Consider the three different, steady two$-$dimensional incompressible flow fields illustrated. The flows represent:

a$)$ a parallel flow in a channel with a linear profile,

b$)$ a flow in an annulus with the liquid rotating like a solid body with the inner and outer walls, c$)$ an inviscid flow in a bend of width $b$ which is small compared with the mean radius of curvature R and the flow is uniform before entering the bend.

A vorticity meter like the one in the NSF fluid dynamic videos is placed in each flow oriented so the arrow is upwards in our drawings. Find the $1)$ clockwise angular rate of rotation $d\frac{}{d}t$ and $2)$ the pressure difference $p_2-p_1.$