Flow in a Corner

The flow of a fluid in the corner is described by the field
defined for
where is a parameter describing the speed of the flow. The potential is
,
and here is a graph of the potential.

Now we plot the vector field together with some representative equipotential curves.

When we work in the corner , we have

and the potential is
.
Its graph is

Now we plot the vector field together with some representative equipotential curves.

Although we have described how these flows vary as the parameter changes, we have not yet described how the vary as the domain varies. In particular, let us consider the flow in the domain , where lies in the range . The cases and have already been discussed.

To begin, the potential in the domain is given by the function
.
The right side is the real part of the complex number . This expression can be simplified when for some integer . Indeed, if is an integer, then

where is the Chebyshev polynomial. The first five Chebyshev polynomials are:

Thus, the potential in the region is
.
We can compare this result to the values previously given for the potentials in and :

Returning to the general case, the velocity field is given by so that

Indeed, we have

We can then plot both the vector field and the potential for a range of values. Here is an animation showing the different fields for .

We can even allow .