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Wetting Effect on Torricelli’s Law

Publication of the Physics Lab

Published on : February 15, 2017
Jets at the exit of the tank for similar conditions except wetting; hole radius = 1.75 mm; initial height = 10 cm. From hydrophilic to hydrophobic plate: (a) Glass, (b) Plexiglas, (c) Hydrophobic-glass. Left side: Zoom in of the jets at the hole exit to display the meniscus profiles for the different wetting conditions (i.e. the small ring of fluid that forms around the outside of the hole), one notices these menisci are different. Photo J. Ferrand©

Make a hole in the bottom of a tank filled with a non-viscous fluid (e.g. water), the liquid it contains will flow out. How quickly the tank empties depends on the shape of the tank, the size of the hole, and, importantly, the initial amount of liquid (the rate at which the tank drains decreases as it gets emptier). This first study of this phenomenon was carried out by Torricelli almost four hundred years ago. Now researchers from the  Laboratoire de Physique at the ENS de Lyon have shown that how the liquid wets the hole’s external surface as it exits the tank also plays a role. This finding helps researchers to explain why identically shaped tanks can empty at different rates.

To demonstrate this effect, the researchers carried out quite simple experiments. They filled a tank with water and unplugged a millimeter-sized hole in the bottom of tank. The size of this hole is slightly smaller or in the order of magnitude than the capillary length*. They filmed the liquid jet exiting the tank, and tracked the drainage rate. To alter the wetting properties of the tank, they switched the bottom glass plate with plates whose external coatings had different affinities for water. They observed that the drainage rate passes through a minimum when the outer surface of the bottom plate of the reservoir changes from hydrophilic to hydrophobic. (Changing the inside coating of the tank had no effect). Water drained fastest for glass or Teflon coatings, materials that have opposite interactions with water—water wets glass (static wetting angle close to 0°), but is repelled by Teflon (static wetting angle larger than 90°). The water drained slowest (up to 20% slower) for wetting properties somewhere in the middle (i.e. for a static wetting angle of about 60°). According to their calculations, the cause of this non-monotonic behavior is the shape of the small ring of fluid that forms around the outside of the hole just as the water exits. In all cases, the presence of this ring causes the draining fluid to accelerate when compared to a completely straight jet, explaining why tanks can sometimes empty more quickly than expected. The degree of acceleration depends on the specific ring shape, leading to the variations seen in the experiments.

* Capillary length: characteristic length scale that compares the respective effects of surface tension and gravity forces; for size below this length capillary forces predominate beyond this is the gravity.

References: Wetting Effect on Torricelli’s Law. Jérémy Ferrand, Lucile Favreau, Sylvain Joubaud, and Eric Freyssingeas, Phys. Rev. Lett. 117, 248002 (2016)

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