Abstract
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.
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References
Antman, S.S.: Nonlinear problems of elasticity. Springer, New York (1995)
Bänsch, E.: Numerical methods for the instationary Navier-Stokes equations with a free capillary surface. PhD thesis, Freiburg University, Freiburg (1998)
Chambolle, E., Desjardins, B., Esteban, M.J., Grandmont, C.: J. Math. Fluid Mech. 7, 368–404 (2005)
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 v2
Cheng, C.H.A., Coutand, D., Shkoller, S.: SIAM J. Math. Anal. 39(3), 742–800 (2007)
Ciarlet, P.G.: Mathematical elasticity, Volume I: three-dimensional elasticity. Studies in mathematics and its applications, vol. 20. North-Holland, Amsterdam (1988)
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)
Heil, M.: J. Fluid. Mech. 353, 285–312 (1997)
Höhn, B.: Numerik für die Marangoni-Konvektion beim Floating-Zone Verfahren. PhD Thesis, Freiburg University, Freiburg (1999)
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)
Liepsch, D.W.: Biorheology 23, 395–433 (1986)
Novozhilov, V.V.: The Theory of thin shells. P. Noordhoff Ltd., Groningen. Translated by Lowe, P.G (1959)
PaÏdoussis M.P.: Fluid-structure interaction. Slender structures and axial flow, vol. I. Academic Press, London (1998)
Perktold, K., Rappitsch, G.: ZAMM 74, T477–T480 (1994)
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)
Pertsev, A.K., Platonov, E.G.: Dynamic of shells and plates. Non-stationary problems. Sudostroenie, Leningrad (in Russian) (1987)
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)
Quarteroni, A., Tuveri, M., Veneziani, A.: Comput. Visual. Sci. 2, 163–197 (2000)
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)
Timoshenko, S.: Vibration problems in engineering. D. Van Nostrand Company, Inc, Toronto, New York, London (1953)
Volmir, A.S.: Nonlinear dynamics of membrans and shells. Moscow, Nauka (1972)
Volmir, A.S.: Shells in a stream of fluid and gas. Problems of aeroelasticity. Moscow, Nauka (in Russian) (1976)
Washizu, K.: Variational methods in elasticity and plasticity. Pergamon Press, Oxford, New York, Toronto, Sydney, Paris, Frankfurt (1982)
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Bänsch, E., Goncharova, O., Koop, A., Kröner, D. (2008). Mathematical and Numerical Modelling of Fluid Flow in Elastic Tubes. In: Krause, E., Shokin, Y.I., Resch, M., Shokina, N. (eds) Computational Science and High Performance Computing III. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 101. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69010-8_9
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DOI: https://doi.org/10.1007/978-3-540-69010-8_9
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