Linear damped interfacial wave theory for an orbitally shaken upright circular cylinder

We present a new theoretical model describing gravity-capillary waves in orbitally shaken cylindrical containers. Our model can account for both one-layer free surface and two-layer interfacial wave systems. A set of modal equations for irrotational waves is formulated that we complement with viscous damping rates to incorporate energy dissipation. This approach allows us to calculate explicit formulas for the phase shifts between wave and shaker which are practically important for the mixing efficiency in orbitally shaken bioreactors. Resonance dynamics are described using eight dimensionless numbers revealing a variety of different effects influencing the forced wave amplitudes. As an unexpected result, the model predicts the formation of novel spiral wave patterns resulting from a damping-induced symmetry breaking mechanism. For validation we compare theoretical amplitudes, fluid velocities and phase shifts with three different and independent experiments and - when using the slightly deviating experimental values of the resonance frequencies - find a good agreement within the theoretical limits.