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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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Particles in Flows

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

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.

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Notes

  1. 1.

    The notation used in this article is quite standard; see e.g. [7].

  2. 2.

    We assume \(\boldsymbol{M}(0)\neq \boldsymbol{0}\), otherwise the motion of the coupled system is physically irrelevant; see Remark 4.9.

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Acknowledgements

Work partially supported by NSF grant DMS-1614011. I would like to thank Professor Jan Prüss and Mr. Jan A. Wein for inspiring conversations.

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Authors and Affiliations

  1. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 607 Benedum Engineering Hall, Pittsburgh, 15621, PA, USA

    G. P. Galdi

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  1. G. P. Galdi
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Correspondence to G. P. Galdi .

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Editors and Affiliations

  1. Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University in Prague, Prague, Czech Republic

    Tomáš Bodnár

  2. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, Pennsylvania, USA

    Giovanni P. Galdi

  3. Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

    Šárka Nečasová

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Galdi, G.P. (2017). Stability of Permanent Rotations and Long-Time Behavior of Inertial Motions of a Rigid Body with an Interior Liquid-Filled Cavity. In: Bodnár, T., Galdi, G., Nečasová, Š. (eds) Particles in Flows. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-60282-0_4

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