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

WHAT are we relating?

We want a single equation linking a fractional area change $dA/A$ to a fractional speed change $dV/V$ .

HOW: Derivation from first principles

We need three physical statements. We will differentiate each, then combine.

Step 1 — Conservation of mass (continuity)

$\rho A V = \text{constant}$

Take the natural log, then differentiate (this turns a product into a sum of fractional changes — the trick that makes everything clean):

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

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

The 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 gives the speed of sound

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

Step 4 — Combine

Put (2) into (3): $d\rho = \frac{-\rho V\,dV}{a^2}$

Divide by $\rho$ : $\frac{d\rho}{\rho} = -\frac{V^2}{a^2}\frac{dV}{V} = -M^2\,\frac{dV}{V} \tag{4}$

Now substitute (4) into the continuity equation (1): $\underbrace{-M^2\frac{dV}{V}}_{d\rho/\rho} + \frac{dA}{A} + \frac{dV}{V} = 0$

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

WHY this explains the de Laval nozzle

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

Read off the sign of $(M^2-1)$ to accelerate the flow ( $dV>0$ ):

Regime	$M^2-1$	To get $dV>0$ we need...	Duct shape
Subsonic $M<1$	negative	$dA<0$	Converging
Sonic $M=1$	zero	$dA=0$	Throat (minimum area)
Supersonic $M>1$	positive	$dA>0$	Diverging

Flashcards

What three conservation/physical laws are combined to derive the area–velocity relation?

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

dp=a^2 d\rho

State the area–velocity relation.

\dfrac{dA}{A}=(M^2-1)\dfrac{dV}{V}

In subsonic flow, to accelerate the gas should the duct converge or diverge?

Converge (

dA<0

), because

M^2-1<0

In supersonic flow, to accelerate the gas should the duct converge or diverge?

Diverge (

dA>0

), because

M^2-1>0

What happens to the relation exactly at

M=1

M^2-1=0

, so

dA=0

— sonic flow can occur only at a throat (minimum area).

Why does log-differentiation help in deriving continuity's differential form?

It converts the product

\rho A V=

const into a sum of fractional changes

d\rho/\rho+dA/A+dV/V=0

What does

a^2=(\partial p/\partial\rho)_s

physically mean?

The speed of sound squared equals how stiffly pressure responds to density at constant entropy.

Why must a de Laval nozzle be converging–diverging?

To accelerate from subsonic (needs converging) through

M=1

at the throat to supersonic (needs diverging).

From Euler's equation, what must pressure do for the flow to speed up?

Pressure must drop:

dp=-\rho V\,dV

, so

dV>0\Rightarrow dp<0

Derive

d\rho/\rho

in terms of

M

and

dV/V

From

dp=-\rho V dV

and

d\rho=dp/a^2

d\rho/\rho=-(V^2/a^2)(dV/V)=-M^2(dV/V)

Recall Feynman: explain it to a 12-year-old

Imagine pushing air through a hose to make it shoot out fast. While the air is moving slower than sound, pinching the hose makes it go faster — just like with water. But once the air is going faster than sound, something weird happens: now you have to make the hose wider to speed it up even more. The air spreads out and thins so quickly that giving it more room actually helps it go faster. That's why a rocket nozzle looks like an hourglass: a narrow neck (where the air hits exactly the speed of sound) and then a wide bell where it goes super fast.

Connections

Speed of Sound in a Gas — supplies $a^2=(\partial p/\partial\rho)_s$ used in Step 3.
Continuity Equation (Compressible Flow) — source of equation (1).
Euler's Equation for Steady Flow — source of equation (2).
Mach Number — the parameter that flips the sign.
De Laval Nozzle and Choked Flow — the engineering payoff of this result.
Isentropic Flow Relations — relates $p,\rho,T$ to $M$ for full nozzle design.
Stagnation Properties — needed to compute throat conditions.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, idea simple hai: ek duct (pipe) me gas chal rahi hai, aur hume decide karna hai ki use tez karne ke liye pipe ko patla (converge) karein ya chauda (diverge) karein. Normal soch ye hoti hai ki "patla karo, gas tez ho jayegi" — jaise paani ke pipe me. Lekin yahaan twist hai: gas compressible hai, density change hoti hai. Iska answer Mach number $M=V/a$ par depend karta hai.

Final formula nikalta hai teen cheezon ko milake: continuity ( $\rho A V$ = constant), Euler ka momentum equation ( $dp=-\rho V\,dV$ ), aur sound speed ka relation ( $dp=a^2 d\rho$ ). In teeno ko jodne par aata hai $\dfrac{dA}{A}=(M^2-1)\dfrac{dV}{V}$ . Trick ye hai ki $(M^2-1)$ ka sign sab kuch decide karta hai.

Agar $M<1$ (subsonic), to $M^2-1$ negative — gas tez karni hai to area kam karo (converge). Agar $M>1$ (supersonic), to $M^2-1$ positive — gas tez karni hai to area badhao (diverge). Aur exactly $M=1$ par bracket zero ho jata hai, matlab $dA=0$ — yahi throat (sabse patli jagah) hai. Isiliye rocket ka nozzle hourglass jaisa hota hai: pehle patla (subsonic ko tez karta), throat par exactly sound speed, fir chauda (supersonic banata). Yaad rakhne ka mantra: "Sub squeezes, Super spreads, Sonic at the slimmest."

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

WHAT are we relating?

HOW: Derivation from first principles

Step 1 — Conservation of mass (continuity)

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

Step 3 — Isentropic relation gives the speed of sound

Step 4 — Combine

WHY this explains the de Laval nozzle

Worked examples

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Compressible Flow & Aerodynamics

Connections