Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 5C (Fluid Dynamics)

The divergence is the trace of the velocity-gradient tensor:Thus the flow is incompressible exactly whenThe constant vector $U_0$ is unrestricted.

The vorticity isHence the flow is irrotational exactly whenequivalently when $\Gamma$ is symmetric. Again, $U_0$ is unrestricted.

A streamline $x(s)=(x(s),y(s),z(s))$ satisfieswith $(x(0),y(0),z(0))=(1,0,0)$. ThereforeFor $\beta \ne 0$ this is a helix, with $x=\cos (\alpha z/\beta )$ and $y=-\sin (\alpha z/\beta )$; for $\beta =0$ it is the unit circle in the plane $z=0$.