In a sample of fluid that is rotating like a solid body with uniform angular velocity ω0, the vorticity lies in the same direction as the axis of rotation, and its magnitude is equal to 2ω0.
If the vorticity is everywhere zero, then so is the circulation around all possible loops, and vice versa.
Its development with time turns out to be described by the partial differential equation In this situation the vorticity, which may be denoted by the symbol Ω, has one nonzero component, directed along the axis perpendicular to the diagram in Figure 14; it is Ω3 = - ∂v1/∂x2).
Once the slightest trace of vorticity is present, it destroys the conditions on which the proof of Thomson's theorem depends.
Moreover, vorticity admitted at interfaces spreads into the fluid in much the same way that a dye would spread, and whether or not the results of potential theory are useful depends on how much of the fluid is contaminated in the particular circumstances under discussion.
Vorticity-free, or potential, flow would be of rather limited interest were it not for the theorem, first proved by Thomson, that, in a body of fluid which is free of vorticity initially, the vorticity remains zero as the fluid moves.
Each small element of fluid outside the core, if examined in isolation for a short interval of time, appears to be undergoing translation without rotation, and the local vorticity is zero.
Rotation is communicated to the fluid, and in the steady state the circulation around any loop that encloses the spindle (and encloses a layer of fluid adjacent to the spindle within which the vorticity is nonzero and potential theory is inapplicable) has some nonzero value K.
This section is concerned with an important class of flow problems in which the vorticity is everywhere zero, and for such problems the Navier-Stokes equation may be greatly simplified.
Well upstream of the obstacle the fluid is certainly vorticity-free, so it should, according to Thomson's theorem, be vorticity-free around the obstacle and downstream as well.
It suggests that vorticity-free water remains vorticity-free if it is squeezed into a narrow pipe, and this too is plainly nonsensical, for the well-established parabolic profile illustrated by Figure 10 is not vorticity-free.
