Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 39C (Fluid Dynamics II)

The incompressible Stokes equations areTaking the curl gives ${\nabla }^2\omega =0$. For a planar stream function, $\omega =-{\nabla }^2\psi$, so the biharmonic stream function for planar Stokes flow satisfies

For $\psi =r^{2}f(\theta )$,The polar rate-of-strain components areHence

Sincebiharmonicity givesNo slip at $\theta =-\alpha$, no penetration at $\theta =0$, and the imposed tangential traction givePut $\Delta =\sin 2\alpha -2\alpha \cos 2\alpha$. Solving,The upper-surface velocity is radial and equals $r{f}^{\prime }(0)$. Therefore the similarity solution for tangentially forced Stokes wedge gives