Analysis of the Small Oscillations of Two Nonmixing Fluids, the Upper Inviscid, the Lower Viscoelastic, in a Fixed Open Container

We study the small oscillations of a system of two nonmixing fluids, the upper inviscid, the lower viscoelastic, in an open container, restricting ourselves for the second to the more simple Oldroyd model. At first, we write the equations of motion of the system. Introducing the displacements potential of the inviscid fluid, that depends on the deflexions of the free surfaces, and a variational formulation for the motion of the viscoelastic fluid, we obtain two equations for the unknown deflexions, from which we can deduce two operatorial equations in a suitable Hilbert space. Finally, we show that these equations can be reduced to a system of four operatorial equations with constant coefficients. We show the existence and the symmetry of the spectrum and the stability of the system. Finally, we prove the existence of two sets of positive real eigenvalues having the first zero, the second the infinity as point of accumulation and, if the viscosity is sufficiently large, of a third set having a suitable point of the real axis as point of accumulation.