Linear theory of Faraday instability in viscous liquids

Abstract

The linear stability of the plane free surface of a viscous liquid on a horizontal plate under vertical sinusoidal oscillation is analysed theoretically. The free surface of a laterally unbounded liquid of any depth h may always be excited to standing waves if the external acceleration is raised above a critical value a$\_{\text{c}}$. For a fixed external frequency $\omega $, solutions are possible only within certain bands of wave numbers k for a given forcing amplitude above a$\_{\text{c}}$, that is, within tongue-like stability zones in the a-k plane. The analysis for a shallow layer of viscous fluids shows new qualitative behaviours compared to the nearly inviscid theory. It predicts a series of bicritical points, where both harmonic and subharmonic solutions exist for the same forcing amplitude and forcing frequency. This makes harmonic solutions possible at the onset in a laterally large container, which is qualitatively different from the results of nearly inviscid theory. For a low viscosity fluid of small depths, the damping coefficient may be considered proportional to ($\nu \omega $)$^{1/2}$/h in contrast to $\nu \kappa ^{2}$ predicted by the nearly inviscid theory. An approximate analytic expression is derived for the lower part of the lowest marginal curve in cases when the depth of the liquid is much larger than the thickness of the viscous boundary layer formed at the bottom plate. This approximate threshold agrees well with that of recent experiments with viscous liquids.