Explain how a streamfunction $\psi$ can be used to represent in Cartesian Coordinates an incompressible flow in two dimensions. Show that the streamlines are given by $\psi=$ const.

Consider the two-dimensional incompressible flow

$$\mathbf{u}(x, y, t)=(x+\sin t,-y)$$

(a) Find the streamfunction, and hence the streamlines at $t=\frac{\pi}{2}$.

(b) Find the path of a fluid particle released at $t=0$ from $\left(x_{0}, 1\right)$. For what value of $x_{0}$ does the particle not tend to infinity?