  We present a long-wavelength approximation to the Navier-Stokes Cahn-Hilliard
equations to describe phase separation in thin films. The equations we derive
underscore the coupled behaviour of free-surface variations and phase
separation. Since we are interested in the long-time behaviour of the
phase-separating fluid, we restrict our attention to films that do not rupture.
To do this, we introduce a regularising Van der Waals potential. We analyse the
resulting fourth-order equations by constructing a solution as the limit of a
Galerkin approximation, and obtain existence and regularity results. In our
analysis, we find a nonzero lower bound for the height of the film, which
precludes the possibility of rupture. The lower bound depends on the parameters
of the problem, and we compare this dependence with numerical simulations. We
find that while the theoretical lower bound is crucial to the construction of a
smooth, unique solution to the PDEs, it is not sufficiently sharp to represent
accurately the parametric dependence of the observed dips in free-surface
height.
