Bernoulli's Princple vs Flow Equation (deltaP=QR)

[PathEnthusiast]

First off, I'm a pathology resident, so it's been a long time since I've done real physics. But I really like physics and I try, whenever possible, to at least develop a basic understanding of the physics underlying physiology. With that as prelude, this question was inspired by blood flow but I suspect it has broader scope.

So in physiology classes we basically model the flow of blood according to an equation formally analogous to Ohm's law:

(Pressure1-Pressure2)=(total resistance)(flow)

Where Pressure1 and Pressure2 are the hydrostatic pressures at the beginning and the end of the circulation respectively.

Okay, so far so good. But today I was trying to understand why, in general terms, red blood cells drift toward the center of a vessel. The general answer seems to be a sort of "lift" force, which forced me to brush up on Bernoulli's principle and to think about laminar flow and viscosity. Because the fluid velocity is highest in the center, Bernoulli's principle seems to suggest the pressure should be lowest there, so there's a net force on a red cell towards the center (very roughly speaking).

But my interest at the moment isn't in the tendency of red cells to drift to the center--my question is how, broadly speaking, I should think about pressures in the flow equation versus pressures in Bernoulli's equation.

Imagine a pipe with fluid flowing through it. This pipe has a narrowing in the center. Bernoulli's equation would therefore suggest that the pressure at the center is lower than the pressure on either end. On the other hand, the flow equation suggests the pressure at one end has to be higher than pressure at the other end, in order to account for flow--and moreover would seem to suggest that there shouldn't be flow from low pressure to high pressure, even though Bernoulli's equation looks like it's saying the pressure in the central segment of pipe should be lower than the pressure in the segment immediately after it.

I must be thinking about this the wrong way, but it's not obvious to me right now how I should change my thinking to reconcile these two equations. What I'm considering now is whether the problem is that Bernoulli's equation assumes a "lossless" system--that is, no energy is "leaking" from the fluid--while the flow equation model, with resistance and so on, assumes that energy is leaving the fluid as one moves along the pipe. Am I on the right track?

 

[anlon]

Imagine a pipe with fluid flowing through it. This pipe has a narrowing in the center. Bernoulli's equation would therefore suggest that the pressure at the center is lower than the pressure on either end. On the other hand, the flow equation suggests the pressure at one end has to be higher than pressure at the other end, in order to account for flow--and moreover would seem to suggest that there shouldn't be flow from low pressure to high pressure, even though Bernoulli's equation looks like it's saying the pressure in the central segment of pipe should be lower than the pressure in the segment immediately after it.

I must be thinking about this the wrong way, but it's not obvious to me right now how I should change my thinking to reconcile these two equations. What I'm considering now is whether the problem is that Bernoulli's equation assumes a "lossless" system--that is, no energy is "leaking" from the fluid--while the flow equation model, with resistance and so on, assumes that energy is leaving the fluid as one moves along the pipe. Am I on the right track?

Blood behaves as a fairly viscous fluid, so it is reasonable to think that it would lose energy due to frictional effects as it flows. Additionally, you're right in that fluid cannot flow against an adverse pressure gradient - not for long, anyway. Consider this: if blood were to flow based entirely on Bernoulli's principle, there would be no need for a heart; blood would flow automatically. In reality, it takes work to pump the blood through the cardiovascular system, which is a process much more complicated than I pretend to understand.
On that note, the phenomenon of red blood cells being shifted towards the center of a blood vessel may be due to the temporary formation of laminar boundary layers during circulation, which would create an area of higher flow velocity and thus decreased static pressure in the center of the vessel. The red blood cells would be drawn into this low-pressure, high-velocity region.

 

[Chestermiller]

Blood is a viscoelastic fluid, and particles within a viscoelastic fluid flowing through a channel are found to migrate to the center of the channel (by virtue of the non-Newtonian fluid rheological behavior), even beyond the hydrodynamic entrance region. For scholarly articles on this, Google "particle migration in viscoelastic flow."

 

[gmax137]

Imagine a pipe with fluid flowing through it. This pipe has a narrowing in the center. Bernoulli's equation would therefore suggest that the pressure at the center is lower than the pressure on either end. On the other hand, the flow equation suggests the pressure at one end has to be higher than pressure at the other end, in order to account for flow--and moreover would seem to suggest that there shouldn't be flow from low pressure to high pressure, even though Bernoulli's equation looks like it's saying the pressure in the central segment of pipe should be lower than the pressure in the segment immediately after it.

There is more than one kind or pressure or "head." All of these are really contributors to the total energy of the fluid. Bernoulli equation is just saying that if the velocity head increases at some point along the streamline, then the pressure head must have decreased. So yes, the static pressure at the restriction is lower than the static pressure at the further downstream end of the pipe. When you think "flow from high to low pressure" it is the total pressure or head you really need to consider, not just the static pressure.

 

[Chestermiller]

I must be thinking about this the wrong way, but it's not obvious to me right now how I should change my thinking to reconcile these two equations. What I'm considering now is whether the problem is that Bernoulli's equation assumes a "lossless" system--that is, no energy is "leaking" from the fluid--while the flow equation model, with resistance and so on, assumes that energy is leaving the fluid as one moves along the pipe. Am I on the right track?

Yes. This is exactly correct. The standard Bernoulli equation neglects pressure losses from viscous friction. In a blood vessel, particularly capillaries, viscous effects tend to dominate.

 

[PathEnthusiast]

Thank you all so much for your replies! They were quite helpful in clarifying the issue. I have some follow-up questions about the way "acceleration" works in fluids, but I'm going to take some time to think about them and see if there's another thread addressing those issues before I start cluttering up the boards. Thanks again!