Circulation Vortex lines We have already written down the general equations for the flow of an incompressible fluid when there may be vorticity. The physical content of these equations has been described in words by Helmholtz in terms of three theorems. First, imagine that in the fluid we were to draw vortex lines rather than streamlines. By vortex lines we mean field lines that have the direction of and have a density in any region proportional to the magnitude of .
Vortex lines are like lines of , they never start or stop, and tend to go in closed loops.
This means that if you mark the fluid particles along some vortex lines then as the fluid moves and carries those particles along, they will always mark the new positions of the vortex lines. That is one description of the laws that also suggests a method for solving any problems. Given the initial flow pattern, say everywhere, then you can calculate . From the you can also tell where the vortex lines are going to be a little later, they move with the speed . With the new you can use I and II to find the new .