I am having a fluid-dynamics problem. I am given the flow:
$$u_{\theta}=a, u_{r}=b$$
in polar coordinates.
I am interested in finding a set of curves which have tangets defined by the vector field $\vec{u} = (u_{\theta},u_r)$.
How can I do that?
What I've tried
I have tried to switch to caartesian coordinates and solve the problem there.
$$u_x=b\cos\phi - a \sin\phi$$ $$u_y=b\sin\phi + a \cos\phi$$
If I have a curve, let's say $\vec{r}(x,y)$, then I express tangentiality by:
$$\frac{d\vec{r}}{ds} \times \vec{u} = 0$$
for arbitrary curve parametrization $s$.
This leads me to:
$$\frac{dr_x}{u_x}=\frac{dr_y}{u_y}$$
($r_x$ and $r_y$ being the $x$ and $y$ components of $\vec{r}$.)
However, I follow through with this method, the expression I end up with is not separable in $x$ and $y$:
$$bydx+axdx=bxdy-aydy$$
So I don't know how to achieve my goal.