On the motion of a liquid-filled heavy body around a fixed point

We study the motion of the coupled system $\mathcal S$ constituted by a heavy rigid body, $\mathcal B$, with an interior cavity entirely filled with a Navier-Stokes liquid. We suppose that $\mathcal B$ is constrained to move around a fixed and frictionless point, $O$, belonging to one of the central axes of inertia, $\mathbf{a}$, of $\mathcal S$. We then show, in a very general class of solutions, that the terminal motion of $\mathcal S$ must be a uniform rigid rotation around the vertical axis, $\mathbf{e}$, passing through $O$ with $\mathbf{a}$ either parallel to $\mathbf{e}$ or, more generally, forming a (constant) non-zero angle, in which case the angular velocity must be sufficiently large. These results are in sharp contrast with the (classical) analogous ones when the cavity is empty, and show a remarkable stabilizing influence exerted by the liquid on the motion of $\mathcal B$. In order to point out these compelling differences, we apply our results to the two significant cases of the spherical pendulum and the heavy top. We show, among other things, the somehow unexpected property that a (frictionless) spherical pendulum with a cavity entirely filled with a viscous liquid may eventually reach the rest configuration with its center of mass in the lowest position.