The physical problem of a body of water in a tank that drains through a hole in the base is a classical problem that has been studied since at least the time of Torricelli. To fixate this in a student’s mind, one could ask them to visualize a bathtub that is being drained through the plughole or a bottle being drained through a tap. This problem is an ideal one to propose at the secondary school level, since it provides a nice example of a physically relevant elementary ODE that can be studied by students with some calculus skills, and it also provides an opportunity to start introducing some concepts from fluid dynamics, as well as general mathematical modelling skills. Like the brachistochrone, the draining vessel and variants of it have been studied many times in the literature. We will focus on a simple pedagogical example, but there are many other possibilities. For example, Castro de Oliveira et al. performed experiments to measure the speed of water flowing through a pinhole at the bottom of a cylindrical bottle as a function of time and the level of water. In the process, they made an empirical correction to Torricelli’s law. Cross studied three different arrangements whereby air and water can flow into or out of two small holes in the side of a water bottle. Hong studied the unsteady draining problem for an incompressible, non-viscous fluid to obtain an exact solution of the unsteady Bernoulli equation, showing that once again Torricelli’s law must be modified in this situation. In a more mathematically involved treatment, Digilov showed that one can obtain an analytic solution for time evolution of a liquid column draining under gravity through a capillary tube in terms of a special function called the Lambert W function. More recently, Durbin studied vessel drainage of fluids of different viscosities through tubes of varying lengths.
Skip Nav Destination
Article navigation
November 2021
PAPERS|
November 01 2021
Vessel Drainage Under the Influence of Gravity Available to Purchase
Hollis Williams
Hollis Williams
University of Warwick
, Coventry, United Kingdom
Search for other works by this author on:
Hollis Williams
University of Warwick
, Coventry, United Kingdom
Phys. Teach. 59, 629–631 (2021)
Citation
Hollis Williams; Vessel Drainage Under the Influence of Gravity. Phys. Teach. 1 November 2021; 59 (8): 629–631. https://doi.org/10.1119/5.0020444
Download citation file:
Pay-Per-View Access
$40.00
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Citing articles via
A “Perpetual Motion Machine” Powered by Electromagnetism
Hollis Williams
Values Reflected in Energy-Related Physics Concepts
Kara E. Gray, Rachel E. Scherr
Related Content
Investigating Torricelli’s Law (and More) with a 19th-Century Bottle
Phys. Teach. (February 2019)
Experimental study of Bernoulli’s equation with losses
Am. J. Phys. (July 2005)
Visualizing Properties of a Quadratic Function Using Torricelli’s Fountain
Phys. Teach. (April 2020)
The Brachistochrone: An Excellent Problem for All Levels of Physics Students
Phys. Teach. (November 2021)
Torricelli's curtain: Morphology of horizontal laminar jets under gravity
Physics of Fluids (June 2021)