**Coalescence characteristics of bulk nanobubbles in water: A molecular
dynamics study coupled with theoretical analysis**

E Bird and E Smith and Z Liang, PHYSICAL REVIEW FLUIDS, 6, 093604 (2021).

DOI: 10.1103/PhysRevFluids.6.093604

The coalescence of two nanobubbles (NBs) in water is a process of great
importance to many industrial applications. In this work, we study the
coalescence of two equal-sized nitrogen NBs in water using molecular
dynamics (MD) simulations and continuum-based theoretical analysis. We
vary the NB diameter from 30 to 50 nm and study the coalescence
characteristics including the expansion speed of the capillary bridge
between two coalescing NBs, the dynamic regime of NB coalescence, the
diameter of fully merged NBs, and the temperature variation of NBs
during the coalescence process. For all cases, we show the MD simulation
results can be well understood by the theoretical models developed in
this work. Due to the large Laplace pressure in the model NBs, the
diameter ratio of fully merged NBs to their daughter NBs is root 2,
which explains the recent experimental result showing that the size of
NBs in water is distributed discretely with a uniform increment factor
of root 2 **Ma et al., J. Phys. Chem. R 124. 5067 (2020)**. The expansion
of gas inside the coalescing NBs and the heat transfer between the gas
NB and surrounding liquid leads to fluctuations of gas temperature
during coalescence. From the theoretical analysis, we find the
coalescence dynamics of NBs is in the crossover regime where neither
viscous stress nor inertial stress in the surrounding liquid dominates
even when the viscous stress is more than ten times higher than inertial
stress. In the range of Ohnesorge number from 0.33 to 0.82, we show the
scaling exponent for the capillary bridge radius vs time at late times
of NB coalescence is around 0.75 +/- 0.05, which is considerably higher
than 0.5 in the viscous-dominated regime.

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