Stagnation Flow - Part II

In my last post, I mentioned that we’d be working with the stream function \(\Psi\), rather than the complex potential \(\Phi\), but I haven’t really defined it well. You’ll remember that \(\Phi\) is related to the velocity field as follows:
$${{d\Phi}\over{dz}}=\overline{V}=u-iv$$
where u and v are the velocities along the x and y axes, respectively. In plotting streamlines, I’ve noted that the imaginary part of \(\Phi\) is \(\Psi\).
$$\Phi =\phi + i\Psi$$
We shall ignore \(\phi\)(little "fee"), as its properties where friction is a factor are the reason we’re tossing \(\Phi\) aside in the first place! So how does \(\Psi\) relate to velocity?
$$\begin{align} u-iv & = {{d\Phi}\over{dz}}\\ & = {{d\Phi}\over{dx}}\\ & = {{d\phi}\over{dx}}+i{{d\Psi}\over{dx}}\\ -iv & = i{{d\Psi}\over{dx}}\\ v & = -{{d\Psi}\over{dx}}\end{align}$$
Above we let dz = dx. Now we let dz = idy.
$$\begin{align} u-iv & = {{d\Phi}\over{dz}}\\ & = {{d\Phi}\over{i dy}}\\ & = {{d\phi}\over{i dy}}+i{{d\Psi}\over{i dy}}\\ & = -i{{d\phi}\over{i dy}}+{{d\Psi}\over{dy}}\\ u & = {{d\Psi}\over{dy}}\end{align}$$
Now we have velocity in terms of the stream function, which will prove to be useful in my next posting, in which we plug the stream function into Navier-Stokes.