Why are pathlines streamlines and streaklines same in a steady flow?

Answer 1

In steady flow (when the velocity vector-field does not change with time), the streamlines, pathlines, and streaklines coincide. This is because when a particle on a streamline reaches a point, , further on that streamline the equations governing the flow will send it in a certain direction . As the equations that govern the flow remain the same when another particle reaches it will also go in the direction . If the flow is not steady then when the next particle reaches position the flow would have changed and the particle will go in a different direction.

Answer 2

In a steady flow, all particles passing through a specific point follow the same path over time. Pathlines, streamlines, and streaklines all represent the path taken by individual particles in the flow, with the only difference being how they are visualized. Therefore, in a steady flow, these representations coincide.