Finding ideal fluid flow with a boundary

Question

I need to find the stream lines of an ideal fluid flow with the given potential of $F(z)=z^2=x^2-y^2+i2xy$ on the upper half of the Cartesian plane.

I have determined that the velocity of the fluid must be $\overline{F'(z)}=2x-i2y$ however this has does not fit the constraint because the fluid has a non-zero imaginary component at the line $\Im(z)=0$ and thus violates the boundary. It seems like you cannot have this potential and this boundary condition because the fluid would be forced to flow through a boundary.

How do I make the fluid fit the boundary? How does one do this for a general potential and boundary condition?

Answer 1

The streamlines should look like as the following:

which belongs to a family of curves $$2xy=c$$

Answer 2

Ok, I figured it out, so heres an answer for anyone confused like me in the future (probably still me).

$F'(z)$ is the derivative with respects to $z$ so the velocity is $\overline{F'(z)} = 2\bar{z}$. This fits our constraint.

Essentially when asked to find the fluid flow of a potential with a constraint, the constraint should not have any impact on the solution other than to provide a domain. Either the potential fits the constraint or it does not. You cannot make a potential fit an arbitrary constraint (unless the potential is zero everywhere), which is where the confusion I had was coming from.