Self-Similar Draining near a Vertical Edge

When a liquid film drains on a vertical plate, the film becomes nonuniform near the vertical edge. Here we experimentally report the three-dimensional (3D) self-similar shape of this film. Based on the well-known 2D self-similar solution of a draining film far from the edge, we identify a new 3D self-similar scaling, which converts the partial differential equation for the film thickness with three independent variables into an ordinary differential equation. Interferometry is performed to measure the film thickness as a function of position and time, and the results are in excellent agreement with the theoretical predictions.

Figure 1

Draining film on a vertical glass substrate. (a) A time series of water (with 0.5 wt% methylene blue hydrate) draining on a glass slide. A pipette moved horizontally along the glass substrate and gradually injected the fluid. The background light is provided by a LED panel. See the movie in the Supplemental Material [18]. (b) The horizontal variation of light intensity $I$ ($=\text{grayscale}/255$) at different times $t$, along the bottom dashed line in (a). (c) A sketch of the interferometric setup for measuring liquid film thickness.

Figure 2

A schematic of a vertical draining film near an edge.

Figure 3

Thickness profile of draining silicone oil films on glass substrates. (a)–(d) Interferometric images for silicone oil films with (a) $\mu =102\mathrm{mPa}\mathrm{s}$ and $t=1\min$, (b) $\mu =102\mathrm{mPa}\mathrm{s}$ and $t=5\min$, (c) $\mu =102\mathrm{mPa}\mathrm{s}$ and $t=25\min$, and (d) $\mu =540\mathrm{mPa}\mathrm{s}$ and $t=25\min$, respectively. The blue dashed lines show the positions of the peaks of the draining films $y_0$. See the movies in the Supplemental Material [18]. (e) The normalized film thickness far from the edge, at $x=3$, 6, and 9 mm and $y=6.3\mathrm{mm}$ [red crosses in (a)–(d)], as a function of time $t$. (f) The normalized film thickness $h/h_J$ as a function of the normalized horizontal position $\eta$. The film thickness profiles on the red dashed lines in (a)–(d) are displayed. The gray solid and dashed lines show the solutions of Eqs. (4) and (6), respectively (details in the Supplemental Material [18]).

Figure 4

The scaled horizontal position of the film peak as a function of (a) time $t$ and (b) vertical position $x$ (note the logarithmic axes). Five experiments with different silicone oils are performed (viscosity $\mu =20.8$, 51.5, 102, 540, and 1081 mPa s, respectively), and the scaled $y_0$, (a) $[(\rho g{)}^{3/8}]/[{\gamma }^{1/4}{\mu }^{1/8}{x}^{3/8}]y_0$ and (b) $[(\rho g{)}^{3/8}{t}^{1/8}]/[{\gamma }^{1/4}{\mu }^{1/8}]y_0$, are displayed, respectively, with (a) vertical positions $x=3$, 6, and 9 mm and (b) times $t=1$, 5, and 25 min. The insets show the unscaled $y_0$ with logarithmic axes.