An analytical study of the motion of a sphere rolling down a smooth inclined plane in an incompressible Newtonian fluid

Abstract

The accelerated motion of a sphere rolling down a smooth inclined plane immersed in a quiescent Newtonian fluid has been investigated theoretically using Jan and Chen's drag and added mass coefficient data [C.D. Jan, J.C. Chen, J. Hydraulic Res. 35 (1997) 689]. A modified expression for the drag coefficient was used to obtain an analytical solution to the equations of motion in the inclined plane. Closed form solutions for the instantaneous acceleration, time and distance required by the sphere to reach a given velocity are presented, and the sphere's settling velocity is determined. The relationship between the plane's angle of inclination and the time and distance required by the sphere to reach 99% of its terminal velocity is also investigated. The analysis reported herein is, however, restricted to dense spheres immersed in light liquids where additional effects arising from the Basset forces can be neglected.