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I'm looking into Reynolds transport theorem (RTT) stated as:

$$ \frac{dB}{dt} = \frac{d}{dt} \int_{CV} b\rho dV+\int_{CS}b\rho(\underline{v}\cdot\underline{n}) dA$$

where B is defined as an extensive property and related to the intensive property b as $b = B/m$. I've seen how this equation is derived from the perspective of looking at a specific control volume, but the extensive property is just assumed from the start, which confuses me a bit.

I've seen how this can be applied to many different extensive properties, such as mass, linear and angular momentum, energy and so forth. My question is, as long as our extensive property is dependent on mass, can we always apply RTT? If so, why / why not? Why can we just use assume we are working with an extensive property in our derivation of the equation for RTT?

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The crucial concept here isn't necessarily the mass--it's the locality of conservation laws. Locality is often taken for granted because it agrees with our intuitive experience.

As a specific example, when we say "momentum is conserved", part of what we mean that "the total momentum of a closed system is constant." You could call this a global conservation law--no matter how the momentum is distributed in that system, as long as the integral of momentum over the whole system is the same, that part of the definition is satisfied.

The local part dictates how the momentum moves around within the system. A local conservation law says that momentum cannot simply vanish from one point and appear somewhere else instantaneously--it has to move continuously from one point to another, passing through any boundaries in the process. That's what the surface integral is reflecting for us here.

So the general answer is that the RTT equation applies to any locally conserved quantity, of which mass is one example in the context of fluid dynamics.

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  • $\begingroup$ Thank you for your nice explanation! So, have I understood you correctly in that RTT can be seen as a more general form of conservation law that describes the local behaviour of the quantity that's conserved? $\endgroup$
    – Tanamas
    Commented Apr 16, 2023 at 18:07

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