= Vorticity
{title2=$\boldsymbol\omega=\nabla\times\mathbf u$}
{wiki}
Use <vorticity> to distinguish the local spin of a <fluid element> from the motion of its center along a <pathline>. Circular <streamlines> can occur both with and without <vorticity>: the two <velocity fields> below even have the same <circulation> around the unit <circle>.
\Image[/vorticity-and-circulation-fields.png]
{title=Solid-body rotation and a potential line vortex: equal unit-circle circulation, different local vorticity}
For a continuously differentiable <velocity field> $\mathbf u=(u_x,u_y,u_z)$, <vorticity> is the <curl> vector
$$
\boldsymbol\omega=\nabla\times\mathbf u=\left(\frac{\partial u_z}{\partial y}-\frac{\partial u_y}{\partial z},\frac{\partial u_x}{\partial z}-\frac{\partial u_z}{\partial x},\frac{\partial u_y}{\partial x}-\frac{\partial u_x}{\partial y}\right).
$$
Each <partial derivative> divides <velocity> by distance, so <vorticity> has units $\mathrm{s^{-1}}$. In a planar <velocity field> independent of $z$, only $\omega_z=\partial_xu_y-\partial_yu_x$ survives. Positive $\omega_z$ points along $+\mathbf e_z$: the <right-hand rule> associates it with counterclockwise spin viewed from positive $z$ toward the plane. https://davidtong.org/pdfs/teaching/fluid-mechanics/fluids1.pdf[David Tong, Fluid Mechanics, section 2.2]{ref}
The local rotational <angular velocity> is half the <vorticity>, $\boldsymbol\Omega_{\rm spin}=\boldsymbol\omega/2$. To see the factor, split the <velocity gradient> $L_{ij}=\partial_ju_i$ into its symmetric <rate-of-strain tensor> $E=(L+L^T)/2$ and antisymmetric <spin tensor> $W=(L-L^T)/2$. A nearby displacement $\delta\mathbf x$ has relative <velocity> $L\delta\mathbf x$; its rotational part satisfies $W\delta\mathbf x=(\boldsymbol\omega/2)\times\delta\mathbf x$. This is a proof outline of the local spin interpretation. The <rate-of-strain tensor> can also change the angle of a particular <material line>, so that line's <angular velocity> need not equal $\boldsymbol\omega/2$.
For <solid-body rotation>, $\mathbf u=(-\Omega y,\Omega x,0)$ gives $\boldsymbol\omega=(0,0,2\Omega)$ and $\boldsymbol\Omega_{\rm spin}=(0,0,\Omega)$. Taking $\Omega=1\ \mathrm{s^{-1}}$, the <vorticity> is $2\mathbf e_z\ \mathrm{s^{-1}}$, although the <angular velocity> is only $1\mathbf e_z\ \mathrm{s^{-1}}$. For the potential <line vortex>, $\mathbf u=K\mathbf e_\theta/r$, the planar <curl> is $r^{-1}\partial_r(ru_\theta)=r^{-1}\partial_rK=0$ wherever $r>0$. Its circular <pathlines> therefore coexist with zero local spin.
This distinction makes <vorticity> useful for recognizing <shear flow> as well as swirling motion; taking the <curl> of the <Euler equations> also removes a <pressure gradient> and leads to the <vorticity equation>. A historical landmark is Hermann von Helmholtz's 1858 paper on <fluid vortex> motion, an early study devoted to <vortex dynamics>. https://zenodo.org/records/1448878[H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen (1858)]{ref}
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