fluid mechanics help

Question

for part a) i get

$$ u = \partial_y \psi, \quad v = - \partial_x \psi$$

I need help with part d, if anyone can show me how to? thanks

Answer 1

Brute force tells us that if $\psi_y = u$, then:

$$ \psi(x,y) = \int u \, \mathrm{d}y + f(x) = \frac{x}{x^2+y^2} + f(x), $$ for some function $f$. Notice now that, on the other hand, if $\psi_x = -v$, then:

$$\psi(x,y) = - \int v \, \mathrm{d}x + g(y) = \frac{x}{x^2+y^2} + g(y), $$ for some function $g$. The computation of the integrals is left to you (check!).

What can we conclude about $f$ and $g$? Is there any obvious choice for them?

Cheers!