YEH swirl kyun measure karta hai:u⋅dℓ tab bada aur positive hota hai jab flow loop ke saath us direction mein chalti hai jis direction mein hum traverse kar rahe hain. Use loop ke around sum karne se pata chalta hai ki fluid kitna poori tarah enclosed area ke around rotate karne ki tendency rakhta hai.
Hum chahte hain DtD∮Cu⋅dℓ. Ek material loop do reasons se change hoti hai: har point par velocity change hoti hai, aur line element dℓ flow ke saath stretch/rotate hota hai.
Step 1 — Loop integral ko differentiate karo.DtD∮Cu⋅dℓ=∮CDtDu⋅dℓ+∮Cu⋅DtD(dℓ)Yeh step kyun? Material derivative ke under product rule — integrand vector aur loop ki geometry dono evolve hoti hain.
Step 2 — Stretching term ko handle karo. Ek material line element neighbouring particles ke position vectors ka difference hota hai, isliye yeh velocity gradient ke saath convect hota hai:
DtD(dℓ)=duKyun? Agar do particles x aur x+dℓ par hain, toh unke separation ki rate of change unki velocities ka difference hai, du. Isliye:
∮Cu⋅du=∮Cd(21∣u∣2)=0Zero kyun? Yeh ek closed loop ke around ek perfect differential ka integral hai — shuru aur end same point par hote hain.
Step 3 — Baaki term ke liye Euler equation use karo. Ek inviscid fluid ke liye:
DtDu=−ρ1∇p−∇Φ
Toh
DtDΓ=∮C(−ρ1∇p−∇Φ)⋅dℓ
Step 4 — Potential term ko khatam karo.∮C∇Φ⋅dℓ=∮CdΦ=0 (perfect differential, closed loop). Kyun? Conservative forces ek loop ke around zero net work karti hain.
Step 5 — Barotropicity use karke pressure term ko khatam karo. Agar ρ=ρ(p) hai, toh pressure function P(p)=∫ρ(p)dp define karo. Tab ρ1∇p=∇P, toh
∮Cρ1∇p⋅dℓ=∮C∇P⋅dℓ=∮CdP=0Barotropic kyun matter karta hai: sirf tab jab ρ akele p par depend kare, ρ1∇p ko pure gradient ke roop mein likha ja sakta hai; warna integral zero nahi ho sakta (yahi baroclinic generation of vorticity hai).
Ek inviscid, barotropic flow mein conservative body forces ke saath ek material loop ke liye, circulation Γ=∮u⋅dℓ constant hai: DΓ/Dt=0.
Three assumptions of Kelvin's theorem
(1) inviscid, (2) conservative body force, (3) barotropic (ρ=ρ(p)).
Why does ∮u⋅du=0
Yeh ∮d(21∣u∣2) hai, ek closed loop ke around ek perfect differential ka integral = 0.
What replaces ρ1∇p in barotropic flow
Ek pure gradient ∇P jahan P=∫dp/ρ(p), toh iska loop integral vanish ho jaata hai.
What kills the body-force term
Conservative force =−∇Φ; ∮∇Φ⋅dℓ=0.
Circulation in terms of vorticity
Γ=∬Sω⋅dA (Stokes' theorem).
What breaks Kelvin's theorem
Viscosity (vorticity diffusion) ya baroclinicity (∇ρ×∇p=0).
Physical analogue of Kelvin's theorem
Conservation of angular momentum; vortex stretching = skater ka arms khinchna.
Connection to aerofoil lift
Total Γ=0 conserved rehta hai, toh −Γbound ka ek starting vortex shed hota hai, jisse lift L=ρUΓbound possible hoti hai.
If ω1A1=Γ and area halves, new vorticity
Double ho jaati hai, kyunki Γ conserved hone ka matlab hai ω∝1/A.
Recall Feynman: ek 12-saal ke bacche ko explain karo
Socho ke tumne ek swirling nadiyan ki surface par ek circle draw kiya jisme line par floating pattiyaan hain. Jab nadi pattiyaanon ko ghumaakar le jaati hai, circle stretch hokar ek weird shape mein badal jaata hai. Kelvin ka rule kehta hai: tumhari pattiyaanon ki loop ke andar trapped total spinning-ness bilkul same rehti hai, jab tak paani smooth ho (koi stickiness nahi) aur nicely behave kare. Agar tumhari loop squeeze hokar choti ho jaati hai, andar ka paani tezi se ghoomta hai total spin same rakhne ke liye — bilkul waisi hi tarah jaise ek spinning skater apni arms khinchne par tez ho jaati hai.
