The motion of a sphere down a liquid-filled tube

Abstract

A previous investigation by Christopherson & Dowson ( Proc. R. Soc. Lond . 251 (1959)) studies the motion of a sphere down a vertical, liquid-filled tube, when there is only a small difference between the diameters of the sphere and the tube. Experimentally they find that the sphere descends in an eccentric position, with its centre displaced from the axis of the tube, and that it rotates in a sense opposite to that which it would have if it were to roll down the line of closest approach. By contrast, one consequence of their theoretical solution is that there is no side force on the sphere, so that no definite lateral displacement of the sphere is predicted and the question of uniqueness is left open. For the purpose of comparison with experiment, the authors speculate that the sphere descends in that position for which its velocity of descent is a maximum. In the present work an alternative explanation is suggested, namely that the eccentric position of descent is determined by a slight displacement of the tube from the vertical position. A modified lubrication theory is described, which predicts a definite side force so that the lateral displacement of the sphere can be calculated.