Analysis of the Small Oscillations of a Heavy Viscous Liquid in a Container Supported by an Elastic Structure

Abstract

We propose here a mathematical study of the small oscillations of a heavy viscous liquid in an arbitrary open rigid container supported by an elastic structure, i.e a spring-mass-damper system. From the equations of motion of the system, we deduce a variational formulation of the problem and after, an operatorial equation in a suitable Hilbert space. Then, we can study the spectrum of the problem. At first, we prove that it is formed by eigenvalues that are located in the right half-plane, so that the equilibrium position is stable. Besides, we show that the operator pencil of the problem is a well-known pencil, whose we prove by a simple method that it has two branches of real eigenvalues having as points of accumulation zero and the infinity and a number at most finite of complex eigenvalues. Finally, we prove the existence and the unicity of the solution of the associated evolution problem by means of Lions method. Afterwards, we consider the case where the damper is removed, that is very different. We prove in this case that the equilibrium position is stable, but the problem is reduced to the study of a Krein-Langer pencil, so that in particular, there exist always non oscillatory eigenmotions.