  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 (DNSs) of lens coalescence, in the two and three
dimensional (2D and 3D) CHNS3, uncover the rich spatiotemporal evolution of the
fluid velocity $\bf u$ and vorticity $\omega$, the concentration fields $c_1,
\, c_2,$ and $c_3$ of the three liquids, and a generalized Laplace pressure
$P^G_\mathcal{L}$, 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 $h(t) \sim t^{\alpha_v}$, with $\alpha_v \simeq 1$
in the viscous regime, and $h(t) \sim t^{\alpha_i}$, with $\alpha_i \simeq 2/3$
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 $P^G_\mathcal{L}$ 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.
