Foundations — Stream function, velocity potential

2.2.11 · D1 · Physics › Fluid Mechanics › Stream function, velocity potential

Parent note 2.2.11 Stream function, velocity potential padhne se pehle, tumhe usmein aane wale har symbol ki solid understanding honi chahiye. Neeche, har symbol ko teen cheezein milti hain: simple words, picture, aur topic ko iske kyun zaroorat hai. Inhe iss order mein arrange kiya gaya hai ki har ek sirf upar wale symbols pe rely kare.

1. Plane mein ek point: $x$ aur $y$

Picture: ek flat graph paper ki sheet. Uske har dot ka ek address $(x,y)$ hai.

Topic ko iske kyun zaroorat hai: fluid ek flat region fill karta hai (hum 2D flow padhte hain — ek thin sheet, jaise ek shallow river upar se dekha jaye). "Paani yahan versus wahan" ki baat karne ke liye hume spots ko name karna aana chahiye. Bas yahi kaam karta hai $(x,y)$.

2. Direction ke saath speed: velocity vector $\vec{V}=(u,v)$

Picture: sheet ke har spot pe ek chhota arrow banao jo dikhata ho ki paani kahan ja raha hai aur kitna fast. Us arrow ka horizontal axis pe shadow $u$ hai; vertical axis pe shadow $v$ hai.

Topic ko iske kyun zaroorat hai: yeh story ka villain hai. Infinitely many points mein se har ek pe hamare paas do numbers hain, $u$ aur $v$. Yahi woh "do functions" hain jo parent note ek mein shrink karna chahta hai. Agar $u$ ya $v$ negative ho sakta hai, toh uska matlab sirf yeh hai ki flow wahan left ya down point kar raha hai — har sign allowed hai.

3. Ek direction mein hill ki slope: partial derivative $\frac{\partial f}{\partial x}$

Yeh pehla tool hai. Isse dheere dheere samjho.

Curly $\partial$ (kaho "partial dee") straight $d$ ki jagah isliye use hota hai taaki yaad rahe: doosre variables ruke hue hain. $\partial f/\partial y$ wahi idea north ki taraf hai.

Topic ko iske kyun zaroorat hai: parent note ka har single formula — $u=\partial \varphi /\partial x$, $v=-\partial \psi /\partial x$, continuity, Laplace — partial derivatives se bana hai. Yeh woh tarika hai jisse hum kehte hain ki "clever scalar is direction mein itna fast change karta hai."

Hum ek shorthand bhi use karenge: $f_x$ ka matlab $\partial f/\partial x$ hai, aur $f_{xy}$ ka matlab hai "pehle $x$ se differentiate karo, phir $y$ se." Same cheez, kam symbols.

4. Upar ki taraf arrow: gradient $\nabla f$

Picture: hill $f$ pe khade ho jao. Daala hua paani slope se neeche bahega; $\nabla f$ woh arrow hai jo opposite direction mein, upar ki taraf, flat contour lines ke perpendicular point karta hai.

Topic ko iske kyun zaroorat hai: velocity potential ko $\vec{V}=\nabla \varphi$ se define kiya jata hai — "flow kisi hill $\varphi$ ka uphill arrow hai." Jab tak $\nabla$ ka matlab kuch nahi, tum woh line nahi padh sakte. Dhyan do: $\nabla f$ ek scalar $f$ ko do numbers (ek arrow) mein turn karta hai — exactly woh "ek se do unpack karo" move jo topic use karta hai.

5. Del ke do kaam: divergence $\nabla \cdot \vec{V}$ aur curl $\nabla \times \vec{V}$

Wahi del symbol do aur kaam karta hai, is baar ek arrow-field khaata hai aur ek rate ugalta hai.

Topic ko dono ki kyun zaroorat hai:

• Incompressible (fluid squish nahi ho sakta, mass conserved) ka matlab hai $\nabla \cdot \vec{V}=0$ — kisi bhi box mein kuch create ya destroy nahi hota. Yeh Continuity Equation hai, aur yahi stream function $\psi$ ko exist karne deta hai.
• Irrotational (koi local spinning nahi) ka matlab hai $\nabla \times \vec{V}=0$ — dekho Vorticity and Irrotational Flow. Yahi velocity potential $\varphi$ ko exist karne deta hai.

Har condition $u$ aur $v$ ko ek equation se bandh karti hai, aur exactly yahi woh tarika hai jisse hum do unknowns se ek pe aa jaate hain.

6. Do star scalars: $\varphi$ aur $\psi$

Ab finally payoff symbols ko name kiya ja sakta hai.

Jab dono exist karte hain, toh $u$ aur $v$ ki definitions ko match karne se Cauchy-Riemann Equations milte hain, aur har scalar Laplace Equation ${\nabla }^2\varphi =0,{\nabla }^2\psi =0$ follow karta hai. Woh jo orthogonal grid banate hain woh ek flow net hai. Potential idea Electrostatic Potential $\vec{E}=-\nabla V$ ko mirror karta hai, aur jab flow solve ho jaata hai, pressures Bernoulli Equation se follow karti hain.

7. Sab kuch saath kaise feed karta hai

Equipment checklist

Right side cover karo aur dekho ki reveal karne se pehle har ek ka jawab de sako.