Vorticity — ω = ∇ × v, circulation Γ — Hinglish notes

2.2.29 · Physics › Fluid Mechanics

WHAT: hum kya define kar rahe hain?

HOW: vorticity = twice the angular velocity derive karna

Hum dikhana chahte hain ki velocity ka curl literally rotation count karta hai. Ek 2D flow socho aur origin par ek tiny fluid square. Do perpendicular material line segments $x$ aur $y$ ke along hain.

Yeh step kyun? Pure rotation dono line segments ko usi taraf rotate karta hai; shear unhe opposite taraf rotate karta hai. Hum average rotation rate chahte hain.

$x$ -segment ka tip extra $y$ -velocity $\partial v_{y} / \partial x$ se move karta hai. Iska angular velocity (counter-clockwise) hai $+ \frac{\partial v _{y}}{\partial x}$ .
$y$ -segment ka tip extra $x$ -velocity $\partial v_{x} / \partial y$ se move karta hai. $y$ -axis arm ko counter-clockwise rotate karne ke liye chahiye $- \frac{\partial v _{x}}{\partial y}$ .

Yeh step kyun? Dono arms ko average karne se shear part hat jaata hai aur pure rotation bachta hai: $Ω_{z} = \frac{1}{2} (\frac{\partial v _{y}}{\partial x} - \frac{\partial v _{x}}{\partial y})$

Lekin curl ka $z$ -component exactly yahi hai: $(\nabla \times v)_{z} = \frac{\partial v _{y}}{\partial x} - \frac{\partial v _{x}}{\partial y} = 2 Ω_{z} .$

HOW: Stokes' theorem $Γ$ aur $ω$ ko link karta hai

$d x, d y$ sides wala ek chhota rectangle lo. Counter-clockwise iske around $v \cdot d l$ sum karte hue chalo:

bottom ( $+ x$ ): $v_{x} (x, y) d x$
top ( $- x$ ): $- v_{x} (x, y + d y) d x$
right ( $+ y$ ): $v_{y} (x + d x, y) d y$
left ( $- y$ ): $- v_{y} (x, y) d y$

Yeh step kyun? Opposite sides ko pair karo aur difference ko Taylor-expand karo: $d Γ = (\frac{\partial v _{y}}{\partial x} - \frac{\partial v _{x}}{\partial y}) d x d y = ω_{z} d A .$

Saare chhote loops ko sum karne par, interior edges cancel ho jaate hain, sirf outer boundary bachti hai:

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

Worked Examples

Common Mistakes

Flashcards

Vorticity kaise define ki jaati hai?

ω = \nabla \times v

, velocity ka curl (units s⁻¹).

Vorticity local angular velocity se kitne guna hoti hai?

Do guna:

ω = 2 Ω

Circulation Γ define karo.

Γ = \oint_{C} v \cdot d l

, ek closed curve ke around velocity ka line integral (units m²/s).

Γ aur ω ko link karne wala Stokes' theorem batao.

Γ = \oint_{C} v \cdot d l = \iint_{S} ω \cdot d A

Kya free vortex

v_{θ} = k / r

rotational hai?

Nahi (r=0 ke alawa); vorticity zero hai, phir bhi circulation

2 π k

≠ 0.

Solid-body rotation

v = Ω (- y, x, 0)

ki vorticity?

ω_{z} = 2Ω

Simple shear

v = (α y, 0, 0)

ki vorticity?

ω_{z} = - α

(seedli streamlines ke bawajood nonzero).

Loop ke around Γ=0 kyun ho sakta hai lekin andar vorticity nonzero?

Sirf ω ka net flux zero hota hai; +/− vorticity cancel ho sakti hai.

Paddle-wheel test kya detect karta hai?

Kya fluid element spin karta hai (vorticity), na ki uska path curve karta hai.

Recall Feynman: ek 12-saal ke baache ko samjhao

Paani mein ek chhoti toy windmill daal do. Agar windmill apni pin par khud ghoomti hai, toh wahan paani "swirly" hai (vorticity hai). Agar windmill bas ek bade circle mein float karti hai lekin uski arms kabhi spin nahi karti, toh surprisingly wahan koi swirl nahi hai — saari swirl bilkul center mein chupi hui hai. Circulation ek alag game hai: ek loop ke around poora chakkar lagao aur add karo ki paani tumhe kitna push karta hai — woh total hi circulation hai. Kamal ka magic trick (Stokes) yeh kehta hai: edge ke around total push = andar ke saare tiny spins ka sum.

Connections

Curl and Divergence (vector calculus)
Stokes' Theorem
Irrotational Flow and Velocity Potential
Kelvin's Circulation Theorem
Bernoulli's Equation (streamlines ke across valid sirf tab jab irrotational ho)
Lift and the Kutta–Joukowski Theorem ( $L = ρ U Γ$ )
Angular Velocity of Rigid Bodies