On the free rotations of rigid bodies with a liquid-filled gap

TL;DR

This paper studies the free rotational dynamics of a hollow rigid body with an internal liquid-filled cavity, proving the existence of solutions and showing that the system's velocities decay to zero over time.

Contribution

It establishes the existence of global weak and local strong solutions for the system's equations and demonstrates the asymptotic decay of velocities.

Findings

Existence of global weak solutions

Existence of local strong solutions

Velocities decay to zero as time approaches infinity

Abstract

We consider the system constituted by a hollow rigid body whose cavity contains a homogeneous rigid ball, and let the gap between the solids be entirely filled by a viscous incompressible fluid. We investigate the free rotations of the whole system, i.e., motions driven only by the inertia of the fluid-solids system once an initial angular momentum is imparted on the whole system. We prove the existence of global weak solutions and local strong solutions to the equations of motion. In addition, we prove that the fluid velocity as well as the inner core angular velocity relative to the outer solid converge to zero as time approaches infinity.