We present a theory on the coalescence of 2 spherical liquid droplets that are initially stationary. The evolution of the radius of a liquid neck formed upon coalescence was formulated as an initial value problem and then solved to yield an exact solution without free parameters, with its 2 asymptotic approximations reproducing the well-known scaling relations in the inertially limited viscous and inertial regimes. The viscous-to-inertial crossover observed in previous research is also recovered by the theory, rendering the collapse of data of different viscosities onto a single curve.

This study employs an improved volume of fluid method and adaptive mesh refinement algorithm to numerically investigate the internal jet-like mixing upon the coalescence of two initially stationary droplets of unequal sizes. The emergence of the internal jet is attributed to the formation of a main vortex ring, as the jet-like structure shows a strong correlation with the main vortex ring inside the merged droplet. By tracking the evolution of the main vortex ring together with its circulation, we identified two mechanisms that are essential to the internal-jet formation: the vortex-ring growth and the vortex-ring detachment. Recognizing that the manifestation of the vortex-ring-induced jet physically relies on the competition between the convection and viscous dissipation of the vortex ring, we further developed and substantiated a vortex-ring-based Reynolds number ( = /4 ) criterion, where is the circulation of the detached main vortex ring and is the liquid kinematic viscosity, to interpret the occurrence of the internal jet at various Ohnesorge numbers and size ratios. For the merged droplet with apparent jet formation, the average mixing rate after jet formation increases monotonically with , which therefore serves as an approximate measure of the jet strength. In this respect, stronger internal jet is responsible for enhanced mixing of the merged droplet. _____________________________a) The first and second authors contributed equally to this work. b) Author to whom correspondence should be addressed. Electronic

The off-center collision of binary bouncing droplets of equal size was studied numerically by a volume-of-fluid method with two marker functions, which has been justified and validated by comparing with available experimental results. A nonmonotonic kinetic energy (KE) recovery with varying impact parameters was discovered. This can be explained by the prolonged entanglement time and the enhanced internal-flow-induced viscous dissipation for bouncing droplets at intermediate impact parameters, compared with those at smaller or larger impact parameters. The distribution of the local viscous dissipation rate (VDR) in the droplet interior shows two major concentration areas, and the competition between these two concentration areas accounts for the nonmonotonic viscous dissipation with varying impact parameters. The nonmonotonic KE recovery with varying impact parameters can also be attributed to the competition between the VDR induced by normal strains and shear strains. The nonmonotonicity was further numerically verified for wider ranges of parameters, and a practically useful formula was proposed to correlate the KE dissipation factor with the impact parameter for various Weber numbers and Ohnesorge numbers.

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