The coalescence of liquid lenses represents a fundamental challenge within the domains of fluid dynamics and statistical physics, particularly in the context of complex multi-phase flows. We demonstrate that the three-phase Cahn–Hilliard–Navier–Stokes (CHNS3) system provides a natural theoretical framework for studying liquid-lens coalescence, which has been investigated in recent experiments. Our extensive direct numerical simulations of lens coalescence, in the two and three dimensional (2D and 3D) CHNS3, uncover the rich spatiotemporal evolution of the fluid velocity u and vorticity ω, the concentration fields c1, c2, and c3 of the three liquids, and an excess pressure PℓG, which we define in terms of these concentrations via a Poisson equation. We find, in agreement with experiments, that as the lenses coalesce, their neck height h(t)∼tαv, with αv≃1 in the viscous regime, and h(t)∼tαi, with αi≃2/3 in the inertial regime. We obtain the crossover from the viscous to the inertial regimes as a function of the Ohnesorge number Oh, a dimensionless combination of viscous stresses and inertial and surface tension forces. We show that a vortex quadrupole, which straddles the neck of the merging lenses, and PℓG play crucial roles in distinguishing between the viscous- and inertial-regime growths of the merging lenses. In the inertial regime, we find signatures of turbulence, which we quantify via kinetic-energy and concentration spectra. Finally, we examine the merger of asymmetric lenses, in which the initial stages of coalescence occur along the circular parts of the lens interfaces; in this case, we obtain power-law forms for the h(t) with inertial-regime exponents that lie between their droplet-coalescence and lens-merger counterparts.
