Circulation measures the accumulated component of fluid velocity tangent to a closed curve.

For an oriented closed curve $C$,

$$\boxed{\Gamma_C=\oint_C\mathbf v\cdot d\mathbf r}.$$

The sign depends on the chosen direction around the curve. Circulation has units

$$[\Gamma]=\mathrm{m^2/s}.$$

A positive circulation means that, on balance, the velocity field points in the chosen positive direction around the loop.

Circulation is a global quantity

Vorticity is defined at a point. Circulation belongs to an entire closed curve.

A large loop can accumulate contributions from many different parts of a velocity field, so circulation answers a different question from the local value of $\boldsymbol\omega$.

Stokes' theorem connects the two

Since

$$\boldsymbol\omega=\nabla\times\mathbf v,$$

Stokes' theorem gives

$$\boxed{ \Gamma_C =\oint_C\mathbf v\cdot d\mathbf r =\iint_S\boldsymbol\omega\cdot\mathbf n,dS },$$

where $S$ is any compatible oriented surface whose boundary is $C$.

Thus circulation equals the flux of vorticity through a spanning surface.

If the vorticity component normal to a flat surface is uniform,

$$\Gamma=\omega_n A.$$

Equivalently,

$$\boxed{\frac{\Gamma}{A}=\omega_n}$$

for that uniform case. This makes precise the interpretation of vorticity as circulation per unit area in the infinitesimal limit.

Worked example: solid-body rotation

Consider a fluid in rigid rotation about the $z$ axis with angular speed $\Omega$.

At radius $R$, the tangential speed is

$$v_\theta=\Omega R.$$

Choose a circular contour of radius $R$ traversed counterclockwise. Velocity is tangent to the contour and constant in magnitude, so

$$\Gamma =\oint_C\mathbf v\cdot d\mathbf r =v_\theta(2\pi R).$$

Therefore

$$\boxed{\Gamma=2\pi\Omega R^2}.$$

The vorticity of solid-body rotation is

$$\boldsymbol\omega=2\Omega\hat{\mathbf z}.$$

The circle spans a disk of area

$$A=\pi R^2,$$

so the vorticity flux is

$$\iint_S\boldsymbol\omega\cdot\mathbf n,dS =(2\Omega)(\pi R^2) =2\pi\Omega R^2.$$

This exactly matches the line-integral calculation.

What zero circulation does and does not mean

If vorticity vanishes everywhere on a smooth spanning surface,

$$\boldsymbol\omega=\mathbf0,$$

then Stokes' theorem gives

$$\Gamma_C=0.$$

However, the hypotheses matter. A curve can surround a singularity or a hole where the velocity field is not defined, so no admissible spanning surface lies entirely inside the smooth flow domain. In that case, local zero vorticity away from the excluded region does not by itself force the circulation around the hole to vanish.

This distinction becomes important for ideal vortex flows.

Circulation is not yet a conservation law

The definition

$$\Gamma_C=\oint_C\mathbf v\cdot d\mathbf r$$

says how to measure circulation at one instant. It does not by itself say whether circulation remains constant as a fluid loop moves.

Conservation or evolution of circulation requires additional dynamical assumptions involving forces, density, and viscosity.

Keeping these ideas separate is useful:

• vorticity describes local rotation;
• circulation accumulates tangential velocity around a closed curve;
• Stokes' theorem relates the two geometrically;
• circulation dynamics determines how that quantity changes in time.