Inertial motions of a rigid body with a cavity filled with a viscous liquid

Galdi, Giovanni P.; Mazzone, Giusy; Zunino, Paolo

doi:10.1016/j.crme.2013.10.001

Cited by 14 publications

(18 citation statements)

References 21 publications

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“…A more complete description of the long-time behavior of S has been given in [6,3,15,5]. It has been proved that each Leray-Hopf solution, as t → ∞, must converge (in a proper topology) to a manifold {v ≡ 0} × A, where A is a compact, connected subset of R 3 constituted by vectors (angular velocities) whose magnitude is compatible with conservation of total angular momentum.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

A Maximal Regularity Approach to the Study of Motion of a Rigid Body with a Fluid-Filled Cavity

Mazzone

Prüß

Simonett

2019

J. Math. Fluid Mech.

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We consider the inertial motion of a rigid body with an interior cavity that is completely filled with a viscous incompressible fluid. The equilibria of the system are characterized and their stability properties are analyzed. It is shown that equilibria associated with the largest moment of inertia are normally stable, while all other equilibria are normally hyperbolic.We show that every Leray-Hopf weak solution converges to an equilibrium at an exponential rate. In addition, we determine the critical spaces for the governing evolution equation, and we demonstrate how parabolic regularization in time-weighted spaces affords great flexibility in establishing regularity of solutions and their convergence to equilibria.2010 Mathematics Subject Classification. Primary: 35Q35, 35Q30, 35B40, 35K58, 76D05.

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Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

A Maximal Regularity Approach to the Study of Motion of a Rigid Body with a Fluid-Filled Cavity

Mazzone

Prüß

Simonett

2019

J. Math. Fluid Mech.

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“…The stabilizing mechanism is the viscosity of the fluid which, due to the assumptions on the initial data and to the coupling at the interface, suffices to stabilize both the fluid and the solid. The coupling of Navier-Stokes equations with rigid body dynamics has been investigated in an "inverted" context in Mazzone et al [14,6], where the rigid body has a cavity filled with a Navier-Stokes fluid. In the situation studied in [14,6] there is no free boundary but, as in our case, it is shown that the viscosity of a fluid stabilizes both the fluid and the solid to one of a finite number of equilibrium states.…”

Section: Discussionmentioning

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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“…Their main achievement was to show that, under suitable hypotheses on the "mass distribution" and for "small" Mach numbers, the system will eventually tend to a steady-state characterized by a rigid, uniform rotation around one of the central axes of inertia [10]. However, unlike [11,5,9], the analysis in [10] is carried out in the class of strong solutions, whose existence is established for initial data that are smooth enough and Here r, p and w are fluid density, pressure and velocity fields, ̟ is the angular velocity of B, and η the velocity of its center of mass C. Moreover, y C denotes the vector position of C, while…”

Section: Introductionmentioning

confidence: 99%

On weak solutions to the problem of a rigid body with a cavity filled with a compressible fluid, and their asymptotic behavior

Galdi

Mácha

Nečasová

2020

International Journal of Non-Linear Mechanics

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We prove the existence of a weak solution to the equations describing the inertial motions of a coupled system constituted by a rigid body containing a viscous compressible fluid. We then provide a weak-strong uniqueness result that allows us to completely characterize, under certain physical assumptions, the asymptotic behavior in time of the weak solution corresponding to smooth data of restricted "size," and show that it tends to a uniquely determined steady-state. 2"small" in size. As explained in the introductory section in [10] the analogous study in the larger class of weak solution corresponding to data possessing just finite energy appears, currently, out of reach. The main reason is ascribed to the fact that a necessarily detailed study of the Ω-limit requires that weak solutions become eventually smooth enough, a property that, while true in the incompressible case [5], need not be generally valid if the fluid is compressible [16].This drawback notwithstanding, one may wonder if, at least, a weak solution with smooth and sufficiently "small" initial data could satisfy the above property. The latter is certainly true, if one can show a weak-strong uniqueness result, which would, in particular, ensure that the weak solution is strong for all positive times.Objective of this article is to prove that this is, in fact, the case. More precisely, we show that, for generic initial data with finite energy, there corresponds at least one weak solution (see Theorem 3.2) suitably defined (see Definition 3). Furthermore, we show that if there is a strong solution (also suitably defined) possessing the same data, then necessarily it should coincide with the weak one (see Theorem 5.2). In view of this result, we may then conclude that, for smooth and "small" initial data, the weak solution must coincide with that constructed and used in [10], and this will ensure that, as established in [10], under the assumption of suitable "mass distribution" and "small" Mach numbers, the ultimate state reached by the weak solution must be a uniform rotation around a central axis of inertia.The plan of the paper is the following. In Section 2 we give a formulation of the problem and recall the relevant equations. Section 3 is dedicated to the definition of weak solutions and the proof of their existence (Theorem 3.2). In achieving the latter, we appropriately adjust to the case at hand an approximation procedure developed in [23,6]. Successively, in Section 4, we state results of local and global existence and uniqueness of strong solutions proved in [10]; see Theorems 4.1 and 4.2. The following Section 5 is devoted to the proof that the weak and the strong solution, dealt with before, corresponding to the same data must, in fact, coincide; see Theorem 5.2. In the proof of this result, we adapt the arguments of [7,18,2]; see also [18]. Once the weak-strong coincidence has been established, we dedicate the last two sections to recall and formulate the results proved in [10] concerning the asymptotic behavior of a strong solution for...

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Inertial motions of a rigid body with a cavity filled with a viscous liquid

Cited by 14 publications

References 21 publications

A Maximal Regularity Approach to the Study of Motion of a Rigid Body with a Fluid-Filled Cavity

A Maximal Regularity Approach to the Study of Motion of a Rigid Body with a Fluid-Filled Cavity

Stabilization of a fluid–rigid body system

On weak solutions to the problem of a rigid body with a cavity filled with a compressible fluid, and their asymptotic behavior

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