- #1
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 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!
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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