42 Reynolds Number

E.F. Redish

[1]The motion of fluids and of objects in fluids (fluid dynamics) is an extremely complex subject, one of the most challenging in physics. Because fluids have so many parts that can move independently and interact strongly with each other, all kinds of interesting phenomena happen — waves, turbulence, eddies, whirlpools, and hurricanes. The well-known difficulty of predicting weather is largely due to the complexity and instabilities that can be found in the motion of fluids. As a result of evolving in a fluid environment, biological systems, from bacteria to water fleas to dolphins to birds, have developed sophisticated interactions with the fluids around them.* One way that we can begin to get a handle on the complex phenomena of fluid dynamics is to determine which of many possible phenomena dominate under particular circumstances.

We can highlight the biological responses to moving fluids by comparing the shapes of aquatic animals having different sizes and different motilities. Look at the similar streamlined shapes of the large fast-moving predators, such as the dolphin (mammal), Ichthyosaur (reptile), and tuna (fish) shown in the figures at the left below. Whenever one sees similar shapes of unrelated organisms, it reveals the possibility that some exciting physics might be lurking in that similarity. Now look at the non-streamlined shapes of much smaller, slow-moving insects and other aquatic animals, such as brine shrimp, mayfly larvae, and water fleas shown on the right. Can physics tell us something about how organisms of different sizes and different mobilities have adapted to their environments?

Fast swimmers

Dolphin:                                                     Ichthyosaur:                                                      Tuna:
Source: E. Redish                                             Source: Wikimedia                                          Source: NOAA
Commons                                                   Public Domain


Slow swimmers

Brine Shrimp:                                              Mayfly larva:                                             Water flea:
Source: Wikimedia                                         Source: NABS                                         Source: British
Commons                                            www.benthos.com                                  Museum Guide

Creating a dimensionless parameter – The Reynolds Number[2]

One powerful method to help our understanding of what matters is to create dimensionless parameters by comparing two influences on the fluid, building a ratio that is a property of the fluid system and does not depend on our choice of units.

An example of this approach is to consider a small object moving in a fluid. And again, to see how this approach works, let’s limit ourselves to a simple model system. For simplicity, let’s assume or object moving through a fluid. In general, the object feels two kinds of resistive forces from the fluid: drag and viscosity.

Inertial drag occurs because in order to keep moving, the object has to push the fluid out of the way in front of it. To do so, it has to exert a force on that fluid and therefore (by Newton 3) the fluid has to exert a force back on the object. Viscosity occurs because fluid tends to stick to the surface of an object, so that when the object moves through the fluid it drags fluid along. The fluid on the object has to slide along neighboring layers of fluid, and the resistance of the layers of fluid to sliding over each other results in viscous drag.

Each of these forces depend on a critical parameter of the fluid. Drag depends on the density of the fluid, since that tells how much mass you have to push out of the way, while viscous forces depend on the viscosity (the stickiness of the fluid).

We define the ratio of these two forces as the Reynolds Number, Re (Yes, it’s written with two characters):

    \[Re = \frac{F_{drag}}{F_{viscosity}}\]

Our Reynolds number ratio tells us the ratio of inertial to viscous drag. A large Re says inertial drag dominates while a small ratio says viscous drag dominates.

For the organisms shown above, you can also estimate the Reynolds number. For the fast swimmers, Re is large, so inertial drag dominates. They have to push the fluid out of the way in front of them. Developing a streamlined shape reduces the drag that is most important for them. For the slow swimmers, as for bacteria, Re is small, so inertial drag is of less importance than viscous drag, and a streamlined shape isn’t needed.



  1. This is from https://www.compadre.org/nexusph/course/Reynolds_number. Accessed 8 November 2023.
  2. Note that the Reynolds number is not only used to think about the motion of objects in a fluid; it has implications for the motion of the fluid itself, since as the fluid moves faster it has to push other fluid out of the way and the viscous drag of neighboring bits of fluid moving at different speeds is going to affect the motion of the fluid.


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