Developing Film Flow on an Inclined Plane With a Critical Point

Viscous, laminar, gravitationally-driven flow of a thin film on an inclined plane is analyzed for moderate Reynolds number under critical conditions. A previous analysis of film flow utilized a momentum integral approach with a semiparabolic velocity profile to obtain an ordinary differential equation for the film thickness for flow over a round-crested weir, and the singularity associated with the critical point for a subcritical-to-supercritical transition was removable. For developing flow on a plane with a supercritical-to-subcritical transition, however, the same approach leads to a nonremovable singularity. To eliminate the singularity, the film equations are modified for a velocity profile of changing shape. The resulting predictions compare favorably with those from the two-dimensional boundary-layer equation obtained by finite differences and with those from the Navier-Stokes equation obtained by finite elements.