It is known that the dripping of a liquid film on the underside of a plate can be suppressed by tilting the plate so as to cause a sufficiently strong flow. This paper uses two-dimensional numerical simulations in a closed-flow framework to study several aspects of this phenomenon. It is shown that, in quasi-equilibrium conditions, the onset of dripping is closely associated with the curvature of the wave crests approaching a well-defined maximum value. When dynamic effects become significant, this connection between curvature and dripping weakens, although the critical curvature remains a useful reference point as it is intimately related to the short length scales promoted by the Rayleigh–Taylor instability. In the absence of flow, when the film is on the underside of a horizontal plate, the concept of a limit curvature is relevant only for small liquid volumes close to a critical value. Otherwise, the drops that form have a smaller curvature and a large volume. The paper also illustrates the peculiarly strong dependence of the dripping transition on the initial conditions of the simulations. This feature prevents the development of phase maps dependent only on the governing parameters (Reynolds number, Bond number, etc.) similar to those available for film flow on the upper side of an inclined plate.