Spectral asymptotics of the Helmholtz model in fluid–solid structures

Authors

  • G. Allaire,

    1. Laboratoire d'Analyse Numérique, Université Paris 6, 4 place Jussieu, 75252 Paris, France
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  • C. Conca,

    Corresponding author
    1. Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170/3-Correo 3, Santiago, Chile
    • Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 170/3-Correo 3, Santiago, Chile
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  • M. Vanninathan

    1. IISc-TIFR Mathematics Programme, TIFR Center, P.O. Box 1234, Bangalore-560012, India
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Abstract

A model representing the vibrations of a coupled fluid–solid structure is considered. This structure consists of a tube bundle immersed in a slightly compressible fluid. Assuming periodic distribution of tubes, this article describes the asymptotic nature of the vibration frequencies when the number of tubes is large. Our investigation shows that classical homogenization of the problem is not sufficient for this purpose. Indeed, our end result proves that the limit spectrum consists of three parts: the macro-part which comes from homogenization, the micro-part and the boundary layer part. The last two components are new. We describe in detail both macro- and micro-parts using the so-called Bloch wave homogenization method. Copyright © 1999 John Wiley & Sons, Ltd.

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