Wetting Effect on Torricelli’s Law

This Letter presents an experimental study on the effect of wetting on the draining of a tank through an orifice set at its bottom. The investigation focuses on flows of liquids in the inertial regime through an orifice the size on the order of magnitude of the capillary length. The results show that although the flows always follow a Torricelli-like behavior, wetting strongly affects the speed of drainage. Surprisingly, this speed goes through a minimum as the outside surface of the tank bottom plate changes from hydrophilic to hydrophobic. The maximum effect in slowing down the flows (up to 20%) is obtained for a static wetting angle ${\theta }_s$ of about 60°. Experiments suggest that the effect of wetting on the exit flows, very likely, is related to the meniscus that forms at the hole’s outlet. A simple model is proposed that estimates the variation of kinetic energy within the meniscus. This model captures the main features of the experimental observations, particularly the nonmonotonic variation of the speed of drainage as a function of ${\theta }_s$ with a minimum for a static wetting angle of about 60°.

Figure 1

Drained volume $V$ versus $t$ for $h_0=10\mathrm{cm}$ (1 L) and 5 cm (0.5 L) [$r_0=1.75\mathrm{mm}$]. The different colors represent the different bottom plates, from dark, the more hydrophobic (coated glass) to light, the more hydrophilic (glass). Dashed curves correspond to the Torricelli’s model. The line thickness is larger than uncertainties on the measured drained volumes.

Figure 2

$r_0^{\mathrm{eff}}/r_0$ and $t_s$ (inset) as a function of ${\theta }_s$ for different initial heights [$r_0=1.75\mathrm{mm}$]; $r_0^{\mathrm{eff}}/r_0$ exhibits a minimum for ${\theta }_s\approx 60°$ where $t_s$ shows a maximum. Lines are fits of $r_0^{\mathrm{eff}}/r_0$ and $t_s$ data to quadratic polynomial functions to point out their nonmonotonic evolution.

Figure 3

Jets at the exit of the tank for similar conditions except wetting [$r=1.75\mathrm{mm}$, $h_0=10\mathrm{cm}$ and $t=90\mathrm{s}$]. From hydrophilic to hydrophobic plate: (a) Glass (${\theta }_s\approx 13°$), (b) Plexiglas (${\theta }_s\approx 64°$), (c) H-glass (${\theta }_s\approx 88°$). Left side: Zoom in of the jets at the hole exit to display the meniscus profiles for the different wetting conditions. Right side: Images of the jets to show their global shape; $\approx 5–6\mathrm{mm}$ beneath the opening exit the three jets look similar. The main notations of the model are reported in Fig. 3.

Figure 4

Calculated values of $d{E}_k/dt$ from Eq. (5) as a function of ${\theta }_s$, for $Q_v=3$, 4 and $5{\mathrm{mL.s}}^{-1}$. Each value of $d{E}_k/dt$ reported here is the average over several $d{E}_k/dt$ values calculated at the same $Q_v$ but for different $h_0$. The lines are guidelines to show the minimum.