Mathematical and Numerical Modelling of Fluid Flow in Elastic Tubes

  • E. Bänsch
  • O. Goncharova
  • A. Koop
  • D. Kröner
Conference paper
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 101)


The study of fluid flow inside compliant vessels, which are deformed under an action of the fluid, is important due to many biochemical and biomedical applications, e.g. the flows in blood vessels.

The mathematical problem consists of the 3D Navier-Stokes equations for incompressible fluids coupled with the differential equations, which describe the displacements of the vessel wall (or elastic structure). We study the fluid flow in a tube with different types of boundaries: inflow boundary, outflow boundary and elastic wall and prescribe different boundary conditions of Dirichlet- and Neumann types on these boundaries. The velocity of the fluid on the elastic wall is given by the deformation velocity of the wall.

In this publication we present the mathematical modelling for the elastic structures based on the shell theory, the simplifications for cylinder-type shells, the simplifications for arbitrary shells under special assumptions, the mathematical model of the coupled problem and some numerical results for the pressure-drop problem with cylindrical elastic structure.


Thin Shell Normal Displacement Shell Theory Middle Surface Elastic Structure 
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.


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  1. 1.
    Antman, S.S.: Nonlinear problems of elasticity. Springer, New York (1995)zbMATHGoogle Scholar
  2. 2.
    Bänsch, E.: Numerical methods for the instationary Navier-Stokes equations with a free capillary surface. PhD thesis, Freiburg University, Freiburg (1998)Google Scholar
  3. 3.
    Chambolle, E., Desjardins, B., Esteban, M.J., Grandmont, C.: J. Math. Fluid Mech. 7, 368–404 (2005)Google Scholar
  4. 4.
    Cheng, C.H.A., Coutand, D., Shkoller, S. (2006) Navier-Stokes equations interacting with a nonlinear elastic fluid shell 22 (November 2006) ArXiv:math.AP/0604313 v2Google Scholar
  5. 5.
    Cheng, C.H.A., Coutand, D., Shkoller, S.: SIAM J. Math. Anal. 39(3), 742–800 (2007)Google Scholar
  6. 6.
    Ciarlet, P.G.: Mathematical elasticity, Volume I: three-dimensional elasticity. Studies in mathematics and its applications, vol. 20. North-Holland, Amsterdam (1988)zbMATHCrossRefGoogle Scholar
  7. 7.
    Formaggia, L., Gerbeau, J.F., Nobile, F., Quarteroni, A.: On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Technical report INRIA RR–3862, 1–26 (2000)Google Scholar
  8. 8.
    Heil, M.: J. Fluid. Mech. 353, 285–312 (1997)Google Scholar
  9. 9.
    Höhn, B.: Numerik für die Marangoni-Konvektion beim Floating-Zone Verfahren. PhD Thesis, Freiburg University, Freiburg (1999)Google Scholar
  10. 10.
    Koiter, W.T.: A consistent first approximation in the general theory of thin elastic shells, part 1: foundations and linear thery. Technical report, Laboratory of Applied Mechanics, Delft (1959)Google Scholar
  11. 11.
    Liepsch, D.W.: Biorheology 23, 395–433 (1986)Google Scholar
  12. 12.
    Novozhilov, V.V.: The Theory of thin shells. P. Noordhoff Ltd., Groningen. Translated by Lowe, P.G (1959)Google Scholar
  13. 13.
    PaÏdoussis M.P.: Fluid-structure interaction. Slender structures and axial flow, vol. I. Academic Press, London (1998)Google Scholar
  14. 14.
    Perktold, K., Rappitsch, G.: ZAMM 74, T477–T480 (1994)Google Scholar
  15. 15.
    Perktold, K., Rappitsch, G.: Mathematical modelling of local arterial flow and vessel mechanics. In: Crolet J, Ohayon R (eds.) Computational methods for fluid structure interaction. Pitman Research Notes in Mathematics (1994)Google Scholar
  16. 16.
    Pertsev, A.K., Platonov, E.G.: Dynamic of shells and plates. Non-stationary problems. Sudostroenie, Leningrad (in Russian) (1987)Google Scholar
  17. 17.
    Quarteroni, A., Veneziani, A.: Modeling and simulation of blood flow problems. In: Bristeau, M.O., Etgen, G., Fitzgibbon, W., Lions, J.L., Periaux, J., Wheeler, M.F. (eds.) Computational science for the 21st Century. J.Wiley & sons, Chichester (1997)Google Scholar
  18. 18.
    Quarteroni, A., Tuveri, M., Veneziani, A.: Comput. Visual. Sci. 2, 163–197 (2000)Google Scholar
  19. 19.
    Quarteroni, A.: Mathematical modelling and numerical simulation of the cardiovascular system. In: Ayache, N. (ed.) Modelling of living systems. Handbook of Numerical Analysis Series. Elsevier, Amsterdam (2002)Google Scholar
  20. 20.
    Timoshenko, S.: Vibration problems in engineering. D. Van Nostrand Company, Inc, Toronto, New York, London (1953)Google Scholar
  21. 21.
    Volmir, A.S.: Nonlinear dynamics of membrans and shells. Moscow, Nauka (1972)Google Scholar
  22. 22.
    Volmir, A.S.: Shells in a stream of fluid and gas. Problems of aeroelasticity. Moscow, Nauka (in Russian) (1976)Google Scholar
  23. 23.
    Washizu, K.: Variational methods in elasticity and plasticity. Pergamon Press, Oxford, New York, Toronto, Sydney, Paris, Frankfurt (1982)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • E. Bänsch
    • 1
  • O. Goncharova
    • 2
    • 3
  • A. Koop
    • 4
  • D. Kröner
    • 5
  1. 1.Institute of Applied Mathematics III, Institute of Applied Mathematics IIIUniversity of Erlangen-NurembergErlangenGermany
  2. 2.Altai State UniversityBarnaulRussia
  3. 3.M.A. Lavrentyev Institute of Hydrodynamics SB RASNovosibirskRussia
  4. 4.  WuppertalGermany
  5. 5.Section of Applied MathematicsUniversity of FreiburgFreiburg i. Br.Germany

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