Flow Around a Cylinder with Circulation

This is the sum of the two previous fields, so

defined for

with the parameters

- radius of the obstacle,

- velocity of the fluid far from the obstacle, and

- which represents the strength of the vortex.

As a consequence, the potential is just the sum of the two previous
potentials. In particular, we can not define a single potential valid
in the entire region , but we can define a potential in the region . Indeed, adding the previous two potentials, we find

if ,

if
,

if .

Here is a graph of this potential.

As before, we clearly see the discontinuity in the potential near . We also note that varies much less rapidly than .

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

Stagnation points in the flow are recognized as points where the equipotential curves cross. Indeed, consider the following.

We see the stagnation point at , .

It is interesting to see how changes in the parameters are reflected in the behavior of the potential and the equipotential curves.