Foundations — Kelvin's circulation theorem

2.2.30 · D1 · Physics › Fluid Mechanics › Kelvin's circulation theorem

Is page pe kuch bhi assumed nahi hai. Parent note Kelvin's circulation theorem mein har arrow, symbol, aur squiggle yahan unpack kiya gaya hai, ek aise order mein jahan har idea sirf usse pehle wale ideas pe rely karta hai.

1. Ek fluid, aur velocity field $\mathbf{u}$

Picture: ek nadi ki kalpana karo jo upar se photograph ki gayi ho, jisme paani ki surface ke har point pe ek tiny arrow chipka ho. Lambe arrows = tez flow; arrow ki direction = us paani ka tukda kahan ja raha hai.

• Bold letters jaise $\mathbf{u}$ ka matlab hai ek vector — ek cheez jisme size aur direction dono hain (ek arrow).
• Plain letters jaise $t$ ya $p$ ka matlab hai ek scalar — sirf ek number, koi direction nahi.

Topic ko iske liye kyun chahiye: Kelvin's theorem ek statement hai ki ek loop ke around arrange kiye gaye arrows time ke saath kaise behave karte hain. Arrows ka field hoga hi nahi toh integrate karne ke liye kuch nahi hoga.

2. Ek closed curve $C$ aur line element $d\mathbf{\ell }$

Picture: paani pe pada ek rubber band. Kisi bhi point pe zoom karo aur band ek seedha tiny arrow jaisa dikhega — woh arrow hi $d\mathbf{\ell }$ hai.

• Symbol ${\oint }_C$ ka matlab hai "closed loop $C$ ke puri tarah ghoom ke kuch add karo." Integral sign pe chhota circle ek reminder hai ki path closed hai.

Topic ko iske liye kyun chahiye: hum ek loop ke around chalenge aur, har tiny step pe, poochhenge "kya fluid mere saath flow kar raha hai ya mere khilaf?" Yeh poochne ke liye, hume step arrows $d\mathbf{\ell }$ chahiye.

3. Dot product $\mathbf{u}\cdot d\mathbf{\ell }$

Hume ek aisa tool chahiye jo ek specific sawaal ka jawab de: "Fluid ki velocity ka kitna hissa mere walking step ke saath point karta hai?" Dot product exactly wahi tool hai — isliye yeh appear hota hai aur, maan lo, lengths ka multiplication nahi.

$\cos \theta$ padhna:

• Agar arrows same way point karein ($\theta =0$, $\cos 0=1$): product sabse bada aur positive hoga.
• Agar woh perpendicular hain ($\theta =90^{\circ }$, $\cos 90^{\circ }=0$): product zero hai — tumhare step ke across flow ka koi count nahi.
• Agar woh opposite point karein ($\theta =180^{\circ }$, $\cos 180^{\circ }=-1$): product negative hai — tum current ke khilaf chal rahe ho.

Topic ko iske liye kyun chahiye: $\mathbf{u}\cdot d\mathbf{\ell }$ tab bada aur positive hota hai jab fluid loop ke saath exactly us tarah run karta hai jis tarah hum traverse karte hain. Yahi "swirl" ka raw ingredient hai.

4. Circulation $\Gamma$ — swirl ko puri tarah add karna

Picture: agar fluid loop ke around drain mein spiral hote paani ki tarah circle kare, toh har step ek positive bit contribute karta hai aur $\Gamma$ bada hota hai. Agar fluid sirf loop ke past seedha slide kare, toh ek taraf ke "mere saath" wale bits doosri taraf ke "mere khilaf" wale bits ko cancel kar dete hain, aur $\Gamma =0$.

Topic ko iske liye kyun chahiye: $\Gamma$ hi conserved quantity hai. Baaki sab machinery hai yeh prove karne ki ki $\Gamma$ nahi badlta.

5. Vorticity $\mathbf{\omega }$ aur curl $\nabla \times \mathbf{u}$

Circulation ek poore loop ke baare mein hai. Vorticity ek single point tak shrunk wahi idea hai: "fluid theek yahan kitni tez spin kar raha hai?"

Picture: flow mein ek tiny paddle-wheel daalo. Agar woh spin kare, toh $\mathbf{\omega }$ wahan nonzero hai; jitna tez spin kare, arrow utna hi lamba; axle direction $\mathbf{\omega }$ ki direction hai.

• $\nabla$ ("nabla" ya "del") derivatives ka ek bundle hai — ek "yeh space mein kitni tez change ho raha hai?" operator. Yahan yeh cross-product form $\nabla \times$ mein appear karta hai; baad mein hum isse $\nabla p$ ("gradient of pressure") ke roop mein milenge.
• Do arrows ka cross product $\times$ teesra arrow deta hai jo dono ke perpendicular hota hai; curl us geometry ko reuse karta hai rotation axis pull out karne ke liye.

Topic ko iske liye kyun chahiye: Kelvin's theorem ko usually "vorticity is conserved along material loops" ke roop mein summarize kiya jaata hai. Vorticity local view hai; circulation loop view hai. Agla section unhe link karta hai.

Deeper details Vorticity and the vorticity equation mein hain.

6. Stokes' theorem — the bridge

Naye symbols padhna:

• $S$ koi bhi surface hai jiska rim loop $C$ hai — rubber band pe tana soap film socho.
• ${\iint }_S$ ka matlab hai "poori surface pe add up karo" (ek double sum, kyunki surface two-dimensional hai).
• $d\mathbf{A}$ us surface ka ek tiny patch hai, ek arrow ke roop mein likha gaya jo patch se seedha bahar point karta hai, jiska length patch ka area hai.
• $\mathbf{\omega }\cdot d\mathbf{A}$ isliye sirf woh spin count karta hai jiska axis patch ke through thread karta hai — dot product phir se aligned part pick out karta hai.

Topic ko iske liye kyun chahiye: yeh "loop ka circulation" aur "loop ke through vorticity flux" ke beech freely swap karne deta hai — aur worked example $\Gamma =\omega A$ kuch nahi hai sirf ek small flat patch ke liye Stokes' theorem. Full statement: Stokes' theorem.

7. Material derivative $\frac{D}{Dt}$

"Kuch kitni tez change ho raha hai?" poochne ke do tarike hain:

• $\frac{\partial }{\partial t}$: ek fixed spot pe baitho aur value ko change hote dekho (Eulerian view).
• $\frac{D}{Dt}$: fluid particle ke saath ride karo aur value ko change hote dekho jaise tum move karte ho (Lagrangian view).

Picture: ek nadi mein float karta thermometer. Chahe naadi ka temperature map time mein frozen ho ($\partial /\partial t=0$), thermometer ki reading phir bhi change hoti hai jaise woh warm patch se cold patch mein drift karta hai — yahi $\mathbf{u}\cdot \nabla$ piece hai.

Topic ko iske liye kyun chahiye: poora theorem ek single statement hai $\frac{D\Gamma }{Dt}=0$. Loop material hai — same particles se bana — toh hume unke saath move karte hue differentiate karna hoga.

8. Pressure $p$, density $\rho$, aur gradient $\nabla p$

• Pressure $p$ (scalar): ek point pe fluid kitna force per unit area bahar push karta hai.
• Density $\rho$ ("rho", scalar): har unit volume mein pack ki gayi mass.
• Gradient $\nabla p$ (vector): ek arrow jo us direction mein point karta hai jahan pressure sabse tez increase hota hai, jiska length kitna steeply rise hota hai uske equal hai. Fluid ko pressure gradient neeche push kiya jaata hai, high $p$ se low $p$ ki taraf — isliye $-\frac{1}{\rho }\nabla p$ mein minus sign hai.

Picture: ek pahaad jiska height pressure hai; $\nabla p$ sabse-steep-uphill arrow hai; ek ball (fluid parcel) opposite way roll karta hai, downhill.

Topic ko iske liye kyun chahiye: Euler equation kehta hai ek fluid parcel pressure gradients aur body forces ki wajah se accelerate karta hai. Kelvin's proof ke do steps entirely $\frac{1}{\rho }\nabla p$ term ko vanish karne ke baare mein hain.

9. Euler equation

Plain words: (ek parcel ka acceleration) = (pressure differences se push) + (body forces jaise gravity se push). Yahan $\Phi$ ("Phi", scalar) ek conservative body force $\mathbf{f}=-\nabla \Phi$ ka potential hai, e.g. gravity deta hai $\Phi =gz$.

• "Inviscid" = frictionless; koi viscosity term nahi hai. Real, sticky fluids ek add karte hain, aur exactly wahi Kelvin's theorem ko break kar sakta hai.

Topic ko iske liye kyun chahiye: proof mein $D\mathbf{u}/Dt$ ko is right-hand side se replace kiya jaata hai, phir har surviving piece ko perfect-differential argument se khatam kiya jaata hai. Full note: Euler equation for ideal fluids.

10. Perfect differentials aur closed-loop trick

Yeh ek idea proof mein teen alag-alag baar heavy lifting karta hai.

Picture: ek mountain trail hike karo jo car park pe wapas loop kare. Tumne chadhai aur utrai ki, lekin tumhara net altitude change exactly zero hai. Altitude hai $f$; har chhoti chadhai/utrai hai $df$.

Topic ko iske liye kyun chahiye: inme se har ek ek perfect differential hai, isliye har loop integral zero hai:

• $\oint \mathbf{u}\cdot d\mathbf{u}=\oint d(\frac{1}{2}|\mathbf{u}{|}^2)=0$ (stretching term),
• $\oint \nabla \Phi \cdot d\mathbf{\ell }=\oint d\Phi =0$ (body-force term),
• $\oint \frac{1}{\rho }\nabla p\cdot d\mathbf{\ell }=\oint dP=0$ (pressure term, sirf barotropic hone pe).

11. Barotropic vs baroclinic

Sirf ek pure gradient ek perfect differential $dP$ ho sakta hai; yahi saari wajah hai ki proof ka step 5 kaam karta hai.

Topic ko iske liye kyun chahiye: yeh assumption (3) hai, aur iska fail hona sabse physically important loophole hai (weather fronts, sea breezes).

Prerequisite map

Equipment checklist

Inhe use karo yeh test karne ke liye ki tum parent theorem ke liye ready ho — right side cover karo aur answer do.