The circulation being always zero round a small plane curve passing through the axis of spin in vortical motion, it follows conversely that a vortex filament is composed always of the same fluid particles; and since the circulation round a cross-section of a vortex filament is constant, not changing with the time, it follows from the previous kinematical theorem that αω is constant for all time, and the same for every cross-section of the vortex filament.
A vortex filament must close on itself, or end on a bounding surface, as seen when the tip of a spoon is drawn through the surface of water.
Denoting the cross-section α of a filament by dS and its mass by dm, the quantity ωdS/dm is called the vorticity; this is the same at all points of a filament, and it does not change during the motion; and the vorticity is given by ω cosεdS/dm, if dS is the oblique section of which the normal makes an angle ε with the filament, while the aggregate vorticity of a mass M inside a surface S is
M−1 ∫ ω cos ε dS.
Employing the equation of continuity when the liquid is homogeneous,
| 2 ( | dζ | − | dη | ) = ∇2u, ... , ∇2 = − | d2 | − | d2 | − | d2 | , |
| dy | dz | dx2 | dy2 | dz2 |
(10)
which is expressed by
∇2 (u, v, w) = 2 curl (ξ, η, ζ), (ξ, η, ζ) = ½ curl (u, v, w).
(11)