Marangoni-flow-induced partial coalescence of a droplet on a liquid/air interface

The coalescence of a droplet and a liquid/air interface of lower surface tension was numerically studied by using the lattice Boltzmann phase-field method. The experimental phenomenon of droplet ejection observed by Blanchette et al. [Phys. Fluids 21, 072107 (2009)] at sufficiently large surface tension differences was successfully reproduced for the first time. Furthermore, the emergence, disappearance, and re-emergence of “partial coalescence” with increasing surface tension difference was observed and explained. The re-emergence of partial coalescence under large surface tension differences is caused by the remarkable lifting motion of the Marangoni flow, which significantly retards the vertical collapse. Two different modes of partial coalescence were identified by the simulation, namely peak injection occurs at lower Ohnesorge numbers and bottom pinch-off at higher Ohnesorge numbers. By comparing the characteristic timescales of the upward Marangoni flow with that of the downward flow driven by capillary pressure, a criterion for the transition from partial to total coalescence was derived based on scaling analysis and numerically validated.

Figure 1

Specifications of computational domain and boundary conditions.

Figure 2

Dependence of surface tension on the fluid composition. BMB 2009 [16].

Figure 3

Experimental verification of droplet ejection at $\left(1-{\sigma }^{*}\right)=0.7, \mathrm{Oh}=0.0018$, and $\mathrm{We}=1.2$. The experimental images are adapted from Ref. [16].

Figure 4

Evolution of different energy components (normalized by the total energy) during the droplet coalescence shown in Fig. 3.

Figure 5

Droplet coalescence at different surface tension differences (a) $\left(1-{\sigma }^{*}\right)=0$, (b) $\left(1-{\sigma }^{*}\right)=0.3$, (c) $\left(1-{\sigma }^{*}\right)=0.5$, (d) $\left(1-{\sigma }^{*}\right)=0.7$, and fixed $\mathrm{Oh}=0.005$ and $\mathrm{We}={10}^{-4}$.

Figure 6

Evolution of the liquid bridge of case (a) in Fig. 5. (a) Schematic of the liquid bridge, whose principal radii of curvature are denoted by $R_a$ in the azimuthal direction and $R_r$ in the radial direction. The representative point (green dot) on the bridge corresponds to the one closest to the axis. (b) Trajectory of the representative point, whose coordinates, $(r,z)$, with respect to the center of the initial liquid/air interface, $(0,z_0)$, are normalized by the droplet diameter $D_0$.

Figure 7

Schematic of the liquid bridge during (a) the initial expansion and (b) the final shrinkage.

Figure 8

Comparison of the vertical trajectory of the droplet apex for the cases in Fig. 5. $z$ denotes the instantaneous vertical position of the droplet apex, and $z_0$ stands for the initial vertical coordinate of the interface. The difference in the Ma number is caused by $(1-{\sigma }^{*})$ since the Oh number is fixed.

Figure 9

Droplet coalescence at different Oh numbers (a) $\mathrm{Oh}=0.003$, (b) $\mathrm{Oh}=0.005$, (c) $\mathrm{Oh}=0.010$, (d) $\mathrm{Oh}=0.015$, and fixed $\left(1-{\sigma }^{*}\right)=0.7$ and $\mathrm{We}={10}^{-4}$.

Figure 10

Dependence of the normalized breakup time on the Oh number.

Figure 11

Time evolution of the flow field for (a) partial coalescence with peak injection, (b) partial coalescence with bottom pinch-off, and (c) total coalescence at fixed $\left(1-{\sigma }^{*}\right)=0.7$ and $\mathrm{We}={10}^{-4}$.

Figure 12

Critical condition for Marangoni-flow-induced partial coalescence. BMB 2009 [16].