In fluid mechanics, the vorticity
is a measure of the local circulation in a fluid.
It is defined as the curl of the flow velocity field :
Just as curves tangent to are called streamlines,
curves tangent to are vortex lines,
which are to be interpreted as the “axes” that is circulating around.
The vorticity is a local quantity,
and the corresponding global quantity is the circulation ,
which is defined as the projection of onto a close curve .
Then, by Stokes’ theorem:
For an inviscid, incompressible fluid,
consider the Bernoulli field , which is defined as:
Where is the gravitational potential,
is the pressure, and is the (constant) density.
We then take the gradient of this scalar field:
the rightmost term is the right-hand side of
the Euler equation.
We substitute the other side of said equation, yielding:
We isolate this equation for ,
and apply a vector identity to reduce it to the following:
Here, the definition of the vorticity is clear to see,
leading us to an equation of motion for :
More about this later.
Now, we take the curl of both sides of this equation, giving us:
On the left, we swap with ,
and on the right, the curl of a gradient is always zero.
We are thus left with the equation of motion of the vorticity :
Let us now return to the equation of motion for .
For steady flows where , in which case
Bernoulli’s theorem applies,
it reduces to:
If a fluid has in some regions, it is known as irrotational.
From this equation, we see that, in that case, ,
meaning that is a constant in those regions,
a fact sometimes referred to as Bernoulli’s stronger theorem.
implies that is the gradient of a potential :
This fact allows us to rewrite the Euler equations in a particularly simple way.
Firstly, the condition of incompressibility becomes the well-known Laplace equation:
And second, the main equation of motion for states
that the quantity is spatially constant
in the irrotational region:
- B. Lautrup,
Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition,