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.

Figure 1

Scheme of the studied three-phase system in a centrifuged cell ($\omega$ is the angular speed): phase 1, lighter (organic) continuous phase; phase 2, droplet (aqueous); and phase 3, heavier (aqueous) continuous phase.

Figure 2

Reynolds-number evolution as a function of normalized distanced covered by the droplet $z^{*}=\frac{z}{d}$ for different mesh resolutions for (a) case 1, ${\text{Re}}_T=20$, ${\text{We}}_{12}=0.11$, and simulation domain $6R\times 12R$ and (b) case 2, ${\text{Re}}_T=180$, ${\text{We}}_{12}=0.31$, and simulation domain $8R\times 16R$.

Figure 3

Simulation of droplets with different values of ${\lambda }_{12}$: aspect ratio as a function of the Weber number ${\text{We}}_{12}$, including numerical results obtained with the same numerical tool from Lalanne et al. [32], experimental results from Wellek et al. [33], and numerical results from Bäumler et al. [34] with another simulation tool.

Figure 4

Mapping of the crossing configuration for solidlike droplets. The red solid line corresponds to the expression (8) and the green solid line corresponds to the expression (9), with an assumed red dashed line that connects the two theoretical predictions. Closed squares correspond to a success in crossing, while open squares are associated with a fail in crossing, in the static case.

Figure 5

Diagram showing the crossing and no-crossing zones as a function of static theory predictions (solid lines), Eqs. (8) and (9). The squares and circles correspond to simulations of solidlike droplets and deformable droplets, respectively, in the dynamic case, with the points labeled by the Archimedes number Ar. Symbols are closed in the case of crossing in the simulation and open otherwise.

Figure 6

Evolution of the Reynolds number $\text{Re}=\frac{{\rho }_{1}Ud}{{\mu }_1}$ ($U$ is the instantaneous drop velocity) as a function of $z^{*}$ (drop center of mass position normalized by $R$), for cases $S_1$ and $S_2$, along with screenshots of the phase indicator function field issued from the simulations. Here $z^{*}=0$ is the interface position. Figures appearing on the curves of the Reynolds number time signal are related to the image sequences; dimensionless time is indicated below the screenshots, using the characteristic timescale $\sqrt{\frac{d}{{\omega }^2{r}_i}}$.

Figure 7

Evolution of the Reynolds number Re and aspect ratio $\chi$ as a function of $z^{*}=z/R$ for cases $D_1$ and $D_2$. Dashed lines correspond to the two-phase flow simulations (without the interface), used as a comparison basis for the three-phase flow case.

Figure 8

Evolution of the Reynolds number as a function of $z^{*}=z/R$ for two cases of deformable droplets $D_1$ and $D_2$ and the case $S_1$ of a solidlike droplet at the same Ar and ${\text{Bo}}_{13}$, along with screenshots of the simulations. The points indicated on the Reynolds number curve are relative to the images shown; dimensionless time is indicated below the screenshots, using the characteristic timescale $\sqrt{\frac{d}{{\omega }^2{r}_i}}$.

Figure 9

From left to right, velocity field of cases $S_1$, $D_1$, and $D_2$ right before the column detachment. The scaling velocity vector is indicated on the rightmost image. Here $x^{*}$ and $z^{*}$ are the coordinates of the calculation domain normalized by the radius of the droplet. In all cases, the viscosity of the two continuous phases is the same (${\lambda }_{13}=1$).

Figure 10

Illustration of the crossing time criterion.

Figure 11

Crossing time $t_{cr}^{*}$ as a function of $2d/(U_{max}+U_{min})$ for both solidlike and deformable droplets (dimensional physical parameters are given in Table 1 for all cases).

Figure 12

Mapping of nondimensional maximal column length $L_{max}^{*}=L_{max}/d$ as a function of the nondimensional inertia excess $I_{ex}$ for both nondeformable and deformable droplets (see Fig. 6 for the definition of $L_{max}$). According to ${\text{Re}}_{\mathrm{col}}$, two regimes of increase of $L_{max}^{*}$ are observed. Based on mesh convergence tests, the uncertainty of the reported values of $L_{max}^{*}$ after column breakup is about 5%.

Figure 13

Dimensionless film volume $V_f^{*}=V_f/V_{\mathrm{drop}}$ as a function of the parameter $F^{*}$ in the case of solidlike droplets and as a function of $F^{*}{\lambda }_{12}^{0.4}$ in the case of deformable droplets. Error bars have been computed according to the criterion discussed in Sec. 2b. A series of screenshots of the phase indicator function at the instant of column detachment is presented for some cases.

Figure 14

Profile of axial velocity (normalized by the instantaneous drop velocity) along the drop equator ($z^{*}=0$), for case $S_9$, in the droplet frame. The covering film (phase 1) is still described by four mesh points at this instant.

Figure 15

Evolution of the Reynolds number $\text{Re}=\frac{{\rho }_{1}Ud}{{\mu }_1}$ as a function of the centroid position for case $D_{13}$. The droplet deforms into a prolate shape when decelerating and then accelerating in phase 3 (${\lambda }_{13}=0.1$). Screenshots correspond to the different positions indicated on the red curve with corresponding values of the aspect ratio.

Figure 16

Kinetic energy variation ratio $\mathrm{\Delta }{E}_k^{*}$ as a function of inertia excess $I_{\mathrm{ex}}$. Orange squares denote solidlike droplets, blue circles denote deformable droplets, and the dashed line is a double exponential fit of all points.

Figure 17

Archimedes number as a function of $\frac{3}{2}\sqrt{{\text{Re}}_{max}(1+0.15{\text{Re}}_{max}^{0.687})}$ for solidlike droplets. Orange squares correspond to all solidlike droplets in the three-phase flow simulations and the yellow square corresponds to the $S_1$ case in a two-phase flow simulation (i.e., without the interface) where the terminal velocity can be reached and fits with Schiller and Naumann's correlation (black dashed line).

Figure 18

Maximum Reynolds number as a function of Archimedes number for deformable droplets. The dashed line represents a linear fit.