# Pressure loss in an infinite pipe

I feel lost in this problem. I know pressure loss occurs in a pipe flow due to friction.

Say given a constant diameter pipe, if we keep lengthening the pipe horizontally, and static pressure will decrease in the downstream.

$Av=\text{constant}$ requires velocity $v$ be constant. [mass conservation]

1. Then we must be able to reach to a point where pressure $p$ drops to zero, is that correct? Then what happens after that? Will pressure keep dropping to negative values?
2. If not, will there be a minimum pressure for which a given fluid and flow scenario pressure will sustain at?
3. If so, energy conservation requires that velocity has to drop as well, so the flow will come to a stop eventually? Do we have such kind of "velocity loss" thing? If there is, what is the relationship between this "pressure loss" and "velocity loss"?
4. If there is such thing as "velocity loss", then what about the mass conservation? Should we then need to discard the "incompressible" assumption?
• Perhaps consider the cause-and-effect the other way. If I have a capped pipe that goes on for hundreds of miles, and I suddenly uncap either end, the water will only start flowing very very slowly. Making the pipe longer will just reduce velocity /everywhere/ to make it self-consistent. Jul 25, 2016 at 2:52

What comes first? Pressure or flow?

Without potential, a pressure differential, there is no flow. It's pressure that needs to be there first. So rather than ask what pressure drop you get, it's better to ask what flow you get from applied pressure.

We can never measure pressure drop across an infinite pipe because we can never reach the infinite point. But just considering a very long pipe, there is allot of resistance. So to move fluid you need lots of inlet pressure over what pressure you have at the outlet. And just increase the pipe a little longer, and more pressure is required. At some point you cannot provide the energy needed - even before you reach infinity. You will have a huge (static) pressure differential but zero flow.

Resistive losses in a pipe only occur when there is velocity. But there is something missing in your model - inertance of the fluid. The potential energy you supply at the inlet must provide sufficient force to first accelerate the fluid, then enough pressure to maintain the flow against the resistance.

Just by turning the question around with regards to cause and effect you see all is consistent with what's expected. Pressure doesn't happen after flow. It was there to begin with, however changes when flow is activated.

• It would be nice to add to your answer that the applied energy gets dissipated into heat. Jul 25, 2016 at 5:17
• Can I understand it in this way, that there is also "velocity loss", however, because the "pressure loss" is more obvious to observe in the beginning section of the pipe, so people historically call it "pressure loss / head loss"? Jul 26, 2016 at 0:30
• Thinking that the limit of resistance exerted by the pipe be equal to the energy of fluid due to pressure differential, the fluid would have the same velocity of inlet (considering constant diameter)? I also struggle on it, @Daniel. May 11, 2021 at 17:37