Sub-harmonic parametric instability in nearly-brimful circular-cylinders: a weakly nonlinear analysis

TL;DR

This paper develops a weakly nonlinear analysis to understand how meniscus waves and static contact angle influence the onset of parametric instability in nearly-brimful circular cylinders, extending classical theories to viscous and contact angle effects.

Contribution

It introduces a numerical weakly nonlinear framework that accounts for viscosity and contact angle effects, filling a gap in the theoretical understanding of meniscus wave instabilities.

Findings

Modified Faraday tongues due to meniscus effects

Quantified bifurcation diagram changes

Predicted impact of contact angle on instability onset

Abstract

In lab-scale Faraday experiments, meniscus waves respond harmonically to small-amplitude forcing without threshold, hence potentially cloaking the instability onset of parametric waves. Their suppression can be achieved by resorting to a contact line pinned at the container brim with static contact angle $\theta_s=90^{\circ}$ (brimful condition). However, tunable meniscus waves are desired in some applications as those of liquid-based biosensors, where they can be controlled adjusting the shape of the static meniscus by slightly under/over-filling the vessel ($\theta_s\ne90^{\circ}$) while keeping the contact line fixed at the brim. Here, we refer to this wetting condition as nearly-brimful. Although classic inviscid theories based on Floquet analysis have been reformulated for the case of a pinned contact line (Kidambi 2013), accounting for (i) viscous dissipation and (ii) static contact angle effects, including meniscus waves, makes such analyses practically intractable and a comprehensive theoretical framework is still lacking. Aiming at filling this gap, in this work we formalize a weakly nonlinear analysis via multiple timescale method capable to predict the impact of (i) and (ii) on the instability onset of viscous sub-harmonic standing waves in both brimful and nearly-brimful circular-cylinders. Notwithstanding that the form of the resulting amplitude equation is in fact analogous to that obtained by symmetry arguments (Douady 1990), the normal form coefficients are here computed numerically from first principles, thus allowing us to rationalize and systematically quantify the modifications on the Faraday tongues and on the associated bifurcation diagrams induced by the interaction of meniscus and sub-harmonic parametric waves.