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

2.2.12 · Physics › Fluid Mechanics

Continuity equation HAI kya?

$ρ A v$ quantity mass flow rate (kg/s) hai. $A v$ quantity volume flow rate $Q$ (m³/s) hai.

YEH sach kyun hai? (Conservation of mass)

Ise first principles se KAISE derive karein

Hum us mass ko track karte hain jo ek tiny time $Δ t$ mein ek cross-section se cross karta hai.

Step 1 — Time $Δ t$ mein ek section se kitna fluid cross karta hai? Cross-section 1 par, fluid $v_{1}$ speed se move karta hai. Time $Δ t$ mein yeh itni distance aage badhta hai: $Δ x_{1} = v_{1} Δ t .$ Yeh step kyun? Speed × time = distance — yahi fluid ka woh slab hai jo boundary se guzarta hai.

Step 2 — Us slab ka Volume. Slab ek cylinder hai jiska area $A_{1}$ aur length $Δ x_{1}$ hai: $Δ V_{1} = A_{1} Δ x_{1} = A_{1} v_{1} Δ t .$ Yeh step kyun? Volume = area × length; yahi woh volume hai jo andar aaya.

Step 3 — Us slab ka Mass. $Δ m_{1} = ρ_{1} Δ V_{1} = ρ_{1} A_{1} v_{1} Δ t .$ Yeh step kyun? Mass = density × volume. Yeh section 1 se andar aane wala mass hai.

Step 4 — Section 2 par wahi logic (bahar jaane wala mass). $Δ m_{2} = ρ_{2} A_{2} v_{2} Δ t .$

Step 5 — Conservation of mass apply karo (steady state). Sections ke beech koi mass accumulate nahi hota, isliye mass in = mass out: $Δ m_{1} = Δ m_{2} \Rightarrow ρ_{1} A_{1} v_{1} Δ t = ρ_{2} A_{2} v_{2} Δ t .$ Yeh step kyun? Sections ke beech ki "box" bhari hui aur unchanging hai, isliye jo bhi andar aata hai woh bahar bhi jaana chahiye.

Step 6 — $Δ t$ cancel karo. $ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2}$ Incompressible flow ke liye $ρ_{1} = ρ_{2}$ , ise divide kar do: $A_{1} v_{1} = A_{2} v_{2} = Q$

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

Worked examples

Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek busy hallway hai jisme bachche kaandhon se kaandha milaake chal rahe hain. Jahan hallway wide hai, bachche aaram se walk karte hain. Jahan yeh ek narrow doorway mein squeeze hoti hai, wahi number of kids per second abhi bhi pass karne hain — isliye unhe jaldi karni padti hai. Pipe mein paani exactly aisa hi hai: koi create nahi hota, koi gayab nahi hota, isliye ek narrow spot fluid ko speed up karne par majboor karta hai. Squeeze karo, speed badhti hai; widen karo, speed ghatti hai.

Active recall

Continuity equation kis physical law ka statement hai?

Conservation of mass (koi mass create/destroy nahi hota).

General continuity equation likho.

ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2} = const

(mass flow rate constant).

Incompressible fluids ke liye kaunsa form hold karta hai aur kyun?

A_{1} v_{1} = A_{2} v_{2}

, kyunki

ρ

constant hai aur cancel ho jaata hai.

Narrowing pipe mein, fluid speed up hota hai ya slow down?

Speed up hota hai;

v \propto 1/ A

Agar radius aadhi ho jaaye (incompressible), speed ka kya hoga?

Speed 4× ho jaati hai (

v \propto 1/ r^{2}

Volume flow rate

Q

kya hai aur iske units kya hain?

Q = A v

, units m³/s.

Compressing gas ke liye

A_{1} v_{1} = A_{2} v_{2}

kyun use nahi kar sakte?

Kyunki

ρ

change hota hai; sirf

ρ A v

(mass flow) conserved hota hai.

Time

Δ t

mein speed

v

par fluid slab ki distance derive karo.

Δ x = v Δ t

, isliye volume

= A v Δ t

, mass

= ρ A v Δ t

SABHI fluids ke liye (compressible bhi) kaunsi quantity conserved hoti hai?

Mass flow rate

\overset{m}{˙} = ρ A v

Connections

Bernoulli's equation — continuity woh speeds deti hai jo Bernoulli ko chahiye.
Volume flow rate and discharge
Incompressible vs compressible flow
Conservation of mass — parent principle.
Venturi meter — $A_{1} v_{1} = A_{2} v_{2}$ ka direct application.
Streamlines and stream tubes