The Concept of Self-similarity

  • Ahmer Mehmood
Part of the Mathematical Engineering book series (MATHENGIN)


One of the important classes of boundary-layer flows comprises the self-similar flows. The concept of self-similarity is equally important in mathematical as well as physical point of views. In this chapter, the concept of self-similarity has been explained in a bit detail. The general procedure for determining the similarity transformations has also been explained by considering suitable examples.


  1. 1.
    G.W. Bluman, S.C. Anco, Symmetry and Integration Methods for Differential Equations (Springer, Berlin, 2002)Google Scholar
  2. 2.
    L. Howarth, On the solution of the laminar boundary layer equations. Proc. R. Soc. Lond. A 164, 547–579 (1938)CrossRefMATHGoogle Scholar
  3. 3.
    S. Lie, Über die Integration durch bestimmte Integrale von einer klasse linearer partialler Differentialgleichungen. Arch. Math. 6, 328–368 (1992); also Gesammelte Abhandlungen, vol. III, pp. 492–523, B.G. Teubner, Leipzig, 1922. Originally in 1881Google Scholar
  4. 4.
    S. Lie, Theorie der Transformationsgruppen, vol. III (B.G. Teubner, Leipzig, 1893)MATHGoogle Scholar
  5. 5.
    A.J.A. Morgan, The reduction by one of the number of independent variables in some systems of partial differential equations. Q. J. Math. Oxford 3, 250–259 (1952)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    R. Manohar, Some similarity solutions of partial differential equations of boundary-layer equations. Math. Res. Center (Uni. Wis.) Tech. Summar Rept. No. 375 (1963)Google Scholar
  7. 7.
    H. Schuh, in Über the ähnlichen Lösungen der instalionaren Laminares Granzschichtgleichungen in inkompressiblen Strömungen, ed. by H. Görtler, W. Tollmien. 50 Jahre Grenzschichtforschung (Vieweg, Braunschweig, 1955), p. 149Google Scholar
  8. 8.
    T. Geis, in Ähnliche Grenzschichten an Rotationskörpern, ed. by H. Göstler, W. Tollmien. 50 Jahre Grenzschichtforschung (Vieweg, Braunschweig, 1955), p. 294Google Scholar
  9. 9.
    A.G. Hansen, Possible similarity solutions of the laminar, incompressible, boundary-layer equations. Trans. ASME 80, 1553–1559 (1958)MathSciNetGoogle Scholar
  10. 10.
    A.J.A. Morgan, Discussion of “possible similarity solutions of the laminar, incompressible, boundary-layer equations”. Trans. ASME 80, 1559–1562 (1958)Google Scholar
  11. 11.
    W.F. Ames, Nonlinear Partial Differential Equations in Engineering (Academic Press, 1965)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Mathematics and StatisticsInternational Islamic University Islamabad (IIUI)IslamabadPakistan

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