  We consider the time evolution of a sessile drop of volatile partially
wetting liquid on a rigid solid substrate. Thereby, the drop evaporates under
strong confinement, namely, it sits on one of the two parallel plates that form
a narrow gap. First, we develop an efficient mesoscopic thin-film description
in gradient dynamics form. It couples the diffusive dynamics of the vertically
averaged vapour density in the narrow gap to an evolution equation for the
profile of the volatile drop. The underlying free energy functional
incorporates wetting, interface and bulk energies of the liquid and gas
entropy. The model allows us to investigate the transition between
diffusion-limited and phase transition-limited evaporation for shallow
droplets. Its gradient dynamics character also allows for a full-curvature
formulation.
  Second, we compare results obtained with the mesoscopic model to
corresponding direct numerical simulations solving the Stokes equation for the
drop coupled to the diffusion equation for the vapour as well as to selected
experiments. In passing, we discuss the influence of contact line pinning.
