Constant mass flow rate in thrust generation of jet/rocket engines

Question

I was watching this video here: https://www.youtube.com/watch?v=MTl5Xy_soFY

on the topic of calculating thrust for jet/rocket engines (where she arrives at $T+P_1A_1-P_2A_2=\frac{\partial(mv)}{\partial t}$). I was somewhat confused by the fact that she was able to assume that the mass flow rate was constant. If your engine is accelerating, the air intake (for a jet engine) will be increasing to some degree and since air is largely compressible, wouldn't you have to deal with the fact that $\frac{dm}{dt}$ isn't necessarily constant in the control volume? In the case of a rocket engine, if you had some design constraint that required a time-dependent release of fuel, would that also have the same effect?

The way I see it, one should apply the product rule to the RHS such that $\frac{\partial(mv)}{\partial t} = m\frac{\partial v}{\partial t} + v\frac{\partial m}{\partial t}$ where some relationship for $\frac{\partial m}{\partial t}$ is required based on the compressbility factor of the working fluid.

Am I completely wrong?

Answer 1

If the mass flow rate is different at entry and exit, you will have to use the product rule for the derivative of momentum. However, such a case would imply that air gets accumulated (or decumulated) in the engine, causing its net mass to increase (or decrease) with time. This usually doesn't happen, as most of the time aircrafts work pretty close to steady-state conditions, making this effect negligible. And about $dm/dt$ varying internally within the control volume: that does not matter because the net thrust force (on the engine as a free body) only depends on the mass flow rates at the entry and exit points.