Inertial Motions of a Rigid Body with a cavity filled with a viscous liquid

We study inertial motions of the coupled system, S, constituted by a rigid body containing a cavity that is completely filled with a viscous liquid. We show that for data of arbitrary size (initial kinetic energy and total angular momentum) every weak solution (a la Leray-Hopf) converges, as time goes to infinity, to a uniform rotation, thus proving a famous "conjecture" of Zhukovskii. Moreover we show that, in a wide range of initial data, this rotation must occur along the central axis of inertia of S that has the largest moment of inertia. Furthermore, necessary and sufficient conditions for the rigorous nonlinear stability of permanent rotations are provided, which improve and/or generalize results previously given by other authors under different types of approximation of the original equations and/or suitable symmetry assumptions on the shape of the cavity. Finally, we present a number of 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.Comment: Some of the main results proved in this paper were previously announced in Comptes Rendus Mecanique, Vol. 341, 760--765 (2013

On the Dynamics of a Rigid Body with Cavities Completely Filled by a Viscous Liquid

This thesis deals with the dynamics of a coupled system comprised of a rigid body containing one or more cavities entirely filled with a viscous liquid. We will present a rigorous mathematical analysis of the motions about a fixed point of this system, with special regard to their asymptotic behavior in time. In the case of inertial motions and motions under the action of gravity, we will show that viscous liquids have a stabilizing effect on the motion of the solid. The long-time behavior of the coupled is characterized by a rigid body motion, and in particular a permanent rotation in the case of inertial motions, and the rest state in the case of a liquid-filled heavy pendulum. Some questions about the attainability and stability of the equilibrium configurations are also answered. Furthermore, we will investigate the time-periodic motions performed by the coupled system liquid-filled rigid body when a time-periodic torque is applied on the solid

Nonlinear Stability Analysis of a Spinning Top with an Interior Liquid-Filled Cavity

Consider the motion of the the coupled system, $\mathscr S$, constituted by a (non-necessarily symmetric) top, $\mathscr B$, with an interior cavity, $\mathscr C$, completely filled up with a Navier-Stokes liquid, $\mathscr L$. A particular steady-state motion $\bar{\sf s}$ (say) of $\mathscr S$, is when $\mathscr L$ is at rest with respect to $\mathscr B$, and $\mathscr S$, as a whole rigid body, spins with a constant angular velocity \bar{\V\omega} around a vertical axis passing through its center of mass $G$ in its highest position ({\em upright spinning top}). We then provide a completely characterization of the nonlinear stability of $\bar{\sf s}$ by showing, roughly speaking, that $\bar{\sf s}$ is stable if and only if |\bar{\V\omega}| is sufficiently large, all other physical parameters being fixed. Moreover we show that, unlike the case when $\mathscr C$ is empty, under the above stability conditions, the top will eventually return to the unperturbed upright configuration

Behavior of bivariate interpolation operators at points of discontinuity of the first kind

We introduce an index of convergence for double sequences of real numbers. This index is used to describe the behaviour of some bivariate interpolation sequences at points of discontinuity of the first kind. We consider in particular the case of bivariate Lagrange and Shepard operators

Rough sound waves in 3D3D compressible Euler flow with vorticity

We prove a series of results tied to the regularity and geometry of solutions to the $3D$ compressible Euler equations with vorticity and entropy. Our framework exploits and reveals additional virtues of a recent new formulation of the equations, which decomposed the flow into a geometric "(sound) wave-part" coupled to a "transport-div-curl-part" (transport-part for short), with both parts exhibiting remarkable properties. Our main result is that the time of existence can be controlled in terms of the $H^{2^+}(\mathbb{R}^3)$-norm of the wave-part of the initial data and various Sobolev and H\"{o}lder norms of the transport-part of the initial data, the latter comprising the initial vorticity and entropy. The wave-part regularity assumptions are optimal in the scale of Sobolev spaces: shocks can instantly form if one only assumes a bound for the $H^2(\mathbb{R}^3)$-norm of the wave-part of the initial data. Our proof relies on the assumption that the transport-part of the initial data is more regular than the wave-part, and we show that the additional regularity is propagated by the flow, even though the transport-part of the flow is deeply coupled to the rougher wave-part. To implement our approach, we derive several results of independent interest: i) sharp estimates for the acoustic geometry, i.e., the geometry of sound cones; ii) Strichartz estimates for quasilinear sound waves coupled to vorticity and entropy; and iii) Schauder estimates for the transport-div-curl-part. Compared to previous works on low regularity, the main new features of the paper are that the quasilinear PDE systems under study exhibit multiple speeds of propagation and that elliptic estimates for various components of the fluid are needed, both to avoid loss of regularity and to gain space-time integrability

Tumor hypoxia does not drive differentiation of tumor-associated macrophages but rather fine-tunes the M2-like macrophage population

Tumor-associated macrophages (TAM) are exposed to multiple microenvironmental cues in tumors, which collaborate to endow these cells with protumoral activities. Hypoxia, caused by an imbalance in oxygen supply and demand because of a poorly organized vasculature, is often a prominent feature in solid tumors. However, to what extent tumor hypoxia regulates the TAM phenotype in vivo is unknown. Here, we show that the myeloid infiltrate in mouse lung carcinoma tumors encompasses two morphologically distinct CD11b(hi)F4/80(hi)Ly6C(lo) TAM subsets, designated as MHC-II(lo) and MHC-II(hi) TAM, both of which were derived from tumor-infiltrating Ly6C(hi) monocytes. MHC-II(lo) TAM express higher levels of prototypical M2 markers and reside in more hypoxic regions. Consequently, MHC-II(lo) TAM contain higher mRNA levels for hypoxia-regulated genes than their MHC-II(hi) counterparts. To assess the in vivo role of hypoxia on these TAM features, cancer cells were inoculated in prolyl hydroxylase domain 2 (PHD2)-haplodeficient mice, resulting in better-oxygenated tumors. Interestingly, reduced tumor hypoxia did not alter the relative abundance of TAM subsets nor their M2 marker expression, but specifically lowered hypoxia-sensitive gene expression and angiogenic activity in the MHC-II(lo) TAM subset. The same observation in PHD2(+/+) → PHD2(+/-) bone marrow chimeras also suggests organization of a better-oxygenized microenvironment. Together, our results show that hypoxia is not a major driver of TAM subset differentiation, but rather specifically fine-tunes the phenotype of M2-like MHC-II(lo) TAM