The Helmholtz vorticity theorem
s (first published by Hermann von Helmholtz
in 1858) are two basic results for understanding the nature of vorticity in a fluid flow
The theorems hold exactly only in the Euler
model, i.e. when there is no viscosity
. They are nevertheless useful for understanding vorticity in fluids with low but non-zero viscosity too.
In fluid dynamics it is impossible to avoid vector calculus altogether, but the vorticity theorems are of such a basic nature that you do not really need any advanced tools to understand them, and I will try to explain them so that they make sense and not just formulas.
But I will need to use some specific terms in order to state the theorems before I explain them. A vortex line is a field line of the vorticity field, i.e. a curve that is everywhere tangent to the vorticity. A vortex tube is a set of vortex lines that pass through the same closed curve. The flux of vorticity across a surface is the surface integral of the vorticity; if the vorticity is constant and normal to the surface it reduces to the area times the magnitude of the vorticity.
First Helmholtz vorticity theorem:
Vortex lines move with the fluid.
Second Helmholtz vorticity theorem:
The flux of vorticity across a cross section of a vortex tube is independent of the cross section and time (this is the strength of the vortex tube).
The idea of the first vorticity theorem is rather easy to understand. That the "vortex lines move with the fluid" means that if we dye all the fluid particles that lie on a vortex line at one instant then the dyed particles will at a later time still lie on a vortex line.
The second vorticity theorem may need some additional explanation.
That the flux of vorticity is independent of the cross section is an immediate consequence of the divergence theorem and the fact that the vorticity field is divergence free.
So it is the fact that the flux of vorticity does not change with time that is interesting. This result can also be thought of as saying that the magnitude of the vorticity in a vortex line increases proportionally as the vortex line is stretched. For consider a very thin vortex tube round the vortex line, so thin that the vorticity is practically constant over its width. As the vortex tube is stretched the cross-sectional area decreases by the same factor, so the vorticity must increase proportionally for the flux across the cross section to remain constant.
One place where this effect can be seen is in your very own sink. There is some vorticity in the water, and when it flows down the drain the vortex lines are stretched out. The vorticity increases and the result is a visible vortex.
The theorems can be derived from the Kelvin circulation theorem.
Below follows another proof which may not be excessively rigourous (but then this is not analysis anyway) but immediately demonstrates that the assertions made about the behaviour of the vortex lines are true.
First we look at how the vorticity w evolves in a fluid with velocity field u. It is easy to show from the Euler momentum equation (which essentially expresses Newton's second law for fluids) that
Dw/Dt = (u.∇)w
(see substantial derivative).
Now we wish to compare this with how curves in the fluid move with the flow. For this purpose we look at material line elements, short displacement vectors dl that we consider to constitute the curve. The material line elements are deformed with time, and it is shown in this writeup that the change governed by
Ddl/Dt = (u.∇)dl
So the equations governing the change in vorticity and material line elements have precisely the same form! Thus it is immediately obvious that the vortex lines move and change in magnitude precisely like the material line elements along them.
Another way to see that the second vorticity theorem must be true is to interpret it as an expression of conservation of angular momentum. The pressure exerts no torque on a thin vortex tube, so in the absence of viscous forces the angular momentum of the vortex tube should be conserved. As the vortex tube is stretched it becomes thinner and its moment of inertia decreases, so its angular velocity (which can be said to be measured by the vorticity) must increase proportionally.