WIAS Preprint No. 1958, (2014)

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



Authors

  • Disser, Karoline

2010 Mathematics Subject Classification

  • 35Q35 35Q30 74F10 76D03 35B40 37L15

Keywords

  • Navier-Stokes equations, asymptotic behaviour of weak solutions, rigid body dynamics, conservation of angular momentum, strict Lyapunov functional

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. Zhukovskiy's Theorem states that in the limit of time going to infinity, 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. We give a rigorous proof of this result. In particular, we prove that every global weak solution in a suitable class is subject to Zhukovskiy's Theorem, and note that existence of these solutions has been established. 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 can show stability of the largest axis and provide criteria on the initial data which are decisive in special cases.

Appeared in

  • Archive for Rational Mechanics and Analysis, 221 (2016) pp. 487--526 under the title ``Inertial motions of a rigid body with a cavity filled with a viscous liquid''

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