,” Computers & Fluids, vol. 73, pp. 202–209, 2013.
  32. J. Zhang and B. Chen, “An iterative method for solving the Falkner-Skan equation,” Applied Mathematics and Computation, vol. 210, no. 1, pp. 215–222, 2009. View at Publisher · View at Google Scholar · View at Scopus
  33. D. Xu and X. Guo, “Fixed point analytical method for nonlinear differential equations,” Journal of Computational and Nonlinear Dynamics, vol. 8, no. 1, 9 pages, 2013. View at Publisher · View at Google Scholar
  34. E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems, Springer, 1986.