A fluid motion has velocity potential $\phi(x, y, t)$ given by

$$\phi=\epsilon y \cos (x-t)$$

where $\epsilon$ is a constant. Find the corresponding velocity field $\mathbf{u}(x, y, t)$. Determine $\nabla \cdot \mathbf{u}$.

The time-average of a quantity $\psi(x, y, t)$ is defined as $\frac{1}{2 \pi} \int_{0}^{2 \pi} \psi(x, y, t) d t$.

Show that the time-average of this velocity field at every point $(x, y)$ is zero.

Write down an expression for the fluid acceleration and find the time-average acceleration at $(x, y)$.

Suppose now that $|\epsilon| \ll 1$. The material particle at $(0,0)$ at time $t=0$ is marked with dye. Write down equations for its subsequent motion and verify that its position $(x, y)$ at time $t>0$ is given (correct to terms of order $\epsilon^{2}$ ) as

$$\begin{aligned} &x=\epsilon^{2}\left(\frac{1}{2} t-\frac{1}{4} \sin 2 t\right) \\ &y=\epsilon \sin t \end{aligned}$$

Deduce the time-average velocity of the dyed particle correct to this order.