On the stability of an equilibrium and the small motions of a rigid body containing a liquid, suspended in a uniform flow

In this paper, we consider a planar motion of a rigid body partially filled with an inviscid liquid and suspended in a uniform horizontal flow. At first, we write the equations of the problem, prove the existence of an equilibrium under a suitable condition and, using a first integral, we give a sufficient condition of stability of this one. Afterwards, we give the equations of the small oscillations of the system about its equilibrium position. Writing these equations in an operatorial form, we prove the existence of a denumerable infinity of complex conjugate pairs of eigenvalues having the infinity as a point of accumulation and obtain the characteristic equation permitting the calculation of the eigenvalues.