On the Slow Motion of Viscous Liquid in Space between Two Eccentric Spheres

Abstract

By using bispherical coordinates, the non-axisymmetrical Stokes flow of an incompressible, homogeneous viscous liquid in space between two eccentric spheres, has been solved. Two different cases have been considered, namely (a) when the spheres are external to each other, and (b) when one sphere encloses the other. In the second case it has been assumed that the inner sphere rotates with uniform angular velocity about an axis perpendicular to the line of centers and the outer sphere is fixed. A method has been developed which reduces the hydrodynamical problem to the solution of two infinite sets of linear algebraic equations for two coefficient sequences, which may be solved numerically to any degree of accuracy. It has been proved that the resultant force acting upon the spheres is at right angles to the axis of rotation and the line of centers. The effect of the stationary sphere on the force and couple exerted by the liquid on the rotating sphere has been discussed and the results are compared with those of the axisymmetrical case of G.B. Jeffery. The limiting case of a sphere and a plane is given in Appendix II.