Flow around a sphere in a circular tube

Abstract

The vector potential for the flow of an ideal fluid through a tube containing a concentric spherical obstacle is found for ratios of sphere radius to tube radius of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95. The flow is confined to the space between sphere and cylinder by thin vortex sheets of variable strength and a table of their circulation intensity on the spherical surface is given. Accuracies vary from about one part in 108 for small spheres to one part in 107 for large ones. The increase in the scalar velocity potential between the ends of the tube caused by the insertion of the sphere is expressed in terms of the effective increase in tube length. This also gives the increase in resistance of a solid conducting cylinder due to the presence of a concentric spherical bubble.