Asymptotic Analysis relating
Spectral Models in
Fluid-Solid Vibrations

by C. Conca, A. Osses and J. Planchard

An asymptotic study of two spectral models which appear in fluid-solid
vibrations is presented in this paper. These two models are derived
under the assumption that the fluid is slightly compressible or
viscous respectively.  In the first case, min-max estimations and a
limit process in the variational formulation of the corresponding
model are used to show that the spectrum of the compressible case
tends to be a continuous set as the fluid becomes incompressible.  In
the second case, we use a suitable family of unbounded non-selfadjoint
operators to prove that the spectrum of the viscous model tends to be
continuous as the fluid becomes inviscid. At the limit, in both cases,
the spectrum of a perfect incompressible fluid model is found. We also
prove that the set of generalized eigenfunctions associated with the
viscous model is dense for the L2-norm in the space of divergence free
vector functions.  Finally, a numerical example to illustrate the
convergence of the viscous model is presented.

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