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

2.2.12 · D5 · Physics › Fluid Mechanics › Continuity equation — derivation (conservation of mass), ρAv

Shuru karne se pehle, do forms ka ek reminder taaki neeche har symbol samajh aaye:

Error pakdo

SE1. "Paani narrow pipe mein speed up hota hai, isliye wahan mass flow rate bhi badh jaati hai."

Galat — mass flow rate

ρ A v

hi conserved quantity hai; speed exactly isliye badhti hai taaki

ρ A v

constant rahe, ye increase nahin hoti.

SE2. "Kyunki

Q = A v

aur

A

kam hota hai, narrow section mein

Q

bhi kam ho jaata hai."

Galat — incompressible flow ke liye

Q

constant hota hai; jaise

A

ghatta hai,

v

badhta hai taaki unka product

Q

fixed rahe.

SE3. "

A \propto r

, isliye radius aadha karne par area bhi aadha ho jaata hai."

Galat — circular cross-section mein

A = π r^{2}

hota hai, isliye area

\propto r^{2}

;

r

aadha karne par

A

ek-chauthaayi ho jaata hai.

SE4. "Bucket example mein

v^{2}

use hota hai, isliye ye Bernoulli problem hai."

Galat — filling time sirf

Q = A v

use karta hai (ek continuity/volume-flow idea); koi pressure ya energy balance appear hi nahin hoti.

SE5. "Mass in ka mass out ke barabar hona zaruri nahin — fluid beech mein pile up ho sakta hai."

Steady incompressible flow mein galat — region already full hai aur

ρ

badh nahin sakta, isliye accumulation impossible hai; in = out.

SE6. "Example 3 mein area same tha, isliye speed mein kuch change nahin hona chahiye tha."

Galat — density double ho gayi, aur

ρ A v

constant ke saath fixed

A

v

ko aadha hona padta hai; density constant area par bhi speed badal sakti hai.

SE7. "Continuity ke liye fluid ka frictionless aur ideal hona zaroori hai."

Galat — continuity pure mass bookkeeping hai; ye viscous, real fluids ke liye bhi hold karta hai. Sirf Bernoulli ko ideal-flow assumptions chahiye.

Why wale questions

WHY1. Hose ka end squeeze karne par paani zyada tez kyun nikalta hai?

Utna hi volume per second ek chhoti opening se pass karna hota hai, isliye

A v = Q = const

v

badhna padta hai jab

A

ghatta hai.

WHY2. Derivation mein

Δ t

ko rakhne ki jagah cancel kyun karte hain?

Dono sides par same

Δ t

tha (mass in

= ρ_{1} A_{1} v_{1} Δ t

, out

= ρ_{2} A_{2} v_{2} Δ t

); ise divide karne par ek relation milta hai jo kisi bhi time interval ke liye sahi hai.

WHY3.

A v

ki jagah

ρ A v

ko "master" statement kyun prefer karte hain?

Kyunki

ρ A v

(mass flow) sabhi fluids ke liye conserved hota hai;

A v

sirf woh special case hai jo tab bachta hai jab incompressible flow mein

ρ

cancel ho jaata hai.

WHY4. Continuity akela narrow section mein pressure kyun nahin bata sakta?

Continuity geometry aur mass se speeds fix karta hai, lekin pressure energy balance se aata hai — uske liye Bernoulli's equation chahiye.

WHY5. Ek nadi rocks ke upar tez aur shallow kyun behti hai lekin pool mein slow aur deep ho jaati hai?

Discharge

Q = A v

roughly constant rehta hai, isliye jahan cross-section (depth × width) chhota hota hai wahan paani speed up karta hai, aur vice-versa.

WHY6. Venturi meter mein jaanbujhkar ek narrowing kyun banate hain?

Narrow throat

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

ke zariye ek jaani-pehchani speed-up force karta hai; resulting pressure drop measure karne par flow rate pata chalta hai.

WHY7. Simple form ke hold karne ke liye "koi mass accumulate nahin hoti" kyun assume karna padta hai?

Agar mass sections ke beech build up hoti, toh inflow outflow se zyada hoti aur

ρ_{1} A_{1} v_{1} = ρ_{2} A_{2} v_{2}

fail ho jaata; steadiness hi in = out banata hai.

Edge cases

EC1. Completely blocked pipe (

v = 0

har jagah) ke liye continuity kya kehti hai?

ρ A v = 0

har section par — ek valid, consistent solution; zero in equals zero out, koi motion nahin.

EC2. Jab outlet area zero ki taraf shrink hoti hai (incompressible), exit speed ka kya hota hai?

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

; model unbounded speed predict karta hai, jo warn karta hai ki real effects (viscosity, cavitation) eventually intervene karenge.

EC3. Agar fluid incompressible HO aur pipe area uniform bhi ho, toh continuity kya ban jaati hai?

Sirf

v_{1} = v_{2}

; ek trivial lekin sahi statement — constant speed, kuch solve karna hi nahin.

EC4. Kya mass flow rate

ρ A v

ek hi steady tube ke do sections ke beech kabhi alag ho sakta hai?

Nahin — steadiness plus no leaks se

ρ A v

har section par identical hota hai; yahi constancy hai continuity equation.

EC5. Ek steadily heated gas jo constant-area duct mein expand karti hai (aur

ρ

girata hai), usme

v

ka kya hona chahiye?

Use badhna chahiye:

A

fixed aur

ρ

girte waqt,

v = \overset{m}{˙} / (ρ A)

badh jaata hai taaki mass flow constant rahe — dekho Incompressible vs compressible flow.

EC6. Do identical hoses ek pipe mein merge hoti hain jiska total area ek hose jaisa hai — kya merged speed ek hose ki speed ke barabar hogi?

Nahin — combined mass flow double ho gayi, isliye agar merged area ek hose ke area ke barabar hai toh speed double ho jaati hai; areas aur flows dono balance karne padte hain.

EC7. Steady flow mein, agar density aur area dono constant hain lekin fluid phir bhi accelerate karti hai, toh kya continuity violate hoti hai?

Haan hogi — constant

ρ

aur

A

se constant

v

force hota hai; ek actual acceleration ka matlab hai ki ya toh

A

ρ

kahin actually change ho raha hai.

Recall Traps ka ek-line summary

Woh single sentence jo yahan almost har trap ko defuse karti hai: == $ρ A v$ woh hai jo nature fix rakhti hai; speed sirf ise protect karne ke liye badlti hai.== Wide⇒slow, narrow⇒fast, aur $ρ$ tab hi drop karo jab pakka ho ki fluid incompressible hai.

Connections

Parent: Continuity equation
Conservation of mass — woh law jo upar ke har answer ke peechhe hai.
Incompressible vs compressible flow — decide karta hai ki kaun sa form use kar sakte ho.
Bernoulli's equation — jahan continuity energy ko hand off karti hai.
Venturi meter · Volume flow rate and discharge · Streamlines and stream tubes

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

Sahi ya galat — justify karo