Critical role of friction for a single particle falling through a funnel

We investigate a single frictional, inelastic, spherical particle falling under gravity through a symmetric funnel. A recent study showed that, for a frictionless particle in such a system, several anomalous phenomena occur: The particle can stay longer, lose more energy, and exert more impulsive force in a funnel with steeper walls. For frictionless particles, such phenomena exist for many small ranges of funnel angles and are a consequence of the many possible repeated patterns in particle trajectories. However, in reality, friction always exists and it is a natural question whether the anomalous phenomena still exist for frictional particles in such systems. We show that, surprisingly, the inclusion of friction in the dynamics actually dramatically enhances the anomalous phenomena. For frictional particles, the anomalous phenomena exist for all funnel angles steeper than 45^{°} and are thus more robust than the frictionless case. Furthermore, instead of many possible complicated repeated patterns in particle trajectories, there is a unique repeated pattern for frictional particles. Moreover, this is the simplest possible repeated pattern. We derive an analytical expression for this unique repeated pattern and provide a theoretical explanation for the anomalous phenomena observed in frictional particle systems. We further show that the friction, no matter how small, plays a critical role in the dynamics, that is, the dynamics of the frictionless particle system is singular.