No abstract

Container motion along a planar circular trajectory at a constant angular velocity, i.e. orbital shaking, is of interest in several industrial applications, e.g. for fermentation processes or in cultivation of stem cells, where good mixing and efficient gas exchange are the main targets. Under these external forcing conditions, the free surface typically exhibits a primary steady-state motion through a single-crest dynamics, whose wave amplitude, as a function of the external forcing parameters, shows a Duffing-like behaviour. However, previous experiments in laboratory-scale cylindrical containers have revealed that, owing to the excitation of super-harmonics, diverse dynamics are observable in certain driving-frequency ranges. Among these super-harmonics, the double-crest dynamics is particularly relevant, as it displays a notably large amplitude response, which is strongly favoured by the spatial structure of the external forcing. In the inviscid limit and with regards to circular cylindrical containers, we formalize here a weakly nonlinear analysis via a multiple-time-scale method of the full hydrodynamic sloshing system, leading to an amplitude equation suitable for describing such a double-crest swirling motion. The weakly nonlinear prediction is shown to be in fairly good agreement with previous experiments described in the literature. Lastly, we discuss how an analogous amplitude equation can be derived by solving asymptotically for the first super-harmonic of the forced Helmholtz–Duffing equation with small nonlinearities.

In labscale 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 adjusting the shape of the static meniscus by slightly underfilling/overfilling the vessel ( $\theta _s\ne 90^{\circ }$ ) 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.

Abstract

We present a physics-inspired mathematical model based on successive linear eigenmode projections to solve the relaxation of small-amplitude and two-dimensional viscous capillary-gravity waves with a phenomenological nonlinear contact line model. We show that each projection eventually induces a rapid loss of total energy in the liquid motion and contributes to its nonlinear damping.