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Assuming laminar flow, the Hagen–Poiseuille equation defines the flow rate as a function of the head pressure. However as the flow rate increases so does the head loss due to friction within a pipe, as stated by the Darcy-Weisbach equation.

My question is this, is there a limit where an increase in head pressure will not result in an increase in flow rate, due to the loss of energy through friction?

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The transition from Hagen-Poiseuille to Darcy-Weisbach behaviour happens when the flow regime switches from laminar to turbulent. Note that the DW equation has a fudge factor that describes the energy loss due to turbulence. This friction factor varies with flow rate.

For an increase in pressure to produce no increase in flow the friction factor would have to go to infinity at some flow rate. As far as I know this doesn't happen so an increase in pressure will always produce some increase in flow rate, though that increase will be proportionally smaller as the flow rate increases.

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    $\begingroup$ And finally, the pipe bursts, spraying all the experimenters and ruining the data... $\endgroup$ – Jon Custer Oct 2 '15 at 13:51
  • $\begingroup$ @JonCuster Mythbusters would approve! $\endgroup$ – Carl Witthoft Oct 2 '15 at 14:36
  • $\begingroup$ Followup question: has anyone found a situation where intense pressure causes friction to decrease? There's the obvious case of pressure-cleaning a sewer pipe to remove detritus, but I'm thinking of a pipe material which does some sort of structural state change. $\endgroup$ – Carl Witthoft Oct 2 '15 at 14:37
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    $\begingroup$ @CarlWitthoft: there are no end of non-Newtonian fluids that behave in this manner. For example anything that is thixotropic. $\endgroup$ – John Rennie Oct 2 '15 at 14:47

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