Asymptotic behaviour of a rigid body with a cavity filled by a viscous liquid

TL;DR

This paper rigorously proves that a rigid body with a cavity filled with viscous fluid eventually rotates steadily around a principal axis, showing the fluid's damping effect on precession regardless of geometry.

Contribution

It provides a rigorous proof of Zhukovskiy's Theorem for viscous fluid-filled bodies, establishing asymptotic steady rotation and stability criteria independent of geometry.

Findings

Fluid damping prevents precession in the long term

The body tends to rotate around a principal axis asymptotically

Stability of the largest inertia axis is demonstrated

Abstract

We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. We give a rigorous proof of Zhukovskiy's Theorem, which states that in the limit of time going to infinity, the relative fluid velocity tends to zero and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. The existence of global weak solutions for this system was established previously. In particular, we prove that every weak solution of this type is subject to Zhukovskiy's Theorem. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we show stability of the largest axis and provide criteria on the initial data which are decisive in special cases.