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

Abstract

We study the motion of the coupled system, $\mathcal{S}$, constituted by a physical pendulum, $\mathcal{B}$, with an interior cavity entirely filled with a viscous, compressible fluid, $\mathcal{F}$. The presence of the fluid may strongly affect on the motion of $\mathcal{B}$. In fact, we prove that, under appropriate assumptions, the fluid acts as a damper, namely, $\mathcal{S}$ must eventually reach a rest-state. Such a state is characterized by a suitable time-independent density distribution of $\mathcal{F}$ and a corresponding equilibrium position of the center of mass of $\mathcal{S}$. These results are proved in the very general class of weak solutions and do not require any restriction on the initial data, other than having a finite energy. We complement our findings with some numerical tests. The latter show, among other things, the interesting property that ``large" compressibility favors the damping effect, since it drastically reduces the time that $\mathcal{S}$ takes to go to rest.