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

3.1.5 · D1 · Physics › Compressible Flow & Aerodynamics › Area-velocity relation — dA - A = (M² − 1)(dV - V) — derivat

Parent derivation padhne se pehle tumhein har letter ka matlab pata hona chahiye. Yeh page har ek cheez zero se build karta hai: plain meaning → ek picture → topic ko iske zaroorat kyun hai, ek aisi order mein jahan har idea apne pehle wale idea pe lean karta hai.

1. Duct aur uska cross-section area $A$

Figure dekho. Duct har jagah same width nahi hai — jahan pinch hoti hai, $A$ chhota hota hai; jahan flare hoti hai, $A$ bada hota hai. Poora topic is baare mein hai ki $A$ duct ke saath saath kaise change hona chahiye, isliye yeh sabse pehli cheez hai jo picture karni hai.

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

Figure mein, same box of gas "dense" (dots crowded) aur "thin" (dots spread) dikhaya gaya hai. Ek normal liquid jaise water ke liye, $\rho$ kuch bhi karo barely change hota hai. Ek fast gas ke liye, $\rho$ bahut zyada change hota hai — aur yahi fact is topic ka poora surprise hai.

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

Woh product kyun? Socho ki gas ek slice ko ek second mein cross kar rahi hai. Yeh length $V$ (usne $V$ metres travel kiye) aur cross-section $A$ ka ek cylinder fill karta hai. Us cylinder ka volume $A\times V$ hai, aur density $\rho$ se multiply karne par uska mass milta hai. Toh $\dot{m}=\rho AV$.

4. Pressure $p$ aur entropy $s$

5. Small-change symbols: $dX$ aur fractional change $\frac{dX}{X}$

Ab hum little-$d$ notation se milte hain, kyunki yahan se aage har cheez iska use karti hai.

6. Speed of sound $a$

7. Mach number $M=V/a$ — the switch

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

8. Euler's law — $\rho VdV$ kahan se aata hai

Blob ko follow karo:

• The force. Uske back face pe pressure $p$ hai, front face pe $p+dp$. Net forward push per unit area $-dp$ hai; face area $A$ se multiply karne pe net force $-Adp$ milti hai. (Pressure mein ek drop blob ko forward push karta hai.)
• The mass. $\rho Adx$, jaise upar.
• The acceleration. Steady flow mein blob ki speed isliye change hoti hai kyunki yeh move karke ek naye spot pe jaata hai jahan $V$ alag hota hai: $\text{accel}=V\frac{dV}{dx}$ (woh "convective" acceleration — speed position ke saath change hoti hai, aur yeh $dx$ ko speed $V$ se cover karta hai).

$F=ma$ mein daalo: $-Adp=(\rho Adx)\left(V\frac{dV}{dx}\right)$ $A$ aur $dx$ cancel ho jaate hain, bacha rehta hai $-dp=\rho VdV⟹dp+\rho VdV=0.$

Isliye $\rho VdV$ term appear hota hai — yeh literally "mass-per-volume × velocity × velocity-change" hai, yaani $F=ma$ ka momentum term. Yeh Euler's Equation for Steady Flow hai, parent ka Step 2.

9. "Isentropic" — clean-flow assumption

Foundations topic ko kaise feed karte hain

Isko top-down padho: teeno raw quantities $\dot{m}$ aur continuity build karti hain; pressure Euler build karta hai aur (density ke saath) speed of sound; sound Mach build karta hai; isentropic tie unhe join karta hai — aur saari streams milke area–velocity relation produce karti hain, jo phir de Laval nozzle shape dictate karta hai.

Equipment checklist

Right side cover karo aur khud test karo. Agar sab jawaab de sako, tum derivation ke liye ready ho.