A note on wall effect on the terminal falling velocity of a sphere in quiescent Newtonian media in cylindrical tubes

The settling velocity of a sphere in a quiescent fluid is significantly reduced by the presence of confining boundaries. The severity of this phenomenon (wall effect) is strongly dependent on the geometrical and kinematic parameters. In this work, the simplest case of a sphere settling on the axis of a long cylindrical tube is considered. The functional dependence of the wall factor on the sphere-to-tube diameter ratio (λ) and the sphere Reynolds number has been examined. The wall factor is a function of λ only both at very low and at very high values of the sphere Reynolds number (Re). On the other hand, it depends on both λ and Re in the intermediate zone. The values of the Reynolds number marking the transitions from the viscous to intermediate and from the transitional to fully inertial flow regimes have been identified as functions of λ. Using these criteria to delineate experimental data, the relative performance of some of the available expressions for the estimation of wall factor has been evaluated against a large body of experimental results.