Drag on a sphere in unsteady motion in a liquid at rest

A sphere was subjected to a simple harmonic motion in an otherwise undisturbed liquid. Records of the resistance of the liquid to the motion for various amplitudes and frequencies were obtained. The resistance was first represented by an equation consisting of three terms with empirical coefficients: the steady-motion drag, a term due to the ‘added mass’ and a term due to the history of the motion. It was found that the data could be correlated only with a large degree of scatter by this type of equation. Subsequently an attempt was made to represent the resistance by means of a single term, with an empirical coefficient C. It was found that C correlated well with the acceleration number Vd/V2 and the Reynolds number Vd/v, where V, V and d are the acceleration, velocity and diameter of the sphere respectively and v is the kinematic viscosity of the liquid. C increased with Vd/V2 and decreased in the limit to the steady-motion drag coefficient Cd when Vd/V2 became very small. The range of the Reynolds number in the experiments was 102 < Vd/v < 104 and the range of the acceleration number was 0 ≤ Vd/V2 ≤ 10·5.