Chapter 6, Problem 25P

Repeat Example 6.1, but with the somewhat more realistic assumption that the flow is similar to a free vortex (irrotational) profile, V_θ = c/r (where c is a constant), as shown in Fig. P6.25. In doing so, prove that the flow rate is given by $Q=k\sqrt{\Delta p}$, where k is

$k=w\text{ln}\left(\frac{r_2}{r_1}\right)\sqrt{\frac{2r_2^2{r}_1^2}{\rho (r_2^2-r_1^2)}}$

and w is the depth of the bend.

P6.25