Asymptotic Analysis Relating Spectral Models in Fluid--Solid Vibrations
Carlos Conca, Axel Osses, Jacques Planchard · SIAM Journal on Numerical Analysis · 1998
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-self-adjoint 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 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.