On the Exact Solution for the Two-Sphere Problem in Axisymmetrical Potential Flow

Abstract

The authors obtain an exact solution for the stream function representing the title flow problem when the gaps, the radii, and the velocities of the two spheres are arbitrary. For spheres spaced apart it is in the form of infinite series. For touching spheres that solution reduces by a suitable limit process to an integral form. A general expression is then obtained for the kinetic energy of the system, and using Lagranges equation, the forces experienced by the spheres are calculated. It is thus found that the interactive force between them does not vanish when one is stationary and the other is accelerating at an infinite distance away. Another interesting result concerns the logarithmically singular dependence of the interactive force on the gap when the latter is very small.