Stability of Permanent Rotations and Long-Time Behavior of Inertial Motions of a Rigid Body with an Interior Liquid-Filled Cavity
Stability of Permanent Rotations and Long-Time Behavior of Inertial Motions of a Rigid Body with an Interior Liquid-Filled Cavity
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内部充满液体腔的刚体的永久旋转稳定性和惯性运动的长期行为
DOI：
10.1007/978-3-319-60282-0_4
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
G. Galdi
中科院分区：
文献类型：
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作者：
G. Galdi
刚体的内部空腔中完全充满了Navier-Stokes液体，它在没有外部力矩的情况下相对于体-液体耦合系统的质心运动(惯性运动)。唯一允许的稳态运动是这样的：系统作为一个完整的刚体，围绕其中一个中心惯性轴均匀旋转(永久旋转)。本文的目的是双重的。一方面，我们给出了永久转动的渐近指数稳定性及其不稳定性的充分条件。另一方面，我们研究了一般运动在弱解类中的渐近行为，证明了在弱解类中存在一个时间，在此之后，所有这些解必然以指数的速度衰减到永久旋转。这一结果为朱可夫斯基的猜想提供了严格的解释，并同样解释了在实验室和数值实验中观察到的其他有趣的现象。
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 timet0after that all such solutions must decay exponentially fast to a permanent rotation. This result provides afulland rigorous explanation of Zhukovsky’s conjecture, and explains, likewise, other interesting phenomena that are observed in both lab and numerical experiments.