fluid dynamicsmaple
I wonder if anybody could help with this. I've been asked to plot the streamlines of the complex potential $\Omega(z)=Uz + \frac{m}{2\pi}ln(z)$ to which I get the stream function $\psi(r,\theta)=rUsin\theta + \frac{m}{2\pi}\theta$.
Hopefully that is correct, but I cannot seem to plot it using polar coords in Maple. I would like to plot a series of streamlines of $\psi = constant.$ Any help appreciated, thanks!
plot3d(x/sqrt(x^2+y^2), x=0..3, y=x/3+2..3+sqrt(9-x^2),
axes=box, orientation=[-120,30,0]);
plot3d(x/sqrt(x^2+y^2), x=0..3, y=x/3+2..3+sqrt(9-x^2),
axes=box, orientation=[-120,30,0], filled=true);
Yes you could do that, but it's probably easier to use the expression of the gradient in polar coordinates:
$$ u_r =\frac{\partial\phi}{\partial r} =\frac{1}{r}\frac{\partial\psi_z}{\partial \theta} , u_\theta=\frac{1}{r}\frac{\partial\phi}{\partial \theta}=-\frac{\partial \psi_z}{\partial r} $$
Both $u_r$ and $u_\theta$ should be zero.
As you probably recognized, this is the potential flow of a source/sink at the origin and a free stream parallel to $x$.
Best Answer
And for $\psi$, I think you can consider it as a function of two variables and plot it by the following code: