### Abstract

In this paper we study a dynamical system consisting of a rigid body and an inviscid incompressible fluid. Two general configurions of the system are considered: (a) a rigid body with a cavity completely filled with a fluid and (b) a rigid body surrounded by a fluid. In the first case the fluid is confined to an interior (for the body) domain and in the second case it occupies an exterior domain, which may, in turn, be bounded by some fixed rigid boundary or may extend Io infinity. The aim of the paper is twofold: (i) to develop Arnold's technique for the system 'body + fluid' and (ii) to obtain sufficient conditions for the stability of steady states of the system. We first establish an energy-type variational principle for an arbitrary steady state of the system. Then we generalize this principle for states that are steady either in translationally moving in some fixed direction or rotating around some fixed axis coordinate system. The second variations of the corresponding functionals are calculated. The general results are applied to a number of particular stability problems. The first is the stability of a steady translational motion of a two-dimensional body in an irrotational flow. Here we have found that (for a quite wide class of bodies) the presence of non-zero circulation about the body does not affect its stability - a result that seems to be new. The second problem concerns the stability of a steady rotation of a force-free rigid body with a cavity containing an ideal fluid. Here we rediscover the stability criterion of Rumyantsev (see Moiseev & Rumyantsev 1965). The complementary problem - when a body is surrounded by a fluid and both body and fluid rotate with constant angular velocity around a fixed axis passing through the centre of mass of the body - is also considered and the corresponding sufficient conditions for stability are obtained.

Original language | English |
---|---|

Pages (from-to) | 43-75 |

Number of pages | 33 |

Journal | Journal of Fluid Mechanics |

Volume | 386 |

Publication status | Published - May 10 1999 |

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### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials
- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

*Journal of Fluid Mechanics*,

*386*, 43-75.

**On the stability of the dynamical system 'rigid body + inviscid fluid'.** / Vladimirov, V. A.; Ilin, K. I.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 386, pp. 43-75.

}

TY - JOUR

T1 - On the stability of the dynamical system 'rigid body + inviscid fluid'

AU - Vladimirov, V. A.

AU - Ilin, K. I.

PY - 1999/5/10

Y1 - 1999/5/10

N2 - In this paper we study a dynamical system consisting of a rigid body and an inviscid incompressible fluid. Two general configurions of the system are considered: (a) a rigid body with a cavity completely filled with a fluid and (b) a rigid body surrounded by a fluid. In the first case the fluid is confined to an interior (for the body) domain and in the second case it occupies an exterior domain, which may, in turn, be bounded by some fixed rigid boundary or may extend Io infinity. The aim of the paper is twofold: (i) to develop Arnold's technique for the system 'body + fluid' and (ii) to obtain sufficient conditions for the stability of steady states of the system. We first establish an energy-type variational principle for an arbitrary steady state of the system. Then we generalize this principle for states that are steady either in translationally moving in some fixed direction or rotating around some fixed axis coordinate system. The second variations of the corresponding functionals are calculated. The general results are applied to a number of particular stability problems. The first is the stability of a steady translational motion of a two-dimensional body in an irrotational flow. Here we have found that (for a quite wide class of bodies) the presence of non-zero circulation about the body does not affect its stability - a result that seems to be new. The second problem concerns the stability of a steady rotation of a force-free rigid body with a cavity containing an ideal fluid. Here we rediscover the stability criterion of Rumyantsev (see Moiseev & Rumyantsev 1965). The complementary problem - when a body is surrounded by a fluid and both body and fluid rotate with constant angular velocity around a fixed axis passing through the centre of mass of the body - is also considered and the corresponding sufficient conditions for stability are obtained.

AB - In this paper we study a dynamical system consisting of a rigid body and an inviscid incompressible fluid. Two general configurions of the system are considered: (a) a rigid body with a cavity completely filled with a fluid and (b) a rigid body surrounded by a fluid. In the first case the fluid is confined to an interior (for the body) domain and in the second case it occupies an exterior domain, which may, in turn, be bounded by some fixed rigid boundary or may extend Io infinity. The aim of the paper is twofold: (i) to develop Arnold's technique for the system 'body + fluid' and (ii) to obtain sufficient conditions for the stability of steady states of the system. We first establish an energy-type variational principle for an arbitrary steady state of the system. Then we generalize this principle for states that are steady either in translationally moving in some fixed direction or rotating around some fixed axis coordinate system. The second variations of the corresponding functionals are calculated. The general results are applied to a number of particular stability problems. The first is the stability of a steady translational motion of a two-dimensional body in an irrotational flow. Here we have found that (for a quite wide class of bodies) the presence of non-zero circulation about the body does not affect its stability - a result that seems to be new. The second problem concerns the stability of a steady rotation of a force-free rigid body with a cavity containing an ideal fluid. Here we rediscover the stability criterion of Rumyantsev (see Moiseev & Rumyantsev 1965). The complementary problem - when a body is surrounded by a fluid and both body and fluid rotate with constant angular velocity around a fixed axis passing through the centre of mass of the body - is also considered and the corresponding sufficient conditions for stability are obtained.

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M3 - Article

VL - 386

SP - 43

EP - 75

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -