Area-velocity relation — dA - A = (M² − 1)(dV - V) — derivation (explains de Laval nozzle)

3.1.5 · Physics › Compressible Flow & Aerodynamics

HUM KYA relate kar rahe hain?

Hum chahte hain ek single equation jo fractional area change $d A / A$ ko fractional speed change $d V / V$ se link kare.

KAISE: Derivation first principles se

Humein teen physical statements chahiye. Hum inhe differentiate karenge, phir combine karenge.

Step 1 — Conservation of mass (continuity)

$ρ A V = constant$

Natural log lo, phir differentiate karo (yeh ek product ko fractional changes ke sum mein convert kar deta hai — woh trick jo sab kuch clean banati hai):

$ln ρ + ln A + ln V = const$ $\boxed{\frac{d\rho}{\rho} + \frac{dA}{A} + \frac{dV}{V} = 0} \tag{1}$

Step 2 — Conservation of momentum (Euler's equation)

1-D steady Euler (momentum) equation:

$dp + \rho V\,dV = 0 \quad\Longrightarrow\quad \frac{dp}{\rho} = -V\,dV \tag{2}$

Step 3 — Isentropic relation speed of sound deti hai

Toh: $d\rho = \frac{dp}{a^2} \tag{3}$

Step 4 — Combine karo

(2) ko (3) mein daalo: $d ρ = \frac{- ρ V d V}{a ^{2}}$

$ρ$ se divide karo: $\frac{d\rho}{\rho} = -\frac{V^2}{a^2}\frac{dV}{V} = -M^2\,\frac{dV}{V} \tag{4}$

Ab (4) ko continuity equation (1) mein substitute karo: $d ρ / ρ - M^{2} \frac{d V}{V} + \frac{d A}{A} + \frac{d V}{V} = 0$

$\frac{d A}{A} = M^{2} \frac{d V}{V} - \frac{d V}{V} = (M^{2} - 1) \frac{d V}{V}$

YEH de Laval nozzle kyun explain karta hai

Figure — Area-velocity relation — dA - A = (M² − 1)(dV - V) — derivation (explains de Laval nozzle)

$(M^{2} - 1)$ ka sign padhkar flow accelerate karo ( $d V > 0$ ):

Regime	$M^{2} - 1$	$d V > 0$ paane ke liye humein chahiye...	Duct shape
Subsonic $M < 1$	negative	$d A < 0$	Converging
Sonic $M = 1$	zero	$d A = 0$	Throat (minimum area)
Supersonic $M > 1$	positive	$d A > 0$	Diverging

Flashcards

Area–velocity relation derive karne ke liye kaunse teen conservation/physical laws combine kiye jaate hain?

Continuity (mass), Euler's momentum equation, aur isentropic speed-of-sound relation

d p = a^{2} d ρ

Area–velocity relation state karo.

\frac{d A}{A} = (M^{2} - 1) \frac{d V}{V}

Subsonic flow mein gas accelerate karne ke liye duct converge karega ya diverge?

Converge (

d A < 0

), kyunki

M^{2} - 1 < 0

Supersonic flow mein gas accelerate karne ke liye duct converge karega ya diverge?

Diverge (

d A > 0

), kyunki

M^{2} - 1 > 0

Exactly

M = 1

par relation ka kya hota hai?

M^{2} - 1 = 0

, isliye

d A = 0

— sonic flow sirf throat (minimum area) par ho sakti hai.

Continuity ke differential form derive karne mein log-differentiation kyun help karta hai?

Yeh product

ρ A V =

const ko fractional changes ke sum

d ρ / ρ + d A / A + d V / V = 0

mein convert karta hai.

a^{2} = (\partial p / \partial ρ)_{s}

physically kya mean karta hai?

Speed of sound ka square equal hota hai us stiffness se jis se pressure constant entropy par density ke response mein react karta hai.

De Laval nozzle converging–diverging kyun hona chahiye?

Subsonic se accelerate karne ke liye (converging chahiye)

M = 1

throat se hote hue supersonic mein (diverging chahiye).

Euler's equation se, flow speed up hone ke liye pressure ko kya karna chahiye?

Pressure drop karna chahiye:

d p = - ρ V d V

, isliye

d V > 0 \Rightarrow d p < 0

d ρ / ρ

M

aur

d V / V

ke terms mein derive karo.

d p = - ρ V d V

aur

d ρ = d p / a^{2}

se:

d ρ / ρ = - (V^{2} / a^{2}) (d V / V) = - M^{2} (d V / V)

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho tum hawa ko ek hose se push kar rahe ho taaki woh tezi se nikal sake. Jab tak hawa sound se dheerae chal rahi hai, hose ko pinch karne se woh tezi ho jaati hai — bilkul paani ki tarah. Lekin jab hawa sound se tez chal rahi hoti hai, kuch ajeeb hota hai: ab tumhe hose ko wide karna padta hai taaki woh aur tez ho sake. Hawa itni tezi se spread aur thin ho jaati hai ki zyada room dene se actually speed badhti hai. Isliye ek rocket nozzle hourglass jaisi dikhti hai: ek narrow neck (jahan hawa exactly sound ki speed pakadti hai) aur phir ek wide bell jahan woh super fast jaati hai.

Connections

Speed of Sound in a Gas — Step 3 mein use hone wala $a^{2} = (\partial p / \partial ρ)_{s}$ supply karta hai.
Continuity Equation (Compressible Flow) — equation (1) ka source.
Euler's Equation for Steady Flow — equation (2) ka source.
Mach Number — woh parameter jo sign flip karta hai.
De Laval Nozzle and Choked Flow — is result ka engineering payoff.
Isentropic Flow Relations — full nozzle design ke liye $p, ρ, T$ ko $M$ se relate karta hai.
Stagnation Properties — throat conditions compute karne ke liye zaroori.