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

2.2.29 · D1 · Physics › Fluid Mechanics › Vorticity — ω = ∇ × v, circulation Γ

Parent note Vorticity padhne se pehle, tumhe har woh symbol apna banana hoga jo woh tumhare saamne phenkta hai. Hum har ek ko zero se build karenge, ek aisi order mein jahan har step sirf pehle se defined cheezein use kare.

1. "Field" kya hota hai? (har cheez ke neeche ki picture)

Bold letter $\mathbf{v}$ ka matlab hai ek vector — ek quantity jisme size aur direction dono hote hain, arrow ke roop mein draw kiya jaata hai. Ek plain letter jaise $\Gamma$ (bold nahi) ka matlab hai ek scalar — sirf ek number, koi direction nahi.

Figure dekho: har grid crossing par ek chota arrow baitha hai. Lambe arrows = tez flow, chhote = slow. Arrows ka yeh carpet hi velocity field hai. Is topic mein sab kuch ek sawaal hai jo tum is carpet ke baare mein poochh rahe ho.

2. Components: ek arrow ko $v_x$ aur $v_y$ mein kaat-na

Chhoti hats $\hat{x},\hat{y},\hat{z}$ unit arrows hain — exactly $1$ length ke arrows jo har axis ke along point karte hain. Yeh woh "rulers" hain jinse hum direction measure karte hain. Likhna $\mathbf{v}=v_x\hat{x}+v_y\hat{y}+v_z\hat{z}$ bas yeh kehna hai: "$v_x$ steps east lo, $v_y$ steps north, $v_z$ steps upar." Woh sum tumhe arrow ki tip par le jaata hai.

3. Partial derivative $\frac{\partial }{\partial x}$ — "koi cheez kitni tezi se change hoti hai jab main sideways step leta hoon?"

Yeh woh single sabse important tool hai jo parent note assume karta hai. Hum ise zero se build karte hain.

Yeh exact tool kyun, ordinary derivative kyun nahi? Ek ordinary derivative $\frac{dv}{dx}$ tab kaam karta hai jab $v$ ek variable par depend karta ho. Humari velocity $x$, $y$, aur $z$ teeno par ek saath depend karti hai. Curly $\partial$ (kaho "partial") ek vaada hai: "Main sirf $x$ badal raha hoon aur baaki ko lock karke rakh raha hoon." Spin detect karne ke liye exactly yahi chahiye — spin is baare mein hai ki east velocity north ki taraf move karne par kaise change hoti hai, versus north velocity east ki taraf move karne par kaise change hoti hai.

Figure mein: $\partial v_y/\partial x$ paane ke liye hum do un points par north-arrows compare karte hain jo sirf apni east position mein differ karte hain (red pair). $\partial v_x/\partial y$ paane ke liye hum un do points par east-arrows compare karte hain jo sirf north position mein differ karte hain (amber pair). Yeh do comparisons vorticity ka poora engine hain.

4. Rotation rate $\Omega$ — ek rigid cheez ka spin

Yeh Angular Velocity of Rigid Bodies se link karta hai: ek solid spinning disc mein har point same $\Omega$ share karta hai. Fluid ke liye, alag-alag regions mein alag $\Omega$ ho sakta hai, isliye yeh bhi ek field ban jaata hai. Parent note ka headline result $\mathbf{\omega }=2\mathbf{\Omega }$ kehta hai ki curl double yeh local turning rate deta hai — $2$ ka factor yaad rakhna.

5. Chota bold omega $\mathbf{\omega }$ — vorticity as a vector

Vector kyun aur ek number kyun nahi? Kyunki spin ko ek axis chahiye — ek paddle-wheel kisi bhi direction ki taraf tilt ho sakta hai, aur $\mathbf{\omega }$ record karta hai uski pin kis taraf point karti hai. Flat 2D flow mein axis hamesha page se seedha bahar hota hai, isliye sirf $z$-part ${\omega }_z$ bachta hai, aur hum ise ek number ki tarah treat kar sakte hain.

6. Curl symbol $\nabla \times$ — spin-detector

$\times$ cross product ki shape lete hai — woh operation jo already matlab rakhta hai "mujhe do aur ke perpendicular ek vector do." Yahan yeh spin-axis arrow produce karta hai. Poori recipe (parent note se), har bracket hamare building-block slope differences mein se ek hai: $\mathbf{\omega }=\nabla \times \mathbf{v}=\left(\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z}\right)\hat{x}+\left(\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x}\right)\hat{y}+\left(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}\right)\hat{z}$

Do slopes ka difference kyun? Figure s02 dekho. Ek pure spin east-velocity ko north ki taraf jaane par climb karata hai aur north-velocity ko east ki taraf jaane par drop karata hai — dono slopes ke opposite sign hain, isliye subtract karne par unke spins add ho jaate hain. Ek pure stretch (no spin) unhe equal banata hai, isliye subtraction cancel ho kar zero ho jaata hai. Minus sign woh filter hai jo non-spinning motion ko throw away karta hai. Ise Curl and Divergence (vector calculus) mein aur gehrayi se samjho.

Figure teen tarah ki motions dikhata hai jo curl ko sort out karni padti hain: pure spin (dono arms same tarah turn karte hain → curl $\ne 0$), pure shear (ek arm turns, curl phir bhi fire karta hai), aur pure stretch (arms sirf lengthen hote hain, koi turn nahi → curl $=0$).

7. Loop integral ${\oint }_C\mathbf{v}\cdot d\mathbf{l}$ — circulation

Yahan notation ke do naye pieces hain; hum dono unpack karte hain.

Inhe saath rakh do: $\Gamma ={\oint }_C\mathbf{v}\cdot d\mathbf{l}$ padhta hai: "loop $C$ ke around ek baar chalo; har tiny step par, record karo ki flow ne tumhe tumhare step ke along kitna push kiya; sab add karo." Woh total circulation $\Gamma$ hai — units ${\text{m}}^2/\text{s}$ (velocity $\times$ length). Yeh "fence ke around feel ki gayi total swirl" hai.

Figure mein, har step par amber component $\mathbf{v}\cdot d\mathbf{l}$ hai: jahan flow tumhari walk ke saath run kare wahan ek positive slice add hoti hai; tumhari walk ke against, ek negative slice. Har slice sum karo $\Gamma$ ke liye.

8. Area, flux, aur bridge $d\mathbf{A}$

Yeh Stokes' Theorem ki machinery hai, jo sections 6 aur 7 ko weld karta hai: $\Gamma ={\oint }_C\mathbf{v}\cdot d\mathbf{l}={\iint }_S\mathbf{\omega }\cdot d\mathbf{A}.$ Words mein: fence ke around total push barabar hai andar ke saare tiny spins add kiye gaye. Local spin (curl) aur global swirl (circulation) do taraf se dekha gaya same fact nikalta hai.

Prerequisite map

Left par sab kuch foundation hai; Stokes theorem par milne wale do arrows payoff hain — dekho Irrotational Flow and Velocity Potential aur Kelvin's Circulation Theorem jahan yeh le jaata hai.

Equipment checklist