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

TL;DR

This paper analyzes the long-term behavior of a rigid body with an internal viscous compressible fluid, showing that under certain conditions, the system stabilizes to a steady state with the fluid at rest and the body rotating uniformly.

Contribution

It provides a rigorous characterization of the asymptotic dynamics of a coupled fluid-body system with compressible fluid and no external forces.

Findings

System tends to a steady state with fluid at rest relative to the body.

The entire system performs a uniform rotation around an axis.

Results hold under small Mach number and initial data assumptions.

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.