Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / K / 2 (K2)

Writing $z=r{e}^{i\theta }$ gives $\varphi =\Gamma \theta /(2\pi )$ and $\psi =-\Gamma \log r/(2\pi )$. Constant streamfunction means constant radius, so streamlines are circles. The tangential velocity is $v_{\theta }=\Gamma /(2\pi r)$, and its anticlockwise circulation is ${\int }_0^{2\pi }{v}_{\theta }rd\theta =\Gamma$.

Use an opposite vortex at $z=-1$ as the image of the vortex at one:On $z=iy$ the two distances are equal, so $\psi =0$ and the wall is a streamline. Its velocity follows from $w^{\prime }=-i\Gamma /[\pi (z^2-1)]$: at the wall $u=0$ and $v=-\Gamma /[\pi (1+y^2)]$.

The Bernoulli equation gives $p-p_{\infty }=-\rho {\Gamma }^2/[2{\pi }^2(1+y^2{)}^2]$. Relative to the ambient pressure force, the fluid's excess force on the wall is $-\int (p-p_{\infty })dy$ in the positive $x$ direction. Using the supplied integral,Thus the wall is attracted toward the fluid and vortex. Subtracting the uniform ambient pressure isolates the finite force due to the vortex.