An amplitude equation modelling the double-crest swirling in orbital-shaken cylindrical containers

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 the 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.

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Caption: (a,c,e,g) Comparison of the dimensionless and phase-averaged wave height measured at the wall $\tilde{\delta}\left(\theta,\pi/\Omega\right)$ (black circles) with the straightforward asymptotic solution rebuilt via (3.14) (grey solid line) and the WNL solution for the DC wave (4.25). Panels correspond to $\tilde{H}=0.52$, $\tilde{d}_s=0.11$ and $D=0.144$ $m$. The experimental measurements, here shown as black circles, are available in Reclari (2013), except for those of (e), which are provided in Reclari et al. (2014). Note that (c) the nonlinear prediction has a very large amplitude. (b,d,f,h) Corresponding three-dimensional free-surface deformation, $\eta\left(r,\theta,\pi/\Omega\right)$, reconstructed via Eq. 4.25 of the paper. The transition from SC to DC swirling via hardening nonlinearity is clearly visible moving from top to bottom, i.e. for increasing frequency.