Asymptotic Analysis relating Spectral Models in Fluid-Solid Vibrations

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-selfadjoint 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 L 2-norm in the space of divergence free vector functions. Finally, a numerical example to illustrate the convergence of the viscous model is presented. 1.-Introduction and main results 1.1.-Introduction. In this paper, we study the asymptotic behavior of some spectral models which represent the vibrations of a bundle of tubes surrounded by a uid. These types of models have considerable importance in engineering as they are used in the design and simulation of various sorts of industrial equipment. In recent decades, much eeort has been devoted to experimental and theoretical research in this subject. For a more detailed treatment of these investigations see, To introduce the physical problem, let us imagine a mobile structure composed of K parallel tubes of constant section R i with rigidity k and mass m, immersed in a uid which occupies a three dimensional region with a constant bounded section. Let ? i be the boundary of each section R i for i = 1; : : :; K and let ? 0 be the exterior boundary of. We assume that all the boundaries are locally Lipschitz continuous and we denote by n the unit normal oriented as in Figure 1.