Well posedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid

Cumsille, Patricio; Takahashi, Takéo

doi:10.1007/s10587-008-0063-2

Cited by 61 publications

(70 citation statements)

References 16 publications

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“…Well-posedness results in which the structure was assumed to be a rigid body immersed in a fluid, or described by a finite number of modal functions, were studied in [5,16,19,20,21,25,26,46]. FSI problems coupling the Navier-Stokes equations with linear elasticity where the coupling was calculated at a fixed fluid domain boundary, were considered in [23], and in [2,3,35] where an additional nonlinear coupling term was added at the interface.…”

Section: A Brief Literature Reviewmentioning

confidence: 99%

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

Muha

Čanić

2013

Communications in Information and Systems

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We study a 3D fluid-structure interaction (FSI) problem between an incompressible, viscous fluid modeled by the Navier-Stokes equations, and the motion of an elastic structure, modeled by the linearly elastic cylindrical Koiter shell equations, allowing structure displacements that are not necessarily radially symmetric. The problem is set on a cylindrical domain in 3D, and is driven by the time-dependent inlet and outlet dynamic pressure data. The coupling between the fluid and the structure is fully nonlinear (2-way coupling), giving rise to a nonlinear, moving-boundary problem in 3D. We prove the existence of a weak solution to this 3D FSI problem by using an operator splitting approach in combination with the Arbitrary Lagrangian Eulerian mapping, which satisfies a geometric conservation law property. We effectively prove that the resulting computational scheme converges to a weak solution of the full, nonlinear 3D FSI problem.

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Section: A Brief Literature Reviewmentioning

confidence: 99%

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

Muha

Čanić

2013

Communications in Information and Systems

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“…We refer to [5] for a survey of this topic, and to [7], [4], [9] and [23] for additional existence and regularity theory. In the absence of external forces, this system is dissipative and both body and fluid must approach the rest state [3], but under the influence of external forces like gravity, many questions regarding asymptotics are still open.…”

Section: Introductionmentioning

confidence: 99%

Inertial Motions of a Rigid Body with a Cavity Filled with a Viscous Liquid

Disser

Galdi

Mazzone

et al. 2016

Arch Rational Mech Anal

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We study inertial motions of the coupled system, (Formula presented.) , constituted by a rigid body containing a cavity entirely filled with a viscous liquid. We show that for arbitrary initial data having only finite kinetic energy, every corresponding weak solution (à la Leray–Hopf) converges, as time goes to infinity, to a uniform rotation, unless two central moments of inertia of (Formula presented.) coincide and are strictly greater than the third one. This corroborates a famous “conjecture” of N.Ye. Zhukovskii in several physically relevant cases. Moreover, we show that, in a known range of initial data, this rotation may only occur along the central axis of inertia of (Formula presented.) with the larger moment of inertia. We also provide necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations, which improve and/or generalize results previously given by other authors under different types of approximation. Finally, we present results obtained by a targeted numerical simulation that, on the one hand, complement the analytical findings, whereas, on the other hand, point out new features that the analysis is yet not able to catch, and, as such, lay the foundation for interesting and challenging future investigation

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“…Early references addressing these difficulties are Conca, San Martín and Tucsnak [1], Desjardins and Esteban [3] and Hoffman and Starovoitov [10]. The case = R 3 has been considered in Galdi and Silvestre [7] and by Cumsille and Takahashi [2]. The global existence and uniqueness of strong solutions have been proved for sufficiently small v 0 in Takahashi [17].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In order to prove the global existence result in Theorem 1.1 we need the following result, which can be proved by slightly adapting the proof of Lemma 4.3 in [2]. …”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Stabilization of a fluid–rigid body system

Takahashi

Tucsnak

2015

Journal of Differential Equations

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We consider the mathematical model of a rigid ball moving in a viscous incompressible fluid occupying a bounded domain , with an external force acting on the ball. We investigate in particular the case when the external force is what would be produced by a spring and a damper connecting the center of the ball h to a fixed point h 1 ∈ . If the initial fluid velocity is sufficiently small, and the initial h is sufficiently close to h 1 , then we prove the existence and uniqueness of global (in time) solutions for the model. Moreover, in this case, we show that h converges to h 1 , and all the velocities (of the fluid and of the ball) converge to zero. Based on this result, we derive a control law that will bring the ball asymptotically to the desired position h 1 even if the initial value of h is far from h 1 , and the path leading to h 1 is winding and complicated. Now, the idea is to use the force as described above, with one end of the spring and damper at h, while other end is jumping between a finite number of points in , that depend on h (a switching feedback law).

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Well posedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid

Cited by 61 publications

References 16 publications

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

Inertial Motions of a Rigid Body with a Cavity Filled with a Viscous Liquid

Stabilization of a fluid–rigid body system

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