Nonlinear Stability Analysis of a Spinning Top with an Interior Liquid-Filled Cavity

Consider the motion of the the coupled system, $\mathscr S$, constituted by a (non-necessarily symmetric) top, $\mathscr B$, with an interior cavity, $\mathscr C$, completely filled up with a Navier-Stokes liquid, $\mathscr L$. A particular steady-state motion $\bar{\sf s}$ (say) of $\mathscr S$, is when $\mathscr L$ is at rest with respect to $\mathscr B$, and $\mathscr S$, as a whole rigid body, spins with a constant angular velocity $\bar{\V\omega}$ around a vertical axis passing through its center of mass $G$ in its highest position ({\em upright spinning top}). We then provide a completely characterization of the nonlinear stability of $\bar{\sf s}$ by showing, roughly speaking, that $\bar{\sf s}$ is stable if and only if $|\bar{\V\omega}|$ is sufficiently large, all other physical parameters being fixed. Moreover we show that, unlike the case when $\mathscr C$ is empty, under the above stability conditions, the top will eventually return to the unperturbed upright configuration.