  Drop coalescence occurs through the rapid growth of a liquid bridge that
connects the two drops. At early times after contact, the bridge dynamics is
typically self-similar, with details depending on the geometry and viscosity of
the liquid. In this paper we analyse the coalescence of two-dimensional viscous
drops that float on a quiescent deep pool; such drops are called liquid lenses.
The analysis is based on the thin-sheet equations, which were recently shown to
accurately capture experiments of liquid lens coalescence. It is found that the
bridge dynamics follows a self-similar solution at leading order, but,
depending on the large-scale boundary conditions on the drop, significant
corrections may arise to this solution. This dynamics is studied in detail
using numerical simulations and through matched asymptotics. We show that the
liquid lens coalescence can involve a global translation of the drops, a
feature that is confirmed experimentally.
