On the Motion of a Body with a Cavity Filled with Compressible Fluid

Abstract

We study the motion of the system, \({\mathcal S}\), constituted by a rigid body, \({\mathcal{B}}\), containing in its interior a viscous compressible fluid, and moving in absence of external forces. Our main objective is to characterize the long time behavior of the coupled system body-fluid. Under suitable assumptions on the “mass distribution” of \({\mathcal{S}}\), and for a sufficiently “small” Mach number and initial data, we show that every corresponding motion (in a suitable regularity class) must tend to a steady state where the fluid is at rest with respect to \({\mathscr{B}}\). Moreover, \({\mathcal{S}}\), as a whole, performs a uniform rotation around an axis parallel to the (constant) angular momentum of \({\mathcal{S}}\), and passing through its center of mass.

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Acknowledgements

Part of this work was carried out when G.P. Galdi was tenured with the Eduard Čech Distinguished Professorship at the Mathematical Institute of the Czech Academy of Sciences in Prague. His work is also partially supported by NSF Grant DMS- 1614011, and the Mathematical Institute of the Czech Academy of Sciences (RVO67985840). The research of V. Mácha is supported by GAČR Project P201-16-032308 and RVO 67985840, and that of Š.Nečasová by GAČR Project P201-16-032308 and RVO67985840.

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Correspondence to Šárka Nečasová.

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Communicated by C. Dafermos

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Galdi, G.P., Mácha, V. & Nečasová, Š. On the Motion of a Body with a Cavity Filled with Compressible Fluid. Arch Rational Mech Anal 232, 1649–1683 (2019). https://doi.org/10.1007/s00205-018-01351-8

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