Interfacial fluid instabilities and Kapitsa pendula

TL;DR

This paper links fluid interface instabilities to inverted pendula, demonstrating how high-frequency vibrations can stabilize or control fluid instabilities across various scales using a control theory approach.

Contribution

It introduces a novel analogy transforming fluid interface equations into pendulum models, enabling the application of control theory to fluid instability management.

Findings

Stability can be induced in fluid interfaces via tuned vibrations.

A 'dictionary' relates classical instabilities to pendulum dynamics.

Applications range from microscopic to galactic scales.

Abstract

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, Rayleigh-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.