Time Optimization of a Draining Tank and Some Similar Problems on Star Graphs

Abstract

We consider a problem of calculus of variations motivated by the model of a tank filled with a given volume of liquid and draining through a small orifice according to Torricelli’s law. We prove that given any length of time, some tank exists which drains in this time. Our main interest in this optimization (i.e., minimization and maximization) problem is that the usual Euler–Lagrange equation may not be used here, at least directly. We consider optimization for some similar physical models where Torricelli’s law has to be modified. We also study optimization problems on the star graphs that are inspired by our physical model.