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

Before you can read the parent derivation you must own every letter it writes. This page builds each one from nothing: plain meaning → a picture → why the topic needs it, in an order where each idea leans on the one before.

1. The duct and its cross-section area $A$

Look at the figure. The duct is not the same width everywhere — where it pinches, $A$ is small; where it flares, $A$ is large. The whole topic is about how $A$ must change along the duct, so this is the very first thing to picture.

2. Flow speed $V$ and density $\rho$

In the figure, the same box of gas is shown "dense" (dots crowded) and "thin" (dots spread). For a normal liquid like water, $\rho$ barely changes no matter what you do. For a fast gas, $\rho$ changes a lot — and that fact is the whole surprise of the topic.

3. Mass flow rate $\dot{m}=\rho AV$ — the iron rule

Why that product? Imagine the gas crossing a slice in one second. It fills a cylinder of length $V$ (it travelled $V$ metres) and cross-section $A$. That cylinder has volume $A\times V$, and multiplying by density $\rho$ gives its mass. So $\dot{m}=\rho AV$.

4. Pressure $p$ and entropy $s$

5. The small-change symbols: $dX$ and fractional change $\frac{dX}{X}$

We now meet the little-$d$ notation, because everything from here on uses it.

6. The speed of sound $a$

7. Mach number $M=V/a$ — the switch

• $M<1$: subsonic — slower than sound.
• $M=1$: sonic — exactly the speed of sound.
• $M>1$: supersonic — faster than sound.

8. Euler's law — where $\rho VdV$ comes from

Follow the blob:

• The force. The pressure on its back face is $p$, on its front face $p+dp$. Net forward push per unit area is $-dp$; times the face area $A$ gives net force $-Adp$. (A drop in pressure pushes the blob forward.)
• The mass. $\rho Adx$, as above.
• The acceleration. In steady flow the blob's speed changes because it moves to a new spot where $V$ is different: $\text{accel}=V\frac{dV}{dx}$ (the "convective" acceleration — speed changes with position, and it covers $dx$ at speed $V$).

Put them into $F=ma$: $-Adp=(\rho Adx)\left(V\frac{dV}{dx}\right)$ The $A$ and $dx$ cancel, leaving $-dp=\rho VdV⟹dp+\rho VdV=0.$

That is why the $\rho VdV$ term appears — it is literally "mass-per-volume × velocity × velocity-change," i.e. the momentum term of $F=ma$. This is Euler's Equation for Steady Flow, Step 2 of the parent.

9. "Isentropic" — the clean-flow assumption

How the foundations feed the topic

Read it top-down: the three raw quantities build $\dot{m}$ and continuity; pressure builds Euler and (with density) the speed of sound; sound builds Mach; the isentropic tie joins them — and all streams meet to produce the area–velocity relation, which then dictates the de Laval nozzle shape.

Equipment checklist

Cover the right side and test yourself. If you can answer all, you're ready for the derivation.