r$=($x$,$y$,$z$)$

Write down the equations that are solved to give the path of a fluid particle in a flow with Eulerian fluid velocity $u(r,t)=(u,v,w).$ Also write down the equations that are solved to give the streamlines of the same flow.

Given the Eulerian velocity field

$u=(xyt,yt,z{t}^2)$

write down the streamline equations and solve these to determine the general form of the streamline at time $t.$ Deduce that at time $t=2$ the streamline through the point $(1,e,e^2)$ is given by

$x=e^{y-e},z=y^2.$