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

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