Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 2 / 7C / Solution

The velocity field is incompressible becauseWith the convention $u_x={\psi }_y$ and $u_y=-{\psi }_x$, a stream function isThe vorticity is vertical:For a steady inviscid incompressible flow, the vorticity equation is $(\mathbf{u}\cdot \nabla )\mathbf{\omega }=(\mathbf{\omega }\cdot \nabla )\mathbf{u}$. Here the right side is zero because the flow is independent of $z$, while the left side is $2(\mathbf{u}\cdot \nabla \psi ){\mathbf{e}}_z=0$ because velocity is tangent to the level sets of $\psi$. Thus the equation is satisfied.

The streamlines are the level curves $\sin x\sin y=\text{constant}$. The lines $x=\pi$ and $y=\pi$, together with the boundary, are separatrices; each of the four cells contains closed nested curves around a centre at $(\pi /2,\pi /2)$, $(3\pi /2,\pi /2)$, $(\pi /2,3\pi /2)$, or $(3\pi /2,3\pi /2)$.