Continuity equation — derivation (conservation of mass), ρAv = const: Visual walkthrough

2.2.12 · D2 · Physics › Fluid Mechanics › Continuity equation — derivation (conservation of mass), ρAv

Step 1 — "Flowing tube" kya hota hai aur hum kya measure kar rahe hain?

KYA. Socho fluid (paani, hawa, shahad) ek pipe ke andar slide kar raha hai. Hum pipe ko do flat imaginary windows se kaatein aur unhe section 1 (inlet) aur section 2 (outlet) bolein. Har window par hum teen numbers ki parwah karte hain.

YEH TEEN HI KYUN? Kyunki jo cheez hum ultimately conserve karte hain woh mass hai, aur ek window par mass exactly inhi se banta hai: kitni jagah ( $A$ ), kitni tezi se refill hoti hai ( $v$ ), har jagah per kitna bhaari ( $ρ$ ). Kuch aur chahiye hi nahi.

PICTURE. Pipe ek wide mouth se slim neck tak narrow hoti hai. Do red windows sections mark karti hain; har ek par apna $A$ , $v$ , $ρ$ label hai.

Figure — Continuity equation — derivation (conservation of mass), ρAv = const

Step 2 — Ek tiny time mein fluid ka ek slab kitni door jaata hai?

KYA. Section 1 par baitha ek patla fluid slab freeze karo. Ab thoda sa time guzarne do — hum ise $Δ t$ kehte hain (symbol $Δ$ , Greek "delta", ka matlab hai "mein ek chhota sa change"; yahan "time ka ek chhota sa chunk"). $Δ t$ mein slab aage drift karta hai ek distance se jise hum $Δ x_{1}$ kehte hain.

YEH STEP KYUN? Yeh motion ka sabse seedha fact hai: distance = speed × time. Yahi woh bridge hai jo ek speed (jo hum measure kar sakte hain) ko ek actual fluid ka chunk (jiska volume aur mass hai) mein convert karta hai. Iske bina hum yeh count hi nahi kar paate ki "kitna cross hua".

YEH UNITS CLEANLY MULTIPLY KYUN HOTE HAIN. $m/s \times s = m$ — seconds cancel ho jaate hain, metres bachte hain. Symbol arithmetic khud confirm karta hai ki hume ek distance milti hai.

PICTURE. Section 1 par slab, do baar drawn: apni starting jagah par pale, $Δ t$ baad bright. Ek magenta arrow jis ki length $Δ x_{1} = v_{1} Δ t$ hai, dono ko link karta hai.

Step 3 — Woh distance ko volume mein badlo

KYA. Jo slab window ke paas se guzra woh ek chhota cylinder hai: uska flat face window hai (area $A_{1}$ ) aur uski length woh distance hai jitna woh move hua ( $Δ x_{1}$ ). Uska volume hai:

YEH STEP KYUN? Kisi bhi seedhe cylinder ka volume = (base ka area) × (uski length). Length hum Step 2 mein pehle hi nikal chuke hain, toh bas multiply karo. Yahi woh fluid ka jagah ka amount hai jo window cross kar gaya.

KUCH ZYADA COMPLICATED KYUN NAHI? Kyunki ek tiny $Δ t$ mein slab chhota aur seedha hota hai — koi curve nahi, koi fancy shape nahi. Cylinder formula chhote $Δ t$ ki limit mein exact hai.

PICTURE. Wahi slab ab ek solid cylinder ki tarah shaded: face $A_{1}$ , length $Δ x_{1}$ , volume violet mein filled.

Step 4 — Woh volume ko mass mein badlo

KYA. Density bataati hai ki har cubic metre mein kitne kilograms hain. Density ko slab ke volume se multiply karo toh slab ka mass milta hai, jise hum $Δ m_{1}$ kehte hain.

YEH STEP KYUN? Mass = density × volume, hamesha. Yeh finally hume woh mass deta hai jo $Δ t$ mein flow hua — woh quantity jis ki conservation of mass ko parwah hai. Notice karo chain: speed → distance → volume → mass. Har step ek measurable cheez ko agla banata gaya.

PICTURE. Step 3 ka violet cylinder, ab uska mass label $Δ m_{1} = ρ_{1} A_{1} v_{1} Δ t$ stamp hua hai, $ρ_{1}$ , $A_{1}$ , $v_{1}$ , $Δ t$ har jagah colour-tagged hain.

Step 5 — Exit par wahi identical cheez karo

KYA. Steps 2–4 mein jo kuch hua woh sab section 2 par repeat hota hai, apne numbers $ρ_{2}, A_{2}, v_{2}$ ke saath. Usi $Δ t$ mein, section 2 se baahaar jaata mass hai:

EKA HI $Δ t$ KYUN? Yeh crucial hai: hum dono windows ko identical stretch of time ke dauran dekh rahe hain. Warna hum apples ki comparison alag size ke apples ke batch se kar rahe hote. Same clock, dono ends par.

PICTURE. Dono windows saath dikhaye gaye hain. Left cylinder (violet) = mass in; right cylinder (orange), patla par lamba, = mass out. Dono par unke mass formulas aur shared $Δ t$ label hain.

Step 6 — Beech wala box fluid hoard ya lose nahi kar sakta

KYA. Dono windows ke beech mein phansa pipe ka region dekho. Woh pehle se fluid se bhara hai aur na stretch ho raha hai na leak kar raha hai. Toh jo bhi mass $Δ t$ mein left side se enter karta hai, woh same $Δ t$ mein right side se baahaar jaana chahiye.

$Δ m_{1} = Δ m_{2} ⟹ ρ_{1} A_{1} v_{1} Δ t = ρ_{2} A_{2} v_{2} Δ t$

YEH ISS SAB KA DIL KYUN HAI. Yeh akela equality hi conservation of mass hai — dekho Conservation of mass. Baaki har line bookkeeping thi yahan tak pahunchne ke liye.

PICTURE. Pipe ka phansa "box" navy mein outlined. Ek magenta arrow mass in label ke saath usme point karta hai; ek barabar orange arrow mass out label ke saath baahaar point karta hai. Beech mein ek balance-scale icon $Δ m_{1} = Δ m_{2}$ padhta hai.

Step 7 — Common time cancel karo aur law padho

KYA. Dono sides mein same factor $Δ t$ hai. Ise divide kar do — yeh sirf ek scaffolding tha jisse hum ek slab count kar sakein, aur yeh dono sides par identically aata hai.

$Δ t$ KYUN GAYAB HOTA HAI. Yeh dono windows par same tiny time tha, isliye usne balance ko affect nahi kiya — sirf dono slabs ki size ko equally affect kiya. Ise drop karne par ek rates ke baare mein statement milti hai, har instant par sach, sirf ek chosen slab ke liye nahi.

Incompressible special case. Agar fluid ki density barely change hoti hai (paani, zyaatar liquids — Incompressible vs compressible flow), toh $ρ_{1} = ρ_{2}$ aur woh bhi cancel ho jaate hain: $A_{1} v_{1} = A_{2} v_{2} = Q \frac{v _{2}}{v _{1}} = \frac{A _{1}}{A _{2}}$ Chhota area ⇒ badi speed. Squeeze karo toh speed aaye.

PICTURE. Equation $ρ_{1} A_{1} v_{1} Δ t = ρ_{2} A_{2} v_{2} Δ t$ jisme dono $Δ t$ magenta mein struck through hain, arrow final boxed law ki taraf.

Step 8 — Degenerate cases: check karo ki law kabhi break na ho

KYA. Ek achcha law apne extreme inputs mein bhi survive karna chahiye. Check karne ke liye chaar cases.

YEH KYUN DIKHAYEIN? Taaki tum kisi bhi pipe geometry se mile, derivation ne use cover kar rakha hoga. Wide, narrow, zero, still, compressing — woh ek boxed law sab handle karta hai.

PICTURE. Chaar mini-panels: (A) straight pipe equal arrows, (B) same width par ek denser packed exit ek chhoti speed arrow ke saath, (C) pipe ek point tak pinching ek badi speed arrow ke saath, (D) ek still pipe crossed-out arrows ke saath.

Ek picture mein summary

Sab kuch ek frame mein: chain speed → distance → volume → mass → balance → law, ek narrowing pipe ke saath drawn.

Recall Feynman retelling — poora walkthrough simple words mein

Paani ek aisi pipe mein beh raha hai jo patli hoti jaati hai. Main fat end par khada hoon aur apni ghadi ki ek tick mein paani ke ek patle slice ko dekhta hoon. Us tick mein woh aage slide karta hai (apni speed × tick) — yeh paani ka ek chhota cylinder hai. Main jaanta hoon cylinder kitna bada hai (uska face pipe ka area hai, uski length kitni door slide hui), aur main jaanta hoon paani ke har bucket mein kitna weight hai (yahi density hai), isliye main exactly jaanta hoon kitne kilograms mere paas se guzre. Ab main skinny end par jaata hoon aur same tick ke dauran wahi measurement karta hoon. Yahi trick hai: mere dono spots ke beech pipe ka chunk pehle se paani se bhara hai aur na phool raha hai na leak ho raha hai — isliye jo kilograms fat side se andar aaye, exactly wahi kilograms skinny side se baahaar gaye. "Mass in = mass out" likh kar common tick of time cross karne par milta hai $ρ A v =$ har jagah same number. Paani ke liye, density barely hilaati hai, isliye yeh simplify hota hai area × speed = constant — isliye pipe dabao toh paani tezi se nikalti hai. Squeeze karo toh speed aaye.

Active recall

Step 7 mein hum dono sides se

Δ t

cancel kyun kar sakte hain?

Yeh dono windows par same tiny time interval hai, isliye dono slabs ko equally multiply karta hai aur drop ho jaata hai, rates ke baare mein ek statement chodta hai.

Step 4 mein speed ko mass mein convert karne wali chain kya thi?

speed → distance (

v Δ t

) → volume (

A v Δ t

) → mass (

ρ A v Δ t

Case B mein (fixed area, density double), speed ka kya hota hai?

Woh halve ho jaati hai;

ρ_{1} v_{1} = ρ_{2} v_{2}

ke saath

ρ_{2} = 2 ρ_{1}

deta hai

v_{2} = v_{1} /2

Jaise area

A_{2} \to 0

(Case C), exit speed kya karta hai?

Woh infinity ki taraf blow up karta hai —

v_{2} = v_{1} A_{1} / A_{2} \to \infty

(ek nozzle jet).

Step 6 kaun sa single physical principle encode karta hai?

Conservation of mass — full, leak-free box exactly utna hi andar aane deta hai jitna baahaar jaata hai.

Connections

Parent topic
Conservation of mass — Step 6 ke peeche ka principle.
Volume flow rate and discharge — incompressible form $Q = A v$ .
Incompressible vs compressible flow — kyun Step 7 ki simplification ko constant $ρ$ chahiye.
Bernoulli's equation — continuity se mili speeds use karta hai.
Venturi meter — Step 8 Case C ek real instrument mein.
Streamlines and stream tubes — woh "imaginary tube" jo hamaari derivation ne quietly assume ki.