Kelvin's circulation theorem

What is circulation?

WHY this measures swirl: $\mathbf{u}\cdot d\boldsymbol\ell$ is large and positive when the flow runs along the loop in the direction we traverse. Summing it around the loop tells us how much the fluid tends to rotate as a whole around the area enclosed.

The theorem

Derivation from scratch

We want $\dfrac{D}{Dt}\oint_C \mathbf{u}\cdot d\boldsymbol\ell$ . A material loop changes for two reasons: the velocity at each point changes, and the line element $d\boldsymbol\ell$ stretches/rotates with the flow.

Step 1 — Differentiate the loop integral. $\frac{D}{Dt}\oint_C \mathbf{u}\cdot d\boldsymbol\ell = \oint_C \frac{D\mathbf{u}}{Dt}\cdot d\boldsymbol\ell + \oint_C \mathbf{u}\cdot \frac{D(d\boldsymbol\ell)}{Dt}$ Why this step? Product rule under the material derivative — both the integrand vector and the geometry of the loop evolve.

Step 2 — Handle the stretching term. A material line element is the difference of position vectors of neighbouring particles, so it convects with the velocity gradient: $\frac{D(d\boldsymbol\ell)}{Dt} = d\mathbf{u}$ Why? If two particles are at $\mathbf{x}$ and $\mathbf{x}+d\boldsymbol\ell$ , the rate of change of their separation is the difference of their velocities, $d\mathbf{u}$ . Therefore: $\oint_C \mathbf{u}\cdot d\mathbf{u} = \oint_C d\!\left(\tfrac12 |\mathbf{u}|^2\right) = 0$ Why zero? It's the integral of a perfect differential around a closed loop — start and end at the same point.

Step 3 — Use the Euler equation for the remaining term. For an inviscid fluid: $\frac{D\mathbf{u}}{Dt} = -\frac{1}{\rho}\nabla p - \nabla\Phi$ So $\frac{D\Gamma}{Dt} = \oint_C \left(-\frac{1}{\rho}\nabla p - \nabla\Phi\right)\cdot d\boldsymbol\ell$

Step 4 — Kill the potential term. $\oint_C \nabla\Phi\cdot d\boldsymbol\ell = \oint_C d\Phi = 0$ (perfect differential, closed loop). Why? Conservative forces do zero net work around a loop.

Step 5 — Kill the pressure term using barotropicity. If $\rho=\rho(p)$ , define the pressure function $P(p)=\displaystyle\int \frac{dp}{\rho(p)}$ . Then $\dfrac{1}{\rho}\nabla p = \nabla P$ , so $\oint_C \frac{1}{\rho}\nabla p\cdot d\boldsymbol\ell = \oint_C \nabla P\cdot d\boldsymbol\ell = \oint_C dP = 0$ Why barotropic matters: only when $\rho$ depends on $p$ alone can $\frac1\rho\nabla p$ be written as a pure gradient; otherwise the integral need not vanish (this is the baroclinic generation of vorticity).

Putting it together: $\frac{D\Gamma}{Dt} = -0 - 0 - 0 = 0\qquad\blacksquare$

Worked examples

Common mistakes

Flashcards

Kelvin's theorem statement

For a material loop in an inviscid, barotropic flow with conservative body forces, the circulation

\Gamma=\oint\mathbf u\cdot d\boldsymbol\ell

is constant:

D\Gamma/Dt=0

Three assumptions of Kelvin's theorem

(1) inviscid, (2) conservative body force, (3) barotropic (

\rho=\rho(p)

Why does

\oint \mathbf u\cdot d\mathbf u=0

It is

\oint d(\tfrac12|\mathbf u|^2)

, the integral of a perfect differential around a closed loop = 0.

What replaces

\frac1\rho\nabla p

in barotropic flow

A pure gradient

\nabla P

with

P=\int dp/\rho(p)

, so its loop integral vanishes.

What kills the body-force term

Conservative force

=-\nabla\Phi

;

\oint\nabla\Phi\cdot d\boldsymbol\ell=0

Circulation in terms of vorticity

\Gamma=\iint_S \boldsymbol\omega\cdot d\mathbf A

(Stokes' theorem).

What breaks Kelvin's theorem

Viscosity (vorticity diffusion) or baroclinicity (

\nabla\rho\times\nabla p\neq0

Physical analogue of Kelvin's theorem

Conservation of angular momentum; vortex stretching = skater pulling arms in.

Connection to aerofoil lift

Total

\Gamma=0

is conserved, so a starting vortex of

-\Gamma_{bound}

is shed, allowing lift

L=\rho U\Gamma_{bound}

\omega_1 A_1=\Gamma

and area halves, new vorticity

Doubles, since

\Gamma

conserved means

\omega\propto1/A

Recall Feynman: explain it to a 12-year-old

Imagine drawing a circle on the surface of a swirling river with floating leaves on the line. As the river drags the leaves around, the circle stretches and twists into a weird shape. Kelvin's rule says: the total spinning-ness trapped inside your loop of leaves stays exactly the same, as long as the water is smooth (no stickiness) and behaves nicely. If your loop gets squeezed smaller, the water inside spins faster to keep the total spin the same — just like a spinning skater speeds up when she pulls her arms in.

Connections

Vorticity and the vorticity equation — Kelvin's theorem in differential form
Euler equation for ideal fluids — the engine of Step 3
Stokes' theorem — links circulation to vorticity flux
Bernoulli's principle — both rely on barotropic + conservative assumptions
Kutta–Joukowski lift theorem — uses conserved circulation for lift
Baroclinic vorticity generation — what happens when barotropicity fails
Helmholtz vortex theorems — vortex lines move with the fluid (corollary)

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Kelvin's circulation theorem ka idea bahut elegant hai. Maan lo tumne fluid ke andar ek closed loop banaya jo paani ke saath-saath move karta hai — yaani loop ke har point pe wahi fluid particle hai (isko material loop kehte hain). Is loop ke around velocity ka line integral $\Gamma=\oint\mathbf u\cdot d\boldsymbol\ell$ ko circulation kehte hain, jo basically loop ke andar ka "total ghoomna" measure karta hai. Theorem kehta hai: agar fluid inviscid (no friction), force conservative, aur flow barotropic ( $\rho$ sirf $p$ pe depend kare) ho, toh ye circulation kabhi change nahi hoti — $D\Gamma/Dt=0$ .

Iska proof simple steps mein nikalta hai. Material derivative lete waqt do cheezein change hoti hain: velocity bhi aur loop ka shape ( $d\boldsymbol\ell$ ) bhi. Stretching wala term $\oint\mathbf u\cdot d\mathbf u$ ek perfect differential hai, isliye closed loop pe zero. Phir Euler equation daalo: pressure term barotropic condition se $\nabla P$ ban jaata hai (zero), aur body force conservative hai toh $\nabla\Phi$ bhi zero. Sab kuch cancel — bachta hai $D\Gamma/Dt=0$ .

Physically socho toh ye angular momentum conservation jaisa hai. Agar loop ka area chhota ho jaaye (vortex stretch ho), toh vorticity badh jaati hai — bilkul skater ki tarah jo haath andar leke tezi se ghoomti hai. Aur ye theorem aeroplane ke lift ka raaz bhi kholta hai: pehle total circulation zero thi, toh jab wing pe bound circulation banti hai, ek starting vortex ulti circulation ke saath pichhe chhoot jaata hai taaki total zero hi rahe.

Important baat: real fluids mein viscosity hoti hai, isliye theorem boundary layers mein exactly apply nahi hota — wahin pe vorticity create/destroy hoti hai. Aur agar flow baroclinic ho (density temperature pe bhi depend kare, jaise weather mein), tab bhi circulation generate ho sakti hai. Toh "I.C.B." — Inviscid, Conservative, Barotropic — ye teen conditions yaad rakhna mandatory hai.

What is circulation?

The theorem

Derivation from scratch

Worked examples

Common mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Fluid Mechanics

Connections