Visual walkthrough — Reynolds transport theorem

2.2.13 · D2 · Physics › Fluid Mechanics › Reynolds transport theorem

Hum kuch bhi assume nahi karte. Agar koi word ya symbol aata hai, pehle usse draw kiya jaata hai. Parent note hai the RTT topic note — yeh uska picture-book companion hai.

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Step 0 — Nadi ko dekhne ke do tarike

Picture mein, pink blob system hai (ek named gang of molecules). Blue dashed box control volume hai — ek region jo humne choose kiya, space mein glued. Nadi (the two views) pink blob ko blue box se hoke le jaati hai.

Hamara poora goal: physics ke laws pink blob ke liye likhe jaate hain (Newton ek fixed mass track karta hai), lekin hum cheezein sirf blue box mein measure kar sakte hain. Humein ek translator chahiye.

Aage badhne se pehle do symbols fix karte hain:

Symbol tally ab tak: $B$ = total stuff, $b$ = stuff per kg, $\rho$ = mass per volume, $dV$ = volume ka tiny chunk, ${\int }_{CV}$ = "poore box par sum".

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Step 1 — Dono clocks ko freeze karo taaki blob box ke barabar ho

KYA. Time $t$ par hum pink blob ko exactly blue box ke andar bithate hain — same shape, same molecules, perfect overlap.

KYUN. Kyunki agar woh same region mein same instant par hain, toh unke andar same stuff hai: $\underset{\text{blob mein stuff}}{\underbrace{B_{sys}(t)}}=\underset{\text{box mein stuff}}{\underbrace{B_{CV}(t)}}.$ Yeh ek akeli equality hi hinge hai. Yeh humein is ek instant par "blob" ko "box" se swap karne deti hai, taaki baad mein hum box (measurable) ki baat kar sakein instead of blob (not measurable) ke.

PICTURE. s02 mein pink aur blue outlines bilkul ek doosre ke upar aate hain — koi bhi sliver kahin se bahar nahi nikalta. Padho: $t$ par, blob = box.

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Step 2 — Ek moment jaane do; blob drift karta hai

Teen regions aate hain, aur unhe naam dena hi poori battle hai:

• Region I (blue, inlet end): yeh ab box ke andar hai lekin pink blob ne ise chhod diya hai. Nayi non-system fluid ne uski jagah le li.
• Region II (mota beech waala): abhi bhi blob aur box dono. Overlap.
• Region III (pink, outlet end): blob box ki wall ke paar bahar push ho gaya hai. Yeh system hai, lekin ab box ke andar nahi.

KYA. Hum after-picture ko I + II + III mein split karte hain.

KYUN. Kyunki "blob ab" aur "box ab" ko compare karne ke liye humein exactly wahan account karna hoga jahan woh disagree karte hain — aur woh sirf I aur III mein disagree karte hain.

PICTURE. s03: box apni jagah rehta hai (dashed blue), blob right mein slide ho gaya hai (pink). Baaya crescent I hai, daaya crescent III hai, lens-shaped overlap II hai.

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Step 3 — Blob ka stuff box ke stuff ke terms mein likho, corrected karke

KYA. Hum blob ka total stuff $t+dt$ par box ke stuff se express karte hain, phir do mismatches ko patch karte hain.

HAR TERM KYUN:

• $B_{CV}(t+dt)$ — box ab jo kuch bhi hold karta hai. Lekin box ab region I (fresh non-system fluid) contain karta hai, toh hum $B_I$ subtract karte hain — yeh hamare blob ka hissa nahi hai.
• Blob region III mein spill ho gaya hai, jo box ke bahar hai, toh box ne use miss kar diya — hum $B_{III}$ add karte hain wapas.
• Region II mein koi correction ki zaroorat nahi: yeh dono ke andar hai, toh $B_{CV}$ mein already correctly counted hai.

PICTURE. s04 ek signed ledger hai: box (blue, $+$), region I (blue, $-$), region III (pink, $+$). Colors ko plus/minus accounting sheet ki tarah padho.

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Step 4 — Rate of change lo (derivative aata hai, aur kyun)

Step 1 ki equation ko Step 3 se subtract karo, phir $dt$ se divide karo aur $dt\to 0$ shrink karo:

$\frac{d{B}_{sys}}{dt}={\lim }_{dt\to 0}\frac{B_{sys}(t+dt)-B_{sys}(t)}{dt}$

$B_{sys}(t)=B_{CV}(t)$ aur Step 3 ka expression plug in karo:

$\frac{d{B}_{sys}}{dt}=\underset{=\frac{d}{dt}{\int }_{CV}\rho bdV}{\underbrace{\underset{dt\to 0}{\lim }\frac{B_{CV}(t+dt)-B_{CV}(t)}{dt}}}+\underset{\text{net stuff leaving per second}}{\underbrace{\underset{dt\to 0}{\lim }\frac{B_{III}(t+dt)-B_I(t+dt)}{dt}}}$

KYA. Do box-terms ek clean time-derivative mein merge ho gaye jo andar stored hai; do crescents I aur III ek alag "net outflow rate" ban gaye.

KYUN. $B_{CV}(t+dt)-B_{CV}(t)$ ko $dt$ se divide karna by definition box ke contents ka time-derivative hai — ise storage term kehte hain (andar ki matra time ke saath kaise bhar-ti ya drain hoti hai). Bache hue I aur III pieces physically wall cross karne waala stuff hai — flux term.

PICTURE. s05: "storage" labelled arrow box ko tank gauge ki tarah fill/empty hote dikhaata hai; "net efflux" labelled arrow do crescents ki taraf point karta hai (III par out, I par in).

Symbol check: $\frac{d}{dt}{\int }_{CV}\rho bdV$ = box ka total stuff kitni tezi se badlta hai; crescent limit = rate jis par stuff boundary cross karta hai.

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Step 5 — Crescent ko surface sweep mein convert karo (dot product aata hai)

Do nayi symbols milte hain, dono drawn hain:

• $\hat{n}$ = outward unit normal: ek length-1 arrow jo wall se seedha bahar nikalta hai (chhoti hat matlab "length 1", "normal" matlab "surface ke perpendicular").
• $\vec{v}$ = us patch par fluid ki velocity arrow.

Time $dt$ mein fluid $|\vec{v}|dt$ distance move karta hai. Woh volume jo patch se sweep hota hai ek slanted cylinder hai. Uska volume base area × perpendicular height hai: $dV=(\vec{v}\cdot \hat{n})dtdA.$

PICTURE. s06: outward normal $\hat{n}$ (yellow), velocity $\vec{v}$ (blue) angle $\theta$ par, aur slanted chalk cylinder jiska skewed height $(\vec{v}\cdot \hat{n})dt$ hai.

Us patch se carry hone waala stuff = $\rho b\times$ (swept volume) $=\rho b(\vec{v}\cdot \hat{n})dtdA$.

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Step 6 — Saare patches jodo: surface integral

$\frac{B_{III}-B_I}{dt}⟶{\oint }_{CS}\rho b(\vec{v}\cdot \hat{n})dA$

KYA. Do-crescent difference ek single loop-integral over the whole closed surface ban gaya ($\oint$ par circle ka matlab hai "closed surface — koi gap nahi").

YEH I AUR III dono automatically kyun handle karta hai: $\vec{v}\cdot \hat{n}$ ki sign sorting khud kar leti hai.

• Outlet par (region III): $\vec{v}$ outward point karta hai, $\vec{v}\cdot \hat{n}>0$ → stuff add karta hai. Yeh hai $+B_{III}$.
• Inlet par (region I): $\vec{v}$ inward point karta hai, $\vec{v}\cdot \hat{n}<0$ → stuff subtract karta hai. Yeh hai $-B_I$.

Hume kabhi bhi inlets aur outlets ko manually label nahi karna padta — geometry khud sign kar leti hai.

PICTURE. s07: closed box wall arrows se studhi hui — green outward arrows (positive, outflow) right face par, red inward arrows (negative, inflow) left face par — sab loop integral se sum hote hain.

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Step 7 — Theorem assemble karo

Term-by-term, ek aakhri baar:

• $\frac{d{B}_{sys}}{dt}$ — Lagrangian rate; yeh woh side hai jis par physics laws bolte hain.
• $\frac{d}{dt}{\int }_{CV}\rho bdV$ — stored stuff time ke saath kaise swell/drain hota hai. Steady flow ke liye Zero.
• ${\oint }_{CS}\rho b(\vec{v}\cdot \hat{n})dA$ — wall cross karne waala net stuff per second; sign in vs out handle karta hai.

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Step 8 — Degenerate cases (reader ko kabhi stranded mat chhodho)

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Ek-picture summary

System = Storage + Surface. Poori derivation ek board par: blob drifts karta hai (left→right), box fixed rehta hai, crescents I aur III inflow/outflow arrows ban jaate hain, tank-gauge storage dikhata hai. Left-to-right padho aur tumne RTT re-derive kar li.

Ab do laws free mein milte hain:

Recall Feynman retelling — poora walkthrough plain words mein

Ek nadi mein hung ek window-box imagine karo. Main paani ke molecules ke ek gang ko pink paint karta hoon aur ek blue box nadi ke takhte se glue karta hoon taaki abhi pink gang box ko perfectly fill kare (Step 1). Ek heartbeat wait karo: pink gang downstream slide ho jaati hai (Step 2). Ab kuch fresh clear water mere box ke baye taraf creep in aa gayi hai (region I — box mein hai lekin meri gang nahi) aur meri pink gang dayi wall se bahar bulge kar gayi hai (region III — meri gang hai lekin box se escape kar gayi) (Step 3). Yeh pata karne ke liye ki meri pink gang ka "stuff" kaise badla, main kehta hoon: box mein stuff ab, minus baaye ka clear intruder, plus daaye ka pink bulge. Use per-second rate mein convert karo (Step 4) aur yeh cleanly split ho jaata hai: box ke contents kaise tezi se badlte hain (storage), plus kaise tezi se stuff walls cross karta hai (flux). Crossing measure karne ke liye main ek patch par zoom karta hoon aur notice karta hoon ki sirf velocity ka straight-through part count karta hai — saath-saath sliding paani kuch nahi cross karta — jo exactly dot product $\vec{v}\cdot \hat{n}$ compute karta hai (Step 5). Har patch par sum karo aur sign inflow ($-$) ko outflow ($+$) se free mein sort kar deta hai (Step 6). Inhe bolt karo (Step 7): meri gang ko follow karte hue change = box ke andar change + walls se bahar net stuff. System = Storage + Surface. Aur agar flow steady hai toh storage zero hai, agar wall paani ke saath ride karti hai toh crossing zero hai — koi case nahi chhuta (Step 8).

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Connections

• Continuity equation — $b=1$ picture (Step 8, Case A dikhata hai jab storage khatam hoti hai).
• Momentum equation (control volume) — $b=\vec{v}$ picture; jet thrust upar work kiya.
• Eulerian vs Lagrangian description — Step 0 ke do viewpoints, drawn.
• Material derivative — is same story ka point-sized ("box ko ek dot tak shrink karo") version.
• Divergence theorem — Step 6 ke surface loop ko volume integral mein collapse karta hai.
• Bernoulli equation — energy branch, $b=e$, restrictions ke under.