CHAPTER 2:Non-Wetting, Stabilization, and Phase Transitions Induced by Vibrations and Spatial Patterns

We study several effects caused by small-amplitude fast vibrations and by small patterns in space. Small fast vibrations can be substituted by an effective force, which affects the equilibrium of a mechanical system. This is a stabilizing “vibro-levitation force” in the case of an inverted single or multiple pendulum on a vibrating foundation (the “Kapitza pendulum”) or in the case of a flexible elastic beam or rope (the “Indian rope trick”) where vibrations affect the critical load of the buckling destabilization. A similar effect of vibrations is found in bouncing droplets of a liquid (oil) on a bath of the same liquid, in non-Newtonian liquids (“cornstarch monsters”), and in granular material which effectively “melts” leading to liquid-like behaviour. Thus, small vibrations do not just affect the stability of a mechanical equilibrium, but also cause an effective phase transition. A mathematical technique for studying the effect of small-amplitude fast vibrations is the method of separation of the fast and slow motions. Since there is an isomorphism between vibrations in time and patterned surfaces in space (including the “Kirchhoff analogy”), surface patterns can also affect the phase stability in the case of micropatterned superhydrophobic surfaces, where surface micropattern preserves a vapour phase or delays boiling. The mathematical techniques of studying small non-linear mechanical vibrations provide a tool to investigate various effects of pattern-induced phase transition in liquids and droplets.