advanced34 min studyLesson 174 in Physics

Local rotation and circulation carried by material

Relate curl to circulation and apply Kelvin’s theorem without confusing a moving loop, a fixed contour, and a constant vorticity.

Before you begin

On this page

The question for today

What you’ll understand

Compute vorticity and circulation in a simple rotating flow.
State the material-loop and force assumptions in Kelvin’s theorem.
Explain how vortex stretching can increase vorticity while preserving circulation.

A small paddle wheel and a large loop integral ask different questions about a flow. Vorticity measures local rotational structure; circulation measures the velocity integrated around a chosen closed curve. They are connected by Stokes’ theorem when the field is smooth on a spanning surface. Kelvin’s theorem adds dynamics by following a curve made of the same moving fluid elements, and its qualifications are essential to understanding when circulation can change.

A curved path need not mean nonzero local curl

In three dimensions the curl of velocity defines the vorticity vector. For rigid rotation, vorticity is twice the angular-velocity vector, while a pure local strain can deform a small element without rotating its principal directions in the same way. A trajectory curving through space is not sufficient evidence of vorticity: an irrotational vortex outside its excluded core has circular streamlines but locally vanishing curl there.

Circulation includes the direction in which the contour is traversed. Reversing orientation reverses its sign. Stokes’ theorem equates it to vorticity flux through a consistently oriented spanning surface when that surface lies in the smooth domain. A loop encircling an excluded singular core may have nonzero circulation even when the curl vanishes everywhere accessible nearby; using a surface through the excluded core would violate the theorem’s hypotheses.

\boldsymbol\omega=\nabla\times\mathbf u,\qquad \Gamma_C=\oint_C\mathbf u\cdot d\boldsymbol\ell=\int_S\boldsymbol\omega\cdot\mathbf n\,dS

The last equality requires a smooth field on an allowed spanning surface S with boundary C and consistent orientation. Circulation has units area per time, while vorticity has inverse-time units.↔ Scroll wide equations horizontally.

The loop in Kelvin’s theorem must move with the fluid

Differentiate the circulation of a material loop. Both the velocity and its moving line element change; omitting the second contribution breaks the proof. Euler’s equation supplies acceleration. For a smooth inviscid flow with a barotropic pressure law and conservative body forces, the terms reduce to line integrals of single-valued gradients, which vanish on the closed loop. Barotropic means density can be expressed as a function of pressure in the region concerned.

The conserved quantity is circulation around that transported material curve, not around an arbitrary fixed circle in the laboratory. Viscous stress, nonconservative forces, baroclinic generation when pressure and density gradients are misaligned, or a shock can invalidate the simple result. A fluid initially irrotational stays so in a smooth region under the theorem’s assumptions, provided its material loops remain within the region of validity.

\frac{d}{dt}\oint_{C(t)}\mathbf u\cdot d\boldsymbol\ell=0\qquad\text{(smooth, inviscid, barotropic; conservative body forces)}

C(t) consists of the same fluid elements at every time. The theorem concerns circulation even though the loop’s shape, enclosed area, and local vorticity may all change.↔ Scroll wide equations horizontally.

Stretch a vortex without destroying its circulation

In constant-density incompressible Euler flow, the material vorticity equation contains a stretching and tilting term. A vortex tube can lengthen and narrow, increasing vorticity magnitude while keeping its flux through a material cross section constant. This is compatible with circulation conservation. In a strictly two-dimensional incompressible flow with no out-of-plane variation or velocity, the stretching term vanishes, giving a different vorticity evolution from a general three-dimensional flow.

A useful diagnostic separates three operations: compute the local curl, integrate around an explicitly identified contour, and state whether that contour is material. Do not infer conservation from a single snapshot or a visually persistent spiral. Boundary layers can generate vorticity at no-slip walls even when the outer region is nearly inviscid, so an ideal outer-flow theorem cannot be extended through a viscous wall region without checking the omitted stresses.

Work it through

Circulation in rigid rotation

In the plane let $\mathbf u=(-\Omega y,\Omega x)$ with $\Omega=2\,\mathrm{s^{-1}}$ . Find vorticity and counterclockwise circulation around a circle of radius $0.5\,\mathrm m$ centered at the origin. Assume the field extends smoothly across the disk.

Differentiate the Cartesian components: $\omega_z=\partial_xu_y-\partial_yu_x=\Omega-(-\Omega)=4\,\mathrm{s^{-1}}$ . The angular rotation rate itself is half the vorticity, not the whole vorticity.
On the circle the tangential speed is $\Omega R=1\,\mathrm{m\,s^{-1}}$ everywhere. Integrating along the counterclockwise direction gives speed times circumference: $\Gamma=1\times2\pi(0.5)=\pi\,\mathrm{m^2\,s^{-1}}$ .
Check with the area integral: $\omega_z\pi R^2=4\pi(0.25)=\pi\,\mathrm{m^2\,s^{-1}}$ . The agreement checks orientation and the factor of two; it does not by itself prove a time-dependent conservation law.

The result

The vorticity is 4 s⁻¹ out of the plane and counterclockwise circulation is π m²/s. Reversing the loop orientation changes only the circulation sign.

The living notebook

Explore the model

Interactive lab

Interactive lab · fields

Separate source arrows from rotational arrows

Set q to zero and increase rotation Ω from zero. Predict that the Field magnitude there readout at (1, 0) grows linearly with |Ω|. The tangential arrows lengthen qualitatively because their visible lengths are compressed; they reverse when Ω changes sign. Then restore q to see a radial component added to the prescribed vector field.

Local field directionSource strength

Fx at (1, 0): 0.833
Fy at (1, 0): 0.5
Field magnitude there: 0.972

Source strength q1Rotation Ω0.5

t = 0 s

The softened illustration is q·r/(r² + 0.2) + Ω(−y, x). Moving markers indicate local direction, not integrated particle paths or a Maxwell solution. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe rotation term is rigid planar rotation, with curl 2Ω. The softened source term is not a globally incompressible source-free flow, and the lab does not evolve a material loop or demonstrate Kelvin conservation.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Circulation is the vorticity flux. Under the stated uniform-normal approximation it is vorticity times area, so halving area doubles vorticity. Incompressibility preserves material volume, allowing length increase and cross-section decrease.

Your turn

Try it yourself

A uniform vortex tube carries circulation $0.6\,\mathrm{m^2\,s^{-1}}$ through a normal area of $0.03\,\mathrm{m^2}$ . Find its initial vorticity and its vorticity after the area becomes $0.02\,\mathrm{m^2}$ with the same circulation. Name one mechanism that could prevent using Kelvin’s simple theorem.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Use circulation equal to uniform normal vorticity times section area; distinguish this calculation from assumptions about the flow dynamics.

Compare with the solution

The initial magnitude is $0.6/0.03=20\,\mathrm{s^{-1}}$ , and the final value is $0.6/0.02=30\,\mathrm{s^{-1}}$ . The increase is consistent with stretching, not evidence that circulation grew. Viscous forces, a shock, a nonconservative body force, or baroclinic pressure-density structure can invalidate the simple Kelvin assumptions. A fixed loop rather than a material loop would also change the question.

Keep the idea

Take it with you

Vorticity is a local curl; circulation is an oriented contour integral.
Kelvin conservation follows a material loop under smooth inviscid barotropic dynamics with conservative forces.
Vorticity can grow through three-dimensional stretching while its material flux remains constant.

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