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

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 Zhukovskiys Theorem [24], which states that in the limit t → ∞, the relative fluid velocity tends to 0 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 in [20]. In particular, we prove that every weak solution of this type is subject to Zhukovskiys 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.