Past exam of the options I course of the University of Oxford / 2012 / K / K1 / c / Solution

The Free surface of a Rankine vortex follows from setting its pressure constant. At a free surface with constant pressure $p_s$, solve the two pressure expressions from part (b) for $z$. With $C=(B-p_s)/(\rho g)$, this givesSubstituting $r^2=x^2+y^2$ recovers the required Cartesian form. The two values coincide at $r=a$, and both radial slopes there equal ${\Gamma }^2/(g{a}^3)$. The surface is parabolic in the core and approaches height $C$ as $r\to \infty$; at the centre its depression below $C$ is ${\Gamma }^2/(g{a}^2)$.