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

- radius of the obstacle,

- 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.