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

V. A. Vladimirov, K. I. Ilin

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 languageEnglish
Pages (from-to)43-75
Number of pages33
JournalJournal of Fluid Mechanics
Volume386
Publication statusPublished - May 10 1999

Fingerprint

Body fluids
rigid structures
dynamical systems
Dynamical systems
Fluids
fluids
two dimensional bodies
ideal fluids
cavities
Io
potential flow
translational motion
body fluids
incompressible fluids
Stability criteria
Angular velocity
variational principles
angular velocity
functionals
infinity

ASJC Scopus subject areas

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

Cite this

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

In: Journal of Fluid Mechanics, Vol. 386, 10.05.1999, p. 43-75.

Research output: Contribution to journalArticle

@article{a704b837a75e4274ab79ab94e0ff9efb,
title = "On the stability of the dynamical system 'rigid body + inviscid fluid'",
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.",
author = "Vladimirov, {V. A.} and Ilin, {K. I.}",
year = "1999",
month = "5",
day = "10",
language = "English",
volume = "386",
pages = "43--75",
journal = "Journal of Fluid Mechanics",
issn = "0022-1120",
publisher = "Cambridge University Press",

}

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.

UR - http://www.scopus.com/inward/record.url?scp=0032627215&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032627215&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032627215

VL - 386

SP - 43

EP - 75

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -