Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 2 / 5A (Fluid Dynamics)

With $u={\psi }_y$, $v=-{\psi }_x$, one may take $\psi =(\alpha +t)y^2/2-x$.

At fixed $t$, streamlines satisfy $(\alpha +t)y^2/2-x=C$: right-opening parabolas, more tightly curved at $t=1$ than at $t=0$.

The relative coefficient change is $t/\alpha$, so for $t\ll \alpha$ the flow is approximately steady and its streamlines are $\alpha y^2/2-x=C$.