It depends on your derivation. (dm/dt)u can be seen as the thrust force which is applied by the mass leaving the system on the system, or it can be seen as the effects of conservation of momentum, as mass leaves the system.
At any rate, F in this case signifies external forces, while (dm/dt)u...
I'm having trouble understanding motion with non-constant mass.
Specifically, if (dm/dt) is positive when the mass of the system is increased, I find that:
\vec{F} = \frac{dm}{dt}\vec{u} + \frac{d\vec{v}}{dt} m
Where u is the velocity of the mass leaving/entering the system relative to the...