Reynolds Transport Theorem

The Reynolds Transport Theorem lets us convert system-based property changes into control-volume based property changes.

Let $B$ represent any of these (or other) fluid parameters and $b$ represent the amount of that parameter per unit mass. That is,

$$B=mb$$

• $b$ is mass-independent
• $B$ is an extensive property
• $b$ is an intensive property

This means that

$$\begin{array}{rl}b=1& ⟶B=m\\ b=V& ⟶B=mV\\ b=\frac{1}{2}{V}^2& ⟶B=\frac{1}{2}m{V}^2\end{array}$$

Now, for each infinitesimal fluid particle in the system with mass $\rho \delta \forall$ and size $\delta \forall$ we can determine $B$ as

$$B_i=\delta m_i\cdot b_i={\rho }_i\delta \forall \cdot b_i$$

For the system, we can then sum over the individual particles:

$$B_{\text{sys}}=\sum B_i=\sum {\rho }_i{b}_i\delta {\forall }_i={\int }_{\text{sys}}\rho bdd\forall$$

Similarly,

$$B_{\text{C.V.}}={\int }_{\text{C.V.}}\rho bdd\forall$$

The Reynold transport theorem states:

$$\frac{D{B}_{\text{sys}}}{Dt}=\frac{\partial }{\partial t}{\int }_{\text{C.V.}}\rho bdd\forall +{\int }_{\text{C.S.}}b\rho \mathbf{V}\cdot \mathbf{n}dA$$