An approach for modeling vibration-induced average flows in two-phase systems

High-frequency vibrations applied to a two-phase fluid system generate steady average flows concentrated near the interface. Direct simulation of these flows is expensive because every vibration period must be resolved, and the number of periods per convective time grows with the frequency. We remove this restriction with an averaged description for the level-set method, obtained by applying the method of multiple scales to the diffuse interface. The averaging is carried out across a transition layer where viscous and vibrational effects balance, so the pulsational flow there is viscous and acquires an out-of-phase component. This flow gives rise to a viscous vibrational stress and a Stokes-drift mass flux. Two exact identities recast the system in Lagrangian-mean variables, in which the velocity that solves the momentum equation also advects the interface. The closed system couples the averaged Navier–Stokes equations, driven by an explicit interfacial vibrational force, to a linear quasi-stationary problem for the pulsation amplitudes. The admissible time step is set by the average flow alone, independent of the vibration frequency. The simulation results are compared with analytical solutions for a vibrated drop and bubble, with a time-resolved simulation of the frozen-wave instability, and with an experiment at high viscosity contrast. The model is shown to predict the structure and the intensity of the generated average flows, which can govern heat and mass transfer in multiphase systems under vibrations. Moreover, since these flows dominate transport whenever buoyancy is absent or weak, their modeling is crucial for materials processing in weightlessness.