Abstract
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.
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Notes
- 1.
Buckingham Pi-theorem is of fundamental importance in dimensional analysis. The interested readers is recommended to consult the Ref. [1].
- 2.
For further detail on this account the reader is referred to follow [1].
- 3.
As in the Falkner–Skan flow, the similarity variables take different forms for different values of m, though the nature of the flow is the same, that is the potential flow past a wedge.
- 4.
This fact can be confirmed in Chap. 5 which, however, does not deny the possibility of any other form.
- 5.
If for a given differential equation, sufficient numbers of auxiliary conditions are known to make the unique solution sure and the solution thus obtained depends continuously upon the given auxiliary data.
- 6.
If for a given differential equation, at least one or more auxiliary conditions are missing, the problem is ill-posed.
References
G.W. Bluman, S.C. Anco, Symmetry and Integration Methods for Differential Equations (Springer, Berlin, 2002)
L. Howarth, On the solution of the laminar boundary layer equations. Proc. R. Soc. Lond. A 164, 547–579 (1938)
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 1881
S. Lie, Theorie der Transformationsgruppen, vol. III (B.G. Teubner, Leipzig, 1893)
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)
R. Manohar, Some similarity solutions of partial differential equations of boundary-layer equations. Math. Res. Center (Uni. Wis.) Tech. Summar Rept. No. 375 (1963)
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. 149
T. Geis, in Ähnliche Grenzschichten an Rotationskörpern, ed. by H. Göstler, W. Tollmien. 50 Jahre Grenzschichtforschung (Vieweg, Braunschweig, 1955), p. 294
A.G. Hansen, Possible similarity solutions of the laminar, incompressible, boundary-layer equations. Trans. ASME 80, 1553–1559 (1958)
A.J.A. Morgan, Discussion of “possible similarity solutions of the laminar, incompressible, boundary-layer equations”. Trans. ASME 80, 1559–1562 (1958)
W.F. Ames, Nonlinear Partial Differential Equations in Engineering (Academic Press, 1965)
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Mehmood, A. (2017). The Concept of Self-similarity. In: Viscous Flows. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55432-7_3
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DOI: https://doi.org/10.1007/978-3-319-55432-7_3
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