= Circulation
{disambiguate=physics}
{title2=$\Gamma_C=\oint_C\mathbf u\cdot d\mathbf l$}
{wiki}
= Circulation
{synonym}
Use <circulation> to summarize how much a <velocity field> runs along an oriented <closed curve> $C$, such as a <circle> around a <fluid vortex>. Its definition is the <line integral>
$$
\Gamma_C=\oint_C\mathbf u\cdot d\mathbf l,
$$
where $d\mathbf l$ is tangent to $C$ in the chosen direction. Reversing that direction reverses the sign of <circulation>. Its units are $\mathrm{m^2\,s^{-1}}$, because <velocity> is multiplied by a length. Counterclockwise <circulation> in the horizontal plane is positive when the chosen <unit normal vector> is $+\mathbf e_z$, according to the <right-hand rule>.
For <solid-body rotation> $\mathbf u=(-\Omega y,\Omega x,0)$, a counterclockwise <circle> of radius $R$ has constant tangential <velocity> $\Omega R$. Multiplying by its circumference gives
$$
\Gamma_C=2\pi\Omega R^2.
$$
For the potential <line vortex> $\mathbf u=K\mathbf e_\theta/r$, the same <circle> instead gives $\Gamma_C=(K/R)(2\pi R)=2\pi K$, independent of $R$. Stipulate $\Omega=1\ \mathrm{s^{-1}}$, $R=1\ \mathrm m$ and $K=1\ \mathrm{m^2\,s^{-1}}$: both <velocity fields> have $\Gamma_C=2\pi\ \mathrm{m^2\,s^{-1}}\approx6.283185\ \mathrm{m^2\,s^{-1}}$, although their <vorticities> in the fluid are $2\mathbf e_z\ \mathrm{s^{-1}}$ and zero respectively. On a <circle> of radius $2\ \mathrm m$, their <circulations> are $8\pi\ \mathrm{m^2\,s^{-1}}$ and $2\pi\ \mathrm{m^2\,s^{-1}}$. The comparison plot in <vorticity> shows this difference in radius dependence.
The bridge between <circulation> and <vorticity> is <Stokes theorem>:
$$
\Gamma_{\partial S}=\int_S\boldsymbol\omega\cdot\mathbf n\,dS.
$$
Here the <velocity field> must be continuously differentiable on a neighborhood of the entire spanning surface $S$ inside the fluid, and its boundary orientation must match the <unit normal vector> $\mathbf n$. For <solid-body rotation>, the disk is admissible and its <surface integral> is $2\Omega\pi R^2$. For the potential <line vortex>, the origin is singular and excluded from the fluid: a disk crossing it is not admissible. On an <annulus>, <Stokes theorem> instead gives $\Gamma_{R_{\rm out}}-\Gamma_{R_{\rm in}}=0$, with both <circulations> here measured counterclockwise; the induced inner boundary direction is clockwise. Nonzero <circulation> therefore does not imply nonzero local <curl> in the available fluid. https://davidtong.org/pdfs/teaching/fluid-mechanics/fluids1.pdf[David Tong, Fluid Mechanics, section 2.2, circular-flow example]{ref}
This is useful because <Stokes theorem> lets a local <vorticity> measurement determine a loop's <circulation> when a suitable surface exists, while <Kelvin's circulation theorem> tracks <circulation> along a moving <material curve>. The beauty is the precise link between a local <derivative> and a global <line integral>, with the domain deciding which conclusions are valid.
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