Vertical fall of a sphere opposed by fluid buoyancy and drag

Abstract A sphere dropped from rest in a stationary fluid will fall under the combined forces of gravity, buoyancy, and drag if its average density is larger than that of the fluid. Introductory physics courses typically model the drag force as varying either with the first power of the sphere’s speed (at low Reynolds numbers) or with its second power (at high Reynolds numbers). The proportionality constants depend on parameters of the sphere and of the fluid that are in principle all known. It is then straightforward to algebraically deduce the terminal speed of the falling sphere. In a calculus-based course, with a bit more effort one can analytically find the complete mathematical dependence of the sphere’s speed and distance of descent as a function of time for either linear or quadratic drag. An issue which is often glossed over, however, is what happens if the Reynolds number varies over a range from low to high values. Then it would seem reasonable to sum together the linear and quadratic drag forces. This paper compares such a two-term model both to experimental data from previous investigators and to a more detailed model which empirically describes the variation in the drag coefficient over a large range of Reynolds numbers.