Motion of spheres falling through fluids

Motion of spheres falling through fluids is a classic problem in fluid mechanics. The problem is solved for steady motion and other special cases such as for small and large Reynolds numbers, but not yet for transitional flow motion because of the complicated drag law. Recently, a series approximation for the transitional motion at zero initial velocity was proposed by applying a nonlinear perturbation method and Rubey's universal drag law to the Boussinesq-Basset-Oseen equation. This research presents a simple closed-form solution to the problem for arbitrary Reynolds numbers at nonzero initial velocity. The proposed solution is confirmed with experimental literature data and reproduces all well-known asymptotic values. The results can be directly used for particle size analysis in hydraulics laboratories; they may also help understanding the fluid-particle interactions in environmental engineering, meteorology, and powder technology.