I would try to answer this with some high school physics because that's all I know. I am in highschool right now.
Okay, lets go in the frame moving parallel to frustum with velocity $v$(which is frustum's initial velocity in frame of still water). Lets call this frame $A$ and still water's frame $B$ . In this frame, frustum is at rest initially, it is effectively inside a river flowing with velocity $v$ .
Our first goal is to prove that the frustum will become one with the river, i.e. it would get dragged along with the river and eventually attain the same speed as water particles, when that happens, no water is coming in or going out of the frustum.
Our second goal is to figure out how water particles gain momentum when they come out of the frustum's smaller end and how it impacts the frustum.
First goal is somewhat intuitive, proof is easy too. Water particles collide with the wall and get deflected. The force applied is normal to surface of frustum, due to the shape of the frustum, force can be divided into two perpendicular components: along the flow of river and perpendicular to it. On closer inspection, perpendicular components cancel out and the frustum doesn't move up or down. Components along the river cause the frustum to move. Particles keep colliding and applying force until frustum doesn't attain same velocity as the river. I could provide a diagram if anybody needs. First goal is proved ! What this implies is that in frame $B$ the frustum actually comes to a halt. So far so good.
Second goal is a bit harder. At some time $t$ , water comes in with $v$ and leaves with $u$ such that $u>v$ (as time goes on $u$ approaches $v$ as frustum speeds up). So, some water has gained momentum where did it come from ? To me it seems that as water particles collide with surface of frustum they get deflected "inwards" i.e they come closer from different directions they have lost some kinetic energy (gave it to frustum) and have a lower speed $v'$, now they are obstructing the way of those particles which were coming straight in without collision with walls. They collide with these. 780
<PERSON> put here a diagram for clarity: These particles keep coming "inwards" i.e. towards the smaller hole. After collisions their velocity actually increases(as expected). I am giving the mathematical proof:
In the diagram above, we can apply conservation of momentum along the horizontal direction i.e. direction of water flow. Resolve $v'$ into $v'{parallel}$ and $v'{perpendicular}$ . It is commonly known result that when particles with same masses collide, their velocities along the lines joining their centers interchange. So, after collision, $v'_{parallel}=v$ which increases overall speed. These collisions keep on happening as we go deeper inside the frustum and speed keeps on increasing until particles come out of the smaller hole.
This is the process which causes momentum of water to increase as it comes out. We can notice a few things, it isn't exactly like a rocket, there is no thrust so it would not move forward i.e. against river flow. The river is actually causing the increased momentum of water. To balance out the river must also be causing some negative momentum change somewhere and that is what happens as water comes out of the smaller hole, after some distance it slows down to match the flow of river. Now this might suggest that it is not being incompressible but actually certain "whirlpools" i.e water currents would be formed near the surface of frustum which would explain it. Honestly, I would have to think more about the exact details but I believe that is true. It is like currents formed when planes fly or submarines move underwater.
Note: To move into frame $B$ you just need to add an opposite $v$ velocity to everything we have done so far. 780