Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the motion of a liquid-filled heavy body around a fixed point

Authors: Giovanni P. Galdi, Giusy Mazzone and Mahdi Mohebbi
Journal: Quart. Appl. Math. 76 (2018), 113-145
MSC (2010): Primary 35Q35, 35Q30, 76D05
DOI: https://doi.org/10.1090/qam/1487
Published electronically: October 4, 2017
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Abstract: We study the motion of the coupled system $ \mathcal S$ constituted by a heavy rigid body, $ \mathcal B$, with an interior cavity entirely filled with a Navier-Stokes liquid. We suppose that $ \mathcal B$ is constrained to move around a fixed and frictionless point, $ O$, belonging to one of the central axes of inertia, $ \boldsymbol {\sf a}$, of $ \mathcal S$. We then show, in a very general class of solutions, that the terminal motion of $ \mathcal S$ must be a uniform rigid rotation around the vertical axis, $ \boldsymbol {\sf e}$, passing through $ O$ with $ \boldsymbol {\sf a}$ either parallel to $ \boldsymbol {\sf e}$ or, more generally, forming a (constant) non-zero angle, in which case the angular velocity must be sufficiently large. These results are in sharp contrast with the (classical) analogous ones when the cavity is empty, and show a remarkable stabilizing influence exerted by the liquid on the motion of $ \mathcal B$. In order to point out these compelling differences, we apply our results to the two significant cases of the spherical pendulum and the heavy top. We show, among other things, the somehow unexpected property that a (frictionless) spherical pendulum with a cavity entirely filled with a viscous liquid may eventually reach the rest configuration with its center of mass in the lowest position.

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Additional Information

Giovanni P. Galdi
Affiliation: Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA 15261
Email: galdi@pitt.edu

Giusy Mazzone
Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240
Email: giusy.mazzone@vanderbilt.edu

Mahdi Mohebbi
Affiliation: Department of Mechanical Engineering, SUNY Korea 119 Songdo Moonhwa-ro, Yeonsu-gu, Incheon, Korea 21985
Email: mahdi.mohebbi@sunykorea.ac.kr

DOI: https://doi.org/10.1090/qam/1487
Received by editor(s): April 28, 2017
Published electronically: October 4, 2017
Additional Notes: The work of the first author is partially supported by NSF DMS Grant-1614011.
Article copyright: © Copyright 2017 Brown University

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