The application of Reynold’s transport theorem to $m$ and $1$ gives the continuity equation for conservation of mass. When the flow is steady, the first term goes to zero.
$$ \frac{Dm}{Dt}=0=\frac{\partial}{\partial t}\int_{sys}\rho \>dV+\int_{cs}\rho \bold u\cdot\hat{\bold n}\>dA $$
Thus, for steady flows, by designing the control volume such that $\bold u\|\hat{\bold n}$ or $\bold u\perp\hat{\bold n}$, the continuity equation gives a relationship between various velocity profiles.
$$ 0={\int_{cs}\rho \bold u\cdot\hat{\bold n}\>dA} $$
The application of Reynold’s transport theorem to $m\bold u$ and $\bold u$ gives an expression for the conservation of linear momentum. The material derivative of linear momentum is equivalent to the net force on the contents of the control volume. When the flow is steady, the first term goes to zero.
$$ \frac{D\bold L}{Dt}=\Sigma \bold F_{cv}=\frac{\partial}{\partial t}\int_{sys}\rho \bold u\>dV+\int_{cs}\bold u \rho\bold u\cdot\hat{\bold n}\>dA $$
Thus, for steady flows, by designing the control volume such that $\bold u\|\hat{\bold n}$ or $\bold u\perp\hat{\bold n}$, we may obtain a simple system of relationships representing the conservation of linear momentum along each axis.
$$ \Sigma \bold F_{cv}=\int_{cs}\bold u\rho\bold u\cdot\hat{\bold n}\>dA $$
The application of Reynold’s transport theorem to $\bold H=\bold r\times m\bold u$ and $\bold r\times \bold u$ gives an expression for the conservation of angular momentum. The material derivative of angular momentum is equivalent to the net moment on the contents of the control volume. When the flow is steady, the first term goes to zero.
$$ \frac{D\bold H}{Dt}=\Sigma\bold M_{cv}=\frac{\partial}{\partial t}\int_{sys}\rho \bold r\times \bold u\>dV+\int_{cs}\bold r\times \bold u\rho \bold u\cdot\hat{\bold n}\>dA $$
Thus, for steady flows, by designing the control volume such that $\bold u\|\hat{\bold n}$ or $\bold u\perp\hat{\bold n}$, we may obtain a simple system of relationships representing the conservation of angular momentum along each axis.
$$ \Sigma\bold M_{cv}=\int_{cs}\bold r\times \bold u\rho \bold u\cdot\hat{\bold n}\>dA $$
The application of Reynold’s transport theorem to $E=em$ and $e$ gives an expression for the conservation of energy. The material derivative of energy is equivalent to the net heat flux into the the control volume plus net power on the contents of the control volume. When the flow is steady, the first term goes to zero.
$$ \frac{DE}{Dt}=\frac{\partial}{\partial t}\int_{sys}\rho e\>dV+\int_{cs}\rho e\bold u\cdot\hat{\bold n}\>dA $$
Incorporating enthalpy and defining $e=u_i+\frac{\bold u^2}2+gz$ gives the following energy equation.
$$ \dot Q+\dot W=\frac{\partial}{\partial t}\int_{sys}e\rho \>dV+\int_{cs}(u_i+\frac p\rho+\frac{\bold u^2}2+gz)\rho\bold u\cdot\hat{\bold n}\>dA $$