Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 16A (Fluid Dynamics)

For a velocity potential, $u=\nabla \varphi$. Writing $\theta =x-t$ givesIts divergence isThus this potential flow is generally compressible.

Both $\sin (x-t)$ and $\cos (x-t)$ have zero average over one period, so $⟨u⟩=0$ at every fixed point.

The material derivative gives the particle accelerationDirect differentiation yieldsTherefore the Eulerian time average at fixed $(x,y)$ is

The dyed particle satisfies the Lagrangian trajectory equationsFor the proposed approximation, $x=O({\epsilon }^2)$ andAlsoThe initial conditions hold, verifyingthrough order ${\epsilon }^2$.

Over one period, the periodic terms return to their initial values while the secular term changes $x$ by ${\epsilon }^2\pi$. Hence the dyed particle has Stokes driftto this order, despite the zero Eulerian mean velocity.