The nonlinear Narimanov-Moiseev multimodal equations are used to study the swirling-type resonant sloshing in a circular base container occurring due to an orbital (rotary) tank motion in the horizontal plane with the forcing frequency close to the lowest natural sloshing frequency. An asymptotic steady-state solution is constructed and the response amplitude curves are analyzed to prove their hard-spring type behavior for the finite liquid depth (the mean liquid depth-to-the-radius ratio ℎ > 1). This behavior type is supported by the existing experimental data. The wave elevations at the vertical wall are satisfactorily predicted except for a frequency range where the model test observations reported wave breaking and/or mean rotational flows.

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An upright circular cylindrical rigid tank performs a small-magnitude prescribed periodic horizontal motion, which is described by the two generalized coordinates 0 1 ( ) r t η and 0 2 ( ) r t η ( 0 r is the tank radius) as shown in fig. 1. Those tank motions are relevant for bioreactors [1]. In contrast to industrial containers whose dimensions are relatively large, the bioreactors have 0 5 10 r ≈ − [cm] that requires accounting for the damping associated with a laminar boundary layer and the bulk viscosity.The problem is studied in the nondimensional statement provided by the characteristic size 0 r and time 1/ σ, where σ is the forcing frequency close to the lowest natural sloshing frequency 11 σ . The nondimensional forcing magnitude is small, i.e. ( ) Fig. 1 illustrates the adopted nomenclature. The unknowns, ς and Φ (the velocity potential), are defined in the tank-fixed coordinate system and can be found from either the corresponding free-surface problem or its equivalent variational formulation. Using the Fourier-type representation (in the cylindrical coordinates), ,