Flow Around a Cylinder with Circulation

We can combine the previous two phenomena and obtain a flow that flows around a cylinder with circulation. The resulting vector field is
defined for
with the parameters
- velocity of the fluid far from the obstacle, and
- which represents the strength of the vortex.
In this case the general structure of the flow varies depending on the values of the parameters , , and . Indeed, we use the following Mathematica commands to view the field.

To see how changing parameters affect the vector field, we examine the following graphs, where we fix and , and let vary in the range .

Stagnation points

One important thing to notice are the points where . These points are called stagnation points, and are points where the fluid is stationary. We can determine these points by simply solving the equation .

The solutions are
,
and
, .
Clearly, the behavior depends on the sign of . If is small enough that , then only the last two solutions are real, and the resulting pair of stagnation points lie on the obstacle. Indeed, direct calculation shows that in this case :

On the other hand, if is large enough that , then only the first two solutions are real. Exactly one of these solutions lies outside the obstacle, yielding precisely one stagnation point; if it will be above the obstacle if and below the obstacle if . Indeed, if and we have
.
Thus, if we make the change of variables , we see that while the requirement implies that . One can verify algebraically that satisfies and for ; indeed

The motion of the stagnation points as varies can be clearly seen in the animations.