An exact solution to the draining reservoir problem of the incompressible and non-viscous liquid

Abstract

The exact expressions for the drain time and the height, velocity and acceleration of the free surface are found for the draining reservoir problem of the incompressible and non-viscous liquid. Contrary to the conventional approximate results, they correctly describe the initial time dependence of the liquid velocity and acceleration. Torricelli's law does not hold in the initial transient region, which imposes restrictions on the validity of the analogy between the drain system and the electric circuit (Ohm's law).

Filling and emptying a tank of liquid

“…neglecting the second-order-small term dm d ẋ. This approach correctly treats the variable-mass problem [2] of liquid in volume 1; equation (6) would not be obtained if one instead wrongly substituted p = m ẋ into equation (5).…”

“…The time required for the tank to fully empty can then be expressed in terms of a ratio of gamma functions as [6]…”