Kinetic Description of Multiscale Phenomena

The Annual Kinetic FRG Meeting
May 23-26, 2011

Van Vleck Hall
Department of Mathematics
University of Wisconsin -Madison
Madison, WI 53706

Decay of an oscillating disk in a collision-less gas: A long-memory effect

Prof. Kazuo Aoki

Kyoto University

Abstract:  A circular disk without thickness is placed in a collision-less 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, with special interest in the manner of its decay. 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 (a special type of the Lorentz gas). This work is a collaboration with Tetsuro Tsuji.