Vorticity — ω = ∇ × v, circulation Γ: Visual walkthrough

2.2.29 · D2 · Physics › Fluid Mechanics › Vorticity — ω = ∇ × v, circulation Γ

Yeh page parent result Vorticity ω = ∇ × v, circulation Γ ko bilkul zero se rebuild karta hai, ek picture at a time. Hum koi bhi symbol use nahi karenge jab tak hum usse draw na kar le. End tak tum dekh paoge ki $ω = \nabla \times v = 2 Ω$ kyun hai aur circulation kyun loop ke andar trapped vorticity ke barabar hota hai.

Step 1 — "Velocity field" kya hota hai?

KYA HAI. Ek second ke liye spinning bhool jao. Ek river ke ek patch par dots ki grid imagine karo. Har dot par ek arrow paint karo: lamba arrow = tez paani, arrow ki direction = paani kidhar ja raha hai.

KYUN. Jo kuch bhi hum banayenge — spin, swirl, circulation — yeh sab inhi arrows se padha jaata hai. Agar hum agree nahi karte ki arrows ka matlab kya hai, toh aage ka kuch bhi sense nahi banega.

PICTURE. Neeche, kale arrows field hain. Dhyan do: paani seedhi lines mein bhi move kar sakta hai aur phir bhi ek floating object ko twist kar sakta hai. Yahi surprise is poore page ka point hai — ise yaad rakho.

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

$x, y$ ::: woh do directions jinse hum position measure karte hain (daayein, aur upar).
$v (x, y) = (v_{x}, v_{y})$ ::: point $(x, y)$ par arrow; $v_{x}$ batata hai ki woh daayein kitni tezi se jaata hai, $v_{y}$ upar kitni tezi se.

Step 2 — Paddle-wheel: hum actually kya measure karna chahte hain

KYA HAI. Hum "swirl" jaise vague word ki jagah ek concrete machine laate hain. Do patli rigid arms: ek $x$ ki taraf point karti hai, ek $y$ ki taraf, beech mein join, rotate karne ke liye free.

KYUN. Kisi number ke liye ek measurement recipe chahiye. "Tiny cross ki rotation rate" measurable aur unambiguous hai — unlike "kya flow swirly lagta hai," jo sab ko fool karta hai (baad mein free vortex dekho).

PICTURE. Lal cross paddle-wheel hai. Har arm ki tip center se thoda alag paani mein hoti hai, isliye har tip drag hoti hai. Yahi tip-drag arm ko turn karata hai.

$Ω$ ::: woh (angular velocity) jo hum chahte hain — kitne radians per second cross ghoomta hai, positive = counter-clockwise (CCW).

Step 3 — Ek arm turn karo: $x$ -arm

KYA HAI. $x$ ke saath wali arm dekho. Uski tip pin se ek chhoti si step $Δ x$ daayein hai. Is arm ko tilt karne wali cheez tip aur pin ke beech vertical velocity ka fark hai — kyunki horizontal arm par ek vertical drag use upar ya neeche swing karata hai.

KYUN. Horizontal arm par horizontal push use sirf slide karata hai, rotate nahi kar sakta. Sirf upar/neeche ka velocity change matter karta hai, aur woh kaise change hota hai jab hum $x$ mein move karte hain. Woh quantity aise likhi jaati hai:

$daayein step karne par u pw a r d speed kitni tezi se badhti hai \frac{\partial v _{y}}{\partial x}$

PICTURE. Lal arm; uski tip extra upward velocity feel karti hai, toh arm CCW rate $+ \frac{\partial v _{y}}{\partial x}$ se rotate hoti hai.

$\partial v_{y} / \partial x$ ::: upward velocity $v_{y}$ mein change jab tum $x$ direction mein thoda move karo.
$+$ sign ::: daayein-badhti-upward velocity horizontal arm ko counter-clockwise tilt karti hai.

Step 4 — Doosri arm turn karo: $y$ -arm, aur woh crucial minus sign

KYA HAI. Ab $y$ ke saath wali arm: uski tip pin se ek step $Δ y$ upar hai. Use tilt karne wali cheez horizontal velocity ka fark hai, aur woh kaise change hoti hai jab hum $y$ mein step karte hain: $\frac{\partial v _{x}}{\partial y}$ .

KYUN. Step 3 jaisi hi logic, 90° rotate karke. Lekin picture rotate karne se "CCW kaunsa taraf hai" ka sign flip ho jaata hai. Ek vertical arm CCW rate $- \frac{\partial v _{x}}{\partial y}$ se ghoomti hai.

PICTURE. Lal vertical arm; uski top par extra rightward velocity top ko daayein push karti hai, yani clockwise — isliye leading minus.

$\partial v_{x} / \partial y$ ::: rightward velocity $v_{x}$ mein change jab tum $y$ mein upar move karo.
$-$ sign ::: upar-badhti-rightward velocity vertical arm ko clockwise turn karati hai, CCW ka ulta.

Step 5 — Dono arms ka average lo: shear cancel hoti hai, spin bachta hai

KYA HAI. Paddle-wheel ek rigid object hai, isliye uska actual spin dono arms ki "want" ka average hai:

$Ω_{z} = \frac{1}{2} x -arm, Step 3 \frac{\partial v _{y}}{\partial x} - y -arm, Step 4 \frac{\partial v _{x}}{\partial y}$

KYUN. Average karna anti-symmetric shear part ko khatam kar deta hai (jo cross ko diamond mein stretch karta hai lekin spin nahi karta) aur woh symmetric rotation rakhta hai jis par dono arms agree karti hain.

PICTURE. Left: pure spin — dono lal arms CCW swing karti hain, reinforce karti hain. Right: pure shear — arms opposite swing karti hain, average zero hai chahe cross deform ho.

$Ω_{z}$ ::: cross ki true spin-rate $z$ -axis ke baare mein (page se bahar).
$\frac{1}{2}$ ::: kyunki humne dono arms ka average liya.

Step 6 — Combination ko naam do: yahi curl hai

KYA HAI. Hum bas bracket ko ek naam dete hain. Step 5 se compare karo:

$ω = \nabla \times v = 2 Ω$

KYUN. History (aur vector calculus, dekho Curl and Divergence (vector calculus)) ne curl ko fluids se pehle naam diya tha. Nikla ki velocity field ka curl exactly paddle-wheel ki spin se do guna hota hai. Factor 2 koi fudge nahi hai — yeh Step 5 ka $\frac{1}{2}$ hai jo doosri taraf move ho gaya.

PICTURE. Ek single dial: left par paddle-wheel $Ω$ se spin karta hai, right par vorticity gauge $2Ω$ read karta hai — same rotation, do scales.

$\nabla \times v$ ::: curl operator velocity field par apply kiya gaya — ek machine jo local spin output karti hai.
factor $2$ ::: vorticity ko angular velocity se do guna bade scale par measure kiya jaata hai.

Step 7 — Ek point se loop tak: circulation

KYA HAI. $d l$ loop ke saath ek tiny step-arrow hai. $v \cdot d l$ hai "paani ka arrow mere step ki taraf kitna point karta hai" (dot product, jo do arrows ke aligned parts ko multiply karta hai). $\oint$ ka matlab hai "poora ghoomkar start tak wapas aa jaao aur sab add karo."

KYUN. Step 6 ne ek point par spin diya. Lekin experiments (lift, drains) poore region ke around swirl ki parwah karte hain. Circulation woh global number hai.

PICTURE. Lal loop; har point par kala flow arrow step direction par project hota hai. Aligned = positive push, opposite = negative.

$\oint_{C}$ ::: closed curve $C$ ke around integral, start par wapas aate hue.
$v \cdot d l$ ::: flow ka woh hissa jo walking direction mein hai, step length se multiply karke.

Step 8 — Loops ko stitch karo: Stokes' theorem

KYA HAI. Ek tiny square ke liye area $d A = d x d y$ , uske chaar sides par $v \cdot d l$ sum karna (opposite sides ko Taylor-expand karke) exactly deta hai:

$d Γ = (\frac{\partial v _{y}}{\partial x} - \frac{\partial v _{x}}{\partial y}) d x d y = ω_{z} d A$

Sab squares add karo; interiors cancel; boundary bachti hai:

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

KYUN. Yahi woh promised bridge hai: edge ke around circulation = andar trapped total vorticity. Local spins, jab sum kiye jaate hain, global swirl ke barabar hote hain.

PICTURE. Chhote CCW squares ki grid; shared arrows cancel (grey), lal survivors outer loop trace karte hain.

$d A$ ::: area element page se bahar point karta hua (CCW loop ke saath right-hand rule).
$\iint_{S}$ ::: poori enclosed surface $S$ par sum.

Step 9 — Woh edge cases jinpar kabhi mat phislo

KYA HAI / KYUN / PICTURE (teeno neeche):

Solid-body $v = Ω (- y, x, 0)$ . Dono arms CCW swing karti hain → $ω_{z} = Ω - (- Ω) = 2Ω$ . Rotational, jaisa expected tha.
Free vortex $v_{θ} = k / r$ . Paani clearly circle karta hai, phir bhi outer arm inner arm se itni slow move karti hai ki wheel ka spin cancel ho jaata hai → $ω_{z} = 0$ for $r \neq = 0$ (saari vorticity singular center par chupi hai). Yeh irrotational flow hai — woh famous trap.
Straight shear $v = (α y, 0, 0)$ . Streamlines bilkul seedhi hain, phir bhi $ω_{z} = 0 - α = - α \neq = 0$ . Koi curving path nahi hone ke bawajood rotational hai.

Circling path $\neq =$ spin (free vortex), straight path $\neq =$ no spin (shear). Sirf wheel decide karta hai.

Ek-picture summary

Yeh single figure saare nau steps compress karta hai: do arms → average → curl → $2Ω$ → loops mein tile karo → Stokes → circulation, saath mein teen edge-case verdicts side par pinned.

Recall Poore walkthrough ki Feynman retelling

Humne moving water mein ek tiny plus-sign toy (paddle-wheel) daala. Ek arm daayein point karti hai, ek upar. Daayein-pointing arm tabhi twist hoti hai jab paani ki upward speed change ho jab tum daayein slide karo — yahi $\partial v_{y} / \partial x$ hai, aur yeh arm ko counter-clockwise twist karta hai. Upar-pointing arm rightward speed ke change se twist hoti hai jab tum upar jaate ho — yahi $\partial v_{x} / \partial y$ hai, aur kyunki "counter-clockwise" flip ho jaata hai jab tum picture rotate karo, yeh doosri taraf twist karta hai, ek minus deta hai. Toy ek solid object hai, isliye woh dono arms ka average spin karta hai — yahan se half aata hai: $Ω = \frac{1}{2} (\partial v_{y} / \partial x - \partial v_{x} / \partial y)$ . Mathematicians ke paas un-halved bracket ke liye already ek naam tha: curl, aur yahan ise vorticity kehte hain, isliye vorticity spin se do guna hai. Phir humne ek point dekhna band kiya aur loop ke around poora ghoom gaye, add karte rahe ki kitna paani hamein saath push karta tha — woh total circulation hai. Agar tum andar ko tiny squares mein chop karo, har inside wall ko do baar opposite directions mein walk kiya jaata hai aur cancel ho jaata hai, isliye edge ke around total push andar saare tiny spins add karne ke barabar hai: yahi Stokes' theorem hai, $Γ = \iint ω d A$ . Finally, woh trap: circles mein jaata paani (free vortex) ka zero spin ho sakta hai, aur seedhi lines mein jaate paani (shear) ka bahut zyada spin ho sakta hai. Apni aankhon par kabhi trust mat karo — toy windmill ke pin par trust karo.

Connections

Curl and Divergence (vector calculus)
Stokes' Theorem
Irrotational Flow and Velocity Potential
Kelvin's Circulation Theorem
Lift and the Kutta–Joukowski Theorem
Angular Velocity of Rigid Bodies