Asymptotic analysis relating spectral models in fluid-solid vibrations

Abstract

An asymptotic study of two spectral models which appear in uid{solid vibrations is presented in this paper. These two models are derived under the assumption that the uid is slightly compressible or viscous, respectively. In the rst 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 uid becomes incompressible. In the second case, we use a suitable family of unbounded non-self-adjoint operators to prove that the spectrum of the viscous model tends to be continuous as the uid becomes inviscid. At the limit, in both cases, the spectrum of a perfect incompressible uid 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.