Bernoulii vs Energy Conserv. Help

[HungryChemist]

Imagine you're are out in the space observing a long pipe. First half of the pipe has Area that are larger than the area of the second half of the pipe.
Further imagine, along the one end of pipe, a chuck of water(or a lot of water) is given a slight push to set in a motion at constant velocity along the pipe(No friction, No compression, Constant Density). Now, the water continues to move along the pipe all the way through and some part of the water now enters the second section of the pipe.

Now, I have a trouble picturing what's going to happen. I thought the mass of water must not change. So the amount of water getting out of the one section of the pipe must enters the other section of the pipe. In order to do that the water has to run faster along the section of pipe with smaller area. So now I have some part of the water running faster than before. Considering the water and pipe as one system, I can see that the energy is not conserved.(since the water now runs faster). But I can not find any source who could possibly do a work on the pipe and the water. For some reason I suspect that it has something to do with Bernouli's equation involving pressure. How should I make of this? It seems like here the Energy is not conserved. Help?

 

[whozum]

The water is compressed into a smaller section of the pipe, and the pressure from the walls of the pipe is pushing the water. The compressed water wants to return to standard pressure, and in doing so travels faster as the rpessure increases. The change in speed using constant velocity is due to the pressure increase.

 

[Andrew Mason]

So now I have some part of the water running faster than before. Considering the water and pipe as one system, I can see that the energy is not conserved.(since the water now runs faster). But I can not find any source who could possibly do a work on the pipe and the water. For some reason I suspect that it has something to do with Bernouli's equation involving pressure. How should I make of this? It seems like here the Energy is not conserved. Help?

Mechanical energy (potential and kinetic) is conserved. It cannot just increase by itself. The kinetic energy of the fluid is drawn from the potential energy it had while under increased pressure. Pressure represents energy density. \rho v^2 also represents energy density. One can be easily converted to the other and back again.

AM

Edit: If, initially, all of the water is in the large diameter section of the pipe some distance from the constriction, it moves along the pipe at speed v. But it slows down when it encounters the narrow constriction and the pressure at the constriction will increase. This pressure is then converted to kinetic energy as the water passing through the constriction increases speed back to the original speed of the flow.

AM

 

[Clausius2]

The water is compressed into a smaller section of the pipe, and the pressure from the walls of the pipe is pushing the water. The compressed water wants to return to standard pressure, and in doing so travels faster as the rpessure increases. The change in speed using constant velocity is due to the pressure increase.

Not quite. The static pressure at the smaller section is lower than the pressure at wider section. As Andrew has pointed, some of the initial pressure energy has been transformed into kinetic energy, globally conserving the total mechanic energy. As "static pressure" I mean precisely static pressure, all the pressure which hasn't a kinetic, gravitational or centripetal origin.

Actually there is no such conservation in this problem. It has no sense to apply Bernoulli equation in such a fast stretching zone. There will be losses of energy mechanical and eventually pressure will decay below the value predicted by "ideal" flow.

 

[HungryChemist]

Not quite. The static pressure at the smaller section is greater than the pressure at wider section. As Andrew has pointed, some of the initial pressure energy has been transformed into kinetic energy, globally conserving the total mechanic energy.

What do you mean by the static pressure at the smaller section is greater than the pressure at wider section? I mean, I don't understand what you mean by 'static pressure'. Could you explain?

I thought the Bernoulii's equation will tell us that the pressure at the smaller section of pipe is lower. So you must mean 'static pressure' different from 'pressure'?

I have another question. When I looked at the derivation of the bernouli's equation from textbook, they first defined a small section or chunk of water moviing through the strem tubes. Then they applied work energy theorem to get the result. But what confused me from their derivation is that the system(the chunk of water) experienced the net force(Pressure from the both sides of the end) there by changing the kinetic energy. I don't see this as a conservation of energy but simply the work was applied to increase the mechanical energy of the system. But the question I asked at first, didn't needed this new pressure to increase the mechanical energy of the system. What's going on here?

 

[Clausius2]

What do you mean by the static pressure at the smaller section is greater than the pressure at wider section? I mean, I don't understand what you mean by 'static pressure'. Could you explain?

I thought the Bernoulii's equation will tell us that the pressure at the smaller section of pipe is lower. So you must mean 'static pressure' different from 'pressure'?

Oooopssss! Sorry. I have made a mistake. Look again at my last post. My apologises.