Water Leaking out of a Tank

The Bernoulli equation predicts the speed at which water exits a small hole in the bottom of a cylindrical tank, a result known as the Torricelli theorem. If the hole is not small, one can use the equation of continuity to correct the prediction for the nonzero rate of decrease of the surface height of water in the tank. However, the corrected result diverges if the cross-sectional area of the hole equals that of the tank (i.e., the water is freely falling down a vertical pipe open at both ends). The issue is that the flow is unsteady and so the standard form of the Bernoulli equation presented in introductory physics courses is not applicable. I will present a simple derivation of the unsteady form of the Bernoulli equation and show that it gives the correct answer for any size hole. Next I will discuss the force that must be applied to the tank to hold it in place while water is escaping from the tank. This is a variable-mass problem and thus care must be taken in applying Newton's second law. I will end with some comments about other shapes of tanks such as a funnel.

reference: J. Otto and C.E. Mungan, "Flow of water out of a funnel," Eur. J. Phys. 45, 055007 (2024)