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

Abstract

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

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

“…We then say that the quadruple (ρ, u, ω, ξ) is a renormalized weak solution to (6.1) if, for some 5 (ii) (ρ, u, ω, ξ) satisfies (6.1) 4,5 , (6.2), 5 Here γ * = 3γ 3+γ for γ > 6, γ * = 9γ 5γ−3 for 3 2 < γ ≤ 6, γ * = 3+ for γ = 3 2 , and γ * = γ γ−1 for γ < 3 2 .…”

“…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].…”

“…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 Motion of a Fluid-Filled Rigid Body with Navier Boundary Conditions

“…The mathematical model features a combination of conservative and dissipative properties as can be observed by the conservation of angular momentum (1.3) and the energy inequality (4.5). These are distinctive characteristics for this type of fluid-solid interactions (see also [5,17,8,18]). We prove the existence of weak solutions á la Leray-Hopf to (1.1) corresponding to initial data with arbitrary (finite) kinetic energy.…”

“…In contrast, there is a large literature dealing with the motion of fluid-filled rigid bodies with no-slip boundary conditions, it spans from the early work by Stokes [38], Zhukovskii [42], Hough [11], Poincaré [26], and Sobolev [35] to more recent contributions mostly concerned with stability problems ( [32,33,21,4,15,13,12,34,16,9,5,17]). A comprehensive study of the motion of fluid-filled rigid bodies has recently been given in [8] (in an L 2 framework), and in [18] (in a more general L q framework).…”