Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / K (Fluid Dynamics and Waves)

For incompressible planar flow define the streamfunction by $u={\psi }_y$, $v=-{\psi }_x$. A velocity potential satisfies $u={\varphi }_x$, $v={\varphi }_y$ when the flow is irrotational; the differential condition is $v_x-u_y=0$. The complex potential of a two-dimensional potential flow is $w=\varphi +i\psi$, with derivative $w^{\prime }=u-iv$.

For $w=m\log (z-z_0)/(2\pi )$, its derivative is $m/[2\pi (z-z_0)]$. This gives outward radial velocity $m/(2\pi r)$, whose flux across a circle is $2\pi r\cdot m/(2\pi r)=m$. It is therefore the stated line source.

Superposition givesFor positive $U,m$, the stagnation points areThey lie on the real axis, one upstream of the source and the other downstream of the sink. Streamlines are level sets of $\psi =Uy+(m/2\pi )[\arg z-\arg (z-1)]$. The dividing streamline passes through both stagnation points and forms a Rankine oval enclosing the source and sink; outside, streamlines tend to uniform horizontal flow.

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.