It is interesting to visualize how/why these lines are different using an example.
Given a time-dependent velocity field such as $v_x = sin(t)$ and $v_y = 1$ it is instructive to look
at how each of these visualization lines look/evolve:
There are a few interesting observations that can be made based on this exercise:
- All three lines are different
- Only the pathline is steady (but, of course, it has to be as it shows the path that a single
particle/fluid element actually takes)
- Looking at the vector field and/or the streamlines give no good impression of what a pathline might actually look
like. This is particularly important to note for mixing applications as many do not realize that mixing should
really be thought of in a Lagrangian sense (like using a pathline) and that a purely Eulerian view (like a streamline) gives
very little insight!