# Reynolds number

#### Prerequisites

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**

**Slow swimmers**

## Creating a dimensionless parameter

One powerful method to help our understanding of what matters is to create *dimensionless parameters* by comparing two influences on the fluid, using dimensional analysis to create 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 is a sphere of radius *R *and that it is not moving so fast that the fluid it's in gets turbulent. In this case, 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. Inertial drag depends on the density of the fluid, $ρ$, since that tells how much mass you have to push out of the way, while viscous drag depends on the coefficient that tells how force is transferred from one layer of the fluid to another, the viscosity of the fluid, $μ$ (mu). The two forces acting on a small object moving through the fluid with a velocity $v$ are

$$F^{drag}_{fluid \rightarrow sphere} = \frac{1}{2} C_{drag} \rho (\pi R^2) v^2$$

$$F^{visc}_{fluid \rightarrow sphere} = 6 \pi \mu R v$$

{For convenience in our analysis of drag, we considered a cylinder. This only affects the drag coefficient and doesn't change the dependence on the parameters.}

Keeping only the dimensioned parameters (and assuming that $*C_{*drag}$ is about 1) we define the ratio of these two forces as the **Reynolds Number,** *Re *(Yes, it's written with two characters):

$$Re = \frac{F^{drag}_{fluid \rightarrow sphere}}{F^{visc}_{fluid \rightarrow sphere}} = \frac{\rho R^2 v^2}{\mu Rv} = (\frac{\rho}{\mu}) Rv$$

## Dimensional analysis

Let's check the dimensionality to be sure that *Re* is dimensionless:

[*ρ*] = M/L^{3} [*μ*] = M/LT [*R*] = L [*v*] = L/T

so

[*ρ/μ*] = MLT/ML^{3} = T/L^{2} [Rv] = L^{2}/T

[Re] = (T/L^{2})(L^{2}/T) = 1

Yep! It works! Note that although we showed that the value of *Re* is dimensionless, we have separated it into two factors: (*ρ/μ*) and (*Rv*). That's because the first factor depends on the properties of the fluid and the second depends on the properties of the object. This suggests that if we are thinking of a particular fluid and many different objects or situations, it might be useful to look at (ρ*/μ*).

## Scales

Let's see what typical values are for some fluids relevant to biology. For air and water, here are the values and ratio for viscosity and density (after Vogel*).

Substance |
Viscosity (μ, Pa-s) |
Density (ρ kg/m^{3}) |
Ratio (ρ/μ, s/m^{2}) |

Air |
1.8 x 10^{-5} |
1.2 | ~ 6 x 10^{4} |

Water |
1.0 x 10^{-3} |
1.0 x 10^{3} |
10^{6} |

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. To get Re we multiply our fluid ratio, ρ*/μ*, times our object product, *Rv*.

Let's consider two objects: a bird flying through air and a bacterium swimming through water. Let's do some estimates to see which resistive fluid force dominates.

For the bird, its size is on the order of 1 m while it flies at a few m/s — call it 1. This means *Rv* ~1 m^{2}/s. Since the fluid ratio for air given in the table is 60,000 s/m^{2}, inertial drag dominates by a large factor.

For the bacterium, its size is on the order of 1 micron (10^{-6} m) and since we can watch it move in a microscope, moving its own length as we watch, its speed will be on the order of 1 micron/s = 10^{-6} m/s. So *Rv* ~10^{-12} m^{2}/s. Even though the density/viscosity ratio is large (about a million s/m^{2}), the *Rv* is 1/million^{2} m^{2}/s, so the Reynolds number is about 10^{-6} — extremely small. This tells us that viscosity is what matters for the bacterium, not inertial drag. (For more discussion of this, see the excellent paper by Edward Purcell, "Life at low Reynolds number".**)

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. (For more discussion of these issues see the references.* **)

{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.}

* For a detailed discussion of some interesting biological examples of the interactions of organisms with fluids, see

- S. Vogel,
*Comparative Biomechanics: Life's Physical World*(Princeton U. Press, 2003) - M. Denny,
*Air and Water: The Biology and Physics of Life's Media*(Princeton U. Press, 1993).

** E. Purcell, "Life at low Reynolds number," *American Journal of Physics* **45** (1977) 3-11.

Joe Redish and Todd Cooke 8/17/15

Last Modified: July 12, 2019