The Problem of Steady Flow over a Two-Dimensional Bottom Obstacle

Part of the International Mathematical Series book series (IMAT, volume 12)


The linear boundary value problem describing a steady flow over a two–dimensional obstacle (bottom protrusion) is considered. This is a mixed problem for a harmonic function in an indented strip of constant width at infinity, where asymmetric conditions are imposed on the gradient. Under rather general assumptions on the obstacle, the existence of a unique solution is proved for all values of the nonnegative parameter (the reciprocal of the Froude number squared) of the problem, except possibly for a sequence of values that tends from above to the critical value.


Green Function Steady Flow Froude Number Unique Solvability Limit Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)CrossRefGoogle Scholar
  2. 2.
    Carleman, T.: Über das Neumann–Poincarésche Problem für ein Gebiet mit Ecken. Ålmqvist & Wiksell, Uppsala (1916)Google Scholar
  3. 3.
    Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Wiley-Intersci., New York etc. (1983)MATHGoogle Scholar
  4. 4.
    Gohberg, I., Krein, M.: Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space. Am. Math. Soc., Providence, RI (1969)Google Scholar
  5. 5.
    Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon, Oxford (1982)MATHGoogle Scholar
  6. 6.
    Kuznetsov, N.: On uniqueness and solvability in the linearized two-dimensional problem of a supercritical stream about a surface-piercing body. Proc. Roy. Soc. Lond. A 450, 233–253 (1995) doi: 10.1098/rspa.1995.0083MATHCrossRefGoogle Scholar
  7. 7.
    Kuznetsov, N.G., Maz'ya, V.G.: On unique solvability of the plane Neumann–Kelvin problem (Russian). Mat. Sb. 135, 440–462 (1988); English transl.: Math. USSR Sb. 63, 425–446 (1989) doi:10.1070/sm1989v063n02abeh003283Google Scholar
  8. 8.
    Kuznetsov, N., Maz'ya, V., Vainberg, B.: Linear Water Waves: A Mathematical Approach. Cambridge Univ. Press, Cambridge (2002)MATHCrossRefGoogle Scholar
  9. 9.
    Lahalle, D.: Calcul des efforts sur un profil portant d'hydroptere par copuplage eléments finis – représentation intégrale. ENSTA Rapport de Recherche. 187 (1984)Google Scholar
  10. 10.
    Motygin, O.V.: Uniqueness and solvability in the linearized two-dimensional problem of a body in a finite depth subcritical stream. Euro. J. Appl. Math. 10, 141–156 (1999) doi: 10.1017/s0956792599003691MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Motygin, O.V., McIver, P.: A uniqueness criterion for linear problems of wave-body interaction. IMA J. Appl. Math. 68, 229–250 (2003) doi: 10.1093/imamat/68.3.229MathSciNetMATHGoogle Scholar
  12. 12.
    Pagani, C.D.: The Neumann–Kelvin problem revisited. Appl. Anal. 85, 277–292 (2006) doi:  10.1080/00036810500276589 MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Pierotti, D.: On unique solvability and regularity in the linearized two-dimensional wave-resistance problem. Quart. Appl. Math. 61, 639–655 (2003)MathSciNetMATHGoogle Scholar
  14. 14.
    Pierotti, D., Simioni, P.: The steady two-dimensional flow over a rectangular obstacle lying on the bottom. J. Math. Anal. Appl. 342, 1467–1480 (2008) doi: 10.1016/j.jmaa.2008.01.020MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Trofimov, V.P.: The root subspaces of operators that depend analytically on a parameter (Russian). Mat. Issledov. 3(9), 117–125 (1968)MathSciNetMATHGoogle Scholar
  16. 16.
    Vainberg, B.R., Maz'ya, V.G.: On the plane problem of the motion of a body immersed in a fluid (Russian). Tr. Mosk. Mat. O-va 28, 35–56 (1973); English transl.: Trans. Moscow Math. Soc. 28, 33–55 (1973)Google Scholar
  17. 17.
    Werner, P.: A Green's function approach to the potential flow around obstacles in a two-dimensional channel. In: Kleinman, R. et al. (eds), Direct and Inverse Boundary Value Problems. Methoden und Verfahren der math. physik. 37, Peter Lang, Frankfurt am Main etc. (1991)Google Scholar
  18. 18.
    Wigley, N.M.: Mixed boundary value problems in plane domains with corners. Math. Z. 115, 33–52 (1970) doi:  10.1007/bf01109747 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Laboratory for Mathematical Modelling of Wave PhenomenaInstitute for Problems in Mechanical Engineering, Russian Academy of SciencesSt.PetersburgRussia

Personalised recommendations