Vorticity — ω = ∇ × v, circulation Γ

HOW: deriving vorticity = twice the angular velocity

We want to show curl of velocity literally counts rotation. Consider a 2D flow and a tiny fluid square at origin. Two perpendicular material line segments live along $x$ and $y$ .

Why this step? A pure rotation rotates both line segments the same way; a shear rotates them oppositely. We want the average rotation rate.

The $x$ -segment tip moves with extra $y$ -velocity $\partial v_y/\partial x$ . Its angular velocity (counter-clockwise) is $+\dfrac{\partial v_y}{\partial x}$ .
The $y$ -segment tip moves with extra $x$ -velocity $\partial v_x/\partial y$ . Rotating the $y$ -axis arm counter-clockwise needs $-\dfrac{\partial v_x}{\partial y}$ .

Why this step? Averaging the two arms removes the shear part and keeps pure rotation: $\Omega_z = \frac{1}{2}\left(\frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}\right)$

But the $z$ -component of the curl is exactly $(\nabla\times\mathbf v)_z = \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} = 2\Omega_z.$

HOW: Stokes' theorem links $\Gamma$ and $\boldsymbol\omega$

Take a small rectangle of sides $dx,dy$ . Walk around it counter-clockwise summing $\mathbf v\cdot d\mathbf l$ :

bottom ( $+x$ ): $v_x(x,y)\,dx$
top ( $-x$ ): $-v_x(x,y+dy)\,dx$
right ( $+y$ ): $v_y(x+dx,y)\,dy$
left ( $-y$ ): $-v_y(x,y)\,dy$

Why this step? Pair opposite sides and Taylor-expand the difference: $d\Gamma = \left(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}\right)dx\,dy = \omega_z\, dA.$

Summing all little loops, interior edges cancel, leaving only the outer boundary:

Figure — Vorticity — ω = ∇ × v, circulation Γ

Flashcards

What is vorticity defined as?

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

, the curl of velocity (units s⁻¹).

Vorticity equals how many times the local angular velocity?

Twice:

\boldsymbol\omega=2\boldsymbol\Omega

Define circulation Γ.

\Gamma=\oint_C\mathbf v\cdot d\mathbf l

, the line integral of velocity round a closed curve (units m²/s).

State Stokes' theorem linking Γ and ω.

\Gamma=\oint_C\mathbf v\cdot d\mathbf l=\iint_S\boldsymbol\omega\cdot d\mathbf A

Is a free vortex

v_\theta=k/r

rotational?

No (except at r=0); vorticity is zero, yet circulation

2\pi k

≠ 0.

Vorticity of solid-body rotation

\mathbf v=\Omega(-y,x,0)

\omega_z=2\Omega

Vorticity of simple shear

\mathbf v=(\alpha y,0,0)

\omega_z=-\alpha

(nonzero despite straight streamlines).

Why can Γ=0 around a loop but vorticity be nonzero inside?

Only the net flux of ω is zero; +/− vorticity can cancel.

What does a paddle-wheel test detect?

Whether the fluid element spins (vorticity), not whether its path curves.

Recall Feynman: explain to a 12-year-old

Drop a tiny toy windmill into water. If the windmill spins on its own pin, the water there is "swirly" (has vorticity). If the windmill just floats around in a big circle but its arms never spin, then surprisingly there's no swirl right there — all the swirl is hiding in the very center. Circulation is a different game: walk all the way around a loop and add up how much the water pushes you along — that total is the circulation. The neat magic trick (Stokes) says: the total push around the edge equals all the tiny spins added up inside.

Connections

Curl and Divergence (vector calculus)
Stokes' Theorem
Irrotational Flow and Velocity Potential
Kelvin's Circulation Theorem
Bernoulli's Equation (valid across streamlines only if irrotational)
Lift and the Kutta–Joukowski Theorem ( $L=\rho U\Gamma$ )
Angular Velocity of Rigid Bodies

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Socho ek chhota sa paddle-wheel (pankha) tum paani me daal do. Agar wo apni keel par ghoomne lage, to wahan flow me vorticity hai. Vorticity ka matlab hai fluid ke chhote element ka local rotation, aur formula hai $\boldsymbol\omega=\nabla\times\mathbf v$ — yaani velocity ka curl. Yaad rakho: $\boldsymbol\omega=2\boldsymbol\Omega$ , vorticity angular velocity ka double hota hai.

Doosri quantity hai circulation $\Gamma=\oint \mathbf v\cdot d\mathbf l$ — ek closed loop ke around ghoom kar velocity ka total tangential push add karte ho. Stokes ka theorem dono ko jodta hai: loop ke around ka total circulation = us area ke andar ki saari vorticity ka sum. Bahut khoobsurat hai — local spin ko area par jod do to boundary ka swirl mil jata hai.

Sabse bada confusion: log sochte hain ki agar streamline curve kar rahi hai to vorticity zaroor hogi. Galat! Free vortex $v_\theta=k/r$ me paani circle me ghoomta hai, par vorticity zero hoti hai (center ko chhod kar). Wahan paddle-wheel center ke around revolve to karta hai par khud spin nahi karta. Ulta, simple shear me streamlines bilkul seedhi hoti hain par vorticity nonzero ( $\omega_z=-\alpha$ ). Toh test hamesha paddle-wheel se karo, streamline ke shape se nahi.

Ye cheez kyun important hai? Aviation me lift ka formula $L=\rho U\Gamma$ circulation par chalta hai, aur Bernoulli ka equation across streamlines tabhi simple hota hai jab flow irrotational ho. Toh vorticity aur circulation samajhna real engineering ke liye core funda hai.

Vorticity — ω = ∇ × v, circulation Γ

WHAT are we defining?

HOW: deriving vorticity = twice the angular velocity

HOW: Stokes' theorem links $\Gamma$ and $\boldsymbol\omega$

Worked Examples

Common Mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Fluid Mechanics

Connections