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.

中文翻译：

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固定开放式容器中两种非混合流体（上部无粘性，下部粘弹性）的小振荡分析

我们研究了在敞开的容器中两种非混合流体（上部无粘性，下部粘弹性）的系统的小振荡，从而将我们自己限制在第二种模式，即更简单的Oldroyd模型中。首先，我们编写系统的运动方程。引入取决于自由表面挠度的无粘性流体的位移潜能以及粘弹性流体运动的变分公式，我们得到了未知挠度的两个方程，从中我们可以推导出两个算术方程合适的希尔伯特空间。最后，我们证明了这些方程可以简化为具有恒定系数的四个算子方程的系统。我们展示了频谱的存在和对称性以及系统的稳定性。最后，