Mathematical Study of the Small Oscillations of Two Nonmixing Fluids, the Lower Inviscid, the Upper Viscoelastic, in an Open Container

The authors study the small oscillations of a system of two nonmixing fluids, the lower inviscid, the upper viscoelastic, in an open container, restricting themselves for the second to the more simple Oldroyd model. From the equations of motion, they obtain a variational formulation of the problem, from which they deduce a variational equation for the viscoelastic fluid only, then a system of operatorial equations in suitable Hilbert space. They show the existence and the symmetry of the spectrum, prove the stability of the system and specify the location of the eigenvalues. They prove the existence of two sets of positive real eigenvalues having, the first l'infinity, the second a point of the real axis, as point of accumulation. Finally, after a suitable transformation of the operatorial equations of motion, they obtain an existence and unicity theorem of the solution of the associated evolution problem by means of the semigroups theory.