Categories: Fluid dynamics, Fluid mechanics, Physics.

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:

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

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. 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 \(\pdv*{\va{v}}{t}\), 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 \(\pdv*{t}\), 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 \(\pdv*{\va{v}}{t} = 0\), in which case Bernoulli’s 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 \(H + \pdv*{\Psi}{t}\) is spatially constant in the irrotational region:

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

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

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