Foundations — Fluid kinematics — Eulerian vs Lagrangian description

2.2.9 · D1 · Physics › Fluid Mechanics › Fluid kinematics — Eulerian vs Lagrangian description

Isse pehle ki tum parent topic ki ek bhi line padh sako, tumhe wo alphabet chahiye jo wo silently assume karta hai. Yeh page har ek symbol ko kuch nahi se build karta hai, ek aisi order mein jahan har ek symbol sirf unhi par tikta hai jo usse pehle aaye hain.

0 — "Fluid particle" kya hota hai? (wo cheez jiske baare mein sab kuch hai)

Yeh idea pehle kyun chahiye? Kyunki poora topic do sawaalon ke beech ek ladaai hai: ek chosen blob ke saath kya hota hai? versus ek chosen jagah par kya hota hai? "Blob" ka ek clear notion ke bina, dono sawaal bekar hain.

Figure mein amber speck dekho. Har instant par uski ek definite position hai, aur wo move karta hai. Woh akela chalti-phirti speck Lagrangian viewpoint ka hero hai. Uske peeche fixed watch-points ki white grid Eulerian viewpoint hai.

1 — Position, aur label $\mathbf{a}$

Alag label kyun invent karen? Kyunki ek particle move karta hai, isliye uski current position $\mathbf{x}$ badhti rehti hai aur naam ka kaam nahi kar sakti. Uski starting position kabhi nahi badlti — ek perfect permanent name tag.

2 — Time $t$, aur reference time $t_0$

Time ko do symbols kyun milte hain? Ek ($t_0$) naming convention set karta hai; doosra ($t$) running variable hai jiske against hum differentiate karte hain. Unhe alag rakhne se tum "jab maine naam diya" aur "jab main dekh raha hoon" ko confuse karne se bacha lete ho.

3 — Velocity $\mathbf{v}=(u,v,w)$

$\mathbf{v}$ ko $u,v,w$ mein todne ki zaroorat kyun hai? Kyunki instruments aur equations ek direction mein ek baar kaam karte hain. Jab parent likhta hai $u\partial \varphi /\partial x+v\partial \varphi /\partial y+w\partial \varphi /\partial z$, toh yeh teeno directions mein alag-alag motion se hone wale change ko jod raha hai.

4 — Ek field: har cheez likhne ka Eulerian tarika

Yeh Eulerian language kyun hai? Kyunki yahan position $\mathbf{x}$ ek input hai jo tum freely choose karte ho — tum pooch rahe ho "is fixed spot par kya ho raha hai?", na ki "is particle ke saath kya ho raha hai?". Alag-alag particles alag-alag times par same spot se guzarte hain.

5 — Partial derivative $\frac{\partial }{\partial t}$, $\frac{\partial }{\partial x}$

Partial, ordinary derivative kyun, partial kyun? Kyunki ek field chaar chezon par depend karta hai ($x,y,z,t$). "Yeh kaise change hota hai?" poochna ambiguous hai jab tak tum na batao ki kaunsa knob tum ghuma rahe ho. Har partial exactly ek knob ghumata hai.

• $\frac{\partial \varphi }{\partial t}$ (space frozen) = field kaise change hoti hai ek fixed spot par jaise time beetha hai → Eulerian / local rate.
• $\frac{\partial \varphi }{\partial x}$ (time & baaki axes frozen) = field kaise vary karta hai jaise tum space mein sideways slide karte ho → convective term ke liye chahiye.

6 — Gradient $\nabla$ aur operator $\mathbf{v}\cdot \nabla$

Bilkul yahi combination kyun — velocity dotted with gradient? Kyunki yeh measure karta hai ki $\varphi$ kitni tezi se change hoti hai jab tum us direction mein ride karte ho jisme particle actually move kar raha hai. Gradient kehta hai "$\varphi$ kis taraf climb karta hai aur kitna steeply"; velocity kehta hai "main kis taraf aur kitni tezi se ja raha hoon." Unka dot product woh rate hai jo $\varphi$ ek $\mathbf{v}$ ke saath travel karne wale ko deliver karta hai. Yahi exactly woh "ek alag region mein move karna" effect hai jise parent convective kehta hai.

7 — Material derivative $\frac{D}{Dt}$ (payoff symbol)

$\partial /\partial t$ ya $d/dt$ reuse karne ki jagah brand-new symbol kyun? Kyunki puraane mein se koi bhi nahi kehta "time run hone ke dauran particle label ko fixed rakho." $\partial /\partial t$ position ko fixed rakhta hai (galat — particle move karta hai); plain $d/dt$ ambiguous hai baare mein ki kya hold hai. Capital $D$ ek deliberate flag hai: tum particle par ride kar rahe ho. Right-hand side ka har ek symbol sections 5 aur 6 mein build hua tha — kuch naya smuggle nahi kiya gaya.

Prerequisite map

Is page par har ek foundation material derivative ko feed karta hai, jo Continuity equation, Euler's equation of motion, Navier-Stokes equations, aur Steady vs unsteady flow ke andar Streamlines, pathlines and streaklines ki geometry ka gateway hai. Yeh page jo bookkeeping set up karta hai use baad mein Reynolds transport theorem formalize karta hai.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle jawab do.