Bernoulli’s equation does work, to a certain degree, in the human body; however, although Bernoulli’s equation the possibility of the amount of change that could occur due to a change in height, it can only approximate the change in pressure in either a person’s translational displacement (i.e. moving a person up into the air or down to the ground) or angular displacement (i.e. having a person sit up or lie down). Why?
- First, the heart does its best to move the blood at a certain speed threshold at every place in the body. Therefore, if there is a different pressure for certain portions of your body than the rest, your heart will look to pump more blood faster in order to raise the speed of your blood in that portion of your body (Linnarsson et al., 1996).
- Second, the blood vessels used for arterial lines are non-Newtonian fluids. Now, blood has two phases: a liquid phase, made up almost entirely of blood plasma, and a solid phase, which are made up of the various particles suspended in the blood. Now, Newtonian fluids are fluids that are pure, one-phase fluids, and thus obey physics laws. In order for a two-phase liquid to be labelled as a Newtonian-like fluid, the two phases inside need to move quickly, and have such a large cross-sectional area that the particles suspended in the blood have a negligible volume. This is the case for a blood vessel like the aorta, which has a cross-sectional area of 7.2 cm², which is extremely large compared to the blood particle within it, and whose blood moves very quickly, as the aorta is directly connected to the heart.
However, in the case of arterial lines, the blood vessels used have a much smaller cross-sectional area, and the blood here is much further from the heart and is therefore much slower, thus suggesting that these blood vessels are more non-Newtonian, and thus, deviate from the ideal fluid behavior. Therefore, it is much harder, if not impossible, to use Bernoulli’s equation to find the blood pressure change in arterial line blood vessels.
- Third, to maintain the blood pressure in the body, our muscles contain skeletal muscle pumps, which help maintain the blood pressure in our veins. Once the blood reaches a place where it can give its oxygen to the cells in our body, the blood moves extremely slow, which is dangerous for our bodies. Therefore, the skeletal muscle pump uses Bernoulli’s equation for our bodies, only, this time, it keeps the change in height constant, which means:
ΔP = ½ ρ(υ² – v²)
What this shows is that, in order for the blood velocity to increase, the pressure in the blood must increase. This further complicates the use of Bernoulli’s equation, as it does not.
With these various physiological complications, it is safe to use Bernoulli’s equation to qualitatively explain what occurs in the body; however, to quantitatively explain the phenomena, we would need much more precise and much more complicated calculations.
References:
Linnarsson, D., Sundberg, C. J., Tedner, B., Haruna, Y., Karemaker, J. M., Antonutto, G., & Di Prampero, P. E. (1996). Blood pressure and heart rate responses to sudden changes of gravity during exercise. Am J Physiol, 270(6 Pt 2), H2132-2142. doi:10.1152/ajpheart.1996.270.6.H2132
Nguyen, Y., & Bora, V. (2021). Arterial Pressure Monitoring. In StatPearls. Treasure Island (FL): StatPearls Publishing
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Saugel, B., Kouz, K., Meidert, A. S., Schulte-Uentrop, L., & Romagnoli, S. (2020). How to measure blood pressure using an arterial catheter: a systematic 5-step approach. Crit Care, 24(1), 172. doi:10.1186/s13054-020-02859-w
Sheriff, D. D., Mullin, T. M., Wong, B. J., & Ladouceur, M. (2009). Does limb angular motion raise limb arterial pressure? Acta Physiol (Oxf), 195(3), 367-374. doi:10.1111/j.1748-1716.2008.01912.x