Mathematical analysis of the small oscillation of a heavy heterogeneous viscous liquid in an open immovable container

Abstract

The aim of this paper is the study of the influence of the viscosity on the oscillations of a heterogeneous liquid in a container. Above all, it is proved that the presence of viscosity removes the essential spectrum which appears in the case of a heterogeneous inviscid liquid.  From the equations of the system container-liquid, we deduce the variational equation of the problem, and then an operatorial equation in a suitable Hilbert space. The study of the normal oscillations is reduced to the study of an operator bundle whose kind is well known. We obtain an infinity of a periodic damped motions and, for sufficiently small viscosity, a finite number of oscillatory damped motions. The existence and uniqueness of the associated evolution problem are then proved using the weak formulation.