Archive of Applied Mechanics

, Volume 65, Issue 5, pp 346–364 | Cite as

Solution of the laminar duct flow problem by a boundary integral equation method

  • T. Zlatanovski


A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.

Key words

laminar flow boundary integral vorticity multiply connected domain singularity 


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  1. 1.
    Shah, R. K.;London, A. L.: Laminar flow forced convection in ducts. Supplement 1 to: Advances in heat transfer. New York: Academic Press 1978Google Scholar
  2. 2.
    Shah, R. K.;Bahtti, M. S.: Laminar convective heat transfer in ducts. In: Kacac, S.; Shah, R. K.; Aung, W. (eds.): Handbook of single phase convective heat transfer, chap. 3. New York: Wiley & Sons 1987Google Scholar
  3. 3.
    Khader, M. S.: A surface integral numerical solution for laminar developed. Trans. ASME/J. Appl. Mech. 48 (1981) 695–700Google Scholar
  4. 4.
    Brebbia, C. A.;Telles, J. C. F.;Wrobel, L. C.: Boundary element techniques. Theory and application in engineering. Berlin: Springer 1984Google Scholar
  5. 5.
    Muskhelishvili, N. I.: Singular integral equations. Groningen: Noordhoff 1972Google Scholar
  6. 6.
    Zlatanovski, T.: Über die Lösung des St.-Venantschen Torsionsproblems durch singuläre Randingtegralgleichungen. ZAMM 71 (1991) 331–340Google Scholar
  7. 7.
    Zlatanovski, T.: First example of an exact analytical solution to laminar flow in straight pipes of triply connected cross-sections. ZAMM 74 (1994) T355-T358Google Scholar
  8. 8.
    Lawrentjew, M. A.: On the construction of the flow around a contour of given shape (in russian). In: Proceedings of the central Aero-Hydrodynamical Inst., Nr. 118. Moscow 1932Google Scholar
  9. 9.
    Albring, W.: Angewandte Strömungslehre, Berlin: Akademie-Verlag 1990Google Scholar
  10. 10.
    Prössdorf, S.;Schmidt, G.: A finite element collocation method for singular integral equations. Math. Nachr. 100 (1981) 33–60Google Scholar
  11. 11.
    Prössdorf, S.;Rathsfeld, A.: On quadrature methods and splines approximation of singular integral equations. In: Boundary element methods IX, vol. 1: Mathematical and computational aspects, pp. 193–211, Berlin: Springer 1987Google Scholar
  12. 12.
    Prössdorf, S.;Rathsfeld, A.: Stabilitätskriterien für Näherungsverfahren bei singulären Integralgleichungen inL p. Zeitschrift für Analysis und ihre Anwendungen 6 (1987) 539–558Google Scholar
  13. 13.
    Hackbusch, W.: Integralgleichungen, Theory und Numerik. Stuttgart: Teubner 1989Google Scholar
  14. 14.
    Jaswon, M. A.;Ponter, A. R.: An integral equation solution of the torsion problem. Proc. Roy. Soc. London. Ser. A 273 (1963) 237–246Google Scholar
  15. 15.
    Athanasiadis, G.: Die Integralgleichungsmethode in der Elastizitätstheorie und ihre Anwendung am Beispiel des Torsionsproblems. Fotschr.-Ber. VDI-Z. Reihe 1, Nr. 90 (1982)Google Scholar
  16. 16.
    Hromadka II, T. V.;Lai, Ch.: The complex variable boundary element method in engineering analysis. Berlin: Springer 1986Google Scholar
  17. 17.
    Hartmann, F.: Methode der Randelemente. Berlin: Springer 1987Google Scholar
  18. 18.
    DeFiguerdo, T. G. B.: A new boundary element formulation in engineering. Berlin: Springer 1991Google Scholar
  19. 19.
    Christiansen, S.: A review of some integral equations for solving the Saint-Venant torsion problem. J. Elasticity 8 (1978) 53–55Google Scholar
  20. 20.
    Athansiadis, G.: Die Untersuchung einiger Integralgleichungen des St.-Venantschen Torsionsproblems. Ing. Arch. 53 (1983) 303–316Google Scholar
  21. 21.
    Klotz, K.: Zur Laminarströmung in einem Rohr mit der Querschnittsform eines Kreisbogen-dreieckes. ZAMM 67 (1987) 249–256Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • T. Zlatanovski
    • 1
  1. 1.Faculty of Mechanical EngineeringSaint Cyril and Methodius UniversitySkopjeMacedonia

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