Two "material" particles can't pass through each other. Since the flow is 1D they can't go around each other either. The problem statement doesn't appear to represent any physically possible flow situation. Perhaps the condition "The velocity of each particle remains constant after firing" must be dropped.

@gbon You are right.

You are right. I have deleted my incorrect answer. Although you have mentioned perfect fluid in your question, the merging video is for viscous fluid (Reynolds number = 100). The counter-rotating vortex video also seems to be for a viscous fluid because overall kinetic energy seems to be going down with time.

@AntoniosSarikas I recommend that you post your doubt as a question.

Is it necessary to start your derivation with an open domain? The derivation goes through just fine if closed domain were used in defining the volume integral.

I can't be sure either, that was my best guess. Whenever two solid objects collide, obviously the air in the gap between them must be completely drained away before actual contact occurs. When one of the objects is fluid, its body can deform around the solid body, and contact can occur in a way that pockets of air are trapped without having been completely drained away.

I am guessing that the formation of air cavity may have to do with the time available for draining away the entrapped air (at the front of the falling sphere), which time is determined primarily by the velocity with which the sphere enters the fluid.

Due to no slip condition, total velocity has to go to zero at the wall. Which means that there will always be a region very close to the wall in which the flow velocities are quite small. Therefore in this region, the inertial terms in the N-S equation is negligible and viscous term is dominant. That is why it is called viscous sublayer.