  We discuss a fluid dynamic variant of the classical Bernoulli's
brachistochrone problem. The classical brachistochrone for a non-dissipative
particle is governed by maximization of the particle's kinetic energy resulting
in a cycloid. We consider a variant where the particle is replaced by a bottle
filled with a viscous fluid and attempt to identify the shape of a curve
connecting two points along which the bottle would move in the shortest time.
We derive the system of integro-differential equations governing system
dynamics for a given shape of the curve. Using these equations, we pose the
brachistochrone problem invoking optimal control formalism and show that (in
general) the curve deviates from a cycloid. This is due to the fact that
increasing the rate of change of bottle kinetic energy is accompanied by
increased viscous dissipation. We show that the bottle motion is governed by a
balance between the desire to minimize travel time and the need to reach the
end point in the face of increased dissipation. The trade-off between these two
physical forces plays a vital role in determining the brachistochrone of a
fluid-filled cylinder. We show that in the two limits of either vanishing or
high viscosity, the brachistochrone for this problem reduces to a cycloid. An
intermediate viscosity range is identified where the fluid brachistochrone is
non-cycloidal. Finally, we show relevance of these results to the dynamics of a
rolling liquid marble.
