Point Vortex Flow Around a Cylinder

The vector field

defined for

represents point vortex flow around a cylinder. It is simple to check that however, cannot be generated by a single-valued potential in the entire region . On the other hand, can be generated by a potential in any simply connected region. For example, in the region , we can set

if ,

if
,

if .

We can verify that is a potential for indeed;

so that is continuous.

so that and agree for . When , we resort to the definition to calculate ∇φ;

which proves that is a potential for .

Here is a graph of this potential, at least for and .

We clearly see the discontinuity in the potential near .

We can plot the vector field, together with a selection of equipotential curves as follows.