Amplitude equation model for prediction of super-harmonic double-crest wave dynamics in orbital shaken cylindrical containers
Date:
A. Bongarzone (speaker) and F. Gallaire
The 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 single-crest dynamics, whose wave amplitude, as a function of the external forcing parameters, displays a Duffing-like behavior. However, previous experiments in lab-scale cylindrical containers have shown that, owing to the excitation of super-harmonics, diverse dynamics are observable in certain driving-frequency range. Among these super-harmonics, the double-crest wave dynamics is particularly relevant. We formalize here a weakly nonlinear (WNL) analysis via multiple timescale method, leading to amplitude equation suitable to describe such a super-harmonic dynamics. The WNL prediction is shown to be in fairly good agreement with previous experimental measurements. Lastly, we show how an analogous amplitude equation can be formally derived by solving asymptotically for the first super-harmonic of the forced Helmholtz-Duffing equation with small nonlinearities.