Physics Letters A 372 (2008) 6742–6749 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Brachistochrones in potential flow and the connection to Darwin’s theorem Roberto Camassa * , Richard M. McLaughlin * , Matthew N.J. Moore, Ashwin Vaidya Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA article info abstract Article history: Received 19 February 2008 Received in revised form 12 June 2008 Accepted 13 June 2008 Available online 6 September 2008 Communicated by C.R. Doering 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.  2008 Published by Elsevier B.V. The hydrodynamics associated with solid bodies moving in flu- ids is fundamental in understanding properties of lift, drag, and global energetics. Calculating the fluid flow generally requires in- tegrating the Navier–Stokes equations with boundary conditions at the surface of a solid body moving under hydrodynamic forcing. Because of the complexity of this problem, the intuition regard- ing the nature of the hydrodynamic forces is often incorrect. Thus, exact mathematical statements regarding such motions are clearly valuable. For example, while aerodynamic lift is correctly under- stood, it is often incorrectly explained in freshman physics courses by use of Bernoulli’s law in which a pressure gradient is inferred by assuming that a particle just above an airfoil will take an equal amount of time to circumvent the body as that of a particle just below the wing. For an interesting discussion of these issues, see the NASA education website [1]. Of course, this equal transit time principle is not justified, and a proper explanation of lift is more complicated (and understood), requiring consideration of the ef- fects of the viscous forces involved. A somewhat related issue concerns assessing the mass of fluid which is dragged along with a moving body. Of course, this “drift” mass should be expected to depend upon the details of viscous forces involved. Nonetheless there is a general proposition due to Taylor [2] and Darwin [3–8] which predicts the drift mass to be the same as the so-called “added” mass, which for a sphere has been shown to be given by the fluid density times half the volume * Corresponding authors. E-mail addresses: camassa@email.unc.edu (R. Camassa), rmm@amath.unc.edu (R.M. McLaughlin). of the body. Darwin’s proposition assumes that the fluid flow is potential and the body has moved an infinite distance through the fluid. These two problems, that of flight time and transport, are in fact mathematically related, as we show in this Letter. Here we prove the existence of a brachistochrone path for a material pla- nar surface oriented normal to a constant far-field wind advected around a fixed sphere under the assumption of potential flow. This brachistochrone path is defined to be the trajectory of that particle in the upstream plane which crosses a fixed (parallel) downstream plane first. Figuratively, this corresponds to a race from an imagi- nary starting line to an imaginary finish line drawn in the fluid. We compute rigorous asymptotics of this path, which allow us to establish the explicit correction formulas for Darwin’s drift vol- ume. We work with the two-dimensional case of a cylinder first, and then present the results for the three-dimensional sphere. Shown in Fig. 1 is a schematic detailing the setup of this study. Several of the features in this figure will be described in greater detail below. The proof of the brachistochrone existence Consider the problem of computing the time of flight for a plane of particles released a distance d 0 upstream of a cylinder to reach the analogous plane a distance d 0 downstream of the cylinder. For this problem, the stream function is ψ = Uy(1 - a 2 /(x 2 + y 2 )), where a is the cylinder radius and U is the far field flow velocity taken in the x-direction. The time of flight will then be T =  d 0 -d 0 1/u(x, y) dx. Here u is the horizontal ve- 0375-9601/$ – see front matter  2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.06.093