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Journal of Hydrodynamics

, Volume 18, Issue 1, pp 101–106 | Cite as

The Lie-group shooting method for boundary layer equations in fluid mechanics

  • Chih-Wen Chang
  • Jiang-Ren Chang
  • Chein-Shan Liu
Session A2

Abstract

In this paper, we propose a Lie-group shooting method to tackle two famous boundary layer equations in fluid mechanics, namely, the Falkner-Skan and the Blasius equations. We can employ this method to find unknown initial conditions. The pivotal point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T) = G(r) we can seek the missing initial conditions through a minimum discrepancy of the target in terms of the weighting factor r(0,1). It is the first time that we can apply the Lie-group shooting method to solve the boundary layer equations. Numerical examples are worked out to persuade that this novel approach has good efficiency and accuracy with a fast convergence speed by searching r with the minimum norm to fit two targets.

Key words

Falkner-Skan equation Blasius equation boundary value problem Lie-group shooting method estimation of missing initial condition 

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References

  1. [1]
    HARTREE, D. R. On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer [J]. Proceedings of the Cambridge Philosophical Society, 1937, 33: 223–239.CrossRefGoogle Scholar
  2. [2]
    WEYL, H. On the differential equations of the simplest boundary-layer problem [J]. Annals of Mathematics, 1942, 43: 381–407.MathSciNetCrossRefGoogle Scholar
  3. [3]
    ROSENHEAD, L. Laminar boundary layer [M]. Oxford: Clarendon Press, 1963.MATHGoogle Scholar
  4. [4]
    NA, T. Y. Computational methods in engineering boundary value problems [M]. New York: Academic Press, 1979.MATHGoogle Scholar
  5. [5]
    CEBECI, T.; KELLER, H. B. Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation [J]. Journal of Computational of Physics, 1971, 7: 289–300.CrossRefGoogle Scholar
  6. [6]
    SMITH, A. M. O. Improved solutions of the Falkner and Skan boundary-layer equation [J]. Institute of the Aeronautical Sciences, 1954, SMF Fund Paper, No. FF-IO.Google Scholar
  7. [7]
    ASAITHAMBI, N. S. A numerical method for the solution of the Falkner–Skan equation [J]. Applied Mathematics and Computation, 1997, 81: 259–264.MathSciNetCrossRefGoogle Scholar
  8. [8]
    ASAITHAMBI, A. A finite-difference method for the Falkner–Skan equation [J]. Applied Mathematics and Computation, 1998, 92: 135–141.MathSciNetCrossRefGoogle Scholar
  9. [9]
    ASAITHAMBI, A. A second-order finite-difference method for the Falkner–Skan equation [J]. Applied Mathematics and Comput ation, 2004, 156: 779–786.MathSciNetCrossRefGoogle Scholar
  10. [10]
    ASAITHAMBI, A. Numerical solution of the Falkner–Skan equation using piece-wise linear functions [J]. Applied Mathematics and Computation, 2004, 159: 267–273.MathSciNetCrossRefGoogle Scholar
  11. [11]
    ASAITHAMBI, A. Solution of the Falkner–Skan equation by recursive evaluation of Taylor coefficients [J]. Journal of Computation and Applied Mathematics, 2005, 176: 203–214.MathSciNetCrossRefGoogle Scholar
  12. [12]
    SCHLICHTING, H. Boundary layer theory [M]. New York, McGraw-Hill, 1979.MATHGoogle Scholar
  13. [13]
    BLASIUS, H. Grenzschichten in Flüssigkeiten mit kleiner Reibung [J]. Zeitschrift für Mathematik und Physik, 1908, 56: 1–37.MATHGoogle Scholar
  14. [14]
    TÖPFER, C. Bemerkungen zu dem Aufsatz von H. Blasius: Grenzschichten in Flüssigkeiten mit kleiner Reibung [J]. Zeitschrift für Mathematik und Physik, 1912, 61: 397–398.MATHGoogle Scholar
  15. [15]
    HOWARTH, L. On the solution of the laminar boundary layer equations [J]. Proceedings of the Royal Society London A, 1938, 164: 547–579.CrossRefGoogle Scholar
  16. [16]
    LOCK, R. C. The velocity distribution in the laminar boundary layer between parallel streams [J]. Quarterly Journal of Mechanics and Applied Mathematics, 1951, 4: 42–63.MathSciNetCrossRefGoogle Scholar
  17. [17]
    LOCK, R. C. Hydrodynamic stability of the flow in the laminar boundary layer between parallel streams [J]. Proceedings of the Cambridge Philosophical Society, 1954, 50: 105–124.MathSciNetCrossRefGoogle Scholar
  18. [18]
    LIAO, S.-J. A kind of approximate solution technique which does not depend upon small parameters–II an application in fluid mechanics [J]. International Journal of Non-Linear Mechanics, 1997, 32: 815–822.MathSciNetCrossRefGoogle Scholar
  19. [19]
    LIAO, S.-J. An explicit, totally analytic approximate solution for Blasius’ viscous flow problems [J]. International Journal of Non-Linear Mechanics, 1999, 34: 759–778.MathSciNetCrossRefGoogle Scholar
  20. [20]
    YU, L.-T.; CHEN, C.-K. The solution of the Blasius equation by the differential transformation method [J]. Mathematical and Computer Modeling, 1998, 28: 101–111.MathSciNetCrossRefGoogle Scholar
  21. [21]
    LIU, C.-S. Cone of non-linear dynamical system and group preserving schemes [J]. International Journal of Non-Linear Mechanics, 2001, 36: 1047–1068.MathSciNetCrossRefGoogle Scholar
  22. [22]
    LIU, C.-S. The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions [J]. CMES: Computer Modeling in Engineering & Sciences, 2006, to appear.Google Scholar
  23. [23]
    CORTELL, R. Numerical solutions of the classical Blasius flat-plate problem [J]. Applied Mathematics and Computation, 2005, 170: 706–710.MathSciNetCrossRefGoogle Scholar

Copyright information

© China Ship Scientific Research Center 2006

Authors and Affiliations

  • Chih-Wen Chang
    • 1
    • 2
  • Jiang-Ren Chang
    • 1
    • 2
  • Chein-Shan Liu
    • 1
    • 2
  1. 1.Department of Systems Engineering and Naval ArchitectureNational Taiwan Ocean UniversityKeelung 202China
  2. 2.Department of Mechanical and Mechatronic EngineeringNational Taiwan Ocean UniversityKeelung 202China

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