ANALYTICAL SOLUTION OF VISCOUS FLOW BY CREEPING MOTION OF A SPHERE IN CIRCULAR TUBE.

The flow field occurring around a spherical particle moving axially at any given location in a quiescent fluid in a circular cylinder under an external force is theoretically investigated. The Navier-Stokes equations of motion are approximated by low Reynolds numbers and are represented in cylindrical coordinates with the continuity equation. A mathematical analysis is made in the same manner as in the free case. Assuming function which satisfies the Laplace equation and is inversely proportional to the distance from the center of the sphere, the solution of the pressure and each of the velocity components is obtained by satisfying the boundary condition at the surface of a sufficiently small sphere. The boundary conditions at the tube wall are satisfied by this assumed function, which is fully derived from the relation between Bessel function and the Legendre function. The final solution for the pressure and each of the velocity components is shown by a series of Bessel functions, which allow for the normal components of the velocity on the tube wall, except for the cross-sectional plane normal to the tube axis contaning the sphere center.