Microorganism billiards in closed plane curves


Recent experiments have shown that many species of microorganisms leave a solid surface at a fixed angle determined by steric interactions and near-field hydrodynamics. This angle is completely independent of the incoming angle. For several collisions in a closed body this determines a unique type of billiard system, an aspecular billiard in which the outgoing angle is fixed for all collisions. We analyze such a system using numerical simulation of this billiard for varying tables and outgoing angles, and also utilize the theory of one-dimensional maps and wavefront dynamics. When applicable we cite results from and compare our system to similar billiard systems in the literature. We focus on examples from three broad classes: the ellipse, the Bunimovich billiards, and the Sinai billiards. The effect of a noisy outgoing angle is also discussed.

Madison S. Krieger, “Microorganism billiards in closed plane curves”,  arXiv, PDF

Interfacial instabilities and Kapitsa pendula


The onset and development of instabilities is one of the central problems in fluid mechanics. Here we develop a connection between instabilities of free fluid interfaces and inverted pendula. When acted upon solely by the gravitational force, the inverted pendulum is unstable. This position can be stabilised by the Kapitsa phenomenon, in which high-frequency low-amplitude vertical vibrations of the base creates a fictitious force which opposes the gravitational force. By transforming the dynamical equations governing a fluid interface into an appropriate pendulum, we demonstrate how stability can be induced in fluid systems by properly tuned vibrations. We construct a “dictionary”-type relationship between various pendula and the classical Rayleigh-Taylor, Kelvin-Helmholtz, Plateau and the self-gravitational instabilities. This makes several results in control theory and dynamical systems directly applicable to the study of “tunable” fluid instabilities, where the critical wavelength depends on the external forces or the instability is suppressed entirely. We suggest some applications and instances of the effect ranging in scale from microns to the radius of a galaxy.

Madison S. Krieger, “Interfacial instabilities and Kapitsa pendula”,  arXiv, PDF