Vorticity — ω = ∇ × v, circulation Γ: Question bank

2.2.29 · D5 · Physics › Fluid Mechanics › Vorticity — ω = ∇ × v, circulation Γ

Shuru karne se pehle, teen words jo neeche baar baar aate hain, simple language mein:

Agla picture woh single visual hai jo tumhe is poore page ke liye mind mein rakhna chahiye — woh paddle-wheel test jo spinning aur sirf going around ko alag karta hai.

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

Neeche do flow families baar baar aati hain. Unki definitions aur coordinates ek baar yahan fix kar lo, taaki baad mein koi symbol mystery na rahe:

Definition Poore note mein use hone wale symbols

Polar coordinates $(r, θ)$ : $r$ ek chosen center se doori hai, $θ$ uske around angle hai. Velocity component $v_{θ}$ radius $r$ ke circle ke saath speed hai (tangential, m/s mein). Radius $r$ ka ek loop $2 π r$ circumference rakhta hai.
Free vortex $v_{θ} = \frac{k}{r}$ : fluid ek center ke around circle karta hai, paas aane par speed badhti hai. Constant $k$ (units m²/s) vortex strength hai; ye circulation se $Γ = 2 π k$ , yaani $k = \frac{Γ}{2 π}$ ke zariye fix hota hai. Toh $k$ bas $2 π$ ke saath packaged circulation hai.
Simple shear $v = (α y, 0, 0)$ : horizontal flow jiska speed height $y$ ke saath linearly badhta hai. Constant $α$ (units s⁻¹) shear rate hai — ek constant velocity gradient $\partial v_{x} / \partial y = α$ . Seedhi streamlines, lekin neighboring layers ek doosre ke past slide karti hain.

True or false — justify

Curved streamline ka matlab hamesha yeh hai ki wahan flow mein vorticity hai.

False. Curving path ko describe karta hai; vorticity describe karta hai ki fluid element apne axis par khud spin karta hai ya nahi. Free vortex

v_{θ} = k / r

mein circular paths hain phir bhi

r \neq = 0

ke liye zero vorticity hai.

Straight streamlines zero vorticity guarantee karti hain.

False. Simple shear

v = (α y, 0, 0)

(constant shear rate

α

) mein bilkul seedhi, parallel streamlines hain lekin vorticity

ω_{z} = - α \neq = 0

hai; shear fluid element ke diagonal ko tilt kar deta hai, isliye woh rotate karta hai.

Agar ek loop ke around circulation zero hai, toh uske andar har jagah vorticity zero hai.

False. Stokes sirf net flux

\iint ω \cdot d A = 0

force karta hai; positive aur negative vorticity ki equal amounts cancel ho sakti hain.

ω = 0

conclude karne ke liye tumhe har loop par

Γ = 0

chahiye, aur region simply connected hona chahiye (koi holes nahi).

Vorticity aur angular velocity ek hi quantity hain.

False. Dono ek hi direction mein point karte hain lekin

ω = 2 Ω

; vorticity twice angular velocity hai, kyunki yeh do perpendicular arms ke rotation ko average karta hai aur har ek

Ω

contribute karta hai.

Vorticity ek vector hai, aur circulation ek scalar hai.

True.

ω = \nabla \times v

ek vector field hai (units s⁻¹) jiska direction right-hand rule se set hota hai;

Γ = \oint v \cdot d l

ek given loop ke liye ek single number hai (units m²/s).

Ek flow kisi point ke around circulate kar sakta hai phir bhi us point ko chhodkar har jagah irrotational ho sakta hai.

True. Free vortex bilkul yahi karta hai:

r > 0

ke liye irrotational, saari vorticity singular center mein jam ke, phir bhi kisi bhi enclosing loop ke around

Γ = 2 π k

Agar ek fluid element deform nahi ho raha, toh flow irrotational hona chahiye.

False. Solid-body rotation

v = Ω (- y, x, 0)

elements ko rigidly around le jaata hai bina unhe deform kiye, phir bhi yeh maximally rotational hai jisme

ω_{z} = 2Ω

hai.

Circulation depend karta hai ki tum loop ke around kis taraf chalte ho.

True. Orientation reverse karne par har

d l

flip ho jaata hai, isliye

Γ

ka sign badal jaata hai. Magnitude same rehta hai; sign track karta hai ki tum swirl ke saath circle kar rahe ho ya uske against (right-hand rule).

Free vortex mein, circulation loop ke radius ke saath badhti hai.

False. Yeh radius se independent hai:

Γ = (k / r) (2 π r) = 2 π k

. Speed

1/ r

ki tarah exactly itni tezi se girta hai ki total swirl constant rahe.

Ek jaisi streamline shape wale do flows mein vorticity bhi same hogi.

False. Streamlines direction dikhati hain, speed nahi. Free vortex aur solid-body rotation dono mein circular streamlines hain lekin vorticity behaviour opposite hai (zero vs.

2Ω

); streamline ke saath speed decide karti hai.

Spot the error

"Streamlines left ko bend karti hain, isliye wahan drop kiya gaya paddle-wheel counter-clockwise spin karega."

Error hai path curvature ko spin se equate karna. Paddle-wheel spin karega ya nahi yeh depend karta hai velocity gradient par (kya opposite arms alag speeds feel karte hain?), streamline ke curve hone par nahi.

Γ = \oint v \cdot d l = 0

, isliye koi vortex enclosed nahi hai."

Galat:

Γ

net enclosed vorticity flux measure karta hai. Andar do counter-rotating vortices

Γ = 0

de sakti hain even though vortices clearly present hain. Net zero ≠ kuch nahi hai.

"Solid-body rotation

v = Ω (- y, x, 0)

ke liye, vorticity

ω_{z} = Ω

hai."

Factor 2 missing hai.

ω_{z} = \partial_{x} v_{y} - \partial_{y} v_{x} = Ω - (- Ω) = 2Ω

. Ise bhool jaane par baad mein jo bhi circulation compute karo woh half ho jaayega.

"Is vortex flow mein Bernoulli's constant har jagah same hai, isliye main koi bhi do points compare kar sakta hoon."

Bernoulli's constant tab hi equal guaranteed hai jab flow irrotational ho, aur woh bhi sirf streamlines ke across. Rotational flow mein yeh streamlines ke beech differ kar sakta hai — dekho Bernoulli's Equation aur Irrotational Flow and Velocity Potential.

"Free vortex mein paddle-wheel center ke around ja raha hai, isliye isko vorticity hai."

Yeh revolve karta hai (center ko orbit karta hai) lekin apne pin par spin nahi karta — faster inner side aur slower outer side isko opposite torque dete hain jo cancel ho jaate hain. Revolving spinning nahi hai; vorticity zero hai.

"Circulation ke units velocity jaisi hain, kyunki yeh velocity ka integral hai."

Nahi — yeh velocity ka length ke upar integral hai: (m/s)·m = m²/s, m/s nahi. Line element

d l

ek extra length dimension laata hai.

"Kyunki vorticity mein velocity ke derivatives hain, poore velocity field ko double karne par vorticity bhi double ho jaayegi."

Is baar reasoning actually sahi hai:

ω = \nabla \times v

linear hai, isliye

v

ko constant se scale karne par

ω

(aur

Γ

) bhi usi constant se scale hoti hai. Koi error nahi — trap hai ek valid linearity ko doubt karna.

Why questions

Hum vorticity ke liye angular velocity ko 2 se kyun multiply karte hain, sirf

Ω

use kyun nahi karte?

Kyunki curl do perpendicular material arms ke rotation ko average karta hai aur har arm

Ω

rate se rotate karta hai; unka sum

2Ω

hai, aur yeh factor

ω

ko exactly curl ke roop mein nikalta hai.

Do perpendicular arms ko average karna shear ko kyun hata deta hai lekin rotation ko rakhta hai?

Pure rotation dono arms ko ek hi taraf ghoomata hai (they add), jabki pure shear unhe opposite taraf ghoomata hai (they cancel on averaging). Toh average genuine rotation ko isolate karta hai, deformation ko discard karta hai.

Stokes' theorem humein loop integral ko area integral se swap kyun karne deta hai?

Kyunki ek subdivided region ka har interior edge do baar opposite directions mein traverse hota hai aur cancel ho jaata hai; sirf outer boundary bachti hai, isliye tiny circulations

ω_{z} d A

ka sum boundary circulation ke barabar hai. Dekho Stokes' Theorem.

Free vortex "classic surprise" example kyun hai?

Kyunki intuition kehta hai circling flow swirly honi chahiye, phir bhi center ko chhodkar har jagah uski vorticity zero hai — yeh dikhata hai ki path curvature aur element spin genuinely alag cheezein hain.

Free vortex ki circulation depend kyun nahi karti ki hum kaunsa loop pick karein (jab tak woh center ko enclose kare)?

Saari vorticity

r = 0

par ek delta-like spike hai; iske around koi bhi loop same total flux capture karta hai, isliye

Γ = 2 π k

size ya shape se regardless rehta hai.

Γ = 0

se irrotational flow conclude karne se pehle domain ko simply connected hona kyun zaroori hai?

Ek hole (jaise vortex core) ek loop ko hidden vorticity ke around circle karne deta hai jise tum shrink nahi kar sakte; sirf tab jab har loop fluid se bahar gaye bina ek point tak contract ho sake,

ω = 0

har jagah follow hota hai. Dekho Irrotational Flow and Velocity Potential.

Wing par lift compute karne ke liye vorticity kyun useful hai?

Lift total bound circulation par depend karti hai via

L = ρ U Γ

; airfoil ke around concentrated vorticity woh

Γ

set karti hai. Dekho Lift and the Kutta–Joukowski Theorem.

Kelvin's theorem vorticity ki parwah kyun karta hai?

Yeh kehta hai ideal flow mein ek material loop ke around circulation conserved rehti hai, matlab vorticity fluid mein "frozen" hai aur uske saath carry hoti hai — viscosity ya forces ke bina net swirl create ya destroy nahi ho sakta. Dekho Kelvin's Circulation Theorem.

Edge cases

Rest mein pade fluid (

v = 0

har jagah) ki vorticity kya hai?

Zero — zero field ka curl zero hota hai, aur koi paddle-wheel spin nahi karta. Trivially irrotational.

Free vortex ki vorticity exactly

r = 0

par kya hoti hai?

Yeh singular (infinite/undefined) hai — saara swirl center par ek point vortex ke roop mein concentrated hai, isliye flow har jagah irrotational hai wahan ko chhodkar.

Agar circulation loop zero area tak shrink ho jaaye, toh

Γ

kya approach karta hai?

Yeh kisi bhi smooth flow ke liye zero approach karta hai, kyunki

Γ \approx ω_{z} d A \to 0

; ratio

Γ/ d A

local vorticity

ω_{z}

approach karta hai, isliye vorticity "circulation per unit area" hai.

Kya vorticity nonzero ho sakti hai jabki kisi particular loop ke around circulation zero ho?

Haan — aisa loop choose karo jo equal positive aur negative vorticity enclose kare taaki fluxes cancel ho jaayein; local

ω

nonzero hai even though us loop ki

Γ = 0

hai.

Kya uniform flow

v = (U, 0, 0)

(har jagah constant) rotational hai?

Nahi — saari velocity derivatives vanish ho jaati hain, isliye

ω = 0

. Ek paddle-wheel sirf translate karta hai bina spin kiye; seedhi parallel equal-speed streamlines koi vorticity nahi carry karti.

Uniform flow aur simple shear mein kya fark hai, kyunki dono mein straight streamlines hain?

Uniform flow mein har particle same speed se move karta hai (koi gradient nahi,

ω = 0

); shear mein speed rate

α

ke saath flow ke across vary karti hai, isliye element ke opposite arms alag speeds feel karte hain aur woh rotate karta hai (

ω_{z} = - α

Solid-body rotation mein, kya circulation loop radius ke saath badhti hai?

Haan —

Γ = ω_{z} \cdot π R^{2} = 2Ω π R^{2}

R^{2}

ke saath badhta hai, kyunki vorticity poore area par uniformly spread hai, free vortex ke unlike jahan yeh sirf center par baithi hai.

Kya ek non-simply-connected region (hole wala fluid) mein flow irrotational ho sakta hai phir bhi nonzero circulation ho sakti hai?

Haan —

r = 0

par punctured free vortex fluid mein har jagah irrotational hai, phir bhi hole ke around loops ki

Γ = 2 π k \neq = 0

hai, precisely kyunki loop hole ke past contract nahi ho sakta.

Recall Yahan har trap ki one-line summary

Path ≠ spin; net flux ≠ pointwise zero; $ω = 2 Ω$ (woh 2 kabhi mat bhoolna); irrotationality ke liye simply connected domain mein sabhi loops par $Γ = 0$ chahiye; free vortex woh exception hai jo sab prove karta hai.

Connections

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