Inertial Motions of a Rigid Body with a cavity filled with a viscous liquid

TL;DR

This paper proves that a rigid body with a viscous liquid-filled cavity eventually reaches a uniform rotation, with conditions for stability and numerical simulations revealing new insights into the dynamics.

Contribution

It establishes convergence to uniform rotation for all weak solutions and provides necessary and sufficient conditions for stability, extending previous results with new analytical and numerical insights.

Findings

Weak solutions converge to uniform rotation over time

Rotation occurs along the axis with the largest moment of inertia

Numerical simulations reveal new features and guide future research

Abstract

We study inertial motions of the coupled system, S, constituted by a rigid body containing a cavity that is completely filled with a viscous liquid. We show that for data of arbitrary size (initial kinetic energy and total angular momentum) every weak solution (a la Leray-Hopf) converges, as time goes to infinity, to a uniform rotation, thus proving a famous "conjecture" of Zhukovskii. Moreover we show that, in a wide range of initial data, this rotation must occur along the central axis of inertia of S that has the largest moment of inertia. Furthermore, necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations are provided, which improve and/or generalize results previously given by other authors under different types of approximation of the original equations and/or suitable symmetry assumptions on the shape of the cavity. Finally, we present a number of results obtained by a targeted numerical simulation that, on the one hand, complement the analytical findings, whereas, on the other hand, point out new features that the analysis is yet not able to catch, and, as such, lay the foundation for interesting and challenging future investigation.