The singular perturbation limit of an elastic structure in a rapidly flowing nearly inviscid fluid
Authors:
Zvi Artstein and Marshall Slemrod
Journal:
Quart. Appl. Math. 59 (2001), 543-555
MSC:
Primary 34C60; Secondary 34C29, 34E15, 74F10, 76B99
DOI:
https://doi.org/10.1090/qam/1848534
MathSciNet review:
MR1848534
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Abstract: The effective forces that govern the vertical oscillations of an elastic structure in a horizontally flowing fluid are displayed for the limit case where fluid velocity and viscosity tend to singular limits. The derived singular perturbation model is based on available representations that model the coupled dynamics as coupled oscillators. The fast dynamics of the system does not, in general, converge on the fast time scale to a stationary point; thus, the classical singular perturbation methods are not applicable. Rather, a method based on Young measures representation of the fast oscillations, and on averaging, is employed.
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- N. E. Kochin, I. A. Kibel’, and N. V. Roze, Theoretical hydromechanics, Interscience Publishers John Wiley & Sons, Inc. New York-London-Sydney, 1964. Translated from the fifth Russian edition by D. Boyanovitch; edited by J. R. M. Radok. MR 0175414
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R. E. O’Malley, Jr., Singular Perturbation Methods of Ordinary Differential Equations, Springer-Verlag, New York, 1991
Z. Artstein and A. Vigodner, Singularly perturbed ordinary differential equations with dynamic limits, Proc. Royal Society Edinburgh 126A, 541–569 (1996)
J. M. Ball, A version of the fundamental theorem for Young measures, in PDEs and Continuum Models of Phase Transitions, M. Rascle, D. Serre, and M. Slemrod, eds., Lecture Notes in Physics, 344, Springer-Verlag, Berlin-New York, 1988, pp. 207–215
Y. N. Chen, Fluctuating lift forces of the Kármán vortex streets on single circular cylinders and in tube bundles, Part 1, J. of Engineering for Industry, Trans. ASME 96, Ser. B, 603–612 (1972)
E. H. Dowell and M. Ilgamov, Studies in Nonlinear Aeroelasticity, Springer-Verlag, New York, 1988
W. D. Iwan and R. D. Belvins, A model for vortex induced oscillation of structure, J. Applied Mechanics 41, 581–586 (1974)
Th. von Kármán, Über den Mechanismus des Flüssigkeitswiderstandes den ein bewegter Körper in einer Flüssigkeit erfährt, Nachrichten Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1912, pp. 508–517
N. E. Kochin, I. A. Kibel’, and N. V. Roze, Theoretical Hydromechanics, John Wiley and Sons, New York, 1964
S. Lefschetz, Differential Equations: Geometric Theory, Dover, New York, 1977
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, 1960
R. E. O’Malley, Jr., Singular Perturbation Methods of Ordinary Differential Equations, Springer-Verlag, New York, 1991
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© Copyright 2001
American Mathematical Society
