Foundations — Potential flow — irrotational, inviscid; superposition of basic flows

2.2.28 · D1 · Physics › Fluid Mechanics › Potential flow — irrotational, inviscid; superposition of ba

Parent note Potential flow padhne se pehle, aapko har woh symbol earn karna hoga jo woh aap par throw karta hai. Hum unhe ek-ek karke build karte hain, har ek pichhle ke upar. Kabhi bhi hum page par koi mark use nahi karte jo aapne pehle define aur draw hote nahi dekha.

1. Vectors aur velocity field $\vec{v}$

Letter ke upar chhota arrow ($\stackrel{⃗}{}$) humara reminder hai ki is quantity ki ek direction hai, sirf size nahi. Ek number jiske paas sirf size ho (jaise temperature) ek scalar hai; ek quantity jiske paas size aur direction dono hon (jaise velocity) ek vector hai.

2D mein hum arrow ko axes ke saath do pieces mein todte hain:

• $u$ = arrow ka kitna hissa horizontal $x$-axis ki taraf point karta hai,
• $v$ = kitna hissa vertical $y$-axis ki taraf point karta hai.

Toh $\vec{v}=(u,v)$ likhna bas yeh kehna hai "arrow $u$ steps right aur $v$ steps up hai."

2. Gradient $\nabla \varphi$ — ek hill ka slope

Maan lo aapke paas ek scalar map hai: har point par ek single number $\varphi$ (maan lo, ek landscape ki height). Gradient is sawaal ka jawab deta hai: "kaun si taraf sabse steep uphill hai, aur kitna steep?"

Yahan do nayi marks aati hain — chaliye unhe earn karte hain.

• Symbol $\partial$ (ek "curly d") ka matlab hai partial derivative: $\varphi$ ka slope jab aap sirf ek coordinate ko thoda hilate hain aur baaki ko freeze karte hain. $\frac{\partial \varphi }{\partial x}$ = "agar main $x$ mein thodi si step leta hun, $y$ ko fixed rakhte hue, toh $\varphi$ kitna change hota hai?" Yeh rate of change hai, steepness measure karne ka hamaara tool — hum derivatives use karte hain (plain subtraction nahi) kyunki slope har point par aur har direction mein alag ho sakta hai.
• Symbol $\nabla$ ("nabla" ya "del") ek bookkeeping device hai: ise kisi scalar par apply karo aur yeh saare partial slopes ka vector wapas deta hai.

3. Dot product aur divergence $\nabla \cdot \vec{v}$

Yeh poochhne ke liye "kya fluid yahan pile up ho raha hai ya spread ho raha hai?" hume ek point par outward flow add karne ka tarika chahiye. Tool hai dot product jo divergence mein jaata hai.

Ab $\nabla$ ko aise apply karo jaise woh vector $\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y}\right)$ ho aur use $\vec{v}$ mein "dot" karo:

4. Cross product aur curl $\nabla \times \vec{v}$ — local spin

Aakhri derivative ek alag sawaal measure karta hai: "kya ek chhota paddle-wheel yahan drop kiya jaaye toh spin karna shuru karega?"

$\nabla$ aur $\vec{v}$ ke beech $\times$ ("cross") vectors combine karne ka ek doosra tarika hai; flat 2D mein yeh upar diye gaye single difference mein collapse ho jaata hai.

5. Curl-zero ko kaam par lagana: potential exist karta hai

Yahan woh pivot hai jis par poora subject balance karta hai:

YEH kya kehta hai: koi local spin nahi $\Rightarrow$ velocity kisi height-map $\varphi$ ka slope hai. YEH sach kyun hai (intuition): agar kisi bhi closed loop ke around chalna aapko hamesha same "height" par wapas laata, toh ek consistent height-map draw kiya ja sakta hai — aur zero curl exactly woh guarantee hai ki loops mein net twist accumulate nahi hota. YEH kya deta hai: teen velocity numbers $(u,v)$ ek single scalar $\varphi$ mein collapse ho jaate hain. Bahut saare ki jagah solve karne ke liye ek function.

§3 aur §5 se do conditions combine karo:

$\underset{\text{incompressible}}{\underbrace{\nabla \cdot \vec{v}=0}}\text{and}\underset{\text{irrotational}}{\underbrace{\vec{v}=\nabla \varphi }}⟹\nabla \cdot (\nabla \varphi )={\nabla }^2\varphi =0.$

Woh final object ${\nabla }^2\varphi$ — "gradient ka divergence" — Laplacian hai, aur ${\nabla }^2\varphi =0$ Laplace's equation hai. Yeh linear hai, jo superposition ke peeche ka secret hai.

6. Polar coordinates $(r,\theta )$ — kyunki flows round hote hain

Parent sources, vortices aur cylinders describe karta hai — sab naturally circular hain. Toh $(x,y)$ ki jagah hum aksar use karte hain:

• $r$ = origin se doori (kitna bahar),
• $\theta$ = positive $x$-axis se anticlockwise measure kiya gaya angle (kitna ghoom ke).

Matching velocity pieces hain $u_r$ (bahar/andar move karna) aur $u_{\theta }$ (circle ke around move karna). Yaad rakhne layak conversions: $x=r\cos \theta ,y=r\sin \theta ,r=\sqrt{x^2+y^2}.$

Yahi reason hai ki parent ek source ko $u_r=\frac{m}{2\pi r}$ (pure outward) aur ek vortex ko $u_{\theta }=\frac{\Gamma }{2\pi r}$ (pure round-and-round) likhta hai.

7. Stream function $\psi$ aur constants $m,\Gamma ,\kappa ,U$

Baaki letters bas dial settings hain — numbers jo kehte hain "kitna strong":

Yeh pressure story ko Bernoulli's equation ke through, lift story ko Kutta–Joukowski theorem aur Magnus effect ke through, aur shape-bending story ko Conformal mapping ke through feed karte hain.

8. Sab kuch kaise fit hota hai

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