On the Motion of Solid Spheres Falling Through Viscous Fluids in Vertical and Inclined Tubes

Abstract

Little is known about the rotational motion of spheres falling through viscous fluids in inclined tubes. Most studies have investigated translational and rotational motions in vertical tubes. These works show that in creeping flow a sphere’s translational and rotational velocities are independent. Rotation is predicted and observed for eccentric spheres while concentric spheres fall without rotation. Experiments were performed by us with steel spheres of radius r falling through glycerine in a tube of variable inclination angle and of radius R such that r/R = 0.882, 0.757, 0.442. For the cases involving two or three spheres falling together various modes of motion were observed. Especially interesting was the finding that the rotation direction of a sphere gradually changes from positive (opposite to downhill rolling) to negative (in the sense of downhill rolling) as the tube inclination angle is increased. This is allowed by the inertia-induced lift force which maintains a sphere at a very small but finite distance from the inclined tube wall. However, by further increasing the inclination angle the lift force eventually becomes smaller than the apparent weight of the sphere which, upon finally contacting the tube wall, descends by rolling along it. Examination of our findings in the light of earlier results for vertical and inclined tubes suggests that, through its effect on sphere eccentricity, inertia indirectly affects the rotational motion of a falling sphere when Rep 10−3 but it does not significantly affect the translational motion when Rep <1. None of the inclined tube studies performed to date has been completely devoid of inertia-induced lift effects.