PROBLEM $11\mathrm{.}3$ HELMHOLTZ VORTEX THEOREMS

As shown, an inviscid incompressible fluid flows through a $4$:$1$ contraction in a two$-$dimensional channel. Upstream and downstream of the contraction, the flow is uniform. A vortex A$,$ initially oriented in the Y direction and of average vorticity $1\mathrm{.}5$ units, is introduced into the flow upstream of the contraction. Reference any theorems you use. $($Note that in inviscid flow, a vortex line may end at a non$-$rotating solid surface, since there is no $\text{'}$no$-$slip' condition.$)$

What theorems apply?

Sketch the vortex as it will appear at locations $B$ and $C.$

Give quantitative estimates of the angle at which the vortex meets the lower and upper walls of the channel at B$.$

Determine the average vorticity at $C.$

Ignoring the no$-$slip condition, at what angle $($in degrees$)$ will the vortex meet the lower solid surface at A$,$ B$,$ and $C$ and the upper free surface.