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Partial coalescence from bubbles to drops

  • F. H. Zhang (a1), M.-J. Thoraval (a1) (a2), S. T. Thoroddsen (a1) and P. Taborek (a3)

The coalescence of drops is a fundamental process in the coarsening of emulsions. However, counter-intuitively, this coalescence process can produce a satellite, approximately half the size of the original drop, which is detrimental to the overall coarsening. This also occurs during the coalescence of bubbles, while the resulting satellite is much smaller, approximately 10 %. To understand this difference, we have conducted a set of coalescence experiments using xenon bubbles inside a pressure chamber, where we can continuously raise the pressure from 1 up to 85 atm and thereby vary the density ratio between the inner and outer fluid, from 0.005 up to unity. Using high-speed video imaging, we observe a continuous increase in satellite size as the inner density is varied from the bubble to emulsion-droplet conditions, with the most rapid changes occurring as the bubble density grows up to 15 % of that of the surrounding liquid. We propose a model that successfully relates the satellite size to the capillary wave mode responsible for its pinch-off and the overall deformations from the drainage. The wavelength of the primary wave changes during its travel to the apex, with the instantaneous speed adjusting to the local wavelength. By estimating the travel time of this wave mode on the bubble surface, we also show that the model is consistent with the experiments. This wavenumber is determined by both the global drainage as well as the interface shapes during the rapid coalescence in the neck connecting the two drops or bubbles. The rate of drainage is shown to scale with the density of the inner fluid. Empirically, we find that the pinch-off occurs when 60 % of the bubble fluid has drained from it. Numerical simulations using the volume-of-fluid method with dynamic adaptive grid refinement can reproduce these dynamics, as well as show the associated vortical structure and stirring of the coalescing fluid masses. Enhanced stirring is observed for cases with second-stage pinch-offs. Numerous sub-satellites are observed when the length of the top protrusion of the drop exceeds the Rayleigh instability wavelength. We also find a parameter regime where the focusing of more than one capillary wave can pinch-off satellites. One realization shows a sequence of three pinch-offs, where the middle one pinches off a toroidal bubble.

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Type Description Title

Zhang et al. supplementary movie
High-speed video of partial coalescence inside the pressure chamber for density ratio of 0.13. Corresponding to Figure 3(b). Video frame rate is about 125 kfps.

 Video (20 KB)
20 KB

Zhang et al. supplementary movie
Formation of multiple satellites, from focusing of subsequent capillary waves. Corresponding to Figure 17(a). Frame rate is about 129 kfps.

 Video (328 KB)
328 KB

Zhang et al. supplementary movie
Formation of multiple satellites. Corresponding to Figure 15(a). Frame rate is about 66 kfps.

 Video (827 KB)
827 KB

Zhang et al. supplementary movie
High-speed video of partial coalescence inside the pressure chamber for density ratio of 0.71. Corresponding to Figure 3(d). Video frame rate is about 81 kfps.

 Video (61 KB)
61 KB

Zhang et al. supplementary movie
High-speed video of partial coalescence inside the pressure chamber for density ratio of 0.03. Corresponding to Figure 3(a). Video frame rate is about 125 kfps.

 Video (42 KB)
42 KB

Zhang et al. supplementary movie
Numerical simulation of partial coalescence for a range of density ratios, $D=\rho_i/\rho_o =$~0.001 (black), 0.015 (green), 0.050 (red), 0.100 (blue), 0.300 (magenta), and 1.000 (cyan). , Corresponds to Figure 5(a).

 Video (16.9 MB)
16.9 MB

Zhang et al. supplementary movie
Numerical simulation of partial coalescence and resulting vorticity structures inside the father bubble, comparing the dynamics for different density ratios, of 0.03, 0.13, 0.35 and 0.71. Same conditions as in Figure 3.

 Video (15.8 MB)
15.8 MB

Zhang et al. supplementary movie
Numerical simulation of partial coalescence, comparing the dynamics for different density ratios, of 0.03, 0.13, 0.35 and 0.71. Corresponds to Figure 3.

 Video (6.0 MB)
6.0 MB

Zhang et al. supplementary movie
High-speed video of partial coalescence inside the pressure chamber for density ratio of 0.35. Corresponding to Figure 3(c). Video frame rate is about 64 kfps.

 Video (24 KB)
24 KB


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