Existence and location of eigenvalues for fluid-solid structures

Abstract

In this paper, we study some existence and location theorems for the eigenvalues of a mathematical model that describes the eigenfrequencies and eigenmotions of a tube bundle immersed in an inviscid compressible fluid. The model is a non-standard eigenvalue problem because it involves a non-local boundary condition, and the eigenvalues of the problem appear in two places: namely, in the equations and in this non-local boundary condition. The problem is treated by using methods from functional analysis. First, the existence of eigenvalues is stated by proving that the original eigenvalue problem is equivalent to that of determining the characteristic values of a linear, compact, selfadjoint operator. Next, the location theorems are established by obtaining explicit bounds for the eigenvalues. While some of these bounds are derived by applying general location theorems of the theory of compact selfadjoint operators, others are deduced by homotopy and continuity arguments.