Our body is a marvellous, if messy tangle of innumerable moving parts. Consider for example, the circulatory system, the network of blood vessels, and the heart that supplies tissues in the body with oxygen and nutrients and removes unnecessary waste products. For the circulatory system to work well, blood must be able to enter through capillaries as small as 3 microns in diameter (a micron is a millionth of a meter!).
How does it accomplish this feat? First, note that blood consists of various types of cells (mostly red blood reds) suspended in a liquid called plasma. The discrete nature of the suspension introduces complications into the modelling of blood as a fluid. For large blood vessels where the plasma carries a huge number of cells, the fluid analogy is a useful simplification, allowing one to apply the laws of physics to study blood flow dynamics. In particular, the Navier-Stokes equation for modelling fluid turbulence comes in handy [1].
An exact solution of the Navier-Stokes equation describes the steady flow of fluid along a uniform pipe of radius r and length l . Let Q denote the axial flow rate measuring the volume of fluid (blood) passing through the pipe (a blood vessel) per unit time. Then, under appropriate assumptions, Q can be expressed as:
This equation is called Poiseuille’s law for laminar (non-turbulent) flow, after the French scientist J. L. Poiseuille (1799–1869), who derived it in an attempt to understand the flow of blood. It shows that for given a viscosity and difference in pressure between the two ends of a pipe or blood vessel, narrow vessels (those with small radius) present very high flow resistance. This of course, is potentially life-threatening. To understand this better, consider the following scenario: suppose the radius of an artery has been reduced by 16%, by what factor has the flow rate of blood in the artery been reduced? The answer from solving equation (1) is a shocking 50%!
In smaller blood vessels of diameters below a hew hundred microns, the discrete nature of the suspension become important and the description of blood as a homogeneous fluid breaks down. Nevertheless, the basic insights of the simplified model remain. Fortunately for us, nature has designed red blood cells to be highly deformable so that they can ‘flatten’ themselves and squeeze through capillaries as small as 3 microns in diameter. In normal circumstances, this enables blood to flow through small capillaries with low viscosity without requiring enormous pressure drops or stresses on the endothelial cells that surround the red blood cells [2]. The circulatory system, unseen and unheard by us as we go through our lives, is indeed a work of wonder.
Notes
[1] Named after French physicist and engineer Claude Louis Navier (1785 – 1836) and Anglo-Irish mathematician, George Babriel Stokes (1819 – 1903).
[2] Leading computational biologist, George Karniadakis (Brown University) uses mathematics to model biological processes including blood flows through the circulatory system. “Blood is a really complex fluid with many yet-unexplored properties”, he says in an interview with the American Scientist, a journal. To visualize the complexity of blood flow, Kaniadakis and his research colleagues have used state-of-the-art visualization tools to quantify both flow dynamics and particle dynamics and their interactions. This video is the result of one of their visualization studies: https://youtu.be/0hibGZi8TWs