Decay of an oscillating disk in a gas:
Case of a collision-less gas and a special Lorentz gas
Kazuo Aoki
Department of Mechanical Engineering and Science
Kyoto University, Kyoto 606-8501, Japan
A circular disk without thickness is placed in a gas, and an external force, obeying
Hooke's law, is acting perpendicularly on the disk. If the disk is displaced
perpendicularly from its equilibrium position and released, then it starts an oscillatory
motion, which decays as time goes on because of the drag exerted by the gas molecules.
This unsteady motion is investigated numerically, under the diffuse reflection condition,
for two kinds of gases, i.e., Case (A): a collision-less gas, and Case (B): a special Lorentz
gas. Special attention is focused on the manner of the decay. In case (A), it is shown that
the decay of the displacement of the disk is slow and is in proportion to an inverse power
of time. The result complements the existing mathematical study in the case of
non-oscillatory decay [S. Caprino, et al., Math. Models. Meth. Appl. Sci. 17, 1369 (2007)].
The slow decay is caused by the long-memory effect produced by the multiple collisions
of gas molecules with the disk. In fact, an exponential decay is observed if the
long-memory effect is destroyed by introducing an interaction of gas molecules with a
background [Case (B)]. This work is a collaboration with Tetsuro Tsuji.