Reynold's Transport Theorem

For simplification the discussion will be focused on one dimensional control volume and it will be generalzed later. The flow through a stream tube is assumed to be one-dimensional so that there isn't any flow except at the tube opening. At the initial time the mass that was in the tube was $m_0$ . The mass after a very short time of $dt$ is $dm$ . For simplicity, it is assumed the control volume is a fixed boundary. The flow on the right through the opening and on the left is assumed to enter the stream tube while the flow is assumed to leave the stream tube.

Supposed that the fluid has a property $\eta$

$$\left. {dN_s} \over {dt} \right) = \lim_{\Delta t \rightarrow 0} {{N_s(t_0+\Delta t) - N_s(t_0)} \over {\Delta t}}$$ (2.1)