Устойчивость сферического движения твердого тела с неоднородной жидкостью, совершающей однородное вихревое движение

Abstract

In this paper, the equations of spherical motion of a solid body with a rotating inhomogeneous incompressible fluid that fills a completely ellipsoidal cavity are obtained and investigated. The stability of rotation of a solid with an inhomogeneous fluid having a linear density distribution is considered. Arbitrary fields of density and velocity of fluid particles are represented as a power series by spatial variables with coefficients that depend only on time. Sufficient conditions for the stability of the rotation of a solid body with a fluid around the vertical axis of dynamic symmetry are presented. The obtained equations of motion make it possible to study the stability of stationary motions of the system under consideration in order to assess the effect of fluid separation on the dynamics of the body. By analogy with the motion of a solid body, it is stated that the obtained conditions are also necessary and sufficient conditions for stationary rotations of an inhomogeneous fluid in an ellipsoidal cavity.