Partial coalescence of drops at liquid interfaces
Blanchette, F., Bigioni, T., Nature 2 (2006).
Navier-Stokes equation, pinch-off, surface tension
Coalescence occurs when two separate masses of the same fluid are brought into contact; to minimize surface energy, they combine into a single larger mass. However, this does not always occur when a drop of fluid comes into contact with a large reservoir of the same fluid. Sometimes, the drop partially coalesces, "pinches off" in the process of merging, and leaves behind a smaller droplet. The authors study the mechanism of this effect.
The authors deposited a liquid onto an identical liquid in air and filmed the process with a high-speed camera. The results are shown in Figure 1. As shown, the drop comes into contact and forms a smaller daughter droplet. The shape of the droplet was numerically simulated by solving the Navier-Stokes equations, including surface tension as a force on the localized interface. The simulation shapes matched well with the experimental results as shown in Figure 1. The authors ruled out static Rayleigh-Plateau instability as a cause for the pinch-off by modifying simulation parameters.
Instead, they suggest that the pinch-off depends on the inward momentum of the collapsing neck as the drop merges. The downward pull of surface tension at the drop's summit is generally larger than the inward horizontal pull of the neck. However, the horizontal collapse may induce pinch-off if the vertical collapse is retarded. The authors suggest that capillary waves generated by the opening of the neck provide the retarding force. The capillary waves stretch the drop by focusing energy on its summit and thereby reduce the vertical collapse. Numerical simulations seem to support this claim, and the phase boundary between partial and total coalescence is found to be characterized by a critical Ohnesorge number of 0.026.
However, these results only apply to systems where gravitational effects are negligible. When the drops are significantly deformed due to gravity, the merging dynamics are more complicated. Furthermore, these experiments were carried out in air; when the same experiment is done in a liquid (different from that of the drop), the critical Ohnesorge number changes.