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The container motion along a planar circular trajectory at a constant angularvelocity, i.e. orbital shaking, is of interest in several industrialapplications, e.g. for fermentation processes or in cultivation of stem cells,where good mixing and efficient gas exchange are the main targets. Under theseexternal forcing conditions, the free surface typically exhibits a primarysteady state motion through a single-crest dynamics, whose wave amplitude, as afunction of the external forcing parameters, shows a Duffing-like behaviour.However, previous experiments in lab-scale cylindrical containers have unveiledthat, owing to the excitation of super-harmonics, diverse dynamics areobservable in certain driving-frequency ranges. Among these super-harmonics,the double-crest dynamics is particularly relevant, as it displays a notablylarge amplitude response, that is strongly favored by the spatial structure ofthe external forcing. In the inviscid limit and with regards to circularcylindrical containers, we formalize here a weakly nonlinear analysis viamultiple timescale method of the full hydrodynamic sloshing system, leading toan amplitude equation suitable to describe such a super-harmonic dynamics andthe resulting single-to-double crest wave transition. The weakly nonlinearprediction is shown to be in fairly good agreement with previous experimentsdescribed in the literature. Lastly, we discuss how an analogous amplitudeequation can be derived by solving asymptotically for the first super-harmonicof the forced Helmholtz-Duffing equation with small nonlinearities.

An amplitude equation modeling the single-to-double crest wave transition in orbital shaken cylindrical containers