Subharmonic parametric instability in nearly brimful circular cylinders: a weakly nonlinear analysis

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 imposing 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 by adjusting the shape of the static meniscus by slightly underfilling/overfilling 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, J. Fluid Mech., vol. 724, 2013, pp. 671–694), 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 time scale method capable of predicting the impact of (i) and (ii) on the instability onset of viscous subharmonic 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, J. Fluid Mech., vol. 221, 1990, pp. 383–409), 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 subharmonic parametric waves.

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Caption: Plot of WNL (black) versus DNS (red) above Faraday threshold (within the unstable Faraday tongue) for $\Omega_d/\omega^{45^{\circ}}=1.0054$ and $F_d =0.675$ $m s^{−2}$ (see Fig. 7 of the paper). (a–c) Comparison in terms of free surface reconstruction for three different time instants: (a) when the centreline elevation is maximum; (b) when it is zero and equal to the static meniscus position; and (c) when it is minimum. For completeness, the shape of the static meniscus for $\theta_s=45^{\circ}$ is reported as a black dotted line. (d–f ) Full three-dimensional visualization extracted from the DNS. (g) Centreline elevation versus time associated with (a–c). Here $t_0$ is an arbitrary time instant. The constant value of the static meniscus elevation at $r=0$ is shown as a black dotted line. (h) Frequency spectrum computed from the time series shown in (g): PSD versus the dimensional oscillation frequency of the system response.