  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 criterion 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 the vortex-ring Reynolds
number, 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.
