Gas-enabled resonance and rectified motion of a piston in a vibrated housing filled with a viscous liquid

Herein, we show how introducing a small amount of gas can completely change the motion of a solid object in a viscous liquid during vibration. We analyze an idealized system exhibiting this behavior: a piston moving in a liquid-filled housing, where the gaps between the piston and the housing are narrow and depend on the piston position. Recent experiments have shown that vibration causes some gas to move below the piston and the piston to subsequently move downward and compress its supporting spring. Herein, we analyze the analogous but simpler situation in which the gas regions are replaced by bellows with similar pressure-volume relationships. We show that these bellows form a spring (analogous to the pneumatic spring formed by the gas regions) which enables the piston and the liquid to oscillate in a mode that does not exist without this spring. This mode is referred to here as the Couette mode because the liquid in the gaps moves essentially in Couette flow (i.e., with almost no component of Poiseuille flow). Since Couette flow by itself produces extremely low damping, the Couette mode has a strong resonance. We show that, near this resonance, the dependence of the gap geometry on the piston position produces a large rectified (net) force on the piston during vibration. As a result, this force can be much larger than the piston weight and the strength of its supporting spring and is in the direction that decreases the flow resistance of the gap geometry.