Teach Poiseuille First — A Call for a Paradigm Shift in Fluid Dynamics Education Author: Brad Moser

Final revisions submitted to “The Physics Teacher” in Nov. 2020 Introduction A classic, life-science-themed fluid dynamics scenario is blood flow through a constriction.1,2 Physics teachers traditionally ask students if the pressure experienced by the blood in the constriction is greater, lesser, or the same as before the constriction. The conventional approach to resolving this question calls upon the equation of continuity, as well as the Bernoulli equation. Biological systems, however, experience a resistance to flow, and a consequential pressure drop, that is often better described by Poiseuille’s law. Within this apparent conflict, which approach is correct? This paper argues that Poiseuille’s law is the more appropriate choice for most biological examples and encourages a Poiseuille-first approach to teaching fluid dynamics in classes designed for life science majors. The Case for a New Approach A review of the chapters on fluids in commonly-used physics textbooks3 shows that, after introducing hydrostatic pressure and buoyancy, each one emphasizes the Bernoulli and continuity equations as the primary descriptors of fluid dynamics. For instance, streamlines of air pass over the wings of an airplane more quickly than under the wings, leading to a pressure gradient that produces a lift force. Mounting evidence4-7 suggests, however, that Bernoulli’s equation alone cannot completely describe airplane lift, curving soccer balls, floating Ping-Pong balls, water-spray, and other classic examples. Coanda and Magnus effects, turbulence, vortices, and entrainment play vital roles in these systems. Bernoulli’s equation, far from playing the lead role in fluid dynamics, is merely a single cog in a larger network of important mechanisms. As teachers are beginning to adopt Introductory Physics for the Life Sciences (IPLS)8-10 curricula, their emphasis shifts towards the physics principles applied to biological systems. Fluid systems receive greater attention in the classroom and biological examples include blood pressure and cell wall tension, flow in circulatory and respiratory systems, transport via diffusion and osmosis, and viscous drag forces that resist the forward motion of swimming organisms. As I will demonstrate, these systems typically have low Reynolds numbers and exhibit pressure drops consistent with Poiseuille. While the Bernoulli principle has a place of its own as a simple, intuitive explanation for various phenomena, it is a limited lens through which to view the rich complexities of the true-to-life fluid dynamics associated with biological cases. The growing emphasis on life science applications heightens the need for teachers to use examples that rely on more realistic viscous and turbulent fluid properties. A Poiseuille-first approach to teaching fluid dynamics should be the natural choice because it is a more accurate representation of these systems. The Physics of Pressure Gradients According to the equation of continuity and Bernoulli’s equation, as blood enters a constriction, its velocity v increases and the pressure P in the vessel decreases (Fig. 1a). If this occurs on a level, horizontal plane, then gravitational effects are ignored and the change in pressure from Region 1 to Region 2 (ΔP21 = P2 – P1) can be written as

∆𝑃!" =

!!𝜌 𝑣!! − 𝑣!! , (1)

where ρ is the mass density of the fluid. In this case, ∆𝑃!" is a negative value, indicating a pressure drop. When the blood exits the constriction into Region 3, its velocity decreases and the pressure in the vessel increases, as indicated by a positive value of ∆𝑃!". The net result is that the pressure is the same at both the entrance and exit of the vessel in the absence of resistive energy losses.

Fig. 1: (a) Flow through a constricted pipe and (b) flow through a non-constricted pipe. Particle velocity is indicated by vector length and a gauge sensor indicates vessel pressure. The pressure and velocity change due to the Bernoulli principle in Fig. 1(a) and the pressure change due to Poiseuille’s law in Fig. 1(b). Real fluids, however, are not resistance-free. For instance, take a straight tube of length L and constant radius r (Fig. 1b). When accounting for resistance, a real fluid experiences a linear pressure drop from one end of the pipe to the other. This is described, in its simplest form, by Poiseuille’s equation (or the Hagen-Poiseuille equation): ∆𝑃 = 𝑄 ∙ !!"

!!!= 𝑄 ∙ 𝑅, (2)

where η is the dynamic viscosity11 of the fluid, Q is the flow rate, and R is the resistance.12 Flow rate Q is a

volume flow per unit time, with the unit 𝑚!𝑠, which relates a cylindrical pipe’s cross-sectional area to the

average fluid velocity 𝑣:13

𝑄 = 𝜋𝑟!𝑣. (3) The typical textbook approach to the example of a blood vessel constriction is to apply Bernoulli’s equation. This approach suggests that the blood pressure before and after the constriction is the same. Yet Poiseuille’s equation dictates that the pressure must drop! How do we resolve this contradiction? Bernoulli is relied upon repeatedly in textbooks, published physics exercises, and classroom demonstrations. Most textbooks emphasize the Bernoulli and continuity equations early in the chapter and place Poiseuille’s law at the end. In fact, some texts relegate viscosity and the associated pressure drop to an optional section, suggesting it is minimally important. We are not the first to note this issue. Hellemans et al.14 introduced a classroom experiment nearly 40 years ago that combined the two laws and showed the limitations of Bernoulli. Ten years later, Badeer and Synolakis15 pleaded with the authors of college physics texts to include a combination of Bernoulli’s and Poiseuille’s equations, as well as a discussion pertaining to them. I revive these prior authors’ arguments in the context of the constricted artery and extend them to the entire cardiovascular system. Two additional mathematical terms can help establish when Bernoulli and Poiseuille are best applied. The late Steven Vogel, an inspiring and insightful educator and biomechanics researcher, lends insight through his love of

the dimensionless ratio.16 He takes a ratio of the dynamic pressure (Bernoulli) to the average Hagen-Poiseuille pressure drop:

𝐵/𝐻𝑃 = !!!!

!"!". (5)

This provides a means of determining which principle dominates. As a rough guideline, if B/HP ≳ 10, then the viscosity-related pressure drop is only a tenth of the dynamic pressure, and Bernoulli’s equation dominates the scene. If B/HP ≲ 0.1, then viscosity dominates the situation and the physics is best described by Poiseuille’s Law. Generally, if the fluid velocity is small, the fluid viscosity is high, and the pipe radius narrow, then Poiseuille’s law will come to the forefront. Another helpful quantity is the Reynolds number Re. Re for fluid flow in a pipe can be written as 𝑅𝑒 = !!!"

!, (4)

with all variables defined as before. Within a cylindrical pipe, if Re ≲ 2000, then the flow is laminar, which is a necessary condition for Poiseuille’s Law. If Re ≳ 4000, then the flow is turbulent, and Poiseuille’s law is no longer applicable.17 The Reynolds number provides an important criterion to the application of Poiseuille’s law.18 The Importance of Poiseuille’s Law in Blood Flow To determine the value of each principle with regards to blood flow, I met with several experts, including many colleagues who teach anatomy and physiology and Dr. Jonathan Lindner19, cardiologist and Professor of Medicine at Oregon Health and Science University. In one respect, physics texts make a wise decision by separating Bernoulli’s principle and Poiseuille’s law into distinct cases. For instance, anatomy and physiology professors emphasizedΔ𝑃 = 𝑄 ∙ 𝑅, and the cardiologist naturally discussed Poiseuille-dominated measurements and Bernoulli-dominated measurements as distinct. One of the most important cardiovascular concerns, as discussed by Dr. Lindner, is stenosis, the constriction or narrowing of a blood vessel or valve in the body (e.g. blood flow through a constriction). A common form of stenosis is the atherosclerotic narrowing of coronary arteries, a primary concern in cardiovascular care. Figure 2 shows typical values20 for both healthy and constricted coronary arteries.

Fig 2: Selected values for a healthy and constricted coronary artery, adapted from Herman20. The constricted radius represents a 75% blockage for the purposes of the following example.

The average velocity of blood flow in a typical, healthy coronary artery is calculated with equation (3):

𝑣 = !!!!

=!!!"!! !

!!

!(!.!!"!)!= 0.08 !

!.

The Vogel number for flow through a coronary artery is then: !!"= !!!!

!"!"=

(!"#" !"!!)(!.!"

!! )(!.!!"!)

!

!"(!.!!" !"∙!)(!.!"!)= 0.27,

and the Reynolds number is

𝑅𝑒 = !!!!=

!"#" !"!! !.!" !! !∙!.!!"!

!.!!" !"∙!= 85.

For a healthy artery, we see that Vogel’s number suggests that Poiseuille’s law is a more appropriate model to consider, and blood flow is laminar, so such a model is applicable. For an artery constricted by 75%, velocity is 1.3 m/s, B/HP remains 0.27, and Re is 340. Flow remains laminar within a constriction, and a Poiseuille-approach remains appropriate. Dr. Lindner confirmed this notion by speaking of an increase in resistance as an artery constricts, where the resistance R is a critical piece in Poiseuille’s equation. To verify these predictions, we should directly compute the pressures involved in each situation. For a healthy artery, the viscosity-related (Poiseuille) pressure drop across the vessel is

∆𝑃 = 𝑄 ∙ !!"!!!

= 1𝑥10!! !!

!∙ ! !.!!" !"∙! !.!"!

! !.!!"! ! = 13𝑃𝑎,

while the dynamic (Bernoulli) pressure is

𝑃 = !!𝜌𝑣! = !

!1060 !"

!!!.!"!!

!= 3𝑃𝑎.

The pressure drop across the vessel is more than 4x greater than the dynamic pressure in a healthy artery. If the artery is constricted, then the Poiseuille drop is about 3300 Pa and the dynamic pressure about 900 Pa, again Poiseuille-dominated. What do these calculations suggest and what conclusions can we draw? The primary pressure change is due to viscosity. A decrease in artery radius leads to a substantial increase in vessel resistance and therefore a large pressure drop that the human body must account for. The reduction in blood vessel area would seemingly lead to an increase in blood pressure as the heart works harder to maintain healthy flow rate (about 4-5 L/min for most adults). This is a consequence of equation (2): if the flow rate Q must remain the same even as resistance R increases, then the driving pressure gradient ΔP must increase proportionally. Fortunately, the body can compensate in other ways; arteries downstream from the occlusion can dilate, increasing the circulatory system’s effective area and thereby reducing the resistance to flow. This self-correction (called autoregulation) is effective until an artery is about 70-80% blocked. At this point, dilation cannot compensate for the blockage and larger health problems are imminent. The human circulatory system, as a whole, is a prime example of Poiseuille’s law in action. The heart needs to provide enough pressure to overcome the resistance of the plumbing that carries blood throughout the body. As arteries branch out to arterioles and then to capillaries, average pipe length and diameter continually change. While this may complicate realistic mathematical expressions, the flow rate remains constant (obeying continuity) and the pressure steadily drops (obeying Poiseuille’s law). Fig. 3 shows a simplified schematic of the circulatory system, where total cross-sectional area of blood vessels, the speed of the blood through those vessels, and the pressure along this route all change. Though individual capillaries are tiny compared to the aorta and major arterial vessels, they are much greater in number, leading to a remarkably large total area and low blood velocity through them. Meanwhile, after the blood is ejected from the heart, the blood pressure steadily drops as blood flow faces continued resistance from each vessel downstream, contradicting the Bernoulli principle and demonstrating a Poiseuille-dominated system. (See Whitmore et al.22 for a kinesthetic laboratory model of the circulatory system that challenges students to grapple with this idea.)

Fig. 3: (a) Blood is ejected from the heart into the aorta, travels through various major vessels in the circulatory system, and returns to the heart via the venae cavae. (b) The total cross-sectional area of each type of blood vessel varies throughout the system. (c) As total area increases, blood velocity decreases by the continuity equation. (d) Pressure steadily drops, demonstrating a Poiseuille-dominated system rather than a Bernoulli-dominated system. Images and graphs adapted from Marieb.21 Conclusion The fluid dynamics of real biological systems demand deeper layers of physics than the typical introductory physics teacher has previously encountered. Though many straightforward Bernoulli principle calculations are being applied to simple life science applications, we may actually be inventing unrealistic problems and, in the process, losing an opportunity to teach our students how to apply a more appropriate model. As teachers, we need to carefully select our example problems and ensure that they come from sources that have sought input from professionals in the fields of biology and medicine. We also need to collaborate with our biology colleagues and learn from them how to create truly authentic biology-based physics questions. The author admits that this is not an easy task. It takes curiosity and commitment to gather knowledge in disciplines we may have previously eschewed. With old explanations faltering and new life science applications taking their place, evidence supports the consideration of a Poiseuille-first approach to teaching fluid dynamics.

Acknowledgements Special thanks for discussions with David Grimm, Dave Sandmire, Jeff Parmelee, Wendy Roberts, and Marlena Koper-Fox, colleagues in the biology department at the University of New England:. Also, gratitude is due to Dr. Jonathan Lindner, professor and cardiologist at Oregon Health and Science University, for his illuminating discussions. Without these individuals’ expertise and insights, this paper could not have been written. Thanks to an anonymous reviewer that encouraged a rewrite of this paper, suggesting that the focus be on introducing readers to less familiar ideas rather than attempting a much more challenging and confusing mathematical treatise. Finally, the author would like to express his deepest gratitude to James Vesenka, a colleague, mentor, and friend who introduced me to the IPLS community and encouraged me to spend far too much time thinking about fluid physics. References 1. E. Mazur, Peer Instruction: A User’s Manual (Prentice Hall, Upper Saddle River, NJ, 1997), pp. 173-174.

2. C. E. Mungan, “Pressure Change in an Arterial Constriction,” Phys. Teach. 53, 561-562 (2015).

http://dx.doi.org/10.1119/1.4935772

3. For example: R. D. Knight, B. Jones, and S. Field, College Physics: A Strategic Approach, 3rd ed. (Pearson, Boston,

2015), pp. 418-423, E. Etkina, M. Gentile, A. Van Heuvelen, College Physics (Pearson, Boston, 2014), pp. 397-

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4. P. Eastwell, “Bernoulli? Perhaps, but what about viscosity?” Sci. Ed. Rev. 6, 1-13 (2007).

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(2018). https://doi.org/10.1119/1.5028240

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scientists: A moment of opportunity,” APS News 19, 3, p. 8 (2010).

11. There are two types of viscosity: dynamic and kinematic. Introductory physics texts typically deal only with

dynamic viscosity η, which relates a fluid’s internal resistance to flow and the force required to move one

plane of liquid over another. Kinematic viscosity υ is often preferred in engineering professions, particularly

for describing lubricating oils.

12. The reference to resistance R provides an enticing analogy to Ohm’s law. Ohm’s and Poiseuille’s laws both

show that a flow rate is simply a driving gradient divided by the resistance to flow. Compare 𝐼 = Δ!!

and

𝑄 = !!!

, respectively. The analogy can only be carried so far, but is quite valuable when considering the basic

principles of fluid flow through pipes or tubes connected in series and parallel networks. Teachers of human

anatomy and physiology use this extensively.

13. Unlike charge flow, which is considered to have a constant drift velocity everywhere within a uniform wire,

fluids experience a velocity gradient within a pipe. Therefore, we refer to an average velocity. This is

modeled nicely in B. Spitznagel, J. Weigal, and J. Rodriguez, “Visualizing Viscous Flow and Diffusion in the

Circulatory System,” Phys. Teach. 57, 529-532 (2019). https://doi.org/10.1119/1.5131117

14. J. Hellemans, P. Forrez, and R. De Wilde, “Experiment illustrating Bernoulli’s equation and Hagen-

Poiseuille’s law,” Am. J. Phys. 48, 254-256 (1980). http://dx.doi.org/10.1119/1.12154

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http://dx.doi.org/10.1119/1.2342887 and C. E. Synolakis and H. S. Badeer, “On combining the Bernoulli and

Poiseuille equation – A plea to authors of college physics texts,” Am. J. Phys. 57, 1013-1019 (1989).

http://dx.doi.org/10.1119/1.15812

16. S. Vogel, Comparative Biomechanics: Life’s Physical World, 2nd ed. (Princeton University Press, New Jersey, 2013),

p. 173.

17. B. R. Munson, D. F. Young, T. H. Okiishi, and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed. (John

Wiley & Sons, Inc., New Jersey, 2009), pp. 410-422.

18. Vogel’s ratio and Reynolds number do not necessarily need to be taught in an introductory physics class of

life science students. These expressions are given primarily to assist the reader in determining whether a

Bernoulli or Poiseuille approach is more appropriate for a given biological example. However, there is a

growing trend to include Reynolds number in the introductory class.

19. You can access the complete interviews with Dr. Jonathan Lindner at https://tinyurl.com/yyclravz. These

interviews were supported by NSF IUSE grant Multimedia Modules for Physics Instruction in a Flipped Classroom

Course for Pre-Health and Life Science Majors (DUE-1431447), where Ralf Widenhorn served as PI.

20. I. P. Herman, Physics of the Human Body (Springer, Berlin Heidelberg New York, 2007).

21. E. N. Marieb and K. Hoehn, Human Anatomy & Physiology, 11th ed. (Pearson, Boston, 2019).

22. E. Whitmore, J. Vesenka, D. Grimm, B. Moser, and R. Lindell, “A kinesthetic circulatory system model for

teaching fluid dynamics”, 2015 PERC Proceedings, 359-362 (2015). http://dx.doi.org/10.1119/perc.2015.pr.085

23. For example: J. R. Cameron, J. G. Skofronick, and R. M. Grant, Physics of the Body, 2nd ed. (Medical Physics

Publishing, Wisconsin, 1999), and J. A. Tuszynski and J. M. Dixon, Biomedical Applications of Introductory

Physics, 1st ed. (John Wiley & Sons, Inc., New Jersey, 2002).