2.2.12 · D1Continuity equation — derivation (conservation of mass), ρAv
Behta hua fluid bas chhote-chhote particles ki bheed hai jo na kahin se aate hain, na kahin gayab hote hain. Isliye har second pipe ke kisi bhi slice se gujarne wale "stuff" ki matra har slice pe same hoti hai — aur jab yeh maan lo, toh puri continuity equation khud-ba-khud likh jaati hai.
Is page pe assume kiya gaya hai ki tumne kuch nahi dekha . Parent derivation padhne se pehle hum har letter ko ek ek karke build karenge, saath mein ek picture aur reason bhi ki yeh exist kyun karta hai.
Fluid woh cheez hai jo behti hai aur apne container ki shape le leti hai — liquids (paani) aur gases (hawa). Ek nadi ki tarah socho jisme chhote-chhote marbles ek doosre se past slide karte hain.
Definition Flow / flow speedFlow ka matlab hai fluid move kar raha hai. Flow speed woh hai kitni tezi se fluid ek jagah se travel karta hai — jaise paani ki surface ki speed jo tum ek patta daalke aur time lekar measure karte ho.
Yeh kyun chahiye: pura topic is baat ke baare mein hai ki pipe ke alag-alag jagahon pe fluid kitni tezi se move karta hai. "Speed at a place" woh cheez hai jise hum track karte hain.
Δ Δ "mein ek chhota sa change" ya "thoda sa". Toh Δ t Δ x multiplication nahi hai — Δ t
Intuition Tiny slices ki zaroorat kyun hai?Fluid ko crossing karte waqt pakadne ke liye, hum ek bahut chhota moment Δ t
Figure mein, fluid chhoti si red distance Δ x Δ t
v Speed v v = Δ t Δ x . Δ x = v Δ t v = 20 Δ t = 3 Δ x = 60
Topic ko yeh kyun chahiye: derivation ke Step 1 mein poochha jaata hai ki fluid ka ek slab Δ t kitni dur jaata hai. Jawab bilkul Δ x = v Δ t
Recall Yeh "v Δ t v /Δ t multiply karna chahiye, divide nahi. Time double karo, distance double ho jaayegi. Zyada time matlab zyada travel ::: distance time ke saath badhti hai, isliye multiply karo
Definition Cross-sectional area, symbolA area A m 2
Radius r A = π r 2 .
r 2 r two-dimensional quantity hai — jab circle bada hota hai toh yeh dono directions mein badhti hai. Radius double karo aur width aur height dono double ho jaati hain, isliye area 2 × 2 = 4 squared ke saath scale karti hai.
Dono circles dekho: daayein wale ka radius double hai lekin shaded area chaar guna hai — ek hi picture mein r 2
Common mistake "Aadha radius ⇒ aadhi area."Kyun sahi lagta hai: radius woh number hai jo hume diya jaata hai. Reality: area r 2 chauthaai area deta hai. Fix: hamesha radius ratio ko square karo: A 1 / A 2 = ( r 1 / r 2 ) 2
Topic ko yeh kyun chahiye: A
Definition Volume, symbolV Volume woh 3-D space ki matra hai jo koi cheez fill karti hai, cubic metres (m 3 A Δ x Δ V = A Δ x . A Δ x
Derivation ke Step 2 ke saath combine karo — kyunki Δ x = v Δ t Δ V = A v Δ t .
Figure mein slab window A Δ x
Topic ko yeh kyun chahiye: yahi slab volume woh fluid hai jo actually ek blink Δ t
Definition Density, symbolρ ρ mass per unit volume hai — har cubic metre mein kitna stuff bhara hai:
ρ = volume mass = V m , units kg/m 3 . m = ρ V ρ
Toh hamare fluid slab ki mass hai:
Δ m = ρ Δ V = ρ A v Δ t .
Intuition Density compressible case mein star kyun haiPaani muskil se squash hota hai — uski ρ ρ mass hai, aur mass = ρ V ρ Incompressible vs compressible flow ).
Topic ko yeh kyun chahiye: law mass conserve karta hai, volume nahi. Density woh cheez hai jo volume slab ko mass slab mein convert karti hai.
m Mass m kabhi create ya destroy nahi hoti (dekho Conservation of mass ). Sealed pipe section mein enter karne wala fluid bahar nikalna chahiye — uske paas jaane ki koi aur jagah nahi hai.
Definition Mass flow rate, symbolm ˙ m "per second" (rate). Toh m ˙ har second ek section se gujarne wali mass :
m ˙ = Δ t Δ m = Δ t ρ A v Δ t = ρ A v ( kg/s ) . Δ t
Topic ko yeh kyun chahiye: m ˙ = ρ A v hi continuity equation hai. Conservation of mass kehta hai yeh number har section pe same hai: ρ 1 A 1 v 1 = ρ 2 A 2 v 2
Definition Volume flow rate, symbolQ Q volume hai jo har second deliver hota hai:
Q = Δ t Δ V = A v ( m 3 / s ) . Q Volume flow rate and discharge mein.
Definition Streamline & stream tubeStreamline woh path hai jo ek fluid particle follow karta hai — jaise flow mein dye ka ek single strand. Streamlines ka ek bundle ek stream tube banata hai, ek imaginary pipe jiske walls se koi fluid cross nahi karta. Dekho Streamlines and stream tubes .
Topic ko yeh kyun chahiye: continuity ke liye real metal pipe ki zaroorat nahi. Koi bhi stream tube sealed "box" ki tarah count hoti hai, isliye same ρ A v = const
1 2 A 1 , v 1 , ρ 1 section 1 pe measure kiya gaya"; subscript 2 do alag jagahon ke labels hain — ek wide jagah aur ek narrow jagah.
ρ A v = constant ρ A v same number milega. Pipe mein bina badla hua travel karne wala woh ek number hi mass flow rate hai.
Speed v = distance over time
Slab distance = v times dt
Volume = A times slab distance
Density rho = mass over volume
Continuity rho A v = const
Ise bottom-up padho: har arrow ek symbol hai jo ab tumhara hai, neeche continuity equation mein feed hota hua.
Khud ko test karo — daayein side cover karo. Agar koi fail ho, parent note se pehle us section ko dubara padho.
Symbol Δ t "Time ka ek tiny slice"; yeh ek single quantity hai, na ki Δ × t
Speed v Δ t Δ x = v Δ t
Radius r A = π r 2
Agar radius aadhi ho jaaye, toh area ka kya hoga? Area chauthaai ho jaayegi (area ∝ r 2
Density ρ Mass per unit volume, ρ = m / V
Face A Δ x Δ V = A Δ x = A v Δ t
Us slab ki mass? Δ m = ρ Δ V = ρ A v Δ t
m ˙ m ˙ "Per second"; mass flow rate m ˙ = ρ A v
Q Volume flow rate Q = A v ρ
SABHI fluids ke liye kaun si quantity conserved hoti hai? Mass flow rate m ˙ = ρ A v
Subscripts 1 2 Ek hi tube mein do alag cross-sections (jagahein).
Parent topic (Hinglish) — woh derivation jisme yeh foundations kaam aati hain.Conservation of mass — yahan ki har cheez ke peeche ka ek law.Volume flow rate and discharge — Q = A v Incompressible vs compressible flow — kab ρ Streamlines and stream tubes — imaginary pipe.Bernoulli's equation — continuity jo speeds deti hai unki zaroorat hai.Venturi meter — continuity action mein. 
