Stability of permanent rotations and long-time behavior of inertial motions of a rigid body with an interior liquid-filled cavity. (English) Zbl 1387.35488
Bodnár, Tomáš (ed.) et al., Particles in flows. Based on the summer course and workshop, Prague, Czech Republic, August 2014. Cham: Birkhäuser/Springer (ISBN 978-3-319-60281-3/hbk; 978-3-319-60282-0/ebook). Advances in Mathematical Fluid Mechanics, 217-253 (2017).
Summary: A rigid body, with an interior cavity entirely filled with a Navier-Stokes liquid, moves in absence of external torques relative to the center of mass of the coupled system body-liquid (inertial motions). The only steady-state motions allowed are then those where the system, as a whole rigid body, rotates uniformly around one of the central axes of inertia (permanent rotations). Objective of this article is twofold. On the one hand, we provide sufficient conditions for the asymptotic, exponential stability of permanent rotations, as well as for their instability. On the other hand, we study the asymptotic behavior of the generic motion in the class of weak solutions and show that there exists a time \( t_0\) after that all such solutions must decay exponentially fast to a permanent rotation. This result provides a \( full\) and rigorous explanation of Zhukovsky’s conjecture, and explains, likewise, other interesting phenomena that are observed in both lab and numerical experiments.
For the entire collection see [Zbl 1381.35003].
For the entire collection see [Zbl 1381.35003].
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B35 | Stability in context of PDEs |
| 35B41 | Attractors |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 76U05 | General theory of rotating fluids |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
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