Circulation and Flux in a Velocity Field
Letbe a smooth path of lengthin the domain of a flow velocity functionand letbe such thatis the equivalent unit speed parametrization of(so that).
The circulation ofalongiswhereis evaluated along the contour
The flux ofacross
For eachhas magnitude 1 sofor somethenand
If for a velocity functionaround every closed curve in the fluid whose motion is described by the velocity field then the velocity field is said to be circulation free.
Example: The flux ofacross the unit circle is
The circulation ofalongis