The transverse force on a spinning sphere moving in a viscous fluid

TLDR

The analysis extends classical Stokes and Oseen theory to study the flow of sphere suspensions in tubes. The authors solve the low‑Reynolds‑number flow around a spinning sphere, deriving expressions for the resulting force and torque and presenting an Oseen‑equation representation in terms of two scalar functions. They discover a transverse force perpendicular to the sphere’s motion, independent of viscosity at low Reynolds numbers, which explains the curved trajectories of spinning baseballs and golf balls. The transverse force is given analytically by FL = (expression) involving the sphere radius, angular velocity, translational velocity, fluid density, and Reynolds number.

Abstract

The flow about a spinning sphere moving in a viscous fluid is calculated for small values of the Reynolds number. With this solution the force and torque on the sphere are computed. It is found that in addition to the drag force determined by Stokes, the sphere experiences a force FL orthogonal to its direction of motion. This force is given by .Here a is the radius of the sphere, Ω is its angular velocity, V is its velocity, ρ is the fluid density and R is the Reynolds number, . For small values of R, the transverse force is independent of the viscosity μ. This force is in such a direction as to account for the curving of a pitched baseball, the long range of a spinning golf ball, etc. It is used as a basis for the discussion of the flow of a suspension of spheres through a tube.The calculation involves the Stokes and Oseen expansions. A representation of solutions of the Oseen equations in terms of two scalar functions is also presented.