Comptes Rendus
no. 11-12
Inertial motions of a rigid body with a cavity filled with a viscous liquid
[Mouvement inertiel dʼun corps rigide avec une cavité remplie dʼun fluide visqueux]
Giovanni P. Galdi1 ; Giusy Mazzone1  ; Paolo Zunino1
1 Department of Mechanical Engineering and Materials Science, University of Pittsburgh, USA
Comptes Rendus. Mécanique, Volume 341 (2013) no. 11-12, pp. 760-765.

Dans cette note, nous décrivons, par une analyse mathématique rigoureuse, le mouvement dʼun système S constitué dʼun corps rigide contenant une cavité remplie dʼun fluide. Les résultats ont été confirmés par des simulations numériques. Le système S nʼétant soumis à aucune force extérieure et le fluide y étant contenu répondant aux équations de Navier–Stokes, nous démontrons que le mouvement de ce corps rigide est inertiel. Notre analyse montre que le mouvement du système S par rapport à son centre de masse est une rotation constante, ce qui démontre totalement la conjecture de N.Ye. Zhukovskii. Dʼautres propriétés du système sont aussi examinées et décrites dans cet article grâce àux résultats de simulations numériques.

In this note we announce a number of analytical and numerical results related to the motion of a system S constituted by a rigid body with a cavity that is completely filled with a Navier–Stokes liquid, and that moves in absence of external forces (inertial motions). Our investigation shows, in particular, that the ultimate motion of S about its center of mass is a permanent rotation, thus proving a longstanding conjecture of N.Ye. Zhukovskii. We also present other interesting features of inertial motions that are emphasized by our numerical tests, but that still lack a rigorous mathematical proof.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2013.10.001
Keywords: Liquid-filled cavity, Navier–Stokes equations, Angular momentum
Mots-clés : Cavité remplie dʼun fluide, Équations de Navier–Stokes, Moment cinétique

Giovanni P. Galdi 1 ; Giusy Mazzone 1 ; Paolo Zunino 1

1 Department of Mechanical Engineering and Materials Science, University of Pittsburgh, USA 
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Giovanni P. Galdi; Giusy Mazzone; Paolo Zunino. Inertial motions of a rigid body with a cavity filled with a viscous liquid. Comptes Rendus. Mécanique, Volume 341 (2013) no. 11-12, pp. 760-765. doi : 10.1016/j.crme.2013.10.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.10.001/

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[4] E. Hairer; Ch. Lubich; G. Wanner Geometric Numerical Integration, Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006

[5] V. Girault; P.-A. Raviart Finite Element Methods for Navier--Stokes Equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986

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