**Coalescence speed of two equal-sized nanobubbles**

E Bird and J Zhou and Z Liang, PHYSICS OF FLUIDS, 32, 123304 (2020).

DOI: 10.1063/5.0030406

In this work, we use molecular dynamics (MD) simulations coupled with
continuum-based theoretical analysis to study the coalescence dynamics
of two equal-sized nanobubbles (NBs). We first derive a governing
equation for the evolution of the capillary bridge radius between two
coalescing NBs from the axisymmetric Navier-Stokes equation. To verify
the prediction from the governing equation, we carry out MD simulations
of the coalescence of two NBs in a Lennard-Jones fluid system and
directly measure the bridge radius, r(b), as a function of time, t. By
varying the bubble diameter, we change the NB Ohnesorge number from 0.46
to 0.33. In all cases, we find the theoretical prediction overestimates
the expansion speed of the capillary bridge at early time of NB
coalescence. However, once we take into account the curvature-dependent
surface tension and restrict the minimum principal radius at the
capillary bridge to the size of the atom in the model liquid, the
theoretical prediction agrees with the MD data very well in both early
time and later time of the coalescence process. From the theoretical
model, we find neither liquid viscous force nor liquid inertial force
dominates at later time of coalescence of the model NBs. In this case,
the MD simulation results show r(b)(t) proportional to t(0.76 +/- 0.04)
with the scaling exponent considerably higher than that in the scaling
law r(b)(t) proportional to t(0.5) for the viscous and inertial
dominated regimes. The diameter ratio of fully merged NB to that of the
original NB is about

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