Foundations — Reynolds transport theorem

2.2.13 · D1 · Physics › Fluid Mechanics › Reynolds transport theorem

Isse pehle ki tum parent note par Reynolds Transport Theorem padho, tumhe ek chhoti si toolkit chahiye. Hum har tool zero se banate hain, ek aisi order mein jahan har tool sirf apne pehle wale tools par depend karta hai. Kuch bhi assume nahi kiya gaya.

1. Ek "field": space ke har point par ek number

Fluids mein pehla idea yeh hai ki ek moving ball ki jagah, humare paas poori space bhari hui hai cheezein se, aur har point par kuch na kuch value hoti hai.

Figure dekho: space ke har dot par ek chhota sa arrow hai. Arrows ka woh collection hi velocity field hai. Yahi wajah hai ki fluids ek single particle se alag lagte hain — follow karne ke liye koi single object nahi hai; room mein arrows ka ek carpet bichha hua hai.

2. $\rho$ — density: "ek chhote box mein kitna mass packed hai"

Picture yeh hai: imagine karo fluid ko laakhon chhote sugar-cubes mein kaatna. $\rho$ batata hai har cube kitna bhaari hai apne size ke har unit ke hisaab se. Paani ka $\rho \approx 1000{\text{kg/m}}^3$ hai: ek cubic metre ka wazan 1000 kg hota hai.

3. $\vec{v}$ — velocity: har point par ek arrow

Figure 1 ka woh arrows ka carpet hi velocity field $\vec{v}$ hai. Point $(x,y,z)$ par time $t$ mein, $\vec{v}(x,y,z,t)$ woh ek arrow hai jo wahan rehta hai.

4. Extensive vs intensive — $B$ aur $b$

Kuch quantities double ho jaati hain jab tum matter ki amount double karte ho (do buckets weighing karo, double mass milta hai). Dusri nahi hoti (paani ke do buckets same temperature ke hote hain jaise ek bucket).

Picture yeh hai: $B$ total pile hai; $b$ hai har kilogram kitna pile carry karta hai. Wapas multiply karo aur tum pile recover karte ho: ek chhota mass $dm$ property ka $bdm=b\rho dV$ carry karta hai.

5. Dekhne ke do tarike: Lagrangian vs Eulerian

Figure mein, Lagrangian observer (blue) downstream drift karta hai usi water blob se chipka hua. Eulerian observer (yellow) ek jagah bolted rehta hai aur paani usse rush karke guzarta hai.

6. System vs control volume — $CV$ aur $CS$

Woh trick jo poori derivation shuru karti hai: ek instant $t$ par, hum system ko $CV$ mein exactly fit hone dete hain. Ek instant baad woh alag drift ho gaye hain — kuch system leak ho gaya, kuch naya fluid andar aaya. Wahi mismatch precisely surface term hai.

7. $\hat{n}$ — outward normal, aur direction kyun matter karta hai

Figure dekho. Right wall par, fluid arrow $\vec{v}$ aur $\hat{n}$ same direction mein point karte hain — fluid ja raha hai. Left wall par, $\vec{v}$ andar point karta hai jabki $\hat{n}$ abhi bhi baahir point karta hai — woh oppose karte hain. Yahi sign hai jisse maths apne aap outflow aur inflow mein farq karta hai.

8. $\vec{v}\cdot \hat{n}$ — dot product: "wall se kitna cross karta hai"

Ab humein ek number chahiye: fluid actually wall ko kitni tezi se pierce karta hai? "Fluid kitna fast hai" nahi — wall ke saath saath sliding karne wala fluid kuch cross nahi karta. Humein $\vec{v}$ ka woh piece chahiye jo $\hat{n}$ ke saath saath point karta ho.

Figure $\vec{v}$ ki tip se $\hat{n}$ line par ek perpendicular giraaता है — woh shadow length $\vec{v}\cdot \hat{n}$ hai. Figure mein teen cases padho:

• Fluid seedha baahir ja raha hai ($\theta =0$): $\cos 0=1$, full speed cross karta hai — maximum outflow.
• Fluid wall ke saath slide kar raha hai ($\theta =90^{\circ }$): $\cos 90^{\circ }=0$, kuch cross nahi karta — exactly jo hum chahte hain, tangential flow kuch pierce nahi karta.
• Fluid seedha andar ja raha hai ($\theta =180^{\circ }$): $\cos 180^{\circ }=-1$, shadow negative hai — inflow, correctly minus sign ke saath.

9. Do integral signs — ${\int }_{CV}dV$ aur ${\oint }_{CS}dA$

Picture yeh hai: ${\int }_{CV}\rho bdV$ interior ko cubes se tile karta hai aur andar stored property ka total karta hai; ${\oint }_{CS}\rho b(\vec{v}\cdot \hat{n})dA$ skin ko tiles se tile karta hai aur jo across leak hota hai uska total karta hai.

10. $\frac{d}{dt}$ vs $\frac{\partial }{\partial t}$ — change ke rates

Storage term $\frac{d}{dt}{\int }_{CV}\rho bdV$ poochtha hai: kya mere fixed box ke andar ka total stuff time ke saath change ho raha hai? Iska flow se koi lena-dena nahi — yeh unsteadiness ke baare mein hai. Steady flow ke liye yeh zero hai, chahe fluid rush karke guzar raha ho.

Prerequisite map

Equipment checklist

Apne aap ko test karo — right side cover karo aur jawab do:

Connections

• Parent: RTT — jahan yeh saare tools assemble hote hain.
• Eulerian vs Lagrangian description — §5 ke do viewpoints.
• Continuity equation — RTT with $b=1$.
• Momentum equation (control volume) — RTT with $b=\vec{v}$.
• Bernoulli equation — RTT with $b=e$ restrictions ke saath.
• Material derivative — storage idea ka point-wise cousin.
• Divergence theorem — §9 ke surface integral ko volume integral se swap karta hai.