Revisiting the discharge time of a cylindrical leaking bucket: or, "one does not simply call dsolve into mordor."

Abstract

In [2] we find an exploration of a new mathematical model of the flow in a leaking bucket suitable for beginning students. The model is derived using the non-steady Bernoulli's Principle, and results in a more sophisticated model than the simple ordinary differential equation [EQUATION] derived using the steady Bernoulli's Principle. The simpler model goes sometimes by the name of Torricelli's Law and is very well studied; indeed it is a favourite example in many textbooks. This present paper provides an alternative derivation of the new model that uses an energy balance, and carefully lays out some numerical issues omitted from the treatment in [2]. We also provide an analytic solution in terms of 2F1 hypergeometric functions, which, while possibly unfamiliar to the student, are available to them via computer algebra systems. Even before that solution, an intermediate equation [EQUATION] is derived, which already explains the similarity of the solutions to the more sophisticated model to the ones from the simple Torricelli's Law. This paper gives a useful example for use of a CAS in a classroom setting.