Bernoulli's Principle and Energetics of Flowing blood

1. Mar 18, 2015

Jmiz

My understanding and application:
Flowing blood with mass m, and velocity v has KE proportional to mean velocity squared
as blood flows inside the vasculature, pressure is also exerted laterally against the walls of the vessels
So, it is then reasonable to use Bernoulli's for the blood and vessel system:
Total E = .5 (density)(v)2 + P + pgy
where P is the hydrostatic pressure
change in P = density*change in height*gravity

So application wise:
I been taught blood flow is driven by the difference in total E between 2 points from high to low
and since most of the cardiovascular system, KE is relatively low, so the pressure difference can be used to drive flow

What I dont understand is if total E gradient is used as the driving force for blood movement, which means that total E is not conserved due to frictional forces, then how can one use the conservation aspect of the Bernoulli's principle to interconvert between kinetic energy and pressure energy for two different points in the blood and vasculature system? For example: a plaque region which decreases the effective diameter of the blood vessel and the region right before the plaque region, with normal diameter.

How can you state that the pressure in the plaque region is lower than the pressure in the non-plaque region if the total E is not conserved between those two points?

Also, in the Bernoulli's principle will the potential energy density be a significant contribution to the total E throughout the circulatory system? Are we measuring y relative to the bottom of the ground or the center of the body?

I have uploaded a practice question that started my series of questions. This question used the conservation of energy of the Bernoulli principle to conclude that hydrostatic pressure was lower in the plaque region that the region right before the plaque region.

The total E if as a gradient will have its highest point starting from the location of the heart (the pump) and lowest as it gets farther away from the pump due to friction.

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2. Mar 18, 2015

Staff: Mentor

The frictional loss is a gradual cumulative effect over longer distances. If you have a short section in which the cross section changes rapidly, the frictional contribution in such a region is negligible, and the pressure variation can be closely approximated by neglecting the frictional contribution.
It doesn't matter where the datum is taken for determining the potential energy. The changes in pressure resulting from changes in elevation are independent of this.
The lowest pressure will be at the location where the blood is about to re-enter the heart (neglecting the elevation difference between the head and the heart).

Incidentally, the item 2 analysis with respect to oxygen makes no sense to me. Dalton's law doesn't apply to gases dissolved in liquids. If there is no gas phase in the blood vessel (hopefully that is the case), then the partial pressure concept does not apply. The gas will stay in solution as long as the total pressure on the blood is higher than the equilibrium vapor pressure of the gas dissolved in the blood (at its concentration in the blood).

Chet

3. Mar 18, 2015

Jmiz

Hi Chet,

Thanks for helping out. I feel like now I understand the plausibility for interconversion between KE and pressure using Bernoulli principle as an approximation.

You mentioned the lowest pressure energy density will be at the veins where the blood is about to reenter the heart, and I have been taught this as well. However, if we were to evaluate the gradient using total energy, instead of the pressure gradient, we would see the same relationship, so can I reason that the KE will also be the lowest at the veins?

And with respect to item 2 analysis, I also had doubts about the application of dalton's law in this example, but then reasoned that a little bit of gas would dissolve in the blood (O2 that were missed by hemoglobin; CO2 that didnt get convered into carbonic acid via enzyme in RBC). But as the overall blood pressure decreased as KE increases substantially in the area of the plaque, the gas pressure in the liquid/solvent/plasma would also decrease.

Also, I have been confused by the effect of vasoconstriction of a blood vessel (decreases cross sec area) on blood pressure. I understand this may not be an area of expertise, but I would appreciate it if you can check my understanding. Vasoconstriction in general of blood vessel leads to increased blood pressure by heart because vasoconstriction increases the total peripheral resistance of the vessel. However, when applying Bernoulli principle and comparing to relatively close points, the position that is constricted will have a relatively lower blood pressure than the position just before with normal vessel diameter. Does those two seem correct to you? I have attached below an equation that was claimed to be an adapted version of Ohm's law that helped me associate an increase in BP as a consequence of vasoconstriction that increases total peripheral resistance of a vessel. Thanks again for your help.

Ohm's law applied to Circulation: pressure differential = cardiac output * total peripheral resistance (Source: Kaplan MCAT 2015 biology chapter 7)

4. Mar 18, 2015

Staff: Mentor

The KE is strictly determined by the flow velocity. The flow velocity is determined by the volumetric flow rate through the blood vessel divided by the cross sectional area.
There's no such thing as gas pressure if all the substances are dissolved in the liquid, and the overall pressure is high enough to keep the dissolved substances in solution. However, if the overall pressure drops enough, the dissolved substances can come out of solution and form a gas phase. While they are in solution, the only thing you can say is that the pressure potential to come out of solution is equal to the Henry's law constant times the mole fraction of the substance in solution. However, this value does not depend on the total pressure.
With vasoconstriction, you presumably have a longer region of the blood vessel involved, and the frictional (viscous) pressure drop can be considerably more significant. Your equation applies in this region. In addition, the increase in kinetic energy entering the constricted region is exactly offset by the kinetic energy decrease exiting the region, so the effects on the pressure cancel.

Chet

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