Understanding Flow Field Continuity and Solving for f(r)

[marklar13]

Homework Statement

A flow field is described by

|V| = f(r) ;

x^2 + y^2 = c (streamlines)

What form must f(r) have if continuity is to be satisfied? Explain your results.

Homework Equations

equation of continuity: div V = d(ur)/dr + (ur)/r = 0

where (ur) is the radial velocity

The Attempt at a Solution

I manipulated the continuity equation to be...
-d(ur)/(ur) = dr/r
Then I integrated both sides and got...
1/(ur) = r
Now I'm not sure what to do next or if I'm even on the right path. Can someone that understands this problem give me a hint?

 

[hunt_mat]

What you have implicitly assumed is that as the modulus of te velocity vector field is independent on the angle then that means that the individual components are, which is not the case. So you have to take:
$$\mathbf{V}=u_{r}(r,\theta )\hat{\mathbf{r}}+u_{\theta}(r,\theta )\hat{\mathbf{\theta}}$$
With the property that:
$$\sqrt{u_{r}^{2}+u_{\theta}^{2}}=f(r)$$