Confusing situation in thrust and conservation of momentum

Question

Imagine I am inside a lake of ideal incompressible, non-viscous fluid. I have a hollow frustrum-like structure with both ends open, basically a tube with one end of a larger radius than the other.
I throw it with some velocity $v$, with the larger opening facing forwards. Now, from the frame of reference of the frustrum, fluid is entering from the front with a velocity $v$. However, as the fluid is incompressible and the other end is smaller, it comes out the back with a greater velocity, so that the volume of fluid entering is same as the volume exiting. In this case, it seems to me that there will be a thrust on the frustrum in the forward direction, as the water itself is gaining momentum in the backward direction.
But this means that the frustrum will keep speeding up forever. This clearly violates conservation of energy and basic logic. What am I missing here?
Edit: I undersand that if you look at the  collisions of the water molecules with the inner wall of the frustrum, they will provide an opposing force. However, while analyzing it from a pure momentum conservation point of view, this doesnt seem to matter. Water gains speed in the negative direction, thus the frustrum has to gain a speed in the positive direction. I want to find out what is wrong with the momentum analysis, not other ways in which analyse the situation so as to get the expected result.

Brain Stroke Patient · Accepted Answer

This is a rough diagram of the same water element at subsequent times as it passes through the frustrum. As you can see a gap is created as water flows through the frustrum. Remember that in the frustrum's frame, all the water outside of it is flowing at velocity $v$. So the gap cannot be filled with fluid coming from behind because fluid behind the element outside of the frustrum is moving at the same speed as the element. On the other hand fluid coming from the smaller hole is travelling at a greater speed than $v$ while the velocity of the fluid outside the smaller hole just when the frustrum started moving is $v$. Because the fluid is incompressible therefore there is an excess of fluid coming out of the smaller hole that need to occupy some extra volume to preserve it's incompressibility. So what these excess fluid do is rush backwards over the outer surface of the frustrum to fill the gaps shown in the picture.
If in a time $dt$, $dm$ mass of water comes out of the smaller hole, then $dm$ mass of water will enter the gaps. But the velocity with which they enter the gap is higher than the velocity with which they leave the smaller hole. The idea is the same as the proof of $A_1 v_1 = A_2 v_2$ for incompressible fluids. The "area" provided by the gap is much much smaller. So to conserve momentum, the frustrum must actually gain momentum in the backward direction which will cause it to slow down.
Edit
Magnified diagram showing the gap. Gap is the region coloured in black. The extra region in blue inside the frustrum that is ahead of the outside water element previously occupied the black region when there was no frustrum.

BioPhysicist · Answer

The increase in velocity of the fluid isn't due to the frustrum pushing on the fluid. It's due to pressure differences in the fluid. Therefore, the fluid speed up will not accelerate the frustrum further upon release.
You can see this by considering the walls of the frustrum. Collisions of the fluid with the walls will push the fluid forwards, and thus those collisions push the frustrum backwards. Therefore, those collisions cannot be responsible for increasing the backwards velocity of the fluid.

BowlOfRed · Answer

Water gains speed in the negative direction, thus the frustum has to gain a speed in the positive direction.

Some water gains speed in the negative direction.  Other portions of the water gain speed in the positive direction.  In particular, fluid immediately in front of the opening is pushed forward.  Also fluid next to the "outside" wall is pulled forward.
By only looking at one particular volume, it seems to be incorrect to suggest that the net momentum change of all the water is rearward.

Physicsa · Answer

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. Lemme 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.

Confusing situation in thrust and conservation of momentum

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