Accelerating motion of a vertically falling sphere in incompressible Newtonian media: an analytical solution

Abstract

The transient motion of a sphere falling through a Newtonian fluid has been investigated using a drag of the form given by Abraham/ Wadell, the average accuracy of which was found to be 7.6% for a sphere with Reynolds number Re in the range 0 < Re ≤ 10⁴, 12% for 10⁴ < Re < 10⁵ and 15% for 0≤Re≤10⁵. The equations of motion have been integrated taking into account the added-mass effects, and exact analytical solutions for the time and distance required by the sphere to reach a given velocity are derived. The acceleration of the sphere and the terminal velocity were also determined. It was found that different properties of the sphere-fluid combination result in different solutions for the time and distance, but not for the acceleration. The analysis reported herein is, however, restricted to dense spheres falling in less dense fluids, when additional effects arising from the Basset forces can be neglected.