Decay of a Linear Pendulum in a Free-Molecular Gas and in a Special Lorentz Gas

Abstract

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 or non-oscillatory unsteady motion, which decays as time goes on because of the drag exerted by the gas molecules. This unsteady motion, i.e., the decay of this linear pendulum, is investigated numerically, under the diffuse reflection condition on the surface of the disk, with special interest in the manner of its decay, for two kinds of gases: one is a collisionless gas (or Knudsen gas) and the other is a special Lorentz gas interacting with a background. It is shown that the decay of the displacement of the disk is slow and is in proportion to an inverse power of time for the collisionless gas. The result complements the existing mathematical study of a similar problem (Caprino et al. in Math. Models Methods Appl. Sci. 17:1369–1403, 2007) in the case of non-oscillatory decay. It is also shown that the manner of the decay changes significantly for the special Lorentz gas.