Energy conservation and Continuity for water travelling up a tube

Question

Suppose I have a tube bent at right angle such that one end points upwards and the other end is immersed in a river where water is flowing with some velocity v. The vertical part of the tube is having a height h.

Now as water travels up the pipe it should have a constant velocity v as the flow is continuous and the tube of uniform cross section.

I wondered where the extra energy came from as the fluid has to rise against gravity. By factoring in a pressure gradient I could explain it by accounting for work done by the pressure forces in the fluid.

It satisfied me, however then I thought about a situation where a nozzle of very small cross section was present at the top end.

Applying energy conservation (Bernoulli's equation) I could see that pressure at the bottom and the top of the tube are both equal to the atmospheric pressure-The bottom as it is exposed to the atmosphere and the top as the pressure sharply falls to the atmospheric pressure around the nozzle. This concludes that some of the energy of the water below was used up to rise against gravity, implying velocity of the water at the nozzle is lesser than at the surface.

However that doesn't seem to hold well with the continuity expression considering the small size of the nozzle.

What's going wrong? Is it wrong to assume the flow to be continuous?

(All conditions are ideal-no friction losses, streamlined motion, incompressible liquid)

Answer 1

The pressure at the entrance to the pipe is NOT equal to the pressure just outside the nozzle. In addition, there is pressure drop in the pipe, due both to the change in elevation and frictional and viscous effects of flowing water (if there is enough pressure drop to allow a flow). The frictional and viscous effects mean that you can't use the Bernoulli equation because it doesn't account for those losses.