A bit of fluid mechanics from scratch not from scratch

> Ok, but if the water is faster at the bottom of the long spout… We could view the top part of this system as an exact copy of the short-spout version. At the interface between the tank bottom and the pipe-spout, the velocity of the water should be the same as in the no-pipe version, right? But that means the water inside the pipe is accelerating inside the pipe:

No, it's not the exact same. In the top part of the long-spout system there's a lack of airpressure holding the water above it back compared to the short-spout, and quite a bit of cohesion in the water pulling the water above it down faster if the lack of air pressure isn't enough. The water in the whole system moves faster as a result.

You'd theoretically get the air (actually vacuum) bubble if you ran the experiment in a vacuum with a liquid that has no cohesion... liquids with no cohesion are otherwise known as gasses though and behave differently in other ways as well.

How much of the energy will get transmitted as opposed to reflected will depend on impedance matching. There are two very important transitions (i) open air to air inside cavern and (ii) cavern air to cavern walls.

Regarding (ii) unless the density transition is eased into, (gravel and sand ?) the energy in the air will get reflected right back. Very little will get transmitted into the rock walls.

Same principle applies for (i). As a resident of such a cavern you would want high impedance mismatch. It makes sense to have well matched labyrinths to dissipate the energy much like silencers and mufflers. One would want all the energy be converted into turbulence, safely and quickly.

I wonder if these considerations inform their design. Probably does.

What kindergarten did you go to? Maybe my public kindergarten education was seriously lacking.

>narrator: he’s assuming that the water at different vertical positions, at a given horizontal position, is moving with the same vector

This assumption is physically impossible, given the first diagram in part #2. Obviously in real life the thing would drain out the side, so obviously there isn't a uniform flow gradient, neither top-to-bottom nor radially. The flow has to bend somewhere, after all. This sometimes happens where students get so wrapped up in the math that they lose common sense and "prove" reality wrong, creating a paradox.

The author does it again with the "top half of an hour glass" shape:

>Well, we know that the velocities are always strictly vertical (and downward).

No, we actually don't know this. This is again physically impossible, given the shape of the container. If there was no horizontal component you'd get a tube-shaped void extending up from the spout, and water would magically fail to drain from the sides as if you'd drilled out a core sample from solid ice. Obviously, this is not what happens in reality, so obviously the assumption that all the velocity is purely vertical is incorrect.

>The water at the bottom of the spout cannot be going faster than at the top! [...] In other words, the water does indeed accelerate, and it also gets narrower; the pipe is partly filled with air.

The author appears to have forgotten that friction and gravity exist. Generally if the flow is steady (which it will be unless the volume of water in the tank is small compared to the volume of water in the outlet pipe), then the pipe will run full of water at all times. Water does not accelerate in the pipe. If you add length to the pipe then the added height is counteracted by added friction and by the weight of the moving fluid.

Grab an undergraduate fluids textbook and do an energy balance.