Drag coefficients for a sphere rolling down an inclined channel

Forces acting on a sphere rolling down an inclined smooth or rough channel include submerged weight of the sphere, fluid drag, fluid lift, and rolling resistance from the channel bed. When the acting forces are in equilibrium, the sphere rolls down the inclined channel with a terminal velocity. The steady movements of such a sphere rolling down smooth and rough channel beds with the terminal velocity are studied through laboratory experiments and a simplified theoretical analysis. A method is proposed to evaluate the rolling resistance that consists of collision and friction parts of resistance. The coefficients of rolling resistance and the drag coefficients are determined by regression analysis with experimental data. The relationships of the drag coefficients and the Reynolds numbers for a sphere rolling over smooth and rough channel beds are also obtained. It is found that at the same Reynolds number; the drag coefficient for a sphere rolling down a rough bed is larger than that for a sphere down a smooth one, and that both are much larger than that of a sphere in the free fall.