Abstract
In this paper we investigate the existence of time-periodic motions of a system constituted by a rigid body with an interior cavity completely filled with a viscous liquid, and subject to a time-periodic external torque acting on the rigid body. We then show that the system of equations governing the motion of the coupled system liquid-filled rigid body, has at least one corresponding time-periodic weak solution. Furthermore if the size of the torque is below a certain constant, the weak solution is in fact strong.
Dedicated, with friendship and admiration, to Professor Yoshihiro Shibata, on the occasion of his 60th birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We adopt summation convention over repeated indices.
- 2.
The same argument does not work for \(\boldsymbol{A}_{n}\), i.e. the solution map does not necessarily lie in \(\mathbb{B}_{R_{1}}\) for all times t ∈ [0, T]. This is the reason for which we have used the linear homotopy for the \(\boldsymbol{A}_{n}\) component.
- 3.
The stated continuity property of \(\mathop{\mathrm{grad}}\boldsymbol{ V }\) follows from classical interpolation results; see, e.g., [13, Théorème 2.1].
References
R.A. Adams, J.J. Fournier, Sobolev Spaces. Pure and Applied Mathematics, 2nd edn. (Elsevier/Academic, Amsterdam, 2003)
V.I. Arnol\(^{{\prime}}\) d, Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, 2nd edn. (Springer, New York, 1989)
G.E. Bredon, Topology and Geometry (Springer, New York, 1993)
F.L. Chernousko, Motion of a Rigid Body with Cavities Containing a Viscous Fluid. (NASA Technical Translations, Moscow, 1972)
G.P. Galdi, An Introduction to the Navier-Stokes Initial-Boundary Value Problem. Fundamental Directions in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2000), pp. 1–70
G.P. Galdi, An Introduction to the Mathematical Theory of Navier-Stokes Equations: Steady-State Problems. Springer Monographs in Mathematics, 2nd edn. (Springer, New York, 2011)
G.P. Galdi, A.L. Silvestre, Existence of time-periodic solutions to the Navier-Stokes equations around a moving body. Pac. J. Math. 223, 251–267 (2006)
G.P. Galdi, G. Mazzone, P. Zunino, Inertial motions of a rigid body with a cavity filled with a viscous liquid. C. R. Méc. 341, 760–765 (2013)
J.G. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)
B.G. Karpov, Dynamics of Liquid-Filled Shell: Resonance and Effect of Viscosity. Ballistic Research Laboratories, Report no. 1279 (1965)
N.D. Kopachevsky, S.G. Krein, Operator Approach to Linear Problems of Hydrodynamics, vol. 2: Nonself-Adjoint Problems for Viscous Fluids (Birkhäuser Verlag, Basel/Boston/Berlin, 2000)
O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, revised 2nd edn. (Gordon and Breach Science Publisher, New York, 1969)
J.L. Lions, Espaces intermédiaires entre espaces Hilbertiens et applications. Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine 2, 419–432 (1958)
N.N. Moiseyev, V.V. Rumiantsev, Dynamic Stability of Bodies Containing Fluids (Springer, Berlin, 1968)
G. Prouse, Soluzioni periodiche dell’Equazione di Navier-Stokes. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8, 443–447 (1963)
P.H. Roberts, K. Stewartson, On the motion of a liquid in a spheroidal cavity of a precessing rigid body II. Proc. Camb. Philos. Soc. 61, 279–288 (1965)
W.E. Scott, The Free Flight Stability of Liquid-Filled Shell, Part 1a. Ballistic Research Laboratories, Report no. 120 (1960)
K. Stewartson, P.H. Roberts, On the motion of a liquid in a spheroidal cavity of a precessing rigid body. J. Fluid Mech. 17, 1–20 (1963)
F.S. Van Vleck, A note on the relation between periodic and orthogonal fundamental solutions of linear systems II. Am. Math. Mon. 71, 774–776 (1964)
Acknowledgements
This work is partially supported by NSF grant DMS-1311983.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Basel
About this chapter
Cite this chapter
Galdi, G.P., Mazzone, G., Mohebbi, M. (2016). On the Motion of a Liquid-Filled Rigid Body Subject to a Time-Periodic Torque. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0939-9_13
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0939-9_13
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0938-2
Online ISBN: 978-3-0348-0939-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)