Abstract

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 is 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.

Highlights

  • Orbital sloshing experiments provide the opportunity to study a broad range of physical effects in the field of fluid mechanics with moderate effort

  • In the paper at hand we develop a new damped sloshing model accounting for both gravity–capillary free-surface and interfacial waves in cylindrical containers

  • We look at the dependence of the resonance dynamics on the different dimensionless numbers (2.1), whereby we focus on the first peak, most relevant in practice

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Summary

Introduction

Orbital sloshing experiments provide the opportunity to study a broad range of physical effects in the field of fluid mechanics with moderate effort. Moisy, Bouvard & Herreman (2018) have investigated the mean flow under the influence of a thin layer of foam and explained the formation of a counter-rotating flow Beside these fundamental approaches, there is currently ongoing research on different flow properties of practical relevance for the design and optimisation of OSBs: wall shear stresses, volumetric mass transfer, mixing times or wave breaking regimes (Discacciati et al 2013; Rodriguez et al 2013; Filipovic et al 2016; Pieralisi et al 2016; Thomas et al 2017; Alpresa et al 2018a,b; Rodriguez, Micheletti & Ducci 2018; Weheliye et al 2018; Zhu et al 2018). A prediction for the nonlinear out-of-phase transition is derived by quantifying the waviness of the free surface

Theory
Statement of the problem
Modal equations and solutions
Viscous damping
R sinh
Dimensionless prediction of resonance dynamics
Limits of applicability
Numerical results
Comparisons with experiments
Concluding remarks
Full Text
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