Paths of shortest time and minimum energy dissipation in viscous fluids

We experimentally and theoretically study the Brachistochronous motion of spherical particles in real fluids with viscous and inertial effects. We observe that either straight ramps or cycloids can give shorter time depending on the experimental conditions.. Following an Euler-Lagrange approach, we derive the dimensionless forms and obtain curves of minimum time and minimum energy dissipation in viscous fluids. Paths of minimum time and minimum energy are distinctly different for the Brachistochronous motion of particles down an inclined path; however they both converge toward a line in the limit of dominant viscous effects.