3.I.8C

State and prove Kelvin's circulation theorem.

Consider a planar flow in the unbounded region outside a cylinder for which the vorticity vanishes everywhere at time $t=0$. What may be deduced about the circulation around closed loops in the fluid at time $t$ :

(i) that do not enclose the cylinder;

(ii) that enclose the cylinder?

Give a brief justification for your answer in each case.

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