Numerical simulation of the crossing of a liquid-liquid interface by a droplet

Numerical simulations of a drop crossing a plane liquid-liquid interface in a centrifugal field are performed using the level-set method. The objective is to characterize the hydrodynamical parameters controlling the coating volume of the droplet, which results from the rupture of the liquid column of the lighter phase entrained by the droplet during the crossing of the interface in the tailing regime. The numerical method is validated first in two-phase flow simulations of a drop rising in a stagnant liquid and then in three-phase flow configurations to reproduce the theoretical critical condition for a drop to cross an interface in static conditions (without initial velocity). Then, in inertial conditions, extensive simulations of crossing droplets are performed in a wide range of flow parameters and phase properties for two types of drop: solidlike droplets (mimicking rigid particles) and deformable drops. The crossing criterion is found to remain very close to that given by the theory in static conditions, for either a spherical or a deformed droplet. For each case studied, the crossing time, the maximum length of the column of liquid pulled by the droplet, and the volume encapsulating the drop after crossing are computed and scaled as a function of an inertia parameter, which is the ratio F* between the inertial stresses pushing on the interface, defined from the minimum drop velocity reached during crossing as the characteristic velocity, and the opposite stress pulling back the entrained column towards the interface. The maximal column length increases with F* (when rescaled by the minimal inertial required for crossing) under two distinct growth rates according to the flow regime in the column. For solidlike drops, the final coating volume is a unique function of F* and grows with F* at large inertia. In the case of deformable droplets, the coating volume evolution can also be well predicted by F* but corrected by the drop-to-film viscosity ratio, which strongly affects the drainage rate of the film along the drop surface during the encapsulation process.