# Control volume and momentum theorem

I'm studying fluid and propulsion mechanics by myself.

I stumbled upon this website from MIT: http://web.mit.edu/16.unified/www/SPRING/propulsion/UnifiedPropulsion2/UnifiedPropulsion2.htm#fallingblock

It states that "Newton’s second law for a control volume of fixed mass" is $$\sum \vec{F}=\int_V\rho \frac{D}{dt}(\vec{u}_0+\vec{u}) dV$$ but it's said that this is valid for a fixed mass control volume. $$\vec{u}_0$$ is the velocity of a reference frame attached to the control volume and $$\vec{u}$$ is the velocity of fluid relative to this moving frame.

The notes then goes on to derive this formula: $$\sum {F}_x-{F}_{0_x}=\int_V \left[\frac{\partial}{\partial t}(\rho{u}_x)\right]dV+\int_S u_x(\rho \vec{u}) \cdot \vec{n} dA$$ where $${F}_{0_x}$$ is basically $$m\vec{a}$$.

So far so good. However, I still don't understand why this equation is only valid for a control volume with fixed mass. Moreover, we're allowing the control volume to change its mass with by having the boundary term.

This is even stressed in the quizz accompanying these notes: https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/thermo-propulsion/q6.PDF

where the solution starts by remarking the validity of this equation depending on this assumption.

This seems to contradict books on Fluid Mechanics, where the mass can vary and they reach this similar equation (or maybe it's not the same equation?). For example Frank White's book equation 3.35: $$\sum \vec{F}=\frac{d}{d t}\int_V \left[(\rho \vec{V})\right]dV+\int_S \vec{V}(\rho \vec{V_r}) \cdot \vec{n} dA$$

I can see that White's equation is not exactly the same but I'm trying to prove they are by expanding V and V_r (V relative to Earth, inertial frame and V_r is a relative velocity).

I think I'm missing something here. Is the requirement of fixed mass even right? Considering that there is a boundary term.

In the term $$\int_V \frac{\partial}{\partial t} (\rho u_x) dV,$$ the time derivative is calculated using the body frame, i.e. the reference frame attached to the control volume. Imagine that you are sitting in the fueltank of a moving vehicle which consumes the liquid fuel, observing the level of the fuel contained in it. The rate of change of the fuel level and consequently, the rate of change of the fuel mass is going to be negligible. If this mass rate of change was not expected to be negligible, the accurate term which would replace this term would be $$\frac{\partial}{\partial t} \int_V \rho u_x dV.$$