Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 1 / 15A (Fluid Dynamics)

Taking the curl of Euler and using incompressibility gives ${\partial }_t\omega +(u\cdot \nabla )\omega =(\omega \cdot \nabla )u$, equivalently $D\omega /Dt=(\omega \cdot \nabla )u$.

The divergence is $-\alpha /2-\alpha /2+\alpha =0$.

For uniform vertical vorticity, advection vanishes and $(\omega \cdot \nabla )u=\alpha \omega \hat{z}$, hence $\dot{\omega }=\alpha (t)\omega$.

$\omega (t)={\omega }_0\exp ({\int }_0^t\alpha (s)ds)$. Positive $\alpha$ stretches material lines vertically; conservation of vortex flux through the shrinking transverse area amplifies vorticity.