How Quickly Can a Submarine Dive?

Abstract

A two-equation model is developed to describe the descent of a submarine as it floods its ballast tanks. The flow into the tanks is assumed to be inviscid, and the drag on the vertical sinking motion of the craft is neglected. The two coupled differential equations are the generalized form of Newton’s second law and the Bernoulli relation. Time derivatives are converted to spatial derivatives to decouple the equations, and the resulting second-order equation is solved using the Euler–Cromer algorithm. The theory and the method of numerical integration are suitable for an intermediate-level undergraduate course in mechanics that includes some basic fluid dynamics.