Reynolds Transport Theorem

Theorem

In real world situation it is easier to easier to analyse a control volume (control volume analysis) rather than keep track of an entire system (control system analysis). This means that we have to meaningfully convert the Lagrangian Derivatives to Eulerian Derivatives.

Say we have a Property $B$. The amount of $B$ in a control volume can be evaluated as follows

$$B_{\text{CV}}={\int }_{\text{CV}}\beta dm={\int }_{\text{CV}}\beta \rho dV$$

Where $\beta \to \frac{dB}{dm}$ The Reynolds Transport theorem states that:

$$\begin{array}{rl}\text{Rate of Change of }B\text{ in the Sys}& =\text{(Accumulation rate of }\beta \text{ inside CV) + (Net Outflux of }\beta \text{ through CS)}\\ \text{Rate of Change of }B\text{ in the Sys}& =\text{(Accumulation rate of }\beta \text{ inside CV )+ }(\beta \text{ Outflow from CV }-\beta \text{ Inflow to CV)}\end{array}$$ $$\frac{d}{dt}(B_{\text{sys}})=\frac{d}{dt}\left({\int }_{\text{cv}}\beta \rho dV\right)+{\int }_{\text{CS}}\beta \rho v\cos \theta d{A}_{out}-{\int }_{\text{CS}}\beta \rho v\cos \theta d{A}_{in}$$

Considering the Surface normal vector $\vec{n}$, above equation can be simplified to

$$\boxed{\frac{d}{dt}(B_{\text{sys}})=\frac{d}{dt}\left({\int }_{\text{CV}}\beta \rho dV\right)+{\int }_{\text{CS}}\beta \rho (\vec{v}\cdot \vec{n})dA}$$

Special Cases

• Steady Flow:

$$\frac{d}{dt}(B_{\text{sys}})={\int }_{\text{CS}}\beta \rho (\vec{v}\cdot \vec{n})dA$$

• Relative Velocity Where ${\vec{v}}_r=\vec{v}-{\vec{v}}_s$

$$\frac{d}{dt}(B_{\text{sys}})=\frac{d}{dt}\left({\int }_{\text{cv}}\beta \rho dV\right)+{\int }_{\text{CS}}\beta \rho ({\vec{v}}_r\cdot \vec{n})dA$$