Coordinates, reference frames, equations, functions, differentials and integrals, yes these can help in mathematical analysis and defining fluid flow or any physical phenomenon. But sometimes visual representation of a fluid flow can provide better understanding of the flow. To visualize a fluid flow Timelines, Pathlines, Streaklines and Streamlines are defined for its velocity field.
These lines are obtained for the actual flow by marking the fluid particles by dye or smoke and then tracing the path of these marked particles as they flow. These flow lines are also defined and obtained mathematically by integrating the velocity field or the vector field for space or time or both with appropriate limits.
Visualization using timelines is started by marking adjacent fluid particles with a dye. Fluid particles are marked in any desired shape or curve at the starting instant called as timeline. Then this timeline is observed in the subsequent motion of the fluid particles in the fluid flow. The changes in the shape timeline as time pass can provide useful information about the variations in the properties of the fluid as time pass.
As its name says a pathline is the path traced by any fluid particle in the flow. To visualize pathline for any fluid particle it is marked with a dye and then observed as it moves. To record the pathline a photograph with prolonged exposure can be taken or one a move of it for analysis. This can help in tracking any particle of interest in the flow.
Streakline is the locus of the positions of the fluid particles, at a particular instant, which have passed through a same fixed point. To obtain a streakline for any given point all the fluid particles passing through that point are marked with a dye or smoke. At any particular instant, after we have started marking fluid particles, these fluid particles can be identified in the flow and the line joining them will be the streakline passing through the given point.
Streamlines are lines which are tangent to the flow velocity vector or the flow direction at every point in the flow field at a particular instant. Streamlines are defined for any given instant and they change with the flow field. There can be no flow across the streamlines as they are tangent to the velocity at every point in the flow. Streamlines are obtained mathematically by integration of the velocity field for space parameter over the region of flow field under consideration.
For a steady flow pathlines, streaklines and streamlines are identical and for an unsteady flow they are not.