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

Abstract

We study the motion of the system, S, constituted by a rigid body, 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 S, and for 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 B. Moreover, S, as a whole, performs a uniform rotation around an axis parallel to the (constant) angular momentum of S, and passing through its center of mass.

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

“…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.…”

“…More recently, the present authors have started to investigate the problem of a rigid body with a fluid-filled interior cavity by relaxing the assumption of incompressibility and, as in [11,5,9], performed their analysis in the case of inertial motions. 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…”

“…On the one hand, we are interested in showing that the above equations possess a weak solution (suitably defined) for initial data of unrestricted size; on the other hand, we want to determine their asymptotic behavior in time with a possible characterization of the associated Ω-limit. The latter will be achieved by reviewing the recent results proved by the authors in [10].…”

“…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 large times, which, by what we proved, must coincide with that of the weak solution corresponding to the same initial data. Thus, after introducing and characterizing the class of renormalized weak steady-state solutions in Section 6, in the last Section 7 we enunciate the main result (proved in [10]) that states that, under suitable assumptions on the mass distribution and for small Mach numbers, every weak solution with sufficiently smooth initial data of restricted size must converge, for large times, to a uniquely determined steady-state where the coupled system uniformly rotates as a whole rigid body around a central axis of inertia.…”

“…Under these circumstances, the center of mass G of the coupled system body-liquid, S, will move by uniform and rectilinear motion with respect to the inertial frame I. Thus, denoting by F the inertial frame with origin in G and axes parallel to those of I, the governing equations of S in F are given by [10,17,19,20]: r (∂ t w + w · ∇w) = div T (w, p) ∂ t r + div(rw) = 0 (y, t) ∈ ∪ t>0 C(t) × {t} , w = ̟(t) × (y − y C ) + η(t) , (y, t) ∈ ∪ t>0 ∂C(t) × {t} , d dt (J C · ̟) = − ∂C(t) (y − y C ) × T (w, p(r)) · N m B η(t) = − C(t) r w    t ∈ (0, ∞) .(2.1)…”