  The transition to dripping in the gravity-driven flow of a liquid film under
an inclined plate is investigated at zero Reynolds number. Computations are
carried out on a periodic domain assuming either a fixed fluid volume or a
fixed flow rate for a hierarchy of models: two lubrication models with either
linearised curvature or full curvature (the LCM and FCM, respectively), and the
full equations of Stokes flow. Of particular interest is the breakdown of
travelling-wave solutions as the plate inclination angle is increased. For any
fixed volume the LCM reaches the horizontal state where it attains a
cosine-shaped profile. For sufficiently small volume, the FCM and Stokes
solutions attain a weak Young-Laplace equilibrium profile, the approach to
which is described by an asymptotic analysis generalising that of Kalliadasis &
Chang (1994) for the LCM. For large volumes, the bifurcation curves for the FCM
and Stokes model have a turning point so that the fully inverted state is never
reached. For fixed flow rate the LCM blows up at a critical angle that is well
predicted by asymptotic analysis. The bifurcation curve for the FCM either has
a turning point or else reaches a point at which the surface profile has an
infinite slope singularity, indicating the onset of multi-valuedness. The
latter is confirmed by the Stokes model which can be continued to obtain
overturning surface profiles. Overall the thin-film models either provide an
accurate prediction for dripping onset or else supply an upper bound on the
critical inclination angle.
