Worked examples — Kelvin's circulation theorem

This is the drill-yard for Kelvin's circulation theorem. The parent note told you what the theorem says and why it is true. Here we hit it from every angle: shrinking loops, stretching loops, zero-swirl loops, loops where the theorem fails, and the exam twists that trap people. Before every worked example you will be asked to forecast the answer — commit to a guess, then check yourself.

Before anything else we pin down every word and symbol the theorem uses, so you never have to hunt back to the parent.

Everything here rests on two facts from the parent:

Recall The two facts we reuse everywhere

Fact A — Circulation is conserved on a material loop. For an inviscid, barotropic flow with conservative body forces, using the material derivative $D/Dt\equiv \partial /\partial t+\mathbf{u}\cdot \nabla$ defined above, $\frac{D\Gamma }{Dt}=0,\Gamma ={\oint }_C\mathbf{u}\cdot d\mathbf{\ell }.$ Here ${\oint }_C\mathbf{u}\cdot d\mathbf{\ell }$ adds up, all the way around the loop, how much the velocity $\mathbf{u}$ points along each line element $d\mathbf{\ell }$. Fact B — Circulation equals vorticity flux (Stokes' theorem): $\Gamma ={\iint }_S\mathbf{\omega }\cdot d\mathbf{A}.$ If the vorticity is roughly uniform over a small flat loop of scalar area $A$ with perpendicular vorticity component $\omega$, this collapses to the pocket formula $\boxed{\Gamma =\omega A}$.

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The scenario matrix

Every problem Kelvin's theorem can throw at you lands in one of these cells. The examples below are labelled by the cell they hit, and together they fill every cell.

#	Cell (case class)	What changes	Theorem's verdict	Example
C1	Area shrinks, positive $\omega$	$A\downarrow$	$\omega \uparrow$ (spin-up)	Ex 1
C2	Area grows	$A\uparrow$	$\omega \downarrow$ (spin-down)	Ex 2
C3	Zero circulation start	$\Gamma =0$ fixed	stays $0$ forever	Ex 3
C4	Sign flip of enclosed swirl	loop turns inside-out	$\Gamma$ keeps its signed value	Ex 4
C5	Degenerate loop (area $\to 0$)	$A\to 0$	$\omega \to \infty$, but is $\Gamma$ finite?	Ex 5
C6	Real-world word problem (aerofoil)	starting vortex shed	${\Gamma }_{\text{total}}=0$	Ex 6
C7	Theorem FAILS — baroclinic	$\nabla \rho \times \nabla p\ne 0$	$D\Gamma /Dt\ne 0$	Ex 7
C8	Theorem FAILS — viscous	friction diffuses $\omega$	$D\Gamma /Dt\ne 0$	Ex 8
C9	Exam twist — fixed vs material loop	wrong loop chosen	trap!	Ex 9

We first define the sign convention we will use in every figure, because a wrong sign is the single most common mistake here.

Figure s01 draws this: the chalk loop with blue walk-arrows running counter-clockwise, the pale-yellow normal $\hat{\mathbf{n}}$ marked as coming out of the board at the centre, and the note that a clockwise flow flips the sign to $\Gamma <0$. Refer to the blue arrows whenever you need to fix which way is "positive".

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C1 — Area shrinks, positive swirl (spin-up)

Figure s02 shows the two states side by side: a big loop (light blue fill, few yellow spin-arrows, ${\omega }_1=5$) shrinking to a small loop (pink fill, four times as many yellow spin-arrows, ${\omega }_2=20$). The density of yellow arrows is the vorticity — denser arrows in the smaller loop is the spin-up made visible.

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C2 — Area grows (spin-down)

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C3 — Zero circulation stays zero

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C4 — Signed circulation and the sign flip

Figure s03 draws the big chalk loop with a blue counter-clockwise patch on the left ($+6$) and a pink clockwise patch on the right ($-6$); the yellow banner across the top states the signed sum $+6+(-6)=0$. The arrow directions — not their count — carry the signs here.

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C5 — Degenerate limit: area $\to 0$

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C6 — Real-world word problem: the aerofoil

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C7 — Where the theorem FAILS: baroclinic generation

Figure s04 draws $\nabla p$ as a blue arrow pointing down (toward higher pressure at the ground), $\nabla \rho$ as a pink arrow pointing right (toward the dense sea air), and the resulting yellow counter-clockwise circulation loop; the caption states $\nabla \rho \times \nabla p$ points out of the page so $\Gamma$ grows.

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C8 — Where the theorem FAILS: viscosity

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C9 — Exam twist: fixed loop vs material loop

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Recall

Recall Which cell am I in? (decision check)

Loop shrinks, positive $\omega$ ::: C1 — spin-up, $\omega$ rises Loop grows ::: C2 — spin-down, $\omega$ falls Flow from rest, ideal ::: C3 — stays irrotational forever Equal-and-opposite swirls in one loop ::: C4 — signed sum, total $\Gamma =0$ Area $\to 0$ ::: C5 — $\omega \to \infty$ but $\Gamma$ finite (point vortex) Wing generates lift ::: C6 — starting vortex $-{\Gamma }_{\text{bound}}$ shed $\nabla \rho \times \nabla p\ne 0$ ::: C7 — baroclinic, Kelvin fails Viscous boundary layer ::: C8 — diffusion, Kelvin fails "Fixed loop violates Kelvin" ::: C9 — wrong loop; theorem needs a material loop