Velocity-Potential Integrals

2.2.1 3D potentials

The velocity fields of the

One complication here is that the arctan( ) polar angle can contain some arbitrary multiple of 2n. This requires introduction of a branch cut extending from the vortex point out to infinity in some direction, as shown in Figure 2.8. The angle jumps by 2n and the potential jumps by Г across the branch cut.

The branch cut also appears for a doublet sheet, which has the following potential in 2D.

However, the branch cut now is only on the doublet sheet itself. It does not need to extend to infinity like with a vortex, unless the doublet sheet itself extends to infinity. Figure 2.9 compares the branch cuts of a point vortex and a doublet sheet, the latter being equivalent to two point vortices of opposite sign.