Reynolds transport theorem — Hinglish notes

2.2.13 · Physics › Fluid Mechanics

Hume iski ZAROORAT hi kyun hai?

Particle mechanics mein tum ek fixed mass $m$ ko hamesha track karte ho. Fluids mein, wahi fluid particles lagaataar move karte rehte hain — unhe track karna (the Lagrangian view) practically impossible hai. Hum prefer karte hain space mein ek window fix karna, the control volume (CV), aur fluid ko usme se guzarte dekhna (the Eulerian view).

Problem yeh hai: conservation laws Lagrangian statements hain.

Toh hume ek machine chahiye jo $\frac{d B _{sy s}}{d t}$ ko control-volume language mein convert kare. Woh machine hai RTT.

OBJECTS kya hain?

Ise SCRATCH se kaise derive karein

Ek aisa system lo jo time $t$ par, bilkul CV ke saath coincide karta hai. Thodi der baad $d t$ mein system drift kar chuka hai: uska kuch hissa bahar flow ho gaya (region III), CV ka kuch hissa ab naye fluid se occupied hai jo andar flow hua (region I), aur bulk overlap karta hai (region II).

Step 1 — $t$ aur $t + d t$ par bookkeeping. Time $t$ par: system bilkul CV ko occupy karta hai, isliye $B_{sy s} (t) = B_{C V} (t) .$ Yeh step kyun? Humne deliberately inhe coincide hone ke liye choose kiya, toh hum is instant par unhe swap kar sakte hain.

Time $t + d t$ par: system region I chhod chuka hai lekin region III mein enter kar gaya hai: $B_{sy s} (t + d t) = B_{C V} (t + d t) - B_{I} (t + d t) + B_{I I I} (t + d t) .$ Kyun? $t + d t$ par CV mein naya inflow (I) hai jo system nahi hai, aur outflow (III) missing hai jo system hai — toh ise correct karo.

Step 2 — Derivative banao.

=\underbrace{\frac{d}{dt}\!\int_{CV}\!\rho b\,dV}_{\text{storage}} +\underbrace{\lim_{dt\to0}\frac{B_{III}-B_{I}}{dt}}_{\text{net efflux}}$$ *Kyun?* Do CV terms $\frac{d}{dt}B_{CV}$ ban jaate hain; I aur III terms surface fluxes ban jaate hain. **Step 3 — Flux ko surface integral mein convert karo.** Time $dt$ mein, surface ke patch $dA$ ke paas ka fluid ek slanted cylinder ka volume sweep karta hai $$dV=(\vec v\cdot\hat n)\,dt\,dA,$$ jahan $\hat n$ **outward** normal hai. Toh carry hone wali property $= \rho b\,(\vec v\cdot\hat n)\,dt\,dA$ hai. *$\vec v\cdot\hat n$ kyun?* Sirf velocity **ka normal ke saath wala component** actually surface cross karta hai; tangential flow uske saath slide karta hai aur kuch cross nahi karta. $dt$ se divide karke aur **poori** closed surface par integrate karke (outflow deta hai $\vec v\cdot\hat n>0$, inflow deta hai $<0$, automatically I aur III handle karte hue): $$\frac{B_{III}-B_{I}}{dt}\to\oint_{CS}\rho b\,(\vec v\cdot\hat n)\,dA.$$ > [!formula] Reynolds Transport Theorem (fixed CV) > $$\boxed{\;\frac{dB_{sys}}{dt}=\frac{d}{dt}\int_{CV}\rho\,b\,dV+\oint_{CS}\rho\,b\,(\vec v\cdot\hat n)\,dA\;}$$ > *(Left = "matter ko follow karte hue". Right = "andar **unsteady storage**" + "surface se **net outflow**".)* **Moving CV?** Surface term mein $\vec v$ ki jagah **relative** velocity $\vec v_r=\vec v-\vec v_{CS}$ use karo, kyunki sirf surface ke *relative* motion hi actually surface cross karta hai. --- ## Laws instantly nikal aate hain > [!example] Continuity (mass): $b=1$ > $\dfrac{dm_{sys}}{dt}=0$ toh > $$0=\frac{d}{dt}\int_{CV}\rho\,dV+\oint_{CS}\rho(\vec v\cdot\hat n)\,dA.$$ > **Yeh step kyun?** Left side ko zero set karna bas encode karta hai "mass conserved hai." Steady incompressible flow ⇒ $\sum(\rho A v)_{out}=\sum(\rho A v)_{in}$. > [!example] Momentum: $b=\vec v$ > $$\sum\vec F=\frac{d}{dt}\int_{CV}\rho\vec v\,dV+\oint_{CS}\rho\vec v(\vec v\cdot\hat n)\,dA.$$ > **Kyun?** $B=m\vec v$ ⇒ $b=\vec v$, aur $\frac{d(m\vec v)_{sys}}{dt}=\sum\vec F$. Yeh jet/thrust force problems ki basis hai. > [!example] Number plug-in (steady jet on a plate) > Water jet, $\rho=1000$, $A=2\times10^{-3}\,\text{m}^2$, $v=10\,\text{m/s}$, wall se takrata hai aur sideways spread ho jaata hai (saara x-momentum kho deta hai). > Mass flow: $\dot m=\rho A v=1000(2\times10^{-3})(10)=20\,\text{kg/s}$. > Steady ⇒ storage term $=0$. x-momentum flux out $=0$, in $=\dot m v=200$. > $$F_x=\oint\rho v_x(\vec v\cdot\hat n)dA = 0-(-\dot m v)=200\ \text{N (force on jet from wall)}.$$ > **Sign kyun?** Inflow mein $\vec v\cdot\hat n<0$ hai; woh minus sign flip ho jaata hai, positive reaction deta hai. Plate $200\,$N se push back karta hai. --- > [!mistake] Classic errors ko steel-man karna > **(1) "Moving CV ke liye actual speed $\vec v$ use karo."** *Sahi lagta hai* kyunki $\vec v$ hi ek wahi speed hai jo tumne measure ki. **Fix:** surface cross karna depend karta hai *surface ke relative* motion par, toh $\vec v_r=\vec v-\vec v_{CS}$ use karo. Ek surface jo fluid ke saath move kar rahi ho uska zero flux hoga chahe $\vec v\ne0$ ho. > **(2) "Jab bhi flow ho toh $\frac{d}{dt}\int_{CV}$ term drop kar do."** *Sahi lagta hai* kyunki clearly kuch flow ho raha hai. **Fix:** woh storage term **time mein unsteadiness** ($\partial/\partial t$) ke baare mein hai, flow ke baare mein nahi. Yeh sirf **steady** flow ke liye vanish hota hai. > **(3) Normal component ki jagah full speed use karna.** *Sahi lagta hai* — "fluid fast hai!" **Fix:** sirf $\vec v\cdot\hat n$ hi surface ko pierce karta hai; tangential flow kuch contribute nahi karta. > **(4) $\rho b$ vs $b$ bhool jaana.** RTT integrate karta hai $\rho b\,dV$, na ki $b\,dV$, kyunki stored amount hai property-per-mass times mass, aur $dm=\rho\,dV$. --- > [!recall]- Feynman: 12-saal ke bache ko explain karo > Ek river mein ek window imagine karo. Tum fish (woh "stuff") count karna chahte ho. **Window box ke andar** fish ki ginti do tarah se change ho sakti hai: fish **andar paida ho ya gum ho jaayein** (storage term — sirf tab matter karta hai jab cheezein time ke saath badlein), ya fish **window ke edges cross karke tairein** (surface term — right se bahar, left se andar). RTT bas yahi kehta hai: *original gang of fish ke liye change = box mein jo change tum dekhte ho + net fish jo tair ke bahar gayi.* Bas itna hi. > [!mnemonic] Structure yaad rakhne ke liye > **"System = Storage + Surface."** (Teen S.) Equals ke left = *System* ko follow karna; right = andar *Storage* + *Surface* se flux. --- ## Active recall #flashcards/physics RTT kis cheez ke beech translate karta hai? ::: System (Lagrangian) rate of change aur control-volume (Eulerian) description ke beech. General RTT formula? ::: $\frac{dB_{sys}}{dt}=\frac{d}{dt}\int_{CV}\rho b\,dV+\oint_{CS}\rho b(\vec v\cdot\hat n)\,dA$ $b$ kya hai? ::: Intensive property, $b=dB/dm$ (property per unit mass). Mass, momentum, energy ke liye $b$? ::: $1$, $\vec v$, $e$ (specific energy). Surface term mein $\vec v\cdot\hat n$ kyun use hota hai? ::: Sirf outward normal ke saath wala velocity component actually surface cross karta hai; tangential flow uske saath slide karta hai. Storage term $\frac{d}{dt}\int_{CV}$ kab vanish hota hai? ::: Steady flow ke liye (CV ke andar koi time dependence nahi). Moving CV ke liye surface term mein $\vec v$ ki jagah kya aata hai? ::: Relative velocity $\vec v_r=\vec v-\vec v_{CS}$. $b=1$ ke saath RTT kaunsa law deta hai? ::: Continuity / conservation of mass. $b=\vec v$ ke saath RTT kaunsa law deta hai? ::: Linear momentum equation, $\sum\vec F=\dots$. System aur CV ko time $t$ par coincide kyun karaate hain? ::: Taaki $B_{sys}(t)=B_{CV}(t)$, jisse woh swap ho sake jo derivation start karta hai. Inflow vs outflow ke liye $\vec v\cdot\hat n$ ka sign? ::: Inflow negative, outflow positive (outward normal convention). --- ## Connections - [[Continuity equation]] — $b=1$ wala special case. - [[Momentum equation (control volume)]] — $b=\vec v$ wala special case; jet thrust. - [[Eulerian vs Lagrangian description]] — do viewpoints jo RTT bridge karta hai. - [[Material derivative]] — RTT ka differential ("point") analogue. - [[Divergence theorem]] — surface flux ko volume integral mein convert karta hai, differential conservation laws deta hai. - [[Bernoulli equation]] — energy form, $b=e$, restrictions ke saath. ## 🖼️ Concept Map ```mermaid flowchart TD CL[Conservation laws are Lagrangian] SYS[System - fixed mass] CV[Control volume - fixed region] RTT[Reynolds Transport Theorem] B[Extensive property B] b[Intensive property b equals dB/dm] BCV[B stored in CV integral rho b dV] COIN[System coincides with CV at time t] STOR[Storage term d/dt B_CV] FLUX[Net efflux term] SURF[Surface integral rho b v dot n dA] CL -->|written for| SYS SYS -->|hard to track| CV CV -->|needs translation| RTT RTT -->|converts| CL B -->|per unit mass gives| b B -->|stored as| BCV b -->|used in| BCV RTT -->|derived from| COIN COIN -->|yields| STOR COIN -->|yields| FLUX FLUX -->|swept volume gives| SURF BCV -->|becomes| STOR ```