---
title: "Vorticity"
date: 2021-04-03
categories:
- Physics
- Fluid mechanics
- Fluid dynamics
layout: "concept"
---

In fluid mechanics, the **vorticity** $\va{\omega}$
is a measure of the local circulation in a fluid.
It is defined as the curl of the flow velocity field $\va{v}$:

$$\begin{aligned}
    \boxed{
        \va{\omega}
        \equiv \nabla \cross \va{v}
    }
\end{aligned}$$

Just as curves tangent to $\va{v}$ are called *streamlines*,
curves tangent to $\va{\omega}$ are **vortex lines**,
which are to be interpreted as the "axes" that $\va{v}$ is circulating around.

The vorticity is a local quantity,
and the corresponding global quantity is the **circulation** $\Gamma$,
which is defined as the projection of $\va{v}$ onto a close curve $C$.
Then, by Stokes' theorem:

$$\begin{aligned}
    \boxed{
        \Gamma(C, t)
        \equiv \oint_C \va{v} \cdot \dd{\va{l}}
        = \int_S \va{\omega} \cdot \dd{\va{S}}
    }
\end{aligned}$$


## Ideal fluids

For an inviscid, incompressible fluid,
consider the *Bernoulli field* $H$, which is defined as:

$$\begin{aligned}
    H
    \equiv \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho}
\end{aligned}$$

Where $\Phi$ is the gravitational potential,
$p$ is the pressure, and $\rho$ is the (constant) density.
We then take the gradient of this scalar field:

$$\begin{aligned}
    \nabla H
    &= \frac{1}{2} \nabla \va{v}^2 + \nabla \Phi + \frac{\nabla p}{\rho}
    \\
    &= \va{v} \cdot (\nabla \va{v}) - \Big( \!-\! \nabla \Phi - \frac{\nabla p}{\rho} \Big)
\end{aligned}$$

Since $-\nabla \Phi = \va{g}$,
the rightmost term is the right-hand side of
the [Euler equation](/know/concept/euler-equations/).
We substitute the other side of said equation, yielding:

$$\begin{aligned}
    \nabla H
    &= \va{v} \cdot (\nabla \va{v}) - \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
    = \va{v} \cdot (\nabla \va{v}) - \pdv{\va{v}}{t} - (\va{v} \cdot \nabla) \va{v}
\end{aligned}$$

We isolate this equation for $\ipdv{\va{v}}{t}$,
and apply a vector identity to reduce it to the following:

$$\begin{aligned}
        \pdv{\va{v}}{t}
        = \va{v} \cdot (\nabla \va{v}) - (\va{v} \cdot \nabla) \va{v} - \nabla H
        = \va{v} \cross (\nabla \cross \va{v}) - \nabla H
\end{aligned}$$

Here, the definition of the vorticity $\va{\omega}$ is clear to see,
leading us to an equation of motion for $\va{v}$:

$$\begin{aligned}
    \boxed{
        \pdv{\va{v}}{t}
        = \va{v} \cross \va{\omega} - \nabla H
    }
\end{aligned}$$

More about this later.
Now, we take the curl of both sides of this equation, giving us:

$$\begin{aligned}
    \nabla \cross \pdv{\va{v}}{t}
    = \nabla \cross (\va{v} \cross \va{\omega}) - \nabla \cross (\nabla H)
\end{aligned}$$

On the left, we swap $\nabla$ with $\ipdv{}{t}$,
and on the right, the curl of a gradient is always zero.
We are thus left with the equation of motion of the vorticity $\va{\omega}$:

$$\begin{aligned}
    \boxed{
        \pdv{\va{\omega}}{t}
        = \nabla \cross (\va{v} \cross \va{\omega})
    }
\end{aligned}$$

Let us now return to the equation of motion for $\va{v}$.
For *steady* flows where $\ipdv{\va{v}}{t} = 0$, in which case
[Bernoulli's theorem](/know/concept/bernoullis-theorem/) applies,
it reduces to:

$$\begin{aligned}
    \nabla H
    = \va{v} \cross \va{\omega}
\end{aligned}$$

If a fluid has $\va{\omega} = 0$ in some regions, it is known as **irrotational**.
From this equation, we see that, in that case, $\nabla H = 0$,
meaning that $H$ is a constant in those regions,
a fact sometimes referred to as **Bernoulli's stronger theorem**.

Furthermore, irrotationality $\va{\omega} = 0$
implies that $\va{v}$ is the gradient of a potential $\Psi$:

$$\begin{aligned}
    \va{v}
    = \nabla \Psi
\end{aligned}$$

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:

$$\begin{aligned}
    0
    = \nabla \cdot \va{v}
    = \nabla^2 \Psi
\end{aligned}$$

And second, the main equation of motion for $\va{v}$ states
that the quantity $H + \ipdv{\Psi}{t}$ is spatially constant
in the irrotational region:

$$\begin{aligned}
    \pdv{\va{v}}{t}
    = \nabla \pdv{\Psi}{t}
    = - \nabla H
    \quad \implies \quad
    \nabla \Big( H + \pdv{\Psi}{t} \Big)
    = 0
\end{aligned}$$


## References
1.  B. Lautrup,
    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
    CRC Press.