When a liquid is supplied through a nozzle onto a relatively non-wetting inclined solid surface, a narrow rivulet forms. This work provides novel physical insights into the following phenomena in the rivulet flow that have not been well understood to date. Firstly, the fundamental mechanism behind the transition of a linear rivulet to a droplet flow is investigated. The experiments show that the droplet flow emerges due to the necking of a liquid thread near the nozzle. Based on the observation, it is argued that when the axial velocity of a liquid is slower than the retraction velocity of its thread, the bifurcation of the liquid thread occurs, and this argument is experimentally verified. Secondly, a discussion on the curled motion of a meandering rivulet is given. This study proposes the contact angle hysteresis as a primary origin of the centripetal force that enables the rivulet`s curved motion. A simple scaling analysis based on this assumption predicts a radius of curvature which agrees with the experimental observation.