Brachistochrones in potential flow and the connection to Darwin's theorem

Abstract

We establish the existence and the asymptotic properties of a path of minimum travel time for a line of particles starting upstream of a sphere or cylinder in potential flow. A connection is made between this brachistochrone path and Darwin's proposition which relates the added mass with the drift volume dragged by a body moving an infinite distance in the fluid. We compute an asymptotic correction to the drift volume for finite distances and show how the brachistochrone path is connected to the reflux volume. We present accurate numerical calculations for the brachistochrone position, point of zero horizontal Lagrangian displacement, reflux and partial drift volumes. These calculations are seen to agree well with the asymptotic predictions even for moderate values of the parameters. In the small Reynolds number regimes, we show that while for the case of Stokes flow past a sphere no brachistochrones exist at finite distances from the sphere, the Oseen correction is sufficient to restore such least-time trajectories. Lastly, the application to a sphere falling in a stratified fluid is discussed using the new drift volume correction formula.