Gravitational drainage on a vertical substrate of a narrow width

The effect of a single, vertical edge on a draining liquid film was studied recently [Phys. Rev. Lett. 125, 064502 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.064502]. In this experimental study, we characterize the structure of a liquid film, draining due to gravity, on a vertical, narrow substrate. We show that surface tension affects the draining film at the two vertical edges. The edge effects propagate into the film to eventually influence the shape over the entire width. Interferometry is performed to measure the film thickness profile. A motorized stage is used to vertically translate the thin film and the substrate, which extends the range of the measurements. Our experiments show that the thickness of the liquid film scales with the well-known Jeffreys' solution, which is the thickness of a draining film on a vertical substrate of infinite width. However, due to the existence of the two vertical edges, near the top contact line, the film thickness changes sharply near the edges and is flat near the middle of the substrate. In contrast, away from the top contact line, the edge effects propagate towards the middle, and the overall horizontal film shape eventually becomes approximately quartic. Further, we identify characteristic length scales in the vertical direction, which combine the effects of the surface tension, viscosity, and gravitational drainage. These length scales, respectively, highlight the effects from the vertical edges and the top contact line, and the experimental results are in agreement with the scaling arguments.