Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 16D (Fluid Dynamics)

The initially irrotational inviscid flow remains irrotational, so $u=\nabla \varphi$; incompressibility gives ${\nabla }^2\varphi =0$. At the plates, normal velocity matches their motion:With $\varphi =r^{2}f(\theta )$, Laplace's equation gives $f^{\prime \prime }+4f=0$. The symmetric solution isHenceA streamfunction isso streamlines are $r^2\sin 2\theta =\text{constant}$ and fluid is expelled radially as the plates close. The outward flux through $r=R$ is