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etotheipi

If we consider a system of fixed mass as well as a control volume which is free to move and deform, then Reynolds transport theorem says that for any extensive property ##B_{S}## of that

My question is, what is the necessary relationship between the control volume and the system in order for that general relation to hold true? I have the feeling that the system and the control volume

Any clarification would be much appreciated, thanks!

*system*(e.g. momentum, angular momentum, energy, etc.) then$$\frac{dB_{S}}{dt} = \frac{d}{dt} \int_{CV} \beta \rho dV + \int_{CS} \beta \rho (\mathbf{V}_r \cdot \mathbf{n}) dA$$where ##\beta := \frac{dB_{S}}{dm}## is the quantity per unit mass and ##\mathbf{V}_r## is the relative velocity of the matter/fluid at the boundary w.r.t. the velocity of the control volume boundary. ##CV## and ##CS## denote control volume and control surface respectively.My question is, what is the necessary relationship between the control volume and the system in order for that general relation to hold true? I have the feeling that the system and the control volume

*must*coincide at time ##t## when the integrals are evaluated (but they will not necessarily coincide at ##t+dt##), but I am not certain of this. My reasoning was because if we consider a case where the control volume and system are completely separated (no overlap), then that relation is definitely wrong.Any clarification would be much appreciated, thanks!

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