My physics has degraded from mathematical to intuitive, always a problem. Asking for help with what appears to be a simple problem that starts here:

fluid flowing in a tube at a given flow rate encounters a constriction reducing the area of the tube. This will create a pressure gradient related to Bernoulli's Principle;

I need clarification on the relationship between that pressure gradient under conditions of increased flow (volume/time)? Is this linear or exponential?

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    $\begingroup$ Might this help? As a first approximation you can model a constriction as a succession of contraction and expansion in the pipe. $\endgroup$ – Deep Apr 4 '18 at 14:12

If a liquid goes through a smaller area, the speed goes up by the ratio of the areas to get all the mass through.

If the speed goes up, the pressure goes down by the square because $\rm{P}+1/2\rho V^2$ doesn’t change.

So 1/2 the area is 1/4 the pressure, 1/3 the area is 1/9th the pressure, etc. no exponentials.

  • $\begingroup$ That is where I've been ending up, so I'm missing something in the model or not explaining it properly. $\endgroup$ – Zenbiker1290 Apr 4 '18 at 11:24
  • $\begingroup$ Assume the system to be a tube with area A1. It is level; h1/h2/g factor to zero. An segment with reduced area (A2) is present. As flow through the system increases, a critical threshold is reached where pressure on the input side of the constriction increases, and continues to rise as flow increases. I presume this is due to a limit of the constriction to allow a given mass segment to transit the constricted segment at the same rate as the input flow. I'm interested in the relationship of how pressure rises relative to increased flow beyond the critical level, and why. $\endgroup$ – Zenbiker1290 Apr 4 '18 at 12:18
  • $\begingroup$ BTW, your explanation of speed/pressure/area is very helpful, thank you. I've read where Bernoulli starts to fall apart in this type of system, but having to relearn old things and add new. This is an initial step in building and explaining a more complex physiologic system. $\endgroup$ – Zenbiker1290 Apr 4 '18 at 12:23
  • $\begingroup$ The increase of pressure on the inlet (pre-restriction) side is related to additional physics: drag due to viscosity. Those calculations depend on the behavior of the pump. It’s it’s providing pressure, drag just results in reduced flow. If it’s providing flow (I.e. fixed volume per second), then the pressure has to rise to provide a fixed flow. Most real cases are in between. $\endgroup$ – Bob Jacobsen Apr 4 '18 at 14:55
  • $\begingroup$ The restriction increases drag from (1) turbulence at the restriction if the flow isn’t smooth there and (2) increased wall drag because a larger of the flow is close enough to the wall to be in the turbulent boundary layer. $\endgroup$ – Bob Jacobsen Apr 4 '18 at 14:58

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