Paths of quickest descent and least energy dissipation for rolling particles in viscous and inertial flow environments

We study the motion of an inertial spherical particle rolling down an incline inside a fluid with both viscous and inertial effects. We reveal experimentally and theoretically how unique curves of shortest time and least energy loss.

We define an effective Stokes number, built on the response time of particle and gravitational time of rolling, that controls the curvature of the optimal ramp for both time and energy minima. With increasing $St_p$, the path of minimum time approaches a straight ramp, beginning and terminating in end-caps of curvature connected to St_p. The paths of energetic minima are oppositely curved as those of time minima. In the limit of least dissipation (or $St_p \gg 1$), we recover the energetic minima approach the familiar parabolas of projectile motion.